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arXiv:2108.01022 · Quantum Science and Technology

Engineering an effective three-spin Hamiltonian in trapped-ion systems for applications in quantum simulation

Bárbara Andrade, Zohreh Davoudi, Tobias Graß, Mohammad Hafezi, Guido Pagano, Alireza Seif

Published Mar 21, 2022• 44 citations

Abstract

Abstract Trapped-ion quantum simulators, in analog and digital modes, are considered a primary candidate to achieve quantum advantage in quantum simulation and quantum computation. The underlying controlled ion–laser interactions induce all-to-all two-spin interactions via the collective modes of motion through Cirac–Zoller or Mølmer–Sørensen schemes, leading to effective two-spin Hamiltonians, as well as two-qubit entangling gates. In this work, the Mølmer–Sørensen scheme is extended to induce three-spin interactions via tailored first- and second-order spin–motion couplings. The scheme enables engineering single-, two-, and three-spin interactions, and can be tuned via an enhanced protocol to simulate purely three-spin dynamics. Analytical results for the effective evolution are presented, along with detailed numerical simulations of the full dynamics to support the accuracy and feasibility of the proposed scheme for near-term applications. With a focus on quantum simulation, the advantage of a direct analog implementation of three-spin dynamics is demonstrated via the example of matter-gauge interactions in the U(1) lattice gauge theory within the quantum link model. The mapping of degrees of freedom and strategies for scaling the three-spin scheme to larger systems, are detailed, along with a discussion of the expected outcome of the simulation of the quantum link model given realistic fidelities in the upcoming experiments. The applications of the three-spin scheme go beyond the lattice gauge theory example studied here and include studies of static and dynamical phase diagrams of strongly interacting condensed-matter systems modeled by two- and three-spin Hamiltonians.

DOI: 10.1088/2058-9565/ac5f5b

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Trapped atomic ion qubits or effective spins are a powerful quantum platform for quantum computation and simulation, featuring densely connected and efficiently programmable interactions between the spins. While native interactions between trapped ion spins are typically pairwise, many quantum algorithms and quantum spin models naturally feature couplings between triplets, quartets or higher orders of spins. Here we formulate and analyze a mechanism that extends the standard Mølmer-Sørensen pairwise entangling gate and generates a controllable and programmable coupling between

N

spins of trapped ions. We show that spin-dependent optical forces applied at twice the motional frequency generate a coordinate-transformation of the collective ion motion in phase-space, rendering displacement forces that are nonlinear in the spin operators. We formulate a simple framework that enables a systematic and faithful construction of high-order spin Hamiltonians and gates, including the effect of multiple modes of motion, and characterize the performance of such operations under realistic conditions.

Universal digital quantum computers are able to approximate any Hamiltonian evolution as a digital quantum circuit consisting of discrete time steps. Strongly interacting physics models, for example in high-energy physics or materials science, make this a challenging task for near-term devices as they place a high demand on the quantum resources required [1]. One alternative is analog quantum simulation where the native evolution of a quantum system can be tailored to mimic that of the model of interest. While more efficient at specific problems, there are limits on the types of evolution available and on the observables that can be measured. In this conference paper we present first results towards unifying these two concepts in a trapped-ion experiment. Trapped atomic ions are one of the most advanced experimental platforms for both analog quantum simulation and digital quantum computing. Typically, the internal spin degrees of freedom are used qubits or for the simulation of spin models. The harmonic oscillator modes given by the motion of the ions in the trap are only employed transiently to create quantum gates or spin-spin interactions. However, these motional modes constitute a large Hilbert space themselves and present a bosonic degree of freedom, which is costly to model with qubits. Many systems of interest, for example Lattice gauge theories, require both spin and boson degrees of freedom. It was recently shown that they can be encoded more efficiently using the harmonic oscillator modes directly [2].

Trapped atomic ion qubits or effective spins are a powerful quantum platform for quantum computation and simulation, featuring densely connected and efficiently programmable interactions between the spins.While native interactions between trapped-ion spins are typically pairwise, many quantum algorithms and quantum spin models naturally feature couplings between triplets, quartets, or higher orders of spins.Here, we formulate and analyze a mechanism that extends the standard Mølmer-Sørensen pairwise entangling gate and generates a controllable and programmable coupling between N spins of trapped ions.We show that spin-dependent optical parametric drives applied at twice the motional frequency generate a coordinate transformation of the collective ion motion in phase space, rendering displacement forces that are nonlinear in the spin operators.We formulate a simple framework that enables a systematic and faithful construction of high-order spin Hamiltonians and gates, including the effect of multiple modes of motion, and characterize the performance of such operations under realistic conditions.

In a recent experiment, Barreiro et al (2011 Nature 470 486) demonstrated the fundamental building blocks of an open-system quantum simulator with trapped ions. Using up to five ions, dynamics were realized by sequences that combined single- and multi-qubit entangling gate operations with optical pumping. This enabled the implementation of both coherent many-body dynamics and dissipative processes by controlling the coupling of the system to an artificial, suitably tailored environment. This engineering was illustrated by the dissipative preparation of entangled two- and four-qubit states, the simulation of coherent four-body spin interactions and the quantum non-demolition measurement of a multi-qubit stabilizer operator. In this paper, we present the theoretical framework of this gate-based ('digital') simulation approach for open-system dynamics with trapped ions. In addition, we discuss how within this simulation approach, minimal instances of spin models of interest in the context of topological quantum computing and condensed matter physics can be realized in state-of-the-art linear ion-trap quantum computing architectures. We outline concrete simulation schemes for Kitaev's toric code Hamiltonian and a recently suggested color code model. The presented simulation protocols can be adapted to scalable and two-dimensional ion-trap architectures, which are currently under development.

While quantum simulators promise to explore quantum many-body physics beyond classical computation, their capabilities are limited by the available native interactions in the hardware. On many platforms, accessible Hamiltonians are largely restricted to one- and two-body interactions, limiting access to multi-body Hamiltonians and to systems governed by simultaneous, non-commuting interaction terms that are central to condensed matter, quantum chemistry, and high-energy physics. We introduce and experimentally demonstrate a hybrid digital-analog protocol that overcomes these limitations by embedding analog evolution between shallow entangling-gate layers. This method produces effective Hamiltonians with simultaneous non-commuting three- and four-body interactions that are generated non-perturbatively and without Trotter error -- capabilities not practically attainable on near-term hardware using purely digital or purely analog schemes. We implement our scheme on a trapped-ion quantum processor and use it to realize a topological spin chain exhibiting prethermal strong zero modes persisting at high temperature, as well as models featuring three- and four-body interactions. Our hardware-agnostic and scalable method opens new routes to realizing complex many-body physics across quantum platforms.

We theoretically investigate the possibility of creating phonon-mediated spin-spin interactions between neutral atoms trapped in optical tweezers. By laser coupling the atoms to Rydberg states, collective modes of motion appear. We show that these can be used to mediate effective spin-spin interactions or quantum logic gates between the atoms in analogy to schemes employed in trapped ions. In particular, we employ Rydberg dressing in a novel scheme to induce the needed interaction, and we show that it is possible to replicate the working of the Mølmer-Sørensen entanglement scheme. The Mølmer-Sørensen gate is widely used in emerging quantum computers using trapped ion qubits and currently features some of the highest fidelities of any quantum gate under consideration. We find arbitrarily high fidelity for the coherent time evolution of the two-atom state even at non-zero temperature.