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

Abstract

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.

Deterministic Preparation of Dicke States

“…Very few results exist on this topic and we are still far from having a comprehensive solution, however, Dicke states form such a class: the Dicke state |D n k has n k non-zero weights, which is not polynomial in n for super-constant k. Among different types of highly entangled states, the family of Dicke states [7] has garnered widespread attention for tasks in quantum networking [24], quantum game theory [34], quantum metrology [30] and as starting states for combinatorial optimization problems via adiabatic evolution [5]. Perhaps most promisingly -Dicke states can be used in the Quantum Alternating Operator Ansatz (QAOA) framework [10,13] for combinatorial optimization problems with hard constraints, as a starting state for the actual QAOA algorithm where they represent a superposition of all feasible solutions (in some problem variations). x∈{0,1} n , wt(x)=k |x .…”

Domain wall encoding of discrete variables for quantum annealing and QAOA

“…While the mixer in Eq. 15 is similar to the controlled-X-rotation mixer discussed at the beginning of 4.2.2 of [22], there is an important distinction, while the controlled-X-rotation mixer is controlled by a single qubit value, whether or not the X is applied in Eq. 15 is actually applied by whether two qubits agree or differ.…”

“…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.…”

“…Let us now consider the more structured problem of maximum graph colouring (referred to as Max-κ-ColorableSubgraph [22], also sometimes referred to as Max-κ-Cut [52,53]), where given a graph and n possible node colours, the goal is to colour the graph to maximize the number of edges which connect vertexes of different colours. Maximum graph colouring is a generalization of the more studied problem of graph colouring, since in graphs which are colourable with n colours, the maximal colouring is a 'proper' colouring of the graph, where no vertexes of the same colour share an edge.…”

Finding spin glass ground states using quantum walks

“…The Ising Hamiltonian is also the basic tool for specialised optimization hardware, such as coherent Ising machines (Inagaki et al 2016, McMahon et al 2016. Optimization using the Ising Hamiltonian can be implemented in digital quantum architectures by using the quantum approximate optimization algorithm (QAOA) (Farhi et al 2014a, 2014b, Marsh and Wang 2019 or quantum alternating operator ansatz (Hadfield et al 2019). Studies by Zhou et al (2018) show how to exploit non-adiabatic effects in QAOA on early quantum hardware.…”