In a recent paper, scientists at the firm Quantinuum have announced the development of a practical quantum algorithm for solving a fundamental problem in a field of mathematics known as knot theory.
Knot theory
Knot theory is a field of mathematics called ‘low-dimensional topology’, with a history, stemming from an idea proposed by Lord Kelvin, who conjectured that chemical elements are different knots formed by vortices in the ether and mathematicians have been classifying and studying knots ever since.
Knot theory is intrinsically linked with many aspects of physics. For example, it naturally shows up in certain spin models in statistical mechanics.
Today, knot theory finds practical uses in fields as diverse as chemistry, robotics, fluid dynamics, and drug design. Measuring the invariants that characterize each knot is a challenge that scales exponentially with the complexity of the knots.
Furthermore, Knot theory plays a role in cryptography, studying DNA behaviour, statistical mechanics, thermodynamics, solar activity as well as topological quantum computing.
H2 quantum system
Using Quantinuum’s current H2 quantum system, the researchers found that a future system could estimate the value of the Jones polynomial, a key ‘fingerprint’ used to distinguish types of knots.
The H2 quantum processor, powered by Honeywell, enables the controlled creation and manipulation of non-Abelian anyons, which are essential for topological qubits. Quantinuum, Harvard University, and Caltech scientists suceeded in demonstrating a new state of matter using the H2 processor.
The H2 system uses an isotope of ytterbium to create qubits for computation and barium ions for cooling. It uses a trapped-ion architecture, called a quantum charged coupled device (QCCD).
New study
With the current study, the scientists identified the resources required for exponential quantum advantage in time-to-solution and energy consumption over the Frontier exascale supercomputer for solving this problem.
Hewlett Packards Enterprise Frontier, or OLCF-5, is the world’s first exascale supercomputer. It is hosted at the Oak Ridge Leadership Computing Facility (OLCF) in Tennessee, U.S. It can calculate at least 10^18 IEEE 754 Double Precision (64-bit) operations (multiplications and/or additions) per second.
The Quantinuum study found advantage crossover will require a system with greater than 85 qubits and 99.99% two-qubit gate fidelity. The H2 currently has 56 qubits with 99.87% two qubit fidelity (fidelity quantifies how closely the actual state or operation aligns with the intended or ideal state or operation).
The researchers believe the algorithm could serve as an easily verifiable benchmark for comparing quantum processors and the progress toward quantum advantage.
Future state
This work additionally shows how a quantum computer can cut through this exponential explosion, indicating that Quantinuum’s next-generation systems will offer practical quantum advantage in solving knot theory problems.
Going forwards, Quantinuum’s hardware roadmap includes even more powerful machines that will come online by the end of the decade. Notably, an advantage in energy consumption emerges for even smaller link sizes.Meanwhile, our teams aim to continue reducing errors through improvements in both hardware and software, thereby moving deeper into quantum advantage territory.
