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Journal of Mathematical Physics

Quantum mechanical Hamiltonian models of discrete processes

Paul A. Benioff

Published Mar 1, 1981• 20 citations
Publisher

Abstract

Here the results of other work on quantum mechanical Hamiltonian models of Turing machines are extended to include any discrete process T on a countably infinite set A. The models are constructed here by use of scattering phase shifts from successive scatterers to turn on successive step interactions. Also a locality requirement is imposed. The construction is done by first associating with each process T a model quantum system M with associated Hilbert space ℋM and step operator UT. Since UT is not unitary in general, M, ℋM, and UT are extended into a (continuous time) Hamiltonian model on a larger space which satisfies the locality requirement. The construction is compared with the minimal unitary dilation of UT. It is seen that the model constructed here is larger than the minimal one. However, the minimal one does not satisfy the locality requirement.

DOI
10.1063/1.524938

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Here the results of other work on quantum mechanical Hamiltonian models of Turing machines are extended to include any discrete process T on a countably infinite set A. The models are constructed here by use of scattering phase shifts from successive scatterers to turn on successive step interactions. Also a locality requirement is imposed. The construction is done by first associating with each process T a model quantum system M with associated Hilbert space ℋM and step operator UT. Since UT is not unitary in general, M, ℋM, and UT are extended into a (continuous time) Hamiltonian model on a larger space which satisfies the locality requirement. The construction is compared with the minimal unitary dilation of UT. It is seen that the model constructed here is larger than the minimal one. However, the minimal one does not satisfy the locality requirement.
A goal of quantum-mechanical models of the computation process is the description of operators that model changes in the information-bearing degrees of freedom. Iteration of the operators should correspond to steps in the computation, and the final state of halting computations should be stable under iteration. The problem is that operators constructed directly from the process description do not have these properties. In general these operators annihilate the halted state. If information-erasing steps are present, there are additional problems. These problems are illustrated in this paper by consideration of operators for two simple one-step processes and two simple Turing machines. In general the operators are not unitary and, if erasing steps are present, they are not even contraction operators. Various methods of extension or dilation to unitary operators are discussed. Here unitary power dilations are considered as a solution to these problems. It is seen that these dilations automatically provide a good solution to the initial- and final-state problems. For processes with erasing steps, recording steps must be included prior to the dilation, but only for the steps that erase information. Hamiltonians for these processes are also discussed. It is noted that H, described by exp(-iH\ensuremath{\Delta})=UT{\mathit{U}}^{\mathit{T}}UT, where UT{\mathit{U}}^{\mathit{T}}UT is a unitary step operator for the process and \ensuremath{\Delta} a time interval, has complexity problems. These problems and those noted above are avoided here by the use of the Feynman approach to constructing Hamiltonians directly from the unitary power dilations of the model operators. It is seen that the Hamiltonians so constructed have some interesting properties.
Quantum Turing machines are discussed and reviewed in this paper. Most of the paper is concerned with processes defined by a step operator T that is used to construct a Hamiltonian H according to Feynman's prescription. Differences between these models and the models of Deutsch are discussed and reviewed. It is emphasized that the models with H constructed from T include fully quantum mechanical processes that take computation basis states into linear superpositions of these states. The requirement that T be distinct path generating is reviewed. The advantage of this requirement is that Schrödinger evolution under H is one dimensional along distinct finite or infinite paths of nonoverlapping states in some basis BT. It is emphasized that BT can be arbitrarily complex with extreme entanglements between states of component systems. The new aspect of quantum Turing machines introduced here is the emphasis on the structure of graphs obtained when the states in the BT paths are expanded as linear superpositions of states in a reference basis such as the computation basis BC. Examples are discussed that illustrate the main points of the paper. For one example the graph structures of the paths in BT expanded as states in BC include finite stage binary trees and concatenated finite stage binary trees with or without terminal infinite binary trees. Other examples are discussed in which the graph structures correspond to interferometers and iterations of interferometers. 03.65.Bz,89.70.+c
In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch's model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97--117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specified. We prove that O(log⁡T)O(\log T)O(logT) bits of precision suffice to support a T step computation. This justifies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class $\BPP$. The class $\BQP$ of languages that are efficiently decidable (with small error-probability) on a quantum Turing machine satisfies $\BPP \subseteq \BQP \subseteq \Ptime^{\SP}$. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.
This paper extends work done to date on quantum computation by association of potentials with different types of steps. Quantum Turing machine Hamiltonians, generalized to include potentials, correspond to sums over tight binding Hamiltonians each with a different potential distribution. Which distribution applies is determined by the initial state. An example, which enumerates the integers in succession as binary strings, is analyzed. It is seen that for some initial states, the potential distributions have quasicrystalline properties and are similar to a substitution sequence.
Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our rst result is the existence of an ecient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97{117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specied. We prove that O(log T) bits of precision suce to support a T step computation. This justies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the rst formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church{Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are eciently decidable (with small error-probability) on a quantum Turing machine satises BPP ⊆ BQP ⊆ P]P. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.
Quantum computation is a subject of much recent interest. In much of the work in the literature quantum computers are described as built up from a sequence of unitary operators where each unitary operator carries out a stage of the overall quantum computation. The sequence and connection of the different unitary operators is provided presumably by some external agent which governs the overall process. However there is no description of a an overall Hamiltonian needed to give the actual quantum dynamics of the computation process. In this talk, earlier work by the author is followed in that simple, time independent Hamiltonians are used to describe quantum computation, and the Schroedinger evolution of the computation system is considered to be quantum ballistic. However, the definition of quantum ballistic evolution used here is more general than that used in the earlier work. In particular, the requirement that the step operator {ital T} associated with a process be a partial isometry, used in, is relaxed to require that {ital T} be a contraction operator. (An operator {ital T} is a partial isometry if the self-adjoint operators T{sup {dagger}}T and TT{sup {dagger}} are also projection operators.{ital T} is a contraction operator if {vert_bar}{vert_bar} {ital T} {vert_bar}{vert_bar} {<=} 1.) The main purpose of this talk is to investigate some consequences for quantum computation under this weaker requirement. It will be seen that system motion along discrete paths in a basis still occurs. However the motion occurs in ,the presence of potentials whose height and distribution along the path depends on {ital T} and the path states.