Fault-Tolerant Logical Measurements via Homological Measurement

Abstract

We introduce homological measurement, a framework for measuring the logical Pauli operators encoded in Calderbank-Shor-Steane stabilizer codes. The framework is based on the algebraic description of such codes as chain complexes. Protocols such as lattice surgery and 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 quantum low density parity-check codes, requiring a number of ancillary qubits growing only linearly with the weight of the logical operator measured, and guarantee that the distance of the code is preserved. We further benchmark our protocol numerically in a photonic architecture based on Gottesman-Kitaev-Preskill qubits, showing that the logical error rates of various codes are on par with other methods requiring more ancilla qubits.