Revisiting the simulation of quantum Turing machines by quantum circuits

TL;DR

This paper improves the simulation of quantum Turing machines by quantum circuits, reducing the circuit depth from quadratic to linear in the number of steps, and extends the method to more complex variants.

Contribution

It introduces a new simulation technique that achieves linear depth circuits for quantum Turing machines and extends the approach to multi-dimensional tape variants.

Findings

Quantum Turing machine simulation depth reduced to linear in steps

Simulation method extended to multi-dimensional tapes

Analysis based on localization of causal unitary evolutions

Abstract

Yao (1993) proved that quantum Turing machines and uniformly generated quantum circuits are polynomially equivalent computational models: $t \geq n$ 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 a method of Arrighi, Nesme, and Werner (2011) that allows for the localization of causal unitary evolutions.