Description
A code defined using a cluster state and a set of \(Z\)-type Pauli strings defined by a binary classical code.
The CWS construction takes in \( \mathcal{Q} = (\mathcal{G},\mathcal{C}) \), where \(\mathcal{G}\) is a graph, and where \(\mathcal{C}\) is an \((n,K,d)\) binary code. From the graph, we form the unique cluster state \( |\mathcal{G} \rangle \). From the binary code, we form Pauli \(Z\)-type operators \( W_i = Z^{c_{i,1}} \otimes \cdots \otimes Z^{c_{i,n}} \), where \(c_{i,j} \) is the \(j\)-th coordinate of the \(i\)-th classical codeword. The CWS codewords are then \( | i \rangle = W_i | \mathcal{G} \rangle \).
The above definition corresponds to the standard form of CWS codes. Since any stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [3][4; Appx. A], any code whose underlying state is a non-cluster stabilizer state is similarly equivalent to a CWS code.
The term CWS was coined in Ref. [2], and their approach is equivalent to another approach [1] based on Boolean functions (see Ref. [5]). In an alternative convention (not used here), CWS codes are defined from an underlying stabilizer state that is not a necessarily a cluster state.
Protection
Code distance \(\mathcal{Q} = ( \mathcal{G},\mathcal{C}) \) is upper bounded by the distance of the classical code \(\mathcal{C} \). A CWS code is degenerate if and only if it is impure [6]. The pure distance is upper bounded by \(\delta + 1\), where \(\delta\) is the minimum degree of \(\mathcal{G}\) [7,8]. Some bounds on the distance are provided in Ref. [9].Encoding
If \( \mathcal{C} \) has an efficient classical encoder, then so does the CWS code \( \mathcal{Q} = (\mathcal{G},\mathcal{C}) \).Sequantial encoder related to MBQC [10].Decoding
There is no known efficient algorithm to decode non-additive (non-stabilizer) CWS codes.Clustered bounded-distance decoder [11–13].Structured error recovery [6], which reduces to syndrome-based recovery for additive (i.e., stabilizer) CWS codes.Notes
See Ref. [14] for an overview of CWS codes.Cousins
- Cluster-state code— A single cluster-state codeword is used to construct a CWS code.
- Movassagh-Ouyang Hamiltonian code— The Movassagh-Ouyang codes overlap the CWS codes but neither family is contained in the other [15].
- Spacetime code (STC)— CWS codes have been considered in the context of spacetime replication of quantum data [16,17], while STCs are designed to replicate classical data.
- Concatenated quantum code— CWS codes can be concatenated by applying generalized local complementation to their underlying graphs [18].
- EA qubit code— EA CWS codes have been formulated [19].
- XP stabilizer code— The orbit representatives of XP codes play a similar role to the word operators of CWS codes.
- \([[4,2,2]]\) Four-qubit code— A \([[4,1,2]]\) subcode can be thought of as a CWS code [20].
- Five-qubit perfect code— The five-qubit perfect code is equivalent via a single-qubit Clifford circuit to a CWS code defined from a five-cycle graph and a classical repetition code [2,21][9; Table I].
- \([[7,1,3]]\) Steane code— The Steane code is equivalent via a single-qubit Clifford unitary to a CWS code for a particular graph and classical code [9; Exam. 4].
- Qubit stabilizer code— CWS codes whose underlying classical code is a linear binary code are qubit stabilizer codes containing a cluster-state codeword [2,20]. Since any stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [3][4; Appx. A], any stabilizer code is similarly equivalent to a CWS code.
Primary Hierarchy
Parents
Any CWS code can be written as a USt whose (\(K=1\)) stabilizer code is the cluster state and whose coset representatives are constructed from the binary classical code. Conversely, USt codes are equivalent to CWS codes via a single-qubit Clifford circuit as follows [11,13][14; Sec. 10.4]. The set of coset representatives of any USt can be extended to a larger set iterating over the underlying stabilizer code such that all codewords can be obtained from a single stabilizer state. Then, one can apply a single-qubit Clifford transformation to map said stabilizer state into a cluster state.
Modular-qudit CWS codes reduce to CWS codes for \(q=2\).
Galois-qudit CWS codes reduce to CWS codes for \(q=2\).
Codeword stabilized (CWS) code
Children
The \(((10,24,3))\) qubit code is a CWS code [21].
References
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- J. Bausch and F. Leditzky, “Error Thresholds for Arbitrary Pauli Noise”, SIAM Journal on Computing 50, 1410 (2021) arXiv:1910.00471 DOI
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- Zeng, Bei. Quantum operations and codes beyond the Stabilizer-Clifford framework. Diss. Massachusetts Institute of Technology, 2009.
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- U. S. Kapshikar, “The Diagonal Distance of CWS Codes”, (2021) arXiv:2107.11286
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- S. Sudevan, S. Das, T. Aswanth, N. Patanker, and N. Kashyap, “Sequentially Encodable Codeword Stabilized Codes”, (2024) arXiv:2405.06142
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- Y. Li, I. Dumer, M. Grassl, and L. P. Pryadko, “Clustered bounded-distance decoding of codeword-stabilized quantum codes”, 2010 IEEE International Symposium on Information Theory 2662 (2010) DOI
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- Li, Yunfan. Codeword Stabilized Quantum Codes and Their Error Correction. Diss. UC Riverside, 2010.
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- M. Grassl and M. Rötteler, “Nonadditive quantum codes”, Quantum Error Correction 261 (2013) DOI
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- R. Movassagh and Y. Ouyang, “Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians”, Quantum 8, 1541 (2024) arXiv:2012.01453 DOI
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- P. Hayden and A. May, “Summoning information in spacetime, or where and when can a qubit be?”, Journal of Physics A: Mathematical and Theoretical 49, 175304 (2016) arXiv:1210.0913 DOI
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- P. Hayden, S. Nezami, G. Salton, and B. C. Sanders, “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016) arXiv:1601.02544 DOI
- [18]
- S. Beigi, I. Chuang, M. Grassl, P. Shor, and B. Zeng, “Graph concatenation for quantum codes”, Journal of Mathematical Physics 52, 022201 (2011) arXiv:0910.4129 DOI
- [19]
- J. Shin, J. Heo, and T. A. Brun, “Entanglement-assisted codeword stabilized quantum codes”, Physical Review A 84, (2011) arXiv:1109.3358 DOI
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- C. Cafaro, D. Markham, and P. van Loock, “Scheme for constructing graphs associated with stabilizer quantum codes”, (2014) arXiv:1407.2777
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- Cross, Andrew William. Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes. Diss. Massachusetts Institute of Technology, 2008.
Page edit log
- Victor V. Albert (2024-03-28) — most recent
- Victor V. Albert (2022-04-21)
- Victor V. Albert (2021-12-16)
- Eric Kubischta (2021-12-15)
Cite as:
“Codeword stabilized (CWS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cws