In the Bloch sphere picture, one finds the coefficients for expanding a single-qubit density operator in terms of the identity and Pauli matrices. A generalization to $n$ qubits via tensor products represents a density operator by a real vector of le
ngth $4^n$, conceptually similar to a statevector. Here, we study this approach for the purpose of quantum circuit simulation, including noise processes. The tensor structure leads to computationally efficient algorithms for applying circuit gates and performing few-qubit quantum operations. In view of variational circuit optimization, we study backpropagation through a quantum circuit and gradient computation based on this representation, and generalize our analysis to the Lindblad equation for modeling the (non-unitary) time evolution of a density operator.
We study the classical Toda lattice with domain wall initial conditions, for which left and right half lattice are in thermal equilibrium but with distinct parameters of pressure, mean velocity, and temperature. In the hydrodynamic regime the respect
ive space-time profiles scale ballisticly. The particular case of interest is a jump from low to high pressure at uniform temperature and zero mean velocity. Thereby the scaling function for the average stretch (also free volume) is forced to change sign. By direct inspection, the hydrodynamic equations for the Toda lattice seem to be singular at zero stretch. In our contribution we report on numerical solutions and convincingly establish that nevertheless the self-similar solution exhibits smooth behavior.
This work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of
integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schrodinger equation and to a classical system on a cylindrical lattice.
Compressible electron flow through a narrow cavity is theoretically unstable, and the oscillations occurring during the instability have been proposed as a method of generating Terahertz radiation. We numerically demonstrate that the endpoint of this
instability is a nonlinear hydrodynamic oscillator, consisting of an alternating shock wave and rarefaction-like relaxation flowing back and forth in the device. This qualitative physics is robust to cavity inhomogeneity and changes in the equation of state of the fluid. We discuss the frequency and amplitude dependence of the emitted radiation on physical parameters (viscosity, momentum relaxation rate, and bias current) beyond linear response theory, providing clear predictions for future experiments.
Dynamical quantum phase transitions (DQPTs) represent a counterpart in non-equilibrium quantum time evolution of thermal phase transitions at equilibrium, where real time becomes analogous to a control parameter such as temperature. In quenched quant
um systems, recently the occurrence of DQPTs has been demonstrated, both with theory and experiment, to be intimately connected to changes of topological properties. Here, we contribute to broadening the systematic understanding of this relation between topology and DQPTs to multi-orbital and disordered systems. Specifically, we provide a detailed ergodicity analysis to derive criteria for DQPTs in all spatial dimensions, and construct basic counter-examples to the occurrence of DQPTs in multi-band topological insulator models. As a numerical case study illustrating our results, we report on microscopic simulations of the quench dynamics in the Harper-Hofstadter model. Furthermore, going gradually from multi-band to disordered systems, we approach random disorder by increasing the (super) unit cell within which random perturbations are switched on adiabatically. This leads to an intriguing order of limits problem which we address by extensive numerical calculations on quenched one-dimensional topological insulators and superconductors with disorder.
We devise a numerical scheme for the time evolution of matrix product operators by adapting the time-dependent variational principle for matrix product states [J. Haegeman et al, Phys. Rev. B 94, 165116 (2016)]. A simple augmentation of the initial o
perator $mathcal{O}$ by the Hamiltonian $H$ helps to conserve the average energy $mathrm{tr}[H mathcal{O}(t)]$ in the numerical scheme and increases the overall precision. As demonstration, we apply the improved method to a random operator on a small one-dimensional lattice, using the spin-1 Heisenberg XXZ model Hamiltonian; we observe that the augmentation reduces the trace-distance to the numerically exact time-evolved operator by a factor of 10, at the same computational cost.
We introduce a graphical user interface for constructing arbitrary tensor networks and specifying common operations like contractions or splitting, denoted GuiTeNet. Tensors are represented as nodes with attached legs, corresponding to the ordered di
mensions of the tensor. GuiTeNet visualizes the current network, and instantly generates Python/NumPy source code for the hitherto sequence of user actions. Support for additional programming languages is planned for the future. We discuss the elementary operations on tensor networks used by GuiTeNet, together with high-level optimization strategies. The software runs directly in web browsers and is available online at http://guitenet.org.
A microscopic understanding of the strongly correlated physics of the cuprates must account for the translational and rotational symmetry breaking that is present across all cuprate families, commonly in the form of stripes. Here we investigate emerg
ence of stripes in the Hubbard model, a minimal model believed to be relevant to the cuprate superconductors, using determinant quantum Monte Carlo (DQMC) simulations at finite temperatures and density matrix renormalization group (DMRG) ground state calculations. By varying temperature, doping, and model parameters, we characterize the extent of stripes throughout the phase diagram of the Hubbard model. Our results show that including the often neglected next-nearest-neighbor hopping leads to the absence of spin incommensurability upon electron-doping and nearly half-filled stripes upon hole-doping. The similarities of these findings to experimental results on both electron and hole-doped cuprate families support a unified description across a large portion of the cuprate phase diagram.
We present determinant quantum Monte Carlo simulations of the hole-doped single-band Hubbard-Holstein model on a square lattice, to investigate how quasiparticles emerge when doping a Mott insulator (MI) or a Peierls insulator (PI). The MI regime at
large Hubbard interaction $U$ and small relative electron-phonon coupling strength $lambda$ is quickly suppressed upon doping, by drawing spectral weight from the upper Hubbard band and shifting the lower Hubbard band towards the Fermi level, leading to a metallic state with emergent quasiparticles at the Fermi level. On the other hand, the PI regime at large $lambda$ and small $U$ persists out to relatively high doping levels. We study the evolution of the $d$-wave superconducting susceptibility with doping, and find that it increases with lowering temperature in a regime of intermediate values of $U$ and $lambda$.
We study the time evolution of quenched random-mass Dirac fermions in one dimension by quantum lattice Boltzmann simulations. For nonzero noise strength, the diffusion of an initial wave packet stops after a finite time interval, reminiscent of Ander
son localization. However, instead of exponential localization we find algebraically decaying tails in the disorder-averaged density distribution. These qualitatively match $propto x^{-3/2}$ decay, which has been predicted by analytic calculations based on zero-energy solutions of the Dirac equation.
