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 \(\mathsf{S}\) [1; Sec. 3.6]. 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][1; Sec. 3.6]. 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 \(k\) logical qudits, with \(K=p^k\).
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]. They correspond to the set of states with positive Wigner functions [6,7] (see [8; Thm. 8.4] for a robust version of Hudson’s theorem for odd prime-dimensional qudits). Stabilizer states 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].
There is a quantum GV bound for modular-qudit stabilizer codes [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].Encoding
Encoder circuits for prime-qudit stabilizer codes [15].Transversal Gates
All qudit stabilizer codes realize modular qudit Pauli transformations transversally.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 [16].Decoding
Trellis decoder for prime-dimensional qudits, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [17].Notes
Distance upper bounds for Galois-qudit stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Refs. [18,19] 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. [6,20].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 [21; Corr. 4-5], which defines CWS codes as admitting an underlying stabilizer state that is not a necessarily a cluster state.
- Linear 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 qudits form 2-designs on complex projective spaces \(\mathbb{C}P^{p^n}\) [22], while the prime-qudit Clifford group is a unitary 2-design on \(U(p^n)\) [23].
- Unitary \(t\)-design— Stabilizer states on \(n\) prime-dimensional qudits form 2-designs on complex projective spaces \(\mathbb{C}P^{p^n}\) [22], while the prime-qudit Clifford group is a unitary 2-design on \(U(p^n)\) [23].
- Complex projective space code— Stabilizer states on \(n\) prime-dimensional qudits form 2-designs on complex projective spaces \(\mathbb{C}P^{p^n}\) [22], while the prime-qudit Clifford group is a unitary 2-design on \(U(p^n)\) [23].
- 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 [24].
- Rotor stabilizer code— By combining the paper’s bounded-phase-space and integer-local-dimension constructions, prime-qudit stabilizer codes can be algebraically imported into rotor-code settings [25; Sec. 3.2].
- Analog stabilizer code— Prime-qudit stabilizer codes can be transformed into analog stabilizer codes on the same number of modes and logical modes, with distance at least as large as that of the original code [25; Thm. 12].
- Majorana stabilizer code— Majorana stabilizer codes can be extended to modular qudits, yielding parafermion stabilizer codes [26].
- 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 \(k=nm-r\) [27]. 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 [28].
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 [29].
Modular-qudit stabilizer code
Children
Qubit stabilizer codeFermion-into-qubit Quantum RM code QLDPC BB Homological 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 [30]. Various bounds exist on the distance of the resulting codes [31,32].
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