Estimates of loss function concentration in noisy parametrized quantum circuits

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Preprint
Report number	arXiv:2410.01893
Title	Estimates of loss function concentration in noisy parametrized quantum circuits
Author(s)	Crognaletti, Giulio (INFN, Trieste ; CERN ; Trieste U.) ; Grossi, Michele (CERN) ; Bassi, Angelo (INFN, Trieste ; Trieste U.)
Imprint	2024-10-02
Number of pages	24
Subject category	quant-ph ; General Theoretical Physics
Abstract	Variational quantum computing provides a versatile computational approach, applicable to a wide range of fields such as quantum chemistry, machine learning, and optimization problems. However, scaling up the optimization of quantum circuits encounters a significant hurdle due to the exponential concentration of the loss function, often dubbed the barren plateau (BP) phenomenon. Although rigorous results exist on the extent of barren plateaus in unitary or in noisy circuits, little is known about the interaction between these two effects, mainly because the loss concentration in noisy parameterized quantum circuits (PQCs) cannot be adequately described using the standard Lie algebraic formalism used in the unitary case. In this work, we introduce a new analytical formulation based on non-negative matrix theory that enables precise calculation of the variance in deep PQCs, which allows investigating the complex and rich interplay between unitary dynamics and noise. In particular, we show the emergence of a noise-induced absorption mechanism, a phenomenon that cannot arise in the purely reversible context of unitary quantum computing. Despite the challenges, general lower bounds on the variance of deep PQCs can still be established by appropriately slowing down speed of convergence to the deep circuit limit, effectively mimicking the behaviour of shallow circuits. Our framework applies to both unitary and non-unitary dynamics, allowing us to establish a deeper connection between the noise resilience of PQCs and the potential to enhance their expressive power through smart initialization strategies. Theoretical developments are supported by numerical examples and related applications.
Other source	Inspire
Copyright/License	preprint: (License: arXiv nonexclusive-distrib 1.0)

$Loss concentration in variational quantum computing. The analytical formulation proposed here employs non-negative matrix theory to describe the interplay between local 2-designs and noise. This allows for precise calculation of loss variances for generic noise maps, from strictly contractive to unitary channels, as illustrated by the coloured band in the Figure. The upper part considers deep circuits, where the loss variance $\Varr^L$ reaches its asymptotic limit. While loss concentration is well-understood for strictly contractive \cite{Schumann23, Wang21} and unitary \cite{Ragone24, Fontana24, Diaz23} channels, \cref{thm:main} provides an analytical solution for the intermediate case, revealing a noise-induced absorption mechanism unique to this regime. Consistency with known limiting cases is verified in \cref{cor:deep_noisy} and \cref{cor:deep_unitary}. The lower part focuses on ``shallow'' circuits, where a lower bound on $\Varr^L$ is established using \cref{thm:lower_bound}. This extends previous works on brickwork circuits \cite{Cerezo21_2, Mele24}, enabling initialization strategies such as small angle initializations \cite{Zhang22, Wang23} to be represented as stochastic unravellings of noise maps. In \cref{sec:illustrative_examples} we expand upon their applicability. For instance, both unitary and non-unitary QResNets can be derived thanks to \cref{prop:noise_small_angle_equiv}.$ $Graphical representations of the stochastic process describing $\Varr^L$. On the left, we show the structure of the general locality transfer matrix (LTM), highlighting the decomposition into irreducible components \cite{Seneta06}. Light blue blocks represent irreducible, essential components of $T$, while red blocks are related to inessential ones. In particular, $Q$ represents the collection of all irreducible, inessential components of $T$ and $R$ their relation with the essential components $T_z$. On the right, the same process is represented graphically, in terms of the local subspaces $\BHk$. Here, each dot represents a single subspace, while the arrows represent the adjoint action of the channel $\mathcal{E}$. Essential and inessential components share the same colour code of $T$.$ $Scaling of $\Varr^L$ as a function of the noise strength and the entangling power of the intermediate channel. The main figure illustrates the scaling of $\Varr^\infty$ with noise strength $p$ for both rapidly entangling (pink) and slowly entangling (light blue) channels, using $L = 8$ and $L = 20$, respectively. The dotted lines represent the theoretical predictions of \cref{eq:quadratic_dependence} and \cref{eq:linear_dependence} The inset verifies the exponential convergence of $\Varr^L$ to $\Varr^\infty$ at $p = 0.1$, justifying the chosen number of layers. The dotted lines represent an exponential fit to the numerical data. All plots are obtained using a $n=10$ qubit system.$ Show more plots

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记录创建於2024-10-09，最後更新在2024-11-11

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