*Article* **Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation**

**Daniil Rabinovich \*, Richik Sengupta, Ernesto Campos, Vishwanathan Akshay and Jacob Biamonte**

> Laboratory of Quantum Algorithms for Machine Learning and Optimisation, Skolkovo Institute of Science and Technology, 3 Nobel Street, 121205 Moscow, Russia; r.sengupta@skoltech.ru (R.S.);

ernesto.campos@skoltech.ru (E.C.); akshay.vishwanathan@skoltech.ru (V.A.); j.biamonte@skoltech.ru (J.B.)

**\*** Correspondence: daniil.rabinovich@skoltech.ru

**Abstract:** The quantum approximate optimisation algorithm is a *p* layer, time variable split operator method executed on a quantum processor and driven to convergence by classical outer-loop optimisation. The classical co-processor varies individual application times of a problem/driver propagator sequence to prepare a state which approximately minimises the problem's generator. Analytical solutions to choose optimal application times (called parameters or angles) have proven difficult to find, whereas outer-loop optimisation is resource intensive. Here we prove that the optimal quantum approximate optimisation algorithm parameters for *p* = 1 layer reduce to one free variable and in the thermodynamic limit, we recover optimal angles. We moreover demonstrate that conditions for vanishing gradients of the overlap function share a similar form which leads to a linear relation between circuit parameters, independent of the number of qubits. Finally, we present a list of numerical effects, observed for particular system size and circuit depth, which are ye<sup>t</sup> to be explained analytically.

**Keywords:** variatonal algorithms; QAOA; quantum circuit optimization

**MSC:** 81P68
