QAOA with Quantum Subspace Expansion

• QAOA is a hybrid quantum-classical algorithm that alternates cost and mixer Hamiltonians to approximate solutions for combinatorial challenges.
• Enhancements using quantum subspace expansion and generator coordinate methods restore broken symmetries and significantly improve approximation ratios and fidelity.
• Resource analysis indicates that, despite additional gate overhead, the QSE-enhanced method offers superior cost-to-solution benefits for larger-scale problems.

The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum–classical variational algorithm designed to tackle combinatorial optimization problems, particularly those that are classically hard. At its core, QAOA approximates solutions by preparing a parameterized quantum state via alternating application of cost and mixer Hamiltonians and optimizing these parameters through a classical feedback loop. While basic QAOA is a flagship for quantum advantage in the near-term and early fault-tolerant regime, significant research addresses improving its performance, resource efficiency, and fidelity through extensions such as quantum subspace expansion and post-processing techniques.

1. Generator Coordinate Method (GCM) and Quantum Subspace Expansion (QSE) Enhancement

The principal advancement described involves supplementing standard QAOA with a generator coordinate method (GCM) implemented using quantum subspace expansion (QSE). After preparing the initial QAOA state, one generates $K$ trial states by real-time evolving the cost Hamiltonian under a set of “generator coordinates” (e.g., time shifts $t_k$). These trial states $\{\chi_k\}$ serve as a basis for a configuration interaction (CI) expansion:

$|\Psi\rangle = \sum_k f_k \chi_k$

This linear combination systematically restores symmetries (such as Hamming weight and parity) broken in the underlying QAOA approximation, and introduces many-body correlations absent in the original variational ansatz. The QSE procedure thus “purifies” the QAOA state, yielding both a higher approximation ratio (i.e., better solution energy) and greater fidelity with the true ground state of the problem Hamiltonian.

2. Performance Metrics and Cost-to-Solution Analysis

Performance evaluation is conducted with three main quantitative metrics:

• Approximation Ratio: Ratio of the energy achieved by the algorithmic state to the ground state (optimal) energy.
• Fidelity ($F$): $$F = |\langle\Psi_\text{ground}|\Psi_\text{alg}\rangle|^2$$, quantifies closeness to the true ground state.
• Cost-to-Solution: Quantified by the number of logical circuit elements—especially CNOT and $T$ gates—required to reach a desired solution quality.

The paper introduces a criterion for cost-effectiveness, where the ratio

$$\frac{F^\text{(GCM)}}{F^\text{(QAOA)}} \gtrsim 4(K-1)\left(1+\frac{1}{L'}\right) \times f$$

provides a threshold; here $K$ is the number of QSE trial states, $L'$ is the circuit depth retained after optimization, and $f$ is a scaling factor determined by the choice of kernel matrix evaluation method (e.g., Pauli expectation value, real-time evolution, LCU approach).

3. Comparative Resource Analysis

The resource requirements for the QSE-enhanced protocol and vanilla QAOA are analyzed methodically. Logical gate counts—CNOTs and $T$ gates—are tallied for evaluation of both standard expectation values and the non-unitary components needed for QSE’s generalized eigenvalue (Hill–Wheeler) equation. While QSE incurs overhead owing to Hadamard test circuits, controlled-$R_{ZZ}$, and multi-controlled gates, this is offset for larger graphs by the rapid convergence of GCM expansion. Numerical extrapolation indicates that for the maximal independent set on Erdős–Rényi graphs, the crossover to lower total logical gate cost for the GCM/QSE-enhanced method compared to QAOA occurs for graphs with more than $\sim75$ nodes, with as few as $K=8$ trial states.

Method	CNOT Gate Count	$T$ Gate Count
Standard QAOA	$O(NL)$	$O(NL)$
QAOA + QSE (Direct Pauli)	$O(K^2 N L')$	$O(K^2 N L')$
QAOA + QSE (RTE/LCU)	Higher per kernel	Higher per kernel, but offset

Here, $N$ is the number of qubits, $L$ the QAOA depth, and $L'$ the reduced depth in the optimized basis.

4. Numerical Results and Symmetry Restoration

The practical benefits of the QSE enhancement are evidenced in numerical experiments on the maximal independent set. On modest-size graphs (up to $N=10$), plain QAOA achieves an approximation ratio of $\sim0.5$ and fidelity as low as $0.15$ with respect to the ground state—often failing to respect required symmetry features. With as few as $K=8$ QSE trial states, the approximation ratio nearly doubles and the fidelity reaches $\sim0.96$, with almost immediate convergence to the correct symmetry sector (parity and Hamming weight). On canonical graphs (e.g. cubes, $K^+_{3,3}$), QSE boosts fidelities from $14-17\%$ to above $98\%$. Extrapolation suggests the GCM-enhanced QAOA outperforms plain QAOA in cost-to-solution for graphs well beyond this size threshold, especially for combinatorics problems on random Erdős–Rényi instances.

5. Generalization and Potential Applications

The GCM approach, while demonstrated for the maximal independent set, is broadly applicable to a range of combinatorial optimization problems with analogous cost Hamiltonian structure. As the methodology is independent of the particular initial ansatz, one can combine QSE/GCM with other quantum algorithms (including analog adiabatic protocols) to accelerate convergence and achieve high-fidelity solutions. Of particular importance for early fault-tolerant quantum devices is the significant reduction in logical resource overhead, as T and CNOT gate counts often set the feasibility boundaries for practical execution.

QSE also inherently restores physically or combinatorially motivated symmetry constraints that QAOA and other variational methods may break, potentially yielding improved solution interpretability in applications where these constraints are essential (e.g., parity, Hamming weight conservation).

6. Summary and Impact

The integration of quantum subspace expansion via the generator coordinate method provides a systematic and resource-efficient improvement to QAOA for combinatorial optimization, as evidenced in the maximal independent set on random graphs (Beaujeault-Taudière, 23 Jun 2025). By constructing a post-QAOA subspace of physically relevant trial states and solving a generalized eigenvalue problem within this subspace, both solution quality (approximation ratio, fidelity) and enforcement of problem constraints are enhanced. Resource estimates indicate that the additional gate cost is justified for moderate to large instance sizes. This class of hybrid quantum algorithms combining variational preparation, subspace expansion, and projection is expected to play a central role in deploying quantum advantage on early fault-tolerant hardware for combinatorial optimization.