Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.02753 · Physical Review X

Fault-Tolerant Logical Measurements via Homological Measurement

Benjamin Ide, Manoj G. Gowda, Priya J. Nadkarni, Guillaume Dauphinais

Published Jun 10, 2025• 10 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevx.15.021088

Related papers

Parallel Logical Measurements via Quantum Code Surgery

Alexander Cowtan, Zhiyang He, Dominic J. Williamson +1 more

Quantum code surgery is a flexible and low-overhead technique for performing logical measurements on quantum error-correcting codes, which generalizes lattice surgery. In this work, we present a code surgery scheme, applicable to any qubit stabilizer 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 ( t ω ( log  t + log 3  ω ) ) ancilla qubits, where ω ≥ 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.

arXiv:2503.05003•PRX Quantum•Apr 7, 2026•2 citations

Magic Quantum Code Surgery

Kathleen Chang, Anasuya Lyons, Yuanjie Ren +5 more

Non-Clifford operations required for universal quantum computation can be implemented by measuring logical Clifford operators. We present a generalized framework to fault-tolerantly measure transversal Clifford logical operators in quantum low-density parity check (QLDPC) codes. Our protocol takes any QLDPC code and deforms the code across additional ancillary qubits such that a product of stabilizers measures the desired logical Clifford operator. We prove that the deformed, Clifford-stabilized code remains LDPC and that its distance is preserved up to a constant factor independent of the number of physical qubits. We further prove that the Clifford measurement has a fault distance against data qubit and measurement errors that grows linearly with the distance of the original code. We apply our methods to implement logical non-Clifford gates in algorithmically relevant subroutines and to prepare magic states in quantum stabilizer codes.

quant-pharXiv:2610.03920•Oct 6, 2026•0 citations

Parsimonious Quantum Low-Density Parity-Check Code Surgery

Andrew C. Yuan, Alexander Cowtan, Zhiyang He +2 more

Quantum code surgery offers a flexible, low-overhead framework for executing logical measurements within quantum error-correcting codes. It encompasses several fault-tolerant logical computation schemes, including parallel surgery, universal adapters and fast surgery, and serves as the key primitive in extractor architectures. The efficiency of these schemes crucially depends on constructing low-overhead ancilla systems for measuring arbitrary logical operators in general quantum Low-Density Parity-Check (qLDPC) codes. In this work, we introduce a method to construct an ancilla system of qubit size $O(W \log W)$ to measure an arbitrary logical Pauli operator of weight $W$ in any qLDPC stabilizer code. This new construction immediately reduces the asymptotic overhead across various quantum code surgery schemes.

quant-pharXiv:2603.05082•Mar 6, 2026•0 citations

Homomorphic Logical Measurements

Shilin Huang, Tomas Jochym-O’Connor, Theodore J. Yoder

Shor and Steane ancilla are two well-known methods for fault-tolerant logical measurements, which are successful on small codes and their concatenations. On large quantum low-density-parity-check (LDPC) codes, however, Shor and Steane measurements have impractical time and space overhead respectively. In this work, we widen the choice of ancilla codes by unifying Shor and Steane measurements into a single framework, called homomorphic measurements. For any Calderbank-Shor-Steane (CSS) code with the appropriate ancilla code, one can avoid repetitive measurements or complicated ancilla state preparation procedures such as distillation, which overcomes the difficulties of both Shor and Steane methods. As an example, we utilize the theory of covering spaces to construct homomorphic measurement protocols for arbitrary $X$- or $Z$-type logical Pauli operators on surface codes in general, including the toric code and hyperbolic surface codes. Conventional surface code decoders, such as minimum-weight perfect matching, can be directly applied to our constructions.

arXiv:2211.03625•Nov 7, 2022•0 citations

Individually Addressed Logical Pauli Measurements by QLDPC Surgery on a Trapped Ion Quantum Computer

Surabhi Luthra, Alex Owens

Quantum low-density parity check (QLDPC) codes may lead to order-of-magnitude reductions in the qubit overhead required for fault-tolerant quantum computation. Achieving this improved overhead requires logical operations that selectively address individual logical qubits encoded within multi-qubit code blocks. Recently proposed architectures achieve this through QLDPC surgery, in which non-destructive Pauli measurements are performed using ancillary ''gadget'' systems transiently entangled with the code block, generalising lattice surgery. Here we implement QLDPC surgery on a [[10,2,3]] generalised bicycle code using Quantinuum's H2-2 trapped ion processor. A single-round surgery measurement using seven gadget qubits measures the $\bar{X}_1$ operator with a 0.72 [0.32, 1.41]% (95% credible interval) error rate and agrees with subsequent transversal measurement with 1.03 [0.60, 1.65]% error. Decoding across three rounds of syndrome extraction suppresses the gadget measurement error, reducing the error rate to below 0.26%. The syndrome after surgery confirms return to the code space and the unaddressed ''spectator'' qubit is preserved with error 0.21 [0.04, 0.67]% during the three-round measurement. We further demonstrate composition of conjugate basis gadgets by sequentially measuring $\bar{X}_1$ then $\bar{Z}_2$, reusing the gadget qubits, and obtain a 1.12 [0.60, 1.94]% error in the $\bar{Z}_2$ measurement. These results constitute a hardware validation of the generalised surgery primitive underlying Pauli-based computing for QLDPC architectures.

quant-pharXiv:2610.39067•Oct 30, 2026•0 citations

QGPU: Parallel logic in quantum LDPC codes

Boren Gu, Andy Zeyi Liu, Armanda O. Quintavalle +3 more

Quantum error correction is critical to the design and manufacture of scalable quantum computing systems. Recently, there has been growing interest in quantum low-density parity-check codes as a resource-efficient alternative to surface codes. Their adoption is hindered by the difficulty of compiling fault-tolerant logical operations. A key challenge is that logical qubits do not necessarily map to disjoint sets of physical qubits, which limits parallelism. We introduce clustered-cyclic codes, a quantum low-density parity-check code family with finite-size instances such as [[136,8,14]] and [[198,18,10]] that are competitive with state-of-the-art constructions. These codes admit a directly addressable logical basis, enabling highly parallel logical measurement layers. To leverage this structure, we propose parallel product surgery for quantum product codes. Using an auxiliary copy of the data patch and an engineered product-connection structure, the protocol performs many logical Pauli-product measurements in a single surgery round with small, fixed overhead. For clustered-cyclic codes, this yields surface-code-style maximal parallelism: up to k/2 disjoint Pauli-product measurements per round under explicit algebraic conditions. We prove that parallel product surgery preserves the code distance for hypergraph product codes and numerically verify distance preservation for the listed clustered-cyclic instances with k = 8. Finally, for the [[24,8,3]] clustered-cyclic code, treating half of the logical qubits as auxiliaries enables arbitrary parallel CNOTs on disjoint pairs; combined with symmetry-derived operations, these gates generate the full Clifford group fault-tolerantly.

quant-pharXiv:2603.05398•Mar 6, 2026•0 citations
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.
Quantum code surgery is a flexible and low-overhead technique for performing logical measurements on quantum error-correcting codes, which generalizes lattice surgery. In this work, we present a code surgery scheme, applicable to any qubit stabilizer 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 ( t ω ( log  t + log 3  ω ) ) ancilla qubits, where ω ≥ 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.
Non-Clifford operations required for universal quantum computation can be implemented by measuring logical Clifford operators. We present a generalized framework to fault-tolerantly measure transversal Clifford logical operators in quantum low-density parity check (QLDPC) codes. Our protocol takes any QLDPC code and deforms the code across additional ancillary qubits such that a product of stabilizers measures the desired logical Clifford operator. We prove that the deformed, Clifford-stabilized code remains LDPC and that its distance is preserved up to a constant factor independent of the number of physical qubits. We further prove that the Clifford measurement has a fault distance against data qubit and measurement errors that grows linearly with the distance of the original code. We apply our methods to implement logical non-Clifford gates in algorithmically relevant subroutines and to prepare magic states in quantum stabilizer codes.
Quantum code surgery offers a flexible, low-overhead framework for executing logical measurements within quantum error-correcting codes. It encompasses several fault-tolerant logical computation schemes, including parallel surgery, universal adapters and fast surgery, and serves as the key primitive in extractor architectures. The efficiency of these schemes crucially depends on constructing low-overhead ancilla systems for measuring arbitrary logical operators in general quantum Low-Density Parity-Check (qLDPC) codes. In this work, we introduce a method to construct an ancilla system of qubit size O(Wlog⁡W)O(W \log W)O(WlogW) to measure an arbitrary logical Pauli operator of weight WWW in any qLDPC stabilizer code. This new construction immediately reduces the asymptotic overhead across various quantum code surgery schemes.
Shor and Steane ancilla are two well-known methods for fault-tolerant logical measurements, which are successful on small codes and their concatenations. On large quantum low-density-parity-check (LDPC) codes, however, Shor and Steane measurements have impractical time and space overhead respectively. In this work, we widen the choice of ancilla codes by unifying Shor and Steane measurements into a single framework, called homomorphic measurements. For any Calderbank-Shor-Steane (CSS) code with the appropriate ancilla code, one can avoid repetitive measurements or complicated ancilla state preparation procedures such as distillation, which overcomes the difficulties of both Shor and Steane methods. As an example, we utilize the theory of covering spaces to construct homomorphic measurement protocols for arbitrary XXX- or ZZZ-type logical Pauli operators on surface codes in general, including the toric code and hyperbolic surface codes. Conventional surface code decoders, such as minimum-weight perfect matching, can be directly applied to our constructions.
Quantum low-density parity check (QLDPC) codes may lead to order-of-magnitude reductions in the qubit overhead required for fault-tolerant quantum computation. Achieving this improved overhead requires logical operations that selectively address individual logical qubits encoded within multi-qubit code blocks. Recently proposed architectures achieve this through QLDPC surgery, in which non-destructive Pauli measurements are performed using ancillary ''gadget'' systems transiently entangled with the code block, generalising lattice surgery. Here we implement QLDPC surgery on a [[10,2,3]] generalised bicycle code using Quantinuum's H2-2 trapped ion processor. A single-round surgery measurement using seven gadget qubits measures the Xˉ1\bar{X}_1Xˉ1​ operator with a 0.72 [0.32, 1.41]% (95% credible interval) error rate and agrees with subsequent transversal measurement with 1.03 [0.60, 1.65]% error. Decoding across three rounds of syndrome extraction suppresses the gadget measurement error, reducing the error rate to below 0.26%. The syndrome after surgery confirms return to the code space and the unaddressed ''spectator'' qubit is preserved with error 0.21 [0.04, 0.67]% during the three-round measurement. We further demonstrate composition of conjugate basis gadgets by sequentially measuring Xˉ1\bar{X}_1Xˉ1​ then Zˉ2\bar{Z}_2Zˉ2​, reusing the gadget qubits, and obtain a 1.12 [0.60, 1.94]% error in the Zˉ2\bar{Z}_2Zˉ2​ measurement. These results constitute a hardware validation of the generalised surgery primitive underlying Pauli-based computing for QLDPC architectures.
Quantum error correction is critical to the design and manufacture of scalable quantum computing systems. Recently, there has been growing interest in quantum low-density parity-check codes as a resource-efficient alternative to surface codes. Their adoption is hindered by the difficulty of compiling fault-tolerant logical operations. A key challenge is that logical qubits do not necessarily map to disjoint sets of physical qubits, which limits parallelism. We introduce clustered-cyclic codes, a quantum low-density parity-check code family with finite-size instances such as [[136,8,14]] and [[198,18,10]] that are competitive with state-of-the-art constructions. These codes admit a directly addressable logical basis, enabling highly parallel logical measurement layers. To leverage this structure, we propose parallel product surgery for quantum product codes. Using an auxiliary copy of the data patch and an engineered product-connection structure, the protocol performs many logical Pauli-product measurements in a single surgery round with small, fixed overhead. For clustered-cyclic codes, this yields surface-code-style maximal parallelism: up to k/2 disjoint Pauli-product measurements per round under explicit algebraic conditions. We prove that parallel product surgery preserves the code distance for hypergraph product codes and numerically verify distance preservation for the listed clustered-cyclic instances with k = 8. Finally, for the [[24,8,3]] clustered-cyclic code, treating half of the logical qubits as auxiliaries enables arbitrary parallel CNOTs on disjoint pairs; combined with symmetry-derived operations, these gates generate the full Clifford group fault-tolerantly.