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arXiv:1409.3585 · New Journal of Physics

Universal quantum computation by scattering in the Fermi–Hubbard model

Ning Bao, Patrick M. Hayden, Grant Salton, Nathaniel Thomas

Published Sep 17, 2015• 7 citations
arXivHTMLPDF

Abstract

The Hubbard model may be the simplest model of particles interacting on a lattice, but simulation of its dynamics remains beyond the reach of current numerical methods. In this article, we show that general quantum computations can be encoded into the physics of wave packets propagating through a planar graph, with scattering interactions governed by the fermionic Hubbard model. Therefore, simulating the model on planar graphs is as hard as simulating quantum computation. We give two different arguments, demonstrating that the simulation is difficult both for wave packets prepared as excitations of the fermionic vacuum, and for hole wave packets at filling fraction one-half in the limit of strong coupling. In the latter case, which is described by the t - J model, there is only reflection and no transmission in the scattering events, as would be the case for classical hard spheres. In that sense, the construction provides a quantum mechanical analog of the Fredkin–Toffoli billiard ball computer.

DOI
10.1088/1367-2630/17/9/093028

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The Hubbard model may be the simplest model of particles interacting on a lattice, but simulation of its dynamics remains beyond the reach of current numerical methods. In this article, we show that general quantum computations can be encoded into the physics of wave packets propagating through a planar graph, with scattering interactions governed by the fermionic Hubbard model. Therefore, simulating the model on planar graphs is as hard as simulating quantum computation. We give two different arguments, demonstrating that the simulation is difficult both for wave packets prepared as excitations of the fermionic vacuum, and for hole wave packets at filling fraction one-half in the limit of strong coupling. In the latter case, which is described by the t - J model, there is only reflection and no transmission in the scattering events, as would be the case for classical hard spheres. In that sense, the construction provides a quantum mechanical analog of the Fredkin–Toffoli billiard ball computer.
Quantum computing provides a novel avenue towards simulating dynamical phenomena, and, in particular, scattering processes relevant for exploring the structure of matter. However, preparing and evolving particle wave packets on a quantum device is a nontrivial task. In this work, we propose a method to prepare Gaussian wave packets with momentum on top of the interacting ground state of a fermionic Hamiltonian. Using Givens rotation, we show how to efficiently obtain expectation values of observables throughout the evolution of the wave packets on digital quantum computers. We demonstrate our technique by applying it to the staggered lattice formulation of the Thirring model and studying the scattering of two wave packets. Monitoring the particle density and the entropy produced during the scattering process, we characterize the phenomenon and provide a first step towards studying more complicated collision processes on digital quantum computers. In addition, we perform a small-scale demonstration on IBM's quantum hardware, showing that our method is suitable for current and near-term quantum devices.
Collider experiments play a central role in understanding the subatomic structure of matter, as well as developing and verifying the fundamental theory of elementary particle interactions. However, comprehending scattering processes at a fundamental level in theory remains a significant challenge. The necessary involved time evolution and the with time rapidly increasing bond dimension in Tensor Networks make simulating the scattering process with this classical method challenging. On the other hand, quantum computers hold great promise to efficiently simulate real-time dynamics of lattice field theories. In this work, we take the first step in this direction toward simulating fermionic scattering using a digital-quantum computing approach. Specifically, we propose a method based on Givens rotation to prepare the initial state of the fermionic scattering process, which consists of two fermionic wave packets with opposite momenta. With a time evolution operator based on the underlying Hamiltonian acting on the initial state, the two fermionic wave packets propagate and interact with each other. Using the lattice Thirring model as the test bed and the Qiskit Statevector simulator, we observe an elastic scattering between fermions and antifermions in the strong interaction region. In addition, we clearly observe an entanglement production in the scattering process. We consider our work also as an indispensable step towards a quantum simulation of a scattering process on a real quantum device.
Quantum simulation holds promise of enabling a complete description of high-energy scattering processes rooted in gauge theories of the Standard Model. A first step in such simulations is preparation of interacting hadronic wave packets. To create the wave packets, one typically resorts to adiabatic evolution to bridge between wave packets in the free theory and those in the interacting theory, rendering the simulation resource intensive. In this work, we construct a wave-packet creation operator directly in the interacting theory to circumvent adiabatic evolution, taking advantage of resource-efficient schemes for ground-state preparation, such as variational quantum eigensolvers. By means of an ansatz for bound mesonic excitations in confining gauge theories, which is subsequently optimized using classical or quantum methods, we show that interacting mesonic wave packets can be created efficiently and accurately using digital quantum algorithms that we develop. Specifically, we obtain high-fidelity mesonic wave packets in the Z 2 and U ( 1 ) lattice gauge theories coupled to fermionic matter in 1+1 dimensions. Our method is applicable to both perturbative and non-perturbative regimes of couplings. The wave-packet creation circuit for the case of the Z 2 lattice gauge theory is built and implemented on the Quantinuum H1-1 trapped-ion quantum computer using 13 qubits and up to 308 entangling gates. The fidelities agree well with classical benchmark calculations after employing a simple symmetry-based noise-mitigation technique. This work serves as a step toward quantum computing scattering processes in quantum chromodynamics.
Scattering of quasiparticles provides direct access to the interactions that govern strongly correlated systems. In this work, we realize the quasiparticle scattering experiment of the Fermi-Hubbard model on a digital quantum processor by colliding a doublon and a holon in the half-filled chain of 21 sites encoded on 42 qubits of the ibm_boston processor. For the first time, we measure the transmission probability of quasiparticles of the Hubbard model on quantum hardware, which agrees with the Bethe-ansatz scattering matrix without free parameters. Beyond the scattering probabilities, the collision generates entanglement by superposing the transmitted and reflected components, which we infer from the particle-number fluctuation. In sharp contrast, a control collision between two doublons shows neither the interaction dependence nor the number fluctuation, as expected for indistinguishable particles. Our work demonstrates that the ability to create quasiparticles of a chosen flavor on demand and program their motion and interactions makes a digital quantum processor a new experimental toolset for probing the entanglement structure generated by quasiparticle scattering.
We study the scattering of a soliton-like propagating particle with a wall of bound particles, in several strongly interacting one-dimensional lattice models with discrete degrees of freedom. We consider spin-polarized fermions (anisotropic Heisenberg spin chain), the fermionic Hubbard model, and the Bose Hubbard model, using precise numerical time dependent Density Matrix Renormalization Group techniques. We show that in all integrable models studied, there is no reflection. Instead, an incoming particle experiences particle-hole transmutation upon entry and exit of the wall, and travels inside the wall as a hole, analoguous to Klein tunneling, even though the dispersion is highly nonlinear and there is no external potential. {\em Two} particles are added to the wall on the incoming side and removed on the opposite side. For spin-polarized fermions a single transmitted particle thus shifts the wall by two lattice sites, in complete contrast to classical physics. For both Hubbard models, the wall shifts by one doubly occupied single site. In the nonintegrable models studied, the same process occurs in linear superposition with backscattering events. We demonstrate a corresponding fermionic quantum Newton's cradle and a metamaterial with "tachyonic" modes travelling faster than in an empty system. We present a possible atomic scale signal counter for spintronics. Our scenario should be realizable in future cold atom experiments.
Quantum circuits solely comprising matchgates can perform nontrivial (but nonuniversal) quantum algorithms. Because matchgates can be mapped to noninteracting fermions, these circuits can be efficiently simulated on a classical computer. Universal quantum computation is attainable by adding any nonmatchgate parity-preserving gate, from which one may infer that interacting fermions are natural candidates for universal quantum computation. We consider the quantum computational power of fermions hopping on a one-dimensional double-well lattice within the context of matchgates. In particular, we show that universal quantum computation can be implemented using spinless (spin-polarized) fermions and nearest-neighbor interactions, as well as with spin-half fermions with on-site interactions (i.e., the Hubbard model). We suggest that these schemes are currently within reach in the context of ultracold atomic gases.