[Paper Review] Average-case hardness of estimating probabilities of random quantum circuits with a linear scaling in the error exponent

Hari Krovi|arXiv (Cornell University)|Jun 12, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper establishes coC=P hardness for classically approximating output probabilities of random quantum circuits—specifically Haar random, p=1 QAOA, and random IQP circuits—under additive error bounds that scale as 2−O(m) for Haar circuits and 2−O(n) for QAOA and IQP circuits. By using truncated Taylor series interpolation and improved robust polynomial reconstruction, the authors achieve linear scaling in the error exponent, strengthening prior results and showing that efficient classical simulation would collapse the polynomial hierarchy.

ABSTRACT

We consider the hardness of computing additive approximations to output probabilities of random quantum circuits. We consider three random circuit families, namely, Haar random, $p=1$ QAOA, and random IQP circuits. Our results are as follows. For Haar random circuits with $m$ gates, we improve on prior results by showing $\mathsf{coC_=P}$ hardness of average-case additive approximations to an imprecision of $2^{-O(m)}$. Efficient classical simulation of such problems would imply the collapse of the polynomial hierarchy. For constant depth circuits i.e., when $m=O(n)$, this linear scaling in the exponent is within a constant of the scaling required to show hardness of sampling. Prior to our work, such a result was shown only for Boson Sampling in Bouland et al (2021). We also use recent results in polynomial interpolation to show $\mathsf{coC_=P}$ hardness under $\mathsf{BPP}$ reductions rather than $\mathsf{BPP}^{\mathsf{NP}}$ reductions. This improves the results of prior work for Haar random circuits both in terms of the error scaling and the power of reductions. Next, we consider random $p=1$ QAOA and IQP circuits and show that in the average-case, it is $\mathsf{coC_=P}$ hard to approximate the output probability to within an additive error of $2^{-O(n)}$. For $p=1$ QAOA circuits, this work constitutes the first average-case hardness result for the problem of approximating output probabilities for random QAOA circuits, which include Sherrington-Kirkpatrick and Erdös-Renyi graphs. For IQP circuits, a consequence of our results is that approximating the Ising partition function with imaginary couplings to an additive error of $2^{-O(n)}$ is hard even in the average-case, which extends prior work on worst-case hardness of multiplicative approximation to Ising partition functions.

Motivation & Objective

• To establish average-case classical hardness for approximating output probabilities of random quantum circuits.
• To improve error scaling in additive approximation hardness results from 2−O(m log m) to 2−O(m) for Haar random circuits.
• To extend average-case hardness to p=1 QAOA and random IQP circuits, where no such results existed prior.
• To reduce the complexity of reductions from BPPNP to BPP by leveraging recent advances in polynomial interpolation.
• To demonstrate that approximating Ising partition functions with imaginary couplings is hard in the average-case, without additional conjectures.

Proposed method

• Uses truncated Taylor series of the output probability as a function of an interpolation parameter to construct a low-degree polynomial of degree O(m/log m).
• Applies robust Berlekamp-Welch algorithm with improved error bounds to reconstruct the polynomial from noisy evaluations.
• Employs Chebyshev-distributed interpolation points to ensure uniform approximation across the interpolation interval.
• Leverages recent results in polynomial interpolation to reduce the required computational power from BPPNP to BPP for the robust reconstruction step.
• Uses the fact that unitary evolution is preserved during interpolation to maintain quantum circuit structure.
• Applies the robustness of the interpolation scheme to show that classical algorithms with additive error δ ≤ 2−O(m) can be used to solve coC=P-hard problems.

Experimental results

Research questions

• RQ1 Can the additive error scaling in classical hardness results for random quantum circuits be improved from 2−O(m log m) to 2−O(m)?
• RQ2 Is average-case hardness of approximating output probabilities provable for p=1 QAOA circuits?
• RQ3 Can the hardness of approximating Ising partition functions with imaginary couplings be established without additional conjectures?
• RQ4 Can the robust polynomial reconstruction used in the proof be implemented in BPP rather than BPPNP?
• RQ5 Does the improved error scaling imply stronger evidence for quantum supremacy in sampling?

Key findings

• For Haar random circuits with m gates, classical additive approximation within error 2−O(m) is coC=P-hard, implying that efficient classical simulation collapses the polynomial hierarchy.
• The error scaling of 2−O(m) is achieved via truncated Taylor series interpolation of the output probability, reducing the polynomial degree to O(m/log m).
• The robust Berlekamp-Welch algorithm is improved to work in BPP rather than BPPNP, strengthening the reduction power.
• For p=1 QAOA circuits, this is the first average-case hardness result for output probability approximation, with hardness holding under additive error 2−O(n).
• For random IQP circuits, the result implies that approximating the Ising partition function with imaginary couplings is coC=P-hard in the average-case, without requiring additional conjectures.
• The paper shows that 2−O(n) error scaling is sufficient to imply classical hardness for QAOA and IQP circuits, which is tight up to constant factors in the exponent.