# 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} }

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

QAOA-based Fair Sampling on NISQ Devices

- Computer ScienceArXiv
- 2021

It is shown that reducing structured errors is necessary to improve fair sampling on NISQ hardware, and a novel metric based on Pearson’s χ test is defined, which indicates that structured errors dominate in this regime, while unstructured errors, which are random and thus inherently fair, dominate in noisier qubits and longer circuits.

Sampling on NISQ Devices: "Who’s the Fairest One of All?"

- Computer Science2021 IEEE International Conference on Quantum Computing and Engineering (QCE)
- 2021

Five fair sampling problems are run that are based both on quantum annealing and on the Grover Mixer-QAOA algorithm for gate-based NISQ hardware to demonstrate what work will still need to be done to achieve a seamless integration of frontend and backend compilation.

Threshold-Based Quantum Optimization

- Computer Science2021 IEEE International Conference on Quantum Computing and Engineering (QCE)
- 2021

Th-QAOA is proposed and study, a variation of the Quantum Alternating Operator Ansatz that replaces the standard phase separator operator with a threshold function that returns a value 1 for solutions with an objective value above the threshold and a 0 otherwise, to arrive at a quantum optimization algorithm.

A Divide-and-Conquer Approach to Dicke State Preparation

- Computer ScienceIEEE Transactions on Quantum Engineering
- 2022

A divide-and-conquer approach to deterministically prepare Dicke states |D k 〉 (i.e. equal-weight superpositions of all n-qubit states with Hamming Weight k) on quantum computers achieves significantly higher state fidelity compared to previous results.

Amplitude Amplification for Optimization via Subdivided Phase Oracle

- Computer Science
- 2022

The algorithm can be used to amplify the probability of measuring the optimal solution to a signiﬁcant degree independent of the search space size to solve combinatorial optimization problems via the use of a subdivided phase oracle.

Classically-Boosted Quantum Optimization Algorithm

- Computer ScienceArXiv
- 2022

The Classically-Boosted Quantum Optimization Algorithm (CBQOA) is proposed that can solve a wide range of combinatorial optimization problems, including all unconstrained problems and many important constrained problems such as Max Bisection, Maximum Independent Set, Minimum Vertex Cover, Portfolio Optimization, Traveling Salesperson and so on.

Encoding trade-offs and design toolkits in quantum algorithms for discrete optimization: coloring, routing, scheduling, and other problems

- Computer ScienceArXiv
- 2022

This manuscript presents an intuitive method for synthesizing and analyzing discrete (i.e., integer-based) optimization problems, wherein the problem and corresponding algorithmic primitives are expressed using a discrete quantum intermediate representation (DQIR) that is encoding-independent.

Fair Sampling Error Analysis on NISQ Devices

- Computer ScienceACM Transactions on Quantum Computing
- 2022

Fair sampling on Noisy Intermediate Scale Quantum devices, in particular the IBM Q family of backends, is studied, and a novel metric based on Pearson’s χ2 test is defined that indicates that structured errors dominate in this regime, while unstructured errors, which are random and thus inherently fair, dominate in noisier qubits and longer circuits.

Multi-angle quantum approximate optimization algorithm

- Computer ScienceScientific reports
- 2022

This work investigates a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters, and finds that good parameters can be found in polynomial time for a test dataset the authors consider.

Multi-round QAOA and advanced mixers on a trapped-ion quantum computer

- Physics
- 2022

Yingyue Zhu, Zewen Zhang, Bhuvanesh Sundar, 4 Alaina M. Green, C. Huerta Alderete, Nhung H. Nguyen, Kaden R. A. Hazzard, 5 and Norbert M. Linke 6 Joint Quantum Institute and Department of Physics,…

## References

SHOWING 1-10 OF 33 REFERENCES

From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

- PhysicsAlgorithms
- 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

- PhysicsScience
- 1996

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 ScienceFCT
- 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

- PhysicsPhysical 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 Science2020 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.