Quadratic Unconstrained D-ary Optimization (QUDO)

• QUDO is defined as minimizing quadratic functions over d-ary variables, generalizing QUBO to model complex combinatorial tasks with native qudit encoding.
• Direct qudit encoding in QUDO eliminates extraneous penalty terms, yielding smoother energy landscapes and enhanced resource efficiency.
• QUDO supports variational quantum and classical algorithms, with tailored tensor network solvers demonstrating polynomial scalability for structured instances.

Quadratic Unconstrained D-ary Optimization (QUDO) refers to the minimization of quadratic functions over $N$ discrete variables, each taking $d$ possible values. QUDO generalizes Quadratic Unconstrained Binary Optimization (QUBO)—the standard formulation with binary variables—to the d-ary case, enabling a more natural and resource-efficient direct encoding of many combinatorial optimization problems. QUDO underpins the representation, analysis, and hardware implementation of a wide class of NP-hard discrete optimization tasks, including Max-$K$-Cut, graph coloring, vehicle routing, and the Traveling Salesman Problem, particularly in quantum and quantum-inspired computational paradigms (Zaborniak et al., 2023, Bhat et al., 7 Feb 2026, Pramanik et al., 2020, Ali et al., 2023).

1. Formal Definition and Model Structure

Quadratic Unconstrained D-ary Optimization is defined as the problem of minimizing a function of the form: $$x_i\in\{0,1,...,d-1\} \quad (i=1,...,N), \quad E(\mathbf x) = \sum_{i<j} J_{ij}^{x_i,x_j} + \sum_{i} h_i^{x_i},$$ where $J_{ij}^{a,b}$ and $h_i^a$ are real coefficients and $E(\mathbf{x})$ encodes quadratic interactions (and local fields) over $N$ d-ary variables. This model includes many classical optimization instances: for $d=2$, it reduces to the standard QUBO formalism; for $d>2$, it encompasses problems such as Max-$d$-Cut, multi-partite graph clustering, and multi-valued assignment (Bhat et al., 7 Feb 2026, Pramanik et al., 2020). The QA/QC implementation typically leverages direct qudit encodings, with each variable represented as a local register in a $d$-dimensional Hilbert space $\mathcal{H}_i \simeq \mathbb{C}^d$ (Bhat et al., 7 Feb 2026).

2. Encodings: QUDO, QUBO, and Penalty-based Reductions

Traditional solution methods for d-ary quadratic optimization cast the problem as a QUBO via explicit encoding. The most prominent mappings are:

• One-hot encoding: Each d-ary variable $x_i$ is encoded by $d$ binary sub-variables, with a constraint enforcing that exactly one bit is set per register:

$\sum_{\alpha=0}^{d-1}b_{i,\alpha}=1.$

The QUBO Hamiltonian includes quadratic penalty terms with tunable weight $\lambda$ to enforce the constraint and penalize invalid (non-one-hot) configurations (Zaborniak et al., 2023).

• Domain-wall (unary) encoding: Each $x_i$ is mapped to $d-1$ binary variables, with a single $1\rightarrow0$ domain wall indicating the variable's value and fixed boundaries. The encoding constraint is enforced via penalty terms constructed to favor valid transitions (Zaborniak et al., 2023).
• Direct QUDO (qudit) encoding: Each $x_i$ is represented directly by a qudit, eliminating the need for auxiliary binary variables and associated penalty Hamiltonians. Constraints such as exclusion or one-of-$d$ are encoded through diagonal projectors or simple local penalties (Bhat et al., 7 Feb 2026, Pramanik et al., 2020).

A compact summary is provided below:

Encoding	Variables per Site	Native Constraint Enforcement	Penalty Required	(Bhat et al., 7 Feb 2026, Zaborniak et al., 2023)
QUBO–One-hot	$d$ qubits	No	Yes ($\lambda$)
QUBO–Domain-wall	$d-1$ qubits	No	Yes ($\lambda$)
QUDO (qudit)	1 $d$-level qudit	Yes (diagonal in basis)	No

Neither one-hot nor domain-wall encodings are free from issues: mapping to binary QUBOs introduces an artificial enlargement of the solution space, producing invalid states requiring additional penalty terms and associated computational overhead (Zaborniak et al., 2023). By contrast, direct QUDO representations natively encode feasible configurations, compacting the Hilbert space and eliminating the need for most penalty constructions (Bhat et al., 7 Feb 2026).

3. Variational Quantum and Classical Algorithms for QUDO

In quantum computing, QUDO is implemented using variational algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) generalized to qudit hardware. The ansatz alternates between:

• Cost unitary: $$U_C(\gamma_\ell) = \exp(-i\gamma_\ell H_{\mathrm{QUDO}})$$, applying the problem Hamiltonian at step $\ell$.
• Mixer unitary: $U_M(\beta_\ell) = \exp(-i\beta_\ell H_M)$, where $H_M$ typically implements global cyclic qudit shifts facilitating exploration.

The initial state is the uniform superposition in the computational basis, prepared via the $d$-dimensional discrete Fourier transform ($F_d$). The variational landscape is then explored and minimized classically (e.g. COBYLA, Nelder–Mead) to optimize the QAOA parameters.

The qudit-based QUDO approach yields several advantages:

• Direct qudit encoding enables native enforcement of cardinality and exclusion constraints, sidestepping the penalty term complexity inherent in QUBO mappings.
• Mixing and cost unitaries leverage the algebraic structure of generalized Pauli operators: $U_d = \sum_j \omega^j|j\rangle\langle j|$ and $V_d = \sum_j |j+1\bmod d\rangle\langle j|$ (with $\omega = e^{2\pi i/d}$) (Pramanik et al., 2020).
• The circuit depth and resource demands scale with $N$ and $|E|$ (number of variables and quadratic couplings), not with the binary expansion size (Bhat et al., 7 Feb 2026, Pramanik et al., 2020).

For specialized QUDO instances with tridiagonal (one-neighbor) structure, polynomial-time tensor network (MPO/MPS) solvers have been developed, applying imaginary-time evolution and sequential partial tracing to efficiently extract optimal assignments. The total complexity is $O(N d^2)$, which is polynomial in both variables and alphabet size (Ali et al., 2023).

4. Solution Landscape, Penalty Tuning, and Hardware Implications

The topology of the optimization landscape is highly sensitive to encoding and penalty parameter selection:

• One-hot encoding: Valid assignments are isolated in Hamming space, separated by at least two bit flips; with sufficiently large penalty, spurious local minima (invalid solutions) can be eliminated. However, increased $\lambda$ steepens the global energy landscape, raising barriers and potentially trapping heuristics or quantum processes in invalid regions (Zaborniak et al., 2023).
• Domain-wall encoding: Reduces resource requirements, but some invalid minima persist regardless of penalty strength due to the topology of the encoding; 1-bit flips may not always increase penalty energy for all invalid states.
• Native QUDO: By construction, assigns nonzero amplitude exclusively to feasible states with no need for penalty landscapes, yielding denser low-energy feasible solutions and smoother optimization (Bhat et al., 7 Feb 2026).

Hardware constraints further affect encoding choices: one-hot encoding is qubit- and coupler-intensive, while domain-wall and QUDO approaches yield substantial savings. Excessively steep penalty terms degrade annealer and variational performance due to hardware dynamic range limits (Zaborniak et al., 2023).

5. Benchmarking, Resource Scaling, and Empirical Observations

Extensive benchmarking across canonical combinatorial problems reveals systematic advantages of QUDO (qudit-level) formulations:

• For TSP (5–6 cities, $p=2$ QAOA layers): QUDO QAOA achieves unity approximation ratio and feasible output fractions orders of magnitude above QUBO QAOA, with runtimes improved by a factor $\sim 10^3$.
• For Max-$K$-Cut ($N=7,K=4$, $p=1$): QUDO produces exact feasible solutions in all runs, compared to $<1\%$ feasibility and significantly degraded approximation in QUBO.
• For graph coloring and job scheduling, QUDO consistently reaches full solution feasibility and high-quality optima at low depth, while QUBO shows steep declines in feasible output and increasingly impractical runtimes (Bhat et al., 7 Feb 2026).

Scalability:

• QUDO requires $N$ qudits, Hilbert space of size $d^N$; QUBO with one-hot encoding requires $N d$ qubits, $2^{N d}$ states.
• As $N$ grows, QUBO’s fraction of feasible states collapses exponentially ($\sim$ $2^{-N d}\,$), while QUDO preserves polynomially small, but robust feasible sample probability ($\sim \mathcal{O}(1/N)$ at worst) (Bhat et al., 7 Feb 2026).
• QUDO’s smoother variational landscapes contribute both to faster convergence in QAOA and superior “classical” solution reach.

6. Specialized Algorithms and Complexity for Structured Instances

For tridiagonal QUDO models (one-neighbor quadratic interactions), explicit quantum-inspired polynomial algorithms are available:

• The quantum-inspired tensor network method constructs the imaginary-time-evolved Boltzmann state as an MPO of bond dimension $d$, then implements sequential partial trace contractions to sample the maximizing assignment $\mathbf{x}^{\text{opt}}$.
• The total runtime is $O(N d^2)$ for $N$ sites and alphabet size $d$ (Ali et al., 2023).
• Degeneracies are handled by tracking maximally valued entries at each step and enumerating solution branches accordingly.
• The provided explicit pseudocode details the assembly of “S-tensors”, (right) environment contraction, marginal maximization, and conditioning steps for reproducibility (Ali et al., 2023).

7. Best Practices, Constraints, and Future Directions

Best-practice recommendations for QUDO problem solving include:

• Encode d-ary variables natively as qudits to maximize resource efficiency and minimize extraneous local minima.
• For QUBO-based workflows, use one-hot encoding when rigorous elimination of invalid minima is necessary and resource budget allows; use domain-wall or hybrid encodings for tight resource constraints, with strong post-processing as required (Zaborniak et al., 2023).
• Penalty strength ($\lambda$) tuning is nontrivial: increment in small steps above heuristic lower bounds, balance shallow feasible minima against suppression of invalid minima, and always post-process near-feasible samples to local optimality (Zaborniak et al., 2023).
• For highly structured problems (linear chains, 1D models), deploy tensor network contraction algorithms for polynomial-classical solution.

Current limitations include the lack of widely available qudit-based quantum hardware and annealers; binary encodings may remain necessary where only qubit platforms are accessible (Pramanik et al., 2020). The QUDO paradigm nevertheless opens opportunities for more expressive variational ansatz, enhanced scalability, and more accurate modeling of constraint-rich combinatorial problems as quantum hardware evolves.

---

Key references substantiating the above:

• Discrete quadratic model solution landscapes, with detailed encoding and penalty landscape analysis (Zaborniak et al., 2023)
• Systematic benchmarking and advantages of QUDO versus QUBO formulations on quantum hardware (Bhat et al., 7 Feb 2026)
• QUDO formalism and variational circuit construction for Max–$d$–cut and related tasks (Pramanik et al., 2020)
• Exact polynomial-time tensor network solvers for tridiagonal QUDO (Ali et al., 2023)