The complexity of Gottesman-Kitaev-Preskill states

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 $|{\mathsf{GKP}}_{\kappa ,\Delta }⟩$. 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 $|{\mathsf{GKP}}_{\kappa ,\Delta }⟩$. 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.