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Journal of Combinatorial Designs

3-Spontaneous Emission Error Designs from PSL(2, q)or PGL(2, q)

Junling Zhou, Yanxun Chang

Published May 18, 2015• 3 citations
Publisher

Abstract

Quantum jump codes are quantum codes that correct errors caused by quantum jumps. A spontaneous emission error design (SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. In this paper, we study the existence of 3-SEEDs from PSL(2, q) or PGL(2, q). By doing this, a large number of 3- SEEDs are derived for prime powers q and all .

DOI
10.1002/jcd.21429

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Quantum jump codes are quantum codes that correct errors caused by quantum jumps. A spontaneous emission error design (SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. In this paper, we study the existence of 3-SEEDs from PSL(2, q) or PGL(2, q). By doing this, a large number of 3- SEEDs are derived for prime powers q and all .
Quantum jump codes were introduced by Alber et al. (2001). Quantum jump codes have a close connection with combinatorial designs called t-SEED (t-spontaneaus emission error design). In this paper, we give a brief survey of quantum jump codes together with some new results. Firstly, fundamental properties of a t-error correcting quantum jump code are described. Secondly, a few examples of jump codes are given and an upper bound on the dimension of a jump code with a fixed length and given error correcting ability is derived. A relation between a t-SEED and a jump code is discussed and various constructions of t-SEEDs are given.
The recently introduced detected-jump-correcting quantum codes are capable of stabilizing qubit systems against spontaneous decay processes arising from couplings to statistically independent reservoirs. These embedded quantum codes exploit classical information about which qubit has emitted spontaneously and correspond to an active error-correcting code embedded in a passive error-correcting code. The construction of a family of one-detected-jump-error-correcting quantum codes is shown and the optimal redundancy, encoding, and recovery as well as general properties of detected-jump-error-correcting quantum codes are discussed. By the use of design theory, multiple-jump-error-correcting quantum codes can be constructed. The performance of one-jump-error-correcting quantum codes under nonideal conditions is studied numerically by simulating a quantum memory and Grover's algorithm.
A new class of embedded error correcting quantum codes is discussed. These detected-jump correcting quantum codes are capable of stabilizing distinguishable qubits against spontaneous decay. Due to their low redundancy and due to the small number of control measurements and recovery operations required these new error-correcting quantum codes offer interesting perspectives for quantum computation. In this contribution main ideas underlying these embedded error correcting quantum codes and their links to fundamental concepts of combinatorial design theory are discussed. In addition, possibilities are explored for implementing universal quantum gates within these code spaces.
A universal set of quantum gates is constructed for the recently developed jump-error correcting quantum codes. These quantum codes are capable of correcting errors arising from the spontaneous decay of distinguishable qubits into statistically independent reservoirs. The proposed universal quantum gates are constructed with the help of Heisenberg- and Ising-type Hamiltonians acting on these physical qubits. This way it is guaranteed that the relevant error correcting code space is not left at any time even during the application of one of these quantum gates. The proposed entanglement gate is particularly well suited for scalable quantum processing units whose elementary registers are based on four-qubit systems.
A new class of error-correcting quantum codes is introduced capable of stabilizing qubits against spontaneous decay arising from couplings to statistically independent reservoirs. These quantum codes are based on the idea of using an embedded quantum code and exploiting the classical information available about which qubit has been affected by the environment. They are immediately relevant for quantum computation and information processing using arrays of trapped ions or nuclear spins. Interesting relations between these quantum codes and basic notions of design theory are established.