A quantum walk-assisted approximate algorithm for bounded NP optimisation problems

The next few years will be exciting as prototype universal quantum processors emerge, enabling the implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation and which have the potential to significantly expand the breadth of applications for which quantum computers have an established advantage. A leading candidate is Farhi et al.'s quantum approximate optimization algorithm, which alternates between applying a cost function based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. 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. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach, in the spirit of the quantum approximate optimization algorithm, to a wide variety of approximate optimization, exact optimization, and sampling problems. In addition to introducing the quantum alternating operator ansatz, we lay out design criteria for mixing operators, detail mappings for eight problems, and provide a compendium with brief descriptions of mappings for a diverse array of problems.

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“…Prior QAOA work: A quantum walk generalization of QAOA is proposed for this problem in Reference [31]. Approximability: APX-complete [51].…”

Section: A23 Minvertexcovermentioning

confidence: 99%

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

et al. 2019

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“…However, higher depth circuits demand a significant optimization task to be performed on a classical computer. In contrast, while reasonably short depth circuits have shown noise-free quantum advantage [16][17][18][19][20][21], less remains known about the depth of circuits required to enable practical advantage using noisy circuits.…”

Section: Introductionmentioning

confidence: 99%

Reachability Deficits in Quantum Approximate Optimization

et al. 2020

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The quantum approximate optimization algorithm (QAOA) has rapidly become a cornerstone of contemporary quantum algorithm development. Despite a growing range of applications, only a few results have been developed towards understanding the algorithms ultimate limitations. Here we report 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 problems). Such reachability deficits persist even in the absence of barren plateaus [1] and are outside of the recently reported level-1 QAOA limitations [2]. Building on general numerical experiments, we compare the presence of reachability deficits with analytic solutions of the variational model of Grover's search algorithm. Comparing QAOA's performance between random 3-SAT (NP-hard) and 2-SAT (efficiently solved) instances, reachability deficits increased with problem density. *

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“…The problem of finding invalid states in finite temperature quantum annealing has been highlighted in [35]. It would therefore be preferable to use a mixing Hamiltonian which only mixes between valid states, as discussed in [22,36,37]. These papers have focused on QAOA, since currently existing quantum annealers use transverse field mixers.…”

Section: Qaoa Mixersmentioning

confidence: 99%

Domain wall encoding of discrete variables for quantum annealing and QAOA

Chancellor

2019

Quantum Sci. Technol.

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In this paper I propose a new method of encoding discrete variables into Ising model qubits for quantum optimization. The new method is based on the physics of domain walls in one dimensional Ising spin chains. I find that these encodings and the encoding of arbitrary two variable interactions is possible with only two body Ising terms. Following on from similar results for the 'one hot' method of encoding discrete variables [Hadfield et. al. Algorithms 12.2 (2019): 34] I also demonstrate that it is possible to construct two body mixer terms which do not leave the logical subspace, an important consideration for optimising using the quantum alternating operator ansatz (QAOA). I additionally discuss how, since the couplings in the domain wall encoding only need to be ferromagnetic and therefore could in principle be much stronger than anti-ferromagnetic couplers, application specific quantum annealers for discrete problems based on this construction may be beneficial. Finally, I compare embedding for synthetic scheduling and colouring problems with the domain wall and one hot encodings on two graphs which are relevant for quantum annealing, the chimera graph and the Pegasus graph. For every case I examine I find a similar or better performance from the domain wall encoding as compared to one hot, but this advantage is highly dependent on the structure of the problem. For encoding some problems, I find an advantage similar to the one found by embedding in a Pegasus graph compared to embedding in a chimera graph.

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A quantum walk-assisted approximate algorithm for bounded NP optimisation problems

Cited by 42 publications

References 36 publications

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

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

Reachability Deficits in Quantum Approximate Optimization

Domain wall encoding of discrete variables for quantum annealing and QAOA

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