A theoretical analysis is given of the equation of motion method, due to Alben et al., to compute the eigenvalue distribution (density of states) of very large matrices. The salient feature of this method is that for matrices of the kind encountered in quantum physics the memory and CPU requirements of this method scale linearly with the dimension of the matrix. We derive a rigorous estimate of the statistical error, supporting earlier observations that the computational efficiency of this approach increases with matrix size. We use this method and an imaginary-time version of it to compute the energy and the specific heat of three different, exactly solvable, spin-1/2 models and compare with the exact results to study the dependence of the statistical errors on sample and matrix size.

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We demonstrate that the Chebyshev expansion method is a very efficient numerical tool for studying spin-bath decoherence of quantum systems. We consider two typical problems arising in studying decoherence of quantum systems consisting of a few coupled spins: (i) determining the pointer states of the system and (ii) determining the temporal decay of quantum oscillations. As our results demonstrate, for determining the pointer states, the Chebyshev-based scheme is at least a factor of 8 faster than existing algorithms based on the Suzuki-Trotter decomposition. For problems of the second type, the Chebyshev-based approach is 3-4 times faster than the Suzuki-Trotter-based schemes. This conclusion holds qualitatively for a wide spectrum of systems, with different spin baths and different Hamiltonians.

We describe portable software to simulate universal quantum computers on massive parallel computers. We illustrate the use of the simulation software by running various quantum algorithms on different computer architectures, such as a IBM BlueGene/L, a IBM Regatta p690+, a Hitachi SR11000/J1, a Cray X1E, a SGI Altix 3700 and clusters of PCs running Windows XP. We study the performance of the software by simulating quantum computers containing up to 36 qubits, using up to 4096 processors and up to 1 TB of memory. Our results demonstrate that the simulator exhibits nearly ideal scaling as a function of the number of processors and suggest that the simulation software described in this paper may also serve as benchmark for testing high-end parallel computers.

We study numerically the damping of quantum oscillations and the increase of entropy with time in model spin systems decohered by a spin bath. In some experimentally relevant cases, the oscillations of considerable amplitude can persist long after the entropy has saturated near its maximum, i.e. when the system has been decohered almost completely. Therefore, the pointer states of the system demonstrate non-trivial dynamics. The oscillations exhibit slow power-law decay, rather than exponential or Gaussian, and may be observable in experiments.PACS numbers: 03.65. Yz, 75.10.Jm, 76.60.Es, 03.65.Ta For a quantum system prepared in a linear superposition of its eigenstates, some observables can oscillate with time. Interaction of the system with its environment leads to a decay of the system's initial pure state into a mixture of several "pointer states"; it causes an increase of the system's entropy (decoherence) and damping of quantum oscillations (dephasing) with time [1,2]. Both effects, decoherence and dephasing, are often considered as equivalent results of the mixed state of the system. But careful analysis shows important differences [3,4] originating from the fact that the same density matrix can describe both an ensemble of similar systems, and a single decohered system. Thus, e.g., dephasing can appear in an ensemble of pure, non-decohered systems with slightly differing dynamics (this is an idealized picture of T 2 processes in NMR). Decoherence and dephasing are hard to distinguish in experiments which employ ensembles of quantum systems. However, recently it has become possible to study single quantum systems, such as trapped ions [5], atoms in cavities [6], or even mesoscopically big Cooper-pair boxes [7]. As a result, theoretical consideration of the relation between dephasing and decoherence has become experimentally relevant and important.In this work, we compare dephasing and decoherence in single systems of interacting s = 1/2 spins coupled to a bath of s = 1/2 spins. We show that in some cases, the oscillations can survive long after the entropy has almost saturated, i.e. that the quantum oscillations can take place for a long time even in an almost completely decohered system. These oscillations do not decay according to usual exponential (exp (−t/τ )) or Gaussian (exp (−t 2 /τ 2 )) law, but exhibit long power-law (1/ √ t or 1/t) tails. This result has interesting consequences. The standard picture of a decoherence process assumes that as soon as the system has decayed into a mixture of pointer states, the fast quantum mechanical motion is over, i.e. the pointer states are essentially static. This has been confirmed by numerous studies of different types of pointer states [1]. However, we observe that after the system has decayed into a mixture of the pointer states, and its entropy has reached maximum, the oscillations still persist. It means that the pointer states are not static: they exhibit non-trivial dynamical behavior. We show this explicitly by analyzing the structure of the densit...