Fault-tolerant logical measurements via homological measurement

TL;DR

This paper introduces homological measurement, a new framework for fault-tolerant logical measurements in CSS stabilizer codes, enabling efficient and scalable measurement protocols that preserve code distance.

Contribution

The paper develops a novel homological measurement framework and a specific edge expanded protocol for fault-tolerant logical measurements in qLDPC codes, with linear ancilla scaling.

Findings

The protocol requires ancillas growing linearly with logical operator weight.

It preserves the code distance during measurement.

Numerical benchmarks show competitive logical error rates in photonic GKP architectures.

Abstract

We introduce homological measurement, a framework for measuring the logical Pauli operators encoded in CSS stabilizer codes. The framework is based on the algebraic description of such codes as chain complexes. Protocols such as lattice surgery some of its recent generalizations are shown to be special cases of homological measurement. Using this framework, we develop a specific protocol called edge expanded homological measurement for fault-tolerant measurement of arbitrary logical Pauli operators of general qLDPC codes, requiring a number of ancillary qubits growing only linearly with the weight of the logical operator measured, and guaranteed that the distance of the code is preserved. We further benchmark our protocol numerically in a photonic architecture based on GKP qubits, showing that the logical error rate of various codes are on par with other methods requiring more ancilla qubits.