General Approach to Error Detection of Bosonic Codes via Phase Estimation

Abstract

We present a general approach to error detection of bosonic quantum error-correction codes via an adaptive quantum phase estimation algorithm assisted by a single ancilla qubit. The approach is applicable to a broad class of bosonic codes whose error syndromes are described by symmetry or stabilizer operators, including the rotation-symmetric codes and Gottesman-Kitaev-Preskill (GKP) codes. The detection precision scales inversely with the total evolution time and thus reaches the Heisenberg limit. We numerically demonstrate the approach for several examples, such as detecting bosonic excitation loss errors in high-order cat or binomial codes and displacement errors in finite-energy GKP codes. We also extend the approach to efficiently generate arbitrary Fock states. Our schemes are feasible in present-day experiments.

Figures (2)

• Figure 1: Schematic of error detection of bosonic codes with adaptive QPE. (a) A single bosonic mode coupled to an ancilla qubit. (b) The quantum circuit of adaptive QPE. Our protocol is applicable to both rotation-symmetric and GKP codes.
• Figure 2: Error detection via adaptive QPE for the (a) cat code and (b) GKP code. The error states are produced by the lossy bosonic channel, and the probability distributions of error detection are shown along with typical detected states. For GKP states, we show the distribution of $\delta(x,p)={\vartheta}(x,p)-\operatorname{round}[{\vartheta}(x,p)]$ to represent the actual displacement, and the original (detected) GKP lattices are labeled by solid (dashed) grey lines. Here we take a truncated Fock space with dimension 100, 701 for the cat and GKP code respectively, and the bosonic loss probability as $\gamma=0.03$.