Quantum field theories are the cornerstones of modern physics, providing relativistic and quantum mechanical descriptions of physical systems at the most fundamental level. Simulating real-time dynamics within these theories remains elusive in classical computing. This provides a unique opportunity for quantum simulators, which hold the promise of revolutionizing our simulation capabilities. Trapped-ion systems are successful quantum-simulator platforms for quantum many-body physics and can operate in digital, or gate-based, and analog modes. Inspired by the progress in proposing and realizing quantum simulations of a number of relativistic quantum field theories using trapped-ion systems, and by the hybrid analog-digital proposals for simulating interacting boson-fermion models, we propose hybrid analog-digital quantum simulations of selected quantum field theories, taking recent developments to the next level. On one hand, the semi-digital nature of this proposal offers more flexibility in engineering generic model interactions compared with a fully-analog approach. On the other hand, encoding the bosonic fields onto the phonon degrees of freedom of the trapped-ion system allows a more efficient usage of simulator resources, and a more natural implementation of intrinsic quantum operations in such platforms. This opens up ways for simulating complex dynamics of, e.g., Abelian and non-Abelian gauge theories, by combining the benefits of digital and analog schemes.
Quantum field theories are the cornerstones of modern physics, providing relativistic and quantum mechanical descriptions of physical systems at the most fundamental level. Simulating real-time dynamics within these theories remains elusive in classical computing. This provides a unique opportunity for quantum simulators, which hold the promise of revolutionizing our simulation capabilities. Trapped-ion systems are successful quantum-simulator platforms for quantum many-body physics and can operate in digital, or gate-based, and analog modes. Inspired by the progress in proposing and realizing quantum simulations of a number of relativistic quantum field theories using trapped-ion systems, and by the hybrid analog-digital proposals for simulating interacting boson-fermion models, we propose hybrid analog-digital quantum simulations of selected quantum field theories, taking recent developments to the next level. On one hand, the semi-digital nature of this proposal offers more flexibility in engineering generic model interactions compared with a fully-analog approach. On the other hand, encoding the bosonic fields onto the phonon degrees of freedom of the trapped-ion system allows a more efficient usage of simulator resources, and a more natural implementation of intrinsic quantum operations in such platforms. This opens up ways for simulating complex dynamics of, e.g., Abelian and non-Abelian gauge theories, by combining the benefits of digital and analog schemes.
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].
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.
Simulating out-of-equilibrium dynamics of quantum field theories in nature is challenging with classical methods, but is a promising application for quantum computers. Unfortunately, simulating interacting bosonic fields involves a high boson-to-qubit encoding overhead. Furthermore, when mapping to qubits, the infinite-dimensional Hilbert space of bosons is necessarily truncated, with truncation errors that grow with energy and time. A qubit-based quantum computer, augmented with an active bosonic register, and with qubit, bosonic, and mixed qubit-boson quantum gates, offers a more powerful platform for simulating bosonic theories. We demonstrate this capability experimentally in a hybrid analog-digital trapped-ion quantum computer, where qubits are encoded in the internal states of the ions, and the bosons in the ions' motional states. Specifically, we simulate nonequilibrium dynamics of a (1+1)-dimensional Yukawa model, a simplified model of interacting nucleons and pions, and measure fermion- and boson-occupation-state probabilities. These dynamics populate high bosonic-field excitations starting from an empty state, and the experimental results capture well such high-occupation states. This simulation approaches the regime where classical methods become challenging, bypasses the need for a large qubit overhead, and removes truncation errors. Our results, therefore, open the way to achieving demonstrable quantum advantage in qubit-boson quantum computing.
Trapped atomic ions are among the most advanced platforms for quantum simulation, computation, and metrology, offering long coherence times and precise, individual control over both internal and motional degrees of freedom. In this review, we present a pedagogical introduction to trapped-ion systems, covering the physics of ion trapping, qubit encodings, and laser-ion interactions. We explain how spin-dependent forces generated by light fields enable both analog and digital quantum simulations of spin and spin-boson models, as well as high-fidelity quantum logic gates. We then highlight an emerging frontier in the simulation of chemical dynamics, summarizing recent experiments that demonstrate the capability of trapped ions to simulate vibronic models and excitation-transfer processes. Finally, we outline future directions in quantum simulation and discuss open challenges in scaling up trapped-ion architectures.
Trapped ions offer long coherence times and high fidelity, programmable quantum operations, making them a promising platform for quantum simulation of condensed matter systems, quantum dynamics, and problems related to high-energy physics. We review selected developments in trapped-ion qubits and architectures and discuss quantum simulation applications that utilize these emerging capabilities. This review emphasizes developments in digital (gate-based) quantum simulations that exploit trapped-ion hardware capabilities, such as flexible qubit connectivity, selective mid-circuit measurement, and classical feedback, to simulate models with long-range interactions, explore non-unitary dynamics, compress simulations of states with limited entanglement, and reduce the circuit depths required to prepare or simulate long-range entangled states.
Trapped ions offer long coherence times and high-fidelity, programmable quantum operations, making them a promising platform for quantum simulation of condensed matter systems, quantum dynamics, and problems related to high-energy physics. We review selected developments in trapped-ion qubits and architectures and discuss quantum simulation applications that utilize these emerging capabilities. This review emphasizes developments in digital (gate-based) quantum simulations that exploit trapped-ion hardware capabilities, such as flexible qubit connectivity, selective midcircuit measurement, and classical feedback, to simulate models with long-range interactions, explore nonunitary dynamics, compress simulations of states with limited entanglement, and reduce the circuit depths required to prepare or simulate long-range entangled states.