The Quantum Theory of Periodic Systems on Modern Computers

Karl E. Friedl

R. Blümel, William P. Reinhardt

Einstein (1917) appreciated early on that within the “old” pre-1925/26 quantum mechanics absence of integrability is a serious obstacle for the quantization of classical systems. Therefore, in retrospect not surprisingly, the quantization problem was not adequately solved until the advent of the “new” quantum mechanics by Heisenberg, Born, Jordan and Schrödinger. The new quantum mechanics did not rely at all on the notion of classical paths, and this way, unwittingly, sidestepped the chaos problem. Within the framework of the new theory, any classical system can be quantized, including classically chaotic systems. But while the quantization of integrable systems is straightforward, the quantization of classically chaotic systems, even today, presents a formidable technical challenge. This is especially true for quantization in the semiclassical regime, where the quantum numbers involved are large. In fact, efficient semiclassical quantization rules for chaotic systems were not known until Gutzwiller (1971, 1990) intoduced periodic orbit expansions. Gutzwiller's method is discussed in Section 4.1.3 below. It is important to emphasize here that the existence of chaos in certain classical systems in no way introduces conceptual problems into the framework of modern quantum theory, although, let it be emphasized again, chaos came back with a vengeance from the “old” days of quantum mechanics. Even given all the modern day computer power accessible to the “practitioner” of quantum mechanics, chaos is the ultimate reason for the slow progress in the numerical computation of even moderately excited states in such important, but chaotic problems as the helium atom.

Dr. Mohit Kumar, Mr. Satish D. Kale, Prof. Prashant Govindrao Ghulaxe

Fenton Ross Clawson

Computational physics has been a backbone of experimental and theoretical physics since the 1940’s, when the first nuclear bomb and ballistic simulations were per- formed at Los Alamos Laboratory. Nowadays, however, simulation of open quan- tum systems has been formalized into coding packages and user-friendly functions, such as QuTiP or Qiskit, which allow users to simulate these systems without hav- ing the in-depth knowledge required to build the computational backbone from the ground-up. With the shift in modern quantum physics towards the realization of quantum computation, a strong computational background is still necessary to lend validity to theoretical results and to help explain experimental results. Here, a new solver is considered which fits well into the regime of many modern quantum com- putational experiments, i.e. one which works well for systems which have Hamil- tonians that are repeating in time. This solver, named FLiMESolve (for Floquet- Lindblad Master Equation Solver), is derived from a Floquet interpretation of the Lindblad master equation, is well suited for these time-periodic Hamiltonians, and will be integrated into QuTiP. This solver maintains the benefits of other solvers used in QuTiP, while superseding their usefulness within its own regime. FLiMESolve is able to efficiently and accurately simulate time-periodic systems while being faster and more accurate than other solvers and with the ability to relax the secular ap- proximation inherent to the Lindblad Equation.

Bart D’Hooghe, Jarosław Pykacz

Nobuyuki Yoshioka, Takeshi Sato, Yuya O. Nakagawa +2 more

We present a quantum-classical hybrid algorithm that simulates electronic structures of periodic systems such as ground states and quasiparticle band structures. By extending the unitary coupled cluster (UCC) theory to describe crystals in arbitrary dimensions, for a hydrogen chain, we numerically demonstrate that the UCC ansatz implemented on a quantum circuit can be successfully optimized with a small deviation from the exact diagonalization over the entire range of the potential energy curves. Furthermore, by using the quantum subspace expansion method, in which we truncate the Hilbert space within the linear response regime from the ground state, the quasiparticle band structure is computed as charged excited states. Our work establishes a powerful interface between the rapidly developing quantum technology and modern material science.