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arXiv:1807.01863 · Physical Review A

Quantum error-correcting code for ternary logic

Ritajit Majumdar, Saikat Basu, Shibashis Ghosh, Susmita Sur‐Kolay

Published May 2, 2018• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.97.052302

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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×\else\texttimes\fi2)(2\ifmmode\times\else\texttimes\fi{}2)(2\ifmmode×\else\texttimes\fi2)-dimensional and (3\ifmmode×\else\texttimes\fi3)(3\ifmmode\times\else\texttimes\fi{}3)(3\ifmmode×\else\texttimes\fi3)-dimensional Pauli errors can occur on qutrits. In this paper, we (i) explore the possible (2\ifmmode×\else\texttimes\fi2)(2\ifmmode\times\else\texttimes\fi{}2)(2\ifmmode×\else\texttimes\fi2)-dimensional as well as (3\ifmmode×\else\texttimes\fi3)(3\ifmmode\times\else\texttimes\fi{}3)(3\ifmmode×\else\texttimes\fi3)-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.
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.
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.
Quantum states have high affinity for errors and hence error correction is of utmost importance to realise a quantum computer. Laflamme showed that 5 qubits are necessary to correct a single error on a qubit. In a Pauli error model, four different types of errors can occur on a qubit. Maximally entangled states are orthogonal to each other and hence can be uniquely distinguished by a measurement in the Bell basis. Thus a measurement in Bell basis and a unitary transformation is sufficient to correct error in Bell states. However, such a measurement is not possible for non-maximally entangled states. In this work we show that the 16 possible errors for a non-maximally entangled two qubit system map to only 8 distinct error states. Hence, it is possible to correct the error without perfect knowledge of the type of error. Furthermore, we show that the possible errors can be grouped in such a way that all 4 errors can occur on one qubit, whereas only bit flip error can occur on the second qubit. As a consequence, instead of 10, only 8 qubits are sufficient to correct a single error. We propose an 8-qubit error correcting code to correct a single error in a non-maximally entangled state. We further argue that for an nnn-qubit non-maximally entangled state of the form a∣0>n+b∣1>na|0>^{n} + b|1>^{n}a∣0>n+b∣1>n, it is always possible to correct a single error with fewer than 5n5n5n qubits, in fact only 3n+23n+23n+2 qubits suffice.
The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to nnn qubits with T-count 3n−33n-33n−3. Our main result is the construction of a novel qutrit [ ⁣[6,2,2] ⁣]\mathopen{[\![} 6,2,2 \mathclose{]\!]}[[6,2,2]] quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decomposition$\unicode{x2014}$composed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverse$\unicode{x2014}$of a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a [ ⁣[48,2,4] ⁣]\mathopen{[\![} 48,2,4 \mathclose{]\!]}[[48,2,4]] code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.
One of the main motivations for using ternary logic systems is the amount of information per circuit line is higher as compared to the corresponding binary logic representation, thereby leading to more compact circuit realizations. This is particularly attractive for quantum computing as qutrits are expensive resources and minimizing their number is one of the main objectives during synthesis. Therefore, ternary reversible logic synthesis has drawn significant attention among researchers. It deals with fundamental unit of information called qutrits that can exist in one of the three states |0>, |1> and |2>. Hence, the aim of this paper is to bridge the knowledge gap for the beginners in this domain than searching the entire space. Therefore, the present work discusses the basic concepts of ternary reversible logic and ternary reversible gates. The detailed discussion of the various ternary reversible logic synthesis will enable the beginners in this domain to understand the ternary reversible logic in a better way.