Efficient classical computation of expectation values in a class of quantum circuits with an epistemically restricted phase space representation

Abstract

We devise a classical algorithm which efficiently computes the quantum expectation values arising in a class of continuous variable quantum circuits wherein the final quantum observable—after the Heisenberg evolution associated with the circuits—is at most second order in momentum. The classical computational algorithm exploits a specific epistemic restriction in classical phase space which directly captures the quantum uncertainty relation, to transform the quantum circuits in the complex Hilbert space into classical albeit unconventional stochastic processes in the phase space. The resulting multidimensional integral is then evaluated using the Monte Carlo sampling method. The convergence rate of the classical sampling algorithm is determined by the variance of the classical physical quantity over the epistemically restricted phase space distribution. The work shows that for the specific class of computational schemes, Wigner negativity is not a sufficient resource for quantum speedup. It highlights the potential role of the epistemic restriction as an intuitive conceptual tool which may be used to study the boundary between quantum and classical computations.

Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable

“…Especially, assuming that ξ characterizes the randomness of each single repetition of the weak measurement, we want to see if Õ(φ n , ξ|ψ) is realizable in a single-shot weak measurement with post-selection and to study its distribution. We also would like to make a detailed comparison between the representation based on c-valued physical quantities with the approach based on quasiprobability, and to elaborate its computational possibility [61]. In particular it is interesting the compare the present formalism with the complex-valued Kirkwood-Dirac quasiprobability [62][63][64], or its real part, i.e., the Terletsky-Margenou-Hill quasiprobability [65,66], which are also deeply related to the concept of weak value [14,15,59,67,68].…”

Monte Carlo simulations for quantum harmonic and anharmonic oscillators within epistemically restricted phase-space representation

“…It opens a wide opportunity to further refine the method to speedup the calculation of quantum computation because the MCER method has other tool under its sleeve to implement, subjected to the near future work. For example, the MCER method can elegantly incorporate the unitary transformation in its algorithm to efficiently find the correct eigen function of a quantum system [7]. Furthermore, the MCER method provides a classical algorithm that is capable to solve broad variants of quantum states even one with a negative Wigner function.…”