Measurement-driven Quantum Approximate Optimization
Tobias Stollenwerk, Stuart Hadfield
TL;DR
This work presents measurement-driven quantum optimization (MDQO), a framework that uses mid-circuit weak measurements mediated by an ancilla to implement a transformed cost Hamiltonian $C=\mathcal{E}(H)$ and amplify high-cost configurations for combinatorial problems. By ensuring $0\le C\le \pi/4$, the per-step success probability $p_1$ is kept strictly above $\tfrac{1}{2}$, enabling iterative improvement and potential integration as post-processing for existing quantum circuits such as QAOA. The authors extend MDQO to constrained optimization via penalty-based and feasibility-preserving strategies, showing the latter preserves feasibility and reduces overhead, with numerical examples on MIS illustrating advantages. They further augment MDQO with feedback-controlled scrambling operators to escape stagnation, providing a practical multi-layer algorithm that remains feasible on NISQ devices and adaptable to future fault-tolerant architectures. Overall, MDQO offers a general, adaptive tool for obtaining samples from improved solution distributions, either standalone or as a subroutine to enhance parameterized quantum circuits.
Abstract
Algorithms based on non-unitary evolution have attracted much interest for ground state preparation on quantum computers. One recently proposed method makes use of ancilla qubits and controlled unitary operators to implement weak measurements related to imaginary-time evolution. In this work we specialize and extend this approach to the setting of combinatorial optimization. We first generalize the algorithm from exact to approximate optimization, taking advantage of several properties unique to classical problems. In particular we show how to select parameters such that the success probability of each measurement step is bounded away from $1/2$. We then show how to adapt our paradigm to the setting of constrained optimization for a number of important classes of hard problem constraints. For this we compare and contrast both penalty-based and feasibility-preserving approaches, elucidating the significant advantages of the latter approach. Our approach is general and may be applied to easy-to-prepare initial states as a standalone algorithm, or deployed as a quantum postprocessing stage to improve performance of a given parameterized quantum circuit. We then propose a more sophisticated variant of our algorithm that adaptively applies a mixing operator or not, based on the measurement outcomes seen so far, as to speeds up the algorithm and helps the system evolution avoid slowing down or getting stuck suboptimally. In particular, we show that mixing operators from the quantum alternating operator ansatz can be imported directly, both for the necessary eigenstate scrambling operator and for initial state preparation, and discuss quantum resource tradeoffs.