IF-QAOA: A Penalty-Free Approach to Accelerating Constrained Quantum Optimization

TL;DR

IF-QAOA provides a penalty-free, oracle-based method to embed inequality constraints into QAOA by evaluating constraint satisfaction with a QPE-based indicator function, yielding a cost tilde f(x) = f(x) Θ[g(x)]. In Knapsack benchmarks up to 22 items, IF-QAOA shows higher solution quality and faster time-to-solution than quadratic penalty baselines and competitive performance versus QTGs, with favorable scaling and robustness to approximate indicator implementations. The approach includes accelerated classical simulation, an approximate indicator for limited ancilla, and extensions to multi-constraint setups, supporting broader COP applicability. Overall, IF-QAOA offers a practical, low-complexity framework for constraint handling in gate-based quantum optimization, enabling more scalable exploration of constrained combinatorial problems on NISQ devices.

Abstract

Traditional methods for handling (inequality) constraints in the Quantum Approximate Optimization Ansatz (QAOA) typically rely on penalty terms and slack variables, which increase problem complexity and expand the search space. More sophisticated mixer-based QAOA variants restrict the search within the feasible assignments but often suffer from prohibitive circuit complexity. This paper presents a low-complexity formalism for incorporating inequality constraints into the cost function of QAOA using an oracle-based subroutine that evaluates constraint satisfaction in an additional register, subsequently called Indicator Function QAOA (IF-QAOA). The IF-QAOA cost function consists of a step-function but does not require a penalty term with additional parameters. Applied to the Knapsack problem, we demonstrate the superior performance of IF-QAOA over conventional penalty-based approaches in simulated experiments. Using advanced QAOA simulation techniques with instances consisting of up to 22 items, we find that IF-QAOA achieves significantly higher solution quality and a faster time-to-solution in 82% of our benchmark cases. Analysis of the scaling behavior shows favorable scaling of IF-QAOA compared to penalty-based methods. Also, benchmarked against the recently developed Quantum Tree Generator QAOA for Knapsack Problems, we demonstrate higher solution quality for circuits of similar complexity. Additionally, the paper introduces a method for approximate indicator function when the number of ancillary qubits is limited. With a specialized simulation algorithm based on projective measurements, we empirically demonstrate that a fixed number of ancillary qubits is sufficient to encode general inequality constraints.

Figures (9)

• Figure 1: Showcasing the search spaces of the various optimization strategies. In (a), the search space is enlarged to $\mathcal{D}^+$ through slack variables in the penalty term. In (b), the search space is exactly the space of the originally defined problem, and in (c), the search space is effectively reduced to the feasible subspace. The shapes of the domains are exemplary.
• Figure 2: Panel (a) shows the IF-QAOA circuit. The cost layer consists of the QPE, evaluating whether constraint $g$ has been satisfied, and a controlled application of the cost Hamiltonian based on the indicator qubit. The projective measurement on the QPE register is only required when $g$ is not resolvable by the QPE register. Panel (b) presents the projection factor $\mathcal{P}$, defined in Eq. \ref{['eq:projector-defined']}, in the approximate application of the indicator function, when $g$ is not resolvable by the QPE-register, here for $M=4$. The shading of the plot shows the probability of successful projective measurements. Clearly, when $g$ is an integer, this probability is always one. The hue indicates the applied phase to the compute register, showing that the application only happens for $g(x) \geq 0$.
• Figure 3: Multi-constraint cost function subroutine. The first qubits of the QPE registers control the application of the cost operator.
• Figure 4: The solution quality measured as RAAR displayed for a selection of results. The top plot shows RAAR depending on problem size $N$ for fixed QAOA layer depth $p = 16$, and the bottom plot shows RAAR w.r.t. $p$ at $N = 16$. Error bars indicate the 50% percentile interval.
• Figure 5: Panel (a) shows the median TTS for all problem instances in various color shades with increasing QAOA layer count $p$. As apparent, IF-QAOA starts with higher TTS than the baseline approach, but starts improving upon it starting with $p \geq 3$. Panel (b) shows the optimal TTS$^*$ depending on the problem size. The dashed line shows the logarithmic regression, with the fitted bases displayed in the legend. Panel (c) shows a scatter plot of all $9 \times 128 = 1152$ problem instances comparing TTS$^*$ of IF-QAOA against the Virtual Penalty. In $82\%$ of the test cases, IF-QAOA arrives at a solution faster. The color shading of the scatter plots indicates the weight ratio $W / \sum_i w_i$ of the Knapsack instance.