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Physical Review A

Decoherence and relevant universality in quantum algorithms via a dynamic theory for quantum measurement

C. P. Sun, Huijuan Zhan, X. F. Liu

Published Sep 1, 1998• 42 citations
Publisher

Abstract

It is well known that environment may decohere a quantum bit (qubit) system immersed in it, making a quantum computation invalid. But the quantitative features of the decoherence seem to depend on both the constitution of the environment and the details of its coupling with the qubit system. In this paper, based on the dynamic approach for quantum measurement developed from the Hepp-Coleman model [K. Hepp, Helv. Phys. Acta 45, 237 (1972)], we generally model the environment as a collection of a large number of subsystems and then consider to what extent and in which way the environment and its coupling with the qubit system may affect a quantum computation process. In the weak-coupling limit, we find that as far as decoherence time is concerned, there is no essential difference between an environment of two-level subsystems and an environment of harmonic oscillators. This implies that there exists some universality independent of specific constitutions of environments. However, it is also shown that this is not true at finite temperature or in the case of strong coupling. So only if the coupling is weak and the temperature low does there exist the possibility of developing a universal scheme of controlling a qubit system such that the decoherence is avoided. The possible effect of environment on the efficiency of a quantum algorithm is also explicitly illustrated through the example of Shor's prime factorization algorithm.

DOI
10.1103/physreva.58.1810

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It is well known that environment may decohere a quantum bit (qubit) system immersed in it, making a quantum computation invalid. But the quantitative features of the decoherence seem to depend on both the constitution of the environment and the details of its coupling with the qubit system. In this paper, based on the dynamic approach for quantum measurement developed from the Hepp-Coleman model [K. Hepp, Helv. Phys. Acta 45, 237 (1972)], we generally model the environment as a collection of a large number of subsystems and then consider to what extent and in which way the environment and its coupling with the qubit system may affect a quantum computation process. In the weak-coupling limit, we find that as far as decoherence time is concerned, there is no essential difference between an environment of two-level subsystems and an environment of harmonic oscillators. This implies that there exists some universality independent of specific constitutions of environments. However, it is also shown that this is not true at finite temperature or in the case of strong coupling. So only if the coupling is weak and the temperature low does there exist the possibility of developing a universal scheme of controlling a qubit system such that the decoherence is avoided. The possible effect of environment on the efficiency of a quantum algorithm is also explicitly illustrated through the example of Shor's prime factorization algorithm.
The possible effect of environment on the efficiency of a quantum algorithm is considered explicitely. It is illustrated through the example of Shor's prime factorization algorithm that this effect may be disastrous. The influence of environment on quantum computation is probed on the basis of its analogy to the problem of wave function collapse in quantum measurement.Techniques from the Hepp-Colemen approach and its generalization are used to deal with decoherence problems in quantum computation including dynamic mechanism of decoherence, quantum error avoiding tricks and calculation of decoherence time.
The dynamical evolution of a quantum register of arbitrary length coupled to an environment of arbitrary coherence length is predicted within a relevant model of decoherence. The results are reported for quantum bits (qubits) coupling individually to different environments (``independent decoherence'') and qubits interacting collectively with the same reservoir (``collective decoherence''). In both cases, explicit decoherence functions are derived for any number of qubits. The decay of the coherences of the register is shown to strongly depend on the input states: We show that this sensitivity is a characteristic of both types of coupling (collective and independent) and not only of the collective coupling, as has been reported previously. A nontrivial behavior (``recoherence'') is found in the decay of the off-diagonal elements of the reduced density matrix in the specific situation of independent decoherence. Our results lead to the identification of decoherence-free states in the collective decoherence limit. These states belong to subspaces of the system's Hilbert space that do not get entangled with the environment, making them ideal elements for the engineering of ``noiseless'' quantum codes. We also discuss the relations between decoherence of the quantum register and computational complexity based on the dynamical results obtained for the register density matrix.
Decoherence induced by coupling a system with an environment may display universal features. We demonstrate that when the coupling to the system drives a quantum phase transition in the environment, the decay of quantum coherences in the system is Gaussian with a width independent of the system-environment coupling strength. We obtain analytical results for a class of solvable models, and present numerical evidence supporting the validity of our results in more general cases. This effect opens the way for a quantum simulation algorithm, where a single qubit is used to detect a quantum phase transition. We discuss possible implementations of such an algorithm and relate our results to available data on universal decoherence in NMR echo experiments.
It is not always justified to model the environment of a realistic quantum system as a collection of harmonic oscillators or two-level systems. To explore interesting physics associated with a composite environment, here we study the dynamics of a qubit coupled to a second two-level system, which is in turn coupled to a harmonic-oscillator bath. That is, the qubit of interest is in the presence of a composite environment consisting of the second two-level system and a conventional harmonic-oscillator bath. We investigate the issue of pointer states of decoherence for the qubit for different coupling strengths between the qubit and such a composite environment. It is shown that for weak-, intermediate-, or strong-coupling strengths, the qubit's reduced density matrix has a coupling-strength-dependent diagonal representation, thus yielding coupling-strength-dependent computational pointer states. Interestingly, the associated decoherence rate can decrease as the coupling strength increases.
This paper analyzes the decoherence induced on a single qubit by the interaction with a spin chain with nontrivial internal dynamics (XYXYXY-type interactions). The aim of the paper is to study the existence and properties of the so-called universal regime, in which the decoherence time scale becomes independent of the strength of the coupling with the environment. It is shown that, although such a regime does exist, as previously established by Cucchietti et al. [Phys. Rev. A 75, 032337 (2007)], it is not a clear signature of a quantum phase transition in the environment. In fact, this kind of universality also exists in the absence of quantum phase transitions. A universal regime can be related to the existence of an energy scale separation between the Hamiltonian of the environment and the one characterizing the system-environment interaction. The results presented also indicate that in the strong-coupling regime the quantum phase transition does not produce an enhancement of decoherence (as opposed to what happens in the weak-coupling regime).
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.