Gate-based quantum simulation of Gaussian bosonic circuits on exponentially many modes

TL;DR

A framework for simulating, on an (n+1)-qubit quantum computer, the action of a Gaussian bosonic circuit on a state over 2^{n} modes is introduced, and a bounded-error-quantum-polynomial time (BQP)-complete GB decision problem is presented, indicating that GB evolutions of Gaussian states on exponentially many modes are as powerful as universal quantum computers.

Abstract

We introduce a framework for simulating, on an $(n+1)$-qubit quantum computer, the action of a Gaussian Bosonic (GB) circuit on a state over $2^n$ modes. Specifically, we encode the initial bosonic state's expectation values over quadrature operators (and their covariance matrix) as an input qubit-state. This is then evolved by a quantum circuit that effectively implements the symplectic propagators induced by the GB gates. We find families of GB circuits and initial states leading to efficient quantum simulations. For this purpose, we introduce a dictionary that maps between GB and qubit gates such that particle- (non-particle-) preserving GB gates lead to real (imaginary) time evolutions at the qubit level. For the special case of particle-preserving circuits, we present a BQP-complete GB decision problem, indicating that GB evolutions of Gaussian states on exponentially many modes are as powerful as universal quantum computers. We also perform numerical simulations of an interferometer on $\sim8$ billion modes, illustrating the power of our framework.

Figures (3)

• Figure 1: Schematic representation of our main results. a) We present a framework for simulating the action of a GB circuit on the first and second moments of quadrature operators of a bosonic state on $2^n$ modes on an $(n+1)$-qubit gate-based quantum computer. b) We show that particle-preserving GB evolutions on Gaussian bosonic states are sufficient to define a problem that is BQP-complete, thus indicating that passive linear optics on exponentially many bosonic modes are as powerful as universal quantum computers.
• Figure 2: Examples of GB gates in the qubit picture. We consider a bosonic system on $M=8$ modes, leading to a circuit on $4$ qubits. The local phase gate acts on the mode $m=6=2^0\times0+2^1\times1+2^2\times1$. The local beamsplitter acts on the modes $m=1$ and $m'=7$, whose binary representations only share the least significant bit. The local squeezing gate acts on mode $m=3=2^0\times1+2^1\times1+2^2\times0$, and is represented by an imaginary time evolution as a linear combination of unitaries with post-selection on two ancillary qubits (which have been added on top). The global phase gate acts on all modes whose index is even. The global beamsplitter is applied to the first half of the modes, pairing each mode with even index $m$ to its nearest neighbor mode with index $m'=m+1$. The global squeezing gate is applied to the first half of the modes.
• Figure 3: Simulation of a structured interferometer on $\sim8$ billion modes. We illustrate \ref{['pbm:interferometer']} by tracking two non-trivial evolutions of $\langle \hat{q}_1\rangle / x$ along a large bit-structured interferometer, the green (red) plot corresponding to a YES (NO) instance. The gray region corresponds to $\frac{1}{3}<\langle \hat{q}_1\rangle / x < \frac{2}{3}$. The simulations were performed with Qiboefthymiou2020qiboefthymiou2022quantum.

Theorems & Definitions (11)

• Definition 1: Bit-structured interferometer
• Theorem 1
• proof
• proof
• proof
• Definition 1: Bit-structured interferometer
• Theorem 1
• proof
• Lemma 1
• proof