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arXiv:1806.06618 · Physical Review A

Probabilistic fault-tolerant universal quantum computation and sampling problems in continuous variables

Tom Douce, Damian Markham, Elham Kashefi, Peter van Loock, Giulia Ferrini

Published Jan 29, 2019• 18 citations
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Abstract

Continuous-variable (CV) devices are a promising platform for demonstrating large-scale quantum information protocols. In this framework we define a general quantum computational model based on a CV hardware. It consists of vacuum input states, a finite set of gates---including non-Gaussian elements---and homodyne detection. We show that this model incorporates encodings sufficient for probabilistic fault-tolerant universal quantum computing. Furthermore, we show that this model can be adapted to yield sampling problems that cannot be simulated efficiently with a classical computer, unless the polynomial hierarchy collapses. This allows us to provide a simple paradigm for experiments to probe quantum advantage relying on Gaussian states, homodyne detection, and some form of non-Gaussian evolution. We finally address the recently introduced model of instantaneous quantum computing in CV, and prove that the hardness statement is robust with respect to some experimentally relevant simplifications in the definition of that model.

DOI
10.1103/physreva.99.012344

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Continuous-variable (CV) devices are a promising platform for demonstrating large-scale quantum information protocols. In this framework we define a general quantum computational model based on a CV hardware. It consists of vacuum input states, a finite set of gates---including non-Gaussian elements---and homodyne detection. We show that this model incorporates encodings sufficient for probabilistic fault-tolerant universal quantum computing. Furthermore, we show that this model can be adapted to yield sampling problems that cannot be simulated efficiently with a classical computer, unless the polynomial hierarchy collapses. This allows us to provide a simple paradigm for experiments to probe quantum advantage relying on Gaussian states, homodyne detection, and some form of non-Gaussian evolution. We finally address the recently introduced model of instantaneous quantum computing in CV, and prove that the hardness statement is robust with respect to some experimentally relevant simplifications in the definition of that model.
Instantaneous quantum computing is a subuniversal quantum complexity class, whose circuits have proven to be hard to simulate classically in the discrete-variable realm. We extend this proof to the continuous-variable (CV) domain by using squeezed states and homodyne detection, and by exploring the properties of postselected circuits. In order to treat postselection in CVs, we consider finitely resolved homodyne detectors, corresponding to a realistic scheme based on discrete probability distributions of the measurement outcomes. The unavoidable errors stemming from the use of finitely squeezed states are suppressed through a qubit-into-oscillator Gottesman-Kitaev-Preskill encoding of quantum information, which was previously shown to enable fault-tolerant CV quantum computation. Finally, we show that, in order to render postselected computational classes in CVs meaningful, a logarithmic scaling of the squeezing parameter with the circuit size is necessary, translating into a polynomial scaling of the input energy.
We introduce a new family of quantum circuits in Continuous Variables and we show that, relying on the widely accepted conjecture that the polynomial hierarchy of complexity classes does not collapse, their output probability distribution cannot be efficiently simulated by a classical computer. These circuits are composed of input photon-subtracted (or photon-added) squeezed states, passive linear optics evolution, and eight-port homodyne detection. We address the proof of hardness for the exact probability distribution of these quantum circuits by exploiting mappings onto different architectures of sub-universal quantum computers. We obtain both a worst-case and an average-case hardness result. Hardness of Boson Sampling with eight-port homodyne detection is obtained as the zero squeezing limit of our model. We conclude with a discussion on the relevance and interest of the present model in connection to experimental applications and classical simulations.
Continuous-variable (CV) systems have shown remarkable potential for quantum computation, particularly excelling in scalability and holding great promise for error correction through bosonic encoding. Within this framework, the foundational notion of computational universality was introduced by Lloyd and Braunstein [], and has proven especially successful since it allows for the identification of finite sets of universal CV gates independent of the encoding scheme. However, realizing the full power of this computational model requires achieving fault tolerance, which in turn demands discretization via some form of encoding—and, to date, no proof exists that universal CV gates can give rise to encoded fault tolerance. We present compelling evidence in this direction by utilizing the Gottesman-Kitaev-Preskill (GKP) encoding. Specifically, we numerically optimize the generation of GKP states from vacua using circuits comprised solely of universal CV gates, and demonstrate that the resulting states are of sufficient quality to exhibit error probabilities below the threshold required for fault-tolerant memory via concatenated GKP-stabilizer codes.
We establish a connection between continuous-variable quantum computing and high-dimensional integration by showing that the outcome probabilities of continuous-variable instantaneous quantum polynomial (CV-IQP) circuits are given by integrals of oscillating functions in large dimensions. We prove two results related to the classical hardness of evaluating these integrals: (i) we show that there exist circuits such that these integrals are approximations of a weighted sum of #P-hard problems and (ii) we prove that calculating these integrals is as hard as calculating integrals of arbitrary bounded functions. We then leverage these results to show that, given a plausible conjecture about the hardness of computing the integrals, approximate sampling from CV-IQP circuits cannot be done in polynomial time on a classical computer unless the polynomial hierarchy collapses to the third level. Our results hold even in the presence of finite squeezing and limited measurement precision, without an explicit need for fault-tolerance.
We introduce a family of quantum circuits in continuous variables and we show that, relying on the widely accepted conjecture that the polynomial hierarchy of complexity classes does not collapse, their output probability distribution cannot be efficiently simulated by a classical computer. These circuits are composed of input photon-subtracted (or photon-added) squeezed states, passive linear optics evolution, and eight-port homodyne detection. We address the proof of hardness for the exact probability distribution of these quantum circuits by exploiting mappings onto different architectures of subuniversal quantum computers. We obtain both a worst-case and an average-case hardness result. Hardness of boson sampling with eight-port homodyne detection is obtained as the zero squeezing limit of our model. We conclude with a discussion on the relevance and interest of the present model in connection to experimental applications and classical simulations.