Our quantum algorithms team has been hard at work exploring solutions to continually optimize our system’s performance. Recently, they’ve invented a novel technique, called the , that can offer significant resource savings in future applications.
The transform takes complex representations and makes them simple, by transforming into a different “basis”. This is like looking at a cube from one angle, then rotating it and seeing just a square, instead. Transformations like this save resources because the more complex your problem looks, the more expensive it is to represent and manipulate on qubits.
You’ve changed
While it might sound like magic, transforms are a commonly used tool in science and engineering. Transforms simplify problems by reshaping them into something that is easier to deal with, or that provides a new perspective on the situation. For example, sound engineers use Fourier transforms every day to look at complex musical pieces in terms of their frequency components. Electrical engineers use Laplace transforms; people who work in image processing use the Abel transform; physicists use the Legendre transform, and so on.
In a new paper outlining the necessary tools to implement the QPT, Dr. Nathan Fitzpatrick and Mr. Jędrzej Burkat explain how the QPT will be widely applicable in quantum computing simulations, spanning areas like molecular chemistry, materials science, and semiconductor physics. The paper also describes how the algorithm can lead to significant resource savings by offering quantum programmers a more efficient way of representing problems on qubits.
Symmetry is key
The efficiency of the QPT stems from its use of one of the most profound findings in the field of physics: that symmetries drive the properties of a system.
While the average person can “appreciate” symmetry, for example in design or aesthetics, physicists understand symmetry as a much more profound element present in the fabric of reality. Symmetries are like the universe’s DNA; they lead to conservation laws, which are the most immutable truths we know.
Back in the 1920’s, when women were largely prohibited from practicing physics, one of the great mathematicians of the century, Emmy Noether, turned her attention to the field when she was tasked with helping Einstein with his work. In her attempt to solve a problem Einstein had encountered, Dr. Noether realized that all the most powerful and fundamental laws of physics, such as “energy can neither be created nor destroyed” are in fact the consequence of a deep simplicity – symmetry – hiding behind the curtains of reality. Dr. Noether’s theorem would have a profound effect on the trajectory of physics.
In addition to the many direct consequences of Noether’s theorem is a longstanding tradition amongst physicists to treat symmetry thoughtfully. Because of its role in the fabric of our universe, carefully considering the symmetries of a system often leads to invaluable insights.
Einstein, Pauli and Noether walk into a bar...
Many of the systems we are interested in simulating with quantum computers are, at their heart, systems of electrons. Whether we are looking at how electrons move in a paired dance inside superconductors, or how they form orbitals and bonds in a chemical system, the motion of electrons are at the core.
Seven years after Noether published her blockbuster results, Wolfgang Pauli made waves when he published the work describing his Pauli exclusion principle, which relies heavily on symmetry to explain basic tenets of quantum theory. Pauli’s principle has enormous consequences; for starters, describing how the objects we interact with every day are solid even though atoms are mostly empty space, and outlining the rules of bonds, orbitals, and all of chemistry, among other things.
Symmetry in motion
It is Pauli's symmetry, coupled with a deep respect for the impact of symmetry, that led our team at to the discovery published today.
In their work, they considered the act of designing quantum algorithms, and how one’s design choices may lead to efficiency or inefficiency.
When you design quantum algorithms, there are many choices you can make that affect the final result. Extensive work goes into optimizing each individual step in an algorithm, requiring a cyclical process of determining subroutine improvements, and finally, bringing it all together. The significant cost and time required is a limiting factor in optimizing many algorithms of interest.
This is again where symmetry comes into play. The authors realized that by better exploiting the deepest symmetries of the problem, they could make the entire edifice more efficient, from state preparation to readout. Over the course of a few years, a team lead Dr. Fitzpatrick and his colleague Jędrzej Burkat slowly polished their approach into a full algorithm for performing the QPT.
The QPT functions by using Pauli’s symmetry to discard unimportant details and strip the problem down to its bare essentials. Starting with a Paldus transform allows the algorithm designer to enjoy knock-on effects throughout the entire structure, making it overall more efficient to run.
“It’s amazing to think how something we discovered one hundred years ago is making quantum computing easier and more efficient,” said Dr. Nathan Fitzpatrick.
Ultimately, this innovation will lead to more efficient quantum simulation. Projects we believed to still be many years out can now be realized in the near term.
Transforming the future
The discovery of the Quantum Paldus Transform is a powerful reminder that enduring ideas—like symmetry—continue to shape the frontiers of science. By reaching back into the fundamental principles laid down by pioneers like Noether and Pauli, and combining them with modern quantum algorithm design, Dr. Fitzpatrick and Mr. Burkat have uncovered a tool with the potential to reshape how we approach quantum computation.
As quantum technologies continue their crossover from theoretical promise to practical implementation, innovations like this will be key in unlocking their full potential.
