Virtual Qudits in Quantum Computing

• Virtual qudits are high-dimensional quantum systems that map multiple logical qubits to a single physical degree of freedom using techniques like time-bin photonics and trapped-ion encodings.
• They enable scalable quantum algorithms by employing specialized gate decompositions and QND measurement strategies on both photonic and qubit-based platforms.
• Resource savings and circuit efficiency improvements are achieved through dimension-lifted mappings and optimized control protocols across various hardware architectures.

Virtual qudits are high-dimensional, abstract quantum systems realized either as mappings within physical qudit architectures or emulated atop qubit-based hardware. These constructs enable efficient encoding and manipulation of multiple logical qubits within a single degree of freedom, as well as simulation of genuinely d-ary quantum algorithms on conventional qubit platforms. The key methodologies span photonic time-bin mappings, trapped-ion internal state encodings, and dimension-lifted circuit constructions. Virtual qudits expand the algorithmic possibilities, facilitate resource reduction in device implementation, and support scalable architectures for quantum information processing.

1. Formal Definitions and Encodings

A virtual qudit of dimension $d$ is typically realized by organizing $\log_2 d$ physical qubits (for $d$ a power of two) or by partitioning the accessible states of a physical system. In hardware, this encompasses:

• Time-bin photonic qudits: Here, the computational basis $\{|0\rangle, \ldots, |d-1\rangle\}$ is identified with a single photon occupying one of $d=2^N$ temporal bins $|t_k\rangle$, permitting encoding of $N$ logical qubits per photon (Delteil, 8 Oct 2024).
• Trapped-ion qudits: Logical qubits are embedded within the low-lying energy levels of an ion. For example, a qutrit ($d=3$) hosts one logical qubit in $\{\ket{0}_Q, \ket{1}_Q\}$, with $\ket{2}_Q$ as an auxiliary state; a ququart ($d=4$) hosts two qubits through explicit binary mapping (Nikolaeva et al., 2023).
• Virtual qudits on qubit hardware: The Hilbert space $\mathbb{C}^d$ is mapped to the tensor product space of $\ell = \log_2 d$ physical qubits, where each basis state $\ket{k}_d$ is realized as $$\ket{k^{(0)}} \otimes \cdots \otimes \ket{k^{(\ell-1)}}$$ (Semre et al., 7 Dec 2025).

These mappings are fundamental to leveraging the high-dimensionality of qudits for quantum computation and simulation purposes.

2. Gate Implementation and Universal Logic

Logical qubit and multi-qubit operations within virtual qudits are engineered using specialized gate decompositions, physical interactions, or modular circuit constructions:

• Time-bin qudit logic: Universal $N$-qubit unitaries are built through a sequence of $d-1$ beam-splitter plus phase-shifter elements, decomposed as

$$U = B_{d,d-1}(\theta_{d,d-1},\phi_{d,d-1}) \cdots B_{2,1}(\theta_{2,1},\phi_{2,1}),$$

where each $B_{k,\ell}$ acts on two time bins (Delteil, 8 Oct 2024). Arbitrary single- and two-qubit gates (CNOT, CZ) are performed with $O(N)$ modules comprising interferometers, delay lines, and fast optical switches.

• Trapped-ion qudit gates: Each ion's multi-level structure supports pairwise rotations $R^{ij}_\phi(\theta)$ and phase gates $\text{Ph}^k(\alpha)$, enabling SU(2) unitaries on logical qubit subspaces. Entangling gates are realized via the Mølmer–Sørensen interaction,

$$\widetilde{\text{MS}}_\phi^{ij,k\ell}(\chi) = \exp[-i \chi \sigma_\phi^{ij} \otimes \sigma_\phi^{k\ell}],$$

supporting both intra- and inter-qudit CNOT, CZ, and generalized Toffoli gates (Nikolaeva et al., 2023).

• Virtual qudit gates on qubit devices: Logical $X_d$ (modular increment), $Z_d$ (qudit phase), controlled operations, and the quantum Fourier transform $\text{QFT}_d$ are effected using standard qubit circuits—ripple-carry adders for $X_d$, single-qubit rotations for $Z_d$, multi-controlled Toffoli/CNOTs for modular control, and staggered Hadamard plus controlled-phase sequences for $\text{QFT}_d$ (Semre et al., 7 Dec 2025).

3. Measurement, Read-Out, and Quantum Non-Demolition Protocols

Measurement of virtual qudits and their logical constituents employs basis mapping and, where available, QND techniques:

• Time-bin photonic qudits: Full computational-basis read-out is accomplished by detecting the arrival time of a photon, directly exposing the binary encoding of each logical qubit. QND measurement of individual qubits within the qudit utilizes a cavity-QED interface: a two-level spin in a cavity applies a conditional phase to reflected time bins, and subsequent spin measurement accesses a chosen logical qubit nondestructively (Delteil, 8 Oct 2024).
• Trapped-ion qudits: State read-out typically operates via fluorescence detection keyed to specific transitions, with leakage errors managed by pulse shaping and level shielding techniques (Nikolaeva et al., 2023).
• Virtual qudits on qubit arrays: Observables corresponding to computational basis states are measured via projective read-out of the underlying qubit layers, providing direct access to the logical qudit value (Semre et al., 7 Dec 2025).

A plausible implication is that QND strategies foster more advanced quantum protocols, especially in scalable photonic and hybrid architectures.

4. Resource Scaling and Circuit Efficiency

Architectural overhead and gate complexity scale favorably in virtual qudit paradigms relative to traditional qubit-based designs:

• Time-bin qudits: Arbitrary state preparation demands $O(2^N)$ elements for full programmability, but all logical single- and two-qubit operations require only $O(N)$ linear-optics modules. The total optical complexity is $aN$ (switches) + $bN$ (interferometers) for $a,b=O(1)$ (Delteil, 8 Oct 2024).
• Trapped-ion qudits: Circuit depth for $N$-qubit Toffoli gates with $d$-level ions is reduced to $O(N)$ or $O(\log N)$ native Mølmer–Sørensen gates, in contrast to $O(N^2)$ for qubit-only approaches. Gate count and error advantage are quantified as $E \approx N e$, with cost-benefit realized when

$$\frac{e_\text{qudit}}{e_\text{qubit}} < \frac{N_\text{qubit}}{N_\text{qudit}} = \kappa$$

and $\kappa \gtrsim 3–6$ in most Toffoli cases (Nikolaeva et al., 2023).

• Virtual qudits on qubits: Simulating a $d$-dimensional qudit incurs $O(\log^2 d)$ gate overhead for QFT preparation, but oracle calls are parallelized. Total circuit depth for algorithms such as Simon's problem is $O(\ell)+\mathrm{Depth}[U_f]$, where $\ell = \log_2 d$ (Semre et al., 7 Dec 2025).

This suggests that virtual qudit schemes provide significant scalability and connectivity improvements.

5. Simulation of Qudit Algorithms on Qubit Hardware

Dimension-lifting and pack/unpack protocols allow simulation of qudit-specific algorithms on qubit-only platforms:

• Hilbert space mapping: For $d=2^{\ell}$, each virtual qudit is represented by $\ell$ qubits, with basis states organized according to binary decomposition.
• Circuit primitives: Qudit gates ($X_d$, $Z_d$, controlled operations, $\text{QFT}_d$) are constructed using standard qubit circuits, preserving logical measurement statistics and algorithmic fidelity.
• Dimension-lifted oracle: The $U_f^{(d)}$ operation relevant to lifted Simon's problem is implemented via $\ell$ parallel applications of the binary Simon oracle on each layer of the virtual qudit (Semre et al., 7 Dec 2025).
• Output statistics: Numerical simulation confirms uniform measurement distributions over the orthogonal hidden subspace $$S^\perp = \{y\in \mathbb{Z}_d^n : y\cdot s\equiv 0\ (\bmod\ d)\}$$, with empirical results for several $(d, n)$ settings validating theoretical predictions.

A plausible implication is that virtual qudit simulation expands the repertoire of testable quantum algorithms beyond current hardware constraints.

6. Comparison of Platforms and Practical Considerations

Platform/Approach	Logical Qubit Packing	Gate Complexity	Unique Challenges
Photonic time-bin qudit	$N$ qubits per photon	$O(N)$ for logic	Fast switching, single-photon detection, QND via cavity-QED
Trapped-ion $d$-level qudit	Up to $3$ qubits per ion ($d=8$)	$O(N), O(\log N)$ for Toffoli	Calibration of transitions, leakage error suppression
Qubit-based virtual qudit	$\ell = \log_2 d$ qubits per logical qudit	$O(\ell^2)$ QFT, parallelized oracle	Layered circuit management, gate depth overhead

These approaches embody the principal strategies for leveraging virtual qudit frameworks in quantum information tasks, with platform-specific optimizations and challenges codified in practice.

7. Scalability, Interfacing, and Future Directions

• Photonic architectures: Interfacing distinct time-bin qudit processors via matter qubits in optical resonators enables advanced functionality—such as single-qubit QND, two-qudit entanglement, and distributed quantum network integration. Delay lengths $2^i T$ and optical switches encode logical structure, while cavity-QED systems mediate scalable entangling gates (Delteil, 8 Oct 2024).
• Trapped-ion arrays: Embedding multiple logical qubits per ion and exploiting higher-dimensional MS-gate interactions reduce total ion count and circuit entangling gate numbers. Ongoing experiments report high-fidelity operations up to $d=8$, supporting large-scale compiler architectures (Nikolaeva et al., 2023).
• Qubit hardware simulation: Simon's algorithm and other d-ary protocols can be tested natively via dimension-lifted mappings, allowing researchers to empirically investigate algorithmic performance in high-dimensional settings without requiring new hardware (Semre et al., 7 Dec 2025).

Editor's term: "dimension-lifted algorithms" captures the strategy of algorithmic simulation via virtual qudit embedding, highlighting its significance for contemporary quantum software research.

In sum, virtual qudit methodologies substantively broaden quantum information processing capabilities both at the hardware and algorithmic levels, with scalable protocols, favorable resource scaling, and demonstrated utility across photonic, ion-trap, and qubit-array platforms.