A qubit regularisation of asymptotic freedom without fine tuning

In quantum electrodynamics, low-energy processes can be calculated in a perturbative way. In quantum chromodynamics (QCD), perturbation theory is applicable at high energies but fails for low-energy processes. A way to overcome this problem is to discretise space-time and study lattice QCD with the help of numerical simulations on large computers. QCD has many interesting properties, such as asymptotic freedom, which can be studied by means of low-dimensional theories giving rise to proofs of concept for new theoretical and computational tools. In recent research published in Physical Review Letters, Professor Marina Marinkovic and co-authors from Duke University in the US and the Saha Institute in India looked into qubit regularisation of asymptotically free quantum field theories (QFT). We spoke with Marinkovic about this ongoing work.

In your paper, you followed a new approach for simulating lattice theories as compared to conventional methods in lattice QCD. How does your strategy differ from conventional approaches?

We're interested in continuum quantum field theories (QFTs) that are known to exhibit infinities at high energies and thus require renormalisation. Discretisation of space-time is one way to regulate this infinity; this leaves behind a residual local infinity (associated with the Fock vacuum). We go a step further to replace the original degrees of freedom by qubits and totally remove all infinities. To our knowledge, this is the first time that the Berezinskii-Kosterlitz-Thouless (BKT) phase transition was proven by using this type of qubit discretisation without fine tuning.

Asymptotic freedom is a fundamental property of the theory that arises at a BKT transition and of several other QFTs, one of them being QCD. In my group, we study the strong interaction by putting QCD on a four-dimensional space-time lattice to understand asymptotic freedom in this particular theory. As a proof of concept, we started to investigate lower-dimensional theories and toy models to develop new algorithms. In our recent paper, we considered a particular model known as the XY-model in two dimensions: it describes two-component classical spins located at all vertices of a space-time lattice. Qubit regularisation replaces the classical spins with qubits so that one can write the Hamiltonian of the theory as an operator on a Hilbert space that is finite-dimensional as long as the space-time lattice is finite. This is not the case for conventional lattice regularisation of QFTs, where one would deal with an infinite-dimensional Hilbert space. Ultimately, the motivation for making Hilbert spaces finite-dimensional is the ability to simulate QFTs on a quantum computer. Therefore, the crucial difference between conventional approaches involving infinite-dimensional Hilbert spaces and qubit discretisation is that the latter guarantees such finite-dimensional Hilbert spaces.

How does the XY-model, which comes from condensed-matter physics, help towards the goal of performing calculations of processes in particle physics?

In condensed-matter physics, statistical models such as the XY-model are used. In particle physics such models are useful, too. We try to understand the strong interactions with the help of lattice QCD. This lattice approach relies on tools from statistical physics. We use statistical-physics methods to first study simpler systems and then extend our insights and methods to four-dimensional theories such as QCD. To be able to apply methods from statistical mechanics, we must replace Minkowski- by Euclidian space-time. For this purpose we render time imaginary, while space remains the same. This enables us to consider theories on a four-dimensional space-time lattice where there is no distinction between space and time, making QFT look more like systems in statistical physics. So yes, we want to understand particle physics but from the statistical physics perspective. This is the only way to study non-perturbative physics with the help of extensive numerical simulations. In our paper we do just that, but for a simpler model.

Would you like to apply this method to explain phenomenological findings in particle physics in the future?

Ultimately yes, but there's still a long way to go. The main goal of the work we recently published is to understand better the concept of universality in QFTs. Concretely, if one starts from different discretisation procedures of the same QFT one eventually ends up reproducing the same underlying physics in the limit where the discretisation is removed. It's well-known that, starting from a conventional lattice regularisation, it's possible to recover continuum physics. What is new in our research is that we showed how the qubit regularisation we used also reproduces the same underlying physics without fine tuning in Euclidean space-time.

Can you explain why it's important to obtain the physics of the underlying QFT without fine tuning?

In the conventional lattice discretisation of this QFT we would have a parameter that describes the interactions of the system and we would have to tune it to a particular value to reach the BKT transition; this is what is called fine tuning. In our approach we don't have to do this: the properties of the BKT transition emerge for a whole range of parameter values, and we believe this is the first time that such a result was obtained in Euclidian space-time. As the system approaches the BKT phase transition, we could reproduce the interactions of this model at all energy scales.

What are other especially notable aspects of your findings?

Another aspect we find important is that our method leads to reduced finite-size effects. These effects are a consequence of performing the calculations on a finite lattice. As we're interested in infinite space-time, we must eventually take the lattice size to infinity. In conventional approaches, simulations of very large lattices of approximately 10⁸ points are required to reduce finite-size effects. In our case, lattices of only a few thousand points suffice to control finite-size errors. In a sense, our approach allowed us to find a faster convergence to the continuum limit compared with conventional methods. We don't yet completely understand why this is so, but it's a relevant aspect.