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Journal of Physics A Mathematical and General

Operation-induced decoherence by nonrelativistic scattering from a quantum memory

Dionisios Margetis, John M. Myers

Published Aug 29, 2006• 3 citations
Publisher

Abstract

Quantum computing involves transforming the state of a quantum memory. We view this operation as performed by transmitting nonrelativistic (massive) particles that scatter from the memory. By using a system of (1+1)-dimensional, coupled Schrödinger equations with point interaction and narrow-band incoming pulse wavefunctions, we show how the outgoing pulse becomes entangled with a two-state memory. This effect necessarily induces decoherence, i.e., deviations of the memory content from a pure state. We describe incoming pulses that minimize this decoherence effect under a constraint on the duration of their interaction with the memory.

DOI
10.1088/0305-4470/39/37/015

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Quantum computing involves transforming the state of a quantum memory. We view this operation as performed by transmitting nonrelativistic (massive) particles that scatter from the memory. By using a system of (1+1)-dimensional, coupled Schrödinger equations with point interaction and narrow-band incoming pulse wavefunctions, we show how the outgoing pulse becomes entangled with a two-state memory. This effect necessarily induces decoherence, i.e., deviations of the memory content from a pure state. We describe incoming pulses that minimize this decoherence effect under a constraint on the duration of their interaction with the memory.
An essential component of any quantum computer is the quantum memory, the content of which is a pure quantum state. A program to study the quantum memory is initiated here, where the spatial variables are of central importance. The presence of the spatial variables makes it possible to apply the powerful and well-developed theory of scattering: The fundamental operations of writing on, reading and resetting the quantum memory are all performed through scattering from the memory. The requirement that the quantum memory must remain in a pure state after scattering implies that the scattering is of a special type, and only certain incident waves are admissible. Models based on the coupled-channel Schrodinger equation for potential scattering are formulated, where there is indeed the required large collection of admissible incident waves. On the basis of these models, certain types of decoherence are unavoidable. Such decoherence and the necessity of using the relativistic Schrodinger equation are discussed. One of the implications of quantum memory is the possible lack of security for the quantum key distribution in quantum cryptography.
The interaction of a traveling pulse of quantum light with a localized quantum system shows non-Markovian features when scattering takes place in a structured environment. Here, we devise a non-Markovian input-output theory by coherently coupling the scatterer to a pseudomode, which decays into a Markovian reservoir. This incorporates memory effects of a structured environment without altering the cascaded nature of the Lindblad master equation, whose solution provides the quantum state of the output field of any desired mode. We apply our theory to the stimulated emission by a two-level atom, and to transmission of a Gaussian pulse through a cavity. We observe that the non-Markovian revival of coherence in the scatterer distorts the single-mode nature of the incoming pulse, thus resulting in a multimode output field.
To a large extent, the power of a quantum computer comes from the possibility of operating on a number of quantum memories simultaneously. For example, if there are n linearly independent quantum states in each of N quantum memories, then the number of linearly independent quantum states is n^N when they are taken together. This exponential increase is the basis of the present interest in quantum computing. In the context of quantum mechanics as described by the Schrodinger equation (which is a partial differential equation), the most explicit model of the quantum memory is one with two linearly independent quantum states described in terms of the Fermi pseudo-potential at one point. In this model, operations on the quantum memory are accomplished through scattering with both symmetrical and anti-symmetrical incident waves. As a first step toward operating on more than one quantum memory, this model is generalized in two directions. (1) With the Fermi pseudo-potential at one point retained, three linearly independent quantum states are used for the quantum memory. This model of the quantum memory requires three external connections. (2) With a more general potential, operations on the quantum memory with two states are accomplished through scattering with either symmetrical or anti-symmetrical incident waves. This second generalization is important because it is not practical to keep the number of external connections equal to the number of linearly independent quantum states.
Decoherence in quantum-computer memory due to the inevitable coupling to the external environment is examined. We make the assumption that all quantum bits (qubits) interact with the same environment rather than the assumption of separate environments for different qubits. It is found that the qubits decohere collectively. For some kinds of entangled input states, no decoherence occurs at all in the memory, even if the qubits are interacting with the environment. Based on this phenomenon, a scheme is proposed for reducing the collective decoherence. We also discuss possible implications of this decoherence model for quantum measurements.
Models of quantum computing rely on transformations of the states of a quantum memory. We study mathematical aspects of a model proposed by Wu in which the memory state is changed via the scattering of incoming particles. This operation causes the memory content to deviate from a pure state, i.e. induces impurity. For nonrelativistic particles scattered from a two-state memory and sufficiently general interaction potentials in 1+1 dimensions, we express impurity in terms of quaternionic commutators. In this context, pure memory states correspond to null hyperbolic quaternions. In the case with point interactions, the scattering process amounts to appropriate rotations of quaternions in the frequency domain. Our work complements a previous analysis by Margetis and Myers (2006 J. Phys. A 39 11567–11581).
Models of quantum computing rely on transformations of the states of a quantum memory. We study mathematical aspects of a model proposed by Wu in which the memory state is changed via the scattering of incoming particles. This operation causes the memory content to deviate from a pure state, i.e. induces impurity. For nonrelativistic particles scattered from a two-state memory and sufficiently general interaction potentials in 1+1 dimensions, we express impurity in terms of quaternionic commutators. In this context, pure memory states correspond to null hyperbolic quaternions. In the case with point interactions, the scattering process amounts to appropriate rotations of quaternions in the frequency domain. Our work complements a previous analysis by Margetis and Myers (2006 J. Phys. A 39 11567--11581).