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Classical Verification of Quantum Learning Advantages with Noises

Yinghao Ma, Jiaxi Su, Dong-Ling Deng

TL;DR

An efficient classical error rectification algorithm is proposed to reconstruct the noise-free results given by the quantum Fourier sampling circuit with practical constant-level noises and it is proved that the error rectification algorithm can restore the heavy Fourier coefficients by using a small number of noisy samples that scales logarithmically with the problem size.

Abstract

Classical verification of quantum learning allows classical clients to reliably leverage quantum computing advantages by interacting with untrusted quantum servers. Yet, current quantum devices available in practice suffers from a variety of noises and whether existed classical verification protocols carry over to noisy scenarios remains unclear. Here, we propose an efficient classical error rectification algorithm to reconstruct the noise-free results given by the quantum Fourier sampling circuit with practical constant-level noises. In particular, we prove that the error rectification algorithm can restore the heavy Fourier coefficients by using a small number of noisy samples that scales logarithmically with the problem size. We apply this algorithm to the agnostic parity learning task with uniform input marginal and prove that this task can be accomplished in an efficient way on noisy quantum devices with our algorithm. In addition, we prove that a classical client with access to the random example oracle can verify the agnostic parity learning results from the noisy quantum prover in an efficient way, under the condition that the Fourier coefficients are sparse. Our results demonstrate the feasibility of classical verification of quantum learning advantages with noises, which provide a valuable guide for both theoretical studies and practical applications with current noisy intermediate scale quantum devices.

Classical Verification of Quantum Learning Advantages with Noises

TL;DR

An efficient classical error rectification algorithm is proposed to reconstruct the noise-free results given by the quantum Fourier sampling circuit with practical constant-level noises and it is proved that the error rectification algorithm can restore the heavy Fourier coefficients by using a small number of noisy samples that scales logarithmically with the problem size.

Abstract

Classical verification of quantum learning allows classical clients to reliably leverage quantum computing advantages by interacting with untrusted quantum servers. Yet, current quantum devices available in practice suffers from a variety of noises and whether existed classical verification protocols carry over to noisy scenarios remains unclear. Here, we propose an efficient classical error rectification algorithm to reconstruct the noise-free results given by the quantum Fourier sampling circuit with practical constant-level noises. In particular, we prove that the error rectification algorithm can restore the heavy Fourier coefficients by using a small number of noisy samples that scales logarithmically with the problem size. We apply this algorithm to the agnostic parity learning task with uniform input marginal and prove that this task can be accomplished in an efficient way on noisy quantum devices with our algorithm. In addition, we prove that a classical client with access to the random example oracle can verify the agnostic parity learning results from the noisy quantum prover in an efficient way, under the condition that the Fourier coefficients are sparse. Our results demonstrate the feasibility of classical verification of quantum learning advantages with noises, which provide a valuable guide for both theoretical studies and practical applications with current noisy intermediate scale quantum devices.

Paper Structure

This paper contains 4 theorems, 5 equations, 1 figure.

Key Result

Theorem 1

Let $(\theta,\delta)\in(0,1)^2$. For $\eta\le \frac{1}{10}\theta$, there exists an efficient classical algorithm that takes $k=O\left(\frac{\log(n/\delta)}{\theta^2}\right)$ noisy samples from $p_{\eta}(s)$ and finds all possibly large components in $p_0(s)$ with success probability at least $1-\del

Figures (1)

  • Figure 1: (a) A sketch of classical verification of quantum learning with noises. We focus our discussion on the scenario where a classical verifier interacts with an untrusted noisy quantum prover, with the verifier and prover accessing classical and quantum data, respectively. The goal of the verifier is to complete the learning task through this interaction. Based on the outcomes of the interaction, the verifier can either output the learned result $h(x)$ or reject the prover. (b) Quantum Fourier sampling (QFS) circuit with measurement noise, where a random bit flip with probability $\eta$ is applied independently on the measurement result for each of the first $n$ qubits. (c) QFS circuit with depolarization noise, where each black node denotes a depolarization channel $\Lambda_{\mathrm{dep}}(\rho)=(1-\eta_{\mathrm{dep}})\rho+\eta_{\mathrm{dep}}I/2$, where $\eta_\text{dep}$ denotes the depolarization strength and $I$ is the identity two-by-two matrix Nielsen2010Quantum.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof