Modular-qudit stabilizer code[1] 

Description

An \(((n,K,d))_q\) modular-qudit code whose logical subspace is the joint eigenspace of commuting qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a qudit Pauli string that implements a nontrivial logical operation in the code.

A modular-qudit stabilizer code encoding an integer number of qudits (\(K=q^k\)) is denoted as \([[n,k]]_{\mathbb{Z}_q}\) or \([[n,k,d]]_{\mathbb{Z}_q}\). For composite \(q\), such codes need not encode an integer number of qudits, with \(K=q^n/|\mathsf{S}|\) [2]. This is because \(|{\mathsf{S}}|\) need not be a power of \(q\), as group generators may have different orders. As a result, \([[n,k,d]]\) notation is often used with non-integer \(k=\log_q K\). Prime-qudit stabilizer codes, where \(q=p\) for some prime \(p\), do not suffer from this issue and encode \(n-k\) logical qudits, with \(K=p^{n-k}\).

Each code can be represented by a check matrix (a.k.a. stabilizer generator matrix) \(H=(A|B)\), where each row \((a|b)\) is the \(q\)-ary symplectic representation of a stabilizer generator. The check matrix can be brought into standard form via Gaussian elimination [2].

One can switch between stabilizer codes by appending another Abelian subgroup of the Pauli group to the stabilizer group and taking the center of the resulting larger group.

Code switching, code deformation, or stabilizer update rule: Code switching is a map between stabilizer codes that is done using a stabilizer group \(\mathsf{F}\) of the \(n\)-qudit Pauli group, \begin{align} \mathsf{S}\to\mathsf{N}_{\left\langle \mathsf{S},\mathsf{F}\right\rangle }\left(\mathsf{F}\right)~, \tag*{(1)}\end{align} where \(\mathsf{Z}\) denotes taking the center of a group (e.g., see [3,4] for proofs). Code switching may not preserve the logical information and instead implement logical measurements; conditions on \(\mathsf{S}\) and \(\mathsf{F}\) such that qubit stabilizer code switching preserves logical information are derived in [5; Prop. II.1]. Clifford operations and Pauli measurements can be expressed as sequences of code switching [6]. In the context of stabilizer codes realizing Abelian topological phases, code switching implements anyon condensation of any anyons represented by operators in the group \(\mathsf{F}\).

Modular-qudit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_q^n\) [7]. Stabilizer codewords for odd qudit dimension have a specific form per the finite-dimensional version of Hudson's theorem [8]; they saturate various uncertainty relations [9]. General modular-qudit stabilizer codes can equivalently [10] be defined using graphs, yielding an analytical form for the codewords [11].

Protection

Detects errors on up to \(d-1\) qudits, and corrects erasure errors on up to \(d-1\) qudits. More generally, define the normalizer \(\mathsf{N(S)}\) of \(\mathsf{S}\) to be the set of all Pauli operators that commute with all \(S\in\mathsf{S}\). A stabilizer code can correct a Pauli error set \({\mathcal{E}}\) if and only if \(E^\dagger F \notin \mathsf{N(S)}\setminus \mathsf{S}\) for all \(E,F \in {\mathcal{E}}\).

Magic

The magic-state yield parameter \(\gamma = \log_d(n/k)\) quantifies the overhead cost of magic-state distillation per the original protocol [12,13].

Gates

Gates in the qudit Clifford hierarchy can be done using qudit gate teleportation, in which a gate can be obtained from a particular qudit magic state. Magic states that are eigenstates of qudit Clifford operators have been classified for prime qudit dimension 3 and 5 [14].

Decoding

The structure of stabilizer codes allows for syndrome-based decoding, where errors are corrected based on the results of stabilizer measurements (syndromes).Trellis decoder for prime-dimensional qudits, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [15].

Notes

A standardized definition of the qudit stabilizer group is developed in [2].The number of modular-qudit stabilizer codes was determined in Refs. [8,16].

Parents

  • Modular-qudit USt code — A modular-qudit stabilizer code with stabilizer group \(\mathsf{S}\) can be thought of as a modular-qudit USt with only the identity coset representative. Conversely, if \(K = q^k\), and if the set of coset representatives of a modular-qudit USt form a \(q\)-ary linear code over \(\mathbb{Z}_q\), then they can be absorbed into a modular-qudit stabilizer group that defines the USt.
  • Stabilizer code
  • Quantum Lego code — Modular-qudit stabilizer codes are quantum Lego codes built out of atomic blocks such as the 2-qudit repetition code, single-qudit trivial stabilizer codes, and tensor-products of the \(|0\rangle\) state [17].

Children

  • Qubit stabilizer code — Modular-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes. Modular-qudit stabilizer codes for prime-dimensional qudits \(q=p\) inherit most of the features of qubit stabilizer codes, including encoding an integer number of qudits and a Pauli group with a unique number of generators. Conversely, qubit codes can be extended to modular-qudit codes by decorating appropriate generators with powers. For example, \([[4,2,2]]\) qubit code generators can be adjusted to \(ZZZZ\) and \(XX^{-1} XX^{-1}\). A systematic procedure extending a qubit code to prime-qudit codes involves putting its generator matrix into local-dimension-invariant (LDI) form [18]. Various bounds exist on the distance of the resulting codes [19,20].
  • \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code
  • 3D lattice stabilizer code
  • Frobenius code
  • Modular-qudit cluster-state code — Modular-qudit cluster-state codes are particular modular-qudit stabilizer codes. Any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit [10] (see also [21,22]).
  • Modular-qudit CSS code — Modular-qudit CSS codes are modular-qudit stabilizer codes whose stabilizer groups admit a generating set of pure-\(X\) and pure-\(Z\) Pauli strings. Any \([[n,k,d]]_{\mathbb{Z}_q}\) stabilizer code can be mapped onto a \([[2n,2k,\geq d]]_{\mathbb{Z}_q}\) two-block CSS code code via symplectic doubling, which preserves geometric locality of a code up to a constant factor.
  • Chiral semion Walker-Wang model code
  • Abelian TQD stabilizer code

Cousins

  • Modular-qudit CWS code — Modular-qudit CWS codes whose underlying classical code is a linear \(q\)-ary code over \(\mathbb{Z}_q\) are modular-qudit stabilizer codes containing a cluster-state codeword; see [23; Corr. 4-5], which defines CWS codes as admitting an underlying stabilizer state that is not a necessarily a cluster state.
  • \(q\)-ary code over \(\mathbb{Z}_q\) — Modular-qudit stabilizer codes are the closest quantum analogues of additive codes over \(\mathbb{Z}_q\) because addition in the ring corresponds to multiplication of stabilizers in the quantum case.
  • Design — Stabilizer states on \(n\) prime-dimensional qubits form complex projective 2-designs [24], while the prime-qudit Clifford group is a unitary 2-design [25].
  • Barnes-Wall (BW) lattice code — The first lattice shell of a BW lattice over a cyclotomic field is formed by stabilizer states [26].
  • Graph quantum code — Graph quantum codes for \(G=\mathbb{Z}_q\) are a subset of modular-qudit stabilizer codes [10]. Any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit [10] (see also [21,22]).
  • Analog stabilizer code — Prime-qudit stabilizer codes can be converted into analog stabilizer codes whose distance is at least as large as that of the original code [27].
  • Subsystem modular-qudit stabilizer code — Subsystem modular-qudit stabilizer codes reduce to modular-qudit stabilizer codes when there are no gauge qudits.
  • Galois-qudit stabilizer code — Recalling that \(q=p^m\), Galois-qudit stabilizer codes can also be treated as prime-qudit stabilizer codes on \(mn\) qudits, giving \(k=nm-r\) [28]. The case \(m=1\) reduces to conventional prime-qudit stabilizer codes on \(n\) qudits.

References

[1]
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
[2]
V. Gheorghiu, “Standard form of qudit stabilizer groups”, Physics Letters A 378, 505 (2014) arXiv:1101.1519 DOI
[3]
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
[4]
D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[5]
D. Aasen et al., “Measurement Quantum Cellular Automata and Anomalies in Floquet Codes”, (2023) arXiv:2304.01277
[6]
M. E. Beverland, S. Huang, and V. Kliuchnikov, “Fault tolerance of stabilizer channels”, (2024) arXiv:2401.12017
[7]
E. Hostens, J. Dehaene, and B. De Moor, “Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic”, Physical Review A 71, (2005) arXiv:quant-ph/0408190 DOI
[8]
D. Gross, “Hudson’s theorem for finite-dimensional quantum systems”, Journal of Mathematical Physics 47, (2006) arXiv:quant-ph/0602001 DOI
[9]
K. Bu, “Extremality of stabilizer states”, (2024) arXiv:2403.13632
[10]
D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
[11]
D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
[12]
S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas”, Physical Review A 71, (2005) arXiv:quant-ph/0403025 DOI
[13]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
[14]
A. Jain and S. Prakash, “Qutrit and ququint magic states”, Physical Review A 102, (2020) arXiv:2003.07164 DOI
[15]
E. Sabo, A. B. Aloshious, and K. R. Brown, “Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes”, (2022) arXiv:2106.08251
[16]
T. Singal et al., “Counting stabiliser codes for arbitrary dimension”, Quantum 7, 1048 (2023) arXiv:2209.01449 DOI
[17]
T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
[18]
L. G. Gunderman, “Local-dimension-invariant qudit stabilizer codes”, Physical Review A 101, (2020) arXiv:1910.08122 DOI
[19]
A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
[20]
L. G. Gunderman, “Degenerate local-dimension-invariant stabilizer codes and an alternative bound for the distance preservation condition”, Physical Review A 105, (2022) arXiv:2110.15274 DOI
[21]
M. Van den Nest, J. Dehaene, and B. De Moor, “Graphical description of the action of local Clifford transformations on graph states”, Physical Review A 69, (2004) arXiv:quant-ph/0308151 DOI
[22]
M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI
[23]
D. F. G. Santiago and G. S. S. Otoni, “A new approach to codeword stabilized quantum codes using the algebraic structure of modules”, (2015) arXiv:1505.00283
[24]
R. Kueng and D. Gross, “Qubit stabilizer states are complex projective 3-designs”, (2015) arXiv:1510.02767
[25]
M. A. Graydon, J. Skanes-Norman, and J. J. Wallman, “Clifford groups are not always 2-designs”, (2021) arXiv:2108.04200
[26]
V. Kliuchnikov and S. Schönnenbeck, “Stabilizer operators and Barnes-Wall lattices”, (2024) arXiv:2404.17677
[27]
L. G. Gunderman, “Stabilizer Codes with Exotic Local-dimensions”, Quantum 8, 1249 (2024) arXiv:2303.17000 DOI
[28]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
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Zoo Code ID: qudit_stabilizer

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“Modular-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_stabilizer
BibTeX:
@incollection{eczoo_qudit_stabilizer, title={Modular-qudit stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudit_stabilizer} }
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“Modular-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/qudit_stabilizer.yml.