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On the Efficiency of Semiclassical and Classical Approximations of Quantum Fidelity in Many-Dimensional Systems

Cesare Mollica

Published Jan 1, 2012• 0 citations
Publisher

Abstract

This thesis is focused on classical and semiclassical approximations of a specific quantum time correlation function, the “quantum fidelity.” Namely, we rigorously study the efficiency of a continuous class of algorithms for the evaluation of its classical limit and both the cost and the error of a particular semiclassical approximation of it. In the physics domain, quantum fidelity can be used to study the cross section of inelastic neutron scattering and phenomena like decoherence and irreversibility in NMR experiments. In chemical physics, quantum fidelity can be employed to compute the spectrum of a quantum state, to evaluate the accuracy of quantum dynamics on an approximate potential energy surface, and to verify the importance of nonadiabatic electronic transitions in molecules. Unfortunately, as for any numerical simulation of a quantum dynamical object, many-dimensional calculations of quantum fidelity are extremely expensive computationally. Classical methods, not including any quantum effect, are generally believed to be efficient approximations for the numerical evaluation of large systems’ properties when quantum effects are not important. We prove that, previous to this research, the computational costs of available algorithms for the simulation of classical fidelity increased exponentially with the number of degrees of freedom and we single out, within a continuous family of algorithms, the only one for which the number of trajectories needed for convergence is independent of the system’s dimensionality. Recently a semiclassical method, called “dephasing representation” (DR), has been proposed to approximate quantum fidelity accurately and efficiently. Although semiclassical methods, approximately including all types of quantum effects, are generally more accurate but much less efficient than classical methods, we prove that for the semiclassical DR there exists at least one algorithm for which the number of trajectories needed for convergence is independent of dimensionality and also that this semiclassical algorithm is even faster than the most efficient classical fidelity algorithm previously singled out. Encouraged by the surprising efficiency of the specific semiclassical method, we investigate the error of the DR compared to the quantum benchmark.

DOI
10.5075/epfl-thesis-5383

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This thesis is focused on classical and semiclassical approximations of a specific quantum time correlation function, the “quantum fidelity.” Namely, we rigorously study the efficiency of a continuous class of algorithms for the evaluation of its classical limit and both the cost and the error of a particular semiclassical approximation of it. In the physics domain, quantum fidelity can be used to study the cross section of inelastic neutron scattering and phenomena like decoherence and irreversibility in NMR experiments. In chemical physics, quantum fidelity can be employed to compute the spectrum of a quantum state, to evaluate the accuracy of quantum dynamics on an approximate potential energy surface, and to verify the importance of nonadiabatic electronic transitions in molecules. Unfortunately, as for any numerical simulation of a quantum dynamical object, many-dimensional calculations of quantum fidelity are extremely expensive computationally. Classical methods, not including any quantum effect, are generally believed to be efficient approximations for the numerical evaluation of large systems’ properties when quantum effects are not important. We prove that, previous to this research, the computational costs of available algorithms for the simulation of classical fidelity increased exponentially with the number of degrees of freedom and we single out, within a continuous family of algorithms, the only one for which the number of trajectories needed for convergence is independent of the system’s dimensionality. Recently a semiclassical method, called “dephasing representation” (DR), has been proposed to approximate quantum fidelity accurately and efficiently. Although semiclassical methods, approximately including all types of quantum effects, are generally more accurate but much less efficient than classical methods, we prove that for the semiclassical DR there exists at least one algorithm for which the number of trajectories needed for convergence is independent of dimensionality and also that this semiclassical algorithm is even faster than the most efficient classical fidelity algorithm previously singled out. Encouraged by the surprising efficiency of the specific semiclassical method, we investigate the error of the DR compared to the quantum benchmark.
While rigorous quantum dynamical simulations of many-body systems are extremely difficult (or impossible) due to exponential scaling with dimensionality, the corresponding classical simulations ignore quantum effects. Semiclassical methods are generally more efficient but less accurate than quantum methods and more accurate but less efficient than classical methods. We find a remarkable exception to this rule by showing that a semiclassical method can be both more accurate and faster than a classical simulation. Specifically, we prove that for the semiclassical dephasing representation the number of trajectories needed to simulate quantum fidelity is independent of dimensionality and also that this semiclassical method is even faster than the most efficient corresponding classical algorithm. Analytical results are confirmed with simulations of fidelity in up to 100 dimensions with 2(1700)-dimensional Hilbert space.
The accurate and efficient simulation of many body systems has been a long-standing challenge for quantum chemists and physicists. At the heart of this challenge lies the correlation between quantum particles, which is crucial in understanding various physical phenomena, but is numerically hard to calculate. Over the decades, there has been remarkable progress in simplifying this problem aided by mathematical and chemical approximations and the availability of powerful computers. However, an exact solution to the problem still remains intractable as it requires exponential re- sources with respect to the system size. Recent advances in the field of quantum computing have shown promise in Feynman’s idea of simulating quantum mechanics on quantum computers. It has been proposed that an ideal fault-tolerant quantum computer can reduce the scaling of these sim- ulations from exponential to polynomial. However, initial quantum algorithms have shown that there is a large gap between the capabilities of current hardware and the resources required for simulating quantum systems of interest. The work of this thesis focuses on reducing this gap by introducing two novel algorithms for quantum xiv chemistry simulations, which are suitable for near-term hardware. First, we intro- duce a hybrid algorithm to decrease the total number of operations required for the quantum simulations. We use a classical computer to generate an effective Hamilto- nian containing O(n2) terms compared to the O(n4) terms of the full Hamiltonian, where n is the number of orbitals. This sparse Hamiltonian can then be used with a high-level method to recover the ground state energies. We demonstrate that this sparsification of the Hamiltonian reduces the number of quantum operations required for the simulation by an order of magnitude, thus making it accessible for the near- term Hardware. Our second algorithm aims to make the calculation of dynamic correlation functions (Green’s functions) more feasible on these quantum machines. This algorithm is designed with the motive of avoiding the use of time-evolution and reduced use of two-qubit gates. We demonstrate that we can reproduce the Green’s function within reasonable error limits using this approach. We have also discussed the use of re-sampling techniques for proper error-propagation of the stochastic data obtained from the quantum machines. The general structure of this thesis is as follows. We begin by motivating this work in Chapter 1. Chapters 2 and 3 provide a general introduction of quantum computing and Green’s function based methods, respectively. In chapter 4, the development of effective Hamiltonian is discussed in detail. Chapter 5 demonstrates the use of these effective Hamiltonians on the quantum machine for time evolution. In chapter 6, we have discussed the algorithm to calculate Green’s functions on a quantum machine. This is followed by conclusions and future directions in Chapters 7 and 8, respectively.
A general semiclassical expression for quantum fidelity (Loschmidt echo) of arbitrary pure and mixed states is derived. It expresses fidelity as an interference sum of dephasing trajectories weighed by the Wigner function of the initial state, and does not require that the initial state be localized in position or momentum. This general dephasing representation is special in that, counterintuitively, all of fidelity decay is due to dephasing and none is due to the decay of classical overlaps. Surprising accuracy of the approximation is justified by invoking the shadowing theorem: twice--both for physical perturbations and for numerical errors. Beyond justifying the approximation, the shadowing theorem makes the dephasing representation practical: without shadowing it would be impossible to find numerically the precise trajectories needed in a semiclassical approximation. It is shown how the general expression reduces to the previously known special forms for localized states. The superiority of the general over the specialized forms is explained and supported by numerical tests for wave packets, nonlocal pure states, and for simple and random mixed states. The tests are done in nonuniversal regimes in mixed phase space where detailed features of fidelity are important. Although semiclassically motivated, the present approach is valid for abstract systems with a finite Hilbert basis provided that the discrete Wigner transform is used. This makes the method applicable, via a phase-space approach, to problems of quantum computation.
We analyze the efficiency of available algorithms for the simulation of classical fidelity and show that their computational costs increase exponentially with the number of degrees of freedom. Then we present an algorithm for which the number of trajectories needed for convergence is independent of the system's dimensionality and show that, within a continuous family of algorithms, this algorithm is the only one with this property. Simultaneously we propose a general analytical approach to estimate efficiency of trajectory-based methods and suggest how to use it to accelerate calculations of other classical correlation functions. Converged numerical results are provided for systems with phase space volume 2^{1700} times larger than the volume of the initial state.
Computing quantum dynamics in condensed matter systems is an open challenge due to the exponential scaling of exact algorithms with the number of degrees of freedom. Current methods try to reduce the cost of the calculation using classical dynamics as the key ingredient of approximations of the quantum time evolution. Two main approaches exist, quantum classical and semi-classical, but they suffer from various difficulties, in particular when trying to go beyond the classical approximation. It may then be useful to reconsider the problem focusing on statistical time-dependent averages rather than directly on the dynamics. In this paper, we discuss a recently developed scheme for calculating symmetrized correlation functions. In this scheme, the full (complex time) evolution is broken into segments alternating thermal and real-time propagation, and the latter is reduced to classical dynamics via a linearization approximation. Increasing the number of segments systematically improves the result with respect to full classical dynamics, but at a cost which is still prohibitive. If only one segment is considered, a cumulant expansion can be used to obtain a computationally efficient algorithm, which has proven accurate for condensed phase systems in moderately quantum regimes. This scheme is summarized in the second part of the paper. We conclude by outlining how the cumulant expansion formally provides a way to improve convergence also for more than one segment. Future work will focus on testing the numerical performance of this extension and, more importantly, on investigating the limit for the number of segments that goes to infinity of the approximate expression for the symmetrized correlation function to assess formally its convergence to the exact result.
Due to the Heisenberg uncertainty principle, various classical systems differing only on the scale smaller than Planck's cell correspond to the same quantum system. We use this fact to find a unique semiclassical representation without the Van Vleck determinant, applicable to a large class of correlation functions expressible as quantum fidelity. As in the Feynman path integral formulation of quantum mechanics, all contributing trajectories have the same amplitude: that is why we denote it the "dephasing representation." By relating our approach to the problem of existence of true trajectories near numerically-computed chaotic trajectories, we make the approximation rigorous for any system in which the shadowing theorem holds. Numerical implementation only requires computing actions along the unperturbed trajectories and not finding the shadowing trajectories. While semiclassical linear-response theory was used before in quasi-integrable and chaotic systems, here its validity is justified in the most generic, mixed systems. Dephasing representation appears to be a rare practical method to calculate quantum correlation functions in nonuniversal regimes in many-dimensional systems where exact quantum computations are impossible.