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  • Open Access

Universal Adapters between Quantum Low-Density Parity Check Codes

Esha Swaroop1,2,*, Tomas Jochym-O'Connor3,4,†, and Theodore J. Yoder3,‡

  • *Contact author: eswaroop@uwaterloo.ca
  • †Contact author: tjoc@ibm.com
  • ‡Contact author: ted.yoder@ibm.com
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PRX Quantum 7, 010324 – Published 4 February, 2026

DOI: https://doi.org/10.1103/1g44-jp62

Abstract

We propose the repetition code adapter as a way to perform joint logical Pauli measurements within a quantum low-density parity check (LDPC) codeblock or between separate such codeblocks, thus providing a flexible tool for fault-tolerant computation with quantum LDPC codes. This adapter is universal in the sense that it works regardless of the LDPC codes involved and the logical Paulis being measured. The construction achieves joint logical Pauli measurement of t weight O(d) operators using O(d) time and O~(td) additional qubits and checks, up to a factor polylogarithmic in d. As a special case, for some geometrically local codes in fixed D≥2 dimensions, only O(td) additional qubits and checks are required instead. By extending the adapter in the case t=2, we also construct a toric code adapter that uses O(d2) additional qubits and checks to perform addressable logical cnot gates on arbitrary LDPC codes via Dehn twists. To obtain these results, we develop a more general form of graph edge expansion and the SkipTree algorithm, which ensures a sparse transformation between different weight-2 check bases for the classical repetition code.

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Popular Summary

Error correction is key to building a quantum computer. This includes both storage of quantum information and computing on it. Quantum low-density parity check (LDPC) codes offer a route to build these devices with low space overhead. The next question is—How do we fault-tolerantly compute on these codes? Recently, there has been a surge of research regarding the nontrivial problem of fault-tolerant computation on qubits encoded in LDPC codes, and one solution that stands out for its general applicability is code deformation through stabilizer measurements, a general scheme referred to as LDPC surgery. In this work, we present a “universal adapter” construction to enable multi-qubit logical measurements within the same code or different codes, with low qubit overhead. This adapter is universal it works regardless of the LDPC codes involved and the logical operators being measured.

The simplest LDPC surgery works by appending a graphlike system, consisting of additional qubits (edges) and checks (vertices and cycles) and tailored to a desired logical measurement. Our adapter construction connects two or more of these auxiliary systems together to measure their corresponding product operator. Connecting them introduces more qubits and checks, but not more than the original systems, and, crucially for practicality, we also guarantee the new code checks remain sparse. We additionally identify some situations in which the surgery overhead can be even further reduced and separately a construction that merges an arbitrary LDPC code and the well-known toric code in order to perform a unitary logical gate.

Future work may focus on generalizing our ideas to LDPC surgery systems that are more time-efficient, by considering auxiliary systems described by hypergraphs instead of graphs.

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Outline

Information

PRX Quantum 7, 010324– Published 4 February, 2026
  • Received 11 April 2025
  • Revised 4 November 2025
  • Accepted 4 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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