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arXiv:2511.08362

An Information-Minimal Geometry for Qubit-Efficient Optimization

Gordon Ma, Dimitris G. Angelakis

quant-phcs.LGSubmitted Nov 11, 2025• Announced Nov 12, 2025• 0 citations• Revised Jan 12, 2026 (v2)
v1 · Nov 11, 2025v2 · Jan 12, 2026
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Abstract

Qubit-efficient optimization studies how large combinatorial problems can be addressed with quantum circuits whose width is far smaller than the number of logical variables. In quadratic unconstrained binary optimization (QUBO), objective values depend only on one- and two-body statistics, yet standard variational algorithms explore exponentially large Hilbert spaces. We recast qubit-efficient optimization as a geometric question: what is the minimal representation the objective itself requires? Focusing on QUBO problems, we show that enforcing mutual consistency among pairwise statistics defines a convex body -- the level-2 Sherali-Adams polytope -- that captures the information on which quadratic objectives depend. We operationalize this geometry in a minimal variational pipeline that separates representation, consistency, and decoding: a logarithmic-width circuit produces pairwise moments, a differentiable information projection enforces local feasibility, and a maximum-entropy ensemble provides a principled global decoder. This information-minimal construction achieves near-optimal approximation ratios on large unweighted Max-Cut instances (up to N=2000) at shallow depth, indicating that pairwise polyhedral geometry already captures the relevant structure in this regime. By making the information-minimal geometry explicit, this work establishes a clean baseline for qubit-efficient optimization and sharpens the question of where genuinely quantum structure becomes necessary.

Comments
39 pages, 9 figures. v2: Revised for clarity

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Qubit-efficient optimization studies how large combinatorial problems can be addressed with quantum circuits whose width is far smaller than the number of logical variables. In quadratic unconstrained binary optimization (QUBO), objective values depend only on one- and two-body statistics, yet standard variational algorithms explore exponentially large Hilbert spaces. We recast qubit-efficient optimization as a geometric question: what is the minimal representation the objective itself requires?
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This information-minimal construction achieves near-optimal approximation ratios on large unweighted Max-Cut instances (up to N=2000) at shallow depth, indicating that pairwise polyhedral geometry already captures the relevant structure in this regime. By making the information-minimal geometry explicit, this work establishes a clean baseline for qubit-efficient optimization and sharpens the question of where genuinely quantum structure becomes necessary.
Combinatorial optimization problems are typically formulated using Quadratic Unconstrained Binary Optimization (QUBO), where constraints are enforced through penalty terms that introduce auxiliary variables and rapidly increase Hamiltonian complexity, limiting scalability on near term quantum devices. In this work, we systematically study Quadratic Unconstrained D-ary Optimization (QUDO) as an alternative formulation in which decision variables are encoded directly in higher dimensional Hilbert spaces. We demonstrate that QUDO naturally captures structural constraints across a range of problem classes, including the Traveling Salesman Problem, two variants of the Vehicle Routing Problem, graph coloring, job scheduling, and Max-K-Cut, without the need for extensive penalty constructions. Using a qudit-level implementation of the Quantum Approximate Optimization Algorithm (qudit QAOA), we benchmark these formulations against their binary QUBO counterparts and exact classical solutions. Our study show consistently improved approximation ratios and substantially reduced computational overhead at comparable circuit depths, highlighting QUDO as a scalable and expressive representation for quantum combinatorial optimization.
We propose and analyze a set of variational quantum algorithms for solving quadratic unconstrained binary optimization problems where a problem consisting of nc classical variables can be implemented on O(log⁡nc) number of qubits. The underlying encoding scheme allows for a systematic increase in correlations among the classical variables captured by a variational quantum state by progressively increasing the number of qubits involved. We first examine the simplest limit where all correlations are neglected, i.e. when the quantum state can only describe statistically independent classical variables. We apply this minimal encoding to find approximate solutions of a general problem instance comprised of 64 classical variables using 7 qubits. Next, we show how two-body correlations between the classical variables can be incorporated in the variational quantum state and how it can improve the quality of the approximate solutions. We give an example by solving a 42 -variable Max-Cut problem using only 8 qubits where we exploit the specific topology of the problem. We analyze whether these cases can be optimized efficiently given the limited resources available in state-of-the-art quantum platforms. Lastly, we present the general framework for extending the expressibility of the probability distribution to any multi-body correlations.
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Quantum computing offers significant potential for solving NP-hard combinatorial (optimization) problems that are beyond the reach of classical computers. One way to tap into this potential is by reformulating combinatorial problems as a quadratic unconstrained binary optimization (QUBO) problem. The solution of the QUBO reformulation can then be addressed using adiabatic quantum computing devices or appropriate quantum computing algorithms on gate-based quantum computing devices. In general, QUBO reformulations of combinatorial problems can be readily obtained by properly penalizing the violation of the problem's constraints in the original problem's objective. However, characterizing tight (i.e., minimal but sufficient) penalty coefficients for this purpose is important and non-trivial for enabling the solution of the resulting QUBO in current and near-term quantum computing devices. Along these lines, we present closed-form characterizations of tight penalty coefficients for two distinct QUBO reformulations of the max kkk-cut problem whose values depend on the (weighted) degree of the vertices of the graph defining the problem. These findings contribute to the ongoing effort to make quantum computing a viable tool for solving combinatorial problems at scale. We support our theoretical results with illustrative examples and simple numerical results.
Quantum computers hold the promise of more efficient combinatorial optimization solvers, which could be game-changing for a broad range of applications. However, a bottleneck for materializing such advantages is that, in order to challenge classical algorithms in practice, mainstream approaches require a number of qubits prohibitively large for near-term hardware. Here we introduce a variational solver for MaxCut problems over m=O(nk)m={{\mathcal{O}}}({n}^{k})m=O(nk) binary variables using only n qubits, with tunable k > 1. The number of parameters and circuit depth display mild linear and sublinear scalings in m, respectively. Moreover, we analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature. Altogether, this leads to high quantum-solver performances. For instance, for m = 7000, numerical simulations produce solutions competitive in quality with state-of-the-art classical solvers. In turn, for m = 2000, experiments with n = 17 trapped-ion qubits feature MaxCut approximation ratios estimated to be beyond the hardness threshold 0.941. Our findings offer an interesting heuristics for quantum-inspired solvers as well as a promising route towards solving commercially-relevant problems on near-term quantum devices. Combinatorial optimization problems might be solvable more efficiently with quantum computing, but near-term quantum optimization solvers are limited in qubit size. Here, the authors propose and demonstrate a variational quantum algorithm that polynomially reduces the qubit overhead for applications like MaxCut.