Qudits and high-dimensional quantum computing

1 Introduction to qudits

• Qudits’ larger state space and ability to perform multiple control operations simultaneously can reduce circuit complexity, simplify experimental setups, and enhance algorithm efficiency.The review notes applications beyond circuit-model computing, including adiabatic and topological quantum systems.Source: §1 · pp. 1–2
• The review covers qudit gate universality and high-dimensional gates, qudit algorithm designs, and experimental methods, comparing qudit designs and platforms with qubit counterparts.Its gate discussion includes universal sets, gate examples, and arbitrary-unitary synthesis; algorithms include oracle decision, hidden non-abelian subgroup, and quantum search problems.Source: §1 · p. 2
• The review surveys implementations on photonic, continuous-spin, ion-trap, nuclear magnetic resonance, and molecular-magnet platforms.It considers potential improvements in computational speed-up and resource saving, and closes with perspectives on future qudit development and applications.Source: §1 · p. 2
• The algorithm survey includes parity of a permutation, Deutsch-Jozsa, Bernstein-Vazirani, quantum Fourier transform, phase estimation, and quantum search algorithms [23, 25, 27].The review also discusses measurement-based, adiabatic, and topological qudit computing models.Source: §1 · p. 2

• A qudit is a quantum version of a d-ary digit whose state lies in a d-dimensional Hilbert space spanned by orthonormal basis vectors.The basis is {|0⟩, |1⟩, …, |d −1⟩}.Source: §2 · p. 3
• A qudit state is a normalized superposition of basis states, can replace a qubit as a basic computational element, and is transformed by qudit gates.The squared magnitudes of its amplitudes sum to 1.Source: §2 · p. 3
• The section reviews qudit universality criteria and gate sets, examples of gates and their advantages over qubit gates, and circuit efficiency for decomposing arbitrary unitary matrices.The circuit-efficiency discussion gives a quantitative boundary on the number of elementary gates needed.Source: §2 · p. 3

2.1.1 Universality

• A universal gate set approximates any unitary on the qudit Hilbert space to acceptable error; exact universality requires zero-error simulation of any unitary and quantum process.Vlasov’s sufficient set uses two noncommuting single-qudit gates and a two-qudit gate, while Brylinski and Brylinski give necessary and sufficient conditions involving random single-qudit gates and an entangling two-qudit gate.Source: §2.1.1 · p. 3

2.1.2 Examples of universal gate sets

• One-qudit general rotations and two-qudit controlled rotation extensions form a physically realizable universal set, proved sufficient to decompose arbitrary SU(d^n) unitaries.The proof uses eigen-decomposition, multi-qudit-controlled gates, and their construction from two-qudit controlled rotations and permutations.Source: §2.1.2 · pp. 4–6
• The construction reports a (log2 d)^2 scaling advantage over the qubit case and an extra factor-n reduction in gate requirement, with primitive qudit gates needing fewer free parameters.Representing an N-dimensional system requires log2 N qubits or logd N qudits, a reduction factor of log2 d [3].Source: §2.1.2 · p. 6
• For topological computing with metaplectic anyons, proposed qutrit and qupit universal sets use SUM, Hadamard, and specified phase gates.The qutrit set uses SUM3, H3, and any P[0]3, P[1]3, or P[2]3; the qupit set uses SUMp, Hp, and Q[i]p for i in [p −1]. The proof uses a dense subgroup of SU(d) and an imprimitive SUM gate; for odd prime p, Q[i]p is a power of P[i]p.Source: §2.1.2 · pp. 6–7

2.2.1 Qudit versions π/8 gate

• A qudit π/8 gate is a non-Clifford gate essential to universal quantum computation, and its generalized form is identical to maximally robust qudit gates for fault-tolerant computation.The gate is also used in teleportation-based universal computation, transversal implementation, gate learning, and assisted quantum computation [49], and in qudit magic-state distillation.Source: §2.2.1 · pp. 8, 11

2.2.2 Qudit SWAP gate

• A Hermitian qudit CNOT variant enables a SWAP circuit using three identical gates, offering a simpler implementation and a way to connect systems with nearest-neighbour interactions.The gate is its own inverse; the construction uses operations realizable on multilevel systems, including multilevel atoms.Source: §2.2.2 · pp. 12–14

2.2.3 Simplified qubit Toffoli gate with a qudit

• A qutrit target reduces the Toffoli circuit from at least 5 to 3 two-qubit gates, and the method extends to n-controlled Toffoli gates with 2n−1 gates.The target needs an extra level for each additional control; this approach also extends to multi-qudit-controlled-unitary gates CnU.Source: §2.2.3 · p. 14
• A tree-topology qudit scheme realizes an n-qubit Toffoli gate with 2n−3 two-qudit CZ gates, and can scale to multi-qubit-controlled unitary gates CnU.The result applies when the qudit dimensions and their connection counts satisfy the stated condition.Source: §2.2.3 · pp. 15–16

2.2.4 Qudit multi-level controlled gate

• Unlike a Muthukrishan-Stroud gate, which acts for one selected control state, a multi-value-controlled gate applies a unique target operation for every control state.The gate fully uses the control qudit’s d states, with each Ui applied when the control is in |i⟩.Source: §2.2.4 · pp. 16–17

2.3 Geometrically quantifying qudit-gate efficiency

• The method assigns a cost to Hamiltonian controls and treats synthesis of U as finding the minimal geodesic from identity I to U; decoherence makes gate-count efficiency important.The qutrit Hamiltonian is expanded using Gell-Mann matrices rather than the Pauli matrices used for qubits.Source: §2.3 · pp. 17, 19
• Generalized Gell-Mann matrices provide an SU(d) basis, and one- and two-body interactions can generate all three-body interactions; higher-body terms receive a penalty in the control cost [5].The resulting cost defines a Riemannian metric on the space of n-qutrit unitary operators.Source: §2.3 · p. 18
• For n-qutrit unitaries, the theorem gives a polynomial-in-distance lower bound on one- and two-qutrit gates, and polynomial-size circuits exist if and only if d(I, U) scales polynomially with n [4].The theorem concerns approximating U by UA with ∥U − UA∥ ≤ c for constant c.Source: §2.3 · p. 19
• For qudits, each unitary in SU(d^n) can be approximated within ε using O(ε^-2) one- and two-qudit gates, with ε^-2 = N^2d^4n^2.Here N is the number of intervals dividing d(I, U), chosen so δ = d(I, U)/N ≤ ε; this relates nonlocal gate cost to approximation error.Source: §2.3 · pp. 19–20

3 Quantum algorithms using qudits

• Qudit algorithms include direct generalizations of qubit algorithms and approaches that use qudit dimensionality in a key subroutine.Qudits can store and process more information than qubits.Source: §3 · p. 20
• The section groups algorithms into oracle-decision problems and hidden Abelian subgroup problems.Source: §3 · p. 20
• Qudit gates can improve quantum search efficiency and reduce the difficulty of its physical setup.Source: §3 · p. 20

3.1.1 Parity determining algorithm

• A single-qutrit oracle query determines permutation parity, and this qudit algorithm uses no entanglement or quantum or classical correlation.The algorithm has been implemented for qutrit and ququart cases using NMR and on a linear-optic system; the higher-dimensional speedup is not exponential and the model problem has limited applications.Source: §3.1.1 · pp. 21–22

3.1.2 Qudit Deutsch-Jozsa algorithm

• The qudit Deutsch–Jozsa algorithm distinguishes constant from balanced functions and recovers nonconstant affine coefficients, while the affine function’s constant term is lost as a global phase.The n-nary algorithm is mainly theoretical, with potential applications in image processing and secure quantum key distribution.Source: §3.1.2 · pp. 22–23

3.1.3 Qudit generalization of the Bernstein-Vazirani algorithm

• The qudit Bernstein–Vazirani algorithm recovers an N-component number string modulo d using one oracle query and is faster than the best classical algorithm by a factor of N [83].The recovered string is specified up to a constant offset.Source: §3.1.3 · pp. 24–25

3.2.1 Quantum Fourier Transform with qudits

• The qudit QFT applies generalized Hadamard gates and multi-value-controlled phase gates, followed by SWAP gates to restore the required digit order.The generalized Hadamard is H_d = F(d, d).Source: §3.2.1 · p. 26
• Qudit QFT offers superior approximations: error magnitude decreases exponentially with dimension, with smaller error bounds than in the binary case.Source: §3.2.1 · p. 26

3.2.2 Phase-estimation algorithm with qudits

• Qudit PEA uses a t-qudit register to control powers of U on an eigenstate register, kicking phase factors back before inverse QFT and measurement recover the phase.The controlled operations can be viewed as a quantum multiplexer.Source: §3.2.2 · pp. 27–28
• For Shor’s factoring algorithm, a metaplectic topological quantum computer can factor an n-bit number with n + 7 logical qutrits, at the cost of greater circuit depth.The section also describes an adiabatic two-qudit proposal using a time-dependent effective Hamiltonian built from rotations between neighboring levels.Source: §3.2.2 · p. 29

3.3 Quantum search algorithm with qudits

• The proposed gate F replaces H to create an equal-weight superposition, simplifying qudit Grover search and improving overall circuit efficiency.The review presents F as easier to realize in a physical system.Source: §3.3 · pp. 29–30
• Other proposed qudit Grover schemes use a discrete Fourier transform [99] or another d-dimensional transformation instead of Hadamard for the diffusion operator.The review also outlines setting up Grover’s algorithm in a qudit system.Source: §3.3 · p. 29
• Grover iterations combine an oracle phase shift on the marked state with reflection about the average, amplifying the target while attenuating other states.The oracle is applied iteratively to solve the unstructured search problem.Source: §3.3 · p. 30
• A single multipod interaction realizes F with equal amplitudes of magnitude d^-1/2 across qudit states, using a common ancilla coupled by two-photon Raman processes.The multipod comprises d degenerate quantum states and a common ancilla state.Source: §3.3 · p. 31
• The multipod construction minimizes the number and duration of algorithmic steps and offers better protection against decoherence or imperfections [27].The review describes it as a conceptually simple, natural realization of qudit Grover search.Source: §3.3 · p. 32

4.1 Measurement-based qudit computing

• Qudit measurement-based computing remains unexplored, with reported work limited to qudit graph-state preparation and a continuous-variable qudit-cluster proposal for error correction.Measurement-based computing prepares an entangled cluster or graph state, then performs computation through sequential single-qubit measurements whose bases depend on a constant number of prior outcomes.Source: §4.1 · p. 32

4.2 Adiabatic qudit computing

• Adiabatic computing encodes a problem’s solution in a Hamiltonian ground state and slowly evolves into an approximation of that state, while current efforts focus on quantum annealing [103].The approach naturally corresponds to quantized satisfiability problems, and quantum annealing generalizes simulated annealing for global optimization.Source: §4.2 · pp. 32–33
• Solid-state annealers treated as qubits may require higher-dimensional dynamics for realistic modeling, while exploiting higher levels and avoided crossings can enable fast, high-fidelity multi-qubit gates.Conditions for sound qubit approximations have been established; another proposal uses a degenerate two-level system to potentially improve the energy gap and success probability.Source: §4.2 · p. 33
• The sole reported qudit adiabatic-algorithm proposal simulates factorization of 35, 21, and 15 using two qudits of dimensions 4 and 3.It uses a time-dependent effective hybrid two-qudit gate Hamiltonian realized with radio-frequency magnetic-field pulses on quadrupole nuclei with spins 3/2 and 1.Source: §4.2 · p. 33

4.3 Topological quantum computing with qudits

• Parafermion braiding generates single-qudit Clifford gates, and an Aharonov–Casher-effect non-Clifford gate completes a universal qudit gate set [119, 120].For d > 2, parafermion braiding is entangling; encoding a dimension-d qudit in a four-parafermion fusion space enables single-qudit Clifford gates modulo phases.Source: §4.3 · pp. 33–34

5 Implementations of qudits and algorithms

• Qudit platforms commonly use three or four levels, and the reviewed implementations show potential for extension to higher dimensions and multi-qudit interactions.The section identifies photons, trapped-ion energy levels, NMR nuclear spins, and molecular-magnet spin states as physical degrees of freedom that can represent qudits.Source: §5 · p. 34

5.1 Time and frequency bin of a photon

• A single-photon time-bin and frequency-bin qutrit system demonstrated phase estimation with 98% fidelity and a multivalue-controlled gate.The two degrees of freedom act as control and target qutrits, avoiding the difficulty of deterministic interactions between photons; for arbitrary phases, repeated measurements support statistical eigenvalue estimation.Source: §5.1 · pp. 35–36

5.2 Ion trap

• Trapped-ion qutrits use arbitrary single-qutrit gates and a controlled-NOT gate as a universal set, with the qutrit states encoded in three ion energy levels.Raman transitions manipulate the single-qutrit states, while a conditional two-qutrit gate uses the ions’ shared center-of-mass motion; measurement distinguishes the three states, and the qutrit computer increases available Hilbert space while requiring the same physical resources.Source: §5.2 · pp. 37–38

5.3 Nuclear magnetic resonance

• NMR implementations realize a single-qudit permutation-parity algorithm, with fidelities of 0.92 and above for the proposed operations.One setup uses a spin in a liquid-crystalline environment; a second treats a single ququart using spin-3/2 nuclei and Zeeman-split energy levels.Source: §5.3 · pp. 38–39

6.1 Summary of the advantages of qudit systems compared to qubit systems

• Qudit gates exploit larger Hilbert spaces to reduce representation costs; Muthukrishnan and Stroud’s universal-computation method has a (log2 d)^2 scaling advantage over the qubit case.The review separately reports that Luo and Wang’s scheme provides an extra factor of n reduction in gate requirements.Source: §6.1 · p. 41
• Qudits can offer shorter computation, lower resource requirements, and greater availability, while using extra physical states and enabling multi-level controlled gates that reduce control-gate requirements.The review describes qudit advantages across circuit design, physical implementation, and potential applications; available extra states are noted for photons, superconducting systems, trapped ions, and magnetic and non-magnetic molecules.Source: §6.1 · pp. 41–42
• Qudits can provide greater noise resilience than qubits in quantum communication, with secret-key rate increasing as Hilbert-space dimension rises at the same noise level.The advantage varies by physical apparatus; higher transmission distance may occur, and increasing dimension can make entanglement more robust by reducing the number of systems exposed to local noise. Photonic OAM implementation is cited as an example.Source: §6.1 · pp. 41–42

6.2 Future outlook of qudit system

• Qudit computing may involve harder-to-implement universal gates, benchmarking, gate characterization, and error-correction challenges tied to the qudit Clifford hierarchy [163] [164].The review presents itself as a summary of recent developments and an introduction for newcomers to qudit quantum computing.Source: §6.2 · p. 42
• Research directions include quantum-resource connections, algorithm improvements, scaling to higher dimensions and more particles, benchmarking, error correction, and links between qudits and continuous-variable computing.The review notes that qudits have received less theoretical and experimental attention than qubits, while mathematical challenges from extending qubits to qudits may also yield new insights.Source: §6.2 · p. 42

Funding

• The National Science Foundation supported the work under award 1839191-ECCS, while BCS received support from NSERC and the Alberta Government.Source: Funding · p. 42