Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes

Grassl, Markus; Rötteler, Martin; Beth, Thomas

doi:10.1142/s0129054103002011

Cited by 85 publications

(82 citation statements)

References 24 publications

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“…We selected code families that are easily accessible by elementary methods; the interested reader can find examples of more intricate algebro-geometric constructions in [6,25,26,56,69] and of binary quantum LDPC codes in [21,65,76]. We did not include constructive aspects of encoding and decoding circuits, since encoding circuits are discussed in [49] and little is known about the decoding of stabilizer codes. We did not include combinatorial aspects, but Kim pointed out that there is a forthcoming book by Glynn, Gulliver, Maks, and Gupta that explores the relation between binary stabilizer codes and finite geometry.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Nonbinary Stabilizer Codes Over Finite Fields

Ketkar

Klappenecker

Kumar

et al. 2006

IEEE Trans. Inform. Theory

567

560

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One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

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Section: Conclusion and Open Problemsmentioning

confidence: 99%

Nonbinary Stabilizer Codes Over Finite Fields

Ketkar

Klappenecker

Kumar

et al. 2006

IEEE Trans. Inform. Theory

567

560

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“…Details of the benchmarks and the latency values for various physical operations in the ion trap technology can be seen in Tables 5 and 6. The benchmarks used in [14] and [26] were chosen from [23] and those used in [14] and [27] were taken from [24]. Therefore, the results are compared with them in Table 7 and Table 8 for the two sets of benchmarks.…”

Section: -Experimental Resultsmentioning

confidence: 99%

A quantum physical design flow using ILP and graph drawing

Yazdani

Zamani

Sedighi

2013

Quantum Inf Process

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Implementing large-scale quantum circuits is one of the challenges of quantum computing. One of the central challenges of accurately modeling the architecture of these circuits is to schedule a quantum application and generate the layout while taking into account the cost of communications and classical resources as well as the maximum exploitable parallelism. In this paper, we present and evaluate a design flow for arbitrary quantum circuits in ion trap technology. Our design flow consists of two parts. First, a scheduler takes a description of a circuit and finds the best order for the execution of its quantum gates using integer linear programming (ILP) regarding the classical resources (qubits) and instruction dependencies. Then a layout generator receives the schedule produced by the scheduler and generates a layout for this circuit using a graph-drawing algorithm. Our experimental results show that the proposed flow decreases the average latency of quantum circuits by about 11% for a set of attempted benchmarks and by about 9% for another set of benchmarks compared with the best in literature.

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“…In [9] we have shown how to compute a quantum circuit consisting of Clifford gates only that transforms any stabilizer S given by the binary (n − k) × 2n matrix (X|Z) into the stabilizer of a trivial code given by (0|I0), where I is an identity matrix of size n − k. The corresponding trivial code corresponds to the mapping |φ → |0 . .…”

Section: Encoding Circuitsmentioning

confidence: 99%

“…Afterwards, the code has been identified as the span of a particular state and its image under five unitary transformations (see also [7]). The recently discovered codes ( (9,12,3)) and ((10, 24, 3)) (see [16], [17]) start with a socalled graph state which corresponds to a stabilizer state, i. e., a stabilizer code with parameters [[n, 0, d]] (see [8], [13]). A basis of the quantum code is obtained by this initial state together with its image under some tensor products of Pauli matrices and identity (Pauli operators).…”

Section: Introductionmentioning

confidence: 99%