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 \(K=p^{n-k}\).

Modular symplectic representation: The single modular-qudit Pauli string \(X_{a} Z_{b}\) for \(a,b\in \mathbb{Z}_q\) is converted to the vector \((a|b)\in \mathbb{Z}_q^2\). The multi modular-qudit version follows naturally.

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 modular 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 modular-qudit 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 \(n\)-modular-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 \(\mathsf{F}\). Code switching can be done using only transversal gates for qubit stabilizer codes [7].

Modular-qudit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_q^n\) [8]. Stabilizer codewords for odd qudit dimension have a specific form per the finite-dimensional version of Hudson's theorem [9]; they saturate various uncertainty relations [10]. General modular-qudit stabilizer codes can equivalently [11] be defined using graphs, yielding an analytical form for the codewords [12].

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 [13,14].

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 [15].

Decoding

Trellis decoder for prime-dimensional qudits, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [16].

Notes

Distance upper bounds for Galois-qudit stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Refs. [17,18] and maintained by M. Grassl at this website, hold for general modular-qudit codes because they are based on linear programming.A standardized definition of the qudit stabilizer group is developed in [2].The number of modular-qudit stabilizer codes was determined in Refs. [9,19].

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
  • Tensor-network 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 [20].

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 modular-qudit 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 [21]. Various bounds exist on the distance of the resulting codes [22,23].
  • \([[5,1,3]]_{\mathbb{Z}_q}\) 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 [11] (see also [24,25]).
  • 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 [26; 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.
  • \(t\)-design — Stabilizer states on \(n\) prime-dimensional qubits form complex projective 2-designs [27], while the prime-qudit Clifford group is a unitary 2-design [28].
  • Barnes-Wall (BW) lattice code — Modular-qudit stabilizer states can be mapped into the first lattice shell of a BW lattice over a cyclotomic field, while the modular-qudit Clifford group is related to the symmetry group of the lattice [29].
  • Graph quantum code — Graph quantum codes for \(G=\mathbb{Z}_q\) are a subset of modular-qudit stabilizer codes [11]. Any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit [11] (see also [24,25]).
  • 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 [30].
  • Majorana stabilizer code — Majorana stabilizer codes can be extended to modular qudits, yielding parafermion stabilizer codes [31].
  • 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 \(k=nm-r\) [32]. The case \(m=1\) reduces to conventional prime-qudit stabilizer codes on \(n\) qudits. A modular-qudit stabilizer code with composite dimension \(q\) contains a subcode that is isomorphic to a \(p\)-dimensional prime-qudit stabilizer code for every prime factor \(p\) of \(q\), and the distance of the full stabilizer code is bounded by the distance of this subcode [33].

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]
S. Heußen and J. Hilder, “Efficient fault-tolerant code switching via one-way transversal CNOT gates”, (2024) arXiv:2409.13465
[8]
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
[9]
D. Gross, “Hudson’s theorem for finite-dimensional quantum systems”, Journal of Mathematical Physics 47, (2006) arXiv:quant-ph/0602001 DOI
[10]
K. Bu, “Extremality of stabilizer states”, (2024) arXiv:2403.13632
[11]
D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
[12]
D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
[13]
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
[14]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
[15]
A. Jain and S. Prakash, “Qutrit and ququint magic states”, Physical Review A 102, (2020) arXiv:2003.07164 DOI
[16]
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
[17]
M. Grassl, “Searching for linear codes with large minimum distance”, Discovering Mathematics with Magma 287 DOI
[18]
M. F. Ezerman et al., “Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes”, (2024) arXiv:2405.15057
[19]
T. Singal et al., “Counting stabiliser codes for arbitrary dimension”, Quantum 7, 1048 (2023) arXiv:2209.01449 DOI
[20]
T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
[21]
L. G. Gunderman, “Local-dimension-invariant qudit stabilizer codes”, Physical Review A 101, (2020) arXiv:1910.08122 DOI
[22]
A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
[23]
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
[24]
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
[25]
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
[26]
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
[27]
R. Kueng and D. Gross, “Qubit stabilizer states are complex projective 3-designs”, (2015) arXiv:1510.02767
[28]
M. A. Graydon, J. Skanes-Norman, and J. J. Wallman, “Clifford groups are not always 2-designs”, (2021) arXiv:2108.04200
[29]
V. Kliuchnikov and S. Schönnenbeck, “Stabilizer operators and Barnes-Wall lattices”, (2024) arXiv:2404.17677
[30]
L. G. Gunderman, “Stabilizer Codes with Exotic Local-dimensions”, Quantum 8, 1249 (2024) arXiv:2303.17000 DOI
[31]
U. Güngördü, R. Nepal, and A. A. Kovalev, “Parafermion stabilizer codes”, Physical Review A 90, (2014) arXiv:1409.4724 DOI
[32]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[33]
Markus Grassl, private communication, 2024
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Zoo Code ID: qudit_stabilizer

Cite as:
“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.