Universality of quantum Turing machines with deterministic control

Abstract

A simple notion of quantum Turing machine with deterministic, classical control is proposed and shown to be powerful enough to compute any unitary transformation which is computable by a finitely generated quantum circuit. An efficient universal machine with the s-m-n property is presented. The BQP class is recovered. A robust notion of plain Kolmogorov complexity of quantum states is proposed and compared with those previously reported in the literature.

Quantum machines with classical control

“…The previous results set the ground to show that quantum automata endowed with classical control recognise a family of regular languages that cannot be recognised by other types of quantum automata and moreover, with exponentially less states than deterministic finite automata [32]. Along the same line, by endowing classical Turing machines with a quantum tape, one can define a deterministic well behaved quantum Turing machine, where the usual classical theorems from computability can be derived [25]. Given the contribution of Amílcar Sernadas to these two elegant results, it is worthwhile to revisit them in this volume.…”

“…First, we focus on the expressivity of quantum automata with both quantum and classical states. We revise the result obtained in [32] where it was proved that such automata are able to recognise, with exponentially less states than deterministic finite automata, a family of regular languages that cannot be recognised by other types of quantum automata.Finally, we revise the concept of quantum Turing machine with classical control introduced in [25]. The novelty of these machines consists in the fact that their termination problem is completely deterministic, in opposition to other notions in the literature.…”

“…Finally, we revise the concept of quantum Turing machine with classical control introduced in [25]. The novelty of these machines consists in the fact that their termination problem is completely deterministic, in opposition to other notions in the literature.…”

A Private Quantum Bit String Commitment

“…We proceed now to prove the security of Protocol 1 in the Abstract Cryptography framework [ 22 ] instantiated with quantum Turing machines [ 23 ]. The equivalences that need to be satisfied are depicted in Figure 4 .…”

Kolmogorov One-Way Functions Revisited

“…In the literature, there are several definitions of (bounded) quantum Kolmogorov complexity [15][16][17][18][19][20] . One can study the adaptation of the results presented in this paper to address directly the characterization of one way-functions that are quantum resilient and provide insight regarding some (quantum) one-way function candidates such as the one presented in [14].…”