Dr Paul Bergold

We introduce a family of variational quantum algorithms, which we coin as quantum iterative power algorithms (QIPAs), and demonstrate their capabilities as applied to global-optimization numerical experiments. Specifically, we demonstrate the QIPA based on a double exponential oracle as applied to ground state optimization of the H2 molecule, search for the transmon qubit ground-state, and biprime factorization. Our results indicate that QIPA outperforms quantum imaginary time evolution (QITE) and requires a polynomial number of queries to reach convergence even with exponentially small overlap between an initial quantum state and the final desired quantum state, under some circumstances. We analytically show that there exists an exponential amplitude amplification at every step of the variational quantum algorithm, provided the initial wavefunction has non-vanishing probability with the desired state and that the unique maximum of the oracle is given by λ1 > 0, while all other values are given by the same value 0 < λ2 < λ1 (here λ can be taken as eigenvalues of the problem Hamiltonian). The generality of the global-optimization method presented here invites further application to other problems that currently have not been explored with QITE-based near-term quantum computing algorithms. Such approaches could facilitate identification of reaction pathways and transition states in chemical physics, as well as optimization in a broad range of machine learning applications. The method also provides a general framework for adaptation of a class of classical optimization algorithms to quantum computers to further broaden the range of algorithms amenable to implementation on current noisy intermediate-scale quantum computers.

Optimization algorithms play a central role in chemistry since optimization is the computational keystone of most molecular and electronic structure calculations. Herein, we introduce the iterative power algorithm (IPA) for global optimization and a formal proof of convergence for both discrete and continuous global search problems, which is essential for applications in chemistry such as molecular geometry optimization. IPA implements the power iteration method in quantics tensor train (QTT) representations. Analogous to the imaginary time propagation method with infinite mass, IPA starts with an initial probability distribution rho(0)(x) and iteratively applies the recurrence relation rho(k+1)(x) = U(x) rho(k)(x)/parallel to U rho(k)parallel to(L1), where U(x) = e(-V(x)) is defined in terms of the potential energy surface (PES) V(x) with global minimum at x = x*. Upon convergence, the probability distribution becomes a delta function delta(x - x*), so the global minimum can be obtained as the position expectation value x* = Tr[x delta(x - x*)]. QTT representations of V(x) and rho(x) are generated by fast adaptive interpolation of multidimensional arrays to bypass the curse of dimensionality and the need to evaluate V(x) for all possible values of x. We illustrate the capabilities of IPA for global search optimization of two multidimensional PESs, including a differentiable model PES of a DNA chain with D = 50 adenine-thymine base pairs, and a discrete non-differentiable potential energy surface, V(p) = mod(N,p), that resolves the prime factors of an integer N, with p in the space of prime numbers {2, 3,..., p(max)} folded as a d-dimensional 2(1) x 2(2) x ... x 2(d) tensor. We find that IPA resolves multiple degenerate global minima even when separated by large energy barriers in the highly rugged landscape of the potentials. Therefore, IPA should be of great interest for a wide range of other optimization problems ubiquitous in molecular and electronic structure calculations.

Methods for efficient simulations of multidimensional quantum dynamics are essential for theoretical studies of chemical systems where quantum effects are important, such as those involving rearrangements of protons or electronic configurations. Here, we introduce the functional tensor-train Chebyshev (FTTC) method for rigorous nuclear quantum dynamics simulations. FTTC is essentially the Chebyshev propagation scheme applied to the initial state represented in a continuous analogue tensor-train format. We demonstrate the capabilities of FTTC as applied to simulations of proton quantum dynamics in a 50-dimensional model of hydrogen-bonded DNA base pairs.

We study the time-sliced thawed Gaussian propagation method, which was recently proposed for solving the time-dependent Schrodinger equation. We introduce a triplet of quadrature-based analysis, synthesis and re-initialization operators to give a rigorous mathematical formulation of the method. Further, we derive combined error bounds for the discretization of the wave packet transform and the time-propagation of the thawed Gaussian basis functions. Numerical experiments in 1D illustrate the theoretical results.

We study the Fourier transform windowed by a bump function. We transfer Jackson's classical results on the convergence of the Fourier series of a periodic function to windowed series of a not necessarily periodic function. Numerical experiments illustrate the obtained theoretical results.

We analyse the Gaussian wave packet transform. Based on the Fourier inversion formula and a partition of unity, which is formed by a collection of Gaussian basis functions, a new representation of square-integrable functions is presented. Including a rigorous error analysis, the variants of the wave packet transform are then derived by a discretization of the Fourier integral via different quadrature rules. Based on Gauss-Hermite quadrature, we introduce a new representation of Gaussian wave packets in which the number of basis functions is significantly reduced. Numerical experiments in 1D illustrate the theoretical results.

In order to alleviate the computational costs of fully quantum nonadiabatic dynamics, we present a mixed quantum-classical (MQC) particle method based on the theory of Koopman wavefunctions. Although conventional MQC models often suffer from consistency issues such as the violation of Heisenberg's principle, we overcame these difficulties by blending Koopman's classical mechanics on Hilbert spaces with methods in symplectic geometry. The resulting continuum model enjoys both a variational and a Hamiltonian structure, while its nonlinear character calls for suitable closures. Benefiting from the underlying action principle, here we apply a regularization technique previously developed within our team. This step allows for a singular solution ansatz which introduces the trajectories of computational particles - the koopmons - sampling the Lagrangian classical paths in phase space. In the case of Tully's nonadiabatic problems, the method reproduces the results of fully quantum simulations with levels of accuracy that are not achieved by standard MQC Ehrenfest simulations. In addition, the koopmon method is computationally advantageous over similar fully quantum approaches, which are also considered in our study. As a further step, we probe the limits of the method by considering the Rabi problem in both the ultrastrong and the deep strong coupling regimes, where MQC treatments appear hardly applicable. In this case, the method succeeds in reproducing parts of the fully quantum results.

While Born-Oppenheimer molecular dynamics (BOMD) has been widely studied by resorting to powerful methods in mathematical analysis, this paper presents a geometric formulation in terms of Hamilton’s variational principle and Euler-Poincaré reduction by symmetry. Upon resorting to the Lagrangian hydrodynamic paths made available by the Madelung transform, we show how BOMD arises by applying asymptotic methods to the variational principles underlying different continuum models and their particle closure schemes. In particular, after focusing on the hydrodynamic form of the fully quantum dynamics, we show how the recently proposed bohmion scheme leads to an on-the-fly implementation of BOMD. In addition, we extend our analysis to models of mixed quantum-classical dynamics.