[Paper Review] Quantum Optimization

Tad Hogg, Dmitriy Portnov|ArXiv.org|Jun 20, 2000

Quantum Computing Algorithms and Architecture23 citations

TL;DR

This paper proposes a quantum heuristic algorithm for combinatorial optimization that shifts amplitude toward low-cost states using cost-dependent phase adjustments and state mixing, without requiring prior knowledge of the minimum cost. Simulations show it concentrates amplitude on low-cost solutions in overconstrained SAT and asymmetric TSP, performing comparably to classical heuristics on small instances despite exponential classical simulation cost.

ABSTRACT

We present a quantum algorithm for combinatorial optimization using the cost structure of the search states. Its behavior is illustrated for overconstrained satisfiability and asymmetric traveling salesman problems. Simulations with randomly generated problem instances show each step of the algorithm shifts amplitude preferentially towards lower cost states, thereby concentrating amplitudes into low-cost states, on average. These results are compared with conventional heuristics for these problems.

Motivation & Objective

• To develop a quantum algorithm that improves search efficiency for optimization problems by leveraging state cost information.
• To extend quantum amplitude amplification techniques beyond decision problems to optimization, where the minimum cost is unknown a priori.
• To evaluate the performance of this heuristic quantum approach on overconstrained satisfiability and asymmetric traveling salesman problems.
• To compare the quantum algorithm’s behavior and efficiency with classical heuristics in terms of solution quality and computational cost.
• To explore whether quantum amplitude mixing can effectively exploit problem structure in optimization, despite limitations in coherence and lack of optimality verification.

Proposed method

• The algorithm initializes a uniform superposition over all search states using equal amplitudes for all 2^n configurations.
• It applies iterative steps combining a phase adjustment matrix P^(h) that encodes state costs via p_c^(h) = e^{iπρ_h c}, and a mixing matrix U^(h) = W T^(h) W that redistributes amplitudes based on Hamming distance.
• The mixing matrix U^(h) depends on the Hamming distance d(r,s) between states r and s, with U^(h)_{rs} = u_d^(h) = (-i tan(πτ_h/2))^d up to phase and normalization.
• The phase parameters ρ_h and τ_h are problem-class specific constants that control the rate of amplitude redistribution toward lower-cost states.
• Each trial consists of j such steps, followed by measurement, yielding a state with probability |ψ_s^(j)|², and multiple trials are used to increase success likelihood.
• The algorithm is heuristic and incomplete: it does not guarantee finding the optimal solution nor verify when one is found.

Experimental results

Research questions

• RQ1 Can quantum amplitude manipulation be effectively used to guide search toward low-cost states in optimization problems without prior knowledge of the minimum cost?
• RQ2 How does the performance of this cost-based quantum optimization algorithm compare to classical heuristics like GSAT and branch-and-bound on overconstrained SAT and ATSP?
• RQ3 To what extent does the algorithm exploit problem structure, particularly in the representation of search states and mixing operations?
• RQ4 What is the trade-off between computational cost and solution quality in this quantum heuristic, especially under coherence constraints?
• RQ5 Can the algorithm’s behavior be improved by re-encoding the problem to better reflect structural similarities, such as common edges in TSP tours?

Key findings

• The algorithm successfully shifts amplitude toward lower-cost states on average, with each iteration preferentially concentrating amplitude in low-cost regions of the search space.
• For overconstrained SAT, the quantum algorithm achieves performance comparable to classical heuristics like GSAT, though classical simulation costs grow exponentially with problem size.
• In asymmetric TSP, the algorithm outperforms random selection and shows no dependence on the standard deviation of intercity distances, indicating robustness to cost distribution.
• The classical simulation cost of the quantum algorithm grows exponentially, precluding evaluation on larger instances, highlighting the need for quantum hardware.
• The algorithm’s performance is sensitive to phase parameter choices and problem representation; non-optimal parameters or poor encoding may reduce effectiveness.
• Despite its heuristic nature, the algorithm requires at most linearly increasing coherent steps with problem size, making it less demanding on coherence than amplitude amplification.