My work from 2020-present is focused developing methods for efficient Hamiltonian Simulation on quantum computers. Beginning during a SULI internship at Oak Ridge National Labs and under the supervision of Dr. Eugene Dumitrescu, I developed the framework for applying Cartan decomposition to Hamiltonian Simulation. As originally formulated in the literature, Cartan decomposition is formulated as a means of producing an efficient gate decomposition of an arbitrary unitary operator on n-qubits. However, this problem is generally has a two-qubit gate cost scaling as \(\mathcal{O}(4^n)\), which means it is not generally useful for gates larger than 4-5 qubits. A key contribution of our work in this area is the reduction of this scaling cost for certain types of Hamiltonian for fermionic and spin systems. Specifically, for the XY and Ising type 1D models, we use Cartan decomposition to efficiently prepare a fast-forwarding circuit that is numerically exact for any input time t. In these cases, compared to Trotter based methods, Cartan decomposition allows for simulations beyond the coherence times of the quantum computer and uses far fewer gates for long-time simulations. The paper is currently published on arXiv, and is in review at Physical Review Letters. This page will be updated if the paper is accepted for publication.
Efekan Kökcü, Thomas Steckmann, JK Freericks, Eugene F Dumitrescu, and Alexander F Kemper, “Fixed depth hamiltonian simulation via cartan decomposition,” (2021) arXiv preprint arXiv:2104.00728
In order to make our work on Cartan Decompositon more accessible to other areas of application, I developed and published the Cartan Quantum Synthesizer python package. The package performs 3 primary steps:
My current work aims to apply Cartan decomposition to a dynamical mean-field calculation for the Hubbard model. The primary goal is to explore methods of minimizing circuit cost for a calculation on the two-site DMFT model, which approximates the dynamics of a Hubbard model of interacting electrons on an infinite dimensional lattice. Using Cartan decomposition, we are able to reduce the gate cost of the simulation to below the usable threshold for current superconducting quantum hardware from IBM. The DMFT calculation maps the Hubbard model the Anderson Impurity model, in this cases consisting of a single impurity site coupled to a single bath site. Using a quantum computer, we compute Green's function for the Anderson model with an initial guess of the mapped parameters: an initial guess gives a better guess for the next computation, which is again run on a quantum computer. This is run until the updated parameters converge to the initial guess. Cartan Decomposition is used to time evolve the system to compute Green's function at different times. Two things of note in the calculation is that Cartan decomposition detaches the circuit error from the simulation time, because it is a circuit to fast-forward the dynamics of the calculation. This means that even through there is significant error within the simulation, the in most cases the error does not disguise the frequencies present in the Green's function calculation. As these frequencies are all that is needed to update the DMFT guess, we are able to iterate the loop even though the quantum system has significantly decohered. Second, this method is used to demonstrate the application of Cartan decomposition and to stretch the capabilities of current hardware. The circuit form used in this work scales exponentially in cost with the number of qubits, meaning it is not generally useful for simulating large systems. Additional work is needed to understand if Cartan decomposition can be truncated efficiently in these cases.
Thomas Steckmann , Trevor Keen, Alexander F. Kemper, Eugene F. Dumitrescu, and Yan Wang, "Simulating the Mott transition on a noisy digital quantum computer via Cartan-based fast-forwarding circuits" https://arxiv.org/abs/2112.05688