Sufficient Conditions for Efficient Classical Simulation of Quantum Optics

We provide general sufficient conditions for the efficient classical simulation of quantum-optics experiments that involve inputting states to a quantum process and making measurements at the output. The first condition is based on the negativity of phase-space quasiprobability distributions (PQDs) of the output state of the process and the output measurements; the second one is based on the negativity of PQDs of the input states, the output measurements, and the transition function associated with the process. We show that these conditions provide useful practical tools for investigating the effects of imperfections in implementations of boson sampling. In particular, we apply our formalism to boson-sampling experiments that use single-photon or spontaneous-parametric-down-conversion sources and on-off photodetectors. Considering simple models for loss and noise, we show that above some threshold for the probability of random counts in the photodetectors, these boson-sampling experiments are classically simulatable. We identify mode mismatching as the major source of error contributing to random counts and suggest that this is the chief challenge for implementations of boson sampling of interesting size.

Popular Summary

It is widely believed that quantum computers will be able to perform certain tasks faster than any classical computer. Identifying the resource that enables this speedup is of great interest in quantum information science. Here, we investigate the circumstances under which generic quantum-optics experiments can be simulated efficiently using only classical resources.

We begin by formulating two sufficient conditions for the efficient classical simulation of quantum-optics experiments using quantum light in an optical network. These conditions provide useful practical tools for investigating the effects of imperfectly implementing quantum-optical protocols such as boson sampling. The goal of our work is to determine whether sampling from the output probability distribution can be efficiently simulated using only classical resources. Using the theory of phase-space quasiprobability distributions, we show that the negativity of these distributions is a necessary resource for experiments not to be efficiently classically simulatable. Considering several sources of error, we show that above some threshold for loss and noise, boson-sampling experiments are classically simulatable. Our analysis identifies mode mismatching—imperfect overlap of optical signals on optical elements—as the chief challenge for implementations of boson-sampling experiments of interesting size.

We expect that our finding will pave the way for future studies of quantum light in optical networks.

Figure 1

Generic quantum-optics sampling problem: The input state ${\mathit{\rho }}_{\mathrm{in}}$ is processed through an $M$-mode quantum process described by quantum operation $\mathcal{E}$, producing the output state ${\mathit{\rho }}_{\mathrm{out}}=\mathcal{E}({\mathit{\rho }}_{\mathrm{in}})$; an output probability distribution $p(\mathit{n})=\mathrm{Tr}[{\mathit{\rho }}_{\mathrm{out}}{\mathrm{\Pi }}_{\mathit{n}}]$ is sampled by measuring a POVM $\{{\mathrm{\Pi }}_{\mathit{n}}\}$.