Dude, what if everything around us was just ... a hologram?
The thing is, it could be—and a University of Michigan physicist is using
quantum computing and machine learning to better understand the idea, called
holographic duality.
Holographic duality is a mathematical conjecture that connects theories of
particles and their interactions with the theory of gravity. This conjecture
suggests that the theory of gravity and the theory of particles are
mathematically equivalent: What happens mathematically in the theory of
gravity happens in the theory of particles, and vice versa.
Both theories describe different dimensions, but the number of dimensions
they describe differs by one. So inside the shape of a black hole, for
example, gravity exists in three dimensions while a particle theory exists
in two dimensions, on its surface—a flat disk.
To envision this, think again of the black hole, which warps space-time
because of its immense mass. The gravity of the black hole, which exists in
three dimensions, connects mathematically to the particles dancing above it,
in two dimensions. Therefore, a black hole exists in a three-dimensional
space, but we see it as projected through particles.
Some scientists theorize our entire universe is a holographic projection of
particles, and this could lead to a consistent quantum theory 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," said Enrico Rinaldi, a research scientist
in the U-M Department of Physics. "Connecting the two different theories is
a longstanding issue in physics—something people have been trying to do
since the last century."
In a study published in the journal PRX Quantum, Rinaldi and his co-authors
examine how to probe holographic duality using quantum computing and deep
learning to find the lowest energy state of mathematical problems called
quantum matrix models.
These quantum matrix models are representations of particle theory. Because
holographic duality suggests that what happens mathematically in a system
that represents particle theory will similarly affect a system that
represents gravity, solving such a quantum matrix model could reveal
information about gravity.
For the study, Rinaldi and his team used two matrix models simple enough to
be solved using traditional methods, but which have all of the features of
more complicated matrix models used to describe black holes through the
holographic duality.
"We hope that by understanding the properties of this particle theory
through the numerical experiments, we understand something about gravity,"
said Rinaldi, who is based in Tokyo and hosted by the Theoretical Quantum
Physics Laboratory at the Cluster for Pioneering Research at RIKEN, Wako.
"Unfortunately it's still not easy to solve the particle theories. And
that's where the computers can help us."
These matrix models are blocks of numbers that represent objects in string
theory, which is a framework in which particles in particle theory are
represented by one-dimensional strings. When researchers solve matrix models
like these, they are trying to find the specific configuration of particles
in the system that represent the system's lowest energy state, called the
ground state. In the ground state, nothing happens to the system unless you
add something to it that perturbs it.
"It's really important to understand what this ground state looks like,
because then you can create things from it," Rinaldi said. "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."
You can think of the numbers in the matrix models as grains of sand, Rinaldi
says. When the sand is level, that's the model's ground state. But if there
are ripples in the sand, you have to find a way to level them out. To solve
this, the researchers first looked to quantum circuits. In this method, the
quantum circuits are represented by wires, and each qubit, or bit of quantum
information, is a wire. On top of the wires are gates, which are quantum
operations dictating how information will pass along the wires.
"You can read them as music, going from left to right," Rinaldi said. "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."
The researchers then wanted to compare using this quantum circuit method to
using a deep learning method. Deep learning is a kind of machine learning
that uses a neural network approach—a series of algorithms that tries to
find relationships in data, similar to how the human brain works.
Neural networks are used to design facial recognition software by being fed
thousands of images of faces—from which they draw particular landmarks of
the face in order to recognize individual images or generate new faces of
persons who do not exist.
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.
Rinaldi's co-authors include Xizhi Han at Stanford University; Mohammad
Hassan at City College of New York; Yuan Feng at Pasadena City College;
Franco Nori at U-M and RIKEN; Michael McGuigan at Brookhaven National
Laboratory and Masanori Hanada at University of Surrey.
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.
Reference:
Enrico Rinaldi et al, Matrix-Model Simulations Using Quantum Computing, Deep
Learning, and Lattice Monte Carlo, PRX Quantum (2022).
DOI: 10.1103/PRXQuantum.3.010324