Modular-qudit stabilizer code

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.

Stabilizer code switching, code deformation, or 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 — 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

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[33]: Markus Grassl, private communication, 2024

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Victor V. Albert (2022-05-19) — most recent
Victor V. Albert (2022-02-16)
Leonid Pryadko (2022-02-16)
Qingfeng (Kee) Wang (2022-01-07)
Victor V. Albert (2021-11-02)

Cite as:

“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.