[Paper Review] Modeling Linear Inequality Constraints in Quadratic Binary Optimization for Variational Quantum Eigensolver

Miguel Paredes Quiñones, Catarina Junqueira|arXiv (Cornell University)|Jul 26, 2020

Quantum Computing Algorithms and Architecture25 references4 citations

TL;DR

This paper proposes tailored variational forms (TVFs) for the Variational Quantum Eigensolver (VQE) that embed linear inequality and equality constraints directly into the quantum circuit, ensuring only feasible solutions are explored. By encoding constraints via controlled rotation and entanglement structures, TVFs maintain a constant number of parameters regardless of problem size, reducing circuit depth and improving convergence—demonstrated on Facility Location and Linear Assignment Problems with fewer gates and faster optimization than 2-Local and QAOA approaches.

ABSTRACT

This paper introduces the use of tailored variational forms for variational quantum eigensolver that have properties of representing certain constraints on the search domain of a linear constrained quadratic binary optimization problem solution. Four constraints that usually appear in several optimization problems are modeled. The main advantage of the proposed methodology is that the number of parameters on the variational form remain constant and depend on the number of variables that appear on the constraints. Moreover, this variational form always produces feasible solutions for the represented constraints differing from penalization techniques commonly used to translate constrained problems into unconstrained one. The methodology is implemented in a real quantum computer for two known optimization problems: the Facility Location Problem and the Set Packing Problem. The results obtained for this two problems with VQE using 2-Local variational form and a general QAOA implementation are compared, and indicate that less quantum gates and parameters were used, leading to a faster convergence.

Motivation & Objective

• To address the challenge of efficiently solving linear constrained quadratic binary optimization (LCQBO) problems on near-term quantum computers.
• To reduce the computational cost of classical optimization in VQE by restricting the search space to feasible solutions only.
• To develop variational forms that inherently satisfy constraints, avoiding the need for penalty methods that can lead to infeasible solutions.
• To demonstrate improved convergence speed and reduced circuit depth compared to standard VQE ansatzes like 2-Local and QAOA.
• To provide a systematic method for constructing quantum circuits that encode specific types of linear constraints (equality and inequality) directly into the ansatz structure.

Proposed method

• The authors design tailored variational forms (TVFs) that embed four common constraint types—equality, inequality, and their combinations—directly into the quantum circuit structure.
• For equality constraints, they use controlled-Ry rotations and CNOT gates to enforce sum-to-one conditions on qubit blocks, ensuring only valid combinations are generated.
• For inequality constraints, they use a recursive construction of controlled rotations and entanglers to represent upper bounds on variable sums, maintaining feasibility throughout the circuit.
• The method ensures the number of parameters remains constant per constraint type, independent of the number of variables, by reusing fixed-depth blocks of rotations and entanglers.
• The TVF is constructed using a two-stage process: first, initialize qubits to |0⟩, then apply Ry(θ) rotations followed by CNOTs to entangle qubits in a way that respects constraint boundaries.
• The approach is validated on two benchmark problems—Facility Location and Linear Assignment—using real quantum hardware and compared against 2-Local and QAOA circuits.

Experimental results

Research questions

• RQ1 Can variational quantum circuits be designed to inherently satisfy linear equality and inequality constraints in LCQBO problems?
• RQ2 Does embedding constraints directly into the ansatz reduce the number of parameters and quantum gates compared to penalty-based methods?
• RQ3 Can constraint-aware variational forms lead to faster convergence in VQE optimization compared to general-purpose ansatzes like 2-Local or QAOA?
• RQ4 What is the trade-off in circuit depth and parameter count when using tailored variational forms versus standard ansatzes?
• RQ5 Is there a systematic way to combine multiple constraint-encoding blocks to handle complex LCQBO problems?

Key findings

• The TVF approach ensures all generated solutions are feasible by construction, unlike penalty-based methods that may converge to infeasible solutions.
• For the Linear Assignment Problem, the TVF required only 4 parameters, 6 single-qubit rotations (SU(2)), and 4 CNOT gates, outperforming 2-Local (8 parameters, 14 SU(2), 3 CNOTs) and QAOA (4 parameters, 24 SU(2), 8 CNOTs).
• The TVF converged in 45 iterations using COBYLA, while QAOA and 2-Local converged to infeasible solutions, as shown in the probability histograms and objective function evolution.
• The TVF achieved the optimal objective value of 15 for the LAP instance, matching the classical optimum, while other methods failed to reach feasibility.
• The TVF used fewer total gates and parameters than QAOA and had comparable or better cost efficiency than 2-Local, with a total cost of 46 vs. 44 for 2-Local.
• The method demonstrates that constraint-aware ansatz design can significantly reduce the Hilbert space exploration burden, improving classical optimization efficiency.