Imposing Constraints on Driver Hamiltonians and Mixing Operators: From Theory to Practical Implementation

TL;DR

This paper develops algebraic methods for constructing constrained driver Hamiltonians and mixing operators in quantum algorithms, analyzes their complexity, and demonstrates their effectiveness through empirical benchmarks on SAT problems.

Contribution

It introduces a general algebraic framework for constrained operators, proves NP-Completeness for their general construction, and provides practical algorithms for tailored quantum ansaetze.

Findings

Tailored ansaetze outperform standard X-mixer in benchmarks.

Quadratic speedup observed with alternative ansaetze.

Algorithms enable construction of constrained operators for complex problems.

Abstract

Driver Hamiltonians and Mixing Operators that satisfy constraints is an important part of ansatz construction for many quantum algorithms. In this manuscript, we give general algebraic expressions for finding Hamiltonian terms and analogously unitary primitives, that satisfy constraint embeddings and use these to give complexity characterizations of the related problems. Finding operators that enforce classical constraints is proven to be NP-Complete in the general case; we also give an algorithmic procedure with worse-case polynomial runtime to find any operators with a constant locality bound - a useful result since many constraints imposed admit local operators to enforce them in practice. We then give algorithmic procedures to turn these algebraic primitives into Hamiltonian drivers and unitary mixers that can be used for Constrained Quantum Annealing (CQA) and Quantum Alternating Operator Ansatz (QAOA) constructions by tackling practical problems related to finding an appropriate set of reduced generators and defining corresponding drivers and mixers accordingly. We apply these concepts to the construction of ansaetze for 1-in-3 SAT instances. We consider the ordinary x-mixer QAOA, a novel QAOA approach based on the maximally disjoint subset, and a QAOA approach based on the disjoint subset as well as higher order constraint satisfaction terms. We empirically benchmark these approaches on instances sized between $ 12 $ and $ 22 $, showing the best relative performance for the tailored ansaetze and that exponential curve fits on the results are consistent with a quadratic speedup by utilizing alternative ansaetze to the X-mixer. We provide general algorithmic prescriptions for finding driver or mixing terms that satisfy embedded constraints that can be utilized to probe quantum speedups for constraints problems with linear, quadratic, or even higher order polynomial constraints.