New publication on Physical Review Letters titled "Sparsity independent Lyapunov exponent in the Sachdev-Ye-Kitaev model" by Dr. Chang Liu and Prof. Antonio M. García-García from Shanghai Jiaotong University, and Prof. Jacobus J. M. Verbaarschot from Stony Brook University

General relativity and quantum mechanics are the two cornerstones of modern physics. General relativity describes physical processes in systems where gravitation is the dominant force, while quantum mechanics describes those where the coherent superpositions of states play an essential role. Unfortunately, these two theories are logically incompatible in a fundamental way: when a gravity-dominated system has non-negligible quantum coherence, both general relativity and quantum mechanics must be applied to describe this system, and mixing them leads to logical paradoxes. Perhaps the most well-known example of this is the Black Hole Information paradox, since black holes are the most important and most well-studied quantum gravity systems. In its most simplistic form, the Black Hole Information paradox can be understood by imagining if an external observer of a black hole throws a book into it. If we assume the Equivalence Principle, which is the fundamental logical assumption of general relativity, Hawking's calculation tells us that the black hole, along with the book that was thrown into it, eventually evaporates into elementary particles that do not carry any information (known as Hawking radiation). However, this is a gross violation of the conservation of information, because the information in the book is essentially lost. Since the conservation of information is a core assumption of quantum mechanics, it follows that naively applying general relativity and quantum mechanics leads to two irreconcilable conclusions.

This is obviously less than ideal, because it implies that the foundation on which modern physics is built is logically unsound. Reconciling general relativity with quantum mechanics has henceforth become one of the most important, if not the most important task for theoretical physicists of today. In recent years, a school of thought has emerged which aims to solve this problem by relegating the spacetime geometry, on which the whole formalism of general relativity is built, to a derived, or emergent notion in a more fundamental quantum theory. This idea and the related theoretical proposals are collectively known as the "holographic principle". If one can construct a quantum theory in which a spacetime geometry naturally emerges, one refers to the theory as having holographic duality, and the spacetime as the holographic dual of the quantum theory. If the idea of holographic principle turned out to be true, one would be able to resolve the Black Hole Information paradox, by keeping the conservation of information as an exact law, while demoting the equivalence principle from a fundamental assumption to a derived statement that only holds true within a certain time-scale. Beyond said time-scale, one would no longer have a consistent spacetime geometry upon which the theory of general relativity could be built. In the example of the black hole system described above, when the book tries to cross the black hole event horizon, within a certain time-scale the shape and content of the book will remain more or less the same to an external observer. However, once past this time-scale, what the black hole event horizon does is that it effectively "scrambles" and re-organizes the material makeups of the book and the information therein into something completely different, which gets radiated out as subtle correlations in the Hawking radiation. However, in this scrambling process none of the original information is lost, nor is any new information added, for the reorganization is a mere reshuffling of the existing information. This process is called "quantum scrambling" and the phenomenon itself is called "quantum chaos", and can in fact (at least in principle) be simulated in a quantum computer. In order to ensure that no quantum cloning can be achieved by having the external observer collect the information encoded in the Hawking radiation, and thus reconstruct the book, followed by jumping into the black hole to meet the in-falling book, this time-scale needs to have a lower bound (the reader is reminded that the notion of information conservation implies that neither elimination nor addition of information can happen --- cloning a quantum state adds information and is therefore a violation of the conservation of information). This lower bound is universal in the sense that no physical process within the universe can scramble quantum information faster than this time-scale. Due to this reason, it is known as the universal bound on quantum chaos, whose reciprocal is known as the Lyapunov exponent. It is possible to show that for a black hole, its Lyapunov exponent is proportional to the temperature of its Hawking radiation.

Of course, as of this writing the holographic principle remains a hypothetical proposal rather than a scientific theory. To construct a quantum theory that describes our universe in a holographic way requires mathematical techniques and formalisms that simply do not exist yet. So theoretical physicists asked for the next best thing, which is a toy model that has essential features required by the holographic principle. The Sachdev-Ye-Kitaev (SYK) model is one such toy model. It was originally proposed by condensed matter physicists as a model that describes the behaviors of electrons in some metals, and was later discovered to be able to describe, in the limit where the temperature of the system is low, a holographically dual spacetime geometry with one spatial dimension and one time dimension and filled with constant negative curvature globally. By analyzing the information scrambling process in this geometry, Juan Maldacena and others found that the scrambling time-scale of this model does have a lower bound, whose reciprocal is proportional to the temperature of the original quantum mechanical system, which is precisely the universal bound on chaos described above.

This is clearly great news. However, if one were to call this lower bound "universal", it ought to work for every quantum mechanical model with holographic duality, and not just one model. In other words, if the lower bound in the SYK model were truly universal, it should not depend on the specific details of the model, but rather represent a feature of the holographic principle itself. This is precisely the question that Dr. Chang Liu and collaborators have sought to answer, which has lead to the consideration that since the original SYK model is fully connected, meaning that every particles interact with one another, if one were to randomly remove most of the interactions and were still able to find the same lower bound on the scrambling time-scale, it would be a strong evidence in favor of the holographic principle. Of course, removing most of the interactions of the original SYK model destroys some of its nice mathematical properties, rendering it non-solvable analytically. One must therefore resort to numerical methods to solve for the scrambling time-scale. However, even numerically simulating the model is a tremendously difficult task, because the amount of computational effort required increases exponentially with the number of particles simulated. Due to the explosive requirement of the computational resources, central processing unit (CPU)-based computing systems would soon start running out of computing power due to the insufficient parallel computing abilities. Therefore, graphics processing unit (GPU)-based computing systems are a much better choice due to its inherently parallel architecture. However, a significant problem is that existing quantum simulation libraries with GPU support are not optimized for GPU architectures specifically, and therefore are not able to utilize the full power of modern GPUs. To this end, they have developed a brand new C++ library for simulation of quantum spin systems, that is heavily optimized for the latest nVidia GPUs. They achieve the optimizations by writing dedicated Compute Unified Device Architecture (CUDA) kernel functions so that computing the action of a quantum operator on a quantum state can be effectively mapped to GPU hardware primitives, on top of which they have implemented Krylov subspace-based algorithm to compute the quantum evolution. They make use of C++'s low-level memory management features to carefully manage video memory allocations and deallocations to minimize the video memory usage as well as the overhead of memory allocations and deallocations. The result is that researchers  are now able to simulate systems with as many as N=64 particles on a system with dual nVidia A100 graphics cards. This is a significant improvement over the previous state of the art N=50 (note the computational difficulty increases exponentially, so this is an improvement by a factor of two to the power of 14). Analyzing the simulation results, they have shown that the SYK model with the majority of its interactions eliminated (called sparse SYK model) still has the same universal lower bound for the information scrambling process, and this lower bound does not depend on the sparsity of the model. This result provides a strong support for the holographic principle, by enlarging the models with potential gravity dual to all sparse SYK models. The paper is published under the title "Sparsity independent Lyapunov exponent in the Sachdev-Ye-Kitaev model" on Physical Review Letters, and is funded by the China National Natural Science Foundation and the United States Department of Energy. Beijing Paratera Co. Ltd. is acknowledged for providing the needed computational resources of this research.