What’s inside a black hole? In the study, the scientists were able to determine the mathematical description of the quantum state of the matrix model.
The American physicist Enrico Rinaldi, along with his team at the University of Michigan, has used quantum computing and computer learning to describe what is believed to be the interior of a black hole.
Scientists relied on the holographic principle, which suggests that the two existing theories – particles and gravity – are equivalent to each other. The complexity of ideas lies in the fact that they are built in different dimensions.
Both theories explain various dimensions, but they differ by one in the number of dimensions they describe. So, for example, gravity exists in three dimensions inside the geometry of a black hole, but particle physics lives in two dimensions on its surface—a flat disk.
Consider the black hole, which, due to its massive mass, warps space-time. The black hole’s gravity, which exists in three dimensions, mathematically connects to the particles dancing above it, which exist in two dimensions. As a result, a black hole exists in three-dimensional space but is perceived as a projection of particles.
Some scientists claim our entire universe is a holographic representation of particles, and this could lead to a consistent quantum explanation of gravity.
“In Einstein’s General Relativity theory, there are no particles—there’s just space-time. And in the Standard Model of particle physics, there’s no gravity, there’s just particles,” says Enrico Rinaldi. “Connecting the two different theories is a longstanding issue in physics—something people have been trying to do since the last century.”
A work by Rinaldi and colleagues, published in the journal PRX Quantum, examines how to probe holographic duality using quantum computing and deep learning to determine the lowest energy state of mathematical problems known as quantum matrix models.
Particle theory is represented by these quantum matrix models. Because holographic duality implies that what happens mathematically in a system representing particle theory will also affect a system representing gravity, solving such a quantum matrix model could yield gravity-related knowledge.
Rinaldi and his colleagues employed two matrix models that are simple enough to solve using traditional methods yet have all of the properties of more difficult matrix models used to describe black holes via holographic duality.
“We hope that by understanding the properties of this particle theory through the numerical experiments, we understand something about gravity,” adds Rinaldi. “Unfortunately it’s still not easy to solve the particle theories. And that’s where the computers can help us.”
These matrix models are numbers that represent objects in string theory, which is a framework in which one-dimensional strings represent particles in particle theory. Researchers are aiming to determine the precise arrangement of particles in the system that represents the system’s lowest energy state, called the ground state, when they solve matrix models like these. Nothing happens to the system in its natural condition unless you add anything to it that causes it to be perturbed.
“It’s really important to understand what this ground state looks like, because then you can create things from it,” Rinaldi says. “So for a material, knowing the ground state is like knowing, for example, if it’s a conductor, or if it’s a super conductor, or if it’s really strong, or if it’s weak. But finding this ground state among all the possible states is quite a difficult task. That’s why we are using these numerical methods.”
Consider the matrix models’ numbers as grains of sand, Rinaldi explains. When the sand is level, that’s the model’s ground state. However, if the sand has ripples, you must find a means to smooth them out. To find a solution, the researchers turned to quantum circuits. The quantum circuits are represented as wires in this manner, and each qubit, or quantum information bit, represents a wire. Gates, which are quantum operations defining how information will move down the wires, are placed on top of the cables.
“You can read them as music, going from left to right,” the author adds. “If you read it as music, you’re basically transforming the qubits from the beginning into something new each step. But you don’t know which operations you should do as you go along, which notes to play. The shaking process will tweak all these gates to make them take the correct form such that at the end of the entire process, you reach the ground state. So you have all this music, and if you play it right, at the end, you have the ground state.”
In Rinaldi’s study, the researchers define the mathematical description of the quantum state of their matrix model, called the quantum wave function. Then they use a special neural network in order to find the wave function of the matrix with the lowest possible energy—its ground state. The numbers of the neural network run through an iterative “optimization” process to find the matrix model’s ground state, tapping the bucket of sand so all of its grains are leveled.
In both approaches, the researchers were able to find the ground state of both matrix models they examined, but the quantum circuits are limited by a small number of qubits. Current quantum hardware can only handle a few dozens of qubits: adding lines to your music sheet becomes expensive, and the more you add the less precisely you can play the music.
“Other methods people typically use can find the energy of the ground state but not the entire structure of the wave function,” Rinaldi said. “We have shown how to get the full information about the ground state using these new emerging technologies, quantum computers and deep learning.
“Because these matrices are one possible representation for a special type of black hole, if we know how the matrices are arranged and what their properties are, we can know, for example, what a black hole looks like on the inside. What is on the event horizon for a black hole? Where does it come from? Answering these questions would be a step towards realizing a quantum theory of gravity.”
The results, says Rinaldi, show an important benchmark for future work on quantum and machine learning algorithms that researchers can use to study quantum gravity through the idea of holographic duality.
Next, Rinaldi is working with Nori and Hanada to study how the results of these algorithms can scale to larger matrices, as well as how robust they are against the introduction of “noisy” effects, or interferences that can introduce errors.
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