A Bottom-up Approach to Constructing Symmetric Variational Quantum Circuits

In the age of noisy quantum processors, the exploitation of quantum symmetries can be quite beneficial in the efficient preparation of trial states, an important part of the variational quantum eigensolver algorithm. The benefits include building quantum circuits which are more compact, with lesser number of paramaters, and more robust to noise, than their non-symmetric counterparts. Leveraging on ideas from representation theory we show how to construct symmetric quantum circuits. Similar ideas have been previously used in the field of tensor networks to construct symmetric tensor networks. We focus on the specific case of particle number conservation, that is systems with U(1) symmetry. Based on the representation theory of U(1), we show how to derive the particle-conserving exchange gates, which are commonly used in constructing hardware-efficient quantum circuits for fermionic systems, like in quantum chemistry, material science, and condensed-matter physics. We tested the effectiveness of our circuits with the Heisenberg XXZ model.Comment: 18 pages, 7 figure

Simulation of braiding anyons using Matrix Product States

Anyons exist as point like particles in two dimensions and carry braid statistics which enable interactions that are independent of the distance between the particles. Except for a relatively few number of models which are analytically tractable, much of the physics of anyons remain still unexplored. In this paper, we show how U(1)-symmetry can be combined with the previously proposed anyonic Matrix Product States to simulate ground states and dynamics of anyonic systems on a lattice at any rational particle number density. We provide proof of principle by studying itinerant anyons on a one dimensional chain where no natural notion of braiding arises and also on a two-leg ladder where the anyons hop between sites and possibly braid. We compare the result of the ground state energies of Fibonacci anyons against hardcore bosons and spinless fermions. In addition, we report the entanglement entropies of the ground states of interacting Fibonacci anyons on a fully filled two-leg ladder at different interaction strength, identifying gapped or gapless points in the parameter space. As an outlook, our approach can also prove useful in studying the time dynamics of a finite number of nonabelian anyons on a finite two-dimensional lattice.Comment: Revised version: 20 pages, 14 captioned figures, 2 new tables. We have moved a significant amount of material concerning symmetric tensors for anyons --- which can be found in prior works --- to Appendices in order to streamline our exposition of the modified Anyonic-U(1) ansat

Numerical simulation of non-Abelian anyons

Two-dimensional systems such as quantum spin liquids or fractional quantum Hall systems exhibit anyonic excitations that possess more general statistics than bosons or fermions. This exotic statistics makes it challenging to solve even a many-body system of non-interacting anyons. We introduce an algorithm that allows to simulate anyonic tight-binding Hamiltonians on two-dimensional lattices. The algorithm is directly derived from the low energy topological quantum field theory and is suited for general Abelian and non-Abelian anyon models. As concrete examples, we apply the algorithm to study the energy level spacing statistics, which reveals level repulsion for free semions, Fibonacci anyons, and Ising anyons. Additionally, we simulate nonequilibrium quench dynamics, where we observe that the density distribution becomes homogeneous for large times - indicating thermalization

Using a biased quantum random walk as a quantum lumped element router

Quantum random walks have received much attention for their intrinsic interest and many possible uses and have been experimentally demonstrated. In this work we look at the possibility of using a biased one-dimensional (1D) quantum walk as an element within a larger quantum device. We ask whether one can use a quantum walk to act as a router with one bias setting engineering the quantum walk to route probability flow one direction while another bias setting routes flow in the opposite direction. Appealing to electrical circuit terminology, we consider a biased quantum walk over a large spatial lattice to act as a single "lumped element" whose routing action depends on the coin bias. We discover that the lumped-element current, when summed over the quantum walk lattice, reaches a steady state and for specific initial states we derive an analytic form for this steady-state lumped-element current. We show that we can control the magnitude and the direction (routing) of the steady-state current. Curiously the control phase and steady-state total current exhibits a sinusoidal current-phase relationship indicating that the lumped element may be similar to that found in Josephson junctions. Finally we illustrate that conservative 1D Hamiltonian systems can also exhibit steady-state dynamics similar to the quantum walk.8 page(s

Trends and transition times in parity progression among women of reproductive age in Nigeria between 1998 and 2012

Context: This study assessed the trends and transition times in parity progression among women of reproductive age in Nigeria between 1998 and 2012. Data Source & Method: This is a descriptive, cross-sectional study which utilized data from the 2003, 2008 and 2013 Nigeria Demographic and Health Survey. Data were analysed using Descriptive statistics, Brass P/F Ratio, Pandey and Suchrinder Parity Progression Method and Feeney and Yu Method of Period parity Progression. Findings: Our findings show that the proportion of women progressing to next parity was fairly equal for periods 1998-2002 and 2003-2007 with a slight decrease for period 2008-2012. There was a relatively consistent reduction in the speed of progression to high order parities from 1998 – 2012. Conclusion: In conclusion, not much progress has been made with respect to fertility reduction in the country. However, a reduction in the pace of progression to high order parities suggests Nigeria is experiencing a consistent but slow total fertility decline