### Universal digital quantum simulations with trapped ions

B. Lanyon, C. Hempel, D. Nigg, M. Müller, R. Gerritsma, F. Zähringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, C. F. Roos

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The difficulty of simulating quantum mechanics to make predictions about physical systems represents a significant obstacle in fundamental and applied science. Simulations of quantum dynamics on a computer face the problem that both the number of parameters required for specifying the state of the quantum system and the number of differential equations increases exponentially with the number of the particles the system is composed of. In general, this problem limits the applicability of simulation techniques to systems of a few ten particles at most.

### Quantum simulation

A fundamentally different approach is to build a highly controllable quantum system for carrying out the simulation of the quantum system of interest. This idea goes back to Richard Feynman who proposed it about thirty years ago. In combination with the work of many others, these ideas have lead to the development of the large field of quantum information science, the primary concern of which is to explore the information encoding and processing possibilities of quantum systems. The centre-piece of this field is the concept of a universal quantum computer, which in any form is equivalent to: a device which encodes information in a register of two-level systems called qubits (the quantum version of classical bits) and processes it using a universal set of quantum logic gates.

In 1996 Seth Lloyd showed that Feynman was correct [1]: a universal quantum computer can be programmed to efficiently simulate any other quantum system that is thought to occur in nature. Around the same time it was also discovered that a universal quantum computer could significantly reduce the complexity of other important problems, including finding the prime factors of composite numbers, and searching databases. However, it is clear that neither of these applications of quantum information processing have a more immediate chance of coming to fruition than quantum simulation: while a device with many tens of thousands of qubits and millions of logic gates would be required to factor practically useful numbers, or search large databases, a device with only 100 or so may already provide new insight through quantum simulation [2].

This often-made statement is largely based, not on the `digital' approach of Lloyd, but a simpler one known as `analog quantum simulation'. Here, rather than replicating the dynamics using a discrete set of logic gates combined in a quantum algorithm, the approach is to build a controllable quantum system specifically to simulate a particular kind of quantum system. A direct mapping is established between controllable interactions in the simulator and those of the system to be simulated - much like simple electric circuits can be mathematically analogous to simple mechanical systems. An example with trapped ions is our early work on simulating the physics of relativistic quantum particles [3]. Following this approach, which is not intrinsically universal, different realisations of a quantum simulator will allow different systems to be simulated. Different approaches for simulating the same systems may also allow a cross-checking of the results.

### Controllable laboratory experiments with trapped ions

Over the last decade, scientists have begun to achieve the required level of control over laboratory-based quantum systems to develop these ideas. Many candidate physical systems are currently being developed including quantum states of light, superconductors, atoms, ions, molecules and solid-state systems. One of the most advanced approaches to general quantum information processing uses strings of cold ions held in radio-frequency electric traps [4]. In these experiments information is typically encoded in long-lived internal states of the ions. In many cases, a universal set of quantum logic gate operations have been achieved leading to the generation of multi-ion entanglement, as well as the implementation of simple quantum algorithms. Trapped ions are recognised as offering an extremely promising approach to large-scale quantum information processing, with unrivalled precision control over quantum degrees of freedom.

Very recently these tools and techniques have been extended and applied to perform the first quantum simulations using trapped-ions. In 2008 it was shown how two magnesium ions can be manipulated to perform an analog simulation of an archetypal condensed-matter physics model of two interacting spin-1/2 particles [5]. This line of work was extended in 2010 by the group of Chris Monroe, who performed analog simulations of a similar lattice model using Yb ions to explore phase changes and entanglement properties in systems with up to nine spins [6]. At the same time our group in Innsbruck performed a series of analog simulations of relativistic quantum particles, employing up to two ions and one harmonic oscillator mode for the simulated wavefunction.

### Digital quantum simulations with trapped ions

In our current work [7], carried out in collaboration with P. Zoller's research group at IQOQI, we demonstrate how a system of precisely controlled trapped ions can be programmed to simulate principle any other local quantum system, using the techniques of universal quantum simulation. In particular, we accurately simulate dynamics due to a range of quantum interactions that are not naturally present in our system. Here an important ingredient is the use of the Trotter decomposition for simulating the action of Hamiltonians - consisting of a sum of noncommuting terms - that we cannot synthesize directly. For example, in order to simulate the action of a Hamiltonian that is the sum of two terms, H=H1+H2, each of which can be realized individually, we apply H1 and H2 many times consecutively for very short periods of time. In this way, we can simulate for example a Ising interaction in the presence of a transverse field as has been done before using analog simulation techniques. But we can also use this technique to build construct a Heisenberg Hamiltonian out of x-x, y-y, and z-z interactions. In this way, we explore a wide range of interactions in strings of two to six ions. The paper also addresses the question of how to characterize the fidelity of the interactions realized in the experiment. Here, a difficulty that we encounter is the characterization of multi-qubit operations that quickly becomes impractible with growing number of qubits if standard quantum process tomography is used. For this reason, we carried out a complete characterization only for two-qubit experiments. For multi-qubit operations, we showed that we could bound the process fidelity from above and below using a restricted set of measurements.

The advantage of the digital approach over analog approaches lies in the flexibility of simulating a large number of different quantum interactions using a small set of basic gate operations. Moreover, it might be easier to bound errors in a digital quantum simulation than in its analog counterpart. In combination with current technological developments that will allow larger numbers of ions to be manipulated, our results demonstrate the imminent feasibility of quantum simulations beyond the capabilities of conventional computers.

### Literature

[1] Universal quantum simulators. S. Lloyd, Science 273, 1073 (1996)

[2] Quantum simulators, I. Buluta and F. Nori, Science 326, 108 (2009)

[3] Quantum simulation of the Dirac equation. R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt and C.F. Roos, Nature 463, 68 (2010)

[4] Quantum computing with trapped ions. H. Häffner, C. F. Roos, R. Blatt, Phys. Rep. 469, 155 (2008)

[5] Simulating a quantum magnet with trapped ions. A. Friedenauer, H. Schmitz, J. Glueckert, D. Porras, T. Schätz, Nat. Phys. 4, 757 (2008)

[6] Onset of a Quantum Phase Transition with a Trapped Ion Quantum Simulator, R. Islam et al., Nat. Comm. 2, 377 (2011)

[7] Universal digital quantum simulations with trapped ions. B. Lanyon, C. Hempel, D. Nigg, M. Müller, R. Gerritsma, F. Zähringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, C. F. Roos, Science **334**, 57 (2011)

### Financial support

We are financially supported by Österreichische Akademie der Wissenschaften, Universität Innsbruck, Fonds zur Förderung der wissenschaftlichen Forschung (FWF) within the program "Foundations and Applications of Quantum Science", the European Union as well as IQI and IARPA.