On the universality of the quantum approximate optimization algorithm

Abstract

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.

Reachability Deficits in Quantum Approximate Optimization of Graph Problems

“…A classical optimization loop iteratively adjusts a quantum state prepared by an adjustable quantum circuit (an ansatz). High depth circuits of fixed structure have been proven to represent universal resources for quantum computation [14,15] and hence can emulate the gate model piece-wise. However, higher depth circuits demand a significant optimization task to be performed on a classical computer.…”

Efficient Quantum Feature Extraction for CNN-based Learning

“…In order to implement the ansatz in a real quantum device, we select {H, R X , R Y , R Z , CN OT } to compose the ansatz In this case, motivated by the universality of QAOA [46,47], we use a QAOA-heuristic circuit as an ansatz. The circuit is as shown in Fig.…”

Variational quantum classifiers through the lens of the Hessian

“…For combinatorial optimization tasks, the quantum approximate optimization algorithm (QAOA) was proposed originally to attain approximate solutions [ 8 ]. These architectures of variational quantum circuits have shown to be computationally universal [ 9 , 10 ].…”

“…It has not received the attention it deserves. The occurrence of barren plateaus issue could abolish the quantum advantage with a parameterized quantum circuit [ 9 ]. The visualization and understanding of the loss landscape of classical machine learning algorithms remains a vital and highly active area of research.…”