arXiv subjects

J. K. Freericks

Publications and source records attributed to J. K. Freericks.

At least 37 records · Page 2Linked to original sources

arXiv2021

Flexibility of the factorized form of the unitary coupled cluster ansatz

The factorized form of the unitary coupled cluster ansatz is a popular state preparation ansatz for electronic structure calculations of molecules on quantum computers. It often is viewed as an approximation (based on the Trotter product formula) for the conventional unitary coupled cluster operator. In this work, we show that the factorized form is quite flexible, allowing one to range from conventional configuration interaction, to conventional unitary coupled cluster, to efficient approximations that lie in between these two. The variational minimization of the energy often allows simpler factorized unitary coupled cluster approximations to achieve high accuracy, even if they do not accurately approximate the Trotter product formula. This is similar to how quantum approximate optimization algorithms can achieve high accuracy with a small number of levels.

physics.chem-ph↗

arXiv2021

Determining quantum phase diagrams of topological Kitaev-inspired models on NISQ quantum hardware

Topological protection is employed in fault-tolerant error correction and in developing quantum algorithms with topological qubits. But, topological protection intrinsic to models being simulated, also robustly protects calculations, even on NISQ hardware. We leverage it by simulating Kitaev-inspired models on IBM quantum computers and accurately determining their phase diagrams. This requires constructing conventional quantum circuits for Majorana braiding to prepare the ground states of Kitaev-inspired models. The entanglement entropy is then measured to calculate the quantum phase boundaries. We show how maintaining particle-hole symmetry when sampling through the Brillouin zone is critical to obtaining high accuracy. This work illustrates how topological protection intrinsic to a quantum model can be employed to perform robust calculations on NISQ hardware, when one measures the appropriate protected quantum properties. It opens the door for further simulation of topological quantum models on quantum hardware available today.

quant-ph↗

arXiv2021

Exact solution of two simple non-equilibrium electron-phonon and electron-electron coupled systems: the atomic limit of the Holstein-Hubbard model and the generalized Hatsugai-Komoto model

One of the challenges in many-body physics is determining the effects of phonons on strongly correlated electrons. The difficulty arises from strong correlations at differing energy scales -- for band metals, Migdal-Eliashberg theory accurately determines electron-phonon coupling effects due to the absence of vertex corrections -- but strongly correlated electrons require a more complex description and the standard Migdal-Eliashberg approach does not necessarily apply. In this work, we solve for the atomic limit Green's function of the Holstein-Hubbard model with both time-dependent electron-electron and electron-phonon couplings. We then examine the photoemission spectra (PES) of this model in and out of equilibrium. Next we use similar methods to exactly solve an extended version of the Hatsugai-Komoto model, and examine its behavior in and out of equilibrium. These calculations lead us to propose using the first moment of the photoemission spectra to signal non-equilibrium changes in electron-electron and electron-phonon couplings.

cond-mat.str-el↗

arXiv2021

Nonequilibrium spectral moment sum rules of the Holstein-Hubbard model

We derive a general procedure for evaluating the ${\rm n}$th derivative of a time-dependent operator in the Heisenberg representation and employ this approach to calculate the zeroth to third spectral moment sum rules of the retarded electronic Green's function and self-energy for a system described by the Holstein-Hubbard model allowing for arbitrary spatial and time variation of all parameters (including spatially homogeneous electric fields and parameter quenches). For a translationally invariant (but time-dependent) Hamiltonian, we also provide sum rules in momentum space. The sum rules can be applied to various different phenomena like time-resolved angle-resolved photoemission spectroscopy and benchmarking the accuracy of numerical many-body calculations. This work also corrects some errors found in earlier work on simpler models.

cond-mat.str-el↗

arXiv2021

Making squeezed-coherent states concrete by determining their wavefunction

With the successes of the Laser Interferometer Gravitational-wave Observatory, we anticipate increased interest in working with squeezed states in the undergraduate and graduate quantum-mechanics classroom. Because squeezed-coherent states are minimum uncertainty states, their wavefunctions in position and momentum space must be Gaussians. But this result is rarely discussed in treatments of squeezed states in quantum textbooks or quantum optics textbooks. In this work, we show three different ways to construct the wavefunction for squeezed-coherent states: (i) a differential equation-based approach; (ii) an approach that uses an expansion in terms of the simple-harmonic oscillator wavefunctions; and (iii) a fully operator-based approach. We do this to illustrate that the concept of the wavefunction can be introduced no matter what methodology an instructor wishes to use. We hope that working with the wavefunction will help demystify the concept of a squeezed-coherent state.

quant-ph↗

arXiv2021

A pedagogical derivation of dynamical susceptibilities

Dynamical two-particle susceptibilites are important for a wide range of different experiments in condensed-matter physics and beyond. Nevertheless, most textbooks avoid describing how to derive such response functions, perhaps because they are viewed as too complex. In the literature, most derivations work with generalized susceptibilities, which are more general, but require an even higher layer of complexity. In this work, we show a more direct derivation in the context of model Hamiltonians which can be mapped directly onto an impurity model. We also present an alternative derivation for the irreducible vertex in the context of the Falicov-Kimball model.

quant-ph↗

arXiv2021

Native multiqubit Toffoli gates on ion trap quantum computers

We examine the detailed scenario for implementing n-control-qubit Toffoli gates and select gates on ion-trap quantum computers, especially those that shuttle ions into interaction zones. We determine expected performance of these gates with realistic parameters for an ion-trap quantum computer and taking into account the time variation of the exchange integrals. This allows us to estimate the errors due to spin-phonon entanglement as well. While there are challenges with implementing these gates, because their performance always has some degree of error, they should be feasible on current hardware, but they may be too slow to be used efficiently in quantum codes on noisy intermediate scale quantum computers.

quant-ph↗

arXiv2020

Many Body Thermodynamics on Quantum Computers via Partition Function Zeros

Interacting quantum systems illustrate complex phenomena including phase transitions to novel ordered phases. The universal nature of critical phenomena reduces their description to determining only the transition temperature and the critical exponents. Numerically calculating these results for systems in new universality classes is complicated due to critical slowing down, requiring increasing resources near the critical point. An alternative approach analytically continues the calculation onto the complex plane and determines the partition function via its zeros. Here we show how to robustly perform this analysis on noisy intermediate scale trapped ion quantum computers in a scalable manner, using the XXZ model as a prototype. We illustrate the transition from XY-like behavior to Ising-like behavior as a function of the anisotropy. While quantum computers cannot yet scale to the thermodynamic limit, our work provides a pathway to do so as hardware improves, allowing the determination of critical phenomena for systems that cannot be solved otherwise.

quant-ph↗

arXiv2020

Driven-dissipative quantum mechanics on a lattice: Simulating a fermionic reservoir on a quantum computer

The driven-dissipative many-body problem remains one of the most challenging unsolved problems in quantum mechanics. The advent of quantum computers may provide a unique platform for efficiently simulating such driven-dissipative systems. But there are many choices for how one can engineer the reservoir. One can simply employ ancilla qubits to act as a reservoir and then digitally simulate them via algorithmic cooling. A more attractive approach, which allows one to simulate an infinite reservoir, is to integrate out the bath degrees of freedom and describe the driven-dissipative system via a master equation, that can also be simulated on a quantum computer. In this work, we consider the particular case of non-interacting electrons on a lattice driven by an electric field and coupled to a fermionic thermostat. Then, we provide two different quantum circuits: the first one reconstructs the full dynamics of the system using Trotter steps, while the second one dissipatively prepares the final non-equilibrium steady state in a single step. We run both circuits on the IBM quantum experience. For circuit (i), we achieved up to 5 Trotter steps. When partial resets become available on quantum computers, we expect that the maximum simulation time can be significantly increased. The methods developed here suggest generalizations that can be applied to simulating interacting driven-dissipative systems.

quant-ph↗

arXiv2020

Algebraic derivation of Kramers-Pasternack relations based on the Schrodinger factorization method

The Kramers-Pasternack relations are used to compute the moments of r (both positive and negative) for all radial energy eigenfunctions of hydrogenic atoms. They consist of two algebraic recurrence relations, one for positive powers and one for negative. Most derivations employ the Feynman-Hellman theorem or a brute-force integration to determine the second inverse moment, which is needed to complete the recurrence relations for negative moments. In this work, we show both how to derive the recurrence relations algebraically and how to determine the second inverse moment algebraically, which removes the pedagogical confusion associated with differentiating the Hamiltonian with respect to the angular momentum quantum number l in order to find the inverse second moment.

quant-ph↗

arXiv2020

Converting translation operators into plane polar and spherical coordinates and their use in determining quantum-mechanical wavefunctions in a representation-independent fashion

Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing to express any operators in position space.

quant-ph↗

arXiv2020

Quantum computation of magnon spectra

We demonstrate quantum computation of two-point correlation functions for a Heisenberg spin chain. Using the IBM Q 20 quantum machines, we find that for two sites the correlation functions produce the exact results reliably. For four sites, results from the IBM Q 20 Tokyo quantum computer are noisy due to read out errors and decoherence. Nevertheless, the correlation functions retain the correct spectral information. This is illustrated in the frequency domain by accurately extracting the magnon energies from peaks in the spectral function.

cond-mat.str-el↗

arXiv2020

Test of the unitary coupled-cluster variational quantum eigensolver for a simple strongly correlated condensed-matter system

The variational quantum eigensolver has been proposed as a low-depth quantum circuit that can be employed to examine strongly correlated systems on today's noisy intermediate-scale quantum computers. We examine details associated with the factorized form of the unitary coupled-cluster variant of this algorithm. We apply it to a simple strongly correlated condensed-matter system with nontrivial behavior---the four-site Hubbard model at half filling. This work show some of the subtle issues one needs to take into account when applying this algorithm in practice, especially to condensed-matter systems.

quant-ph↗

arXiv2019

A completely algebraic solution of the simple harmonic oscillator

We present a full algebraic derivation of the wavefunctions of the simple harmonic oscillator in coordinate and momentum space. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the representation-dependent wavefunctions from the representation-independent energy eigenfunctions. It is simple to incorporate into the undergraduate and graduate curricula. This new derivation begins with the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schroedinger factorization method), and then supplements it by employing the translation (or boost) operator to determine the wavefunctions algebraically, without any derivatives. In addition, we provide a summary of the history of this approach, which seems to have been neglected by most historians of quantum mechanics, until now.

quant-ph↗

arXiv2019

An introduction to many-body Green's functions in and out of equilibrium

This is an introductory chapter on how to calculate nonequilibrium Green's functions via dynamical mean-field theory for the Autumn School on Correlated Electrons: Many-Body Methods for Real Materials, 16-20 September 2019, Forschungszentrum Juelich. It is appropriate for graduate students with a solid state physics and advanced quantum mechanics background.

cond-mat.str-el↗

arXiv2019

Comparison Between the f-Electron and Conduction-Electron Density of States in the Falicov-Kimball Model at Low Temperature

The spinless Falicov-Kimball model is one of the simplest models of many-body physics. While the conduction-electron density of states is temperature independent in the normal state, the f-electron density of states is strongly temperature dependent---it has an orthogonality catastrophe singularity in the metallic phase and is gapped in the insulating phase. The question we address here is whether the spectral gap is the same for both electron species as T->0. We find strong evidence to indicate that the answer is affirmative.

cond-mat.str-el↗

arXiv2019

Bang-bang shortcut to adiabaticity in the Dicke model as realized in a Penning trap experiment

We introduce a bang-bang shortcut to adiabaticity for the Dicke model, which we implement via a 2-D array of trapped ions in a Penning trap with a spin-dependent force detuned close to the center-of-mass drumhead mode. Our focus is on employing this shortcut to create highly entangled states that can be used in high-precision metrology. We highlight that the performance of the bang-bang approach is comparable to standard preparation methods, but can be applied over a much shorter time frame. We compare these theoretical ideas with experimental data which serve as a first step towards realizing this theoretical procedure for generating multi-partite entanglement.

quant-ph↗

arXiv2018

Calculating spherical harmonics without derivatives

The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom. It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In this work, we show how one can determine spherical harmonics in a more natural way based on operators and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but little used elsewhere). This new strategy follows naturally after one has introduced Dirac notation, computed the angular momentum algebra, and determined the action of the angular momentum raising and lowering operators on the simultaneous angular momentum eigenstates (under $\hat L^2$ and $\hat L_z$).

quant-ph↗