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 \(q\)th root of unity forms a group called the modular-qudit Pauli group (on \(n\) modular qudits.
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.
Rate
Non-stabilizer states yield higher quantum capacity of the discrete beamsplitter channel [10].Gates
The normalizer of the modular-qudit Pauli group is the modular-qudit Clifford group [1,11–13]. Universal computing can be achieved using qudit Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate [14]. 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 [15]. Optimizing non-Clifford-gate count can be done using various procedures; see Refs. [16–19] for qutrit codes.Qudit Clifford hierarchy: The modular-qudit Clifford hierarchy [20–23] 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 [24].
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 [25].Notes
Review of qudit quantum computation [13].Weight distribution of a code depends on the average entanglement of codewords [26,27].Qudit Cirq library [28].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 [29,31–33][30; 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.
Primary Hierarchy
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]
- K. Bu and A. Jaffe, “Magic Resource Can Enhance the Quantum Capacity of Channels”, Physical Review Letters 134, (2025) arXiv:2401.12105 DOI
- [11]
- 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
- [12]
- 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
- [13]
- Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, “Qudits and High-Dimensional Quantum Computing”, Frontiers in Physics 8, (2020) arXiv:2008.00959 DOI
- [14]
- F. H. E. Watson, E. T. Campbell, H. Anwar, and D. E. Browne, “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
- [15]
- A. Jain, A. R. Kalra, and S. Prakash, “A Normal Form for Single-Qudit Clifford+\(T\) Operators”, (2020) arXiv:2011.07970
- [16]
- A. Bocharov, X. Cui, V. Kliuchnikov, and Z. Wang, “Efficient topological compilation for a weakly integral anyonic model”, Physical Review A 93, (2016) arXiv:1504.03383 DOI
- [17]
- A. R. Kalra, D. Valluri, and M. Mosca, “Synthesis and Arithmetic of Single Qutrit Circuits”, (2024) arXiv:2311.08696
- [18]
- S. Evra and O. Parzanchevski, “Arithmeticity, thinness and efficiency of qutrit Clifford+T gates”, (2024) arXiv:2401.16120
- [19]
- A. R. Kalra, M. Saikia, D. Valluri, S. Winnick, and J. Yard, “Multi-qutrit exact synthesis”, (2024) arXiv:2405.08147
- [20]
- 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
- [21]
- M. Howard and J. Vala, “Qudit versions of the qubitπ/8gate”, Physical Review A 86, (2012) arXiv:1206.1598 DOI
- [22]
- F. Pastawski and B. Yoshida, “Fault-tolerant logical gates in quantum error-correcting codes”, Physical Review A 91, (2015) arXiv:1408.1720 DOI
- [23]
- S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
- [24]
- N. de Silva and O. Lautsch, “The Clifford hierarchy for one qubit or qudit”, (2025) arXiv:2501.07939
- [25]
- W. van Dam and M. Howard, “Noise thresholds for higher-dimensional systems using the discrete Wigner function”, Physical Review A 83, (2011) arXiv:1011.2497 DOI
- [26]
- 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
- [27]
- L. Schatzki, G. Liu, M. Cerezo, and E. Chitambar, “Hierarchy of multipartite correlations based on concentratable entanglement”, Physical Review Research 6, (2024) arXiv:2209.07607 DOI
- [28]
- S. Kushnarev and H. J. Asghar, “The Qudit Cirq Library: An Extension of Google’s Cirq Library for Qudits”, (2025) arXiv:2501.07812
- [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
- [31]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [32]
- Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
- [33]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
- [34]
- J. Keppens, Q. Eggerickx, V. Levajac, G. Simion, and B. Sorée, “Qudit vs. Qubit: Simulated performance of error correction codes in higher dimensions”, (2025) arXiv:2502.05992
- [35]
- P. Gokhale, J. M. Baker, C. Duckering, N. C. Brown, K. R. Brown, and F. T. Chong, “Asymptotic improvements to quantum circuits via qutrits”, Proceedings of the 46th International Symposium on Computer Architecture 554 (2019) arXiv:1905.10481 DOI
- [36]
- E. T. Campbell, H. Anwar, and D. E. Browne, “Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes”, Physical Review X 2, (2012) arXiv:1205.3104 DOI
- [37]
- E. T. Campbell, “Enhanced Fault-Tolerant Quantum Computing ind-Level Systems”, Physical Review Letters 113, (2014) arXiv:1406.3055 DOI
Page edit log
- Victor V. Albert (2022-05-07) — most recent
- Victor V. Albert (2021-10-29)
Cite as:
“Modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudits_into_qudits