A universal quantum computer promises to tackle a wide range of problems such as materials design and molecular modelling, with the ultimate goal of addressing general classes of hard problems. A quantum simulator is a restricted type of quantum computer that uses qubits to study a specific many-body system. One of the main challenges in the development of such devices is scalability, namely the ability to increase the number of qubits while exerting individual control on each of them. In this work we performed the largest spin model quantum simulation to date, using 53 qubits.
Our trapped-ion quantum simulator consists of individual ytterbium ions—charged atoms trapped in place by gold-coated electrodes—which are used to study quantum magnetism in out-of-equilibrium systems. In particular, we studied a dynamical phase transition that occurs after a sudden change of the system parameters, a.k.a. a quantum quench. The system is described by the following Hamiltonian:
where σxi is the Pauli matrix acting on the ith spin along the x direction, Jij the Ising coupling between spins i and j, and Bz the transverse magnetic field. The spin–spin interaction is long range and falls off approximately as a power law Jij~J0/|i–j|α . We studied the response of the system as a function of the ratio of the two competing energy scales in the Hamiltonian, namely J0 and B. The experiment we had in mind was very simple: prepare the spins along the x-axis, quench the Hamiltonian and then measure the magnetization of the spins along the x-basis over long times. The question we wanted to answer was: is there a dynamical phase transition, namely a non-analytic change in the properties of the system, as we vary the ratio B/ J0?
Ideally, to answer this question and observe a non-analytic response of the system, we should have taken the thermodynamic limit both numerically and experimentally. Numerically this is possible for those few cases where the system can be solved analytically, but experimentally it was definitely out of question to put an infinite number of ions in the trap!
We decided more modestly to perform finite-size scaling, namely to measure how the properties of the system changed as the number of particles increased and try to observe non-analytic behaviors smoothed in a crossover by finite-size effects.
Therefore, we tried to perform the experiment looking at the long-time average magnetization of the systems, but our system sizes were not large enough to see any significant signature of the phase transition. At some point, we had the idea to look for the second-order correlations at long times and there we found something very interesting in the data: at what we thought to be the critical point, we observed a dip in the correlations! We checked in the numerics and had the confirmation that the dip—which is a signature of a dynamical phase transition—was physical. We numerically checked that the correlation dip went to zero in the thermodynamic limit of a toy model with all-to-all interaction (α=0 , which is analytically solvable), and it did. We had finally the first evidence of the phase transition! Since the finite scaling of this signal was not really satisfactory for systems with up to 16 ions, we tried to increase the signal-to-noise ratio as much as possible by going to larger and larger system sizes, and eventually we managed to take data with 53 ions.
This experiment offered a very concrete perspective on doing experiments with a very large number of qubits, putting us on the cusp of exploring physics that is unreachable by even with the fastest modern supercomputers.