PhysRevA 106 062610-Quansistor

PHYSICAL REVIEW A 106, 062610 (2022)
Universal quantum computation with symmetric qubit clusters coupled to an environment
Christian Boudreault ,1,2,* Hichem Eleuch ,3,4,† Michael Hilke ,5,‡ and Richard MacKenzie 2,§
1
Département des sciences de la nature, Collège militaire royal de Saint-Jean 15 Jacques-Cartier Nord,
Saint-Jean-sur-Richelieu, Quebec, Canada, J3B 8R8
2
Département de physique, Université de Montréal, Complexe des Sciences, C.P. 6128, succursale Centre-ville,
Montreal, Quebec, Canada, H3C 3J7
3
Department of Applied Physics and Astronomy, University of Sharjah, Sharjah, United Arab Emirates
4
College of Arts and Sciences, Abu Dhabi University, Abu Dhabi 59911, United Arab Emirates
5
Department of Physics, McGill University, Montreal, Quebec, Canada, H3A 2T8
(Received 3 June 2021; revised 10 October 2021; accepted 9 November 2022; published 15 December 2022)
One of the most challenging problems for the realization of a scalable quantum computer is to design a
physical device that keeps the error rate for each quantum processing operation low. These errors can originate
from the accuracy of quantum manipulation, such as the sweeping of a gate voltage in solid state qubits or the
duration of a laser pulse in optical schemes. Errors also result from decoherence, which is often regarded as more
crucial in the sense that it is inherent to the quantum system, being fundamentally a consequence of the coupling
to the external environment. Grouping small collections of qubits into clusters with symmetries may serve to
protect parts of the calculation from decoherence. In this work, we use four-level cores with a straightforward
generalization of discrete rotational symmetry, called ω-rotation invariance, to encode pairs of coupled qubits and
universal two-qubit logical gates. We include quantum errors as a main source of decoherence, and show that
symmetry makes logical operations particularly resilient to untimely anisotropic qubit rotations. We propose
a scalable scheme for universal quantum computation where cores play the role of quantum-computational
transistors, or quansistors for short. Initialization and readout are achieved by tunnel-coupling the quansistor to
leads. The external leads are explicitly considered and are assumed to be the other main source of decoherence.
We show that quansistors can be dynamically decoupled from the leads by tuning their internal parameters, giving
them the versatility required to act as controllable quantum memory units. With this dynamical decoupling,
logical operations within quansistors are also symmetry-protected from unbiased noise in their parameters. We
identify technologies that could implement ω-rotation invariance. Many of our results can be generalized to
higher-level ω-rotation-invariant systems, or adapted to clusters with other symmetries.
DOI: 10.1103/PhysRevA.106.062610
I. INTRODUCTION cation of “simple” quantum universal sets like single-qubit

gates with CNOT [8], and finite quantum universal sets like
Quantum information theory has become a mature field
Toffoli with Hadamard and π4 -gate [9], or SWAP with almost
of research over the last three decades, equipped with its
any two-qubit gate [10,11]. The deep theorem of Solovay
own objectives towards quantum computation and commu-
and Kitaev showed that it is possible to translate between
nication [1], as well as quantum simulation [2], while at the
strictly universal sets with at most polylogarithmic overhead
same time allowing entirely novel perspectives on other es-
[9]. Alongside strict universality, encoded universality [12,13]
tablished fields, in particular an algorithmic approach to quan-
and computational universality [14] allow even more systems
tum systems, a structure-of-entanglement characterization of
to qualify as universal quantum computers.
large classes of many-body quantum states (matrix product
Circumstantial evidence suggests that quantum computers
states, tensor networks) [3], and quantum-enhanced mea-
might achieve superpolynomial speedups over probabilis-
surements reaching the Heisenberg precision limit (quantum
tic classical ones. Lloyd’s universal quantum simulator and
metrology) [4].
Shor’s algorithms for integer factorization and for discrete
Quantum information processing departed from its classi-
logarithms are prominent examples of efficient quantum so-
cal counterpart with the proof that two-qubit gates [5–7] can
lutions for problems suspected to be not computable in
simulate arbitrary unitary matrices, followed by the identifi-
polynomial time classically [15,16]. Quantum communication
protocols are provably exponentially faster than classical-
probabilistic ones for specific communication complexity
*
christian.boudreault@cmrsj-rmcsj.ca problems [17,18], and there exist problems that space-
†
heleuch@fulbrightmail.org bounded quantum algorithms can solve using exponentially
‡
hilke@physics.mcgill.ca less work space than any classical algorithm [19]. Nonethe-
§
richard.mackenzie@umontreal.ca less, large classes of quantum tasks involving highly entangled
2469-9926/2022/106(6)/062610(23) 062610-1 ©2022 American Physical Society

BOUDREAULT, ELEUCH, HILKE, AND MACKENZIE PHYSICAL REVIEW A 106, 062610 (2022)
states are efficiently simulatable classically. Quantum tele- source of decoherence. Indeed, in solid-state-based qubits
portation, superdense coding and computation using only electric leads are often the main source of decoherence, partic-
Hadamard, CNOT, and measurements fall into this category ularly in superconducting qubits and semiconductor quantum
according to the Gottesman-Knill theorem [1,20]. Fermionic dots [45]. While our model is not limited to a particular im-
linear optics with measurements, and more generally match- plementation, we will use the coupled quantum dot geometry
gate computation, are also known to be classically simulatable as an illustration of our quantum processor unit.
in polynomial time [21–23]. [This is in contrast to universal We will restrict ourselves to examining the fidelity and
bosonic linear optics with measurements [24] and univer- robustness of two double-qubit gates in the presence of a
sal fermionic nonlinear optics with measurements [25]. The selected error set, and observe what appears to be an improve-
computational difference between particle-number-preserving ment in the results arising due to symmetry. Our results are of
fermions and bosons arises as a result of the easy task of com- immediate relevance to the study of noisy intermediate-scale
puting a (Slater) determinant in the case of fermions versus quantum (NISQ) devices, which could realize useful versions
the hard task (P-complete) of computing a permanent [26] in of quantum supremacy in the very near future, long before
the case of bosons.] fully operational fault-tolerant architectures become available
The physical realization of quantum computers and quan- [46]. Here we do not discuss logical error decay under the
tum communication channels is a major endeavor. Most consumption of a resource, nor do we discuss fault tolerance
building blocks of quantum computers are based on qubits, beyond a few remarks in the Outlook.
which are quantum two-level systems. They form the unit
cells that allow us to exploit the potential of quantum infor- II. PRELIMINARIES
mation processing, when many of these qubits are coherently
coupled and manipulated so as to perform various coherent Let us consider first a physical system formed by a core
quantum operations. While many different types of qubits of four coupled quantum dots with on-site energies i . Each
have been developed, such as semiconductor technologies in quantum dot interacts with all the other dots via complex
quantum dots [27], including silicon [28,29], or GaAs [30], in couplings (we will discuss in Sec. III F how it is possible
superconducting technologies [31,32], in all-optical technolo- to realize complex couplings physically). The corresponding
gies [33], and in hybrid technologies such as ion traps [34], isolated Hamiltonian is
cold atoms, and nitrogen-vacancy centers in diamond [35] 1 † 1
4 4
that require quantum systems and a laser for control. Topo- Hcore = i ai ai + hi j ai† a j + H.c. (1)
logical technologies can also form the basis of qubits [36], 2 i=1 2 i, j=1
though their experimental realization is much harder. These
Each dot is now made to interact with a semi-infinite chain
technologies all share the same basic principle of operating as
consisting of a semi-infinite hopping Hamiltonian with hop-
a quantum two-level system.
ping parameter set to unity (thus setting the scale for all
In this work we explore a quantum processing unit based
energies). The leads have scattering eigenstates with energies
on a four-level system. While there have been some earlier
−2 < E < 2. Tunnel couplings between dot and chain are
works on such higher-level systems, including multilevel su-
initially all identical and are chosen real, positive and small
perconducting circuits as single qudits and two-qubit gates
(0 < tc 1). As in the case of double and triple dots, the
[37,38], here we consider a special four-level system with ω-
Feshbach projector method shows that the effect of each lead
rotation invariance, defined and discussed below, that we will
is to modify the self-energies of the dots. In this work, we
compare to a pair of qubits in order to address one of the major
will study a similar core system formed by four sites (though
challenges in quantum information processing, namely, the
not necessarily quantum dots) tunnel-coupled to semi-infinite
fidelity of two-qubit operations against environment-induced
leads, but with a crucial additional core symmetry.
effects [39].
Specifically, we will consider the single-particle sector of a
Indeed, one of the biggest obstructions for a competitive
class of tunable systems possessing a simple geometric sym-
quantum computation is to keep the error rate low for each
metry, dubbed ω-rotation invariance, to be defined in the next
quantum operation [40]. These errors can stem from the pre-
section. A diagram of the model used throughout the paper
cision of the quantum manipulation, like the sweeping of a
is displayed in Fig. 1. It consists of a completely connected
gate voltage in solid-state qubits or the duration of a laser
four-site core system tunnel-coupled to four identical semi-
pulse in optical schemes [41]. In addition, there are errors due
infinite leads (a simple physical example being four quantum
to decoherence [42]. These are often considered more funda-
dots tunnel-coupled to semi-infinite leads). The Hamiltonian
mental in the sense that they don’t depend on the precision
is
of the instrumentation but are intrinsic to the quantum system
considered. They are a reflection of the coupling to the outside H = Hcore + Hint + Hlead
environment. Sources of decoherence can be leads, nuclear ∞
1
4 4 4
spins, optical absorption, phonons, and nonlinearities. Most = hi j ai† a j + tc ai† bi,1 + b†i, j bi, j+1 + H.c.
of these environments fall into the category of fermionic or 2 i, j=1 i=1 i=1 j=1
bosonic baths [43,44].
(2)
In our basic quantum information unit, based on a four-
level system, untimely single-qubit and double-qubit unitaries restricted to the single-particle sector of Hilbert space. The
will correspond to environment-induced logical errors. We core couplings hi j are chosen to satisfy relationships ensuring
will also consider the effect of external leads as the other main ω-rotation invariance (see Sec. III). The coupling between site
062610-2
UNIVERSAL QUANTUM COMPUTATION WITH SYMMETRIC … PHYSICAL REVIEW A 106, 062610 (2022)
tuning. Our ability to fully control the (potentially complex)

eigenenergies will result in the possibility of transmitting an
eigenstate through the leads or else of protecting it from
decoherence, independently of the other eigenstates. In that
sense, the four-level core may be used as a two-qubit quantum
memory unit.
Finally, in Sec. V we propose a scalable scheme for uni-
versal quantum computation based on four-level cores as the
elementary computational units. The number of cores required
scales linearly in the number of qubits. Because cores play
a role similar to that of transistors in classical computation,
we propose to call them quantum-computational transistors,
FIG. 1. Depiction of the model, consisting of a four-site core or more succinctly quansistors.
system (disks and dark arrows) tunnel-coupled to four identi- Rotation-invariant (circulant) 4 × 4 Hamiltonians have re-
cal semi-infinite leads (rectangular boxes). The core system is cently been advocated [48] as a way to implement the
ω-rotation-invariant (see Sec. III) for any ω4 = 1. The tunable pa- adiabatic Fourier transform on two qubits, with gate fidelities
rameters are , γ ∈ R and τ ∈ C. The hopping parameters linking
and entanglement benefitting from a symmetry that pro-
the sites are generally complex. The coupling constants between the
tects against decoherence. The proposal includes a possible
core’s sites and the leads are equal to tc . All hopping amplitudes
physical implementation of circulant symmetry by tuning
along the leads are set to unity.
spin-spin interactions in ion traps. Although our work also uti-
lizes (generalized) circulant symmetry for protection against
i and lead i is tc , which can be taken real and positive without decoherence, the aim and scope of the present article are
loss of generality. The Hamiltonian has been normalized such somewhat different. We put forward a blueprint for scalable
that the hopping parameter within the semi-infinite chains is universal quantum computation based on symmetry-protected
unity. The operators ai and bi, j are annihilation operators, qubit clusters, with ω-rotation invariance standing out as the
acting respectively on site i of the core system and on site j of prototype of a symmetry which is provably universal, and re-
the ith lead. Since we work in the single-particle sector, these alistically implementable physically on a variety of platforms.
operators could be fermionic or bosonic. (An example of each
would be a single electron and a Cooper pair, respectively.
III. CORE SYSTEM
Cold atoms can realize either choice.) Our choice of a four-
level core is motivated by our desire to describe two coupled For a 4 × 4 matrix, we make a slight generalization of
qubits. The four semi-infinite leads simulate individual con- the notion of discrete rotational invariance (which can also
tact with the environment and enable us to reveal selective be viewed as cyclic permutation of the sites) to ω-rotation
protection from decoherence. Most of our results can be gen- invariance: M is ω-rotation-invariant if
eralized to an arbitrary number of sites in the core system with
corresponding identical leads. The required modifications will Jω† MJω = M, ω4 = 1, (3)
be discussed briefly in the Outlook and in the Appendixes.
with a modified shift matrix
⎛ ⎞
A. Outline of the paper 0 1 0 0
⎜0 0 ω 0⎟
In Sec. III we focus on the core system. We define ω- Jω = ⎜⎝0 0 0
⎟, Jω4 = ω2 1. (4)
ω2 ⎠
rotation invariance as an obvious generalization of discrete
rotation invariance, and show that the tunable parameters of an ω3 0 0 0
ω-rotation-invariant system give full control over its eigenen-
Rotational invariance obviously corresponds to the case ω =
ergies while the energy eigenstates remain fixed. Independent
1. The matrices J1 and Jeiπ/2 , and their higher dimensional
control over the energy levels will be used frequently and is
versions, have been discussed in discrete quantum mechanics
the main motivation for implementing ω-rotation invariance.
under the name of Weyl’s X and Y matrices, and in quantum
Systems with this symmetry could be realized by applying
information under the name of generalized Pauli X and Y
the technique of synthetic gauge fields on a tight-binding
matrices (see Appendix A). In the 4 × 4 case we have
Hamiltonian [47]. Selecting a representative from two distinct
ω classes and following the scheme of Deutsch et al. [10], we ⎛ ⎞ ⎛ ⎞
0 1 0 0 1 0 0 0
show that our four-level core system is strictly universal for ⎜0 0 1 0 ⎟ ⎜0 i 0 0 ⎟
quantum computation. We then consider one possible two- X =⎜ ⎟
⎝0 0 0 1⎠ = J1 , Z = ⎝0 0 i2 0 ⎠,
⎜ ⎟
qubit logical basis and discuss single-pulse logical gates as 1 0 0 0 0 0 0 i3
well as symmetry protection against errors, and qubit initial-
ization and readout. Y = ZX = Jeiπ/2 . (5)
In Sec. IV we consider the effect of the four identical leads
on the core. The effective Hamiltonian of the core will in gen- (Note that the matrix X is sometimes called X † in the
eral be non-Hermitian but will remain ω-rotation-invariant, literature.) Just as rotation-invariant matrices are precisely
and as a consequence will still allow independent energy circulant matrices, ω-rotation-invariant matrices correspond
062610-3
to ω-circulant matrices, which we will write Hamiltonian from each class. We will now consider each of
these classes in turn.

3
Circω (z0 , . . . , z3 ) = zs Jωs , (6)
s=0 A. Symmetry class X (ω = 1)
with zs ∈ C. The terms “ω-rotation-invariant” and “ω- Class X is rotation-invariant in the position eigenbasis:
circulant” will be used interchangeably. For Hermitian
ω-circulant 4 × 4 matrices the number of independent real
3
H pos (g, 1) = zs (g)X s (zs ∈ C, g ∈ R4 ). (13)
parameters is reduced to four. Such matrices constitute what
s=0
we propose to call a flat class: mutually commuting ma-
trices {H (g) | g ∈ R4 } with common eigenbasis independent The most general form of the Hamiltonian matrix is
of g, and real eigenvalues λ1 (g), . . . , λ4 (g) in one-to-one ⎡ ⎤
τ γ τ†
correspondence with the values of the parameters g ∈ R4 . ⎢τ † τ γ ⎥
Each fourth root of unity ω corresponds to a flat class. (See H pos (g, 1) = ⎢
⎣γ
⎥ (14)
τ† τ ⎦
Appendix A for details.)
τ γ τ†
We consider four-level cores with the ability to take a
nonsymmetric form (the off mode), and a symmetric form with , γ ∈ R and τ = |τ |eiθ = α + iβ giving the four real
(the computational mode). In the off mode, the Hamiltonian parameters embodied in g. For any value of g the normalized
is almost diagonal in the single-particle position eigenbasis: eigenstates of H pos (g, 1) are

Hoff = − Ki j ai† a j + i ai† ai , (7) 1 imk(π/2)
i j i
|φk = Fm,k
†
|m = e |m (15)
m
2 m
where large energy offsets |i − j | Ki j > 0 effectively
for k = 1, . . . , 4, with eigenenergies
suppress spontaneous transitions. The logical states
{|00, |01, |10, |11} are naturally chosen to coincide kπ
with the position basis eigenstates {|m}m=1,...,4 . We will λk = + 2|τ | cos θ + + (−1)k γ (16)
2
come back to this mode later.
or
In the computational mode, the core system is ω-rotation-
invariant in the position basis for all values of its parameters. λ1 = − 2β − γ , λ2 = − 2α + γ ,
The matrix of core couplings {hi j } in (2) will thus form a class
of Hermitian ω-circulant Hamiltonian matrices λ3 = + 2β − γ , λ4 = + 2α + γ . (17)

3 These can be inverted, giving
H pos
(g, ω) = zs (g)Jωs , (8) λ1 λ2 λ3 λ4
s=0 = + + + ,
4 4 4 4
with zs ∈ C, g ∈ R4 , and ω4 = 1. [Recall that the complex λ2 λ4
coefficients zs are constrained by Hermiticity H pos = (H pos )† , α=− + ,
4 4
leaving only four real independent parameters g.] For each
q = 0, . . . , 3 the class ω = eiqπ/2 is flat and diagonalized by λ1 λ3
β=− + ,
a modified quantum Fourier transform FDq , where F is the 4 4
regular quantum Fourier transform λ1 λ2 λ3 λ4
γ =− + − + . (18)
Fk,m = 21 e−imk(π/2) (9) 4 4 4 4
Any path in the R4 manifold of eigenenergies (λ1 , λ2 , λ3 , λ4 )
and of the class corresponds to a unique path in the R4 manifold
D = diag(e−iπ/4 , 1, −e−iπ/4 , 1). (10) of parameters (, α, β, γ ), giving full control over the energy
levels of the class.
The eigenstates are
1 † q imk(π/2)
ϕkq = (FDq )†m,k |m = (Dm,m ) e |m (11) B. Symmetry class Y (ω = eiπ/2 )
m
2 m Class Y is eiπ/2 -rotation-invariant in the position eigenba-
for q, k = 0, . . . , 3. Note that the first index of a matrix corre- sis:
sponds to a dual vector component, whereas the second index
3
corresponds to a vector component: H pos (g, eiπ/2 ) = zs (g)Y s (zs ∈ C, g ∈ R4 ). (19)
s=0
m ϕkq = (FDq )†m,k , ϕkq |m = (FDq )k,m . (12)
The most general form of the Hamiltonian is
For the purpose of universal quantum computation, two ⎡ ⎤
classes of ω-circulant Hamiltonians are necessary and suffi- τ −γ iτ †
cient: for instance, the class X of circulant Hamiltonians (ω = ⎢ τ† iτ γ ⎥
H (g, e ) = ⎢
pos iπ/2
⎣ −γ
⎥, (20)
1), and the class Y of i-circulant Hamiltonians (ω = eiπ/2 ). In −iτ † −τ ⎦
Sec. III C we will build a universal set comprising only one −iτ γ −τ †
062610-4
TABLE I. Two possible logical bases. Binary code will be used

throughout, unless explicitly stated otherwise.
Position Binary-coded Gray-coded

|1 |00 |00
|2 |01 |01 FIG. 2. Commutative diagram of the four working dual bases and
|3 |10 |11 their relationships. The quantum Fourier transform F and the matrix
|4 |11 |10 D are defined in Eqs. (9) and (10), respectively. Top: Position basis
{m|}. Middle left: Class X energy basis {φk |}. Middle right: Class
Y energy basis {χk |}. Bottom: Logical basis {|}.
where the parameters , γ ∈ R and τ ∈ C are chosen to have
exactly the same form as those of (14). The entries in class X an acceptable cost in a fault-tolerant setting, such is not the
and Y are seen to differ by at most a prefactor. For any value case in a NISQ setting, where the number of qubits is a severe
of g the normalized eigenstates of H pos (g, eiπ/2 ) are constraint and even constant factor overheads are typically
1 † imk(π/2) important. The four working bases {|m}, {|φk }, {|χk }, {|},
|χk = (FD)†m,k |m = D e |m (21) and their relationships defined in (15), (21), and Table I, are
2 m m,m
m summarized in the commutative diagram of Fig. 2. The dia-
F
for k = 1, . . . , 4, where the coefficients Dm,m
†
are given in bases, where an arrow like {m|} →
gram actually uses dual
Eq. (10). The eigenenergies are {φk |} means φk | = m Fk,m m|. Composition of arrows
√ √ agrees with conventional matrix composition.1
λ1 = + 2α − 2β − γ ,
√ √
λ2 = − 2α − 2β + γ , C. Strict universality on two qubits
√ √
λ3 = − 2α + 2β − γ , The system H pos (g, ω) of (8), with ω equal to either 1 or
√ √ eiπ/2 , generates a strictly universal set of two-qubit gates. In
λ4 = + 2α + 2β + γ , (22) fact we prove the stronger result that the finite set {V, W} ⊂
which can be inverted, giving U(4) is strictly universal, where the unitaries V and W, de-
fined below, belong to classes X and Y , respectively. (A word
λ1 λ λ λ about notation: sans-serif symbols, like V and W, will always
= + 2 + 3 + 4,
4 4 4 4 denote logical gates, other more common examples being the
λ
λ
λ λ π -phase shift Z, the qubit flip X or NOT, the Hadamard
α = √1 − √2 − √3 + √4 , gate H, the swapping gate SWAP, and the controlled-not
4 2 4 2 4 2 4 2
CNOT.) We use the scheme of Ref. [10] to prove our claim.
λ λ λ λ We construct sixteen Hermitian 4 × 4 matrices H1 , . . . , H16
β = − √1 − √2 + √3 + √4 ,
4 2 4 2 4 2 4 2 whose evolution unitaries are all within our repertoire, mean-
λ λ λ λ ing that those unitaries can be approximated with arbitrary
γ = − 1 + 2 − 3 + 4. (23) accuracy by repeatedly applying the gates V and W. The set
4 4 4 4
{H1 , . . . , H16 } is linearly independent over R so it spans the
Again, any path in the R4 manifold of eigenenergies 16-dimensional R-space of Hermitian 4 × 4 matrices, which
(λ1 , λ2 , λ3 , λ4 ) of the class corresponds to a unique path in are evolved to generate all 4 × 4 unitaries. Our repertoire
parameter space (, α, β, γ ), giving full control over the en- therefore coincides with U(4), or in other words, is strictly
ergy levels of the class. universal on two qubits.
Independent control over the energy levels will be used We first define
later and is a prime motivation for using ω-rotation invariance,
1
but we stress that this choice of symmetry is not unique. (See H1 = 1 + (π + i)X + X 2 + H.c.
Sec. A for details.) We can now distinguish four “natural” 2
⎛ ⎞
bases for the system, namely, the position basis {|m}, the 1 π +i 2 π −i
⎜π − i (24)
energy bases {|φk } and {|χk }, and the logical basis | = 1 π +i 2 ⎟
=⎜ ⎝ 2
⎟
{|00, |01, |10, |11}, defined by identification with the posi- π −i 1 π + i⎠
tion eigenstates |m. Throughout, we will mostly consider the π +i 2 π −i 1
binary-code-ordered logical basis, but at times it will be con-
venient to consider also the Gray-code-ordered logical basis. and the unitary V = e−iH1 , both of class X . We also define
Both bases are illustrated in Table I. Our choice of these bases
1 i
is motivated by simplicity, and also by the proposal for quan- H̃ = 1 + π 1 + Y + H.c. (25)
sistor interaction, to be discussed later. The universality result 2 4
discussed in the next section is independent of this choice.
However, we should point out that the choice of basis is not
immaterial. Indeed, the Solovay-Kitaev theorem teaches us 1
To agree with conventional matrix composition,
from right to
that simulating one universal set with another will produce, at left, the arrow corresponding to |φk = m (F † )m,k |m should be
worst, polylogarithmic overhead. And while this is considered F † : {|φk } → {|m}, which is somewhat counterintuitive.
062610-5
√
and the unitary W = e−iH̃ / 2 , both of class Y . All unitaries performing a logical operation in either ω class, its eigenstates
of the form Vs = e−isH1 for s ∈ [0, 2π ) are in our repertoire, are independent of the (real) parameters g = (, α, β, γ ) in
because integers mod 2π can be found arbitrarily close to the Hamiltonian. As a consequence, when these parameters
s. The repertoire also comprises WVW† , and more generally evolve,
WVs W† for s ∈ [0, 2π ), which are generated by the Hamilto- (, α, β, γ ) → ((t ), α(t ), β(t ), γ (t )), t ∈ [0, T ], (30)
nian
the corresponding logical gate unitary U (T ) is a function of
H2 = WH1 W† . (26) the parameters’ time averages only
(Note that whether or not H2 can be obtained from the sys-
U (T ) = U (, α, β, γ ), (31)
tem’s Hamiltonian is irrelevant. It is sufficient that the unitary

WVs W† be in the repertoire for any s.) We finally define T
with · = T1 0 dt (·). This is easily seen by recognizing that
H j = i[H1 , H j−1 ], j = 3, . . . , 14, H (, α, β, γ ) is diagonalized by a common unitary V for all
values of the parameters:
H15 = i[H2 , H3 ],
V † H (g)V = diag[λ1 (g), . . . , λ4 (g)]. (32)
H16 = i[H2 , H5 ]. (27)
Thus
Any unitary generated by H j , for j ∈ {1, . . . , 16}, is in the
repertoire because of the identity U (T ) = V exp −i dt V † H (g)V V †
√ √ √ √
e[P,Q] = lim (e−iP/ n eiQ/ n eiP/ n e−iQ/ ) ,
n n
(28)
n→∞
= V diag(e−iT λ1 , . . . , e−iT λ4 )V † . (33)
which ultimately boils down to a sequence of V’s and W’s.
Unitaries generated by real linear combinations of the H j ’s From (17) and (22) we get immediately that U (T ) =
are in the repertoire as well because of the following identity U (, α, β, γ ). Accordingly, any parameter noise h(t )
for noncommuting matrices: without bias, h = 0, will leave the unitary evolution operator
U (t ) unaffected:
ei(xP+yQ) = lim (eixP/n eiyQ/n )n . (29)
n→∞ U (g(t ) + h(t )) = U (g + h) = U (g). (34)
To show that {H1 , . . . , H16 } is linearly independent over R If the quansistor interacts with the environment in such a
we consider the 4 × 4 matrices H j as 16-component vectors way that the dominant effect of the latter on the quansistor is
(obtained by stacking the columns of the matrix one on top unbiased noise in the parameters, then logical operations in-
of the next from left to right), and compute the determinant ternal to the quansistor are protected from those influences by
of the 16 × 16 matrix [H1 | · · · |H16 ] whose columns are made symmetry. And if the bias has a nonzero but known value, it is
of these 16-component vectors. We find det[H1 | · · · |H16 ] = easily compensated for. The argument is valid for any flat class
P(π ), where P is a polynomial of high order with nontran- √ (see Appendix A), i.e., generalizing from four states to N, any
scendental coefficients (specifically, coefficients
√ in Q[ 2, i], class of Hamiltonians of the form V diag[λ1 (g), . . . , λN (g)]V †
that is to say, linear combinations of 2 and i with ratio- for some unitary V , and functions λr (g) = s gs λsr with
nal coefficients). Since π is transcendental we conclude that det[λsr ] = 0. Of course, the symmetry itself, being the key
P(π ) = 0—actually |P(π )| ∼ 1073 —so {H1 , . . . , H16 } spans ingredient here, must be enforced.
the space of 4 × 4 Hermitian matrices, as required. We have On a more interesting level, we now consider the robust-
thus proven that the repertoire of {V, W} is all of U(4). ness of our logical gates against genuine quantum errors. We
We conclude with a few comments about the Hamiltoni- empirically find that the universal set {V, W}, defined in the
ans H1 , H̃ chosen to generate the gates V, W in the above previous section, is particularly resilient to small single-qubit
construction. First, these Hamiltonians were chosen to pro- x and z rotations, i.e., errors of the form
√ a sequence of matrices H1 , . . . , H16 with coefficients in
duce
Q[ 2, π , i], a property used in the proof of linear indepen- Ex(1) (τ ) = e−iτ (σx ⊗1) , Ex(2) (τ ) = e−iτ (1⊗σx ) , (35)
dence of the H j ’s. This condition is by no means necessary
for linear independence, and many sets of gates other than and
{V, W} would qualify as universal. Second, the Hamiltonians Ez(1) (τ ) = e−iτ (σz ⊗1) , Ez(2) (τ ) = e−iτ (1⊗σz ) , (36)
H1 , H̃ were also chosen to be nondegenerate, with spectra
π
{3 ± 2π , −3, 1} and {1 ± 2√5 2 π , 1 ± 2√3 2 π }, respectively, a for small τ . For particular values τk = + kπ , k ∈ N, the
2
property also shared with the off mode, Eq. (7). Nondegener- unitaries Ex(1,2) produce single-qubit flips, while Ez(1,2) generate
acy plays no role in the above argument but is desirable in any phase shifts,
physical implementation in order to avoid spurious transitions Ex(1) (τk ) ∝ X ⊗ 1, Ex(2) (τk ) ∝ 1 ⊗ X,
due to coupling with external degrees of freedom.
Ez(1) (τk ) ∝ Z ⊗ 1, Ez(2) (τk ) ∝ 1 ⊗ Z. (37)
D. Symmetry-protected logical operations As a first figure of merit, we have numerically evaluated
Now let us consider how robust our proposal is against the average fidelity of computational sequences belonging to
errors. We first mention a somewhat obvious fact about pa- the set {V, W}, when affected at each computational step by
rameter noise. When a quansistor is in its symmetric form, an error randomly chosen among {Ex(1,2) (τ ), Ez(1,2) (τ )}, for a
062610-6
small fixed value of τ . (In certain situations, static imper-

fections are known to dominate random fluctuation errors
[49], to be considered next.) For comparison, we have re-
peated the same steps with two other computational sequences
belonging respectively to two other strictly universal sets,
namely, the Kitaev set {H ⊗ 1, CP(i), SWAP} [9], and the set
{A, SWAP}. Here H ⊗ 1 is the first-qubit Hadamard gate, and
CP(i) is the controlled i-phase gate

1 1 1
H⊗1= √ ⊗ 1, CP(i) = diag(1, 1, 1, i).
2 1 −1
(38)
The A gate is a variant of the CNOT gate, and is part of
a class of unitaries A(φ, α, θ ) known to be strictly universal
individually (in combination with the SWAP gate) for many xz
values of the parameters [7,10,50]. Specifically, the A gate is FIG. 3. Average fidelity Favg , Eq. (43), against single-qubit x and
⎛ ⎞ z rotations, Eqs. (35) and (36), as a function of scaled computa-
1 0 0 0 tional length m̃, for three strictly universal sets: {V, W} (upper),
⎜0 1 0 0 ⎟ {A, SWAP} (lower), and {H ⊗ 1, CP(i), SWAP} (middle). The cou-
A=⎜ ⎝0 0 i cos(1) − sin(1)⎠.
⎟ (39) pling τ is set to 5 × 10−4 . We find power-law best fits 1 − Favg xz
=
0 0 − sin(1) i cos(1) 1 − α m̃β with respective powers β1 = 1.10, β2 = 1.95, β3 = 1.37.
The scaled computational length takes account of scaling effects,
Let us describe the method more precisely. For each uni- as discussed in the text. The maximum value m̃ = 403 corresponds
versal set S considered, and for integers m ∈ [1, 400], r ∈ to sequences of length m = 403 for {V, W} and {A, SWAP}, and
[1, 100], we generate the sequence GS1,r , . . . , GSm,r of gates m = 253 for Kitaev.
picked randomly from S (with equal probabilities). We call
m the computational length. We also generate sequences relevant from the point of view of quantum-coherent compu-
E1,r , . . . , Em−1,r of errors picked randomly (with equal prob- xz
tation. The power-law best fit Favg = 1 − α m̃β gives
abilities) from {Ex(1,2) (τ ), Ez(1,2) (τ )}, with τ = 5 × 10−4 . The ⎧
ideal computations are then ⎨1 − 3.0 × 10−7 m̃1.10 for {V, W},
Favg ≈ 1 − 3.0 × 10−8 m̃1.95
xz
for {A, SWAP},
Gm,r
S
= m
i=1 Gi,r ,
S
(40) ⎩
1 − 3.0 × 10−7 m̃1.37 for {H ⊗ 1, CP, SWAP}.
while the noisy computations are (44)
Manifestly, the set {V, W} fares much better than the other
S = GS m−1 Ei,r GS .
G (41) two against this type of error, with an almost linear decay of
m,r m,r i=1 i,r
fidelity (depending on the degree of error anisotropy). When
(The symbol “” indicates composition from right to left.) As
equiprobable x-, y-, and z-rotation errors are considered, with
an illustration, a noisy computation of length m = 5 from our
= VE4 VE3 WE2 VE1 W. x, z coupling τ = 5 × 10−4 and variable y coupling τy , we
universal set {V, W} could be, say, G
S being find that the advantage of {V, W} over the Kitaev set narrows
The average fidelity of the rth computation G m,r down with increasing τy , vanishing at around τy /τ = 0.55
1 S,† S 2 (not shown). And {V, W} still outperforms the set {A, SWAP}
xz
F̃avg (S, m, r) = Tr Gm,r Gm,r , (42) when τy /τ = 1 (not shown). This hard-y-axis, easy-xz-axes
4 anisotropy is a nontrivial property of the set {V, W}. (Ad-
we finally average over all computations ditional numerical results point to the special role of the W
gate.) It is worth emphasizing that, while being more sensi-
1 xz
100
tive to y rotations, the set {V, W} outperforms the other two
xz
Favg (S, m) = F̃ (S, m, r). (43)
100 r=1 avg universal sets with respect to both x- and z rotations.
Let us also mention that the selected type of noise is by
While the simplest comparison of the three gate sets would be no means exhaustive, and was primarily chosen for ease of
to plot average fidelity as a function of computational length comparison with more common, qubit-based universal sets. It
m, this is not necessarily a fair comparison since the Kitaev could nevertheless be realistic in certain implementations. For
set, consisting of three rather than two gates, can presumably example, in a square of charge quantum dots with Gray-code
approximate a given unitary operator to the desired precision logical basis (see Fig. 4), a thermal photon could stimulate
in fewer gates by a factor log3 2 ≈ 0.63. Accordingly, we tunneling events along the sides of the square parallel to
define a scaled computational length m̃ which is m for {V, W} polarization, increasing the likeliness of the corresponding x
and {A, SWAP}, and m log2 3 ≈ 1.59m for Kitaev. A plot of rotation, of which X ⊗ 1 and 1 ⊗ X are particular instances.
xz
Favg as a function of the scaled computational length m̃ is In the same setup, the presence of a resonator near one side
given in Fig. 3 for each universal set considered, {V, W}, of the square (as discussed in Sec. V B) could modify the
{A, SWAP}, and {H ⊗ 1, CP(i), SWAP}. The region shown effective self-energies of the two closest dots. In a first-order
lies within the stage of polynomial decay, and does not show treatment, the corresponding logical states would be affected
the decaying exponential behavior of the saturation stage, less by an identical phase factor, resulting in the dephasing of
062610-7
(a) (b)
FIG. 4. (Fractional) X ⊗ 1 and Z ⊗ 1 errors in quantum dots

with Gray-code logical basis. (a) Fractional X ⊗ 1 induced by an
environmental thermal photon. (b) Fractional Z ⊗ 1 induced by the
presence of a resonator near the right-hand side of the square. States
|11 and |10 acquire an extra phase factor eiφ .
parallel sides of the square, i.e., a z rotation, of which Z ⊗ 1

xyz
and 1 ⊗ Z are particular instances. FIG. 5. Average fidelity Favg , Eq. (46), against the equiprobable
We should add that, through the technique of circuit ran- Pauli error set (45), as a function of scaled computational length
domization, coherent Markovian noise can be tailored into m̃, for three strictly universal sets: {V, W} (upper), {A, SWAP}
effective stochastic Pauli noise with the same error rate (lower), and {H ⊗ 1, CP(i), SWAP} (middle). The couplings τa are
normally distributed with standard deviation 10−4 . The means are
[51–53]. Coherent error rate, although not explicitly consid-
τx = τz = 10−3 , and τy = 10−4 . We find power-law best fits
ered here, is well quantified by gate fidelity against stochastic
1 − Favg
xyz
= 1 − α m̃β with respective powers β1 = 1.10, β2 = 1.81,
Pauli errors. The technique was also experimentally observed
β3 = 1.31. The scaled computational length takes account of scaling
to largely suppress signatures of non-Markovian errors [52]. effects, as discussed in the text. The maximum value m̃ = 403 corre-
As a second figure of merit, let us consider fidelity against sponds to sequences of length m = 403 for {V, W} and {A, SWAP},
a fluctuating noise corresponding to the larger error set and m = 253 for Kitaev.
Ea(1) = e−iτa (σa ⊗1) ,
set (45), but we now assume that interactions with the envi-
Eb(2) = e−iτb (1⊗σb ) , (45)
ronment are such as to make the system more prone to x and
Eab = e−iτa τb (σa ⊗σb ) , z rotations. For definiteness, the eight errors Ea(1) , Ea(1) , Eab ,
for a, b ∈ {x, z}, are picked randomly with probability 3/31,
for a, b ∈ {x, y, z}, and normally distributed couplings τa , τb . while the seven remaining errors, each containing at least one
The 15 corresponding generators constitute, along with the y rotation, are picked with probability 1/31. The couplings
identity, a basis for all 4 × 4 Hamiltonians. For now, these 15 τa are now identically distributed without bias, τa = 0, and
errors are picked with equal probability 1/15 to generate the with standard deviation 10−3 . The average fidelity is
noisy computations of Eq. (41), and the average fidelity
1 xyz
2500
1
900 xyz
Favg (S, m) = F̃ (S, m, r), (48)
xyz
Favg (S, m) = xyz
F̃avg (S, m, r) (46) 2500 r=1 avg
900 r=1
where r enumerates 50 random generations of τx , τy , τz times
is evaluated as a function of computational length m. Here 50 random noisy sequences for each generation. In Fig. 6 we
r enumerates 30 random generations of τx , τy , τz times 30 xyz
plot Favg as a function of the scaled computational length m̃,
xyz
random noisy sequences for each generation. In Fig. 5 we defined below Eq. (43). The power-law best fit Favg =1−
xyz β
plot Favg as a function of the scaled computational length α m̃ gives
m̃, defined below Eqn. (43). If the interactions with the en- ⎧
vironment are such as to produce a stronger bias on x and z ⎨1 − 2.2 × 10−7 m̃1.24 for {V, W},
rotations, then the set {V, W} is at an advantage. This is seen Favg ≈ 1 − 1.1 × 10−7 m̃1.63
xyz
for {A, SWAP},
⎩
in Fig. 5 where τx and τz have average τx = τz = 10−3 , 1 − 5.0 × 10−7 m̃1.14 for {H ⊗ 1, CP, SWAP}.
while τy has on average one order smaller, τy = 10−4 . All (49)
standard deviations are equal to 10−4 . The power-law best fit In spite of the fact that the Kitaev set has a smaller β ex-
xyz
Favg = 1 − α m̃β gives ponent than {V, W}, we find that {V, W} is at an advantage,
⎧ up to m̃ ∼ 3500, in the presence of a hard-y-axis anisotropy.
⎨1 − 2.1 × 10−7 m̃1.10 for {V, W}, Evaluating the performance of {V, W} on a larger error set,
Favg ≈ 1 − 2.0 × 10−8 m̃1.81
xyz
for {A, SWAP}, generated by linear combinations of Pauli tensors, is the object
⎩
1 − 1.5 × 10−7 m̃1.31 for {H ⊗ 1, CP, SWAP}. of a future work. The use of fractional V and W operations is
(47) also likely to help reducing errors in variational algorithms,
where small-angle rotations typically abound. Error-divisible
For this degree of error anisotropy, the set {V, W} presents gates implement these small-angle operations directly, with-
an almost linear decay of fidelity. out using long, noisy sequences of full rotations [54].
As our third and last figure of merit, we once more consider Although we have been concerned with the short-m̃ stage
fidelity against a fluctuating noise corresponding to the error of polynomial decay, it should be mentioned that for larger m̃,
062610-8
xz
xyz FIG. 7. Average fidelity Favg against single-qubit x and z rotations
FIG. 6. Average fidelity Favg , Eq. (48), against the Pauli error
as a function of asymmetry parameter , for on-site energy asymme-
set (45), as a function of scaled computational length m̃, for three
try A0 , nearest-neighbor asymmetry A1 , and second-nearest-neighbor
strictly universal sets: {V, W} (upper), {A, SWAP} (lower), and
asymmetry A2 . [A0 , A1 , and A2 are defined in Eqs. (50) and (51).] The
{H ⊗ 1, CP(i), SWAP} (middle, just below upper). Errors without a
computational length is m = 100. The average is obtained over 200
y rotation have probability 3/31; other errors have probability 1/31.
iterations.
The couplings τa are normally distributed without bias, and with
standard deviation 10−3 . The scaled computational length takes ac-
count of scaling effects, as discussed in the text. The maximum value form h jk = |h jk |eiθ jk . In the lattice picture, vertices | j stand
m̃ = 403 corresponds to sequences of length m = 403 for {V, W} and θ jk
{A, SWAP}, and m = 253 for Kitaev. for the matter field, and the phases on the links | j −→ |k
k
correspond to a U(1) gauge potential θ jk = j A · dr. Paths
some of the curves plotted in Figs. 3, 5 and 6 present fidelity around elementary triangular plaquettes yield gauge-invariant
revivals (“echoes” in the Loschmidt echo [55–57], not shown) plaquette fluxes,
before reaching the large-m̃ saturation stage.
For implementation purposes in realistic, nonideal plat- j = A · dr, (52)
forms, it is important to understand the effect of slightly j

breaking the ω-invariance symmetry of the set {V, W}. For which may be written A · dr = j (∇ × A) · ds. A Her-
simplicity’s sake, we consider once more four quantum dots j
mitian Hamiltonian has θ jk = −θk j , and it is straightforward
arranged in a square, and perform fidelity simulations in the
to check directly that the field B = ∇ × A is divergence-free
presence of a systematic asymmetry A in the Hamiltonians.
in the tetrahedron,
Specifically, ω-invariant Hamiltonians H are replaced with
H + A for on-site energy asymmetry, j = 0. (53)
⎛ ⎞
0 0 0 j
⎜0 0 0 0⎟
A0 = ⎜⎝0 0 0 0⎠, 0,
⎟ (50) As a consequence, only three plaquette fluxes are linearly
independent, and the planar and tetrahedral models are
0 0 0 0 completely equivalent. (The four-level quansistor is essen-
and asymmetry in nearest-neighbor coupling strengths, and in tially two-dimensional. This is in contrast to higher-level
second-nearest-neighbor coupling strengths, respectively,
⎛ ⎞ ⎛ ⎞
0 0 0 0 0 0 (a) (b)
⎜ 0 0 0⎟ ⎜0 0 0 0 ⎟
A1 = ⎜ ⎟ ⎜
⎝0 0 0 0⎠, A2 = ⎝ 0 0 0⎠. (51)
⎟
0 0 0 0 0 0 0 0
xz
For each asymmetry type, average fidelity Favg against single-
qubit x and z rotations is plotted as a function of in Fig. 7. We
detect no singular effect of the asymmetries, with fluctuations
well within 10−5 , and a minor dependence on .
E. Gauge potentials for ω classes FIG. 8. Core displayed as either planar or tetrahedral. The
parameters are as in Fig. 1, with τ = |τ |eiθ , γ ∈ R, and ω =
We now provide a “lattice gauge field” description of ω- eikπ /2 . (a) Transition amplitudes. The part of the Hamiltonian in
rotation invariance. In Fig. 8 the core is displayed so that span(Jω , Jω3 ) is proportional to τ (solid black). The part of the Hamil-
it can be visualized as either planar (as shown) or tetrahe- tonian in span(Jω2 ) is proportional to γ (dotted red). (b) Gauge phases
dral (by raising the central point |4). The complex hopping on links for γ > 0. If γ ∈ R<0 , there is an additional phase of π on
parameters linking the sites (the link variables) have the θ13 and θ24 (dotted red). If γ = 0, then both θ13 and θ24 vanish.
062610-9
F. Physical implementation
In the previous section, we have argued that Hamil-
tonians from different ω classes may have different flux
structures, with three linearly independent plaquette fluxes.
They could therefore be realized by applying magnetic fields
onto two-dimensional or three-dimensional charged systems
with initial Hamiltonians in the form of the off-mode Hamilto-
nian, Eq. (7). The topological (rightmost) flux structure from
Fig. 9, for instance, could be produced from a very long
and thin solenoid penetrating a tetrahedron through one face,
FIG. 9. Flux structure modulo 2π for H ∈ span(Jω , Jω3 ). The pa-
and isotropically releasing a flux of 2π at the center of the
rameters are τ = |τ |eiθ , and ω = eikπ /2 . The first structure depends
tetrahedron. This flux structure properly belongs to class Y ,
on the coupling τ . The second structure is topological, and depends
and cannot be realized in class X .
only on the class ω.
In this section we sketch how the classes X and Y could be
implemented in a wide range of physical systems, comprised
of either charged or neutral levels, using the techniques of
synthetic gauge fields. The appearance of gauge structures
quansistors, which are intrinsically higher-dimensional, as ex- in systems with parameter-dependent Hamiltonians [58,59] or
plained in the Outlook.) Of the six gauge phases θ jk , j < k, time-periodic Hamiltonians [60] is well known. In the former
three are independent and generate a manifold of Hamiltoni- case, and when the adiabatic approximation holds, the dy-
ans for each given flux structure. It is convenient to distinguish namics of an adiabatically evolving particle can be projected
ω-circulant Hamiltonians by their flux structure or “magnetic” onto the subspace spanned by the mth eigenstate ψm [R(t )].
field, whether fundamental or synthetic. Hamiltonians with The resulting effective Schrödinger equation for ψm [R(t )]
different flux structures belong to gauge-inequivalent classes, involves a Berry connection A(R) playing the role of a
and are measurably different. Figure 8 displays (a) transition gauge potential, through the substitution p → p − A in the
amplitudes and (b) gauge phases. As before, the parameters effective
are τ = |τ |eiθ , γ ∈ R, and ω = eikπ/2 . The corresponding flux Hamiltonian, or equivalently, as a geometric phase
exp i dR · A acquired by ψm [R(t )] over the displacement.
structures (modulo 2π ) are represented in Figs. 9 and 10. For This has been shown to occur in mechanical systems [61],
any Hamiltonian of class X , the flux structure is as in the left molecular systems [62], and condensed matter systems [63].
diagram of Fig. 9. Therefore, the topological flux structure Similarly, for systems driven by fast time-periodic modula-
j ≡ − π2 observed in the right diagram of Fig. 9, character- tions (Floquet engineering), one may consider the evolution
istic of Hamiltonians H ∈ span(Y, Y 3 ), cannot be realized by at stroboscopic times tN = NT , where T is the driving period
any Hamiltonian of class X . Symbolically, span(X, X 2 , X 3 ) [47,64]. Here again, the resulting effective dynamics has been
span(Y, Y 3 ). Similarly, the topological flux structure observed shown to yield nontrivial gauge structures in different plat-
in Fig. 10 cannot be realized by Hamiltonians of class X , forms such as condensed matter systems [65,66], photonics
hence span(X, X 2 , X 3 ) span(Y 2 ). On the other hand, by [67,68], ultracold atoms in optical lattices [69–74], and ions
combining the diagrams of Figs. 9 and 10 we see that in microfabricated traps [75,76]. In a lattice with coordination
span(X, X 2 , X 3 ) ∼
= span(Y, Y 2 , Y 3 ). Indeed, a matrix of class number d, nearest-neighbor hopping terms Km,m+u |mm +
X with X coefficient τ = |τ |eiθ is gauge-equivalent to a ma-
1
u| act on wave functions as

trix of class Y with Y 1 coefficient τ = |τ |eiθ if and only if
θ ≡ θ − π /4 mod 2π . In particular, the Hamiltonians H1 and ψ (m) → Km,m+u ψ (m + u) = Km,m+u e−iu·p ψ (m), (54)
H̃ from the universality proof, Eqs. (24) and (25), belong to
where naturally p is the momentum operator and u is a vector
inequivalent flux structures (although it can be shown that this
of unit norm in Zd . In the presence of an effective gauge po-
inequivalence is not generic).
tential A(m), the Peierls substitution p → p − A(m) amounts
to the complexification of real hopping parameters
Km,m+u → Km,m+u eiu·A(m) = Km,m+u eiθm,m+u . (55)
The Peierls phases θm,m+u may also depend on internal de-
grees of freedom (pseudospin) and can then be thought of
as resulting from an artificial or synthetic non-Abelian gauge
field [47]. For the implementation of the classes X and Y , we
need to realize the gauge-invariant flux structures described in
Sec. III E, whether fundamental or artificial. One possibility is
to Floquet engineer Peierls phases as in the Hamiltonians (14)
and (20). In the former, we have Peierls phases θ j, j+1 ≡ θ ,
and all others zero. In the latter, we have instead θ j, j+1 =
FIG. 10. Flux structure modulo 2π for H ∈ span(Jω2 ). The pa- θ + (π /2) j−1 and θ13 = π , and all others zero.
rameters are γ ∈ R>0 , and ω = eikπ /2 . If γ < 0, there is an additional In [71], for instance, lattice shaking is used to prompt a fast
flux π in each plaquette. If γ = 0, all fluxes vanish. periodic modulation of the on-site energies of a tight-binding
062610-10
Hamiltonian analogous to our off-mode Hamiltonian, Eq. (7): with effective self-energies
∞ (E ) = + tc2 (E )
H (t ) = − Ki j ai† a j + [i + vi (t )]ai† ai , (56) !
E (E + i0+ )2 − 4
i j i = + tc2 − . (62)
2 2
where Ki j > 0, vi (t ) = vi (t + T ), and vi T =

1 T
dt vi (t ) = 0. Using Floquet analysis, the resulting Because ω-rotation invariance is preserved, the eigenstates are
T 0
effective time-independent Hamiltonian proves to be of the energy-independent, and still given by (15). The correspond-
form ing effective eigenvalues are obtained from the isolated levels
λk , (16), by the replacement → ∞ (E ):
Heff = − j e ai a j +
Kieff iθi j † eff
ai† ai , (57)
kπ
i j i λk,∞ (E ) = ∞ (E ) + 2|τ | cos θ + + (−1)k γ
2
with complex tunneling amplitudes
= tc2 (E ) + λk (63)
i(w j −wi )/h̄
Kieff e iθi j
= Ki j e T , (58)
j
for k = 1, . . . , 4. But these are not effective eigenenergies as
t t can be seen from the Green’s function:
where wi (t ) = − t0 dt vi (t ) + t0 dt vi (t )T . As long as the

driving functions break certain symmetries, the Peierls phases |φk φk |
Gcore (E ) = [E − H∞ (E )]−1 = ; (64)
can be varied smoothly to any value between 0 and 2π .
k
E − λk,∞ (E )
Producing nontrivial Peierls phases that cannot be gauged
away may require additional static structure, like large energy the effective energy levels of the core-with-leads are fixed
offsets | j − i | Ki j [71]. In our setup, these large energy points Ek = λk,∞ (Ek ). (From now on the symbol “” will
offsets are already present in the off-mode Hamiltonian to always indicate an effective energy due to the presence of the
effectively suppress spontaneous transitions between logical leads.) From (62), and the convention used for the definition
states (position eigenstates). of the complex square root,
" $
1 tc2 tc2 # 2
IV. COUPLING TO LEADS Ek = 1− λk − λk − 4(1 − tc )
2
1 − tc2 2 2
" $
tc2 #
We now consider the effect of the semi-infinite leads on the
core system. As indicated in the Hamiltonian (2) and in Fig. 1, 1 tc2
= 1− λk − i 4(1 − tc ) − λk . (65)
2 2
each site (for example, a quantum dot) is tunnel-coupled to its 1 − tc2 2 2
own lead (which could be, for example, a semi-infinite spin
Since each k mode is decoupled from the others, we have
chain) but the parameters and coupling constants of the four
chosen λk 0 with no loss of generality. Equation (65) and
leads are chosen to be identical. The transition amplitudes
the corresponding expression for negative λk is obtained
in the leads are set to unity, and the lead-to-site coupling tc
in Appendix C by analytically solving the core-with-leads
can be chosen real and positive with no loss of generality.
Schrödinger equation. Because the effective eigenstates do
Coupling the core to the leads may serve to model the core’s
not depend on the scattering energy, it is easy to define
immersion in its immediate environment, and that is the point
a first-order effective core Hamiltonian which is energy-
of view adopted in Sec. IV A. Alternatively, the leads may
independent:
represent designed transmission wires between the core and
distant devices. This perspective is explored in Sec. IV B. |φk φk |
Gcore (E ) ≈ (E − Heff )−1 = . (66)
It is shown in Appendix B that the effect of the leads on
k
E − Ek
the core Hamiltonian (8) can be summarized in an effective,
energy-dependent diagonal offset: In the ordered eigenbasis {|φ1 , |φ2 , |φ3 , |φ4 } we have
φ j |Heff |φk = E j δ jk . All the results from Sec. III can now be
H∞ (E ) = H pos (g, ω) + tc2 (E )1, (59)
modified by the replacement λk → Ek . Note that Ek is real if
where (E ) is the surface Green’s function of a semi-infinite and only if |λk | 2 1 − tc2 + O(tc4 ). (More precisely, |λk |
lead: 2 − tc2 , as shown in Appendix C [see (C22) and (C23)]. Since
any path in (λ1 , λ2 , λ3 , λ4 )-space corresponds to a unique path
E (E + i0+ )2 − 4
(E ) = − . (60) in parameter space (, α, β, γ ), each Ek can be made real or
2 2 complex independently of the other three. Thus, each eigen-
It follows that ω circulation is preserved. For instance, when state can be made to evolve unitarily or not by adjusting the
ω = 1 (class X ) we have the circulant effective Hamiltonian internal parameters of the core, permitting exquisite control
⎡ ⎤ over (partial) decoherence [77].
∞ (E ) τ γ τ† For class Y (i.e., ω = eiπ/2 ), expressions identical to (65)
⎢ ⎥ and (66) hold with the replacements φk → χk (21) and λk →
⎢ τ† ∞ (E ) τ γ ⎥
⎢
H∞ (E ) = ⎢ ⎥ (61) λk (22). Again, full control over the core energies allows one
⎣ γ τ† ∞ (E ) τ ⎥ ⎦ to make each Ek real or complex independently of the other
τ γ τ †
∞ (E ) three.
062610-11
The same is true, with a caveat, when the core is in the off mode offers comparatively less flexibility because of the
nonsymmetric off mode, condition |i − j | Ki j > 0, although crossing levels is ill-
advised in any mode.) The dissipation of the kth eigenstate is
Hoff = − Ki j ai† a j + i ai† ai , (67) characterized by the tunable dynamical rate
i j i
τ −1 (λk ) = |Im(Ek )|, (70)
where large energy offsets |i − j | Ki j > 0 ef-
fectively suppress spontaneous transitions, so that which is found from (65) to be a continuous function of
position i is almost a good quantum number. Then
√ controllable) energy λk , with values in the interval
the (fully
again Gcore (E ) ≈ (E − Heff )−1 = m |mm|
E −m
with m = [0, 2/3 3] ≈ [0, 0.385]. The ability to prevent the eigen-
2 2
1 t t
[(1 − 2c )m − 2c m2 − 4(1 − tc2 )]. This time care must states from escaping to the leads allows us to consider the
1−tc2
core as a versatile quantum memory unit that can protect a
be taken to maintain the large-offset condition when lowering
state k ak |φk for a long time, and then release it entirely or
the m ’s below the escape threshold, in order to prevent
partially at a later time. (In what follows |φk will stand for an
spurious logical transitions.
eigenstate of either class X or Y , unless specified otherwise.)
This is the perspective that we adopt in this section, and to do
A. Leakage-free logical operations so it is convenient to go beyond the first-order Green’s func-
In this section we show that, even in the presence of leads, tion analysis that we have employed so far, which loses track
all logical gates can be realized without leakage, and are of effective core eigenstates as they escape, with no possibility
still symmetry-protected from unbiased parameter noise. It of ever coming back. We emphasize that our proposal assumes
is sufficient to consider single-pulse, ω-circulant gates, since nothing other than the existence of a flat symmetry class in the
they are universal for quantum computation. We could use physical support of information, an aspect which to the best
only the gates {V, W} from Sec. III C, for instance, which of our knowledge has not been exploited in quantum memory
are especially resilient against x- and z-rotation errors. Let technologies [78–82].
a single pulse producing the gate U be given in the time We consider the following finite system: it consists of two
interval t ∈ [0, T ] by the path [λ1 (t ), . . . , λ4 (t )] in the R4 ω-circulant four-level cores standing face to face, and con-
manifold of eigenenergies of the bare core. With a lead nected by four identical leads, each comprised of L sites. One
coupled to each site, eigenenergies λk are modified to pos- may think of it as a square prism of height L, with the cores
sibly complex effective eigenenergies Ek . We must make sure as top and bottom faces. The cores act as memory storage
that |λk | 2 − tc2 at all times to keep Ek real and prevent units. We will identify when and why a state localized on
escape through the leads. If not, rescaling the energies by one core will scatter within characteristic time τs , eventually
n and the T time by 1/n leaves unchanged the unitary U = reaching the other core. Alternatively, we discuss how the lo-
exp(−i 0 dt Hcore (t )). Avoiding the energy band of the leads calized state can be protected from scattering over a timescale
is therefore not an issue. Moreover, for these values of λk , τb τs . The index b stands for bound states, whose presence
the function Ek (λk ) increases monotonically, and hence is or absence determines the dissipation regime.
one-to-one. As an immediate consequence, there is a (unique) In Appendix C we show that the ω-circulant system decou-
path [p1 (t ), . . . , p4 (t )] lying entirely outside the band of the ples into four identical modes, each in the form of two sites of
leads such that self-energies λ, μ connected by a finite lead of L sites. The
single-particle Hamiltonian of one mode is
Ek [pk (t )] = λk (t ), k ∈ {1, . . . , 4}, t ∈ [0, T ]. (68) ⎡ ⎤
λ tc,1
We thus obtain ⎢ tc,1
∗ ⎥
T ⎢ 0 1 ⎥
⎢ ... ⎥
Ueff = exp −i dt Heff (t ) = U. (69) H1P = ⎢ 1 0 ⎥. (71)
⎢ . . ⎥
0
⎣ . . . . tc,2 ⎦
Having reproduced the ideal gate U in the presence of ∗
tc,2 μ
the leads, and recalling that identical leads preserve ω-
rotation invariance, we conclude that the effective gate Ueff (The argument generalizes in a straightforward manner if
is symmetry-protected from unbiased noise in the effective the four-level cores are replaced by N-level cores.) The cor-
parameters Ek . And since the functions Ek (λk ) are very nearly responding Schrödinger equation is easily solved, yielding
linear outside the band, unbiased noise in Ek is equivalent eigenenergies E in implicit form
to unbiased noise in the bare parameters. This completes our
claim that, even in the presence of leads, logical gates can E
˜ −− ˜ UL−2 (E /2)
= , (72)
be realized without leakage and are still protected against − 1
˜ UL−1 (E /2)
unbiased noise in the parameters (whether bare or effective). where
E −λ ˜ = E − μ,
B. Core as quantum memory = , (73)
|tc,1 |2 |tc,2 |2
Within each ω class the effective eigenenergies can be
chosen real or complex independently of one another, and and Un (x) is a Chebyshev polynomial of the second kind.
as a consequence each energy eigenstate can independently The L + 2 solutions of (72) are the system’s eigenenergies.
be made to dissipate in the leads or remain stationary. (The Unsurprisingly, this equation cannot be solved analytically; a
062610-12
FIG. 11. Left-hand (LHS) and right-hand (RHS) sides of the energy constraint equation (72) as a function of the dimensionless energy
E for (L, tc,1 , tc,2 ) = (10, 0.1, 0.1). The LHS curve crosses the RHS curve L + 2 times. (a) For λ = μ = 2.5, the crossings correspond to L
continuum states in the band, plus two bound states near λ. (b) Zoom-in of the neighborhood of E = λ for λ = μ = 2.5. The nearly degenerate
bound state energies E = 2.5049765 and E = 2.5049904 are not yet resolved, but we plot these states in Fig. 12. (c) For λ = μ = 1.4 the
LHS curve crosses the RHS curve L + 2 times within the band. (d) Zoom-in of the neighborhood of E = λ for λ = μ = 1.4, showing one
continuous scattering mode (leftmost) and two hybridized scattering modes with energy separation ∼10−2 .
graphical solution is displayed in Fig. 11. Note that the right- the lead will only be approximately stationary. If the system
hand side (RHS; blue curves) is independent of the couplings evolves for a long time, the state localized on one end will
to the cores and pertains to the spectrum of the leads whereas eventually tunnel through the lead.
the left-hand side (LHS; yellow curves) is independent of All other single-particle solutions, Eq. (74), fall within the
the lead parameters and pertains to the coupling. L solutions band of the leads, E ∈ [−2, 2], with coefficients given by
always belong to the energy band [−2, 2], and form the (per- " $
E −λ
turbed) continuous spectrum of the leads. The two remaining βj = N sin jθ − sin( j − 1)θ , (76)
solutions may lie outside the band (bound states) or within |tc,1 |2
the band (hybridized scattering states). We now discuss these where θ = cos−1 (E /2). For any finite L, there are L scattering
cases in turn. states from the (perturbed) continuous spectrum of the leads.
Let us write single-particle states as A generic feature of these L states is their small amplitude
β0 L
βL+1 at the endpoints. Two continuum states for λ = μ = 2.5 are
|E = ∗
|0 + β j | j + |L + 1, (74) displayed in Figs. 12(c) and 12(d).
tc,1 j=1
tc,2 When bound states are not present, in addition to the con-
tinuum states there will be two hybridized scattering states:
where | j is the state with one particle on site j. The coef-
ficients of bound states, with energies |E | > 2, are given by
(a) (b)
" $
|E | ± λ
β j = (±1) j N sinh jξ − sinh( j − 1)ξ , (75)
|tc,1 |2
for 0 j L + 1. Here ± = sgn(E ), ξ = cosh−1 (|E |/2),
and N is a normalization factor. These states are localized
around both endpoints, decaying exponentially over the char-
acteristic length scale ξ −1 from the endpoints.
Throughout this section, all calculations will be done using (c) (d)
L = 10 and tc,1 = tc,2 ≡ tc = 0.1. The bound states for λ =
μ = 2.5 are displayed in Figs. 12(a) and 12(b). We see that
there is a symmetric state |bS and an antisymmetric state |bA ,
a consequence of the Schrödinger equation symmetry j ↔
L + 1 − j resulting from = ˜ (see Appendix C). When
L → ∞, the limiting expression for β j describes a bound
state localized at the left endpoint and decaying exponentially FIG. 12. The two bound states (a), (b) and two states from the
with distance. Similarly, the limiting expression for βL+1− j continuum (c), (d) for (L, tc,1 , tc,2 ) = (10, 0.1, 0.1) and λ = μ =
describes a state localized at the right endpoint. In that limit, 2.5. (a) Antisymmetric bound state, E = 2.5049765. (b) Symmetric
the eigenvalue equation (72) is equivalent to the fixed-point bound state, E = 2.5049904. The energy separation is ∼10−5 . A
relations E = λ ± |tc,1 |2 (E ) and E = μ ± |tc,2 |2 (E ) for generic feature of bound states is their large amplitude at the end-
the states localized on the left and right, respectively. In the points (sites 0 and L + 1). (c) A symmetric continuum state, E =
Green’s function treatment of the core with semi-infinite leads 0.282. (d) An antisymmetric continuum state, E = 0.827. A generic
(see Appendix B), the same fixed-point relation appears as the feature of continuum states is their small amplitude at the endpoints.
effective self-energy of the core once the leads are traced out. In each graph, the red dot represents a consistency condition on βL+1 .
In the finite-L case, states localized around a single end of See Appendix C, Eq. (C6) for details.
062610-13
√
(a) (b) |ψ = (|bS + |bA )/ 2, which is well localized around the
left-hand dot. Then

|ψ|e−itH |ψ|2 = 21 1 + cos ESb − EAb t . (81)
With the canonical parameter values (L, tc ) = (10, 0.1), this
goes slowly to zero with rate τb−1 = ESb − EAb ∼ 10−5 (and
oscillates back and forth unless L is infinite).
(c) (d) If bound states are not available (e.g., if λ = μ ∈ [−2, 2]),
then a localized state necessarily has an f (E ) with large
support E , and |ψ|e−itH |ψ|2 will decay within time t ∼
O(1/E ) to the oscillating steady state

|ψ|e−itH |ψ|2 ∼ |cS |4 + |cA |4 + 2|cS |2 |cA |2 cos ESh −EAh t.
(82)
FIG. 13. The two hybridized scattering states (a), (b) and two With the same canonical parameter values, this oscillates with
states from the continuum (c), (d) for (L, tc,1 , tc,2 ) = (10, 0.1, 0.1) rate τs−1 = ESh − EAh ∼ 10−2 . With these values, the charac-
and λ = μ = 1.4. (a) Antisymmetric hybridized state, E = 1.4044. teristic escape time of localized states is reduced by a factor
(b) Symmetric hybridized state, E = 1.4226. The energy separation of 103 when effective eigenenergies become complex and
is ∼10−2 . A generic feature of hybridized states is their relatively bound states are no longer available. The overall rate of decay,
large amplitude at the endpoints (sites 0 and L + 1). (c) A symmetric max(E , ESh − EAh ) 4, can be as large as O(1). The above
continuum state, E = 0.282. (d) An antisymmetric continuum state,
analysis shows that the core in either ω class has the ability to
E = 0.827. Continuum states are visually indistinguishable from
receive and release states over the timescale τs or shorter, and
those of Fig. 12.
to store a state over the much larger timescale τb . Switching
between these two coupling regimes is performed by tuning
one symmetric |hS and one antisymmetric |hA . A generic the internal parameters of the core. The same procedure is
feature of these states is their relatively large amplitude at the also possible in the nonsymmetric off mode, with somewhat
endpoints. Such states are displayed in Figs. 13(a) and 13(b) less flexibility due to the large-offset condition.
for λ = μ = 1.4; for this same case example scattering states
with E ∈ [−2, 2] are displayed in Figs. 13(c) and 13(d). C. Qubit initialization and readout
Decay rates in the presence of bound states |bS and |bA
are compared with decay rates in the presence of hybridized Initializations and measurements are naturally performed
states |hS and |hA . Let the state be through position measurements or energy measurements of
either ω class. Position eigenstates correspond to binary-code

L
logical states, as given in Table I, whereas energy eigenstates
|ψ = an |n + cS |ϕS + cA |ϕA , (77) of classes X and Y correspond to columns of F † and (FD)† ,
1 respectively:
where |n is a state from the continuous spectrum, and |ϕS,A
is bound or hybridized. Then |φk = F,k
†
|, |χk = (FD)†,k |. (83)

L
Define, for instance, the POVM {E0 , E1 } with elements the
ψ|e−itH |ψ = |an |2 e−itEn + |cS |2 e−itES + |cA |2 e−itEA .
joint-position projectors
n=1
(78) E0 = |11| + |22|, E1 = |33| + |44|. (84)
Let f (Em ) = |am |2 and assume Em − En ≈ (m − n) where
= 4/(L − 1). In the continuous limit, These operators correspond to the measurement or initializa-
tion of the first qubit only:
ψ|e−itH |ψ ∼ f˜(t ) + |cS |2 e−itES + |cA |2 e−itEA , (79)
E0 = |0000| + |0101| = |00| ⊗ 1,
where f˜ is the inverse Fourier transform of f , and
E1 = |1010| + |1111| = |11| ⊗ 1. (85)
|ψ|e−itH |ψ|2
Similarly, the POVM {E 0 , E 1 } with elements
∼ | f˜(t )|2 + 2 f˜(t )(|cS |2 sin ES t + |cA |2 sin EAt )
E 0 = |11| + |33|, E 1 = |22| + |44| (86)
+ |cS | + |cA | + 2|cS | |cA | cos(ES − ES )t.
4 4 2 2
(80)
corresponds to the measurement or initialization of the second
If f (E ) has support of width E (in [−2, 2]), its in- qubit:
verse Fourier transform f˜(t ) will decay within time t ∼
O(1/E ). A small decay rate implies that |ψ has overlap E 0 = |0000| + |1010| = 1 ⊗ |00|,
almost zero with most states of the continuous spectrum. This E 1 = |0101| + |1111| = 1 ⊗ |11|. (87)
is possible for a localized |ψ only if bound states |bS and
|bA are available, e.g., if λ = μ ∈
/ [−2, 2]. An example is Note that Gray code produces the same output.
062610-14
V. SCALABILITY the other. Notice that

We now propose a scalable technology for universal quan- span{|1, |2} = {|0|ψ | ψ any second qubit state},
tum computation. The idea is strongly reminiscent of classical
span{|3, |4} = {|1|ψ | ψ any second qubit state} (90)
computer architecture, in which operations are decomposed
into elementary two-bit steps to be performed on large arrays (see Fig. 14). Similarly,
of transistors. Here we use the fact that any unitary U on d
qubits can be approximated to arbitrary accuracy by finite span{|1, |3} = {|ψ|0 | ψ any first qubit state},
products of elementary two-qubit unitaries, i.e., operations span{|2, |4} = {|ψ|1 | ψ any first qubit state}, (91)
of the form S = 1⊗m ⊗ K ⊗ 1⊗d−m−2 , where K is a 4 × 4
unitary [6]. For any > 0 there exists a finite sequence of (see Fig. 15). We now demand that the off-mode on-site ener-
such two-qubit unitaries S1 , . . . , Sk achieving gies satisfy
3 − 1 = h̄ν1 = 4 − 2 , (92)
max (U − S1 S2 · · · Sk )|ψ < , (88)
|ψ
2 − 1 = h̄ν2 = 4 − 3 . (93)
where |ψ is any normalized d-qubit state. We thus consider
the possibility of realizing quantum computation on scalable Equations (90) and (92) imply that setting the quansistor
grids of four-level cores. By analogy with the role played into resonance at frequency ν1 prompts the onset of first-
by transistors in classical computation, we may consider the qubit oscillations |0|ψ ↔ |1|ψ with frequency ν1 . By the
cores to be quantum-computational transistors, or more suc- same token, Eqs. (91) and (93) imply that quansistor reso-
cinctly, quansistors. nance at frequency ν2 corresponds to second-qubit oscillations
|ψ|0 ↔ |ψ|1 at frequency ν2 . For this reason, the frequen-
cies ν1 and ν2 may be called qubit splitting and will be used
A. Quansistors
to exchange single qubits between distant quansistors. When
The core, or quansistor, is a four-level tight-binding sys- coupled to a single-mode resonator, the quansistor resonating
tem with the ability to become ω-rotation-invariant for ω = at one of these frequencies νq will effectively look like a single
1, eiπ/2 (classes X, Y ). Until now we have mostly considered qubit coupled to the oscillator as described by the Jaynes-
the quansistor in its symmetric form, performing computation Cummings Hamiltonian
on its double qubit. From the universality proof of Sec. III C,
h̄νq
we know that the set of gates {V, W}, constructed from the HJC = h̄νr a† a + σz + h̄g(a† σ + + aσ − ), (94)
Hamiltonians (24) and (25), is universal on two qubits. This 2
set contains one representative from each class, X and Y , where σz = |11| − |00|. The frequencies ν1 ± ν2 , on
and these representatives realize inequivalent flux structures. the other hand, are not qubit splitting, and correspond to the
The next step is to allow interactions between quansistors. oscillations |00 ↔ |11 and |01 ↔ |10, respectively. The
We choose the most basic two-quansistor interaction, involv- same results can be obtained in Gray code with minor modifi-
ing one qubit from each quansistor. To that end, we need cations.
to materialize the tensor product structure inherent to the
quansistor logical basis {|00, |01, |10, |11}. Thus far, these B. Scalable architecture
qubit states merely label the states of the quansistor, and
need to be factored into pairs of spatially separable qubits Interactions between quansistors are to be performed by
before they can be shared with distinct target quansistors. bringing their qubit-splitting frequencies into resonance. It
Let us devote some attention to the nonsymmetric form of seems desirable to mediate the coupling with single-mode res-
the quansistor, which is also the off mode for computation. onators, allowing distributed circuit elements, and to work in
Note that the off-mode Hamiltonian cannot be the ω-circulant the dispersive regime where two quansistors A, B are mutually
matrix 1, because the degenerate eigenstates of the latter are resonant, but far-detuned from the resonator:
unstable to perturbations. νA = νB = νr . (95)
In the off mode, the Hamiltonian is simply
The interaction then proceeds through virtual photon ex-
Hoff = − Ki j ai† a j + i ai† ai , (89) change, as opposed to real photons in the resonant regime,
i j i alleviating the major drawback of the latter, namely, the
resonator-induced decay due to energy exchange with the
where Ki j > 0, and large energy offsets |i − j | Ki j ef- resonator (Purcell effect) [83]. In the rotating wave approx-
fectively suppress spontaneous transitions, so that position imation, the Hamiltonian in the absence of direct coupling
i is almost a good quantum number in the off mode. (As between the quansistors is
before, the symbol “” indicates an effective energy due to the h̄νq
presence of the leads.) Logical states {|00, |01, |10, |11} H = h̄νr a† a + σz, j + h̄g j (a† σ j− + aσ j+ ),
coincide with position eigenstates {|1, |2, |3, |4}, respec- j=A,B
2 j=A,B
tively. Section III F illustrates how the system can be switched (96)
from (89) to ω-circulant Hamiltonians of class X or Y and where νq is the common frequency of A and B. We use the
back using electrically charged levels and magnetic fields on Baker-Campbell-Hausdorff formula eS He−S = H + [S, H] +
the one hand, and neutral levels and synthetic gauge fields on 1
2!
[S, [S, H]] + · · · to perform the unitary transformation U =
062610-15
(a) (b)
FIG. 14. Quansistor’s off mode. (a) Real effective on-site energies i and negligible transition amplitudes 0 < Ki j |i − j | (not shown).
Position is almost a good quantum number, and logical states (position eigenstates) are almost stationary. Resonance at the qubit-splitting
frequency ν1 corresponds to first qubit oscillation. The quansistor is then effectively a single qubit with basis states |0|ψ and |1|ψ (light
green and dark green, respectively). (b) Energy diagram satisfying the qubit-splitting conditions (92) and (93).
g
exp j j (a† σ j− − aσ j+ ), with detuning = |νr − νq |. To quansistor would be coupled to four identical leads (not shown
first order in g j / we get in the figure) for initialization and readout.
h̄νq
H = h̄νr a† a + σz, j + h̄K (σA+ σB− + σA− σB+ ), (97) VI. OUTLOOK
j=A,B
2
Error-correcting codes are a promising avenue to make
where K = 2h̄gA gB /. The qubit-cavity interaction terms quantum computation fault-tolerant. In our technology, quan-
cancel out exactly, leaving only an effective qubit-qubit in- sistors are the support of the physical qubits, whose states and
teraction with coupling K. interactions are symmetry-protected, to some extent, against
√ This term, when evolved for a
time π /4K, generates the iSWAP gate, which is entangling external influences. Once we identify the dominant errors af-
and equivalent to the CNOT gate [27,83], up to single-qubit fecting them, we can find k-qubit states (typically k = 5, 7, 9)
operations already available within the quansistors. The dis- that are invariant under those errors. These would be encoded
persive regime thus allows for the possibility of long-distance logical qubits. (In the standard Hamming notation, we get a
entangling interactions between quansistors. code of type [kn, n] using kn physical qubits, and having 2n
One possible architecture for universal computation on logical codewords [84].) Having physical universality at hand
2d qubits consists in a closed linear array of d quansistors gives the freedom to encode qubits and operations yielding
coupled through resonators. Each quansistor represents a pair encoded universality. Because our technology is scalable (Q
of qubits, and every qubit is represented exactly once (see quansistors making 2Q qubits), the encoding uses k identical
Fig. 16). Additional qubit-splitting frequencies may be added “processors” instead of one.
to prevent untimely higher-order, mth-nearest-neighbor cou- Crucially, we must determine whether the dominant errors
plings. To cut down space and running time costs, a physical are constrained by the symmetry of the quansistors. We have
implementation of the array would likely be folded, and a res- already observed a significant robustness of the universal set
onator would couple every adjacent pair of quansistors, thus {V, W} against single-qubit and double-qubit x and z rotation
reducing considerably the need for qubit-shuffling operations. errors. As a second source of decoherence, we considered the
Figure 17 schematically depicts a planar-grid computer. Each
FIG. 16. A d-quansistor linear array for universal computation

on 2d qubits. The numbers schematically represent the qubits. Quan-
sistors (larger, blue) perform two-qubit universal operations on qubit
pairs (2 j, 2 j + 1), j = 0, . √ . . , d − 1 (mod d ). Couplers (smaller,
green) perform entangling iSWAP gates on qubit pairs (2 j −
1, 2 j), j = 0, . . . , d − 1 (mod d ). Leads are attached to the lower
faces of the quansistors. With only three qubit-splitting frequencies
in the system, ν1 , ν2 , ν3 , untimely second-nearest-quansistor interac-
FIG. 15. Quansistor’s off mode. Resonance at the qubit-splitting tions occur at higher orders. [For instance, pairs (2,3) and (6,7) both
frequency ν2 corresponds to second qubit oscillation. The quansistor respond to ν1 .] More frequencies may be added to suppress these
is then effectively a single qubit with basis states |ψ|0 and |ψ|1 unwanted exchanges. The required physical resources scale linearly
(light green and dark green, respectively). in the number of qubits.
062610-16
gued in the previous point. Error-correcting codes, notably,

have been developed for qudits, which use d × d Weyl ma-
trices (or equivalently the d × d version of our matrices X
and Y ) [86,87]. Note, however, that the diagram of transition
amplitudes of the general qudit (Kd in graph theory terminol-
ogy) is nonplanar for d 5 [88], in contrast to the essentially
planar diagram K4 of the four-level system (Fig. 8). A gauge
potential implementation of ω-rotation invariant qudits, in the
spirit of Secs. III E and III F, might then be essentially three-
dimensional.
For the purpose of reconstructing the final state of an N-
qubit system, a symmetry based on Pauli tensors of dimension
d = 2N is better suited than ω-rotation invariance (based on
Weyl matrices) as it allows for the maximal number (2N + 1)
of mutually unbiased bases, and complete state characteriza-
tion via state tomography [89,90]. By contrast, the four-level
FIG. 17. A 36-quansistor grid for universal computation on 72 quansistor (N = 2) based on ω-rotation invariance has only
qubits. Quansistors (large blue squares) perform universal two-qubit three unbiased bases: {|m}, {|φk }, {|χk }, the respective
operations.
√ Couplers (smaller green squares) perform entangling eigenbases of Z, X, Y . As for reconstructing the final state
iSWAP between qubits of adjacent quansistors. Each quansistor
of an N-qupit system (where p is an odd prime), the Weyl-
is coupled to four identical leads (not shown) for initialization and
based scheme of dimension d = pN allows for the maximal
readout. The required physical resources scale linearly in the number
number (pN + 1) of mutually unbiased bases, and complete
of qubits.
state characterization via quantum tomography [89,91]. This
could be of value for encrypted communication using a higher
coupling to semi-infinite leads, and have omitted other factors (prime) alphabet (see previous point), and could be based on
such as the effect of a heat bath on the system, the types of p-site quansistors with ω-rotation invariance.
errors that it would produce, and the extent to which it would To perform interquansistor interactions, we have used the
destroy symmetry. simplest possible scenario involving a single qubit from each
Throughout, we have used ω-rotation invariance as the quansistor. It would be worth investigating whether a com-
prototype of a symmetry of flat classes, universal for quan- bined use of resonators and symmetry could make possible the
tum computation, and realistically implementable physically. implementation of robust three- and four-qubit gates, or even
Other flat classes would perform equally well at protecting interactions soliciting three quansistors or more. However,
information and operations, as long as leads (or any other this is beyond the scope of this work.
immediate environment of the clusters) do not break the
corresponding symmetry. Universal sets of gates originating
from nondegenerate Hamiltonians are symmetry-specific, but VII. CONCLUSION
should not be too difficult to find given the relative scarcity
of nonuniversal sets and the completeness of flat classes. The In this work, we have put forward a blueprint for scalable
possibilities of physical implementation, on the other hand, universal quantum computation based on two-qubit clusters
will strongly depend on the chosen symmetry and would have (quansistors) protected by symmetry (ω-rotation invariance).
to be found on a case-by-case basis. We find a significant robustness of the proposed universal
There might be additional value to using larger qubit set against single-qubit and double-qubit x- and z-rotation
clusters, i.e., k-qubit quansistors realized as 2k sites with sym- errors. Embedding in the environment, initialization and read-
metries, for k = 3, 4, 5. All observations from the previous out are achieved by tunnel-coupling each quansistor to four
point regarding protection, universality, and implementation, identical semi-infinite leads. We show that quansistors can
apply here as well. Larger clusters and symmetry classes be dynamically decoupled from the leads by tuning their
would offer protection to k-qubit operations. It could also internal parameters, giving them the versatility required to
allow the encoding of a logical qubit within a single k-qubit act as controllable quantum memory units. With this dy-
quansistor. Encodings with k = 3 (eight sites) can already namical decoupling, universal two-qubit logical operations
correct some single-qubit errors, while some encodings with within quansistors are also symmetry-protected against un-
k = 5 (32 sites) can correct any single-qubit error [1]. biased noise in their parameters. Two-quansistor entangling
There is provable surplus value to using higher alphabets operations are achieved by resonator-coupling their √ qubit-
(ququarts, specifically), instead of qubit pairs, in encrypted splitting frequencies to effectively carry out the iSWAP
communications [85]. Since ω-circulant four-level quansis- gate, with one qubit coming from each quansistor. We have
tors are universal in U(4), i.e., universal for single-ququart also identified a variety of platforms that could implement
operations, and have the ability to dynamically decouple from ω-rotation invariance.
the leads, a quansistor-with-leads could be a versatile mem- The complete tunability of ω-circulant quansistors can
ory unit for ququarts, and become an essential component of be exploited to build highly expressive and trainable pa-
quantum-secure communications. Generalizations to d-level rameterized quantum circuits, to be used as the noisy
quansistors (qudits), with d > 4, is also conceivable, as ar- intermediate-scale quantum (NISQ) component of a quantum-
062610-17
classical hybrid machine learning model [92–94]. These ideas any unitary U to the matrix class {diag[λ1 (g), . . . , λN (g)] |
will be explored in detail in a future publication. g ∈ RN } will produce a commutative family of (Hermitian)
Hamiltonian matrices H(g) with eigenstates independent of
ACKNOWLEDGMENTS
g, and linearly controlled eigenenergies λr (g), i.e., a flat class.
If symmetries other than ω-rotation invariance were preferred
This work was supported in part by the Natural Science for practical reasons, one would replace the diagonalizing
and Engineering Research Council of Canada and by the unitaries F and FD, defined in Eqs. (9) and (10), with the
Fonds de Recherche Nature et Technologies du Québec via the appropriate operations.
INTRIQ strategic cluster grant. C.B. thanks the Department of It is instructive to recast ω rotations in terms of Sylvester’s
National Defence of Canada for financial support to facilitate clock-and-shift matrices (also called Weyl’s matrices or gen-
the completion of his Ph.D. eralized Pauli matrices)
⎛ ⎞ ⎛ ⎞
1 0 1 0 0
⎜ ω ⎟ ⎜ ⎟
APPENDIX A: MATHEMATICAL FRAMEWORK
Z4 = ⎜ ⎟, X4 = ⎜0 0 1 0⎟,
⎝ ω 2 ⎠ ⎝ 0 0 0 1⎠
This section describes some mathematical aspects of ω-
ω 3 1 0 0 0
rotation invariance. Although this paper has focused on a
(A2)
four-level system, many of the results are easily generalized.
with ω = eiπ/2 . For k = 0, . . . , 3 we have the identity
Here we consider the more general case of an N-level system.
For multiple reasons it may be desirable to have full control Z4k X4 = Jeikπ/2 . The matrices {Z4k X4j }k, j=0,...,3 constitute a non-
over the N eigenenergies of the system. In what follows we Hermitian trace-orthogonal basis for gl(4, C), and in fact the
consider what we propose to call flat classes of Hamiltoni- same is true of their obvious N-dimensional generalization,
ans: classes of Hermitian matrices {H (g) | g ∈ RN } with a with ω = ei2π/N , which span gl(N, C) and are orthogonal
common eigenbasis, and real eigenenergies λ1 (g), . . . , λN (g) under the Hilbert-Schmidt inner product. If N = 2, they re-
in one-to-one linear correspondence with the values of the duce (up to a factor) to the Pauli matrices. The matrices
XN and ZN are central to Weyl’s formulation of periodic
parameters, that is, λr (g) = s gs λsr with det[λsr ] = 0. (The
term flat refers to vanishing Berry curvature in g space.) A finite-dimensional quantum mechanics where they respec-
class of Hamiltonians is flat in this sense if and only if it can tively correspond to finite position and momentum shifts:
be represented as a sum of N × N Hermitian matrices, XN = ei(2π/N ) p̂ , XN |x = |x − 1 mod N,

N ZN = ei(2π/N )x̂ , ZN | p = |p + 1 mod N, (A3)
H (g1 , . . . , gN ) = gs Hs , [Hs , Hr ] = 0 (∀s, r), (A1)
s=1 where of course |x and | p are position and momentum
eigenstates, respectively. Thus ω-rotation invariance is a sym-
and the N × N matrix [λsr ] of all eigenvalues of the Hs ’s metry in quantum (or optical) phase space, and is not found
is nonsingular, det[λsr ] = 0. (The key observation is the in other, internal-space generalizations of Pauli matrices like
−1
action
of the diagonalizing unitary: U ( s g s Hs )U = Pauli tensor products or Gell-Mann matrices.
s g s diag(λ s1 , . . . , λ sN ) = diag[λ 1 (g), . . . , λ N (g)].) Flat In any dimension 2, the eigenbases of ZN , XN , and ZN XN
classes therefore coincide with N-dimensional commutative are mutually unbiased: a measurement in one (orthonormal)
algebras of N × N Hermitian matrices, the nonsingularity basis {|ψr } provides no information about measurements in
of [λsr ] being equivalent to the linear independence of the another (orthonormal) basis {|ηs } because |ψr |ηs | = √1N for
Hs ’s. Because each flat class is diagonalized by a common any r, s. In particular when N = 4, the flat classes of Z4 , X4
unitary U (unique up to permutations), the set of all flat (the matrix X in the main text), and Y4 (the matrix Y in the
classes that correspond to the same [λsr ] is in one-to-one main text) have mutually unbiased eigenbases {|m}, {|φk },
correspondence with unitaries modulo permutations. (In and {|χk } respectively [see (15) and (21)].
this paper, we do not consider nonunitarily diagonalizable
matrices, like non-Hermitian PT -symmetric Hamiltonians,
for instance.) The exponentials of a flat class also share the APPENDIX B: EFFECTIVE CORE HAMILTONIAN
common eigenbasis of the class, and their eigenenergies are Consider an N-level core tunnel-coupled to N identical
in (nonlinear) one-to-one correspondance with the values of semi-infinite leads:
the parameters g. There seems to be an interesting connection
between flat classes, on the one hand, and commuting H = Hcore + Hint + Hlead
bases of unitary matrices [89] and stabilizers of quantum ∞
1
4 4 4
error-correcting codes [86], on the other. = hi j ai† a j + tc,i ai† bi,1 + b†i, j bi, j+1 + H.c.
For the purpose of quantum computation a single flat 2 i, j=1 i=1 i=1 j=1
class is clearly not enough because it is commutative. The (B1)
ω-circulant matrices defined in (6) are of the form (A1), and
constitute a flat class for each ω. Independent control over For the time being, the matrix h is not required to be Her-
the energy levels is a prime motivation for using ω-rotation mitian, and the couplings tc,i need not be equal, but could
invariance, but we stress again that this choice of symme- be chosen real positive with no loss of generality since the
try is far from unique. Starting from any nonsingular matrix Hamiltonian is invariant under tc,i → tc,i eiθi , bi, j → bi, j e−iθi .
[λsr ], defining the functions λr (g) = s gs λsr , and applying We let them be complex anyways. Let us restrict the system’s
062610-18
dynamics to the single-particle sector of Hilbert space. For effective Hamiltonian h∞ (E ) is also ω-circulant:
illustration purposes, our examples below will use a core with

3
' (
3
N = 3 levels, but all the results go over to general N. The h= zs Jωs −→ h∞ (E ) = z0 + tc2 (E ) 1 + zs Jωs .
one-particle Hamiltonian matrix H with N = 3 is s=0 s=1
⎡ ⎤ (B11)
h11 h12 h13 tc,1
⎢h21 h22 h23 tc,2 ⎥ In particular, when h is 4 × 4 Hermitian we obtain expression
⎢ ⎥ (59). Again, the fact that h∞ (E ) is unaffected by the presence
⎢h31 h32 h33 tc,3 ⎥
⎢∗ ⎥ of phase factors, dynamical or stochastic, in the core-to-lead
⎢tc,1 0 1 ⎥
⎢ ... ⎥ couplings tc,i is a consequence of (B1) being invariant under
⎢ ⎥
⎢ 1 0 ⎥ tc,i → tc,i eiθi , bi, j → bi, j e−iθi .
⎢ ... ... ⎥
⎢ ⎥
⎢ ⎥
⎢ t ∗
0 1 ⎥, APPENDIX C: ANALYTICAL SOLUTION: TWO CORES
⎢ c,2 ⎥
⎢ . ⎥
⎢ 1 0 .. ⎥ CONNECTED BY FINITE LEADS
⎢ . .
.. .. ⎥
⎢ ⎥ We analytically solve the Schrödinger equation of two
⎢ ⎥
⎢ t ∗
0 1 ⎥ ω-circulant N-level cores (with the same ω, but possibly dif-
⎢ c,3 ⎥
⎢ . . .⎥ ferent core Hamiltonians h1 and h2 ) connected face-to-face by
⎣ 1 0 ⎦
... ... N identical leads of L sites. The Hamiltonian is
1 †
2
(B2) H= a · hs · as + tc,1 a1† · b1 + tc,2 a2† · bL
2 s=1 s
which we write as
" $
L−1
h V + b†j · b j+1 + H.c.,
H= (B3) (C1)
V† 13 ⊗ hlead j=1
in obvious notation. The corresponding Green’s function is where as = (as,1 , . . . , as,N ) and b j = (b1, j , . . . , bN, j ). Leads
G(E ) = (E 1 − H )−1 have hopping energies all equal, and set to unity (thus setting
the scale for all energies). Eigenvalues of the unitary symme-
" $−1 (B4)
E 13 − h −V try (ω-rotation) correspond to superselection sectors. Without
= ,
−V † 13 ⊗ (E 1lead − hlead ) loss of generality, energy eigenstates may be chosen to have
support in exactly one sector k. If h1 , h2 are ω-circulant with
where 1lead is the identity on the single-lead space. The top left ω = eiqπ/2 and q = 0, . . . , 3, the change of basis [Eqs. (9) and
3 × 3 block of the Green’s function, Gcore (E ), can be obtained (10)]
using the blockwise inversion formula
" $−1 " −1 $ as → ãs = FDq as , b j → b̃ j = FDq b j , (C2)
A B
= A − BD−1C ··· . (B5) decouples the system into N identical modes, each in the form
C D ··· ··· of two dots of self-energies λk , μk connected by a finite lead
We obtain of L sites. The single-particle Hamiltonian for one of these
% &−1 modes is
Gcore (E ) = E 13 − h − V [13 ⊗ (E 1lead − hlead )]−1V † ⎡ ⎤
λk tc,1
= [E 13 − h∞ (E )]−1 . (B6) ⎢ tc,1
∗
0 1 ⎥
⎢ ⎥
⎢ ... ⎥
Noticing that H1P = ⎢ 1 0 ⎥. (C3)
⎢ ... ... ⎥
⎣ tc,2 ⎦
[13 ⊗ (E 1lead − hlead )]−1 = 13 ⊗ Glead (E ), (B7) ∗
tc,2 μk
Glead (E ) being the Green function of a single lead, we obtain L
from (B6) Let |ψ = α|0 + j=1 β j | j + γ |L + 1 be a single-particle
eigenstate in sector k, where | j is the state with a single parti-
h∞ (E ) = h + V [13 ⊗ Glead (E )]V † . (B8) cle in the jth site. The Schrödinger equation E |ψ = H1P |ψ
A straightforward calculation yields yields the relations
⎡ ⎤ β1 = β0 , (C4)
|tc,1 |2
h∞ (E ) = h + ⎣ |tc,2 |2 ⎦(E ), (B9)
|tc,3 |2 β j + β j+2 = E β j+1 (0 j L − 1), (C5)
where (E ) is the surface Green’s function of a single lead: βL = β

˜ L+1 , (C6)

E (E + i0+ )2 − 4 E −λ
(E ) = (Glead )00 (E ) = − . (B10) = , (C7)
2 2 |tc,1 |2
The analogous version of (B9) for general N is now obvious. ˜ = E − μ,
(C8)
Remarkably, if h is ω-circulant and |tc,i | = tc for all i, the |tc,2 |2
062610-19
FIG. 18. LHS and RHS of (C15) for L = 11, and |tc,1 | = |tc,2 | = 0.1. Energy E is dimensionless. The LHS curve crosses the RHS curve
L + 2 times. One solution has energy ∼O(λ|tc,1 |2 ), a consequence of weakly broken spectrum symmetry E ↔ −E due to simultaneous nonzero
tc,1 and nonzero λ. (a) For λ = μ = 2.5, the crossings correspond to L continuum states in the band, plus two bound states near λ. (b) Zoom-in
of the neighborhood of E = λ for λ = μ = 2.5. The nearly degenerate bound state energies E = 2.504980 and E = 2.504987 are not resolved
yet, but we plot these states in Fig. 19. (c) For λ = μ = 1.2 the LHS curve crosses the RHS curve L + 2 times within the band. (d) Zoom-in
of the neighborhood of E = λ for λ = μ = 1.2, showing one continuous scattering mode (leftmost) and two hybridized scattering modes with
energy separation ∼10−2 .
∗
where we have assumed that tc,i = 0, and defined β0 = tc,1 α Substituting in (C9) gives βL+1− j = ±β j .We must consider
and βL+1 = tc,2 γ . The bulk equation (C5) is translation- two cases: when E is outside the energy band of the leads,
invariant with general solution and when it is within this energy band.
E
β j = Aei jθ + Be−i jθ , θ = cos−1
. (C9) 1. Bound states (E outside the band)
2
The boundary conditions (C4) and (C6) give the ratio When E is outside the energy band of the leads, we have
˜ θ = iξ if E > 2, and θ = π + iξ if E < −2, where ξ =
A − e−iθ − eiθ cosh−1 (|E |/2) is a real parameter. Then
= iθ = e−2iθ (L+1) −iθ , (C10)
B e − e − ˜ " $
|E | ± λ
and the eigenenergies, E = 2 cos θ , in implicit form β j = (±1) j N sinh jξ − sinh( j − 1)ξ (C17)
|tc,1 |2
( + )
˜ sin θ L −
˜ sin θ (L + 1) − sin θ (L − 1) = 0. with ± = sgn(E ), where N is a normalization factor, and
(C11) where the eigenenergies satisfy the constraint
In terms of Chebyshev polynomials of the second kind

˜ sinh(L + 1)ξ ∓ ( + )
˜ sinh Lξ + sinh(L − 1)ξ = 0.
sin(n + 1)θ
Un (cos θ ) = , (C12) (C18)
sin θ
and recalling the recursion relation Alternatively, we can write the solution as
" $
Un (x) = 2xUn−1 (x) − Un−2 (x), (C13) |E | ± μ
βL+1− j = (±1) Nj
sinh jξ − sinh( j − 1)ξ .
|tc,2 |2
we find
(C19)
E
˜ −− ˜ UL−2 (E /2)
= (C14)
− 1
˜ UL−1 (E /2) A generic feature of these bound states is their large ampli-
tude at the dots. Because there are at least L states from the
as in Eq. (72). In the simplest case where λ = μ and |tc,1 |2 = continuous spectrum (see next section), there are at most two
|tc,2 |2 (identical dots, identical couplings), we have bound states. The bound states for L = 10 and λ = μ = 2.5
(E − λ)(E (E − λ) − 2|tc,1 |2 ) UL−2 (E /2) are displayed in Figs. 12(a) and 12(b). The bound states for
= . (C15) L = 11 are plotted in Figs. 19(a) and 19(b).
(E − λ + |tc,1 | )(E − λ − |tc,1 | )
2 2 UL−1 (E /2) In the limit L → ∞, the constraint is equivalent to
The system’s L + 2 eigenvalues coincide with the solutions
1 1 ξ 1 1 1
of the above equation. Notice that for small |tc,1 | the LHS e2ξ ∓ + e + = eξ ∓ eξ ∓ = 0,
is ∼E + O(|tc,1 |2 ) when E is not in the vicinity of the sin- ˜ ˜ ˜
gularities λ ± |tc,1 |2 . This is illustrated in Fig. 11 for L even (C20)
(L = 10), and in Fig. 18 for L odd (L = 11). In this symmetric yielding e−ξ = ± or e−ξ = ±. ˜ The resulting expressions
case for which = , ˜ all eigenstates are either symmetric for β j and βL+1− j describe bound states localized at either
or antisymmetric. This is seen from (C10), which becomes dot and decaying exponentially with distance over the charac-
A/B = e−2iθ (L+1) (A/B)−1 , implying teristic length ξ −1 . Moreover, the constraint equations for e−ξ
are equivalent to the fixed-point relations E = λ ± |tc,1 |2 (E )
A A and E = μ ± |tc,2 |2 (E ), respectively. These relations can be
= 1 and e−iθ (L+1) = ± . (C16)
B B obtained as normalizability conditions on the eigenstates of
062610-20
(a) (b) or equivalently

" $
1 |tc |2 |tc |2 # 2
E= 1 − λ k − λ − 4(1 − |t c | 2) ,
1 − |tc |2 2 2 k
(C22)
with λk 2 − |tc | , in agreement with (65). For E −2, the
2
condition on gives instead

" $
|tc |2 # 2
(c) (d)
1 |tc |2
E= 1+ λk + λk − 4(1 + |tc | ) ,
2
1 + |tc |2 2 2
(C23)
with λk −2 + |tc |2 . Similar relations hold for the bound
state condition on .
˜
FIG. 19. States from the case L = 11, and |tc,1 | = |tc,2 | = 0.1.
2. Scattering states (E within the band)
(a), (b) The two bound states for λ = μ = 2.5 (a) Antisymmet-
ric bound state, E = 2.504980. (b) Symmetric bound state, E = When E ∈ [−2, 2], θ is real and we find
2.504987. The energy separation is ∼10−5 . (c), (d) The two hy- " $
bridized states for λ = μ = 1.2. (c) Symmetric hybridized state,
E −λ
βj = N sin jθ − sin( j − 1)θ , (C24)
E = 1.1990. (d) Antisymmetric hybridized state, E = 1.2142. The |tc,1 |2
energy separation is ∼10−2 . In each graph, the red dot represents the
where N is a normalization factor. For any finite L, there are
value of βL+1 from the consistency condition (C6).
L scattering states from the (perturbed) continuous spectrum
of the lead. A generic feature of these L states is their small
a dot connected to a semi-infinite lead, by solving the cor-
amplitude at the quantum dots. Additionally, there can be two
responding Schrödinger equation. Alternatively, the Green’s
scattering hybridized states. A generic feature of these states
function treatment of the core with semi-infinite leads,
is their relatively large amplitude on the dots. The hybridized
Appendix B, gives the same fixed-point relations as effective
states for L = 10 and λ = μ = 1.4 are displayed in Figs. 13(a)
self-energies of the core once the leads are traced out. The
and 13(b). States from the (perturbed) continuous spectrum
bound state condition on for E 2 amounts to
) are displayed in Figs. 12(c) and 12(d) and 13(c) and 13(d). The
hybridized states for L = 11 and λ = μ = 1.2 are displayed in
|tc |2 E 2
λk = 1 − E + |tc | 2
− 1, (C21) Figs. 19(c) and 19(d).
2 2
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PhysRevA 106 062610-Quansistor

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PHYSICAL REVIEW A 106, 062610 (2022)

Universal quantum computation with symmetric qubit clusters coupled to an environment

I. INTRODUCTION cation of “simple” quantum universal sets like single-qubit

2469-9926/2022/106(6)/062610(23) 062610-1 ©2022 American Physical Society

tuning. Our ability to fully control the (potentially complex)

TABLE I. Two possible logical bases. Binary code will be used

Position Binary-coded Gray-coded

small fixed value of τ . (In certain situations, static imper-

FIG. 4. (Fractional) X ⊗ 1 and Z ⊗ 1 errors in quantum dots

parallel sides of the square, i.e., a z rotation, of which Z ⊗ 1

V. SCALABILITY the other. Notice that

FIG. 16. A d-quansistor linear array for universal computation

gued in the previous point. Error-correcting codes, notably,

where (E ) is the surface Green’s function of a single lead: βL = β

(a) (b) or equivalently

condition on gives instead

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PhysRevA 106 062610-Quansistor

Uploaded by

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Available Formats

PhysRevA 106 062610-Quansistor

Uploaded by

Document Information

Original Description:

Original Title

Copyright

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Share this document

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PhysRevA 106 062610-Quansistor

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PHYSICAL REVIEW A 106, 062610 (2022)

Universal quantum computation with symmetric qubit clusters coupled to an environment

I. INTRODUCTION cation of “simple” quantum universal sets like single-qubit

2469-9926/2022/106(6)/062610(23) 062610-1 ©2022 American Physical Society

tuning. Our ability to fully control the (potentially complex)

TABLE I. Two possible logical bases. Binary code will be used

Position Binary-coded Gray-coded

small fixed value of τ . (In certain situations, static imper-

FIG. 4. (Fractional) X ⊗ 1 and Z ⊗ 1 errors in quantum dots

parallel sides of the square, i.e., a z rotation, of which Z ⊗ 1

V. SCALABILITY the other. Notice that

FIG. 16. A d-quansistor linear array for universal computation

gued in the previous point. Error-correcting codes, notably,

where (E ) is the surface Green’s function of a single lead: βL = β

(a) (b) or equivalently

condition on  gives instead

You might also like

where (E ) is the surface Green’s function of a single lead: βL = β

condition on gives instead