Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian circuits

Cameron Calcluth1, Alessandro Ferraro2,3, and Giulia Ferrini1

1Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, SE-412 96 Göteborg, Sweden
2Centre for Theoretical Atomic, Molecular and Optical Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom
3Dipartimento di Fisica ``Aldo Pontremoli,'' Università degli Studi di Milano, I-20133 Milano, Italy

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Abstract

We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements. For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems such as the Gottesman-Knill theorem can be employed to assess the simulatability. We first develop a method to evaluate the probability density function corresponding to measuring a single GKP state in the position basis following arbitrary squeezing and a large set of rotations. This method involves evaluating a transformed Jacobi theta function using techniques from analytic number theory. We then use this result to identify two large classes of multimode circuits which are classically efficiently simulatable and are not contained by the GKP encoded Clifford group. Our results extend the set of circuits previously known to be classically efficiently simulatable.

Quantum computers – devices in which quantum information can be encoded, processed, and read out – are expected to solve certain computational tasks exponentially faster than classical computers. This property is referred to as quantum advantage and has recently motivated a global effort toward building a quantum computer. But which quantum computing architectures are able to provide quantum advantage and which are not? Beyond the fundamental interest, being able to trace a boundary separating computationally useful quantum computing architectures, capable of providing computational speed-up, from classically efficiently simulatable ones is of technological importance in order to design devices capable of outperforming classical computation. Our work contributes to tracing this boundary.

We focus on quantum computer architectures whereby the information is encoded into continuous variables (CVs). This approach relies on quantized variables with a continuous spectrum, such as the position and momentum quadratures of the electromagnetic field. An example of such an encoding procedure is known as the Gottesman-Kitaev-Preskill (GKP) encoding, using GKP states. Architectures using this encoding allow for increased resilience to noise, with respect to architectures using discrete-variable systems.

Our work demonstrates that a large class of quantum circuits with input GKP states prepared to encode computational states such as 0 and 1 is efficiently simulatable with classical computers. We, therefore, demonstrate that these circuits are not capable of achieving quantum advantage. Our findings contribute therefore to drawing a divide between computationally useful and useless architectures of quantum computers.

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[1] Eric R. Anschuetz, Hong-Ye Hu, Jin-Long Huang, and Xun Gao, "Interpretable Quantum Advantage in Neural Sequence Learning", arXiv:2209.14353, (2022).

[2] Ulysse Chabaud and Mattia Walschaers, "Resources for bosonic quantum computational advantage", arXiv:2207.11781, (2022).

[3] Cameron Calcluth, Alessandro Ferraro, and Giulia Ferrini, "The vacuum provides quantum advantage to otherwise simulatable architectures", arXiv:2205.09781, (2022).

[4] Giacomo Pantaleoni, Ben Q. Baragiola, and Nicolas C. Menicucci, "The Zak transform: a framework for quantum computation with the Gottesman-Kitaev-Preskill code", arXiv:2210.09494, (2022).

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