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:1906.11137

Optimal Error Correcting Code For Ternary Quantum Systems

Ritajit Majumdar, Susmita Sur‐Kolay

Published Jun 26, 2019• 3 citations
arXivHTMLPDF

Abstract

Multi-valued quantum systems can store more information than binary ones for a given number of quantum states. For reliable operation of multi-valued quantum systems, error correction is mandated. In this paper, we propose a 5-qutrit quantum error-correcting code and provide its stabilizer formulation. Since 5 qutrits are necessary to correct a single error, our proposed code is optimal in the number of qutrits. We prove that the error model considered in this paper spans the entire $(3 \times 3)$ operator space. Therefore, our proposed code can correct any single error on the codeword. This code outperforms the previous 9-qutrit code in (i) the number of qutrits required for encoding, (ii) our code can correct any arbitrary $(3 \times 3)$ error, (ii) our code can readily correct bit errors in a single step as opposed to the two-step correction used previously, and (iii) phase error correction does not require correcting individual subspaces.

DOI
10.48550/arxiv.1906.11137

Related papers

Quantum error-correcting code for ternary logic

Ritajit Majumdar, Saikat Basu, Shibashis Ghosh +1 more

Ternary quantum systems are being studied because they provide more computational state space per unit of information, known as qutrit. A qutrit has three basis states, thus a qubit may be considered as a special case of a qutrit where the coefficient of one of the basis states is zero. Hence both $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional and $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors can occur on qutrits. In this paper, we (i) explore the possible $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional as well as $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors in qutrits and show that any pairwise bit swap error can be expressed as a linear combination of shift errors and phase errors, (ii) propose a special type of error called a quantum superposition error and show its equivalence to arbitrary rotation, (iii) formulate a nine-qutrit code which can correct a single error in a qutrit, and (iv) provide its stabilizer and circuit realization.

arXiv:1807.01863•Physical Review A•May 2, 2018•26 citations

Exploiting degeneracy to construct good ternary quantum error correcting code

Ritajit Majumdar, Susmita Sur‐Kolay

Quantum error-correcting code for higher dimensional systems can, in general, be directly constructed from the codes for qubit systems. What remains unknown is whether there exist efficient code design techniques for higher dimensional systems. In this paper, we propose a 7-qutrit error-correcting code for the ternary quantum system and show that this design formulation has no equivalence in qubit systems. This code is optimum in the number of qutrits required to correct a single error while maintaining the CSS structure. This degenerate CSS code can (i) correct up to seven simultaneous phase errors and a single bit error, (ii) correct two simultaneous bit errors on pre-defined pairs of qutrits on eighteen out of twenty-one possible pairs, and (iii) in terms of the cost of implementation, the depth of the circuit of this code is only two more than that of the ternary Steane code. Our proposed code shows that it is possible to design better codes explicitly for ternary quantum systems instead of simply carrying over codes from binary quantum systems.

arXiv:2008.00713•Aug 3, 2020•0 citations

Qutrit Error Correction Codes for Quantum Communications and Computations

Muhammad Imansyah Basudewa, Khoirul Anwar, Melinda Br Ginting

This paper proposes simple quantum [[9],[1],[3]] error correction codes for quantum systems beyond qubits, focusing on quantum ternary digits (qutrits) that represent the simplest higher-dimensional systems for future quantum technologies. Although qutrit-based systems offer larger capacity information over 2-dimensional systems, the challenge of identifying commuting stabilizers for constructing error correction codes becomes more difficult as the dimensionality increases or when working with a large number of qutrits. We design qutrit codes based on the generalized Pauli group and investigate the underlying structure, including shift, phase, and rotation operators. We successfully create suitable stabilizer codes that detect and correct qutrit errors. Furthermore, we (i) identify operators in qutrit, which are selected from various matrix interpretations for higher-dimensional systems, (ii) propose stabilizers for 9-qutrit error correction codes, and (iii) design a new 9-qutrit quantum circuit for communications. Our study is expected to contribute a foundational basis for possible future theoretical and experimental advancements in quantum computing, communications, and code-based cryptography in higher dimensional systems.

IEEE Asia-Pacific Conference on Applied Electromagnetics (APACE)•Dec 21, 2024•2 citations

Graph-theoretic approach to quantum error correction

Robert R. Vandermolen, Duncan Wright

We investigate a class of quantum error-correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum errors. In this framework, we represent the algebraic conditions for error correction in terms of edge avoidance between graphs providing a visual representation of the interplay between errors and error-correcting codes. Most importantly, this framework supports the development of quantum codes that correct against a predetermined set of errors, in contrast to current methods. A heuristic algorithm is presented, providing steps to develop codes that correct against an arbitrary noisy channel. We benchmark the correction capability of reflexive stabilizer codes for the case of single-qubit errors by comparison to existing stabilizer codes that are widely used. In addition, we present two instances of optimal encodings: an optimal encoding for fully correlated noise, which achieves a higher encoding rate than previously known, and a minimal encoding for single-qudit errors on a four-state system..

arXiv:2110.08414•Physical Review A•Mar 29, 2022•5 citations

Special Session: Quantum Error Correction in Near Term Systems

Ritajit Majumdar, Susmita Sur‐Kolay

Large-scale quantum computers mandate error correction and fault tolerance. Due to constraints on the number of qubits, fault tolerance is difficult to achieve in near-term quantum systems. Therefore, error correction should require minimal resources. Gates in the near-term devices are also noisy. Quantum error correction code blocks built with these noisy gates can inject further error in the circuit. The goals for error correction in near-term systems are as follows: (i) using a small number of qubits for encoding, and (ii) keeping cost of circuits for encoding and decoding low. In this paper, we propose two techniques to achieve these mutually orthogonal goals. For a binary quantum system we propose an error estimation method that can aid in reducing the number of error correcting blocks via sparse scheduling. For ternary quantum systems, we propose an approximate code that can correct errors with high probability while significantly reducing the circuit cost. These techniques are expected to be helpful for error mitigation in near-term systems in the absence of fault tolerance.

IEEE International Conference on Computer Design: VLSI in Computers and Processors (ICCD)•Oct 1, 2020•6 citations

Approximate Ternary Quantum Error Correcting Code with Low Circuit Cost

Ritajit Majumdar, Susmita Sur‐Kolay

Quantum error correcting code (QECC) with encoding and decoding circuits having low gate count, is an important criterion for realizing quantum computing systems. CSS QECCs are known to have simple circuits. Shaw et al. showed that it is not possible to have a 6-qubit CSS type QECC without sharing entanglement between the encoder and the decoder. In this paper, we propose a 6-qutrit approximate QECC (AQECC) of CSS structure which can simultaneously correct phase errors in upto six qutrits, and one bit error in only four of the six qutrits, without sharing prior entanglement. Our AQECC corrects a single error with probability 0.75 for symmetric error model, and probability 0.9988 for asymmetric error model. It also maintains CSS structure without sharing prior entanglement. Furthermore, the quantum cost of a circuit for this AQECC is 55.72% less than that for the 9-qutrit exact QECC. Overall, low qutrit count, low cost circuit realization with low depth and the ability to correct multiple phase errors make our proposed AQECC a suitable candidate for real life quantum channels.

IEEE International Symposium on Multiple-Valued Logic (ISMVL)•Nov 1, 2020•6 citations
Multi-valued quantum systems can store more information than binary ones for a given number of quantum states. For reliable operation of multi-valued quantum systems, error correction is mandated. In this paper, we propose a 5-qutrit quantum error-correcting code and provide its stabilizer formulation. Since 5 qutrits are necessary to correct a single error, our proposed code is optimal in the number of qutrits. We prove that the error model considered in this paper spans the entire (3×3)(3 \times 3)(3×3) operator space. Therefore, our proposed code can correct any single error on the codeword. This code outperforms the previous 9-qutrit code in (i) the number of qutrits required for encoding, (ii) our code can correct any arbitrary (3×3)(3 \times 3)(3×3) error, (ii) our code can readily correct bit errors in a single step as opposed to the two-step correction used previously, and (iii) phase error correction does not require correcting individual subspaces.
Ternary quantum systems are being studied because they provide more computational state space per unit of information, known as qutrit. A qutrit has three basis states, thus a qubit may be considered as a special case of a qutrit where the coefficient of one of the basis states is zero. Hence both $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional and $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors can occur on qutrits. In this paper, we (i) explore the possible $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional as well as $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors in qutrits and show that any pairwise bit swap error can be expressed as a linear combination of shift errors and phase errors, (ii) propose a special type of error called a quantum superposition error and show its equivalence to arbitrary rotation, (iii) formulate a nine-qutrit code which can correct a single error in a qutrit, and (iv) provide its stabilizer and circuit realization.
Quantum error-correcting code for higher dimensional systems can, in general, be directly constructed from the codes for qubit systems. What remains unknown is whether there exist efficient code design techniques for higher dimensional systems. In this paper, we propose a 7-qutrit error-correcting code for the ternary quantum system and show that this design formulation has no equivalence in qubit systems. This code is optimum in the number of qutrits required to correct a single error while maintaining the CSS structure. This degenerate CSS code can (i) correct up to seven simultaneous phase errors and a single bit error, (ii) correct two simultaneous bit errors on pre-defined pairs of qutrits on eighteen out of twenty-one possible pairs, and (iii) in terms of the cost of implementation, the depth of the circuit of this code is only two more than that of the ternary Steane code. Our proposed code shows that it is possible to design better codes explicitly for ternary quantum systems instead of simply carrying over codes from binary quantum systems.
This paper proposes simple quantum [[9],[1],[3]] error correction codes for quantum systems beyond qubits, focusing on quantum ternary digits (qutrits) that represent the simplest higher-dimensional systems for future quantum technologies. Although qutrit-based systems offer larger capacity information over 2-dimensional systems, the challenge of identifying commuting stabilizers for constructing error correction codes becomes more difficult as the dimensionality increases or when working with a large number of qutrits. We design qutrit codes based on the generalized Pauli group and investigate the underlying structure, including shift, phase, and rotation operators. We successfully create suitable stabilizer codes that detect and correct qutrit errors. Furthermore, we (i) identify operators in qutrit, which are selected from various matrix interpretations for higher-dimensional systems, (ii) propose stabilizers for 9-qutrit error correction codes, and (iii) design a new 9-qutrit quantum circuit for communications. Our study is expected to contribute a foundational basis for possible future theoretical and experimental advancements in quantum computing, communications, and code-based cryptography in higher dimensional systems.
We investigate a class of quantum error-correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum errors. In this framework, we represent the algebraic conditions for error correction in terms of edge avoidance between graphs providing a visual representation of the interplay between errors and error-correcting codes. Most importantly, this framework supports the development of quantum codes that correct against a predetermined set of errors, in contrast to current methods. A heuristic algorithm is presented, providing steps to develop codes that correct against an arbitrary noisy channel. We benchmark the correction capability of reflexive stabilizer codes for the case of single-qubit errors by comparison to existing stabilizer codes that are widely used. In addition, we present two instances of optimal encodings: an optimal encoding for fully correlated noise, which achieves a higher encoding rate than previously known, and a minimal encoding for single-qudit errors on a four-state system..
Large-scale quantum computers mandate error correction and fault tolerance. Due to constraints on the number of qubits, fault tolerance is difficult to achieve in near-term quantum systems. Therefore, error correction should require minimal resources. Gates in the near-term devices are also noisy. Quantum error correction code blocks built with these noisy gates can inject further error in the circuit. The goals for error correction in near-term systems are as follows: (i) using a small number of qubits for encoding, and (ii) keeping cost of circuits for encoding and decoding low. In this paper, we propose two techniques to achieve these mutually orthogonal goals. For a binary quantum system we propose an error estimation method that can aid in reducing the number of error correcting blocks via sparse scheduling. For ternary quantum systems, we propose an approximate code that can correct errors with high probability while significantly reducing the circuit cost. These techniques are expected to be helpful for error mitigation in near-term systems in the absence of fault tolerance.
Quantum error correcting code (QECC) with encoding and decoding circuits having low gate count, is an important criterion for realizing quantum computing systems. CSS QECCs are known to have simple circuits. Shaw et al. showed that it is not possible to have a 6-qubit CSS type QECC without sharing entanglement between the encoder and the decoder. In this paper, we propose a 6-qutrit approximate QECC (AQECC) of CSS structure which can simultaneously correct phase errors in upto six qutrits, and one bit error in only four of the six qutrits, without sharing prior entanglement. Our AQECC corrects a single error with probability 0.75 for symmetric error model, and probability 0.9988 for asymmetric error model. It also maintains CSS structure without sharing prior entanglement. Furthermore, the quantum cost of a circuit for this AQECC is 55.72% less than that for the 9-qutrit exact QECC. Overall, low qutrit count, low cost circuit realization with low depth and the ability to correct multiple phase errors make our proposed AQECC a suitable candidate for real life quantum channels.