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We construct a quantum algorithm that creates the Laughlin state for an arbitrary number of particles $n$ in the case of filling fraction one. This quantum circuit is efficient since it only uses $n(n-1)/2$ local qudit gates and its depth scales as $2n-3$. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. Finally, we discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary filling fraction.
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We study lattice wave functions obtained from the SU(2)$_1$ Wess-Zumino-Witten conformal field theory. Following Moore and Reads construction, the Kalmeyer-Laughlin fractional quantum Hall state is defined as a correlation function of primary fields. By an additional insertion of Kac-Moody currents, we associate a wave function to each state of the conformal field theory. These wave functions span the complete Hilbert space of the lattice system. On the cylinder, we study global properties of the lattice states analytically and correlation functions numerically using a Metropolis Monte Carlo method. By comparing short-range bulk correlations, numerical evidence is provided that the states with one current operator represent edge states in the thermodynamic limit. We show that the edge states with one Kac-Moody current of lowest order have a good overlap with low-energy excited states of a local Hamiltonian, for which the Kalmeyer-Laughlin state approximates the ground state. For some states, exact parent Hamiltonians are derived on the cylinder. These Hamiltonians are SU(2) invariant and nonlocal with up to four-body interactions.
An analytical expression for the von Neumann entropy of the Laughlin wave function is obtained for any possible bipartition between the particles described by this wave function, for filling fraction nu=1. Also, for filling fraction nu=1/m, where m is an odd integer, an upper bound on this entropy is exhibited. These results yield a bound on the smallest possible size of the matrices for an exact representation of the Laughlin ansatz in terms of a matrix product state. An analytical matrix product state representation of this state is proposed in terms of representations of the Clifford algebra. For nu=1, this representation is shown to be asymptotically optimal in the limit of a large number of particles.
We study numerically the geometric entanglement in the Laughlin wave function, which is of great importance in condensed matter physics. The Slater determinant having the largest overlap with the Laughlin wave function is constructed by an iterative algorithm. The logarithm of the overlap, which is a geometric quantity, is then taken as a geometric measure of entanglement. It is found that the geometric entanglement is a linear function of the number of electrons to a good extent. This is especially the case for the lowest Laughlin wave function, namely the one with filling factor of $1/3$. Surprisingly, the linear behavior extends well down to the smallest possible value of the electron number, namely, $ N= 2$. The constant term does not agree with the expected topological entropy.
In this paper we used the Fredholm method in Schroedingers integral equation in the investigation of the scattering effect near the center of it between a stationary quantum wave function and an electrostatic potential. Two potentials are studied one Coulombian and the other Podolsky. The result shows the importance of the proposal of Podolsky to regularize the effect near the scattering center in the quantum wave function. Being that the coulombian potential presents with strong variation in the amplitude of the wave after the scattering. In the case of Podolskys potential, this is corrected by adopting a constant that removes this strong variation.
We propose a quantum inverse iteration algorithm which can be used to estimate the ground state properties of a programmable quantum device. The method relies on the inverse power iteration technique, where the sequential application of the Hamiltonian inverse to an initial state prepares an approximate groundstate. To apply the inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.
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