Parallel computers are finding major applications in almost all scientific and engineering disciplines. An interesting area that has received attention is quantum scattering. Algorithms for studying quantum scattering are computation intensive and hence suitable for parallel machines. The state-of-the-art methods developed for uniprocessors require the computation of two Fast Fourier Transforms (FFTs) at each time step. However, the communication overhead in implementing FFTs make them an expensive operation on distributed memory parallel machines;The focus of this dissertation is the development of efficient parallel methods for studying the phenomenon of time-dependent quantum-wave scattering. The methods described belong to the class of integral equation methods, which involve the application of a repeated sequence of very short time step propagations. Free propagation of a wavepacket is most easily handled in the so-called momentum representation whereas the effect of the potential is most easily obtained in the coordinate representation. The two representations are Fourier Transforms of each other. The algorithm presented eliminates the computation of FFTs by performing the propagation totally within the coordinate representation. The communication required is only with the nearest neighbors and is load balanced, thus making the algorithm suitable for distributed memory parallel machines. Implementation results on the nCUBE hypercube and comparison with standard FFT methods are also presented.
Parallel computers are finding major applications in almost all scientific and engineering disciplines. An interesting area that has received attention is quantum scattering. Algorithms for studying quantum scattering are computation intensive and hence suitable for parallel machines. The state-of-the-art methods developed for uniprocessors require the computation of two Fast Fourier Transforms (FFTs) at each time step. However, the communication overhead in implementing FFTs make them an expensive operation on distributed memory parallel machines;The focus of this dissertation is the development of efficient parallel methods for studying the phenomenon of time-dependent quantum-wave scattering. The methods described belong to the class of integral equation methods, which involve the application of a repeated sequence of very short time step propagations. Free propagation of a wavepacket is most easily handled in the so-called momentum representation whereas the effect of the potential is most easily obtained in the coordinate representation. The two representations are Fourier Transforms of each other. The algorithm presented eliminates the computation of FFTs by performing the propagation totally within the coordinate representation. The communication required is only with the nearest neighbors and is load balanced, thus making the algorithm suitable for distributed memory parallel machines. Implementation results on the nCUBE hypercube and comparison with standard FFT methods are also presented.
For many years it has been recognized that fundamental physical constraints such as the speed of light will limit the ultimate speed of single processor computers to less than about three billion floating point operations per second (3 GFLOPS). This limitation is becoming increasingly restrictive as commercially available machines are now within an order of magnitude of this asymptotic limit. A natural way to avoid this limit is to harness together many processors to work on a single computational problem. In principle, these parallel processing computers have speeds limited only by the number of processors one chooses to acquire. The usefulness of potentially unlimited processing speed to a computationally intensive field such as quantum chemistry is obvious. If these methods are to be applied to significantly larger chemical systems, parallel schemes will have to be employed. For this reason we have developed distributed-memory algorithms for a number of standard quantum chemical methods. We are currently implementing these on a 32 processor Intel hypercube. In this paper we present our algorithm and benchmark results for one of the bottleneck steps in quantum chemical calculations: the four index integral transformation.
While scattering phenomena are discussed in most quantum mechanics textbooks, the analysis is usually performed using time-independent techniques. These techniques have certainly been successful in predicting the results of scattering events but fail to present an intuitive picture of what happens to the physical system during a collision. A remedy to this situation could be affected by the introduction of time-dependent techniques, utilizing the conceptually simple idea of a wavefunction evolving over time, but these methods have traditionally been computationally unmanageable. This paper describes a novel numerical technique, the split-operator Fourier transform method, which greatly reduces the amount of computational effort required to accurately solve the time-dependent Schrödinger equation. This method is applied to the Rutherford scattering problem in two dimensions and it is shown that the method is not only efficient enough to yield results in less than 15 min when implemented on a standard personal computer, but is also simple enough to be employed by an advanced undergraduate. The method also has the advantage of reinforcing the quantum-mechanical formalism in the student’s mind; both the evolution operator and the complementary relationship between operators and wavefunctions in coordinate and momentum space are exploited. The results of the calculations are presented as color graphs portraying the magnitude and phase of wave packets as they evolve through time.
Simulating quantum many-body dynamics is important both for fundamental understanding of physics and practical applications for quantum information processing. Therefore, classical simulation methods have been developed so far. Specifically, the Trotter-Suzuki decomposition can analyze a highly complex quantum dynamics, if the number of qubits is sufficiently small so that main memory can store the state vector. However, simulation of quantum dynamics via Trotter-Suzuki decomposition requires a huge number of steps, each of which accesses the state vector, and hence the simulation time becomes impractically long. To settle this issue, we propose a technique to accelerate simulation of quantum dynamics via simultaneous diagonalization of mutually commuting Pauli groups, which is also attracting a lot of attention to reduce the measurement overheads in quantum algorithms. We group the Hamiltonian into mutually commuting Pauli strings, and each of them is diagonalized in the computational basis via a Clifford transformation. Since diagonal operators are applied on the state vector simultaneously with minimum memory access, this method successfully uses performance of highly parallel processors such as Graphics Processing Units (GPU). Compared to both CPU and GPU implementations using fast quantum circuit simulator “qulacs,” the numerical experiments have shown that our method provides a few tens of times acceleration.
The time-dependent form of the Lippmann–Schwinger integral equation is used as the basis for a novel wave-packet propagation scheme. The method has the advantage over a previous integral equation treatment in that it does not require extensive matrix inversions involving the potential. This feature will be important when applications are made to systems where in some degrees of freedom the potential is expressed in a basis expansion. As was the case for the previous treatment, noniterated and iterated versions of the equations are given; the iterated equations, which are much simpler in the present new scheme than in the old, eliminate a matrix inversion that is required for solving the earlier noniterated equations. In the present noniterated equations, the matrix to be inverted is a function of the kinetic energy operator and thus is diagonal in a Bessel function basis set (or a sine basis set, if the centrifugal potential operator is incorporated into an effective potential). Transition amplitudes for various orbital angular momentum quantum numbers can be obtained from: (1) Fourier transform of the amplitude density from the time to the energy domain, and (2) direct analysis of the scattered wave packet. The approach is illustrated by an application to a standard potential scattering model problem.
Electromagnetic waves are an inherent part of all plasmas—laboratory fusion plasmas or astrophysical plasmas. The conventional methods for studying properties of electromagnetic waves rely on discretization of Maxwell equations suitable for implementing on classical, present day, computers. The traditional methodology is not efficient for quantum computing implementation—a future computational source offering a tantalizing possibility of enormous speed up and a significant reduction in computational cost. This paper addresses two topics relevant to implementing Maxwell equations on a quantum computer. The first is on formulating a quantum Schrödinger representation of Maxwell equations for wave propagation in a cold, inhomogeneous, and magnetized plasma. This representation admits unitary, energy preserving, evolution and conveniently lends itself to appropriate discretization for a quantum computer. Riding on the coattails of these results, the second topic is on developing a sequence of unitary operators which form the basis for a qubit lattice algorithm (QLA). The QLA, suitable for quantum computers, can be implemented and tested on existing classical computers for accuracy as well as scaling of computational time with the number of available processors. In order to illustrate the QLA for Maxwell equations, results are presented from a time evolving, full wave simulation of propagation and scattering of an electromagnetic wave packet by non-dispersive dielectric medium localized in space.
The unique principles of quantum mechanics may one day enable computers to perform operations that would be impossible on a classical computer. Although no one knows whether it will be possible to build a large-scale, functional, and stable quantum computer, researchers can study quantum-mechanical systems and develop algorithms and circuits by simulating quantum systems in software. Performance and memory bottlenecks prevent most current quantum computer simulators from being able to simulate quantum systems that are large enough to be useful. In this thesis, we develop a matrix-free sequential quantum computer simulator to vastly improve both time and memory performance of sequential code on a single processor. Next, we distribute the matrix-free algorithm over multiple parallel processors using the Message Passing Interface in order to simulate quantum systems that are too large to reside wholly within the memory of a single processor. Finally, we simulate various quantum circuits using the Hamming high-performance computing cluster in order to conduct algorithmic analysis.