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 \(K=q^n/|\mathsf{S}|\) [3]. 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 \(k=\log_q K\), and the code dimension can be inferred from the prime decomposition of \(q\) [4]. 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 [3].
Modular-qudit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_q^n\) [5]. There exists an analogue of the Wigner function for modular qudits [6,7], and modular-qudit stabilizer states correspond to the set of states with positive Wigner functions [8,9] (see [10; Thm. 8.4] for a robust version of Hudson's theorem for odd prime-dimensional qudits). Stabilizer states saturate various uncertainty relations [11]. General modular-qudit stabilizer codes can equivalently [12] be defined using graphs, yielding an analytical form for the codewords [13].
There is a quantum GV bound for modular-qudit stabilizer codes [14].
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 [15,16].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 [17].Decoding
Trellis decoder for prime-dimensional qudits, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [18].Notes
Distance upper bounds for Galois-qudit stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Refs. [19,20] 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 [3].The number of modular-qudit stabilizer codes was determined in Refs. [8,21].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 [22; 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 [23], while the prime-qudit Clifford group is a unitary 2-design [24].
- 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 [25].
- Graph quantum code— Graph quantum codes for \(G=\mathbb{Z}_q\) are a subset of modular-qudit stabilizer codes [12]. Any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit [12] (see also [26,27]).
- 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 [28].
- Majorana stabilizer code— Majorana stabilizer codes can be extended to modular qudits, yielding parafermion stabilizer codes [29].
- 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 \(k=nm-r\) [30]. 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 [31].
Primary Hierarchy
Parents
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.
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 [32].
Modular-qudit stabilizer code
Children
Qubit stabilizer codeBB Homological Fermion-into-qubit Quantum RM code Color Twist-defect color CDSC Twist-defect surface Kitaev surface
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 [33]. Various bounds exist on the distance of the resulting codes [34,35].
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.
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Page edit log
- 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