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Quantum simulation of fermionic models

J. M. Kreula

Published Jan 1, 2017• 0 citations
Publisher

Abstract

This work is a theoretical study of fermionic models. We focus on problems where highly controllable quantum simulators of these models have an important role, and we utilise both the analogue and the digital paradigm of quantum simulation. In the part on analogue quantum simulation, we focus on the proposed 'spin-asymmetric' Josephson effect where Cooper-paired spins display frequency synchronized Josephson oscillations with spin-dependent amplitudes. We consider different scenarios where the phenomenon could manifest in ultracold atomic Fermi gases. We study a Fermi gas Josephson junction in the recently realized Josephson plasma oscillation regime with an additional spin-dependent potential and show that the asymmetry in the resulting spin-dependent plasma oscillation amplitudes is on the order of a couple of per cent. We also demonstrate numerically that spin-asymmetric Josephson-like currents occur in a one-dimensional spin-dependent optical superlattice, with amplitude asymmetries up to 39%. Finally, we show that at zero temperature the tunable critical current in ferromagnetic Josephson junctions can be explained by the spin-asymmetric Josephson effect. In the part where digital quantum simulation is used, we propose a hybrid quantum-classical approach to studying strongly correlated fermion models. In this approach, a digital quantum simulator works in conjunction with a classical feedback loop to solve the infinite-dimensional Hubbard model directly in the thermodynamic limit. The scheme implements the well-established dynamical mean-field theory (DMFT) method, such that the digital quantum simulator solves the classically hard DMFT impurity problem and self-consistency is taken care of in a classical computer. We first present a few-qubit proof-of-principle setup for equilibrium systems that implements the simplified 'two-site' DMFT. This few-qubit setup is used for a qualitative description of the Mott transition in the half-filled infinite-dimensional Hubbard model. We then describe a scalable setup for simulating non-equilibrium many-body quantum dynamics by proposing the implementation of the non-equilibrium extension of DMFT with the hybrid device.

DOI
10.5287/ora-wvbkgaojp

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This work is a theoretical study of fermionic models. We focus on problems where highly controllable quantum simulators of these models have an important role, and we utilise both the analogue and the digital paradigm of quantum simulation. In the part on analogue quantum simulation, we focus on the proposed 'spin-asymmetric' Josephson effect where Cooper-paired spins display frequency synchronized Josephson oscillations with spin-dependent amplitudes. We consider different scenarios where the phenomenon could manifest in ultracold atomic Fermi gases. We study a Fermi gas Josephson junction in the recently realized Josephson plasma oscillation regime with an additional spin-dependent potential and show that the asymmetry in the resulting spin-dependent plasma oscillation amplitudes is on the order of a couple of per cent. We also demonstrate numerically that spin-asymmetric Josephson-like currents occur in a one-dimensional spin-dependent optical superlattice, with amplitude asymmetries up to 39%. Finally, we show that at zero temperature the tunable critical current in ferromagnetic Josephson junctions can be explained by the spin-asymmetric Josephson effect. In the part where digital quantum simulation is used, we propose a hybrid quantum-classical approach to studying strongly correlated fermion models. In this approach, a digital quantum simulator works in conjunction with a classical feedback loop to solve the infinite-dimensional Hubbard model directly in the thermodynamic limit. The scheme implements the well-established dynamical mean-field theory (DMFT) method, such that the digital quantum simulator solves the classically hard DMFT impurity problem and self-consistency is taken care of in a classical computer. We first present a few-qubit proof-of-principle setup for equilibrium systems that implements the simplified 'two-site' DMFT. This few-qubit setup is used for a qualitative description of the Mott transition in the half-filled infinite-dimensional Hubbard model. We then describe a scalable setup for simulating non-equilibrium many-body quantum dynamics by proposing the implementation of the non-equilibrium extension of DMFT with the hybrid device.
We propose a non-linear, hybrid quantum-classical scheme for simulating non-equilibrium dynamics of strongly correlated fermions described by the Hubbard model in a Bethe lattice in the thermodynamic limit. Our scheme implements non-equilibrium dynamical mean field theory (DMFT) and uses a digital quantum simulator to solve a quantum impurity problem whose parameters are iterated to self-consistency via a classically computed feedback loop where quantum gate errors can be partly accounted for. We analyse the performance of the scheme in an example case.
This work considers the theory of ultracold Fermi gases. We study the spin-asymmetric Josephson effect where the spin components of a Cooper pair feel spin-dependent potentials during the tunneling process and as a result oscillate at the same Josephson frequency but with different amplitudes. This contrasts the conventional interpretation of the Josephson effect in which the Cooper-paired spins 'tunnel together'. Here, we propose a new experimental arrangement to detect the yet-unobserved spin-asymmetric Josephson effect. We investigate two-component fermions in a one-dimensional spin-dependent optical superlattice described by a Fermi-Hubbard Hamiltonian and simulate the dynamics of the set-up with the essentially exact time-evolving block decimation (TEBD) numerical method. We demonstrate that the characteristics of the spin-asymmetric Josephson effect, frequency-synchronized particle number oscillations with spin-asymmetric critical currents, are present in the system. We show that the Josephson frequency depends only on the sum of the spin-dependent potential offsets between two adjacent sites in the super-lattice. We further exhibit that significant asymmetries of over 10% in the critical currents of the spin components are feasible. We also evince that the properties of the particle currents in the super-lattice system agree with the analytical results given by a Hubbard dimer model consisting of only one unit cell of the super-lattice in the strongly interacting limit.
The spin-asymmetric Josephson effect is a proposed quantum-coherent tunneling phenomenon where Cooper-paired fermionic spin-12\frac{1}{2}21​ particles, which are subjected to spin-dependent potentials across a Josephson junction, undergo frequency-synchronized alternating-current Josephson oscillations with spin-dependent amplitudes. Here, in line with present-day techniques in ultracold Fermi gas setups, we consider the regime of small Josephson oscillations and show that the Josephson plasma oscillation amplitude becomes spin dependent in the presence of spin-dependent potentials, while the Josephson plasma frequency is the same for both spin components. Detecting these spin-dependent Josephson plasma oscillations provides a possible means to establish the yet-unobserved spin-asymmetric Josephson effect with ultracold Fermi gases using existing experimental tools.
Among the most compelling targets for quantum simulation are strongly correlated phases and electronic-structure problems, where interactions organize electrons into collective states and produce intricate low-temperature phase diagrams. Cuprate superconductors are a canonical example: at low doping, superconductivity emerges from a metallic regime that exhibits a pseudogap above the critical temperature. Ultracold atoms in optical lattices provide a well-controlled, tunable setting to probe the microscopic mechanisms behind this behavior and, in fermion-native architectures, to implement electronic structure dynamics directly. This thesis demonstrates both large scale systematic analog quantum simulations in the most interesting regime of doped antiferromagnets and elementary high-fidelity gates as the basis for a digital or hybrid fermionic quantum processors in an optical lattice. Working with ultracold Lithium-6 in a two–dimensional optical superlattice, we first map the parameter space of the Fermi–Hubbard model over a wide range of temperatures and dopings. More than 40000 single-site spin- and density-resolved snapshots, spanning over a dense grid of doping levels and temperatures in a numerically challenging parameter space, reveal antiferromagnetic correlations extending across the system size and into the pseudogap regime. Temperatures down to T/t=0.2 are realized, determined by benchmarking the data to state-of-the-art numerical methods. By evaluating spin–charge correlators up to fifth order, we experimentally uncover a universal scaling behavior across the pseudogap crossover, governed by a single, doping-dependent temperature scale. These findings provide the first microscopic evidence of universal scaling in multi-point correlators and establish stringent benchmarks for future theoretical developments. On the same apparatus, we demonstrate that the controlled motion of fermions in double wells enables high-fidelity entanglement. Double-well potentials are used to engineer coherent pair-tunnelling and spin-exchange interactions that act as native SWAP^α operations. We demonstrate the collisional gate fidelities up to 99.8% and Bell-state lifetimes exceeding ten seconds, limited mainly by residual inhomogeneities in the optical potential. A composite pulse sequence isolates the pair-exchange channel, realizing the elementary building block required for fermionic quantum-chemistry algorithms. With the addition of site-selective addressing, our results mark a key step towards a fully digital, scalable fermionic quantum computer based on the controlled motion and entanglement of fermionic neutral atoms. The work establishes a hybrid analog-digital architecture: analog evolution under the Hubbard Hamiltonian yields thermodynamic states whose magnetic properties match and extend theoretical benchmarks, while digital collisional gates provide building blocks for error-corrected quantum computation and complex state preparation.
We study a proof-of-principle example of the recently proposed hybrid quantum-classical simulation of strongly correlated fermion models in the thermodynamic limit. In a ‘two-site’ dynamical mean-field theory (DMFT) approach we reduce the Hubbard model to an effective impurity model subject to self-consistency conditions. The resulting minimal two-site representation of the non-linear hybrid setup involves four qubits implementing the impurity problem, plus an ancilla qubit on which all measurements are performed. We outline a possible implementation with superconducting circuits feasible with near-future technology.
We experimentally study the two-dimensional Fermi-Hubbard model using a Rydberg-based quantum processing unit in the analog mode. Our approach avoids encoding directly the original fermions into qubits and instead relies on reformulating the original model onto a system of fermions coupled to spins and then decoupling them in a self-consistent manner. We then introduce the auxiliary spin solver: this hybrid quantum-classical algorithm handles a free-fermion problem, which can be solved efficiently with a few classical resources, and an interacting spin problem, which can be naturally encoded in the analog quantum computer. This algorithm can be used to study both the equilibrium Mott transition as well as non-equilibrium properties of the original Fermi-Hubbard model, highlighting the potential of quantum-classical hybrid approaches to study strongly correlated matter.