Encoding of Quantum Stabilizer Codes Over Qudits with $d=p^{k}$

Nadkarni, Priya J.; Garani, Shayan Srinivasa

doi:10.1109/glocomw.2018.8644150

Cited by 5 publications

(6 citation statements)

References 4 publications

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“…Mathematically, each basis state |β p k of the p kdimensional quantum system is equal to a tensor product of k basis states of p-dimensional systems based on its F p -ary representation, i.e., |β p k = |b k−1 p ⊗ |b k−2 p ⊗ • • • ⊗ |b 0 p [14]. We define each of these components as a subqudit of the qudit.…”

Section: A Quantum States and Operators Over Qudits 1) Quantum Statementioning

confidence: 99%

“…Similar to the basis states, a qudit basis operator from G p k is equal to a tensor product of k subqudit basis operators from G p as follows [14]:…”

Section: A Quantum States and Operators Over Qudits 1) Quantum Statementioning

confidence: 99%

“…Our procedure to obtain the encoding operation is given in Algorithm 1 [14]. 1) Expand the elements of H Q in F p k in terms of F p as in equation ( 14), to obtain the matrix H…”

Section: Encoding Algorithmmentioning

confidence: 99%

“…To practically implement small and moderate length 1 coded quantum communication systems, it is necessary to develop an encoding circuit architecture to encode the quantum information, motivating the work in this paper. In [14], we provided the basic idea of the encoding procedure for entanglement-unassisted stabilizer codes over qudits.…”

Section: Introductionmentioning

confidence: 99%

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Encoding of Nonbinary Entanglement-Unassisted and Assisted Stabilizer Codes

Nadkarni

Garani

2021

IEEE Trans. Quantum Eng.

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Quantum coding schemes over qudits using pre-shared entanglement between the encoder and decoder can provide better error correction capability than without it. In this paper, we develop procedures for constructing encoding operators for entanglement-unassisted and entanglement-assisted qudit stabilizer codes over F p k , with p prime and k ≥ 1 from first principles, generalizing prior works on qubit based codes and codes that work over F p . We also provide quantum encoding architectures based on the proposed encoding procedures using one and two qudit gates, useful towards realizing coded quantum computing and communication systems using qudits.INDEX TERMS Quantum error correction, Entanglement-assisted codes, Encoding architecture, Qudit stabilizer codes, Non-binary codes. 1 The encoding circuit complexity linearly increases with codelength. To realize the practical implementation of large length codes, we need to construct all the stabilizers or their extensions from a carefully chosen group of quantum operators, making it a computationally challenging problem.2 Preliminary version on the encoding procedure for entanglementunassisted stabilizer codes over qudits was presented at IEEE Globecom 2018, Abu Dhabi [14].

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Section: A Quantum States and Operators Over Qudits 1) Quantum Statementioning

confidence: 99%

“…Similar to the basis states, a qudit basis operator from G p k is equal to a tensor product of k subqudit basis operators from G p as follows [14]:…”

Section: A Quantum States and Operators Over Qudits 1) Quantum Statementioning

confidence: 99%

“…Our procedure to obtain the encoding operation is given in Algorithm 1 [14]. 1) Expand the elements of H Q in F p k in terms of F p as in equation ( 14), to obtain the matrix H…”

Section: Encoding Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Encoding of Nonbinary Entanglement-Unassisted and Assisted Stabilizer Codes

Nadkarni

Garani

2021

IEEE Trans. Quantum Eng.

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“…For β ∈ F p m , the qudit state and the operators acting on it can be represented as a tensor product of m states and operators for p-dimensional qudits, respectively [19]. Let α be the primitive element of…”

Section: B Stabilizer Codes Over Quditsmentioning

confidence: 99%

$\mathbb {F}_p$-Linear and $\mathbb {F}_{p^m}$-Linear Qudit Codes From Dual-Containing Classical Codes

Nadkarni

Garani

2021

IEEE Trans. Quantum Eng.

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Quantum code construction from two classical codes D 1 [n, k 1 , d 1 ] and D 2 [n, k 2 , d 2 ] over the field F p m (p is prime and m is an integer) satisfying the dual containing criteria D ⊥ 1 ⊂ D 2 using the Calderbank-Shor-Steane (CSS) framework is well-studied. We show that the generalization of the CSS framework for qubits to qudits yields two different classes of codes, namely, the F p -linear CSS codes and the well-known F p m -linear CSS codes based on the check matrix-based definition and the coset-based definition of CSS codes over qubits Our contribution to this work are three-folds (a) We study the properties of the F p -linear and F p m -linear CSS codes and demonstrate the trade-off for designing codes with higher rates or better error detection and correction capability, useful for quantum systems. (b) For F p m -linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances d x and d z are equal to d 2 and d 1 , respectively, if and only if the code is non-degenerate. (c) We propose two classes of quantum codes obtained from the codes D 1 and D 2 , where one code is an F p l -linear code (l divides m) and the other code is obtained from a particular subgroup of the stabilizer group of the F p m -linear CSS code. Within each class of codes, we demonstrate the trade-off between higher rates and better error detection and correction capability. INDEX TERMSQuantum error correction, F p m -linear CSS codes, F p m -linear CSS codes, Dual containing codes, Qudit Codes, CSS-like codes.

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