Simulating fermionic particles with superconducting quantum hardware

Digital quantum simulation is one of the key applications of a future, viable quantum computer. Researchers around the world hope that quantum computing will not only be able to process certain calculations faster than any classical computer, but also help simulate nature more accurately and answer longstanding questions with regard to high temperature superconductivity, complex quantum materials, and applications in quantum chemistry.

A crucial part in describing nature is simulating electrons. Without electrons, you cannot describe metals and their conductivity, or the interatomic bonds which hold molecules together. But simulating systems with many electrons makes for a very tough problem on classical computers, due to some of their peculiar quantum properties.

Electrons are fermionic particles, and as such obey the well-known Pauli exclusion principle which states that no fermions in a system can occupy the same quantum state. This is due to a property called anticommutation, an inherent quantum mechanical behavior of all fermions, that makes it very tricky to fully simulate anything that is composed of complex interactions between electrons. The upshot of this anticommutative property is that if you have identical electrons, one at position A and another at position B, and you swap them, you end up with a different quantum state. If your simulation has many electrons you need to carefully keep track of these changes, while ensuring all the interactions between electrons can be completely, yet separately tunable.

Add to that the memory errors caused by fluctuation or noise from their environment and the fact that quantum physics prevents one from directly monitoring the superconducting quantum bits (“qubits”) of a quantum computer directly to account for those errors, and you've got your hands full. However, earlier this year we reported on some exciting steps towards Quantum Error Correction - as it turns out, the hardware we built isn't only useable for error correction, but can also be used for quantum simulation.

In Digital quantum simulation of fermionic models with a superconducting circuit, published in Nature Communications, we present digital methods that enable the simulation of the complex interactions between fermionic particles, by using single-qubit and two-qubit quantum logic gates as building blocks. And with the recent advances in hardware and control we can now implement them.

We took our qubits and made them act like interacting fermions. We experimentally verified that the simulated particles anticommute, and implemented static and time-varying models. With over 300 logic gates, it is the largest digital quantum simulation to date, and the first implementation in a solid-state device.

Left: Model picture with four fermionic modes in two sites. The modes are occupied or unoccupied. For example, we can start with two fermionic particles in the right well, by occupying the blue and green mode. If the particles repel each other, there's a good chance that one of the them will hop to the left well through the process of quantum tunneling through the barrier. It will then occupy the red or purple mode. This interplay of on-site interaction and hopping lies at the core of describing processes in physics and chemistry, ranging from the conductivity of metals to the binding between atoms in molecules. Right: The false-colored cross-shaped structures are the superconducting quantum bits. The colors correspond to the modes, so if we have two fermionic particles in the blue and red modes, the rightmost two quantum bits are excited.

Coming up with an efficient sequence of logic gates that can accurately model the interactions for systems of fermions wasn’t easy. So we teamed up with Dr. Lucas Lamata, M.Sc. Laura García-Álvarez, and Prof. Enrique Solano from the QUTIS group at the University of the Basque Country (UPV/EHU) in Bilbao, Spain, who are experts in constructing algorithms and translating them into the streams of logic gates we can implement with our hardware.

For the future, digital quantum simulation holds the promise that it can be run on an error-corrected quantum computer. But before that, we foresee the construction of larger testbeds for simulation with improvements in logic gates and architecture. This experiment is a critical step on the path to creating a quantum simulator capable of modeling fermions as well as bosons (particles which can be interchanged, as opposed to fermions), opening up exciting possibilities for simulating physical and chemical processes in nature.