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arXiv:2301.05273 · Physical Review A

Improving the speed of variational quantum algorithms for quantum error correction

Fabio Zoratti, Giacomo De Palma, Bobak T. Kiani, Quynh T. Nguyen, Milad Marvian, Seth Lloyd, Vittorio Giovannetti

Published Aug 18, 2023• 15 citations
arXivHTMLPDF

Abstract

We consider the problem of devising suitable quantum error correction (QEC) procedures for a generic quantum noise acting on a quantum circuit. In general, there is no analytic universal procedure to obtain the encoding and correction unitary gates, and the problem is even harder if the noise is unknown and has to be reconstructed. The existing procedures rely on variational quantum algorithms (VQAs) and are very difficult to train since the size of the gradient of the cost function decays exponentially with the number of qubits. We address this problem using a cost function based on the quantum Wasserstein distance of order 1 ($Q{W}_{1}$). At variance with other quantum distances typically adopted in quantum information processing, $Q{W}_{1}$ lacks the unitary invariance property which makes it a suitable tool to avoid getting trapped in local minima. Focusing on a simple noise model for which an exact QEC solution is known and can be used as a theoretical benchmark, we run a series of numerical tests that show how, guiding the VQA search through the $Q{W}_{1}$, can indeed significantly increase both the probability of a successful training and the fidelity of the recovered state, with respect to the results one obtains when using conventional approaches.

DOI
10.1103/physreva.108.022611

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We consider the problem of devising suitable quantum error correction (QEC) procedures for a generic quantum noise acting on a quantum circuit. In general, there is no analytic universal procedure to obtain the encoding and correction unitary gates, and the problem is even harder if the noise is unknown and has to be reconstructed. The existing procedures rely on variational quantum algorithms (VQAs) and are very difficult to train since the size of the gradient of the cost function decays exponentially with the number of qubits. We address this problem using a cost function based on the quantum Wasserstein distance of order 1 (QW1Q{W}_{1}QW1​). At variance with other quantum distances typically adopted in quantum information processing, QW1Q{W}_{1}QW1​ lacks the unitary invariance property which makes it a suitable tool to avoid getting trapped in local minima. Focusing on a simple noise model for which an exact QEC solution is known and can be used as a theoretical benchmark, we run a series of numerical tests that show how, guiding the VQA search through the QW1Q{W}_{1}QW1​, can indeed significantly increase both the probability of a successful training and the fidelity of the recovered state, with respect to the results one obtains when using conventional approaches.
Variational quantum algorithms (VQAs) are expected to become a practical application of near-term noisy quantum computers. Although the effect of the noise crucially determines whether a VQA works, the heuristic nature of VQAs makes it difficult to establish analytic theories. Analytic estimations of the impact of the noise are urgent for searching for quantum advantages, as numerical simulations of noisy quantum computers on classical computers are heavy and quite limited to small-scale problems. In this paper, we establish analytic estimations of the error in the cost function of VQAs due to the noise. The estimations are applicable to any typical VQAs under Gaussian noise, which is equivalent to a class of stochastic noise models. Notably, depolarizing noise is included in this model. As a result, we obtain estimations of the noise level to guarantee a required precision. Our formulas show how the Hessian of the cost function, the spectrum of the target operator, and the geometry of the ansatz affect the sensitivity to the noise. This insight implies trade-off relations between the trainability and the noise resilience of the cost function. We also obtain rough estimations which can be easily calculated without detailed information of the cost function. As a highlight of the applications of the formula, we propose a quantum error mitigation method which is different from the extrapolation and the probabilistic error cancellation.
Variational quantum algorithms (VQAs) represent the most promising approach for achieving quantum advantage in near-term supervised learning, yet they face two fundamental challenges that severely limit their practical deployment: device noise degrades classification performance, and barren plateaus render gradient-based training ineffective. We address both challenges simultaneously through a novel hybrid architecture that selectively applies stabilizer quantum error correction only to the most noise-sensitive circuit components while preserving the variational structure essential for NISQ-era feasibility. Our approach combines Steane [7, 1, 3] encoding around entangling layers with a learned variational recovery mechanism (VarQEC). We establish rigorous theoretical foundations through two main contributions: First, we prove an explicit error-suppression theorem showing that selective protection reduces logical error scaling from O(p) to O(p 2) in the physical depolarizing rate p, with the exact leading constant p L ≤ 147 9 p 2 + O(p 3) derived through deterministic combinatorial enumeration of harmful Pauli patterns. Second, we demonstrate enhanced trainability by proving that selective encoding mitigates barren plateaus (reducing exponential suppression from physical to logical qubit scaling) while simultaneously improving noise-induced gradient attenuation from O(p) to O(p 2) through rigorous Weingarten calculus analysis. The manuscript provides complete implementation details including gate-level circuits, unambiguous training algorithms, and reproducible numerical experiments, positioning this work as both a theoretical advance and a practical framework for noise-resilient quantum machine learning.
Exploring quantum applications of near-term quantum devices is a rapidly growing field of quantum information science with both theoretical and practical interests. A leading paradigm to establish such near-term quantum applications is variational quantum algorithms (VQAs). These algorithms use a classical optimizer to train a parameterized quantum circuit to accomplish certain tasks, where the circuits are usually randomly initialized. In this work, we prove that for a broad class of such random circuits, the variation range of the cost function via adjusting any local quantum gate within the circuit vanishes exponentially in the number of qubits with a high probability. This result can unify the restrictions on gradient-based and gradient-free optimizations in a natural manner and reveal extra harsh constraints on the training landscapes of VQAs. Hence a fundamental limitation on the trainability of VQAs is unraveled, indicating the essential mechanism of the optimization hardness in the Hilbert space with exponential dimension. We further showcase the validity of our results with numerical simulations of representative VQAs. We believe that these results would deepen our understanding of the scalability of VQAs and shed light on the search for near-term quantum applications with advantages.
Variational quantum algorithms (VQAs) provide a promising approach to achieve quantum advantage in the noisy intermediate-scale quantum era. In this era, quantum computers experience high error rates and quantum error detection and correction is not feasible. VQAs can utilize noisy qubits in tandem with classical optimization algorithms to solve hard problems. However, VQAs are still slow relative to their classical counterparts. Hence, improving the performance of VQAs will be necessary to make them competitive. While VQAs are expected perform better as the problem sizes increase, increasing their performance will make them a viable option sooner. In this work we show that circuit-level concurrency provides a means to increase the performance of variational quantum algorithms on noisy quantum computers. This involves mapping multiple instances of the same circuit (program) onto the quantum computer at the same time, which allows multiple samples in a variational quantum algorithm to be gathered in parallel for each training iteration. We demonstrate that this technique provides a linear increase in training speed when increasing the number of concurrently running quantum circuits. Furthermore, even with pessimistic error rates concurrent quantum circuit sampling can speed up the quantum approximate optimization algorithm by up to 20x with low mapping and run time overhead.
Variational quantum algorithms (VQAs) offer the most promising path to obtaining quantum advantages via noisy intermediate-scale quantum (NISQ) processors. Such systems leverage classical optimization to tune the parameters of a parameterized quantum circuit (PQC). The goal is minimizing a cost function that depends on measurement outputs obtained from the PQC. Optimization is typically implemented via stochastic gradient descent (SGD). On NISQ computers, gate noise due to imperfections and decoherence affects the stochastic gradient estimates by introducing a bias. Quantum error mitigation (QEM) techniques can reduce the estimation bias without requiring any increase in the number of qubits, but they in turn cause an increase in the variance of the gradient estimates. This work studies the impact of quantum gate noise on the convergence of SGD for the variational eigensolver (VQE), a fundamental instance of VQAs. The main goal is ascertaining conditions under which QEM can enhance the performance of SGD for VQEs. It is shown that quantum gate noise induces a nonzero error-floor on the convergence error of SGD (evaluated with respect to a reference noiseless PQC), which depends on the number of noisy gates, the strength of the noise, as well as the eigenspectrum of the observable being measured and minimized. In contrast, with QEM, any arbitrarily small error can be obtained. Furthermore, for error levels attainable with or without QEM, QEM can reduce the number of required iterations, but only as long as the quantum noise level is sufficiently large, and a sufficiently large number of measurements is allowed at each SGD iteration. Numerical examples for a max-cut problem corroborate the main theoretical findings.
The variational quantum algorithm (VQA) is a hybrid classical-quantum algorithm. It can actually run in an intermediate-scale quantum device where the number of available qubits is too limited to perform quantum error correction, so it is one of the most promising quantum algorithms in the noisy intermediate-scale quantum era. In this paper, two ideas for solving the learning with errors problem (LWE) using VQA are proposed. First, after reducing the LWE problem into the bounded distance decoding problem, the quantum approximation optimization algorithm (QAOA) is introduced to improve classical methods. Second, after the LWE problem is reduced into the unique shortest vector problem, the variational quantum eigensolver (VQE) is used to solve it, and the number of qubits required is calculated in detail. Small-scale experiments are carried out for the two LWE variational quantum algorithms, and the experiments show that VQA improves the quality of the classical solutions.