Sponsor: ERC EMC2
Supervisor: Y. Maday
See pdf for more details.
Context
It is widely agreed that the accurate simulation of quantum many-body physics is one of the most exciting applications of sophisticated quantum algorithms such as quantum phase estimation (QPE) [3]. Unfortunately, despite promising experimental advances in recent years, present quantum devices continue to suffer from the issue of decoherence, which prevents the practical execution of such algorithms.
Hybrid quantum algorithms such as the variational quantum eigensolver (VQE) [4] in which a quantum subroutine is incorporated within a largely classical numerical method were originally developed as an alternative that would, perhaps, be better suited for implementation on near-term quantum devices. Despite initial excitement, recent research indicates that the VQE suffers from at least two major issues that prevent its execution on near-term quantum devices for quantum systems of chemical and physical interest. These issues are (i) the huge number of measurements that must be performed on the quantum device to obtain statistical averages of the variational energy [8] and (ii) the difficulty of the associated optimisation problem which involves a cost function that may become exponentially flat with increasing system size [1].
As a consequence of this situation, two broad research directions have developed. The first advocates for a return to quantum algorithms such as QPE and focuses on the optimisation of the individual steps within such algorithms to reduce the quantum resources required for their execution while waiting for the development of fault-tolerant, error-corrected quantum hardware. The second direction focuses instead on the continued development of alternative hybrid quantum algorithms suited to near-term quantum devices with the hope that some form of quantum advantage can still be leveraged from these algorithms in the short term. Note that advocates of this later approach have an additional justification: a necessary pre-requisite for algorithms such as QPE is the availability - on the quantum computer - of a trial state that is a reasonable approximation of the sought-after ground state of the quantum system under study [3]. Therefore, even after the advent of error-corrected quantum computers, hybrid algorithms could find use as state-preparation algorithms required to successfully run a QPE routine.
In the present project, we choose to take the second route mentioned above by attempting to answer some basic mathematical questions regarding the so-called projective quantum eigensolver proposed by Stair and Evangelista [6].
References
[1] Marco Vinicio Sebastian Cerezo de la Roca, Michael Ragone, Bojko Bakalov, Martin Larocca, Frederic Antoine Sauvage, Alexander F Kemper, and Carlos Ortiz Marrero. A unified theory of barren plateaus for deep parametrized quantum circuits. Nature Communications, 15(LA-UR-23-30483), 2024.
[3] Yu Kitaev. Quantum measurements and the abelian stabilizer problem. arXiv preprint quant-ph/9511026, 1995.
[4] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Al´an Aspuru-Guzik, and Jeremy L O’brien. A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5(1):4213, 2014.
[6] Nicholas H Stair and Francesco A Evangelista. Simulating many-body systems with a projective quantum eigensolver. PRX Quantum, 2(3):030301, 2021.
[8] Dave Wecker, Matthew B Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A, 92(4):042303, 2015.