Parallel Logical Measurements via Quantum Code Surgery

TL;DR

This work tackles the challenge of performing fault-tolerant, parallel logical Pauli measurements on quantum LDPC codes by introducing a code-surgery framework that combines brute-force branching, gauging logical measurements, and universal adapters. The scheme achieves parallel measurement of disjoint Pauli products with time complexity independent of the number of terms, and ancilla overhead scaling as $O(t \omega(\log t + \log^3 \omega))$, without requiring ancillary logical code blocks. It generalizes beyond CSS codes to arbitrary stabiliser codes and supports measuring Pauli products and arbitrary commuting sets via twist-free surgery, including mixed Y terms without costly ancilla states. The approach leverages hypergraph-product structures and carefully chosen logical-basis representations to preserve LDPC properties and code distance, offering a path toward low-overhead, Pauli-based quantum computation. Future work includes threshold analyses, finite-size optimizations, and empirical evaluations of circuit-level performance.

Abstract

Quantum code surgery is a flexible and low overhead technique for performing logical measurements on quantum error-correcting codes, which generalises lattice surgery. In this work, we present a code surgery scheme, applicable to any qubit stabiliser low-density parity check (LDPC) code, that fault-tolerantly measures many logical Pauli operators in parallel. For a collection of logically disjoint Pauli product measurements supported on $t$ logical qubits, our scheme uses $O\big(t ω(\log t + \log^3ω)\big)$ ancilla qubits, where $ω\geq d$ is the maximum weight of the single logical Pauli representatives involved in the measurements, and $d$ is the code distance. This is all done in time $O(d)$ independent of $t$. Our proposed scheme preserves both the LDPC property and the fault-distance of the original code, without requiring ancillary logical codeblocks which may be costly to prepare. This addresses a shortcoming of several recently introduced surgery schemes which can only be applied to measure a limited number of logical operators in parallel if they overlap on data qubits.

Figures (10)

• Figure 1: An example logical circuit with three timesteps of Pauli product measurements. Each timestep contains logically disjoint Pauli products, which are measured in parallel.
• Figure 2: Schematic depiction of logical operator representatives with overlapping supports.
• Figure 3: A Tanner graph for the $\llbracket 9,1,3 \rrbracket$ Shor code.
• Figure 4: Examples of scalable Tanner graphs.
• Figure 5: Illustration of brute-force branching logical representatives with overlapping support.

Theorems & Definitions (45)

• Remark 1.1
• Definition 1.2
• Remark 1.3: Weights of logical representatives
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7