Mixer-Phaser Ansätze for Quantum Optimization with Hard Constraints

Abstract

We introduce multiple parametrized circuit ansätze and present the results of a numerical study comparing their performance with a standard Quantum Alternating Operator Ansatz approach. The ansätze are inspired by mixing and phase separation in the QAOA, and also motivated by compilation considerations with the aim of running on near-term superconducting quantum processors. The methods are tested on random instances of a weighted quadratic binary constrained optimization problem that is fully connected for which the space of feasible solutions has constant Hamming weight. For the parameter setting strategies and evaluation metric used, the average performance achieved by the QAOA is effectively matched by the one obtained by a "mixer-phaser" ansatz that can be compiled in less than half-depth of standard QAOA on most superconducting qubit processors.

Portfolio Optimization via Quantum Zeno Dynamics on a Quantum Processor

“…Corollary 1. Let U Z-QAOA (β, γ) denote the QAOA circuit on n qubits with N measurements added to each mixing operator as defined in (15). Let the initial state ρ 0 = |s s| be in-constraint.…”

“…In general, constraint-preserving mixers are difficult to implement, even when constructions are available [13,14]. The cost of implementing the algorithm on hardware can be reduced for a restricted class of problems by combining the phase and mixing operators [15]. If a uniform superposition of in-constraint states can be prepared efficiently, a Grover operator can be used as the mixer [16][17][18].…”

Benchmarking the performance of portfolio optimization with QAOA

“…Finally, we modify the full mixer according to the recently proposed quantum alternate mixer-phase ansatz (QAMPA) [18]. Instead of applying the operator e −iγ F to all qubits before applying the mixer, we reorder the terms such that, for each pair of qubits, the gates corresponding to the mixer and the phase separation are contracted as follows:…”

Graph neural network initialisation of quantum approximate optimisation

“…Due to this potential, there has been a rapid development in the study of the QAOA algo-rithm and its components, including (but not limited to) theoretical observations and limitations [13][14][15][16][17][18][19][20], variations on the circuit structure (ansatz) [21][22][23][24][25] used, the cost function [26][27][28] and initialisation and optimisation methods [29][30][31][32][33][34] used for finding optimal solutions. Since the algorithm is suitable for near-term devices, there has also been substantial progress in experimental or numerical benchmarks [34][35][36][37][38] and the effect of quantum noise on the algorithm [39,40].…”