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Sequentially Encodable Codeword Stabilized Codes

Sowrabh Sudevan, Sourin Das, Thamadathil Aswanth, Navin Kashyap

Published Jul 7, 2024• 1 citation
Publisher

Abstract

$n$-qubit,$m$-uniform states are$[[n,\ 0,\ m+1]]$nonde-generate quantum error correcting codes (QECC). The codeword stabilized (CWS) codes use [$n, k, d\geq m+1$] classical codes and$m$-uniform graph states to construct$[[n,\ k,\ m+1]]$QECCs. We provide a set of conditions on the classical codes to obtain CWS codes, which can be encoded in a proposed measurement-based approach. We arrive at an explicit construction of a CWS code when the starting$m$-uniform graph state is a cluster state of qubits on a one-dimensional lattice. The proposed encoding protocol admits sequential encoding of logical qubits, making it suitable for quantum memory applications.

DOI
10.1109/isit-w61686.2024.10591769

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•Apr 1, 2015•0 citations
nnn-qubit,mmm-uniform states are[[n, 0, m+1]][[n,\ 0,\ m+1]][[n, 0, m+1]]nonde-generate quantum error correcting codes (QECC). The codeword stabilized (CWS) codes use [n,k,d≥m+1n, k, d\geq m+1n,k,d≥m+1] classical codes andmmm-uniform graph states to construct[[n, k, m+1]][[n,\ k,\ m+1]][[n, k, m+1]]QECCs. We provide a set of conditions on the classical codes to obtain CWS codes, which can be encoded in a proposed measurement-based approach. We arrive at an explicit construction of a CWS code when the startingmmm-uniform graph state is a cluster state of qubits on a one-dimensional lattice. The proposed encoding protocol admits sequential encoding of logical qubits, making it suitable for quantum memory applications.
An m-uniform quantum state on n qubits is an entangled state in which every m-qubit subsystem is maximally mixed. Starting with an m-uniform state realized as the graph state associated with an m-regular graph, and a classical n,k,d≥m+1 binary linear code with certain additional properties, we show that pure [n,k,m+1]]2 quantum error-correcting codes (QECCs) can be constructed within the codeword stabilized (CWS) code framework. As illustrations, we construct pure [[22r-1,22r-2r-3,3]]2 and [[(24r-1)2,(24r-1)2-32r-7,5]]2 QECCs. We also give measurement-based protocols for encoding into code states and for recovery of logical qubits from code states.
In this paper, we introduce a unified framework to construct entanglement-assisted quantum error-correcting codes, including additive and nonadditive codes, based on the codeword stabilized framework on subsystems. The codeword stabilized (CWS) framework is a scheme to construct quantum error-correcting codes (QECCs) including both additive and nonadditive codes, and gives a method to construct a QECC from a classical error-correcting code in standard form. Entangled pairs of qubits (ebits) can be used to improve capacity of quantum error correction. In addition, it gives a method to overcome the dual-containing constraint. Operator quantum error correction (OQEC) gives a general framework to construct quantum error-correcting codes. We construct OQEC codes with ebits based on the CWS framework. This new scheme, entanglement-assisted operator codeword stabilized (EAOCWS) quantum codes, is the most general framework we know of to construct both additive and nonadditive codes from classical error-correcting codes. We describe the formalism of our scheme, demonstrate the construction with examples, and give several EAOCWS 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)].
There is a connection between classical codes, highly entangled pure states (calledkkk-uniform or absolutely maximally entangled (AME) states), and quantum error correcting codes (QECCs). This leads to a systematic method to construct stabilizer QECCs by starting from akkk-uniform state or the corresponding classical code and tracing out one party at each step. We provide explicit constructions for codewords, encoding procedure and stabilizer formalism of the QECCs by describing the changes that partial traces cause on the corresponding generator matrix of the classical codes. We then modify the method to produce another set of stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. This construction produces quantum codes starting from an AME state without tracing out any party. Therefore, quantum stabilizer codes with larger codespace can be constructed.
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.
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.