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arXiv:1206.1566

Non-Pauli Observables for CWS Codes

Douglas Frederico Guimarães Santiago, Renato Portugal, Nolmar Melo

Published Jun 7, 2012• 0 citations
arXivHTMLPDF

Abstract

It is known that nonadditive quantum codes are more optimal for error correction when compared to stabilizer codes. The class of codeword stabilized codes (CWS) provides tools to obtain new nonadditive quantum codes by reducing the problem to finding nonlinear classical codes. In this work, we establish some results on the kind of non-Pauli operators that can be used as decoding observables for CWS codes and describe a procedure to obtain these observables.

DOI
10.48550/arxiv.1206.1566

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It is known that nonadditive quantum codes are more optimal for error correction when compared to stabilizer codes. The class of codeword stabilized codes (CWS) provides tools to obtain new nonadditive quantum codes by reducing the problem to finding nonlinear classical codes. In this work, we establish some results on the kind of non-Pauli operators that can be used as decoding observables for CWS codes and describe a procedure to obtain these observables.
Codeword stabilized (CWS) codes form a general class of quantum codes that includes stabilizer codes and many families of nonadditive codes with good parameters. Similar to classical nonlinear codes, a CWS code can be decoded by screening all possible errors. For an n-qubit quantum code correcting up to t errors, this brute-force approach consecutively tests different errors of weight t or less, and employs a separate n-qubit measurement in each test. To simplify decoding, we propose an algorithm that employs a single measurement to process all errors located on a given cluster of t qubits. Compared to an exhaustive error screening, this reduces the total number of measurements required for error correction about 3ttimes.
The codeword-stabilized (CWS) quantum code formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes [see A. Cross et al., e-print arXiv:0708.1021], but only for binary states. Here we generalize the CWS framework to the nonbinary case (of both prime and nonprime dimensions) and map the search for nonbinary quantum codes to a corresponding search problem for classical nonbinary codes with specific error patterns. We show that while the additivity properties of nonbinary CWS codes are similar to the binary case, the structural properties of the nonbinary codes differ substantially from the binary case, even for prime dimensions. In particular, we identify specific structure patterns of stabilizer groups, based on which efficient constructions might be possible for codes that encode more dimensions than any stabilizer codes of the same length and distance; similar methods cannot be applied in the binary case. Understanding of these structural properties can help prune the search space and facilitate the identification of good nonbinary CWS codes.
The codeword stabilized (CWS) quantum code formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes [IEEE Trans. Inf. Theory 55, 433 (2009)]. This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state. Finding such a classical code can be very difficult. Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we provide three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find that the re does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS-search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank [IEEE Trans. Inf. Theory 54, 1700 (2008)].
We discuss a method to adapt the codeword stabilized (CWS) quantum code framework to the problem of finding asymmetric quantum codes. We focus on the corresponding Pauli error models for amplitude damping noise and phase damping noise. In particular, we look at codes for Pauli error models that correct one or two amplitude damping errors. Applying local Clifford operations on graph states, we are able to exhaustively search for all possible codes up to length 999. With a similar method, we also look at codes for the Pauli error model that detect a single amplitude error and detect multiple phase damping errors. Many new codes with good parameters are found, including nonadditive codes and degenerate codes.
Quantum decoherence and errors represent some of the major challenges arising in quantum computations. Quantum error correcting codes protect quantum states against these errors and make quantum computing more reliable. One of the main problems of quantum error correction is the design of feasible decoding algorithms that can simplify error-correction for general quantum codes. The dissertation addresses decoding of general Codeword Stabilized (CWS) codes. This class of quantum codes also includes some other important classes such as additive Stabilizer codes and non-additive Union Stabilizer (USt) codes.We first design a generic error-correcting algorithm for CWS codes and analyze the number of decoding measurements and quantum gates. This algorithm performs exhaustive screening of different error patterns, similar to decoding of classical non-linear codes. For an n-qubit quantum code correcting up to t erroneous qubits, this brute-force approach consecutively tests all correctable error patterns and employs a separate n-qubit measurement in each test.The main result is a new error grouping technique that enables simultaneous testing of large groups of errors in a single measurement. To achieve this reduction, we first proceed with a new error-correction algorithm for the USt codes. Secondly, we design an algorithm that converts generic non-linear CWS codes into the simpler quasi-linear USt codes. Each decoding measurement can then either locate the actual error in a given group of errors or entirely eliminate this group. This technique yields a much simpler algorithm that exponentially reduces the number of measurements about 3^t times in any t-error-correcting CWS code.
In this study we present a method that adapts the codeword stabilized (CWS) quantum code framework to the problem of finding asymmetric quantum codes. Making use of the the corresponding Pauli error models for amplitude and phase-damping models, we focus on codes that correct one or two amplitude-damping errors. As a result, we are able to exhaustively search for all possible codes up to length 9 by applying local Clifford operations on graph states. With a similar method, we also look at codes for the Pauli error models that detect a single amplitude error and detect multiple phase damping errors. Many new codes with good parameters are found, including non-additive codes and degenerate codes.