Simulating Particle Physics with Quantum Computers

Preface

The references included vary in difficulty. Those followed by * can be mostly understood by undergraduates with some background in physics, while those with ** probably require a part III course on quantum field theory and/or quantum computing! Those with no stars should be accessible to anyone. I am grateful to Johannes Bausch for useful comments.

Introduction

The study of fundamental particles and how they interact, for example, electrons interacting via the electromagnetic force, is fascinating but challenging. The mathematics involved in particle physics has its share of difficulties and the experiments typically involve huge technological challenges. On the other hand, the theories that we currently have to describe particle physics are incredibly successful, and the predictions they make have been verified to an incredible degree of accuracy. (For a nice discussion of this, see reference 1.) Experiments conducted in particle physics, like those at the large hadron collider at CERN, are immensely complicated from an engineering point of view, and the equipment used isn’t cheap to build!  But theory without experiments does not allow progress: we need to find out if our theories are right, and, when we don’t even have a theory, we need experiments to obtain data that can help enhance our understanding of what is going on in our experimental systems.

To help increase our understanding of particle physics we can perform simulations on computers. The particle physics simulations typically conducted (called lattice quantum field theory simulations) are computationally difficult; part of the reason for this is that the simulations are carried out on conventional or classical computers. It is strongly believed that quantum computers should be better for simulating nature. After all, nature is quantum, not classical. We’ll get to what exactly a quantum computer is soon, but first let’s take a further look at particle physics.

Particle Physics

Particle physics involves fundamental particles, like electrons or photons, and their interactions. The most familiar type of experiment in particle physics involves smashing fast moving particles into each other and seeing what types of particles are created. The faster the particles are moving, the higher the energy in the collision, and, as you increase the energy, you can create heavier particles (remember , so the bigger the energy is, the bigger the mass you can have). For this reason, albeit not the only reason, some particles took longer to find than others. Two particles, the top and bottom quark, were predicted to exist in 1973.  Four years later the bottom quark was found, but it wasn’t until 1995 that the top quark was detected.  This isn’t surprising given that the top quark is around 40 times heavier, so a collision with a much higher energy was needed to find it.

Figure 1: A simulation of a collision where the famous Higgs particle is produced and decays into smaller particles.  Image taken from Wikimedia Commons.

The maths used to describe all of this is called quantum field theory, which involves combining Einstein’s special relativity ( and all that) and quantum mechanics (This is where Schrödinger’s cat comes in: things can be in multiple states at once, called superpositions. For example, the cat is neither alive nor dead, but both until you open the box.). A consequence of the combination of quantum mechanics and special relativity is that particle numbers aren’t fixed; an electron and positron (the electron’s anti-particle) can be created or annihilate each other.

Quantum field theory is really useful for describing this kind of situation. We say that there are fields that exist everywhere in space and these fields have different values at different points.  A simple example is temperature: it has different values in different places.  Another example is the magnetic field, which, very roughly, has larger values when there is a strong magnetic field, which happens close to a magnet, for example.  The fields we are interested in describe the number of particles at different places.

Quantum Simulators

To simulate fields, and hence particle physics, the approximation is made that space (and often time) is discrete, so finite volumes have finite numbers of points, like a lattice. Then the information describing the state of the field at each point is stored on the computer. Classical computers store information as bits, which can be either zero or one. Quantum computers, on the other hand, store information on qubits, which can be either zero, one or both (like Schrödinger’s cat). (A good book on quantum computing is listed as reference 2*.)  Fields in particle physics are quantum systems, and it stands to reason that quantum systems would be easier to represent by qubits rather than classical bits, since qubits are quantum systems themselves and storing quantum states on classical bits is inefficient (quantum systems (qubits, for example) can be in many more states than classical ones, so you need more classical bits to store the state of a qubit). This is one of the reasons quantum computers should be better for simulating nature.

There are two kinds of quantum simulations that we can perform: analog and discrete. Analog means setting up a quantum system in a lab that evolves over time in a way that looks similar to the physical system we want to study. Discrete simulations evolve over discrete timesteps in a way that approximates our physical system. Each approach has its drawbacks. Analog simulators are not really computers: for example there is no error correction but they are more feasible to make in a lab. Discrete simulators run on fully fledged quantum computers, but are much further from being built in a lab.

Let us first discuss analog simulators. There is a really nice proposal for an experiment that would simulate a simple model of interacting electrons and positrons³**. The setup for the experiment involves cold atoms trapped on an optical lattice (see figure 3). An optical lattice is made by crossed laser beams which trap the atoms. By tuning the laser beams, you can change the height of the energy barrier between sites. This changes the likelihood that atoms will hop from one site to the next; a higher barrier means hopping is less likely. The state of the field is represented by the atoms: if there is an atom at some site there is an excitation in the field at that site. Through clever engineering, the atoms mimic interacting electrons and positrons. Although the model is fairly simple, it still exhibits interesting phenomena that occur in the standard model (that’s the best theory we have for how fundamental particles interact, although it doesn’t include gravity). There are also some interesting proposals for analog simulators^4,5** using trapped ions.

Figure 2: An optical lattice. Atoms are trapped by the laser beams, but can still hop between sites.  Image taken from Wikipedia.

Discrete quantum simulators essentially run on quantum computers. The idea that quantum computers could be useful for simulating physics goes back at least as far as Feynman (1982) (see reference 6 and reference 7*). In fact, simulating physics was one of the original reasons for the development of quantum computers in the first place.

Figure 3: Apparatus for ion trapping in a lab in Innsbruck. Ion traps are one possibility for constructing a quantum computer.  Image taken from Wikimedia Commons.

Some models in particle physics are known to be easy to simulate on a quantum computer. For example, Jordan et al ⁸** look at a model involving a single type of particle that has interactions (it’s called theory). Although conceptually simple (well, as conceptually simple as quantum field theories get), demonstrating that this model could be efficiently simulated on a quantum computer was a complicated task. This suggests that it may be possible for the whole standard model of particle physics to be simulated using a quantum system. This presents a fascinating question: can we build some kind of computer that can efficiently simulate nature?

What does the Future hold?

A further motivating factor for building quantum computers is that they can be used to solve computational problems believed to be hard to solve with classical computers. For example, it is known that you can efficiently factor large numbers on a quantum computer, something that is famous for being hard to do with a classical computer ²*. However, building a quantum computer is a big challenge because of the many technical issues that need to be dealt with. For example, you need to be able to isolate the quantum computer from the environment, otherwise the qubits will interact with it and their states will change.
 
There are many experimental proposals for making quantum computers ⁹*, each of which has its advantages and disadvantages. Luckily, however, for discrete simulations a much smaller quantum computer can perform difficult simulations of physics than you would need to solve other difficult computational problems ⁷*. Furthermore, analog simulators do not require the same level of control as discrete simulators and are already feasible. So, even if very large quantum computers useful for solving other problems (like factoring) never become a reality, simulating physics using smaller quantum computers (computers with less memory and fewer logical operations resulting in a physically smaller machine that should be easier to build) should still be possible.

References

1. R. Feynman.  QED: The strange theory of light and matter.  Princeton University Press, 1985.
2. M. A. Nielsen and I. L. Chuang.  Quantum Computation and Quantum Information.  Cambridge University Press, Cambridge, 2000.
3. J. I. Cirac, P. Maraner, and J. K. Pachos.  Cold atom simulation of interacting relativistic quantum field
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4. J. Casanova, L. Lamata, I. L. Egusquiza, R. Gerritsma, C. F. Roos, J. J. García-Ripoll, and E. Solano.  Quantum simulation of quantum field theories in trapped ions.  Phys. Rev. Lett., 107:260501, 2011.
5. J. Casanova, A. Mezzacapo, L. Lamata, and E. Solano.  Quantum simulation of interacting fermion lattice models in trapped ions.  Phys. Rev. Lett., 108:190502, 2012.
6. R. Feynman.  Simulating physics with computers.  International Journal of Theoretical Physics, 21:467–488,
   1982.

li>S. Lloyd.  Universal quantum simulators.  Science, 273(5278):1073–1078, 1996.

7. S. P. Jordan, K. S. M. Lee, and J. Preskill.  Quantum algorithms for quantum field theories.  Science, 336(6085):1130–1133, 2012.
8. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien.  Quantum computers.  Nature, 464:45, 2010.