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Chinese Physics Letters

Systematic Algebraic Method to Identify Clifford Operations for Quantum Error-Correction Codes

Sheng-Chen Liu, Yu-Xuan Lin, Ying-Xiang Wang, Liang-You Peng

Published Mar 10, 2026• 0 citations
Publisher

Abstract

Abstract In the domain of quantum error correction, a critical task involves identifying logical operations on logical qubits for various quantum codes. However, owing to the inherent complexity of many quantum codes, devising an efficient method to implement the desired logical operations utilizing the structure of these codes presents a significant challenge. In previous studies, several methods were used to realize specific logical operations for certain quantum codes; however, they usually do not work for other quantum codes. In this study, we propose a systematic method to find solutions to any Clifford logical operation for Calderbank–Shor–Steane codes, which represent a broad category of many quantum codes. In contrast to previous geometrical methods, our method is based on an algebraic approach. The solved logical operation can be further decomposed into a sequence of single-qubit and two-qubit gates that can be directly realized using quantum hardware. By applying our method, we can find many geometrically difficult logical operations for two different types of quantum code, that is, toric and hyperbolic surface codes. We believe that the present study will find wide applications in the design of logical operations for many quantum low-density parity-check codes, which can play an important role in future fault-tolerant quantum computing.

DOI
10.1088/0256-307x/43/4/040603

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Abstract In the domain of quantum error correction, a critical task involves identifying logical operations on logical qubits for various quantum codes. However, owing to the inherent complexity of many quantum codes, devising an efficient method to implement the desired logical operations utilizing the structure of these codes presents a significant challenge. In previous studies, several methods were used to realize specific logical operations for certain quantum codes; however, they usually do not work for other quantum codes. In this study, we propose a systematic method to find solutions to any Clifford logical operation for Calderbank–Shor–Steane codes, which represent a broad category of many quantum codes. In contrast to previous geometrical methods, our method is based on an algebraic approach. The solved logical operation can be further decomposed into a sequence of single-qubit and two-qubit gates that can be directly realized using quantum hardware. By applying our method, we can find many geometrically difficult logical operations for two different types of quantum code, that is, toric and hyperbolic surface codes. We believe that the present study will find wide applications in the design of logical operations for many quantum low-density parity-check codes, which can play an important role in future fault-tolerant quantum computing.
Abstract Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli X and Z gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.
Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli X and Z gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉\lceil d/2\rceil⌈d/2⌉ for distance-ddd codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed TTT gate on the [[16,4,4]][[16,4,4]][[16,4,4]] BB code, TTT-type rotations on the [[64,8,8]][[64,8,8]][[64,8,8]] BB code, and CCZ-, CS-, and TTT-type operations on the [[36,6,6]][[36,6,6]][[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields Cr{}^rrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.
We study the implementation of fault-tolerant logical Clifford gates on stabilizer quantum error-correcting codes based on their symmetries. Our approach is to map the stabilizer code to a binary linear code, compute its automorphism group, and impose constraints based on the Clifford operators permitted. We provide a rigorous formulation of the method for finding automorphisms of stabilizer codes and generalize the Z X dualities previously introduced for Calderbank-Shor-Steane (CSS) codes to non-CSS codes. We provide a package implementing our algorithms that uses the computational-algebra software system for certain subroutines. Our algorithms map automorphism-group generators to physical circuits, calculate Pauli corrections based on the destabilizers of the code, and determine their logical action. We discuss the fault tolerance of the circuits and include examples of gates through automorphisms for the [ [ 4 , 2 , 2 ] ] and perfect [ [ 5 , 1 , 3 ] ] codes, bivariate bicycle codes, and the best-known distance codes.
Abstract Fault-tolerant quantum computation (FTQC) schemes using large block codes that encode k > 1 qubits in n physical qubits can potentially reduce the resource overhead to a great extent because of their high encoding rate. However, the fault-tolerant (FT) logical operations for the encoded qubits are difficult to find and implement, which usually takes not only a very large resource overhead but also long in situ computation time. In this paper, we focus on Calderbank–Shor–Steane [[ n , k , d ]] (CSS) codes and their logical FT Clifford circuits. We show that the depth of an arbitrary logical Clifford circuit can be implemented fault-tolerantly in O (1) steps in situ via either Knill or Steane syndrome measurement circuit, with the qualified ancilla states efficiently prepared. Particularly, for those codes satisfying k / n ∼ Θ(1), the resource scaling for Clifford circuits implementation on the logical level can be the same as on the physical level up to a constant, which is independent of code distance d . With a suitable pipeline to produce ancilla states, our scheme requires only a modest resource cost in physical qubits, physical gates, and computation time for very large scale FTQC.
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.