Complexity of Gottesman-Kitaev-Preskill States

Authors:
Open Access
Publisher website
Google Scholar
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 and 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 approximation to the state in trace distance. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state | G K P κ , Δ ⟩ . Here, κ − 2 is the variance of the envelope, and Δ 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 ( κ , Δ ) to the state | G K P κ , Δ ⟩ . The size of our circuit is linear in ( log 1 / κ , log 1 / Δ ) . 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.
All Related Versions
Funding Information
  • European Research Council (101001976)
  • Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

This publication has 35 references indexed in Scilit: