Quantum Computing’s Dimensionality Curse: Distributed Computing and Cavity QED Model Offer Solutions

The curse of dimensionality, a problem in quantum computing, particularly affects computational fields like polymer chemistry and macromolecular biology. It arises when simulating chemical and biological reactions involving a large number of particles, causing the dimension of the quantum system to increase exponentially. This issue, first described by R.E. Bellman, is a major obstacle in studying high-dimensional quantum systems. However, the use of distributed computing, aided by Cannon’s algorithm, can help solve this problem. This paper also introduces the cavity quantum electrodynamics (QED) model and the Tavis-Cummings model, providing a new paradigm for studying light-matter interaction.

What is the Curse of Dimensionality in Quantum Computing?

The curse of dimensionality is a problem that arises in quantum computing, particularly in the field of computational mathematics. This issue is especially prevalent in computational fields involving polymer chemistry and macromolecular biology. When simulating chemical and biological reactions, a large number of particles are often involved. The dimension of the quantum system composed of these particles increases exponentially as the number of particles increases, thus causing the curse of dimensionality. This problem was first described by R.E. Bellman to describe a number of events that occur while organizing and evaluating data in high-dimensional areas.

The curse of dimensionality has always been a major obstacle to the study of high-dimensional quantum systems. However, the study of the structure of biological macromolecules is one of the most important frontier studies of quantum computing, thus the curse of dimensionality is an urgent issue to be solved. With the development of supercomputers in recent decades, the use of distributed computing can solve a series of memory and efficiency problems caused by the curse of dimensionality to a certain extent. Moreover, some distributed computing algorithms can be applied to quantum unitary evolution.

How Can Distributed Computing Help Solve the Curse of Dimensionality?

A distributed computing approach to solve the curse of dimensionality caused by the complex quantum system modeling is discussed. With the help of Cannon’s algorithm, the distributed computing transformation of numerical method for simulating quantum unitary evolution is achieved. Based on the Tavis-Cummings model, a large number of atoms are added into the optical cavity to obtain a high-dimensional quantum closed system implemented on the supercomputer platform. The comparison of time cost and speedup of different distributed computing strategies is discussed.

What is the Cavity Quantum Electrodynamics (QED) Model?

A key contribution of this paper is the cavity quantum electrodynamics (QED) model, which is easy to implement in the laboratory and offers a unique scientific paradigm for studying light-matter interaction. According to this paradigm, impurity two-level systems, also known as atoms, are connected to fields of cavities. The cavity QED model includes the Jaynes-Cummings model (JCM) and the Tavis-Cummings model (TCM), as well as their generalizations. Many studies have been conducted recently in the field of these models.

How is the Tavis-Cummings Model Used in this Study?

In this paper, the Tavis-Cummings model (TCM) involving a large number of atoms is introduced. The basic states of the TCM is as follows: Ψ(p,n,O) = i1^li, where p=0, n = number of free photons, n = number of atoms, li = i1^ n = electronic state, li = 0 = ist electron in ground state, li = 1 = ist electron in excited state. Since the closed system dissipation process is disabled, thus we can omit the qubit involving photons: Ψ(n,O) = i1^li.

Interaction between atom and field is explained in detail in Fig. 1. In a, an electron in the ground state absorbs a photon and transfer to the excited state, which is called excitation. In b, an electron in the excited state transfer to the ground state after releasing a photon, this process is called de-excitation. Initial state is shown in Fig. 1 c, where exist n-excited atoms and no photons. The excited atom will de-excite and become a ground state atom, at which time a photon will be released. When all atoms change to the ground state, n-photons are present in the optical cavity. In a closed system, the de-excitation and excitation of atom exist at the same time, so a certain regularity will appear in the light-matter system, which is embodied in the periodic oscillation of the quantum states.

What is the Numerical Method and its Distributed Computing Transformation?

The quantum master equation (QME) is a numerical method used in this study. The distributed computing transformation of this numerical method for simulating quantum unitary evolution is achieved with the help of Cannon’s algorithm. This approach allows for the simulation of the cavity QED model with a large number of two-level artificial atoms on a supercomputing platform. The time cost and speedup under different distributed computing strategies are compared with that under the situation without distributed computing. This study provides valuable insights into the efficiency of distributed computing methods in the field of quantum computing.