# Grover Mixers for QAOA: Shifting Complexity from Mixer Design to State Preparation

@article{Brtschi2020GroverMF,
title={Grover Mixers for QAOA: Shifting Complexity from Mixer Design to State Preparation},
author={Andreas B{\"a}rtschi and Stephan Johannes Eidenbenz},
journal={2020 IEEE International Conference on Quantum Computing and Engineering (QCE)},
year={2020},
pages={72-82}
}
• Published 30 May 2020
• Computer Science
• 2020 IEEE International Conference on Quantum Computing and Engineering (QCE)
We propose GM-QAOA, a variation of the Quantum Alternating Operator Ansatz (QAOA) that uses Grover-like selective phase shift mixing operators. GM-QAOA works on any NP optimization problem for which it is possible to efficiently prepare an equal superposition of all feasible solutions; it is designed to perform particularly well for constraint optimization problems, where not all possible variable assignments are feasible solutions. GM-QAOA has the following features: (i) It is not susceptible…
31 Citations

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## References

SHOWING 1-10 OF 33 REFERENCES
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
• Physics
Algorithms
• 2019
The essence of this extension, the quantum alternating operator ansatz, is the consideration of general parameterized families of unitaries rather than only those corresponding to the time evolution under a fixed local Hamiltonian for a time specified by the parameter.
Universal Quantum Simulators
Feynman's 1982 conjecture, that quantum computers can be programmed to simulate any local quantum system, is shown to be correct.
XY mixers: Analytical and numerical results for the quantum alternating operator ansatz
• Computer Science
• 2020
This paper explores strategies for enforcing hard constraints by using $XY$ Hamiltonians as mixing operators (mixers) and demonstrates 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.
$XY$-mixers: analytical and numerical results for QAOA
• Computer Science
• 2019
Despite the complexity of simulating the XY model, it is demonstrated that for problems represented through one-hot-encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth O(κ) where κ is the number of assignable colors.
A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem
• Mathematics, Computer Science
• 2014
This paper applies the recent Quantum Approximate Optimization Algorithm to the combinatorial problem of bounded occurrence Max E3LIN2 and shows that the level one QAOA will efficiently produce a string that satisfies $\left(\frac{1}{2} + 1}{101 D^{1/2}\, l n\, D}\right)$ times the number of equations.
The Quantum Alternating Operator Ansatz on Max-k Vertex Cover
• Computer Science
• 2019
This paper studies Max-k Vertex Cover under this ansatz due to its modest complexity, while still being more complex than the well studied problems of Max-Cut and Max E3-LIN2.
Deterministic Preparation of Dicke States
• Computer Science
FCT
• 2019
This work presents a deterministic quantum algorithm for the preparation of Dicke states and yields a quasilinear-depth circuit for efficient compression of quantum information in the form of symmetric pure states, improving on existing work requiring quadratic depth.
Reachability Deficits in Quantum Approximate Optimization
• Physics
Physical review letters
• 2020
It is reported that QAOA exhibits a strong dependence on a problem instances constraint to variable ratio-this problem density places a limiting restriction on the algorithms capacity to minimize a corresponding objective function (and hence solve optimization problem instances).
The Quantum Alternating Operator Ansatz on Maximum k-Vertex Cover
• Computer Science
2020 IEEE International Conference on Quantum Computing and Engineering (QCE)
• 2020
This paper studies Maximum k-Vertex Cover under this ansatz due to its modest complexity, while still being more complex than the well studied problems of Max-Cut and Max E3-LIN2.
A quantum algorithm to count weighted ground states of classical spin Hamiltonians
• Computer Science, Physics
• 2019
This work modifications AQO and QAOA to count the ground states of arbitrary classical spin Hamiltonians, including counting ground states with arbitrary nonnegative weights attached to them, an important step in approaching general ground-state counting problems beyond those that can be solved with Grover's algorithm.