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• W3118956135 abstract "We consider the task of estimating the expectation value of an $n$-qubit tensor product observable $O_1otimes O_2otimes cdots otimes O_n$ in the output state of a shallow quantum circuit. This task is a cornerstone of variational quantum algorithms for optimization, machine learning, and the simulation of quantum many-body systems. Here we study its computational complexity for constant-depth quantum circuits and three types of single-qubit observables $O_j$ which are (a) close to the identity, (b) positive semidefinite, (c) arbitrary. It is shown that the mean value problem admits a classical approximation algorithm with runtime scaling as $mathrm{poly}(n)$ and $2^{tilde{O}(sqrt{n})}$ in cases (a,b) respectively. In case (c) we give a linear-time algorithm for geometrically local circuits on a two-dimensional grid. The mean value is approximated with a small relative error in case (a), while in cases (b,c) we satisfy a less demanding additive error bound. The algorithms are based on (respectively) Barvinok's polynomial interpolation method, a polynomial approximation for the OR function arising from quantum query complexity, and a Monte Carlo method combined with Matrix Product State techniques. We also prove a technical lemma characterizing a zero-free region for certain polynomials associated with a quantum circuit, which may be of independent interest." @default.
• W3118956135 created "2021-01-18" @default.
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• W3118956135 date "2021-01-04" @default.
• W3118956135 modified "2023-10-16" @default.
• W3118956135 title "Classical algorithms for quantum mean values" @default.
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• W3118956135 doi "https://doi.org/10.1038/s41567-020-01109-8" @default.
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