Turing machines based on quantum logic can solve undecidable problems. In this paper we will give recursion-theoretical characterization of the computational power of this kind of quantum Turing machines. In detail, for the unsharp case, it is proved that Σ01 ∪Π01⊆ LTd(ε,Σ)(LTw(ε,Σ))⊆Π02 when the truth value lattice is locally finite and the operation ∧ is computable, where LTd(ε,Σ)(LTw(ε,Σ))denotes the class of quantum language accepted by these Turing machine in depth-first model (respectively, width-first model); for the sharp case, we can obtain similar results for usual orthomodular lattices.
Turing machines based on quantum logic can solve undecidable problems. In this paper we will give recursion-theoretical characterization of the computational power of this kind of quantum Turing machines. In detail, for the unsharp case, it is proved that Σ01 ∪Π01⊆ LTd(ε,Σ)(LTw(ε,Σ))⊆Π02 when the truth value lattice is locally finite and the operation ∧ is computable, where LTd(ε,Σ)(LTw(ε,Σ))denotes the class of quantum language accepted by these Turing machine in depth-first model (respectively, width-first model); for the sharp case, we can obtain similar results for usual orthomodular lattices.
Quantum computing allows us to solve some problems much faster than existing classical algorithms. Yet, the quantum computer has been believed to be no more powerful than the most general computing model—the Turing machine. Undecidable problems, such as the halting problem, and unrecognizable inputs, such as the real numbers, are beyond the theoretical limit of the Turing machine. I suggest a model for a quantum computer, which is less general than the Turing machine, but may solve the halting problem for any task programmable on it. Moreover, inputs unrecognizable by the Turing machine can be recognized by the model, thus breaking the theoretical limit for a computational task. A quantum computer is not just a successful design of the Turing machine as it is widely perceived now, but is a different, less general but more powerful model for computing, the practical realization of which may need different strategies than those in use now.
In this thesis, we introduce a new quantum Turing machine (QTM) model that supports general quantum operators, together with its pushdown, counter, and finite automaton variants, and examine the computational power of classical and quantum machines using small space bounds in many different cases. The main contributions are summarized below. Firstly, we consider QTMs in the unbounded error setting: (i) in some cases of sublogarithmic space bounds, the class of languages recognized by QTMs is shown to be strictly larger than that of classical ones; (ii) in constant space bounds, the same result can still be obtained for restricted QTMs; (iii) the complete characterization of the class of languages recognized by realtime constant space nondeterministic QTMs is given. Secondly, we consider constant space-bounded QTMs in the bounded error setting: (i) we introduce a new type of quantum and probabilistic finite automata (QFAs and PFAs, respectively,) with a special two-way input head which is not allowed to be stationary or move to the left but has the capability to reset itself to its starting position; (ii) the computational power of this type of quantum machine is shown to be superior to that of the probabilistic machine; (iii) based on these models, two-way PFAs and two-way classical-head QFAs are shown to be more succinct than two-way nondeterministic finite automata and their one-way variants; (iv) we also introduce PFAs and QFAs with postselection with their bounded error language classes, and give many characterizations of them. Thirdly, the computational power of realtime QFAs augmented with a write-only memory is investigated by showing many simulation results for different kinds of counter automata. Finally, some lower bounds of realtime classical Turing machines in order to recognize a nonregular language are shown to be tight.
We present results on quantum Turing machines and on prover-verifier interactions. \n \n \nIn our work on quantum Turing machines, we continue the line of research opened by Yao (1993), who 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 then first revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, 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 easily to many 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, involving abstract lemmas that might be of independent interest. \n \n \nWe also consider the more complex extension of our variant to the circuit simulation of multi-tape quantum Turing machines, where our variant provides a circuit with O(t^k) size and O(t^{k-1}) depth for the simulation of t steps of a machine with k tapes. This can be contrasted with the O(t^{k}) depth corresponding to the generalization of Yao's simulation by Nishimura and Ozawa (2002). Our usage of abstract techniques regarding the localization of causal unitary evolutions allows again for a simplification of the algebraic manipulation aspects of the construction. We also discuss the further extension to the case of oracle quantum Turing machines. \n \n \nIn our work on prover-verifier interactions, we first consider a protocol under the name of perfect/conclusive quantum state exclusion. This means to be able to discard with certainty at least one out of n possible quantum state preparations by performing a measurement of the resulting state. When all the preparations correspond to pure states and there are no more of them than their common dimension, it is an open problem whether POVMs give any additional power for this task with respect to projective measurements. This is the case even for the simple case of three states in three dimensions, which is discussed by Caves, Fuchs and Schack (2002) as unsuccessfully tackled. In our work, we give an analytical proof that in this case POVMs do indeed not give any additional power with respect to projective measurements. We also discuss possible generalizations of our work, including an application of Quadratically Constrained Quadratic Programming that might be of special interest. \n \n \nWe additionally consider the problem of quantum hedging, a particular kind of quantum correlation that arises between parallel instances of prover-verifier interactions. M. and Watrous (2012) studied a protocol that exhibited a perfect form of quantum hedging, where the risk for the prover of losing a first game can completely offset the corresponding risk for a second game. We take a step towards a better understanding of this hedging phenomenon by giving a characterization of the prover's optimal behavior for a natural generalization of this protocol. Furthermore, we discuss how the usage of the logarithmic utility principle to analyze prover-verifier interactions could justify further study of quantum hedging.
We study the observation of quantum Turing machines by allowing interactions between a quantum machine and its environment during the computation, whereas a quantum Turing machine --original model introduced by Deutsch-- remains isolated. We show that the introduction of observations leads to a weakening of the well formedness conditions of quantum Turing machines such that any (reversible or not) classical Turing machine is a special instance of a quantum machine. Moreover, observation of quantum Turing machines provides a formal solution to the halting process problem: the impossibility to know whether a given quantum Turing machine has actually reached its halting state. It also provides a more realistic abstract architecture of a quantum computer while most of the physical proposals of quantum computers are based on an hybrid classical-quantum architecture. However, we show that a natural formalisation of an observable quantum Turing machines leads to an over-powerful model solving undecidable problems. As a consequence, we introduce a restricted version of observable quantum Turing machines and we show that, under this restriction, any observable quantum Turing machine can be efficiently simulated by a quantum Turing machine. Finally, we discuss the potential application of the observable quantum Turing machine in the quest of a universal quantum Turing machine while recent papers have pointed out that a classical control is a key feature of a universal quantum machine.
Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not been fully investigated. In particular, there are features of QTMs that have not been exploited, a notable example being the intrinsic infinite nature of any quantum computation. In this paper, we propose a definition of QTM, which extends and unifies the notions of Deutsch and Bernstein & Vazirani. In particular, we allow both arbitrary quantum input, and meaningful superpositions of computations, where some of them are “terminated” with an “output”, while others are not. For some infinite computations an “output” is obtained as a limit of finite portions of the computation. We propose a natural and robust observation protocol for our QTMs, which does not modify the probability of the possible outcomes of the machines. Finally, we use QTMs to define a class of quantum computable functions—any such function is a mapping from a general quantum state to a probability distribution of natural numbers. We expect that our class of functions, when restricted to classical input-output, will not be different from the set of the recursive functions.
Abstract According to the Church-Turing Thesis, a Turing machine, and thus the programming languages and computers in existence, can simulate any reasonable computing device. However, there are physical devices that are exponentially more efficient than a Turing machine, and hence would prove exponentially difficult to simulate using such a machine. This chapter explores the concept of quantum computers and their potential not only to simulate quantum physics exponentially faster than classical computers but also to solve other problems exponentially faster. It considers algorithms that break cryptosystems, find needles in haystacks, and predict who can win a game. It first describes particles, waves, and amplitudes in quantum computation before turning to states and operators. It then examines how interference lets a quantum computer learn some things about a function, how quantum computers can solve factoring in polynomial time, the link between factoring and modern cryptography, graph isomorphism and the hidden subgroup problem, and quantum walks and scattering.