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IEEE Asia-Pacific Conference on Applied Electromagnetics (APACE)

Qutrit Error Correction Codes for Quantum Communications and Computations

Muhammad Imansyah Basudewa, Khoirul Anwar, Melinda Br Ginting

Published Dec 21, 2024• 2 citations
Publisher

Abstract

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.

DOI
10.1109/apace62360.2024.10877324

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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.
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.
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.
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.
In this paper, we propose quantum [[10, 10, 2]] stabilizer codes constructed from the existing quantum [[5, 5, 1]] Hamming codes. The proposed quantum [[10, 10, 2]] codes satisfy the symplectic inner product (SIP) to ensure stabilizer generators commute each other for successful detections and corrections. A comprehensive end-to-end construction from parity-check matrix (PCM) of classical codes is provided to ensure quantum error correction codes (QECC) with higher information. We perform a series of computer simulations to evaluate proposed quantum [[10, 10, 2]] codes in terms of quantum word error rate (QWER) performances under the depolarizing channels. The results confirm that the proposed quantum [[10, 10, 2]] codes can successfully detect and correct all single quantum bit (qubit) errors, which is validated by the curve of QWER performances that coincide with the theoretical QWER performances. The results are expected to contribute to the development of QECC in quantum computing and communication systems.
Quantum technology offers transformative potential for future applications, including secure communications and high-speed computing. However, quantum systems are highly susceptible to decoherence caused by environment interactions, making error correction essential for reliable operation. This paper proposes a new 7-qubit quantum error correction code constructed from classical Hamming codes through a distinct derivation of the parity matrix. Unlike the well-known Steane code, our scheme employs a different parity matrix within the stabilizer framework, resulting in a novel code structure. We detail the step-by-step construction of codewords, syndrome extraction using stabilizer measurements, and the corresponding quantum circuits. Performance evaluation is conducted through simulations under depolarizing channels across error probabilities from to 0, using Quantum Word Error Rate (QWER) as the metric. Simulation results confirm that the proposed stabilizers can uniquely identify all single-qubit error patterns of Pauli X, Y, and Z. Moreover, the scheme achieves a significant error reduction of approximately 28 times (about 96%) compared to uncoded transmission. These findings demonstrate that deriving quantum codes from modified classical matrices offers an effective strategy for enhancing the reliability of future quantum communication systems.
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⟩|0\rangle∣0⟩and∣1⟩|1\rangle∣1⟩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⟩|0\rangle∣0⟩and∣1⟩|1\rangle∣1⟩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[[9,5,3]]_{3}[[9,5,3]]3​,[[5,1,3]]3[[5,1,3]]_{3}[[5,1,3]]3​, and[[7,1,3]]3[[7,1,3]]_{3}[[7,1,3]]3​codes, and a 9–21% reduction for the ququint (d = 5)[[10,6,3]]5[[10,6,3]]_{5}[[10,6,3]]5​code when compared to prior work. We also achieved circuit depth reductions up to 42%.