Solution of the time-dependent Schrödinger equation for two-dimensional spin-1/2 Heisenberg systems

Simulating Hydrodynamics on Noisy Intermediate-Scale Quantum Devices with Random Circuits

In a recent milestone experiment, Google's processor Sycamore heralded the era of "quantum supremacy" by sampling from the output of (pseudo-)random circuits. We show that such random circuits provide tailor-made building blocks for simulating quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. Specifically, we propose an algorithm consisting of a random circuit followed by a trotterized Hamiltonian time evolution to study hydrodynamics and to extract transport coefficients in the linear response regime. We numerically demonstrate the algorithm by simulating the buildup of spatiotemporal correlation functions in one- and two-dimensional quantum spin systems, where we particularly scrutinize the inevitable impact of errors present in any realistic implementation. Importantly, we find that the hydrodynamic scaling of the correlations is highly robust with respect to the size of the Trotter step, which opens the door to reach nontrivial time scales with a small number of gates. While errors within the random circuit are shown to be irrelevant, we furthermore unveil that meaningful results can be obtained for noisy time evolutions with error rates achievable on near-term hardware. Our work emphasizes the practical relevance of random circuits on NISQ devices beyond the abstract sampling task.

Selected applications of typicality to real-time dynamics of quantum many-body systems

Loosely speaking, the concept of quantum typicality refers to the fact that a single pure state can imitate the full statistical ensemble. This fact has given rise to a rather simple but remarkably useful numerical approach to simulate the dynamics of quantum many-body systems, called dynamical quantum typicality (DQT). In this paper, we give a brief overview of selected applications of DQT, where particular emphasis is given to questions on transport and thermalization in low-dimensional lattice systems like chains or ladders of interacting spins or fermions. For these systems, we discuss that DQT provides an efficient means to obtain time-dependent equilibrium correlation functions for comparatively large Hilbert-space dimensions and long time scales, allowing the quantitative extraction of transport coefficients within the framework of, e. g., linear response theory (LRT). Furthermore, it is discussed that DQT can also be used to study the far-from-equilibrium dynamics resulting from sudden quench scenarios, where the initial state is a thermal Gibbs state of the pre-quench Hamiltonian. Eventually, we summarize a few combinations of DQT with other approaches such as numerical linked cluster expansions or projection operator techniques. In this way, we demonstrate the versatility of DQT.

From Linear to Nonlinear Responses of Thermal Pure Quantum States

We propose a self-validating scheme to calculate the unbiased responses of quantum many-body systems to external fields of arbitrary strength at any temperature. By switching on a specified field to a thermal pure quantum state of an isolated system, and tracking its time evolution, one can observe an intrinsic thermalization process driven solely by many-body effects. The transient behavior before thermalization contains rich information on excited states, giving the linear and nonlinear response functions at all frequencies. We uncover the necessary conditions to clarify the applicability of this formalism, supported by a proper definition of the nonlinear response function. The accuracy of the protocol is guaranteed by a rigorous upper bound of error exponentially decreasing with system size, and is well implemented in the simple ferromagnetic Heisenberg chain, whose response at high fields exhibits a nonlinear band deformation. We further extract the characteristic features of excitation of the spin-1/2 kagome antiferromagnet; the wave-number-insensitive linear responses from the possible spin liquid ground state, and the significantly broad nonlinear peaks which should be generated from numerous collisions of quasiparticles, that are beyond the perturbative description.

Random phase vector for calculating the trace of a large matrix

We derive an estimate of the statistical error in calculating the trace of a large matrix by using random vectors, and show that the random phase vector gives the results with the smallest statistical error for a given basis set. This result supports use of random phase vectors in the calculation of density of states and linear response functions of large quantum systems.

Efficient scheme for numerical simulations of the spin-bath decoherence

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.

Fast algorithm for finding the eigenvalue distribution of very large matrices

A theoretical analysis is given of the equation of motion method, due to Alben et al. [Phys. Rev. B 12, 4090 (1975)], 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 the matrix size. We use this method and an imaginary-time version of it to compute the energy and 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.