Efficient Classical Simulation of Continuous Variable Quantum Information Processes

Abstract

We obtain sufficient conditions for the efficient simulation of a continuous variable quantum algorithm or process on a classical computer. The resulting theorem is an extension of the Gottesman-Knill theorem to continuous variable quantum information. For a collection of harmonic oscillators, any quantum process that begins with unentangled Gaussian states, performs only transformations generated by Hamiltonians that are quadratic in the canonical operators, and involves only measurements of canonical operators (including finite losses) and suitable operations conditioned on these measurements can be simulated efficiently on a classical computer.

Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

“…Natural bosonic analogues of the fermionic results also exist. As a result it is possible to efficiently simulate quantum computational models in which coherent states are acted on by linear optical circuits, and measured via homodyne detection [6], and of models with initial multimode squeezed states and squeezing gates as well as linear ones [20]. Like LQC with the second measurement strategy, these involve the efficient simulation, in the dimension of a Lie algebra, of a computational model in which coherent states of a Lie group with gates generated by the algebra constitute the initial states and computation.…”

“…also [25,26]) concerning the efficient simulatability of Clifford-group computational models, and of results on the simulatability of certain multimode coherentstate and squeezed-state computational models [6,20]. One might hope for a treatment, perhaps based on Lie groups and groups of Lie type, that will unify these results, specifically those based on (1) finite dimensional semisimple Lie algebras, (2) Bosonic linear optics with homodyne detection (tied to an infinite-dimensional irreducible representation of a solvable Lie algebra) and possibly squeezing (involving a nilpotent Lie algebra), and (3) Clifford groups and semigroups.…”

“…Typically, the size of a problem instance is the number of bits required to represent it, and the relevant resources are time and space. In the last few years it has been shown that many pure-state quantum algorithms can be efficiently simulated on a CC when the extent of entanglement is limited (e.g., [2,3]) or when the quantum gates available are far from allowing us to build a set of universal gates [4,5,6]. Here, we focus on a Lie algebraic analysis to obtain other situations where quantum algorithms can be efficiently simulated by CCs.…”

Coherent Ising machines—optical neural networks operating at the quantum limit

“…A continuous variable (harmonic oscillator) version of the above theorem was developed by Bartlett et al 47 The statement of this theorem runs as follows:…”

Resonant cascaded down-conversion

“…These states are not general enough for universal quantuminformation operations [3]. For instance, it has been shown that quantum computation based solely on Gaussian CV states can be efficiently simulated by a classical computer [4]. Also, CV entanglement purification requires a Kerr-nonlinearity-based quantum nondemolition measurement [5] or, in general, a non-Gaussian state [6].…”