Revisiting the simulation of quantum Turing machines by quantum circuits

Yao's 1995 publication 'Quantum circuit complexity' in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, pp. 352-361, proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t >= n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t, and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t, rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of method described by Arright, Nesme and Werner in 2011 in Journal of Computer and System Sciences 77, 372-378. (doi:10.1016/j.jcss.2010.05.004), that allows for the localization of causal unitary evolutions.