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

Bärtschi, Andreas; Eidenbenz, Stephan

doi:10.1109/qce49297.2020.00020

Cited by 80 publications

(88 citation statements)

References 19 publications

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“…The portfolio optimization simulations indicate that WS-QAOA finds better solutions than standard QAOA. Here, future work could investigate tying budget constraints into the quantum circuit of WS-QAOA [95].…”

Section: Discussionmentioning

confidence: 99%

Warm-starting quantum optimization

Egger

Mareček

Woerner

2021

Quantum

153

111

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There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.

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Section: Discussionmentioning

confidence: 99%

Warm-starting quantum optimization

Egger

Mareček

Woerner

2021

Quantum

153

111

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“…1. In some sense, this is the reverse of the method in the general ansatz, and it was shown to be applicable to many relevant optimization problems [4], but there also exists a range of constrained problems where it is ruled out under complexity-theoretic assumptions [5].…”

Section: Fair Sampling In the Uantum Alternating Operator Ansatzmentioning

confidence: 99%

“…This property holds for all feasible states (not only ground states) and is independent of the number of QAOA levels or the choice of angles for U M , U P . For a complete proof, see [4]. Impact of noise: While the original based Quantum Approximate Optimization Algorithm ofers non-trivial provable performance guarantees already for a single for certain problems such as MaxCut on (d=3)-regular graphs [8], a higher number of rounds ś p ≥ 6 [33] or even p ∈ Ω(log(n)/d ) [3] ś might be necessary to outperform the best-known classical approximation ratios.…”

Section: Fair Sampling In the Uantum Alternating Operator Ansatzmentioning

confidence: 99%

“…The most widely-used quantum optimization framework is the Quantum Alternating Operator Ansatz (QAOA), which alternates between a mixer operator that mixes the amplitudes of all feasible solutions and a phase separator operator that separates the phases of the feasible solutions [13]. Most mixers do not exhibit fair sampling, however, the recently discovered Grover Mixer [4] has the property that all feasible solutions with the same objective value have identical amplitudes. This novel Grover Mixer implementation of QAOA thus solves the theoretical side of fair sampling for gate-based quantum computing.…”

Section: Introductionmentioning

confidence: 99%

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Fair Sampling Error Analysis on NISQ Devices

Golden

Bärtschi

O’Malley

et al. 2022

ACM Transactions on Quantum Computing

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We study the status of fair sampling on Noisy Intermediate Scale Quantum (NISQ) devices, in particular the IBM Q family of backends. Using the recently introduced Grover Mixer-QAOA algorithm for discrete optimization, we generate fair sampling circuits to solve six problems of varying difficulty, each with several optimal solutions, which we then run on twenty backends across the IBM Q system. For a given circuit evaluated on a specific set of qubits, we evaluate: how frequently the qubits return an optimal solution to the problem, the fairness with which the qubits sample from all optimal solutions, and the reported hardware error rate of the qubits. To quantify fairness, we define a novel metric based on Pearson’s χ 2 test. We find that fairness is relatively high for circuits with small and large error rates, but drops for circuits with medium error rates. This 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. Our results show that fairness can be a powerful tool for understanding the intricate web of errors affecting current NISQ hardware.

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“…QAOA tunes a set of classical parameters to optimize the cost function expectation value for a quantum state prepared by well-defined sequence of operators acting on a known initial state. Variations to the original algorithm include alternative operators and initial states [23][24][25][26][27][28][29][30][31] while purely classical aspects such as the parameter optimization and problem structure have been tested as well [32][33][34][35][36]. However, an outstanding concern is that practical implementations of QAOA require large numbers of qubits and deep circuits [37,38].…”

Section: Introductionmentioning

confidence: 99%

Multi-angle Quantum Approximate Optimization Algorithm

Herrman¹,

Lotshaw²,

Ostrowski³

et al. 2021

Preprint

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The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we introduce a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.

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

Cited by 80 publications

References 19 publications

Warm-starting quantum optimization

Warm-starting quantum optimization

Fair Sampling Error Analysis on NISQ Devices

Multi-angle Quantum Approximate Optimization Algorithm

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