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Modular-qudit code

Alternative names: \(\mathbb{Z}_q\)-qudit code, Modular-qudit subspace code.
Root code for the Modular-qudit Kingdom

Description

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.

Protection

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 modular-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. Tensor products of \(X\) (\(Z\)) modular-qudit Paulis acting on different qudits are called \(X\)-type (\(Z\)-type) modular-qudit Pauli strings. Combining the \(X\)-type and \(Z\)-type strings with a primitive \(2q\)th root of unity forms a group called the modular-qudit Pauli group [3].

Modular-qudit Pauli matrices [4,5] are also known as Weyl operators [6], Sylvester-t'Hooft generators [7,8], shift and boost operators [9], or clock and shift matrices [10]; they are special cases of Manin's quantum plane [11–14].

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.

Rate

Non-stabilizer states yield higher quantum capacity of the discrete beamsplitter channel [15].

Gates

The normalizer of the modular-qudit Pauli group is the modular-qudit Clifford group [1,3,16–20]. There is a standard form for modular-qudit Clifford-group operators [3; Lemma 4], and any modular-qudit gate can be constructed phase-shift and quantum Fourier transform gates [21]. Universal computing can be achieved using qudit Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate [22]. 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 [23]. Optimizing non-Clifford-gate count can be done using various procedures; see Refs. [24–27] for qutrit codes.

Qudit Clifford hierarchy: The modular-qudit Clifford hierarchy [28–31] 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. Gates for one prime-dimensional qudit have been classified [32].

Decoding

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.

Threshold

There is a threshold against depolarizing noise for any modular-qudit gate that determines if the gate is non-Clifford [33].

Notes

Review of qudit quantum computation [20].Weight distribution of a code depends on the average entanglement of codewords [34,35].Qudit Cirq library [36].

Cousins

  • 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; see [37,39–41][38; Sec. 5.3]. 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.

Member of code lists

Primary Hierarchy

Quantum Domain
Modular-qudit Kingdom
Parents
Block quantum codeQECC Quantum
Group-based quantum codeQECC Quantum
Group quantum codes whose physical spaces are constructed using modular-integer groups \(\mathbb{Z}_q\) are modular-qudit codes.
Category-based quantum codeQECC Quantum
Category quantum codes whose physical spaces are constructed using the group \(\mathbb{Z}_q\) as the category are modular-qudit codes.
Modular-qudit code
Children
Qubit codeHastings-Haah Floquet Fermion Self-complementary qubit BB Homological Fermion-into-qubit Quantum RM code Color Twist-defect color CDSC Twist-defect surface Kitaev surface
Modular-qudit quantum codes for \(q=2\) correspond to qubit codes. Modular-qudit codes [42], circuits [43], and magic-state distillation schemes [44,45] can have advantages over their qubit counterparts. There are several ways to embed one or more qubits into a single modular qudit, yielding efficient qubit gate decompositions [46].
Modular-qudit dynamical codeHastings-Haah Floquet
\(((3,6,2))_{\mathbb{Z}_6}\) Euler code
Modular-qudit USt codeFermion-into-qubit Twist-defect color Twist-defect surface Fracton stabilizer Color BB Homological Quantum RM code CDSC Kitaev surface

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Zoo Code ID: qudits_into_qudits

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/qudits_into_qudits.yml.