For decades, materials scientists have attempted to model the complex dance of electrons in strongly interacting systems. However, uncovering the secrets behind phenomena like unconventional superconductivity and exotic magnetism remains a significant challenge, even with today's most sophisticated experimental techniques and classical computational models.
In collaboration with researchers at NNF Quantum Computing Programme (NQCP) at the University of Copenhagen, our new research paper, Fault-tolerant quantum simulation of generalized Hubbard models, introduces a technique called “Tile Trotterization”. This method allows efficient quantum simulation of various complex states of matter, including superconductors and spin liquids. By gaining a deeper understanding of these systems at the quantum level, researchers may eventually be able to utilize this knowledge to design materials with valuable properties, such as new superconducting materials that operate at higher temperatures.
Building on Earlier Breakthroughs: Quantum Simulation in Early Fault-Tolerance
The motivation for our study was a paper by Prof. Earl Campbell, which introduced a method called “Plaquette Trotterization”. Trotterization is a quantum simulation method that we expect to be particularly important for early error-corrected devices. The circuits to implement Trotterization are simple and can be efficiently parallelised, and the number of qubits required is lower than that of many alternative quantum simulation methods.
Campbell demonstrated that classically non-trivial models of materials can be studied using approximately one million Toffoli gates, making this combination of Trotterization and model systems suitable for early fault tolerance. However, Plaquette Trotterization was only developed for a specific system, limiting its broader applicability.
Tile Trotterization
Our paper extends this work by introducing Tile Trotterization, which generalises the methodology of Plaquette Trotterization to a much larger range of lattices and model systems. In doing so, we extend the potential applicability of early error-corrected quantum computers to a larger range of materials problems.
We perform resource estimation for Tile Trotterization as a subroutine in quantum phase estimation – a method used to find the energies of a quantum system. However, Trotterization can be used in a range of problems of interest in early fault tolerance, including quantum dynamics simulations.
In addition to generalising the scheme from Campbell's paper 'Early fault-tolerant simulations of the Hubbard model', we also introduce a combination of analytic and numerical approaches to accurately bound Trotter error. While Trotterization is a powerful method, estimating the error in this method is a challenging task, and, as a result, the cost of performing Trotterization is often overestimated. We hope that the techniques in our paper will help to improve this situation.
This research is an important step towards unlocking the potential of MegaQuOp-scale quantum computers – those which can perform around one million error-free operations. Both the simulation technique and numerical tools introduced can be applied to a class of systems that is believed to support exotic phenomena, including high-temperature superconductivity, spin liquids, and non-conventional forms of magnetism. We therefore hope that MegaQuOp-scale machines could be used in the near future to accelerate the discovery of materials with exotic states of matter.
The Road Ahead: Paving the Way to MegaQuOp
There are a number of natural next steps for tile Trotterization. While our paper demonstrates that valuable model systems can be studied with around one million non-Clifford gates, this does not yet imply MegaQuOp-scale applications just yet. This is because the total quantum operations count must include not just non-Clifford gates, but also Clifford gate and idling costs.
Therefore, an important next step will be to take algorithms such as tile Trotterization and perform compilation for the architectures that will be used by the first generation of error-corrected quantum computers. This would provide a better understanding of the requirements of these early devices to perform useful quantum computation. This includes optimising implementations of logical operations for specific quantum architectures and understanding the interplay between algorithm design and hardware limitations. It will also require further developing algorithms optimised for the MegaQuOp regime. Performing this research will give a better understanding of what tasks MegaQuOp-scale devices will be able to perform, as well as informing how to best build such an architecture.
At Riverlane, we are building the tools and technology to help the quantum community scale to the MegaQuOp. Research on applications for early fault tolerance not only accelerates our understanding of what will be possible at that milestone, but it also informs how we design and refine our own technology.