Quantum Robots and Environments

Quantum robots and their interactions with environments of quantum systems are described and their study justified. A quantum robot is a mobile quantum system that includes a quantum computer and needed ancillary systems on board. Quantum robots carry out tasks whose goals include specified changes in the state of the environment or carrying out measurements on the environment. Each task is a sequence of alternating computation and action phases. Computation phase activities include determination of the action to be carried out in the next phase and possible recording of information on neighborhood environmental system states. Action phase activities include motion of the quantum robot and changes of neighborhood environment system states. Models of quantum robots and their interactions with environments are described using discrete space and time. To each task is associated a unitary step operator T that gives the single time step dynamics. T = T_{a}+T_{c} is a sum of action phase and computation phase step operators. Conditions that T_{a} and T_{c} should satisfy are given along with a description of the evolution as a sum over paths of completed phase input and output states. A simple example of a task carrying out a measurement on a very simple environment is analyzed. A decision tree for the task is presented and discussed in terms of sums over phase paths. One sees that no definite times or durations are associated with the phase steps in the tree and that the tree describes the successive phase steps in each path in the sum.Comment: 30 Latex pages, 3 Postscript figures, Minor mathematical corrections, accepted for publication, Phys Rev

A lambda calculus for quantum computation with classical control

The objective of this paper is to develop a functional programming language for quantum computers. We develop a lambda calculus for the classical control model, following the first author's work on quantum flow-charts. We define a call-by-value operational semantics, and we give a type system using affine intuitionistic linear logic. The main results of this paper are the safety properties of the language and the development of a type inference algorithm.Comment: 15 pages, submitted to TLCA'05. Note: this is basically the work done during the first author master, his thesis can be found on his webpage. Modifications: almost everything reformulated; recursion removed since the way it was stated didn't satisfy lemma 11; type inference algorithm added; example of an implementation of quantum teleportation adde

Entanglement production in quantum decision making

The quantum decision theory introduced recently is formulated as a quantum theory of measurement. It describes prospect states represented by complex vectors of a Hilbert space over a prospect lattice. The prospect operators, acting in this space, form an involutive bijective algebra. A measure is defined for quantifying the entanglement produced by the action of prospect operators. This measure characterizes the level of complexity of prospects involved in decision making. An explicit expression is found for the maximal entanglement produced by the operators of multimode prospects.Comment: Latex file, 7 page

Transmission and Spectral Aspects of Tight Binding Hamiltonians for the Counting Quantum Turing Machine

It was recently shown that a generalization of quantum Turing machines (QTMs), in which potentials are associated with elementary steps or transitions of the computation, generates potential distributions along computation paths of states in some basis B. The distributions are computable and are thus periodic or have deterministic disorder. These generalized machines (GQTMs) can be used to investigate the effect of potentials in causing reflections and reducing the completion probability of computations. This work is extended here by determination of the spectral and transmission properties of an example GQTM which enumerates the integers as binary strings. A potential is associated with just one type of step. For many computation paths the potential distributions are initial segments of a quasiperiodic distribution that corresponds to a substitution sequence. The energy band spectra and Landauer Resistance (LR) are calculated for energies below the barrier height by use of transfer matrices. The LR fluctuates rapidly with momentum with minima close to or at band-gap edges. For several values of the parameters, there is good transmission over some momentum regions.Comment: 22 pages Latex, 13 postscript figures, Submitted to Phys. Rev.

The Representation of Natural Numbers in Quantum Mechanics

This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This work is limited to k-ary representations of length L and to the axioms for arithmetic modulo k^{L}. A model of the axioms is described based on states in and operators on an abstract L fold tensor product Hilbert space H^{arith}. Unitary maps of this space onto a physical parameter based product space H^{phy} are then described. Each of these maps makes states in H^{phy}, and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's Algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This conditions states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.Comment: Much rewrite, including response to comments. To Appear in Phys. Rev.

On the Interpretation of Energy as the Rate of Quantum Computation

Over the last few decades, developments in the physical limits of computing and quantum computing have increasingly taught us that it can be helpful to think about physics itself in computational terms. For example, work over the last decade has shown that the energy of a quantum system limits the rate at which it can perform significant computational operations, and suggests that we might validly interpret energy as in fact being the speed at which a physical system is "computing," in some appropriate sense of the word. In this paper, we explore the precise nature of this connection. Elementary results in quantum theory show that the Hamiltonian energy of any quantum system corresponds exactly to the angular velocity of state-vector rotation (defined in a certain natural way) in Hilbert space, and also to the rate at which the state-vector's components (in any basis) sweep out area in the complex plane. The total angle traversed (or area swept out) corresponds to the action of the Hamiltonian operator along the trajectory, and we can also consider it to be a measure of the "amount of computational effort exerted" by the system, or effort for short. For any specific quantum or classical computational operation, we can (at least in principle) calculate its difficulty, defined as the minimum effort required to perform that operation on a worst-case input state, and this in turn determines the minimum time required for quantum systems to carry out that operation on worst-case input states of a given energy. As examples, we calculate the difficulty of some basic 1-bit and n-bit quantum and classical operations in an simple unconstrained scenario.Comment: Revised to address reviewer comments. Corrects an error relating to time-ordering, adds some additional references and discussion, shortened in a few places. Figures now incorporated into tex

A Quantum Broadcasting Problem in Classical Low Power Signal Processing

We pose a problem called ``broadcasting Holevo-information'': given an unknown state taken from an ensemble, the task is to generate a bipartite state transfering as much Holevo-information to each copy as possible. We argue that upper bounds on the average information over both copies imply lower bounds on the quantum capacity required to send the ensemble without information loss. This is because a channel with zero quantum capacity has a unitary extension transfering at least as much information to its environment as it transfers to the output. For an ensemble being the time orbit of a pure state under a Hamiltonian evolution, we derive such a bound on the required quantum capacity in terms of properties of the input and output energy distribution. Moreover, we discuss relations between the broadcasting problem and entropy power inequalities. The broadcasting problem arises when a signal should be transmitted by a time-invariant device such that the outgoing signal has the same timing information as the incoming signal had. Based on previous results we argue that this establishes a link between quantum information theory and the theory of low power computing because the loss of timing information implies loss of free energy.Comment: 28 pages, late

Error Rate of the Kane Quantum Computer CNOT Gate in the Presence of Dephasing

We study the error rate of CNOT operations in the Kane solid state quantum computer architecture. A spin Hamiltonian is used to describe the system. Dephasing is included as exponential decay of the off diagonal elements of the system's density matrix. Using available spin echo decay data, the CNOT error rate is estimated at approsimately 10^{-3}.Comment: New version includes substantial additional data and merges two old figures into one. (12 pages, 6 figures

Geophysical studies with laser-beam detectors of gravitational waves

The existing high technology laser-beam detectors of gravitational waves may find very useful applications in an unexpected area - geophysics. To make possible the detection of weak gravitational waves in the region of high frequencies of astrophysical interest, ~ 30 - 10^3 Hz, control systems of laser interferometers must permanently monitor, record and compensate much larger external interventions that take place in the region of low frequencies of geophysical interest, ~ 10^{-5} - 3 X 10^{-3} Hz. Such phenomena as tidal perturbations of land and gravity, normal mode oscillations of Earth, oscillations of the inner core of Earth, etc. will inevitably affect the performance of the interferometers and, therefore, the information about them will be stored in the data of control systems. We specifically identify the low-frequency information contained in distances between the interferometer mirrors (deformation of Earth) and angles between the mirrors' suspensions (deviations of local gravity vectors and plumb lines). We show that the access to the angular information may require some modest amendments to the optical scheme of the interferometers, and we suggest the ways of doing that. The detailed evaluation of environmental and instrumental noises indicates that they will not prevent, even if only marginally, the detection of interesting geophysical phenomena. Gravitational-wave instruments seem to be capable of reaching, as a by-product of their continuous operation, very ambitious geophysical goals, such as observation of the Earth's inner core oscillations.Comment: 29 pages including 8 figures, modifications and clarifications in response to referees' comments, to be published in Class. Quant. Gra

Quantum Computation in Quantum-Hall Systems

We describe a quantum information processor (quantum computer) based on the hyperfine interactions between the conduction electrons and nuclear spins embedded in a two-dimensional electron system in the quantum-Hall regime. Nuclear spins can be controlled individually by electromagnetic pulses. Their interactions, which are of the spin-exchange type, can be possibly switched on and off pair-wise dynamically, for nearest neighbors, by controlling impurities. We also propose the way to feed in the initial data and explore ideas for reading off the final results.Comment: 12 pages in LaTeX + 1 PostScript figur