Simulating quantum computers with probabilistic methods
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TL;DR: It is shown that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes, and it is proved that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.
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Abstract: We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove that sparse quantum circuits as well as circuits composed of CNOT and exp[itheta;X] gates can be simulated classically. In a second part, we apply our results to the simulation of quantum algorithms. It is shown that a recent quantum algorithm, concerned with the estimation of Potts model partition functions, can be simulated efficiently classically. Finally, we show that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes. Specifically, we prove that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.
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Citations
Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
TL;DR: The class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection is introduced, and it is proved first that post- IQP equals the classical class PP, and that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, then the infinite tower of classical complexity classes known as the polynomial hierarchy would collapse to its third level.
Practical characterization of quantum devices without tomography.
TL;DR: It is demonstrated that fidelity can be estimated from a number of simple experiments that is independent of the system size, removing an important roadblock for the experimental study of larger quantum information processing units.
Simulation of quantum circuits by low-rank stabilizer decompositions
Sergey Bravyi,Dan E. Browne,Padraic Calpin,Earl T. Campbell,David Gosset,David Gosset,Mark Howard +6 more
- 02 Sep 2019
TL;DR: A comprehensive mathematical theory of the stabilizerRank and the related approximate stabilizer rank is developed and a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art are presented.
Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
TL;DR: In this article, the authors consider quantum computations comprising only commuting gates, known as IQP computations, and provide compelling evidence that the task of sampling their output probability distributions is unlikely to be achievable by any efficient classical means.
294
Using Quantum Computers for Quantum Simulation
TL;DR: The theoretical and experimental development of quantum simulation using quantum computers is surveyed, from the first ideas to the intense research efforts currently underway.
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