Classical simulation of restricted cluster state quantum circuits

Abstract

A fundamental open problem in quantum computing is to understand when quantum systems can or cannot be efficiently classically simulated. In this thesis, we study when cluster state quantum circuits with alternative input states (parameterised by a radius r) and measurements in the Z basis and XY-plane, can or cannot be efficiently classically simulated. In the first part of this thesis, we study when such a system can be efficiently classically simulated. The main technical tool we consider is a generalised notion of separability in terms of cylindrical operators. We first show that if a CZ gate acts on cylindrical operators with radius r, then the output can be given a separable decomposition provided the radius of the cylindrical operators in the decomposition grows by a constant λ > 0. By combining this with a modified version of a previous algorithm, we find that this enables an efficient classical simulation algorithm that can sample from the output distribution to within additive error. We then use a coarse-graining approach to increase the range of input states that can be efficiently classically simulated. We then compute the equivalent of λ for arbitrary diagonal two-qubit gates. Lastly, we use alternative notions of entanglement to show that there are state spaces that can improve the region of input states that can be classically simulated. In the second part, we examine potential obstacles that may arise when attempting to efficiently classically simulate an increased range of quantum input states. Using a percolation based approach, we show that if the input states are permitted with sufficient radius, then BQP can be supported and efficient classical simulation is unlikely. Furthermore, using conjectures about the polynomial hierarchy, we show that there is a threshold for which cluster states with alternative inputs cannot be efficiently classically simulated with multiplicative error.