arXiv:quant-ph/9607030 · Physical Review A

Efficient computations of encodings for quantum error correction

Richard Cleve, Daniel Gottesman

Published Jul 1, 1997• 122 citations

Abstract

We show how, given any set of generators of the stabilizer of a quantum code, an efficient gate array that computes the code words can be explicitly constructed. For an $n$-qubit code whose stabilizer has $d$ generators, the resulting gate array consists of $O(\mathrm{nd})$ operations, and converts $k$-qubit data (where $k=n\ensuremath{-}d)$ into $n$-qubit code words.

DOI: 10.1103/physreva.56.76

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arXiv:quant-ph/9607030 · Physical Review A

Efficient computations of encodings for quantum error correction

Richard Cleve, Daniel Gottesman

Published Jul 1, 1997• 122 citations

arXiv HTML PDF

Abstract

We show how, given any set of generators of the stabilizer of a quantum code, an efficient gate array that computes the code words can be explicitly constructed. For an

n

-qubit code whose stabilizer has

d

generators, the resulting gate array consists of

O(\mathrm{nd})

operations, and converts

k

-qubit data (where

k=n\ensuremath{-}d)

into

n

-qubit code words.

DOI: 10.1103/physreva.56.76

Related papers

Efficient one-way quantum computations for quantum error correction

Wei Huang, Zhaohui Wei

We show how to explicitly construct an $O(nd)$ size and constant quantum depth circuit which encodes any given $n$ -qubit stabilizer code with $d$ generators. Our construction is derived using the graphic description for stabilizer codes and the one-way quantum computation model. Our result demonstrates how to use cluster states as scalable resources for many multi-qubit entangled states and how to use the one-way quantum computation model to improve design of quantum algorithms.We show how to explicitly construct an $O(nd)$ size and constant quantum depth circuit which encodes any given $n$-qubit stabilizer code with $d$ generators. Our construction is derived using the graphic description for stabilizer codes and the one-way quantum computation model. Our result demonstrates how to use cluster states as scalable resources for many multi-qubit entangled states and how to use the one-way quantum computation model to improve design of quantum algorithms.

arXiv:0712.2295Journal of Physics A Mathematical and TheoreticalJun 30, 20095 citations

Codeword-stabilized quantum codes on subsystems

Jeonghwan Shin, Jun Heo, Todd A. Brun

Codeword-stabilized quantum codes provide a unified approach to constructing quantum error-correcting codes, including both additive and nonadditive quantum codes. Standard codeword-stabilized quantum codes encode quantum information into subspaces. The more general notion of encoding quantum information into a subsystem is known as an operator (or subsystem) quantum error-correcting code. Most operator codes studied to date are based in the usual stabilizer formalism. We introduce operator quantum codes based on the codeword-stabilized quantum code framework. Based on the necessary and sufficient conditions for operator quantum error correction, we derive an error-correction condition for operator codeword-stabilized quantum codes. Based on this condition, the word operators of a operator codeword-stabilized quantum code are constructed from a set of classical binary errors induced by generators of the gauge group. We use this scheme to construct examples of both additive and nonadditive codes that encode quantum information into a subsystem.Codeword-stabilized quantum codes provide a unified approach to constructing quantum error-correcting codes, including both additive and nonadditive quantum codes. Standard codeword-stabilized quantum codes encode quantum information into subspaces. The more general notion of encoding quantum information into a subsystem is known as an operator (or subsystem) quantum error-correcting code. Most operator codes studied to date are based in the usual stabilizer formalism. We introduce operator quantum codes based on the codeword-stabilized quantum code framework. Based on the necessary and sufficient conditions for operator quantum error correction, we derive an error-correction condition for operator codeword-stabilized quantum codes. Based on this condition, the word operators of a operator codeword-stabilized quantum code are constructed from a set of classical binary errors induced by generators of the gauge group. We use this scheme to construct examples of both additive and nonadditive codes that encode quantum information into a subsystem.

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Bounds on Quantum Error Correcting Codes

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The previous two chapters laid out the theoretical framework for quantum error correcting codes, and especially for quantum stabilizer codes. Following Cleve and Gottesman [1], the present chapter will explain how to construct quantum circuits that efficiently encode and decode a quantum stabilizer code. Before entering into that discussion, it proves advantageous to introduce a standard form for quantum stabilizer codes. This is done in Section 4.1. Assuming that a quantum stabilizer code in standard form has been given, Sections 4.2 and 4.3 explain how to, respectively, encode and decode its code words. A brief summary of quantum circuits is given in Appendix C.The previous two chapters laid out the theoretical framework for quantum error correcting codes, and especially for quantum stabilizer codes. Following Cleve and Gottesman [1], the present chapter will explain how to construct quantum circuits that efficiently encode and decode a quantum stabilizer code. Before entering into that discussion, it proves advantageous to introduce a standard form for quantum stabilizer codes. This is done in Section 4.1. Assuming that a quantum stabilizer code in standard form has been given, Sections 4.2 and 4.3 explain how to, respectively, encode and decode its code words. A brief summary of quantum circuits is given in Appendix C.

CRC Press eBooksFeb 7, 20080 citations

Implementation of standard quantum error-correction codes for solid-state qubits

Tetsufumi Tanamoto

In quantum error-correcting code (QECC), many quantum operations and measurements are necessary to correct errors in logical qubits. In the stabilizer formalism, which is widely used in QECC, generators ${G}_{i}\phantom{\rule{0.28em}{0ex}}(i=1,2,...)$ consist of multiples of Pauli matrices and perform encoding, decoding, and measurement. In order to maintain encoding states, the stabilizer Hamiltonian ${H}_{\mathrm{stab}}=\ensuremath{-}{\ensuremath{\sum}}_{i}{G}_{i}$ is suitable because its ground state corresponds to the code space. On the other hand, Hamiltonians of most solid-state qubits have two-body interactions and show their own dynamics. In addition solid-state qubits are fixed on substrate and qubit-qubit operation is restricted in their neighborhood. The main purpose of this paper is to show how to directly generate the stabilizer Hamiltonian ${H}_{\mathrm{stab}}$ from conventional two-body Hamiltonians with Ising interaction and $XY$ interaction by applying a pulse control method such as an NMR technique. We show that generation times of ${H}_{\mathrm{stab}}$ for nine-qubit code, five-qubit code, and Steane code are estimated to be less than 300 ns when typical experimental data of superconducting qubits are used, and sufficient pulse control is assumed. We also show how to prepare encoded states from an initial state $|0,...,0\ensuremath{\rangle}$. In addition, we discuss an appropriate arrangement of two- or three-dimensional arrayed qubits.In quantum error-correcting code (QECC), many quantum operations and measurements are necessary to correct errors in logical qubits. In the stabilizer formalism, which is widely used in QECC, generators ${G}_{i}\phantom{\rule{0.28em}{0ex}}(i=1,2,...)$ consist of multiples of Pauli matrices and perform encoding, decoding, and measurement. In order to maintain encoding states, the stabilizer Hamiltonian ${H}_{\mathrm{stab}}=\ensuremath{-}{\ensuremath{\sum}}_{i}{G}_{i}$ is suitable because its ground state corresponds to the code space. On the other hand, Hamiltonians of most solid-state qubits have two-body interactions and show their own dynamics. In addition solid-state qubits are fixed on substrate and qubit-qubit operation is restricted in their neighborhood. The main purpose of this paper is to show how to directly generate the stabilizer Hamiltonian ${H}_{\mathrm{stab}}$ from conventional two-body Hamiltonians with Ising interaction and $XY$ interaction by applying a pulse control method such as an NMR technique. We show that generation times of ${H}_{\mathrm{stab}}$ for nine-qubit code, five-qubit code, and Steane code are estimated to be less than 300 ns when typical experimental data of superconducting qubits are used, and sufficient pulse control is assumed. We also show how to prepare encoded states from an initial state $|0,...,0\ensuremath{\rangle}$. In addition, we discuss an appropriate arrangement of two- or three-dimensional arrayed qubits.

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Encoder Circuit Optimization for Nonbinary Quantum Error Correction Codes in Prime Dimensions: An Algorithmic Framework

Aditya Sodhani, Keshab K. Parhi

Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically operate on qubits—quantum information elements that can occupy superpositions of the basis $|0\rangle$ and $|1\rangle$ states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis $|0\rangle$ and $|1\rangle$ states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3) $[[9,5,3]]_{3}$ , $[[5,1,3]]_{3}$ , and $[[7,1,3]]_{3}$ codes, and a 9–21% reduction for the ququint (d = 5) $[[10,6,3]]_{5}$ code when compared to prior work. We also achieved circuit depth reductions up to 42%.Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically operate on qubits—quantum information elements that can occupy superpositions of the basis$|0\rangle$and$|1\rangle$states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis$|0\rangle$and$|1\rangle$states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3)$[[9,5,3]]_{3}$,$[[5,1,3]]_{3}$, and$[[7,1,3]]_{3}$codes, and a 9–21% reduction for the ququint (d = 5)$[[10,6,3]]_{5}$code when compared to prior work. We also achieved circuit depth reductions up to 42%.

arXiv:2509.25587IEEE Transactions on Quantum EngineeringJan 1, 20260 citations

Strategy for implementing stabilizer-based codes on solid-state qubits

Tetsufumi Tanamoto, Vladimir M. Stojanović, Christoph Bruder +1 more

We present a method for implementing stabilizer-based codes with encoding schemes of the operator quantum error correction paradigm, e.g., the ``standard'' five-qubit and CSS codes, on solid-state qubits with Ising or $XY$ -type interactions. Using pulse sequences, we show how to dynamically generate the effective dynamics of the stabilizer Hamiltonian, the sum of an appropriate set of stabilizer operators for a given code. Within this approach, the encoded states (ground states of the stabilizer Hamiltonian) can be prepared without measurements and preserved against both the time evolution governed by the original qubit Hamiltonian, and errors caused by local sources.We present a method for implementing stabilizer-based codes with encoding schemes of the operator quantum error correction paradigm, e.g., the ``standard'' five-qubit and CSS codes, on solid-state qubits with Ising or $XY$-type interactions. Using pulse sequences, we show how to dynamically generate the effective dynamics of the stabilizer Hamiltonian, the sum of an appropriate set of stabilizer operators for a given code. Within this approach, the encoded states (ground states of the stabilizer Hamiltonian) can be prepared without measurements and preserved against both the time evolution governed by the original qubit Hamiltonian, and errors caused by local sources.

arXiv:1301.4796Physical Review AMay 6, 201312 citations

We show how, given any set of generators of the stabilizer of a quantum code, an efficient gate array that computes the code words can be explicitly constructed. For an

n

-qubit code whose stabilizer has

d

generators, the resulting gate array consists of

O(\mathrm{nd})

operations, and converts

k

-qubit data (where

k=n\ensuremath{-}d)

into

n

-qubit code words.

We show how to explicitly construct an

O(nd)

size and constant quantum depth circuit which encodes any given

n

-qubit stabilizer code with

d

generators. Our construction is derived using the graphic description for stabilizer codes and the one-way quantum computation model. Our result demonstrates how to use cluster states as scalable resources for many multi-qubit entangled states and how to use the one-way quantum computation model to improve design of quantum algorithms.

{G}_{i}\phantom{\rule{0.28em}{0ex}}(i=1,2,...)

consist of multiples of Pauli matrices and perform encoding, decoding, and measurement. In order to maintain encoding states, the stabilizer Hamiltonian ${H}_{\mathrm{stab}}=\ensuremath{-}{\ensuremath{\sum}}_{i}{G}_{i}$ is suitable because its ground state corresponds to the code space. On the other hand, Hamiltonians of most solid-state qubits have two-body interactions and show their own dynamics. In addition solid-state qubits are fixed on substrate and qubit-qubit operation is restricted in their neighborhood. The main purpose of this paper is to show how to directly generate the stabilizer Hamiltonian

{H}_{\mathrm{stab}}

from conventional two-body Hamiltonians with Ising interaction and

XY

interaction by applying a pulse control method such as an NMR technique. We show that generation times of

{H}_{\mathrm{stab}}

for nine-qubit code, five-qubit code, and Steane code are estimated to be less than 300 ns when typical experimental data of superconducting qubits are used, and sufficient pulse control is assumed. We also show how to prepare encoded states from an initial state $|0,...,0\ensuremath{\rangle}$. In addition, we discuss an appropriate arrangement of two- or three-dimensional arrayed qubits.

|0\rangle

and

|1\rangle

states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis

|0\rangle

and

|1\rangle

states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3)

[[9,5,3]]_{3}

[[5,1,3]]_{3}

, and

[[7,1,3]]_{3}

codes, and a 9–21% reduction for the ququint (d = 5)

[[10,6,3]]_{5}

code when compared to prior work. We also achieved circuit depth reductions up to 42%.

XY

-type interactions. Using pulse sequences, we show how to dynamically generate the effective dynamics of the stabilizer Hamiltonian, the sum of an appropriate set of stabilizer operators for a given code. Within this approach, the encoded states (ground states of the stabilizer Hamiltonian) can be prepared without measurements and preserved against both the time evolution governed by the original qubit Hamiltonian, and errors caused by local sources.