[Paper Review] On the Classical Hardness of Spoofing Linear Cross-Entropy Benchmarking

Scott Aaronson, Sam Gunn|arXiv (Cornell University)|Oct 26, 2019

Advancements in Photolithography Techniques17 citations

TL;DR

This paper establishes that spoofing Linear Cross-Entropy Benchmarking (Linear XEB) is classically hard under a new assumption called XQUATH, which posits that estimating the output probability of a random quantum circuit for a specific string (e.g., $0^n$) with better-than-trivial accuracy is infeasible. The authors reduce the problem of spoofing XEB to XQUATH, proving that any efficient classical algorithm for XEB spoofing would imply a solution to XQUATH, thus linking the hardness of spoofing to a strong, sampling-free assumption.

ABSTRACT

Recently, Google announced the first demonstration of quantum computational supremacy with a programmable superconducting processor. Their demonstration is based on collecting samples from the output distribution of a noisy random quantum circuit, then applying a statistical test to those samples called Linear Cross-Entropy Benchmarking (Linear XEB). This raises a theoretical question: how hard is it for a classical computer to spoof the results of the Linear XEB test? In this short note, we adapt an analysis of Aaronson and Chen [2017] to prove a conditional hardness result for Linear XEB spoofing. Specifically, we show that the problem is classically hard, assuming that there is no efficient classical algorithm that, given a random n-qubit quantum circuit C, estimates the probability of C outputting a specific output string, say 0^n, with variance even slightly better than that of the trivial estimator that always estimates 1/2^n. Our result automatically encompasses the case of noisy circuits.

Motivation & Objective

• To establish the classical computational hardness of spoofing the Linear Cross-Entropy (XEB) benchmark, a key test in quantum supremacy experiments.
• To formulate a new, sampling-free assumption—XQUATH—that captures the difficulty of estimating output probabilities of random quantum circuits.
• To show that spoofing XEB is as hard as solving XQUATH, thereby transferring the hardness of XEB spoofing to a well-defined complexity-theoretic assumption.
• To provide a conditional proof of hardness for XEB spoofing that applies even to noisy quantum circuits.

Proposed method

• Introduces XQUATH as a new complexity assumption: no polynomial-time classical algorithm can estimate the output probability of a random $n$-qubit quantum circuit for a fixed string (e.g., $0^n$) with success probability $>1/2 + ext{poly}(1/2^n)$.
• Constructs a reduction from XQUATH to the problem of spoofing Linear XEB, showing that any efficient XEB spoofing algorithm implies a solution to XQUATH.
• Uses a random string $z$ and a transformed circuit $C'$ obtained by applying NOT gates on qubits where $z_i = 1$, preserving circuit distribution and linking $\bra{0^n}C\ket{0^n}$ to $\bra{z}C'\ket{0^n}$.
• Employs a probabilistic argument based on the expected value of a quadratic expression $\mathbb{E}[X]$, showing that if XEB is spoofed with high probability, then the estimator must outperform the trivial one, contradicting XQUATH.
• Derives a lower bound on the number of samples $k$ required for a successful XEB spoofing algorithm: $k \geq 1/((2s-1)b - 1)(b-1)$, where $s$ is the success probability and $b$ is the XEB threshold.
• Demonstrates that for $b \approx 2$, spoofing XEB with high probability requires $k = \Omega((b-1)^{-2})$ samples, which grows rapidly as noise increases.

Experimental results

Research questions

• RQ1 Is spoofing Linear XEB classically hard under plausible complexity assumptions?
• RQ2 Can the hardness of XEB spoofing be reduced to a sampling-free assumption rather than one involving full sampling?
• RQ3 What is the relationship between XQUATH and the earlier QUATH assumption in terms of computational strength and implications?
• RQ4 How does the number of samples required for XEB spoofing scale with the fidelity threshold $b$ and success probability $s$?

Key findings

• Spoofing Linear XEB with success probability $s > \frac{1}{2} + \frac{1}{2b}$ is classically hard under the XQUATH assumption.
• The number of samples $k$ required for a classical algorithm to spoof XEB with high probability is bounded below by $k \geq \frac{1}{((2s-1)b - 1)(b - 1)}$.
• For $b = 1 + \delta$ and $s = \frac{1}{2} + \frac{1}{2b} + \epsilon$, the required sample count is approximately $1/(2\epsilon\delta)$.
• Even with $s = 1$, the number of samples must exceed $(b - 1)^{-2}$, which becomes large as $b \to 1^+$, corresponding to high noise.
• The reduction shows that spoofing XEB is no easier than estimating output amplitudes of random quantum circuits, implying no special shortcut exists for XEB spoofing.
• The result holds for noisy circuits, as the XQUATH assumption is independent of noise model, and the proof applies to depolarizing noise and other realistic settings.