Quantum Simulation

We present a detailed study of the topological Schwinger model [Phys. Rev. D 99, 014503 (2019)], which describes (1+1) quantum electrodynamics of an Abelian U(1) gauge field coupled to a symmetry-protected topological matter sector, by means of a class of ZN lattice gauge theories. Employing density-matrix renor-malization group techniques that exactly implement Gauss' law, we show that these models host a correlated topological phase for different values of N, where fermion correlations arise through inter-particle interactions mediated by the gauge field. Moreover, by a careful finite-size scaling, we show that this phase is stable in the large-N limit, and that the phase boundaries are in accordance to bosonization predictions of the U(1) topolog-ical Schwinger model. Our results demonstrate that ZN finite-dimensional gauge groups offer a practical route for an efficient classical simulation of equilibrium properties of electromagnetism with topological fermions. Additionally, we describe a scheme for the quantum simulation of a topological Schwinger model exploiting spin-changing collisions in boson-fermion mixtures of ultra-cold atoms in optical lattices. Although technically challenging, this quantum simulation would provide an alternative to classical density-matrix renormalization group techniques, providing also an efficient route to explore real-time non-equilibrium phenomena.

We study the out-of-equilibrium properties of 1 + 1 dimensional quantum elec-trodynamics (QED), discretized via the staggered-fermion Schwinger model with an Abelian Zn gauge group. We look at two relevant phenomena: first, we analyze the stability of the Dirac vacuum with respect to particle/antiparticle pair production , both spontaneous and induced by an external electric field; then, we examine the string breaking mechanism. We observe a strong effect of confinement, which acts by suppressing both spontaneous pair production and string breaking into quark/antiquark pairs, indicating that the system dynamics displays a number of out-of-equilibrium features.

A particular family of time- and space-dependent discrete-time quantum walks (QWs) is considered in onedimensional
physical space. The continuous limit of these walks is defined through a procedure discussed here
and computed in full detail. In this limit, the walks coincide with the propagation of a massless Dirac fermion
in an arbitrary gravitational field. A QW mimicking the radial propagation of a fermion outside and inside the
event horizon of a Schwarzschild black hole is explicitly constructed and simulated numerically. Thus, the family
of QWs considered in our manuscript provides an analog system to study experimentally coherent quantum
propagation in curved spacetime.

We show that indirect spin-spin interactions between effective spin-1/2 systems can be realized in two parallel one-dimensional optical lattices loaded with polar molecules and/or Rydberg atoms. The effective spin can be encoded into low-energy rotational states of polar molecules or long-lived states of Rydberg atoms, tightly trapped in a deep optical lattice. The spin-spin interactions can be mediated by Rydberg atoms, placed in a parallel shallow optical lattice, interacting with the effective spins by charge-dipole (for polar molecules) or dipole-dipole (for Rydberg atoms) interaction. Indirect XX, Ising, and XXZ interactions with interaction coefficients J ⊥ and J zz sign varying with interspin distance can be realized, in particular, the J 1-J 2 XXZ model with frustrated ferro-(antiferro-)magnetic nearest (next-nearest) neighbor interactions.

The key dynamic properties of fermionic systems, like controllability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory. It just requires knowing drift and control Hamiltonians of an experimental setup. Then one can easily determine all the states that can be reached from any given initial state. Likewise all the quantum operations that can be simulated with a given setup can be identified. Observing the parity superselection rule, we treat the fully controllable and quasifree cases of fermions, as well as various translation-invariant and particle-number conserving cases. We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds.

We introduce a scheme for the quantum simulation of many-body decoherence based on the unitary evolution of a stochastic Hamiltonian. Modulating the strength of the interactions with stochastic processes, we show that the noise-averaged density matrix simulates an effectively open dynamics governed by k-body Lindblad operators. Markovian dynamics can be accessed with white-noise fluctuations; non-Markovian dynamics requires colored noise. The time scale governing the fidelity decay under many-body decoherence is shown to scale as N −2k with the system size N. Our proposal can be readily implemented in a variety of quantum platforms including optical lattices, superconducting circuits, and trapped ions.

The interplay of symmetry, topology, and many-body effects in the classification of phases of matter poses a formidable challenge in condensed-matter physics. Such many-body effects are typically induced by inter-particle interactions involving an action at a distance, such as the Coulomb interaction between electrons in a symmetry-protected topological (SPT) phase. In this work we show that similar phenomena also occur in certain relativistic theories with interactions mediated by gauge bosons, and constrained by gauge symmetry. In particular, we introduce a variant of the Schwinger model or quantum electrodynamics (QED) in 1+1 dimensions on an interval, which displays dynamical edge states localized on the boundary. We show that the system hosts SPT phases with a dynamical contribution to the vacuum θ-angle from edge states, leading to a new type of topological QED in 1+1 dimensions. The resulting system displays an SPT phase which can be viewed as a correlated version of the Su-Schrieffer-Heeger topological insulator for polyacetylene due to non-zero gauge couplings. We use bosonization and density-matrix renormalization group techniques to reveal the detailed phase diagram, which can further be explored in experiments of ultra-cold atoms in optical lattices. Global and local symmetries play a crucial role in our understanding of Nature at very different energy scales [1, 2]. At high energies, they govern the behavior of fundamental particles [3], their spectrum and interactions [4, 5]. At low energies [6], spontaneous symmetry breaking and local order parameters characterize a wide range of phases of matter [7] and a rich variety of collective phenomena [8]. There are, however, fundamental physical phenomena that can only be characterized by non-local order parameters, such as the Wil-son loops distinguishing confined and deconfined phases in gauge theories [9], or hidden order parameters distinguishing topological phases in solids [10]. The former, requiring a non-perturbative approach to quantum field theory (e.g. lattice gauge theories (LGTs)), and the latter, demanding the introduction of mathematical tools of topology in condensed matter (e.g. topological invariants), lie at the forefront of research in both high-energy and condensed-matter physics. The interplay of symmetry and topology can lead to a very rich, and yet partially-uncharted, territory. For instance, different phases of matter can arise without any symmetry breaking: symmetry-protected topological (SPT) phases. Beyond the celebrated integer quantum Hall effect [11-14], a variety of SPT phases have already been identified [15-17] and realized [18]. Let us note that some representative models of these SPT phases [19] can be understood as lower-dimensional versions of the so-called domain-wall fermions [20], introduced in the context of chiral symmetry in lattice field theories [21]. A current problem of considerable interest is to understand strong-correlation effects in SPT phases as interactions are included [22], which may, for instance, lead to exotic fractional excitations [23, 24]. So far, the typical interactions considered involve an action at a distance (e.g. screened Coulomb or Hubbard-like nearest or next-to-nearest neighbor interactions). To the best of our knowledge, and with the recent exception [25], the study of correlated SPT phases with mediated interactions remains a largely-unexplored subject. In this work, we initiate a systematic study of SPT phases with interactions dictated by gauge symmetries focusing on a 2a x c 2n+1 c † 2n (1 − )U 2n U 2n−1 c † 2n−1 c 2n a b U n c n+1 +m s −m s c † n c n n+ + +1 +m m s −m m m m s fermion fields gauge fields FIG. 1. Discretizations for standard and topological QED 2 : (a) Staggered-fermion approach to the massive Schwinger model. The relativistic Dirac field is discretized into spinless lattice fermions subjected to a staggered on-site energy ±m s , represented by filled/empty circles in a 1D chain with alternating heights. The gauge field is discretized into rotor-angle operators that reside on the links, depicted as shaded ellipses with various levels representing the electric flux eigenbasis. The gauge-invariant term c † n U n c n+1 involves the tunneling of neighboring fermions, dressed by a local excitation of the gauge field in the electric-flux basis U n | = | + 1, represented by the zigzag grey arrow joining two neighboring fermion sites, via an excitation of the link electric-flux level. (b) Dimerized-tunneling approach to the topological Schwinger model. The previous staggered mass is substituted by a gauge-invariant tunneling with alternating strengths (1 − δ n)c † n U n c n+1 , where δ n = 0, ∆ for even/odd sites. This dimerization of the tunneling matrix elements is represented by alternating big/small ellipses at the odd/even links. the lattice Schwinger model, an Abelian LGT that regularizes quantum electrodynamics in 1+1 dimensions (QED 2) [26]. We show that a discretization alternative to the standard lattice approach [27] leads to a topological Schwinger model, and derive its continuum limit referred to as topological QED 2. This continuum quantum field theory is used to predict a phase diagram that includes SPT, confined, and fermion-condensate phases, which are then discussed in the context of the afore-mentioned domain-wall fermions in LGTs.

We study the ground-state properties of a class of Zn lattice gauge theories in 1 + 1 dimensions, in which the gauge fields are coupled to spinless fermionic matter. These models, stemming from discrete representations of the Weyl commutator for the U(1) group, preserve the unitary character of the minimal coupling, and have therefore the property of formally approximating lattice quantum electrodynamics in one spatial dimension in the large-n limit. The numerical study of such approximated theories is important to determine their effectiveness in reproducing the main features and phenomenology of the target theory, in view of implementations of cold-atom quantum simulators of QED. In this paper we study the cases n = 2 ÷ 8 by means of a DMRG code that exactly implements Gauss' law. We perform a careful scaling analysis, and show that, in absence of a background field, all Zn models exhibit a phase transition which falls in the Ising universality class, with spontaneous symmetry breaking of the CP symmetry. We then perform the large-n limit and find that the asymptotic values of the critical parameters approach the ones obtained for the known phase transition the zero-charge sector of the massive Schwinger model, which occurs at negative mass.

In digital quantum simulation of fermionic models with qubits, one requires the use of non-local maps for encoding. Such maps require linear or logarithmic overhead in circuit depth which could render the simulation useless, for a given decoherence time. Here we show how one can use a cavity-QED system to perform digital quantum simulation of fermionic models. In particular, we show that highly nonlocal Jordan-Wigner (JW) or Bravyi-Kitaev (BK) transformations can be efficiently implemented through a hardware approach. The key idea is using ancilla cavity modes, which are dispersively coupled to a qubit string, to collectively manipulate and measure qubit states. Our scheme reduces the circuit depth in each Trotter step of JW (BK) encoding, by a factor of N 2 (N log N), where N is the number of orbitals for a generic two-body Hamiltonian. Additional analysis for the Fermi-Hubbard model on an N × N square lattice results in a similar reduction. We also discuss a detailed implementation of our scheme with superconducting qubits and cavities.

The continuous limit of quantum walks (QWs) on the line is revisited through a recently developed method. In all cases but one, the limit coincides with the dynamics of a Dirac fermion coupled to an artificial electric and/or relativistic gravitational field. All results are carefully discussed and illustrated by numerical simulations.

Atomic three-grating Mach-Zehnder interferometry constitutes an important tool to probe fundamental aspects of the quantum theory. There is, however, a remarkable gap in the literature between the oversimplified models and robust numerical simulations considered to describe the corresponding experiments. Consequently, the former usually lead to paradoxical scenarios, such as the wave-particle dual behavior of atoms, while the latter make difficult the data analysis in simple terms. Here these issues are tackled by means of a simple grating working model consisting of evenly-spaced Gaussian slits. As is shown, this model suffices to explore and explain such experiments both analytically and numerically, giving a good account of the full atomic journey inside the interferometer, and hence contributing to make less mystic the physics involved. More specifically, it provides a clear and unambiguous picture of the wavefront splitting that takes place inside the interferometer, illustrating how the momentum along each emerging diffraction order is well defined even though the wave function itself still displays a rather complex shape. To this end, the local transverse momentum is also introduced in this context as a reliable analytical tool. The splitting, apart from being a key issue to understand atomic Mach-Zehnder interferometry, also demonstrates at a fundamental level how wave and particle aspects are always present in the experiment, without incurring in any contradiction or interpretive paradox. On the other hand, at a practical level, the generality and versatility of the model and methodology presented, makes them suitable to attack analogous problems in a simple manner after a convenient tuning.

by David S. Simon, Casey A. Fitzpatrick, Shuto Osawa, and Alexander V. Sergienko.

It is shown that quantum walks on one-dimensional arrays of special linear optical units allow the simulation of discrete-time Hamiltonian systems with distinct topological phases. In particular, a slightly modified version of the Su-Schrieffer-Heeger (SSH) system can be simulated, which exhibits states of nonzero winding number and has topologically-protected boundary states. In the large-system limit this approach uses quadratically fewer resources to carry out quantum simulations than previous linear-optical approaches and can be readily generalized to higher-dimensional systems. The basic optical units that implement this simulation consist of combinations of novel optical multiports that allow photons to reverse direction.

Photon-based strongly correlated lattice models like the Jaynes–Cummings and Rabi lattices differ from their more conventional relatives like the Bose–Hubbard model by the presence of an additional tunable parameter: the frequency detuning between the pseudo-spin degree of freedom and the harmonic mode frequency on each site. Whenever this detuning is large compared to relevant coupling strengths, the system is said to be in the dispersive regime. The physics of this regime is well-understood at the level of a single Jaynes–Cummings or Rabi site. Here, we extend the theoretical description of the dispersive regime to lattices with many sites, for both strong and ultra-strong coupling. We discuss the nature and spatial range of the resulting qubit–qubit and photon–photon coupling, demonstrate the emergence of photon-pairing and squeezing and illustrate our results by exact diagonalization of the Rabi dimer.

In this paper, we present the unique features exhibited by a proposed structure of coaxially gated carbon nanotube field-effect transistor (CNTFET) with doped source and drain extensions using the self-consistent and atomistic scale simulations, within the nonequilibrium Green's function formalism. In this novel CNTFET structure, three adjacent metal cylindrical gates are used, where two side metal gates with lower workfunction than the main gate as an extension of the source/drain on either side of the main metal gate are biased, independent of the main gate, to create virtual extensions to the source and the drain and also to provide an effective electrical screen for the channel region from the drain voltage variations. We demonstrate that the proposed structure of CNTFET shows improvement in device performance focusing on leakage current, on-off current ratio, and voltage gain. In addition, the investigation of short-channel effects for the proposed structure shows improved drain-induced barrier lowering, hot-carrier effect, and subthreshold swing, all of which can affect the reliability of CMOS devices.

Recent cold atom experiments have realized models where each hyperfine state at an optical lattice site can be regarded as a separate site in a synthetic dimension. In such synthetic ribbon configurations, manipulation of the transitions between the hyperfine levels provide direct control of the hopping in the synthetic dimension. This effect was used to simulate a magnetic field through the ribbon. Precise control over the hopping matrix elements in the synthetic dimension makes it possible to change this artificial magnetic field much faster than the time scales associated with atomic motion in the lattice. In this paper, we consider such a magnetic flux quench scenario in synthetic dimensions. Sudden changes have not been considered for real magnetic fields as such changes in a conducting system would result in large induced currents. Hence, we first study the difference between a time varying real magnetic field and an artificial magnetic field using a minimal six site model. This minimal model clearly shows the connection between gauge dependence and the lack of on site induced scalar potential terms. We then investigate the dynamics of a wavepacket in an infinite two or three leg ladder following a flux quench and find that the gauge choice has a dramatic effect on the packet dynamics. Specifically, a wavepacket splits into a number of smaller packets moving with different velocities. Both the weights and the number of packets depend on the implemented gauge. If an initial packet, prepared under zero flux in a n-leg ladder, is quenched to Hamiltonian with a vector potential parallel to the ladder; it splits into at most n smaller wavepackets. The same initial wavepacket splits into up to n 2 packets if the vector potential is implemented to be along the rungs. Even a trivial difference in the gauge choice such as the addition of a constant to the vector potential produces observable effects. We also calculate the packet weights for arbitrary initial and final fluxes. Finally, we show that edge states in a thick ribbon are robust under the quench only when the same gap supports an edge state for the final Hamiltonian. * firat.yilmaz@bilkent.edu.tr 2

Here, we conduct a quantum simulation of a particle in a harmonic oscillator potential using a digital quantum simulator provided by IBM quantum experience platform. The simulation is carried out in two spatial dimensions and the algorithm used is generalized for n-spatial dimensions which can be used to simulate n-dimensional harmonic oscillator. We implement the unitary operator on an arbitrary quantum state to show that the probability amplitudes of position oscillate in time. We propose a quantum circuit to effectuate the unitary operator and design it on the simulator. The proposed circuit is then generalized for a n-qubit system that can be used to realize more meticulous simulations.

There exists a Hamiltonian formulation of the factorisation problem which also needs the definition of a factorisation ensemble (a set to which factorable numbers, N = x y , having the same trivial factorisation algorithmic complexity, belong). For the primes therein, a function E, that may take only discrete values, should be the analogous of the energy from a confined system of charges in a magnetic trap. This is the quantum factoring simulator hypothesis connecting quantum mechanics with number theory. In this work, we report numerical evidence of the existence of this kind of discrete spectrum from the statistical analysis of the values of E in a sample of random OpenSSL n-bits moduli (which may be taken as a part of the factorisation ensemble). Here, we show that the unfolded distance probability of these E's fits to a Gaussian Unitary Ensemble, consistently as required, if they actually correspond to the quantum energy levels spacing of a magnetically confined system that exhibits chaos. The confirmation of these predictions bears out the quantum simulator hypothesis and, thereby, it points to the existence of a liaison between quantum mechanics and number theory. Shor's polynomial time complexity of the quantum factorisation problem, from pure quantum simulation primitives, was obtained.

Quantum computers are the promising candidates for simulation of large quantum systems, which is a daunting task to perform in a classical computer.  Here we report the experimental realization of quantum tunneling of a single particle through different types of potential barriers by simulating it in the IBM quantum computer, which here acts as a universal quantum simulator.  We consider two-qubit and three-qubit systems for visualizing the tunneling process illustrating its unique quantum nature.  We clearly observe the tunneling and oscillations of the particles in a step potential and in double- and multi-well potentials through our experimental results.  The proposed quantum circuits and simulational techniques used here can be extended for observing the tunneling phenomena for multi-particle systems in different shaped potentials.

In this article we underline and compare different quantum and deterministic models of computation by means of a probabilistic point of view. The main models used for this are deterministic, probabilistic and quantum Turing machines (DTM, PTM and QTM). Then we suggest that extending PTM with only part of quantum features does not is useful to implement real quantum supremacy if we don't incorporate them fully. This is done reviewing some notorious quantum physical experiment and quantum algorithms and showing that PTM as it is does not fit for them.

A universal quantum simulator would enable efficient simulation of quantum dynamics by implementing quantum-simulation algorithms on a quantum computer. Specifically the quantum simulator would efficiently generate qubit-string states that closely approximate physical states obtained from a broad class of dynamical evolutions. I provide an overview of theoretical research into universal quantum simulators and the strategies for minimizing computational space and time costs. Applications to simulating many-body quantum simulation and solving linear equations are discussed

Arguably one of the most important applications of quantum computers is the simulation of quantum systems. In the case where the Hamil- tonian consists of a sum of interaction terms between small subsystems, the simulation is thought to be exponentially more efficient than classical simula- tion. More generally, evolution under suitably specified sparse Hamiltonians may be efficiently simulated. In recent work we have shown that the complex- ity of simulating evolution under a Hamiltonian is very close to linear in the evolution time. In addition, we have shown that in the general case of a sparse Hamiltonian the complexity grows slowly with respect to the number of qubits. In this chapter we review these results.

Carbon Nanotube Field Effect Transistors (CNTFET) are promising nano-scaled devices for implementing high performance, very dense and low power circuits. The core of a CNTFET is a carbon nanotube. Its conductance property is determined by the chirality of the tube. In this paper, the current-voltage characteristics of coaxial type CNTFET is studied by quantum simulations using the non-equilibrium Green's function formalism with the self-consistent born approximation. The simulation shows that the current-voltage characteristics of CNTFET vary depends on the chirality and thickness of the oxide layer. This analysis is applicable to primitive and complex digital and analog circuit design based on CNTFETs. The CNTFET simulation is carried out using Nanohub.org, which is an online based java platform engine.