Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity

Abstract

Abstract-We show that there exists a universal quantum Turing machine (UQTM) that can simulate every other QTM until the other QTM has halted and then halt itself with probability one. This extends work by Bernstein and Vazirani who have shown that there is a UQTM that can simulate every other QTM for an arbitrary, but preassigned number of time steps.As a corollary to this result, we give a rigorous proof that quantum Kolmogorov complexity as defined by Berthiaume et al. is invariant, i.e. depends on the choice of the UQTM only up to an additive constant.Our proof is based on a new mathematical framework for QTMs, including a thorough analysis of their halting behaviour. We introduce the notion of mutually orthogonal halting spaces and show that the information encoded in an input qubit string can always be effectively decomposed into a classical and a quantum part.

An overview of quantum computation models: quantum automata

“…[34], Deutsch [6] also proved the existence of a universal QTM simulating any given QTM, however, with exponential slowdown.) Recently, Müller [39] gave a type of strongly universal QTM, extending the work by Bernstein and Vazirani [34]. Also in Ref.…”

“…Quantum automata involve various types, as classical automata [10], and include quantum finite automata (QFA) [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], quantum sequential machines (QSM) [29][30][31], quantum pushdown automata (QPDA) [26,32,33], quantum Turing machines (QTM) [6,[34][35][36][37][38][39], quantum multi-counter machines (QMCM) [38], quantum multi-stack machines (QMSM) [38], quantum cellular automata (QCA) [1,40,41], and automata theory based on quantum logic, called orthomodular latticevalued finite automata (l-VFA) [42][43][44][45][46][47][48][49][50] as well as the other models. In this paper, we only review QFA, QSM, QPDA, QTM, and l-VFA.…”

Universality of quantum Turing machines with deterministic control

“…In [3,12], the Kolmogorov complexity of a qubit string is defined as the length of the shortest quantum bit string that given to a given quantum universal Turing machine produces the qubit string with high fidelity. This notion has a similar flavor to Vitányi's approach, but, rather than considering only classical programs, it considers the possibility of quantum ones.…”

Quantum Dynamical Entropies and Gács Algorithmic Entropy

“…into itself: these maps are interpreted as Quantum Turing Machines (see [22] for a detailed mathematical characterization and a discussion of the halting time in the context of the quantum superposition principle).…”