Gauge theories underpin the standard model of particle physics, but are difficult to study using conventional computational methods. An experimental quantum system opens up fresh avenues of investigation. **See Letter ** p.516

There are many questions still to be answered about the standard model of particle physics, which describes the fundamental forces and interactions of nature. On page 516, Martinez *et al*.^{1} report a pioneering experiment in which calcium ions that are trapped and controlled by electromagnetic fields form a quantum simulator of elementary particle physics. This is a first experimental step towards the use of quantum simulators to answer some of those outstanding questions.

Theoretical physics often involves problems that do not have a simple mathematical solution. This quandary is usually overcome using numerical calculations performed by conventional (classical) computers. Some problems, however, cannot be solved by these techniques, and require other methods, especially when direct experimental study is also impossible or difficult.

Physicist Richard Feynman suggested that, to simulate the quantum behaviour of physical systems, other quantum systems must be used — quantum computers^{2}. This concept, called quantum simulation^{3}, is beautifully simple. Consider a quantum system, A, that cannot be studied by conventional theoretical and experimental methods, and another quantum system, B, that can be built and controlled with high precision in the laboratory. If the physical components of B, and the interactions between them, mimic and behave like those of A, then B is a quantum simulator of A. Once B is built, tuned and operated, experimental study of it effectively serves as a study of A.

Quantum simulators can be either analog, in which system B simply 'behaves' like system A because its dynamics and interactions exactly or approximately map those of A, or digital, in which a sequence of operations acts on the components of B and possibly on some auxiliary elements, generating dynamics that are equivalent to those of A with controlled precision. The simulating systems are often atomic or optical, and have included systems of cold atoms or ions trapped by electromagnetic fields. These have been designed (and some have been built) to simulate many areas of physics, ranging from condensed-matter physics to gravitational effects^{3}.

The interactions between elementary particles are a great candidate for quantum simulation. In the standard model of particle physics, such interactions are mediated by vector fields known as gauge fields, thanks to a special type of symmetry called local gauge invariance. Electrons, for example, interact according to the quantum theory of electrodynamics (QED, the simplest gauge theory) through the electromagnetic gauge field. Other fundamental constituents of matter are quarks, whose interactions through the strong force are described by another gauge theory, quantum chromodynamics (QCD).

QCD has several open questions. One is the phenomenon of confinement, in which quarks are bound together by the strong force to form composite particles called hadrons (which include protons and neutrons). The strong force prevents quarks from being isolated experimentally. The theoretical study of confinement is also difficult, and has been a subject of research for decades. A highly successful avenue for studying gauge theories is called lattice gauge theory^{4}, but using it for conventional computer simulations is still problematic for the study of several questions.

The quantum simulation of lattice gauge theories has been a rapidly growing area of study over the past few years, and several proposals have been made for how such simulators could be realized^{5,6}. These simulators quantitatively map the simulated system — which is typically highly energetic — onto low-energy atomic and optical experimental systems. Martinez *et al*. report the first experimental realization of just such a quantum simulator.

The authors simulated lattice QED in a one-dimensional space (a lattice Schwinger model) using a digital quantum simulator — a tailored quantum computer that consists of four trapped calcium ions controlled and manipulated by lasers (Fig. 1). Two energy levels of each ion form a quantum bit (a qubit), which represents the presence or absence of a particle of matter in the corresponding simulated theory. The gauge field is represented as interactions between the ions that are direct and exotic, yet experimentally implementable. This is achieved using a theoretical transformation available in one dimension that eliminates direct manifestations of the field in the simulated model and allows it to be expressed in terms of matter.

The quantum simulation of complicated gauge theories requires a non-trivial combination of advanced technologies in atomic and optical physics. Martinez and colleagues therefore investigated a small version of a 1D lattice QED model, a relatively simple system that enabled their results to be compared with predictions, but that still demonstrates important features of more-complicated models. The authors' quantum simulator did indeed reproduce the expected physical behaviour of the simulated model with great accuracy.

In future work, larger systems should be simulated that have a greater number of dimensions (to reveal further non-trivial types of interaction) and involve more-complicated simulated models such as QCD. Quantum simulators for many of these models have already been proposed — both analog^{5,6} and digital^{7} — for various gauge theories in different dimensions, mostly using cold atoms, but also trapped ions and superconducting qubits. The experimental requirements and feasibility of these proposals vary, because the simulators use different approaches and involve various simulated models, but they mostly require combinations of existing experimental techniques. Technological developments will help to make such experiments more achievable, even for the simulation of complicated models. As the first quantum simulator of a lattice gauge theory to be built, Martinez and co-workers' system serves as a beacon that will lead gauge-theory physicists to the promised land of experimental realization.

The authors' work proves that it is indeed realistic to use quantum-optics techniques to study particle physics and fundamental forces. Further theoretical and experimental advances might enable quantum simulators to solve challenges such as study of the exotic phases of QCD, and to observe new phenomena. More generally, this realization of the great power of quantum simulation reminds us how wonderfully multidisciplinary physics is.

## References

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Martinez, E. A.

*et al.**Nature***534**, 516–519 (2016). - 2.
Feynman, R. P.

*Int. J. Theor. Phys.***21**, 467–488 (1982). - 3.
Nature Physics Insight: Quantum simulation

*Nature Phys.*www.nature.com/nphys/insight/quantum-simulation (2012). - 4.
Wilson, K. G.

*Phys. Rev. D***10**, 2445–2459 (1974). - 5.
Wiese, U.-J.

*Ann. Phys. (Leipz.)***525**, 777–796 (2013). - 6.
Zohar, E., Cirac, J. I. & Reznik, B.

*Rep. Prog. Phys.***79**, 014401 (2016). - 7.
Tagliacozzo, L., Celi, A., Zamora, A. & Lewenstein, M.

*Ann. Phys.***330**, 160–191 (2013).

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### Erez Zohar is in the Theory Division, Max Planck Institute of Quantum Optics, 85748 Garching, Germany.

- Erez Zohar

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Correspondence to Erez Zohar.

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