Quantum supremacy and hardness of estimating output probabilities of quantum circuits
Yasuhiro Kondo, Ryuhei Mori, Ramis Movassagh
Abstract
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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\exp(-\Omega(m\log m))$</tex> additive error is hard for any classical computer, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> is the number of gates in the quantum computation. More precisely, we show that the above problem is #P-hard under BPP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NP</sup> reduction. In the recent experiments, the quantum circuit has n-qubits and the architecture is a two-dimensional grid of size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sqrt{n}\times\sqrt{n}$</tex> [1]. Indeed for constant depth circuits approximating the output probabilities to within <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^{-\Omega(n\log n)}$</tex> is hard. For circuits of depth <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\log n$</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sqrt{n}$</tex> for which the anti-concentration property holds, approximating the output probabilities to within <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^{-\Omega(n\log^{2}n)}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^{-\Omega(n^{3/2}\log n)}$</tex> 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.