Resource-Scalable Fully Quantum Metropolis-Hastings for Integer Linear Programming

TL;DR

This work addresses the NP-hard problem of integer linear programming by introducing a fully quantum Metropolis–Hastings algorithm that operates coherently over the discrete feasible region using reversible quantum circuits with no QRAM. The core idea is a Szegedy-type quantum walk with step operator $W = R\,P^{\dagger}\,F\,P$ that, under an annealing schedule, biases amplitudes toward low-cost feasible solutions and yields a final state close to the Boltzmann distribution $\pi_{\beta}(\mathbf{x}) \propto e^{-\beta f(\mathbf{x})}$. The authors provide explicit logical resource scaling: the total number of qubits scales as $\mathcal{O}(n \log_2 N)$ and the Toffoli-equivalent cost per Metropolis step scales as $\mathcal{O}(k)$, with simulations confirming linear scaling across thousands of random ILP instances. This framework establishes a principled, resource-characterized baseline for fully quantum constraint programming and opens pathways for integrating additional quantum speedups, such as amplitude amplification, as quantum hardware advances. Practically, the approach enables coherent exploration of an exponentially large discrete search space while maintaining polynomial resource growth and robust convergence toward high-quality feasible solutions.

Abstract

Integer linear programming (ILP) remains computationally challenging due to its NP-complete nature despite its central role in scheduling, logistics, and design optimization. We introduce a fully quantum Metropolis-Hastings algorithm for ILP that implements a coherent random walk over the discrete feasible region using only reversible quantum circuits, without quantum-RAM assumptions or classical pre/post-processing. Each walk step is a unitary update that prepares coherent candidate moves, evaluates the objective and constraints reversibly -- including a constraint-satisfaction counter to enforce feasibility -- and encodes Metropolis acceptance amplitudes via a low-overhead linearized rule. At the logical level, the construction uses $\mathcal{O}(n\log_2 N)$ qubits to represent $n$ integer variables over the interval $[-N,\,N-1]$, and the Toffoli-equivalent cost per Metropolis step grows linearly with the total logical qubit count. Using explicit ripple-carry adder constructions, we support linear objectives and mixed equality/inequality constraints. Numerical circuit-level simulations on a broad ensemble of randomly generated instances validate the predicted linear resource scaling and exhibit progressive thermalization toward low-cost feasible solutions under the annealing schedule. Overall, the method provides a coherent, resource-characterized baseline for fully quantum constraint programming and a foundation for incorporating additional quantum speedups in combinatorial optimization.

Figures (10)

• Figure 1: Heat map of the probability distribution obtained after applying the Randomized Quantum Metropolis algorithm (see Sec. \ref{['sec:overview']}) for $20$ steps. Color intensity encodes the probability of measuring the quantum state $\ket{x_1, x_2}$, where each grid cell corresponds to a discrete point in the feasible region defined by the constraint $x_1 + x_2 \ge 0$. The optimization function is given by $f(x_1, x_2) = -2x_1 - x_2$, and the distribution reflects the thermalized process. The problem is solved for 2 qubits of discretization per variable, leaving the possible search space as the square $[-2,1]\times[-2,1]$.
• Figure 2: One iteration of the quantum Metropolis walk. Registers (top to bottom) hold the current position $S$, its cost $F$, a candidate position $S'$, its cost $F'$, the constraint counter $R$ and a coin qubit $C$. The red shaded region encloses the walk operator $W = R\,P^{\dagger}\,F\,P$.
• Figure 3: Quantum circuit implementing two's-complement negation of an $n$-qubit register ($n=3$ shown). The circuit comprises two stages: (i) bit-wise NOT operations (left-hand $X$ gates) that flip all qubits, and (ii) a cascade of multi-controlled NOT gates (shaded region) that increments the result by one, together realizing the two's-complement negation $- x = \overline{x} + 1$.
• Figure 4: Synthesis of the unitary $U_f$ for the two-variable linear form $f(x_{1},x_{2}) = c_{0}-c_{1}x_{1}+c_{2}x_{2}$ with $c_{i}>0$. Register $F$ is first loaded with the constant term $c_{0}$ (red box). Orange block: subtraction of $c_{1}x_{1}$. The value qubits of $x_{1}$ and the global carry-in are flipped with $X$ gates, the sign bit is fan-out via CNOTs to sign-extend $x_{1}$ up to the number of qubits of register $F$, and the CDKM full adder is applied $c_{1}$ times, yielding $c_{0}-c_{1}x_{1}$. Blue block: addition of $c_{2}x_{2}$. Because $c_{2}$ is positive, only sign extension is required before invoking the adder $c_{2}$ times. Upon completion the accumulator holds $c_{0}-c_{1}x_{1}+c_{2}x_{2}$ and all ancillas (carry and extension lines) are returned to $\ket{0}$, ensuring that $U_f$ is reversible.
• Figure 5: Explicit circuit for the operator $V$ of the one-step operator $W$. For every candidate vector $\mathbf{x'}$ in $S'$ the unitary first computes the constraint values in the workspace register $F'$ and then updates the counter register $R$. Controls on the equality blocks $H_j$ act on all qubits of $F'$, whereas the inequality blocks $G_i$ need only the sign qubit. The final state of $R$ encodes the number of satisfied constraints, being $m{+}p$ for feasible points.

Theorems & Definitions (11)

• Theorem 1: Resource-efficient equality elimination
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Corollary 1
• proof
• Theorem 3
• Theorem 4