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arXiv:quant-ph/9605022 · Physical Review A

Quantum ballistic evolution in quantum mechanics: Application to quantum computers

Paul A. Benioff

Published Aug 1, 1996• 14 citations
arXivHTMLPDF

Abstract

Quantum computers are important examples of processes whose evolution can be described in terms of iterations of single-step operators or their adjoints. Based on this, Hamiltonian evolution of processes with associated step operators T is investigated here. The main limitation of this paper is to processes which evolve quantum ballistically, i.e., motion restricted to a collection of nonintersecting or distinct paths on an arbitrary basis. The main goal of this paper is proof of a theorem which gives necessary and sufficient conditions that T must satisfy so that there exists a Hamiltonian description of quantum ballistic evolution for the process, namely, that T is a partial isometry and is orthogonality preserving and stable on some basis. Simple examples of quantum ballistic evolution for quantum Turing machines with one and with more than one type of elementary step are discussed. It is seen that for nondeterministic machines the basis set can be quite complex with much entanglement present. It is also proven that, given a step operator T for an arbitrary deterministic quantum Turing machine, it is decidable if T is stable and orthogonality preserving, and if quantum ballistic evolution is possible. The proof fails if T is a step operator for a nondeterministic machine. It is an open question if such a decision procedure exists for nondeterministic machines. This problem does not occur in classical mechanics. Also the definition of quantum Turing machines used here is compared with that used by other authors. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1106

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Quantum computers are important examples of processes whose evolution can be described in terms of iterations of single-step operators or their adjoints. Based on this, Hamiltonian evolution of processes with associated step operators T is investigated here. The main limitation of this paper is to processes which evolve quantum ballistically, i.e., motion restricted to a collection of nonintersecting or distinct paths on an arbitrary basis. The main goal of this paper is proof of a theorem which gives necessary and sufficient conditions that T must satisfy so that there exists a Hamiltonian description of quantum ballistic evolution for the process, namely, that T is a partial isometry and is orthogonality preserving and stable on some basis. Simple examples of quantum ballistic evolution for quantum Turing machines with one and with more than one type of elementary step are discussed. It is seen that for nondeterministic machines the basis set can be quite complex with much entanglement present. It is also proven that, given a step operator T for an arbitrary deterministic quantum Turing machine, it is decidable if T is stable and orthogonality preserving, and if quantum ballistic evolution is possible. The proof fails if T is a step operator for a nondeterministic machine. It is an open question if such a decision procedure exists for nondeterministic machines. This problem does not occur in classical mechanics. Also the definition of quantum Turing machines used here is compared with that used by other authors. \textcopyright{} 1996 The American Physical Society.
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.
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
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.
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.
ABSTRACT: The main aim of this article is to present the time-evolution as the unitary and self-adjoint operators on Hilbert spaces and to describe its application in the development of quantum mechanics. The initial value problems associated with the quantum mechanical Schrodinger equation, ( ) = ( ) in the Hilbert space is solved by the use of timeevolution. The importance of time-evolution is also seen as the operation of turning machine in quantum mechanics, which is regarded as the time-evolution of the machine control state. Time-evolution of a quantum mechanical system is observed to be unitary and self-adjoint operators of Hilbert space, since it corresponds to an observable, specifically energy position and momentum. It was observed that time-evolution of a quantum mechanical system is generated by a self-adjoint operator, called Hamiltonian, V K H ˆ ˆ ˆ   , expressed by the Schrodinger equation, above.
The Unitary Dynamical Group in Schrödinger's Picture Up to now we have been concerned with the description of a physical system at a fixed instant of time. The time evolution of quantum systems constitutes a many-faced, wide issue which, however, is somewhat tangential to the main aim of this volume. We shall touch upon it only briefly, restricting ourselves to the more familiar aspects of the problem. We assume, to begin with, that the physical system has no superselection rules, so that states and physical quantities correspond one-to-one to density operators and self-adjoint operators, respectively, on the Hilbert space. To describe the dynamical evolution of the system one has to specify the way the representatives of states and physical quantities evolve in, which is fixed in time, since the physical system is supposed to preserve its identity. There is an old tenet according to which there are two kinds of time evolution: the discontinuous, nondeterministic evolution undergone by the quantum system when a measurement is made on it, and the continuous, deterministic evolution caused by the interaction with external forces or with other quantum systems. The first kind of evolution will be dealt with in Chapter 8, where we shall find it less alarming than is often claimed.