Quantum Circuits for Measuring Levin-Wen Operators

TL;DR

This paper develops quantum circuits for measuring key operators in the Levin-Wen model for Fibonacci anyons, enabling syndrome measurements and code verification with optimized gate sets.

Contribution

It introduces quantum circuits for measuring Levin-Wen operators and simplifies verification of the Fibonacci code's self-consistency conditions.

Findings

Measurement circuits become more efficient with n-qubit Toffoli gates for n=3,4,5

Quantified circuit complexity using different universal gate sets

Provided simplified circuits for code verification and pentagon equation

Abstract

We construct quantum circuits for measuring the commuting set of vertex and plaquette operators that appear in the Levin-Wen model for doubled Fibonacci anyons. Such measurements can be viewed as syndrome measurements for the quantum error-correcting code defined by the ground states of this model (the Fibonacci code). We quantify the complexity of these circuits with gate counts using different universal gate sets and find these measurements become significantly easier to perform if n-qubit Toffoli gates with n = 3,4 and 5 can be carried out directly. In addition to measurement circuits, we construct simplified quantum circuits requiring only a few qubits that can be used to verify that certain self-consistency conditions, including the pentagon equation, are satisfied by the Fibonacci code.