Modular-qudit code 

Also known as \(\mathbb{Z}_q\)-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 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.

Bounds on code parameters

Bounds on code performance include the quantum Singleton bound, quantum Hamming bound, quantum GV bound, various quantum linear programming (LP) bounds [10,11] (see the book [12]), and other bounds [13]. A code whose parameters attain the quantum Hamming bound (quantum Singleton bound) is called a perfect quantum code (a quantum MDS code).

Gates

The normalizer of the modular-qudit Pauli group is the modular-qudit Clifford group [1,14–16]. Universal computing can be achieved using qudit Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate [17]. 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 [18]. Optimizing non-Clifford-gate count can be done using various procedures; see Refs. [19–22] for qutrit codes.

Qudit Clifford hierarchy: The modular-qudit Clifford hierarchy [23–26] 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.

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.

Notes

Weight distribution of a code depends on the average entanglement of codewords [27,28].

Parents

Children

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 [29]; see Sec. 5.3 of Ref. [30]. 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.

References

[1]
“Full length article”, Chaos, Solitons & Fractals 10, 1749 (1999) arXiv:quant-ph/9802007 DOI
[2]
R. Sarkar and T. J. Yoder, “The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems”, Quantum 8, 1307 (2024) arXiv:2302.07966 DOI
[3]
J. v. Neumann, “Die Eindeutigkeit der Schrödingerschen Operatoren”, Mathematische Annalen 104, 570 (1931) DOI
[4]
J. Schwinger, Quantum Kinematics and Dynamics (CRC Press, 2018) DOI
[5]
Weyl, Hermann. The theory of groups and quantum mechanics. Courier Corporation, 1950.
[6]
Sylvester, James Joseph. The Collected Mathematical Papers of James Joseph Sylvester... Vol. 3. University Press, 1909.
[7]
G. ’t Hooft, “On the phase transition towards permanent quark confinement”, Nuclear Physics B 138, 1 (1978) DOI
[8]
C. Zachos, “Hamiltonian Flows, SU(∞), SO(∞), USp(∞), and Strings”, Differential Geometric Methods in Theoretical Physics 423 (1990) DOI
[9]
Yu. I. Manin, “Some remarks on Koszul algebras and quantum groups”, Annales de l’institut Fourier 37, 191 (1987) DOI
[10]
E. M. Rains, “Quantum shadow enumerators”, (1997) arXiv:quant-ph/9611001
[11]
A. Ashikhmin and S. Litsyn, “Upper Bounds on the Size of Quantum Codes”, (1997) arXiv:quant-ph/9709049
[12]
D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
[13]
Keqin Feng, San Ling, and Chaoping Xing, “Asymptotic bounds on quantum codes from algebraic geometry codes”, IEEE Transactions on Information Theory 52, 986 (2006) DOI
[14]
E. Hostens, J. Dehaene, and B. De Moor, “Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic”, Physical Review A 71, (2005) arXiv:quant-ph/0408190 DOI
[15]
S. Clark, “Valence bond solid formalism ford-level one-way quantum computation”, Journal of Physics A: Mathematical and General 39, 2701 (2006) arXiv:quant-ph/0512155 DOI
[16]
Y. Wang et al., “Qudits and High-Dimensional Quantum Computing”, Frontiers in Physics 8, (2020) arXiv:2008.00959 DOI
[17]
F. H. E. Watson et al., “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
[18]
A. Jain, A. R. Kalra, and S. Prakash, “A Normal Form for Single-Qudit Clifford+\(T\) Operators”, (2020) arXiv:2011.07970
[19]
A. Bocharov et al., “Efficient topological compilation for a weakly integral anyonic model”, Physical Review A 93, (2016) arXiv:1504.03383 DOI
[20]
A. R. Kalra, D. Valluri, and M. Mosca, “Synthesis and Arithmetic of Single Qutrit Circuits”, (2024) arXiv:2311.08696
[21]
S. Evra and O. Parzanchevski, “Arithmeticity and covering rate of the \(9\)-cyclotomic Clifford+\(\mathcal{D}\) gates in \(PU(3)\)”, (2024) arXiv:2401.16120
[22]
A. R. Kalra et al., “Multi-qutrit exact synthesis”, (2024) arXiv:2405.08147
[23]
D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations”, Nature 402, 390 (1999) arXiv:quant-ph/9908010 DOI
[24]
M. Howard and J. Vala, “Qudit versions of the qubitπ/8gate”, Physical Review A 86, (2012) arXiv:1206.1598 DOI
[25]
F. Pastawski and B. Yoshida, “Fault-tolerant logical gates in quantum error-correcting codes”, Physical Review A 91, (2015) arXiv:1408.1720 DOI
[26]
S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[27]
A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
[28]
L. Schatzki et al., “A Hierarchy of Multipartite Correlations Based on Concentratable Entanglement”, (2022) arXiv:2209.07607
[29]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[30]
A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
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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

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