Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Circuits

TL;DR

The paper addresses the challenge of spoofing Google's Linear XEB benchmark for shallow quantum circuits by presenting a classical algorithm that directly targets the linear-XEB metric rather than full circuit simulation. The core idea is to compute marginals for a carefully chosen set of output qubits within their light cones and sample the remaining outputs uniformly, achieving a fidelity lower bound of $\mathbb{E}_{C}[\mathcal{F}_C(A_C)] \ge (1+15^{-d})^{m}-1$ with running time polynomial in $n$ and $2^{L}$, where $L$ is the light-cone size and $m$ is the number of outputs used. The analysis hinges on a single-qubit Haar-integration step that yields a Markov-chain interpretation, which then extends to multiple outputs; the work also analyzes the sample complexity via variance and collision-probability bounds, showing favorable results for 1D logarithmic-depth circuits and providing conjectures for 2D cases. Overall, the results offer evidence that fooling the linear XEB benchmark can be easier than full quantum circuit simulation, with implications for the interpretation of quantum supremacy experiments and the hardness assumptions underlying Linear XEB. The techniques combine tensor networks, Haar integration, Markov chains, and spin-model methods to connect circuit structure with empirical performance in a way that informs both theory and potential practical spoofing strategies.

Abstract

The linear cross-entropy benchmark (Linear XEB) has been used as a test for procedures simulating quantum circuits. Given a quantum circuit $C$ with $n$ inputs and outputs and purported simulator whose output is distributed according to a distribution $p$ over $\{0,1\}^n$, the linear XEB fidelity of the simulator is $\mathcal{F}_{C}(p) = 2^n \mathbb{E}_{x \sim p} q_C(x) -1$ where $q_C(x)$ is the probability that $x$ is output from the distribution $C|0^n\rangle$. A trivial simulator (e.g., the uniform distribution) satisfies $\mathcal{F}_C(p)=0$, while Google's noisy quantum simulation of a 53 qubit circuit $C$ achieved a fidelity value of $(2.24\pm0.21)\times10^{-3}$ (Arute et. al., Nature'19). In this work we give a classical randomized algorithm that for a given circuit $C$ of depth $d$ with Haar random 2-qubit gates achieves in expectation a fidelity value of $Ω(\tfrac{n}{L} \cdot 15^{-d})$ in running time $\textsf{poly}(n,2^L)$. Here $L$ is the size of the \emph{light cone} of $C$: the maximum number of input bits that each output bit depends on. In particular, we obtain a polynomial-time algorithm that achieves large fidelity of $ω(1)$ for depth $O(\sqrt{\log n})$ two-dimensional circuits. To our knowledge, this is the first such result for two dimensional circuits of super-constant depth. Our results can be considered as an evidence that fooling the linear XEB test might be easier than achieving a full simulation of the quantum circuit.

Figures (4)

• Figure 1: An example of 1D circuit skeleton with $n=8$ and $d=3$. In this example the permutations are $\pi^{(0)}=\pi^{(4)}=\textsf{id},\ \pi^{(1)}=\pi^{(3)}=(18765432),\ \pi^{(2)}=(81234567)$.
• Figure 2:
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Theorems & Definitions (25)

• theorem 1.0: Linear XEB for circuits with small light cones
• corollary 1.1: Constant dimensional circuits
• proof
• definition 2.1: Circuit skeleton and light cone
• definition 2.2: Light cone
• lemma 2.3: Marginal probability and light cone
• proof : Proof of \ref{['lem:marginal light cone']}
• definition 2.4: Random quantum circuits
• theorem 3.0: Linear XEB for circuits with small light cones.
• theorem 4.1: Linear XEB of a single output qubit