Modular-qudit code 

Also known as \(\mathbb{Z}_q\)-qudit code.
Root code for the Modular-qudit Kingdom


Encodes \(K\)-dimensional Hilbert space into a \(q^n\)-dimensional (\(n\)-qudit) Hilbert space, with canonical qudit states \(|k\rangle\) labeled by elements \(k\) of the group \(\mathbb{Z}_q\) of integers modulo \(q\). Usually denoted as \(((n,K))_{\mathbb{Z}_q}\) or \(((n,K,d))_{\mathbb{Z}_q}\), whenever the code's distance \(d\) is defined, and with \(q=p\) when the dimension is prime.


Modular-qudit Pauli-string error basis

A convenient and often considered error set is the modular-qudit analogue [1,2] of the Pauli string basis for qubit codes.

Modular-qudit Pauli strings: For a single qudit, this set consists of products of powers of the qudit Pauli matrices \(X\) and \(Z\), which act on computational basis states \(|k\rangle\) for \(k\in\mathbb{Z}_q\) as \begin{align} X\left|k\right\rangle =\left|k+1\right\rangle \,\,\text{ and }\,\,Z\left|k\right\rangle =e^{i\frac{2\pi}{q}k}\left|k\right\rangle ~, \tag*{(1)}\end{align} with addition performed modulo \(q\). For multiple qudits, error set elements are tensor products of elements of the single-qudit error set. Modular-qudit Pauli matrices [3,4] are also known as Weyl operators [5], Sylvester-t'Hooft generators [6,7], or clock and shift matrices [8]; they are special cases of Manin's quantum plane [9]

The Pauli error set is a unitary basis for linear operators on the multi-qudit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a nice error basis. The distance associated with this set is often the minimum weight of a qudit Pauli string that implements a nontrivial logical operation in the code.


The normalizer of the modular-qudit Pauli group is the modular-qudit Clifford group. Universal computing can be achieved using qudit Clifford gates [1,1012] and a single type of non-Clifford gate, such as the \(T\) gate [13]. Non-Clifford gates are typically more difficult to implement than Clifford gates and so are treated as a resource. There is a normal form for Clifford+\(T\) operators for odd prime qudits [14]. Optimizing non-Clifford-gate count can be done using various procedures; see Refs. [1518] for qutrit codes.

Qudit Clifford hierarchy: The modular-qudit Clifford hierarchy [1922] is a tower of gate sets which includes modular-qudit Pauli and modular-qudit Clifford gates at its first two levels, and non-Clifford qudit gates at higher levels. The \(k\)th level is defined recursively by \begin{align} C_k = \{ U | U P U^{\dagger} \in C_{k-1} \}~, \tag*{(2)}\end{align} where \(P\) is any modular-qudit Pauli matrix, and \(C_1\) is the modular-qudit Pauli group.


For few-qudit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood (ML) decoder.


Weight distribution of a code depends on the average entanglement of codewords [23,24].




  • Entanglement-assisted (EA) QECC — Pure modular-qudit codes can be used to make EA-QECCs with the same distance and dimension; see Thm. 10 of Ref. [25].
  • Bosonic \(q\)-ary expansion — The bosonic \(q\)-ary expansion allows one to map between prime-dimensional qudit states and a Fock subspace of a single mode.
  • Subsystem modular-qudit code — Subsystem modular-qudit codes reduce to (subspace) modular-qudit codes when there is no gauge subsystem.
  • Galois-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [26]; see Sec. 5.3 of Ref. [27]. The two coincide when \(q\) is prime, and reduce to qubits when \(q=2\). However, Pauli matrices for the two types of qudits are defined differently.


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“Modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_qudits_into_qudits, title={Modular-qudit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.