The complexity of Gottesman-Kitaev-Preskill states

TL;DR

This paper studies the complexity of preparing Gottesman-Kitaev-Preskill states in continuous-variable quantum systems, providing a new circuit with provable fidelity guarantees and establishing optimality of its size.

Contribution

It introduces a novel circuit for approximate GKP state preparation with fidelity guarantees and proves the optimality of its linear size dependence.

Findings

The circuit prepares GKP states with polynomial closeness in parameters.

The circuit size is linear in the logarithm of inverse parameters.

This work characterizes the complexity of GKP states fully.

Abstract

We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations, single- and two-qubit operations, as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an $L^1$-norm approximation to the state. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state $|\mathsf{GKP}_{\kappa,\Delta}\rangle$. Here $\kappa^{-2}$ is the variance of the envelope and $\Delta^2$ is the variance of the individual peaks. We show that the circuit accepts with constant probability and -- conditioned on acceptance -- the output state is polynomially close in $(\kappa,\Delta)$ to the state $|\mathsf{GKP}_{\kappa,\Delta}\rangle$. The size of our circuit is linear in $(\log 1/\kappa,\log 1/\Delta)$. To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states.