Characterization of errors in a CNOT between surface code patches

TL;DR

This paper analyzes the errors in a lattice-surgery-based CNOT operation between surface code patches, optimizing measurement rounds and characterizing the logical error channel under physical error models.

Contribution

It introduces an optimized measurement protocol and provides a detailed characterization of the logical error channel for surface code CNOTs.

Findings

Optimal measurement rounds vary with error rates and patch separation.

Logical error channel exhibits symmetry and error correlations are suppressed.

Provides guidelines for fault-tolerant quantum circuit implementation.

Abstract

As current experiments already realize small quantum circuits on error corrected qubits, it is important to fully understand the effect of physical errors on the logical error channels of these fault-tolerant circuits. Here, we investigate a lattice-surgery-based CNOT operation between two surface code patches under phenomenological error models. (i) For two-qubit logical Pauli measurements -- the elementary building block of the CNOT -- we optimize the number of stabilizer measurement rounds, usually taken equal to $d$, the size (code distance) of each patch. We find that the optimal number can be greater or smaller than $d$, depending on the rate of physical and readout errors, and the separation between the code patches. (ii) We fully characterize the two-qubit logical error channel of the lattice-surgery-based CNOT. We find a symmetry of the CNOT protocol, that results in a symmetry of the logical error channel. We also find that correlations between X and Z errors on the logical level are suppressed under minimum weight decoding.