Quantum supremacy and hardness of estimating output probabilities of quantum circuits
Quantum supremacy and hardness of estimating output probabilities of quantum circuits
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DOI：
10.1109/focs52979.2021.00126
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发表时间：
2021-02
期刊：
影响因子：
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通讯作者：
Yasuhiro Kondo;R. Mori;R. Movassagh
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文献类型：
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作者：
Yasuhiro Kondo;R. Mori;R. Movassagh
由量子至上的最新实验演示的动机，证明随机量子圆的输出的硬度是一个急需的近期目标。量子圆到$ \ exp（ - \ omega（m \ log M））$对于任何经典计算机而言都很难，其中$ m $是量子计算中的大门数量。更确切地说，我们表明上面的问题是在bppnp降低下的＃p-hard。 SQRT {n} $ [1]，对于$ 2^{ - \ omega（n \ log n）}的恒定深度电路很难。 {n} $抗浓缩属性所拥有的$，将输出的可能性近似于$ 2^{ - \ omega（n \ log^{2} n）} $和$ 2^{ - \ omega（n^{3/2} \ log n）} $然后表明，硬度结果扩展到了带有身份门的琐碎电路的任意（固定）电路的任何开放式社区。例如berlekamp – Welch算法，通常的Paturi的引理和Rakhmanov的结果。
Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within $\exp(-\Omega(m\log m))$ additive error is hard for any classical computer, where $m$ is the number of gates in the quantum computation. More precisely, we show that the above problem is #P-hard under BPPNP reduction. In the recent experiments, the quantum circuit has n-qubits and the architecture is a two-dimensional grid of size $\sqrt{n}\times\sqrt{n}$ [1]. Indeed for constant depth circuits approximating the output probabilities to within $2^{-\Omega(n\log n)}$ is hard. For circuits of depth $\log n$ or $\sqrt{n}$ for which the anti-concentration property holds, approximating the output probabilities to within $2^{-\Omega(n\log^{2}n)}$ and $2^{-\Omega(n^{3/2}\log n)}$ is hard respectively. We then show that the hardness results extend to any open neighborhood of an arbitrary (fixed) circuit including the trivial circuit with identity gates. We made an effort to find the best proofs and proved these results from first principles, which do not use the standard techniques such as the Berlekamp–Welch algorithm, the usual Paturi's lemma, and Rakhmanov's result.