In this paper, we present the pm-ary entanglement-assisted(EA) stabilizer formalism, where p is a prime and m is a positive integer. Given an arbitrary non-abelian "stabilizer", the problem of code construct...
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In this paper, we present the pm-ary entanglement-assisted(EA) stabilizer formalism, where p is a prime and m is a positive integer. Given an arbitrary non-abelian "stabilizer", the problem of code construction and encoding is settled perfectly in the case of m = 1. The optimal number of required maximally entangled pairs is discussed and an algorithm to determine the encoding and decoding circuits is proposed. We also generalize several bounds on p-ary EA stabilizercodes, such as the BCH bound, the G-V bound and the linear programming bound. However, the issue becomes tricky when it comes to m > 1, in which case, the former construction method applies only when the non-commuting "stabilizer" satisfies a sophisticated limitation.
Using pre-shared entangled states between the encoder and the decoder, we provide a previously unreported coding-theoretic framework for constructing entanglement-assisted stabilizercodes over qudits of dimension p(k...
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Using pre-shared entangled states between the encoder and the decoder, we provide a previously unreported coding-theoretic framework for constructing entanglement-assisted stabilizercodes over qudits of dimension p(k) from first principles, where p is prime and k is an element of Z(+). We introduce the concept of mathematically decomposing a qudit of dimension p(k) into k subqudits, each of dimension p. Our contributions toward the entanglement-assisted stabilizer coding framework over qudits are multi-fold as follows: (a) We study the properties of the code and derive an analytical expression for the minimum number of pre-shared entangled subqudits required to construct the code. (b) We provide a code construction procedure that involves obtaining the explicit form of the stabilizers of the code. (c) We show that the proposed entanglement-assisted qudit stabilizercodes are analogous to classical additive codes over F-pk. (d) We provide the quantum coding bounds, such as the quantum Hamming bound, the quantum Singleton bound, and the quantum Gilbert-Varshamov bound for non-degenerate entanglement-assisted stabilizercodes over qudits. (e) We finally demonstrate that the error correction capability of the code can be increased with entanglement assistance. The proposed framework is useful for realizing coded quantum computing and communication systems over p(k)-dimensional qudits.
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