Computational quantum-classical boundary of noisy commuting quantum circuits.

Keisuke Fujii,Shuhei Tamate

doi:10.1038/srep25598

Abstract

It is often said that the transition from quantum to classical worlds is caused by decoherence originated from an interaction between a system of interest and its surrounding environment. Here we establish a computational quantum-classical boundary from the viewpoint of classical simulatability of a quantum system under decoherence. Specifically, we consider commuting quantum circuits being subject to decoherence. Or equivalently, we can regard them as measurement-based quantum computation on decohered weighted graph states. To show intractability of classical simulation in the quantum side, we utilize the postselection argument and crucially strengthen it by taking noise effect into account. Classical simulatability in the classical side is also shown constructively by using both separable criteria in a projected-entangled-pair-state picture and the Gottesman-Knill theorem for mixed state Clifford circuits. We found that when each qubit is subject to a single-qubit complete-positive-trace-preserving noise, the computational quantum-classical boundary is sharply given by the noise rate required for the distillability of a magic state. The obtained quantum-classical boundary of noisy quantum dynamics reveals a complexity landscape of controlled quantum systems. This paves a way to an experimentally feasible verification of quantum mechanics in a high complexity limit beyond classically simulatable region.

Highlights

Even in the case of depth-four circuits, we show that the computational quantum-classical (CQC) boundary is sharply upper and lower bounded by 14.6% and 13.4%, respectively
We directly show that commuting quantum circuits being subject to decoherence themselves are classically intractable if a strength of noise is smaller than a certain constant threshold value
We have shown that if the noise strength q is smaller than a threshold value, the corresponding noisy quantum circuits cannot be simulated by classical computer unless the polynomial hierarchy (PH) collapses at the third level

Summary

A Sharp CQC Boundary

We derive a CQC boundary that sharply divides the classically simulatable and intractable regions of noisy commuting quantum circuits. The measurement outcomes are reinterpreted as m i i i If we consider the classical simulatability by using the postselection argument, the threshold, i.e. CQC boundary, is given solely by the distillation threshold of the magic state This result is quite reasonable since the magic state distillation is an essential ingredient for universal quantum computation. If ps > (1 − 2 /2)/2, the noisy magic state becomes a convex mixture of the Pauli basis states This indicates that if p1 =p2 > 0.0998 for the depolarizing noise model, the noisy commuting circuits become classically simulatable. Note that while we here randomly choose the angle θij =π/4 ±φ to depolarize a commuting gate into a correlated dephsing, we can calculate the intractable region for θij =π/4 −φ by taking e−φZZ as a noise and evaluating its diamond norm

Discussion

Method