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

Hannes Leipold, Federico M. Spedalieri, Stuart Hadfield, Eleanor Rieffel

TL;DR

This work develops a comprehensive algebraic framework for enforcing classical constraints in quantum optimization by constructing constraint-commuting drivers and mixers. It proves general NP-hardness for finding such operators in the general case while showing polynomial-time solvability for locality-bounded constraints, and provides practical backtracking and generator-sequencing algorithms to build constrained QAOA/CQA ansatze. The authors apply the theory to 1-in-3 SAT, introducing three tailored ansatz constructions (X-mixer baseline, maximum disjoint subset, and neighborhood-symmetric mixers) and demonstrate a quadratic improvement in scaling over problem sizes $12$–$22$ in simulations, indicating substantial potential speedups for constrained optimization on quantum hardware. The work offers algorithmic prescriptions and a detailed pipeline from algebraic conditions to implementable unitaries, with implications for improved noise resilience, trainability, and resource efficiency in constrained quantum optimization. These methods pave the way for systematic exploration of quantum speedups in constraint-rich problems across QA, QAOA, and beyond.

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.

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

TL;DR

This work develops a comprehensive algebraic framework for enforcing classical constraints in quantum optimization by constructing constraint-commuting drivers and mixers. It proves general NP-hardness for finding such operators in the general case while showing polynomial-time solvability for locality-bounded constraints, and provides practical backtracking and generator-sequencing algorithms to build constrained QAOA/CQA ansatze. The authors apply the theory to 1-in-3 SAT, introducing three tailored ansatz constructions (X-mixer baseline, maximum disjoint subset, and neighborhood-symmetric mixers) and demonstrate a quadratic improvement in scaling over problem sizes – in simulations, indicating substantial potential speedups for constrained optimization on quantum hardware. The work offers algorithmic prescriptions and a detailed pipeline from algebraic conditions to implementable unitaries, with implications for improved noise resilience, trainability, and resource efficiency in constrained quantum optimization. These methods pave the way for systematic exploration of quantum speedups in constraint-rich problems across QA, QAOA, and beyond.

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 and , 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.
Paper Structure (29 sections, 24 theorems, 123 equations, 16 figures, 4 tables, 4 algorithms)

This paper contains 29 sections, 24 theorems, 123 equations, 16 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Driver Hamiltonians and Mixing Operators in Quantum Computing
  3. Algebraic Conditions for Imposing Constraints
  4. A Frame for Classical Symmetries of Quantum Operators
  5. A Single Constraint, its Feasible Subspaces, and their Embeddings
  6. From Singular Constraints to Collection of Constraints
  7. The Commutation Space of Constraints
  8. An Algebraic Condition for Commuting with an Embedded Constraint Operator
  9. Example: Maximum Independent Set
  10. Wrap Up: Simplification for Linear Constraints and Sufficient Conditions for Ising Spin-z Constraints
  11. The Complexity of Imposing Constraints on Quantum Operators
  12. The Matrix Ring of Constraint Drivers and Their Generators
  13. The Matrix Ring of Commutators
  14. From Theory to Practice: Tractably Finding Commutators, Generator Selection, and Sequencing into Mixers
  15. Finding Commutators with Backtracking
  16. ...and 14 more sections

Key Result

Theorem 3.1

Given a constraint $C(x)$ written as Eq. eq:def_constraint and an embedded constraint operator $\hat{C}$ written as Eq. eq:defconop, for any computational basis state $\left\lvert\,x\,\right\rangle$ for $x \in \{0,1\}^n$, $\left\langle\,x\,\right\rvert \hat{C} \left\lvert\,x\,\right\rangle = C(x)$.

Figures (16)

  • Figure 1: Symmetry Imposed Ansätze in QML. Control parameters of a quantum computer are trained via a classical learner in the cloud to minimize a cost function. Before the training can begin, a parameterized unitary $U(\theta)$, must be selected during the preprocessing. Classical symmetries inform the selection of an ansatz; that is, the classical invariance should be imposed on the ansatz. The ansatz determines what gates will be applied during a computation on the quantum computer and how these gates are parameterized.
  • Figure 2: Constraint's Feasibility Decomposition and Commutation Space. Two example decompositions, each visualizing the embedded constraint operator down the diagonal and the associated commutation space. (a) visualizes for a constraint $C = x_{1} x_{2} + x_{2} x_{3} + x_{1} x_{3}$ leading to $\hat{C} = \sigma_{1}^{1} \sigma_{2}^{1} + \sigma_{2}^{1} \sigma_{3}^{1} + \sigma_{1}^{1} \sigma_{3}^{1} = 0 \, \hat{F}_{0} + 1 \, \hat{F}_{2} + 3 \, \hat{F}_{3}$. Then the associated commutation space is $\mathcal{F}_{C} = \mathcal{F}_{0} \oplus \mathcal{F}_{2} \oplus \mathcal{F}_{3}$. (b) visualizes for two constraints $C_1 = x_1 + x_2$, $C_2 = x_2 + x_3$ with $\hat{C}_{1} = \sigma_{1}^{1} + \sigma_{2}^{1} = 0 \, \hat{F}_{0}^{1} + 1 \, \hat{F}_{1}^{1} + 2 \, \hat{F}_{2}^{1}$, $\hat{C}_{2} = \sigma_{2}^{1} + \sigma_{3}^{1} = 0 \, \hat{F}_{0}^{2} + 1 \, \hat{F}_{1}^{2} + 2 \, \hat{F}_{2}^{2}$ with the joint spectrum $(0,0) \, \hat{F}_{(0,0)} + (0,1) \, \hat{F}_{(0,1)} + (1,0) \, \hat{F}_{(1,0)} + (1,1) \, \hat{F}_{(1,1)} + (2,1) \, \hat{F}_{(2,1)} + (1,2) \, \hat{F}_{(1,2)} + (2,2) \, \hat{F}_{(2,2)}$ associated with the commutation space $\mathcal{F}_{C_1,C_2} = \mathcal{F}_{(0,0)} \oplus \mathcal{F}_{(1,0)} \oplus \mathcal{F}_{(0,1)} \oplus \mathcal{F}_{(1,1)} \oplus \mathcal{F}_{(2,1)} \oplus \mathcal{F}_{(1,2)} \oplus \mathcal{F}_{(2,2)}$. For readability we suppress $\mathcal{F}_{(b_1,b_2)}$ for (b) when it is equivalent to $\text{span}(\hat{F}_{(b_1,b_2)})$
  • Figure 3: Embedded Constraint Operators and their Commutators. Utilizing the frame $\{\mathbbm{1}, \sigma^{0}, \sigma^{1}, \sigma^{+}, \sigma^{-}\}^{\otimes n}$, depicted on the left, we can embed any polynomial constraint as an embedded constraint operator, depicted in the middle, and then valid driving or mixing generators commute with these constraints, depicted on the right. On the right, we depicted the unique commutator up to coefficient, with $\left[ \alpha \, \sigma_{1}^{+} \, \sigma_{2}^{-} + \alpha^{\dagger} \sigma_{1}^{-} \, \sigma_{2}^{+}, \sigma_{1}^{0} + \sigma_{2}^{0} \right] = 0$ and $\left[ \alpha \, \sigma_{1}^{+} \, \sigma_{2}^{+} + \alpha^{\dagger} \, \sigma_{1}^{-} \, \sigma_{2}^{-}, \sigma_{1}^{0} + \sigma_{2}^{1} \right] = 0$ for any $\alpha \in \mathbb{C}$.
  • Figure 4: Imposing Set Independence. By constructing drivers that include (or revert inclusion) of a node in the set based on no neighbors being in the set, we can explore the entire feasible state space of valid independent sets of a graph. Such drivers satisfy the algebraic condition in Thm. \ref{['thm:gencom']} and are the lowest locality drivers for the problem.
  • Figure 5: Complexity of Imposing Constraints. A venn diagram showing the classification of important problems associated with imposing constraints, under the assumption that $NP\neq P$ and CP-IRREDUCIBLECOMMUTE-GIVEN-k is higher in the polynomial hierarchy (and it does not collapse).
  • ...and 11 more figures

Theorems & Definitions (66)

  • Definition : Phase-separation Operator
  • Definition : Mixing Operator
  • Definition : Generator (of Mixing or Driving)
  • Definition : k-Locality
  • Definition : Tensor Product for Matrices
  • Definition : Polynomial Constraint
  • Definition : Embedded Constraint Operator
  • Theorem 3.1: Eigenvalue Correspondence for Constraints
  • proof
  • Definition : Feasible Subspace (single equality constraint and constraint value)
  • ...and 56 more