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Asia-Pacific Conference on Communications (APCC)

Identification Of Quantum Encoder Matrix From A Collection Of Pauli Errors

Gilles Burel, Hugo Pillin, El-Houssaïn Baghious, Paul Baird, Roland Gautier

Published Nov 1, 2019• 2 citations
Publisher

Abstract

Quantum information processing is a rapidly evolving field, due to promising applications in communications, cryptography, and computing. In this framework, there is a need to protect quantum information against errors, using quantum error-correcting codes. Efficient quantum codes, based on the stabilizer formalism (that exploits elements of the Pauli group), have been proposed. The stabilizer formalism allows one to simulate quantum codes and quantum errors using operations inside the Pauli group only, leading to huge gains in simulation time. However, to deeply study and simulate unconventional quantum errors and devices, there is a need to know the true quantum operator (represented by a unitary matrix). In this paper, we propose an algorithm, based on linear algebra, to identify the quantum encoder matrix from a collection of Pauli errors. The approach is two-steps. First, from a collection a Pauli errors whose matrix representation is diagonal, a search of common eigenvectors identifies the encoder matrix up to phase indeterminates. Second, additional Pauli errors with nondiagonal matrix representations are used to eliminate the remaining indeterminations. Simulation results are also provided to illustrate and validate the approach.

DOI
10.1109/apcc47188.2019.9026518

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Quantum information processing is a rapidly evolving field, due to promising applications in communications, cryptography, and computing. In this framework, there is a need to protect quantum information against errors, using quantum error-correcting codes. Efficient quantum codes, based on the stabilizer formalism (that exploits elements of the Pauli group), have been proposed. The stabilizer formalism allows one to simulate quantum codes and quantum errors using operations inside the Pauli group only, leading to huge gains in simulation time. However, to deeply study and simulate unconventional quantum errors and devices, there is a need to know the true quantum operator (represented by a unitary matrix). In this paper, we propose an algorithm, based on linear algebra, to identify the quantum encoder matrix from a collection of Pauli errors. The approach is two-steps. First, from a collection a Pauli errors whose matrix representation is diagonal, a search of common eigenvectors identifies the encoder matrix up to phase indeterminates. Second, additional Pauli errors with nondiagonal matrix representations are used to eliminate the remaining indeterminations. Simulation results are also provided to illustrate and validate the approach.
Mapping the error syndromes to error operators is the core of quantum decoding network and the key step to realize quantum error correction. The definitions of the bit flip error syndrome matrix and the phase flip error syndrome matrix are presented, and then the error syndromes of Pauli errors are expressed in terms of the columns of the bit flip error syndrome matrix and the phase flip error syndrome matrix. It is also shown that the error syndrome matrix of a stabilizer code is determined by its check matrix, which is similar to the relationship between the classical error and the parity check matrix of classical codes. So, the techniques of error detection and error correction for classical linear codes can be applied to quantum stabilizer codes after some modifications. The error correction circuits are constructed based on the relationship between the error operator and error syndrom. The decoding circuit is constructed by reversing the encoding circuit because the encoding operators are unitary.
We study mathematical, physical and computational aspects of the stabilizer formalism arising in quantum information and quantum computation. The measurement process of Pauli observables with its algorithm is given. It is shown that to detect genuine entanglement we need a full set of stabilizer generators and the stabilizer witness is coarser than the GHZ (Greenberger–Horne–Zeilinger) witness. We discuss stabilizer codes and construct a stabilizer code from a given linear code. We also discuss quantum error correction, error recovery criteria and syndrome extraction. The symplectic structure of the stabilizer formalism is established and it is shown that any stabilizer code is unitarily equivalent to a trivial code. The structure of graph codes as stabilizer codes is identified by obtaining the respective stabilizer generators. The distance of embeddable stabilizer codes in lattices is obtained. We discuss the Knill–Gottesman theorem, tableau representation and frame representation. The runtime of simulating stabilizer gates is obtained by applying stabilizer matrices. Furthermore, an algorithm for updating global phases is given. Resolution of quantum channels into stabilizer channels is shown. We discuss capacity achieving codes to obtain the capacity of the quantum erasure channel. Finally, we discuss the shadow tomography problem and an algorithm for constructing classical shadow is given.
One of the fundamental barriers to continued advances in quantum computing, quantum cryptography, and quantum communication is the basic problem of controlling and maintaining features of quantum systems as, they evolve in time. The common ground for these investigations is "quantum error correction." One of the most important approaches to quantum error correction is the "stabilizer formalism." In this thesis we give the basics of quantum computing, stabilizer formalism and quantum error correction, and eventually show that the stabilizer formalism can be generalized to subsystems through operator algebras. Key aspects of the analysis include Pauli groups, completely positive maps and quantum channels, and operators on Hilbert space.
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.
Jacket matrices motivated by the center weight Hadamard matrices have played an important role in signal processing, communications, image compression, cryptography, etc. In this paper, we suggest a design approach for the Pauli block jacket matrix achieved by substituting some Pauli matrices for all elements of common matrices. Since, the well-known Pauli matrices have been widely utilized for quantum information processing, the large-order Pauli block jacket matrix that contains commutative row operations are investigated in detail. After that some special Abelian groups are elegantly generated from any independent rows of the yielded Pauli block jacket matrix. Finally, we show how the Pauli block jacket matrix can simplify the coding theory of quantum error-correction. The quantum codes we provide do not require the dual-containing constraint necessary for the standard quantum error-correction codes, thus allowing us to construct quantum codes of the large codeword length. The proposed codes can be constructed structurally by using the stabilizer formalism of Abelian groups whose generators are selected from the row operations of the Pauli block jacket matrix, and hence have advantages of being fast constructed with the asymptotically good behaviors.
Quantum error correction codes were introduced as a means to protect quantum information from decoherance and operational errors. Based on their approach to error control, error correcting codes can be divided into two different classes: block codes and convolutional codes. There has been significant development towards finding quantum block codes, since they were first discovered in 1995. In contrast, quantum convolutional codes remained mainly uninvestigated. In this thesis, we develop the stabilizer formalism for quantum convolutional codes. We define distance properties of these codes and give a general method for constructing encoding circuits, given a set of generators of the stabilizer of a quantum convolutional stabilizer code, is shown. The resulting encoding circuit enables online encoding of the qubits, i.e., the encoder does not have to wait for the input transmission to end before starting the encoding process. We develop the quantum analogue of the Viterbi algorithm. The quantum Viterbi algorithm (QVA) is a maximum likehood error estimation algorithm, the complexity of which grows linearly with the number of encoded qubits. A variation of the quantum Viterbi algorithm, the Windowed QVA, is also discussed. Using Windowed QVA, we can estimate the most likely error without waiting for the entire received sequence.