Explore Some Core Ingredients of Topological Quantum Computing with Qiskit
By Robert Davis with contributions from Oles Shtanko
Researchers have long sought to build quantum hardware that is robust against errors caused by environmental noise — in other words, to make quantum computers that are “fault tolerant.” Quantum computers employing techniques taken from the field of topology might one day achieve this goal. But to realize them, we’ll need a much deeper understanding of the exotic particles known as topological Majorana fermions.
Topological Majorana fermions are collective behaviors of excited-state electrons that appear in the atomic structure of topological materials, an exotic phase of matter that is the subject of immense scientific interest. Researchers believe these strange collective excitations could provide useful properties to exploit in materials science, or even form the basis of a new quantum computing architecture that would be stable against environmental noise called topological quantum computing. And now, with a recent publication in Nature Communications, IBM Quantum researchers are debuting techniques that present exciting new avenues of study for these exotic materials. You can explore these techniques using Qiskit.
So…what makes topological Majorana fermions so useful?
Topological materials have the potential to be useful in quantum computing because they naturally produce non-local quantum correlations within their atomic structure. This makes them remarkably robust against local distortions caused by environmental noise. Specifically, topological Majorana fermions —special collections of electrons that appear only at the surface layer of topological materials — are valuable for research and application purposes.
The individual electrons that make up Majorana fermions are particles themselves, but really there’s no such thing as an “individual” Majorana fermion. They act like particles, but they are always built from collective excitations of many electrons. Together, that collective exhibits the paradoxical ability to function as one half of a single electron.
These topological Majorana modes, as they’re called, could be very useful for storing quantum information because they naturally form two-level quantum systems that can represent the 1 and 0 states that are essential to quantum computation. But we’re still a long way off from understanding enough about topological materials to use them in a functioning quantum computer. Building simulations to enable deeper, more quantitative analyses of topological system properties has proven to be a significant challenge.
The new research by IBM’s Nikhil Harle, Oles Shtanko, and Ramis Movassagh demonstrates novel quantitative methods for not only identifying, but also verifying and even manipulating topological Majorana modes at scale. What’s more, these methods are available today via two resources that are freely available to anyone in the quantum community: (1) superconducting quantum hardware to act as a quantum simulator, and (2) Qiskit.
Wrangling topological states through quantum simulation
In their experiment, Harle, Shtanko and Movassagh set out to build a quantitative simulation of topological quantum matter, specifically setting their sights on a one-dimensional topological superconducting material that is known to host pairs of “half-electron” Majorana modes at its boundaries.
Their goal was to develop methods for reliably determining the structure of these Majorana modes, methods for distinguishing topological modes from non-topological modes, and methods for manipulating Majorana fermions in one dimension.
In their simulations, the researchers started by initializing qubits in the product state, the simplest state you can prepare without needing to create entanglement. Then, they applied a circuit designed to emulate the dynamics of the topological system. To simulate those dynamics, the researchers used a mathematical trick called the Jordan-Wigner transformation, which allowed them to interpret the dynamics of qubits as the motion of fermions.
Theoretical predictions, including the team’s own 2020 paper, suggest that local observables in the bulk of the simulated material — i.e., in the bulk of the qubit chain, far from the edges — should be rather chaotic. This is true for both topological and non-topological regimes. The researchers quickly found that their simulation matched these predictions.
But their simulation also matched another theoretical prediction: In topological regimes, their system exhibited a very strong oscillating signal at its edges, corresponding to the oscillations of the local observables — the quantifiable features of the system — at the surface of the simulated material. In non-topological regimes, their system exhibited no such signal, meaning they could use the presence of these oscillations as a reliable signature for identifying when a system enters the topological phase.
The next step was to build a mathematical model connecting this oscillation with the amplitude of the Majorana mode’s wavefunction. This would allow the researchers to quantitatively characterize the structure of the Majorana mode.
By measuring the oscillations of key operators at the edges of their topological systems, they found they could mathematically derive the wavefunction for the entire Majorana mode. The researchers believe this is the first experiment to ever use these oscillations to calculate some useful quantitative information about a topological system.
Verifying and manipulating topological modes
Observing and quantitatively characterizing the oscillating signal in topological boundary states is a noteworthy achievement itself, but for the researchers running this experiment, that was only half the battle. Any number of accidental phenomena — like a gate miscalibration, for example — can create oscillations at the edges of the qubit chain that an observer might mistake for a topological mode. The team wanted to devise a way to verify that the signature they saw was real.
To do this, they designed a measurement to distinguish between topological Majorana modes and other non-topological modes. Their method took advantage of the fact that non-topological modes are always “full electron” modes, meaning local measurements taken in the simulation will have a clear effect on them. This is not true for topological Majorana modes which, due to their non-local correlations, are not affected by local measurements.
The researchers also wanted to develop a new technique for manipulating the Majoranas in their system and changing their quantum states in a controlled way. Specifically, they created a new method of “braiding” Majorana modes, or exchanging their positions in space.
Braiding Majorana modes not only reveals useful information about the topological system they inhabit; it also forms the theoretical basis of topological quantum computation. If you have four Majorana modes and you braid two of them, you can generate entanglement between two logical qubits encoded in these Majorana modes, effectively performing quantum computation. This is the most fundamental operation of quantum information processing in a topological system.
There are plenty of techniques for executing this kind of braiding, but most involve a very slow change in Majorana mode positions, which we accomplish by slowly modifying the system’s Hamiltonian, the mathematical description of the system’s total energy.
Simulations of such methods are very hard to implement in moderately noisy settings, as noise usually destroys the system’s state entirely before the braiding is completed. These methods also impose additional restrictions on the hardware — e.g., we cannot perform them in one-dimensional systems.
The IBM Quantum researcher’s new method, by contrast, can be implemented relatively quickly and also does not require changing the system’s Hamiltonian itself.
Useful quantum computing and condensed-matter physics
The new braiding method developed by the IBM team isn’t just some theoretical proposal. The researchers say it works, and that any Qiskit user can experiment with it and the other techniques they demonstrated in their paper, using the code and data they’ve made accessible via GitHub.
These methods could become a useful tool for other researchers studying not just topological Majoranas, but any kind of topological material. Perhaps the methods could be used to study parafermions — a more general class of topological matter. With the newfound ability to make quantitative predictions about topological systems, condensed-matter physicists can thinking more seriously about designing experiments that would challenge today’s classical systems.
As for the goal of building a topological quantum computer, the IBM team says their work represents only a small step on a very long road. However, with quantum computers now solving increasingly challenging simulation problems, it could well be the case that simulations run on the noisy quantum computers of today could be key to unlocking the fault tolerant quantum computation of tomorrow.
