An Adaptive Mixer Allocation Algorithm for the Quantum Alternating Operator Ansatz

TL;DR

This paper tackles the gate-cost challenge of QAOA+ for constrained combinatorial optimization by proposing AMA-QAOA+, an adaptive mixer allocation strategy that selectively applies mixer operations to a subset of qubits per layer while constraining the search to feasible MIS solutions. It combines a pre-trained initial circuit, an evaluation function that fuses average initial energy $F_{fun}$ and average gradient $F_{grad}$, and intermittent optimization to build successive mixer layers, discarding the target unitary after the first layer to reduce parameter complexity. Numerical results on MIS instances from Erdős–Rényi and 3-regular graphs show that AMA-QAOA+ achieves higher mean optimal and average approximation ratios and dramatically lowers CNOT gate counts compared with QAOA+, Adaptive-QAOA+, and PNU, demonstrating improved solution quality and circuit efficiency. The approach is potentially generalizable to other CCOPs, offering a scalable, resource-efficient path toward practical quantum optimization on near-term devices.

Abstract

Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA-QAOA+. Unlike QAOA+, AMA-QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA-QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA-QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by $5.30\%$ on ER random graphs and $5.41\%$ on 3-regular graphs. Moreover, AMA-QAOA+ significantly reduces the CNOT gate consumption, requiring only $15.30\%$ and $25.18\%$ of the CNOT gates used by QAOA+ on ER and 3-regular random graphs, respectively. These results demonstrate that AMA-QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource-efficient quantum circuits.

Figures (13)

• Figure 1: The implementation of a single-layer QAOA+ ansatz for the given ER graph. (a) An ER graph with $n = 8$ vertices. (b) The circuits with the green and purple undertone correspond to $\mathrm{e}^{-\mathrm{i}\beta_{1}H_{M}}$ and $\mathrm{e}^{-\mathrm{i}\gamma_{1}H_{C}}$, respectively. Each qubit $q_v$ corresponds to vertex $v$, where $v = 0, 1, \dots, n-1$.
• Figure 2: The quantum circuit architectures of QAOA+ and AMA-QAOA+, where $H_C = -\sum_{v=1}^{n} \frac{I - \sigma_v^z}{2}$. (a) In QAOA+, the PQC is built by $p$-layer QAOA+ ansatz, where one layer of QAOA+ ansatz has two unitary operators, and there are two optimized parameters in each layer. Notably, each qubit is acted on by the mixer when implementing each layer of the mixer unitary operator. (b) In the AMA-QAOA+ algorithm, an initial PQC is first built by unitary operators $U_{C}(\gamma_{1})$ and $U_{M}(\beta_{1})$. In the subsequent layers, there is only one layer of mixer unitary operation $U_{M}(\beta_{l})$. For $l \ge 1$, several mixers are randomly or adaptively added into $U_{M}(\beta_{l})$.
• Figure 3: Schematic illustration of the adaptive circuit expansion process in AMA-QAOA+. The variable $M_{\text{add}}$ is initialized to 0 to track the number of mixer operators added to the current layer of the mixer unitary operator. Two predefined thresholds, $\delta_{\text{add}}$ and $\delta_{\text{grad}}$, are given to control the number of added mixer operators. At each expansion, the evaluation function $C(U_{M_j})$ is computed for every candidate mixer operator $U_{M_j}$ in the operator pool, where $C(U_{M_j})$ incorporates both the average initial expectation value and the average gradient. The mixer operator with the highest evaluation score is selected and appended to the current circuit $U_{\text{init}}$, after which the operator pool is updated and $M_{\text{add}}$ is incremented by one. If the stopping conditions (i.e., $M_{\text{add}} \ge \delta_{\text{add}}$ or max($F_\text{grad}(U_{M_j}))<\delta_{\text{grad}}$ are not yet satisfied, the updated circuit $U_{\text{init}}$, operator pool, and $M_{\text{add}}$ are passed into the next round of evaluation. This expansion process is repeated until the stop criterion is met, and these selected mixer operators commonly construct a new layer of the mixer unitary operator. Then, optimize the entire circuit, where AMA-QAOA+ randomly generates $\beta_{i}$, and reuses the optimized parameters $(\gamma_{1}^{*},\boldsymbol{\beta}_{1:i-1}^{*})$. If the stop condition in Equation \ref{['eq:stopping_condition']} is not met, re-initialize the relevant variables and continue constructing the next layer of the mixer unitary operator.
• Figure 4: The mean OAR and AAR obtained by various algorithms vary with graph size $n$ on ER graphs with an edge probability of 0.5. Here, "Adaptive-QAOA+ without $U_{C}$" is the variant of Adaptive-QAOA+, which is without the target unitary operator in $i(\ge 2)$-th layer of ansatz.
• Figure 5: The mean number of CNOT gates and iterations required by different algorithms per run varies with the graph size $n$ on ER graphs with an edge probability of 0.5. To facilitate a more direct comparison of how the evaluation function and optimization method influence resource consumption, this figure also includes a variant of Adaptive-QAOA+ that excludes the target unitary operator in the $i$-th layer for $i \geq 2$.