XY mixers: Analytical and numerical results for the quantum alternating operator ansatz
Open accessAbstract
The quantum alternating operator ansatz (QAOA) is a promising gate-model metaheuristic for combinatorial optimization. Applying the algorithm to problems with constraints presents an implementation challenge for near-term quantum resources. This paper explores strategies for enforcing hard constraints by using $XY$ Hamiltonians as mixing operators (mixers). Despite the complexity of simulating the $XY$ model, we demonstrate that, for an integer variable admitting $\ensuremath{\kappa}$ discrete values represented through one-hot encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth $O(\ensuremath{\kappa})$. We also specify general strategies for implementing QAOA circuits on all-to-all connected hardware graphs and linearly connected hardware graphs inspired by fermionic simulation techniques. Performance is validated on graph-coloring problems that are known to be challenging for a given classical algorithm. The general strategy of using $XY$ mixers is borne out numerically, demonstrating a significant improvement over the general $X$ mixer, and moreover the generalized $W$ state yields better performance than easier-to-generate classical initial states when $XY$ mixers are used.
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The usage of XY-mixers, which allow rotations around the X- and Y-axis, has been explored.
“In this context, the usage of XY-mixers, which allow rotations around the X- as well as the Y-axis, have been explored [116].”
From Quantum Combinatorial Optimization in the NISQ Era: A Systematic Mapping Study (2024) · cites this paper as [116]
The paper explores XY Hamiltonians as mixers (p. 1). In “XY mixer: Enforcing evolution in the feasible subspace” (p. 4), it says they drive rotations within the {(0,1),(1,0)} subspace; it does not describe them as allowing rotations around the X- and Y-axes individually. (LLM, conf=0.86)
Experimental — accuracy not yet measuredAnsatz adaptations can occur as a byproduct, as in the Quantum Alternating Operator Ansatz where the primary goal is to find a mixing Hamiltonian that ensures staying in the feasible subspace.
“Besides the mentioned ansatz search, as the driving force behind ansatz adaptation, there is also the possibility of having these adaptations as a byproduct, like it happens, e.g., within the Quantum Alternating Operator Ansatz where the primary goal is to find a mixing Hamiltonian that ensures to stay in the feasible subspace [11, 58, 95, 112, 116, 117, 120, 127].”
From Quantum Combinatorial Optimization in the NISQ Era: A Systematic Mapping Study (2024) · cites this paper as [116]
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- Authors
- Wang, Z., Rubin, N. C., Dominy, J. M., & Rieffel, E. G.
- Affiliations
- Ames Research Center (US)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2020-01-14
- Volume
- 101
- Issue
- 1
- ISSN
- 0556-2791 (ISSN-L)
- 2469-9926
- 2469-9934
- OpenAlex
- W2939513517
- Cited by
- 190
- References
- 35
- Citations by year
- 2026: 22 · 2025: 31 · 2024: 37 · 2023: 29 · 2022: 26 · 2021: 29 · 2020: 12 · 2019: 4
- FWCI
- 9.43
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Ames Research Center
- Air Force Research Laboratory (F4HBKC4162G00)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Metaheuristic Optimization Algorithms Research
- Keywords
- Quantum Alternating Operator Ansatz, mixing operators, one-hot encoding, Trotter error, graph coloring, combinatorial optimization, W state
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