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. *

The quantum approximate optimization algorithm (QAOA) is considered to be one of the most promising approaches towards using near-term quantum computers for practical application. In its original form, the algorithm applies two different Hamiltonians, called the mixer and the cost Hamiltonian, in alternation with the goal being to approach the ground state of the cost Hamiltonian. Recently, it has been suggested that one might use such a set-up as a parametric quantum circuit with possibly some other goal than reaching ground states. From this perspective, a recent work (Lloyd, arXiv:1812.11075) argued that for one-dimensional local cost Hamiltonians, composed of nearest neighbour ZZ terms, this set-up is quantum computationally universal and provides a universal gate set, i.e. all unitaries can be reached up to arbitrary precision. In the present paper, we complement this work by giving a complete proof and the precise conditions under which such a one-dimensional QAOA might produce a universal gate set. We further generalize this type of gate-set universality for certain cost Hamiltonians with ZZ and ZZZ terms arranged according to the adjacency structure of certain graphs and hypergraphs.

Given a parameterized quantum circuit such that a certain setting of these real-valued parameters corresponds to Grover's celebrated search algorithm, can a variational algorithm recover these settings and hence learn Grover's algorithm? We studied several constrained variations of this problem and answered this question in the affirmative, with some caveats. Grover's quantum search algorithm is optimal up to a constant. The success probability of Grover's algorithm goes from unity for two-qubits, decreases for three-and four-qubits and returns near unity for five-qubits then oscillates ever-so-close to unity, reaching unity in the infinite qubit limit. The variationally approach employed here found an experimentally discernible improvement of 5.77% and 3.95% for three-and four-qubits respectively. Our findings are interesting as an extreme example of variational search, and illustrate the promise of using hybrid quantum classical approaches to improve quantum algorithms. This paper further demonstrates that to find optimal parameters one doesn't need to vary over a family of quantum circuits to find an optimal solution. This result looks promising and points out that there is a set of variational quantum problems with parameters that can be efficiently found on a classical computer for arbitrary number of qubits.Grover's algorithm [2] is one of the most celebrated quantum algorithms, enabling quantum computers to quadratically outperform classical computers at database search provided database access is restricted to a 'black box' -called the oracle model. In addition to the wide application scope of database search, Grover's algorithm has further applications as a subroutine used in a variety of other quantum algorithms.Variational hybrid quantum/classical algorithms have recently become an area of significant interest [3][4][5][6][7][8][9][10]. These algorithms have shown several advantages such as robustness to quantum errors and low coherence time requirements [11], which make them ideal for implementations in current quantum computer architectures. Here we take inspiration from algorithms such as the variational quantum eigensolver (VQE) [3] and the quantum approximate optimization algorithm (QAOA) [4]. The general procedure of these variational hybrid quantum/classical algorithms is the following:

A contemporary technological milestone is to build a quantum device performing a computational task beyond the capability of any classical computer, an achievement known as quantum adversarial advantage. In what ways can the entanglement realized in such a demonstration be quantified? Inspired by the area law of tensor networks, we derive an upper bound for the minimum random circuit depth needed to generate the maximal bipartite entanglement correlations between all problem variables (qubits). This bound is lattice geometry dependent and makes explicit a nuance implicit in other proposals with physical consequence. The hardware itself should be able to support superlogarithmic ebits of entanglement across some poly(n) number of qubit bipartitions; otherwise the quantum state itself will not possess volumetric entanglement scaling and full-lattice-range correlations. Hence, as we present a connection between quantum advantage protocols and quantum entanglement, the entanglement implicitly generated by such protocols can be tested separately to further ascertain the validity of any quantum advantage claim.

Parameterized complexity theory was developed in the 1990s to enrich the complexitytheoretic analysis of problems that depend on a range of parameters. In this paper we establish a quantum equivalent of classical parameterized complexity theory, motivated by the need for new tools for the classifications of the complexity of real-world problems. We introduce the quantum analogues of a range of parameterized complexity classes and examine the relationship between these classes, their classical counterparts, and well-studied problems. This framework exposes a rich classification of the complexity of parameterized versions of QMA-hard problems, demonstrating, for example, a clear separation between the Quantum Circuit Satisfiability problem and the Local Hamiltonian problem.