Abstract

Background and Motivation When Feynman first proposed the notion of quantum computers in 1982, his primary motivation was efficient computation of quantum physics and quantum chemistry. On a classical computer we encounter exponential slowdown when attempting to simulate multi-particle systems due to the large Hilbert space such quantum systems reside in. A universal quantum computer should, in principle, be able to overcome this curse of dimensionality by utilising entanglement.

However, there are two major reasons why advances in classical algorithms for solving equations of quantum mechanics are desperately needed.

Firstly and ironically, the process of creating quantum computers will itself require routinely solving equations of quantum mechanics such as the time-dependent Schrodinger equation for predicting the dynamics of qubits and designing controls and gates using lasers, magnetic fields etc. These will need to be solved on the only devices available to us—the classical computers.

Secondly, a major stumbling block will still remain even once quantum computers become available: Trotter splitting based quantum computing algorithms have already been devised and, unsurprisingly, they need a very small time step for a reasonably accurate simulation. The Trotter splitting is the lowest order method among exponential splittings which allow us to propagate the actions of different components of a Hamiltonian separately. We need higher-order exponential splittings for overcoming the small time step barrier.

Higher-Order Methods and Algebraic Structures

The main focus of my talk will be on some higher-order exponential splitting algorithms for efficiently solving the time-dependent Schrödinger equation. These algorithms are entirely in the classical language but it should be possible to utilise them to devise corresponding quantum algorithms.

In particular, I will talk about the Zassenhaus splittings whose costs grow quadratically in contrast to the exponential growth in cost of competing Yosida splittings. This exponential speedup, along with unitarity, stability, convergence and error bounds of these splittings is traced to the structural properties of a Z2 graded Lie algebra.

We will see how these Lie algebras arise in the context of any associative algebra with a commutative subalgebra and its non-trivial Lie idealiser. The generality of this framework suggests that the results should extend to many other equations of quantum mechanics.