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Hamiltonian models of quantum computers which evolve quantum ballistically

Paul A. Benioff

Published Dec 31, 1996• 1 citation
OpenAlex

Abstract

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.

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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 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.
Validation of a presumably universal theory, such as quantum mechanics, requires a quantum mechanical description of systems that carry out theoretical calculations and sytems that carry out experiments. The description of quantum computers is under active development. No description of systems to carry out experiments has been given. A very small step in this direction is taken here by giving a description of quantum robots as mobile systems with on board quantum computers that interact with environments of quantum systems. The dynamics of these systems are described in terms of tasks that consist of sequences of computation and action phases. For each task a step operator is defined which can then be used to define finite time interval step dynamics or a task Hamiltonian. A specific task carried out on a very simple environment is used to illustrate the models.
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
We have seen how information can be encoded onto a quantum system by selecting the state in which it is prepared. Retrieving the information is achieved by performing a measurement, and the optimal measurement in any given situation is usually a generalized measurement. In between preparation and measurement, the information resides in the quantum state of the system, which evolves in a manner determined by the Hamiltonian. The associated unitary transformation may usefully be viewed as quantum information processing; if we can engineer an appropriate Hamiltonian then we can use the quantum evolution to assist in performing computational tasks. Our objective in quantum information processing is to implement a desired unitary transformation. Typically this will mean coupling together a number, perhaps a large number, of qubits and thereby generating highly entangled states. It is fortunate, although by no means obvious, that we can realize any desired multiqubit unitary transformation as a product of a small selection of simple transformations and, moreover, that each of these need only act on a single qubit or on a pair of qubits. The situation is reminiscent of digital electronics, in which logic operations are decomposed into actions on a small number of bits. If we can realize and control a very large number of such operations in a single device then we have a computer. Similar control of a large number of qubits will constitute a quantum computer. It is the revolutionary potential of quantum computers, more than any other single factor, that has fuelled the recent explosion of interest in our subject. We shall examine the remarkable properties of quantum computers in the next chapter. In digital electronics, we represent bit values by voltages: the logical value 1 is a high voltage (typically +5 V) and 0 is the ground voltage (0 V). The voltage bits are coupled and manipulated by transistor-based devices, or gates. The simplest gates act on only one bit or combine two bits to generate a single new bit, the value of which is determined by the two input bits. For a single bit, with value 0 or 1, the only possible operations are the identity (which does not require a gate) and the bit flip.
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.