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Neural-network quantum states for many-body physics

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Abstract

Variational quantum calculations have borrowed many tools and algorithms from the machine learning community in the recent years. Leveraging great expressive power and efficient gradient-based optimization, researchers have shown that trial states inspired by deep learning problems can accurately model many-body correlated phenomena in spin, fermionic and qubit systems. In this review, we derive the central equations of different flavors variational Monte Carlo (VMC) approaches, including ground state search, time evolution and overlap optimization, and discuss data-driven tasks like quantum state tomography. An emphasis is put on the geometry of the variational manifold as well as bottlenecks in practical implementations. An overview of recent results of first-principles ground-state and real-time calculations is provided.

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Data Availability Statement

There is no data associated with the manuscript.

Notes

  1. By efficiently we mean with an amount of computational resources that scales polynomially with the system size

  2. In the context of NQS for fermionic systems, universal representation claims are based on lookup-table arguments, thus not providing any practical information on the classes of states that can be represented by these ansatze

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Acknowledgements

M. M. and J. R. M acknowledge support from the CCQ graduate fellowship in computational quantum physics. The Flatiron Institute is a division of the Simons Foundation. The authors acknowledge useful discussions with Antoine Georges, Christopher Roth, Schuyler Moss, Agnes Valenti, Alev Orfi, Anna Dawid and Anirvan Sengupta.

Funding

Simons Foundation (Matija Medvidović, Javier Robledo Moreno).

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Authors and Affiliations

  1. Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY, 10010, USA

    Matija Medvidović & Javier Robledo Moreno

  2. Department of Physics, Columbia University, New York, NY, 10027, USA

    Matija Medvidović

  3. Department of Physics, Center for Quantum Phenomena, New York University, 726 Broadway, New York, NY, 10003, USA

    Javier Robledo Moreno

  4. IBM Quantum, IBM Research, 1101 Kitchawan Rd, Yorktown Heights, NY, 10598, USA

    Javier Robledo Moreno

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Correspondence to Matija Medvidović.

Appendix A: Derivation of the projected equations of motion

Appendix A: Derivation of the projected equations of motion

In Eq. 15 in the main text, a simple update scheme for variational parameters is presented:

$$\begin{aligned} |\Psi _{\theta + \delta \theta }\rangle = e^{-\delta \tau H} |\Psi _\theta \rangle \end{aligned}$$
(A1)

where \(|\Psi _\theta \rangle = |\psi \rangle / \sqrt{\langle {\psi }|{\psi } \rangle }\) is the normalized version of our variational state of choice, turning the proportionality sign in Eq. 15 into an equality. We note that the differential version of Eq. A1, \(\frac{\textrm{d}}{\textrm{d}\tau } |\Psi _\theta \rangle = - H |\Psi _\theta \rangle\) is known as the Bloch equation and can serve as an equivalent starting point for the following analysis.

In this appendix, we derive the projected equations of motion given in Eq. 16 for variational parameters \(\theta\). We note that the following derivation plays out almost identically for the case of real-time evolution, upon substitution \(\delta \tau \rightarrow i \delta t\). Borrowing notational conventions set up in Sect. 2, we begin by Taylor-expanding both sides of Eq. 15 in small parameters:

$$\begin{aligned} \begin{gathered} e ^{-\delta \tau H} |\Psi _{\theta }\rangle = \left( \mathbbm {1} - \delta \tau H \right) |\Psi _{\theta }\rangle + \cdots \\ \quad |\Psi _{\theta + \delta \theta }\rangle = |\Psi _{\theta }\rangle + \sum _\mu \delta \theta ^\mu \partial _\mu |\Psi _{\theta }\rangle = \left( \mathbbm {1} + \sum _\mu \delta \theta ^\mu {\mathcal {D}} _\mu \right) |\Psi _{\theta }\rangle + \cdots \, . \end{gathered} \end{aligned}$$
(A2)

where we exploit the following chain of identities

$$\begin{aligned} \partial _\mu |\Psi _{\theta }\rangle = \sum _{{\textbf {x}} }\partial _\mu \Psi _\theta ({{\textbf {x}} }) |{{\textbf {x}} }\rangle = \sum _{{\textbf {x}} }\Psi _\theta ({{\textbf {x}} }) \, \partial _\mu \ln \Psi _\theta ({{\textbf {x}} }) |{{\textbf {x}} }\rangle = \left( \sum _{{{\textbf {x}} }^{'}} \partial _\mu \ln \Psi _\theta ({{\textbf {x}} }) |{\textbf {x}}^{'}\rangle \langle {\textbf {x}}^{'}| \right) \left( \sum _{{\textbf {x}} }\Psi _\theta ({{\textbf {x}} }) |{{\textbf {x}} }\rangle \right) = {\mathcal {D}} _\mu |\psi _\theta \rangle \end{aligned}$$
(A3)

to construct a convenient representation of derivative operators

$$\begin{aligned} \partial _\mu \mapsto {\mathcal {D}} _\mu = \sum _{{{\textbf {x}} }} \partial _\mu \ln \Psi _\theta ({{\textbf {x}} }) |{\textbf {x}}\rangle \langle {\textbf {x}}| \end{aligned}$$
(A4)

on the variational manifold \({\mathcal {M}} _\psi\). Truncating the Taylor expansions in Eq. A2, we get

$$\begin{aligned} \sum _\mu {\mathcal {D}} _\mu |\Psi _\theta \rangle {\dot{\theta }} ^\mu = -H |\Psi _\theta \rangle \end{aligned}$$
(A5)

and reduce the original equation to finding \({\dot{\theta }} ^\mu = {{\,\textrm{argmin}\,}}\Vert \sum _\mu {\mathcal {D}} _\mu {\dot{\theta }} ^\mu - H\Vert\). To build a convenient way of solving for \({\dot{\theta }}\), we left-multiply by \(\langle \Psi _\theta | {\mathcal {D}} ^\dagger _\nu\) and relabel indices:

$$\begin{aligned} \sum _\nu \left\langle {\mathcal {D}} _\mu ^\dagger {\mathcal {D}} _\nu \right\rangle _{\Psi _\theta } {\dot{\theta }} ^\nu = - \left\langle {\mathcal {D}} _\mu ^\dagger H\right\rangle _{\Psi _\theta } \; , \end{aligned}$$
(A6)

yielding a linear system.

Finally, we need to unpack the fact that we usually do not have access to the full normalized state \(|\Psi \rangle\). Instead, we need to substitute \(|\Psi \rangle \mapsto |\psi \rangle / \sqrt{\langle {\psi } | {\psi } \rangle }\). The expectation value changes trivially into the form that can be evaluated using Monte Carlo sampling, as discussed in the main text. However, variational derivative operators \({\mathcal {D}} _\mu\) pick up an extra term because

$$\begin{aligned} \partial _\mu \ln \Psi = \partial _\mu \ln \psi _\theta - \frac{1}{2} \frac{\langle \partial _\mu \psi _\theta |\psi _\theta \rangle + \langle \psi _\theta |\partial _\mu \psi _\theta \rangle }{\langle {\psi _\theta } | {\psi _\theta } \rangle } = \partial _\mu \ln \psi _\theta - \text{ Re }\left\langle {\mathcal {O}}_\mu \right\rangle _{\psi _\theta } \end{aligned}$$
(A7)

assuming that \(|\partial _\mu \psi _\theta \rangle = \partial _\mu |\psi _\theta \rangle\) and that all of the variational parameters \(\theta\) are real: \(\theta \in \mathbbm {R} ^P\). Therefore, we are free to substitute \({\mathcal {D}}_\mu \mapsto {\mathcal {O}}_\mu - \text{ Re }\left\langle {\mathcal {O}}_\mu \right\rangle\) in Eq. A6, resulting in

$$\begin{aligned} \text{ Re }\sum _\nu \left\{ \left\langle {\mathcal {O}}_\mu ^\dagger {\mathcal {O}}_\nu \right\rangle _{\psi _\theta } - \left\langle {\mathcal {O}}_\mu ^\dagger \right\rangle _{\psi _\theta } \left\langle \vphantom{{\mathcal {O}}_\mu ^\dagger} {\mathcal {O}}_\nu \right\rangle _{\psi _\theta } \right\} {\dot{\theta }} ^\nu = - \text{ Re }\left\{ \left\langle {\mathcal {O}}_\mu ^\dagger H\right\rangle _{\Psi _\theta } - \left\langle {\mathcal {O}}_\mu ^\dagger \right\rangle _{\psi _\theta } \left\langle \vphantom{{\mathcal {O}}_\mu ^\dagger} H \right\rangle _{\psi _\theta } \right\} \; , \end{aligned}$$
(A8)

which is identical to expressions in Eq. 16 in the main text.

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Medvidović, M., Moreno, J.R. Neural-network quantum states for many-body physics. Eur. Phys. J. Plus 139, 631 (2024). https://doi.org/10.1140/epjp/s13360-024-05311-y

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