Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Circuits
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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 ℱ_C(p) = 2^n 𝔼_x ∼ p q_C(x) -1 where q_C(x) is the probability that x is output from the distribution C|0^n⟩. A trivial simulator (e.g., the uniform distribution) satisfies ℱ_C(p)=0, while Google's noisy quantum simulation of a 53 qubit circuit C achieved a fidelity value of (2.24±0.21)×10^-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(nL· 15^-d) in running time poly(n,2^L). Here L is the size of the 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(√(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.
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