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

The next few years will be exciting as prototype universal quantum processors emerge, enabling 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 quantum computing applications. 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 parametrized 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 to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape

Adaptive Search and Constraint Optimisation in Engineering Design

The dissertation presents the investigation and development of novel adaptive computational techniques that provide a high level of performance when searching complex high-dimensional design spaces characterised by heavy non-linear constraint requirements. The objective is to develop a set of adaptive search engines that will allow the successful negotiation of such spaces to provide the design engineer with feasible high performance solutions. Constraint optimisation currently presents a major problem to the engineering designer and many attempts to utilise adaptive search techniques whilst overcoming these problems are in evidence. The most widely used method (which is also the most general) is to incorporate the constraints in the objective function and then use methods for unconstrained search. The engineer must develop and adjust an appropriate penalty function. There is no general solution to this problem neither in classical numerical optimisation nor in evolutionary computation. Some recent theoretical evidence suggests that the problem can only be solved by incorporating a priori knowledge into the search engine. Therefore, it becomes obvious that there is a need to classify constrained optimisation problems according to the degree of available or utilised knowledge and to develop search techniques applicable at each stage. The contribution of this thesis is to provide such a view of constrained optimisation, starting from problems that handle the constraints on the representation level, going through problems that have explicitly defined constraints (i.e., an easily computed closed form like a solvable equation), and ending with heavily constrained problems with implicitly defined constraints (incorporated into a single simulation model). At each stage we develop applicable adaptive search techniques that optimally exploit the degree of available a priori knowledge thus providing excellent quality of results and high performance. The proposed techniques are tested using both well known test beds and real world engineering design problems provided by industry.British Aerospace, Rolls Royce and Associate