Imagine you’re shown two identical objects and then asked to close your eyes. When you open your eyes, you see the same two objects in the same position. How can you determine if they have been swapped back and forth? Intuition and the laws of quantum mechanics agree: If the objects are truly identical, there is no way to tell.

While this sounds like common sense, it only applies to our familiar three-dimensional world. Researchers have predicted that for a special type of particle, called an anyon, that is restricted to move only in a two-dimensional (2D) plane, quantum mechanics allows for something quite different. Anyons are indistinguishable from one another and some, non-Abelian anyons, have a special property that causes observable differences in the shared quantum state under exchange, making it possible to tell when they have been exchanged, despite being fully indistinguishable from one another. While researchers have managed to detect their relatives, Abelian anyons, whose change under exchange is more subtle and impossible to directly detect, realizing “non-Abelian exchange behavior” has proven more difficult due to challenges with both control and detection.

In “Non-Abelian braiding of graph vertices in a superconducting processor”, published in *Nature*, we report the observation of this non-Abelian exchange behavior for the first time. Non-Abelian anyons could open a new avenue for quantum computation, in which quantum operations are achieved by swapping particles around one another like strings are swapped around one another to create braids. Realizing this new exchange behavior on our superconducting quantum processor could be an alternate route to so-called topological quantum computation, which benefits from being robust against environmental noise.

## Exchange statistics and non-Abelian anyons

In order to understand how this strange non-Abelian behavior can occur, it’s helpful to consider an analogy with the braiding of two strings. Take two identical strings and lay them parallel next to one another. Swap their ends to form a double-helix shape. The strings are identical, but because they wrap around one another when the ends are exchanged, it is very clear when the two ends are swapped.

The exchange of non-Abelian anyons can be visualized in a similar way, where the strings are made from extending the particles’ positions into the time dimension to form “world-lines.” Imagine plotting two particles’ locations vs. time. If the particles stay put, the plot would simply be two parallel lines, representing their constant locations. But if we exchange the locations of the particles, the world lines wrap around one another. Exchange them a second time, and you’ve made a knot.

While a bit difficult to visualize, knots in four dimensions (three spatial plus one time dimension) can always easily be undone. They are trivial — like a shoelace, simply pull one end and it unravels. But when the particles are restricted to two spatial dimensions, the knots are in three total dimensions and — as we know from our everyday 3D lives — cannot always be easily untied. The braiding of the non-Abelian anyons’ world lines can be used as quantum computing operations to transform the state of the particles.

A key aspect of non-Abelian anyons is “degeneracy”: the full state of several separated anyons is not completely specified by local information, allowing the same anyon configuration to represent superpositions of several quantum states. Winding non-Abelian anyons about each other can change the encoded state.

## How to make a non-Abelian anyon

So how do we realize non-Abelian braiding with one of Google’s quantum processors? We start with the familiar surface code, which we recently used to achieve a milestone in quantum error correction, where qubits are arranged on the vertices of a checkerboard pattern. Each color square of the checkerboard represents one of two possible joint measurements that can be made of the qubits on the four corners of the square. These so-called “stabilizer measurements” can return a value of either + or – 1. The latter is referred to as a plaquette violation, and can be created and moved diagonally — just like bishops in chess — by applying single-qubit X- and Z-gates. Recently, we showed that these bishop-like plaquette violations are Abelian anyons. In contrast to non-Abelian anyons, the state of Abelian anyons changes only subtly when they are swapped — so subtly that it is impossible to directly detect. While Abelian anyons are interesting, they do not hold the same promise for topological quantum computing that non-Abelian anyons do.

To produce non-Abelian anyons, we need to control the degeneracy (i.e., the number of wavefunctions that causes all stabilizer measurements to be +1). Since a stabilizer measurement returns two possible values, each stabilizer cuts the degeneracy of the system in half, and with sufficiently many stabilizers, only one wave function satisfies the criterion. Hence, a simple way to increase the degeneracy is to merge two stabilizers together. In the process of doing so, we remove one edge in the stabilizer grid, giving rise to two points where only three edges intersect. These points, referred to as “degree-3 vertices” (D3Vs), are predicted to be non-Abelian anyons.

In order to braid the D3Vs, we have to move them, meaning that we have to stretch and squash the stabilizers into new shapes. We accomplish this by implementing two-qubit gates between the anyons and their neighbors (middle and right panels shown below).

Non-Abelian anyons in stabilizer codes.** a:** Example of a knot made by braiding two anyons’ world lines. **b: **Single-qubit gates can be used to create and move stabilizers with a value of –1 (red squares). Like bishops in chess, these can only move diagonally and are therefore constrained to one sublattice in the regular surface code. This constraint is broken when D3Vs (yellow triangles) are introduced.** c: **Process to form and move D3Vs (predicted to be non-Abelian anyons). We start with the surface code, where each square corresponds to a joint measurement of the four qubits on its corners (**left** panel). We remove an edge separating two neighboring squares, such that there is now a single joint measurement of all six qubits (**middle** panel). This creates two D3Vs, which are non-Abelian anyons. We move the D3Vs by applying two-qubit gates between neighboring sites (right panel).

Now that we have a way to create and move the non-Abelian anyons, we need to verify their anyonic behavior. For this we examine three characteristics that would be expected of non-Abelian anyons:

The “fusion rules” — What happens when non-Abelian anyons collide with each other?

Exchange statistics — What happens when they are braided around one another?

Topological quantum computing primitives — Can we encode qubits in the non-Abelian anyons and use braiding to perform two-qubit entangling operations?

## The fusion rules of non-Abelian anyons

We investigate fusion rules by studying how a pair of D3Vs interact with the bishop-like plaquette violations introduced above. In particular, we create a pair of these and bring one of them around a D3V by applying single-qubit gates.

While the rules of bishops in chess dictate that the plaquette violations can never meet, the dislocation in the checkerboard lattice allows them to break this rule, meet its partner and annihilate with it. The plaquette violations have now disappeared! But bring the non-Abelian anyons back in contact with one another, and the anyons suddenly morph into the missing plaquette violations. As weird as this behavior seems, it is a manifestation of exactly the fusion rules that we expect these entities to obey. This establishes confidence that the D3Vs are, indeed, non-Abelian anyons.

Demonstration of anyonic fusion rules (starting with **panel I**, in the **lower left**). We form and separate two D3Vs (yellow triangles), then form two adjacent plaquette violations (red squares) and pass one between the D3Vs. The D3Vs deformation of the “chessboard” changes the bishop rules of the plaquette violations. While they used to lie on adjacent squares, they are now able to move along the same diagonals and collide (as shown by the red lines). When they do collide, they annihilate one another. The D3Vs are brought back together and surprisingly morph into the missing adjacent red plaquette violations.

## Observation of non-Abelian exchange statistics

After establishing the fusion rules, we want to see the real smoking gun of non-Abelian anyons: non-Abelian exchange statistics. We create two pairs of non-Abelian anyons, then braid them by wrapping one from each pair around each other (shown below). When we fuse the two pairs back together, two pairs of plaquette violations appear. The simple act of braiding the anyons around one another changed the observables of our system. In other words, if you closed your eyes while the non-Abelian anyons were being exchanged, you would still be able to tell that they had been exchanged once you opened your eyes. This is the hallmark of non-Abelian statistics.

Braiding non-Abelian anyons. We make two pairs of D3Vs (**panel II**), then bring one from each pair around each other (**III-XI**). When fusing the two pairs together again in panel XII, two pairs of plaquette violations appear! Braiding the non-Abelian anyons changed the observables of the system from panel I to panel XII; a direct manifestation of non-Abelian exchange statistics.

## Topological quantum computing

Finally, after establishing their fusion rules and exchange statistics, we demonstrate how we can use these particles in quantum computations. The non-Abelian anyons can be used to encode information, represented by logical qubits, which should be distinguished from the actual *physical* qubits used in the experiment. The number of logical qubits encoded in *N* D3Vs can be shown to be *N*/2–1, so we use *N*=8 D3Vs to encode three logical qubits, and perform braiding to entangle them. By studying the resulting state, we find that the braiding has indeed led to the formation of the desired, well-known quantum entangled state called the Greenberger-Horne-Zeilinger (GHZ) state.

Using non-Abelian anyons as logical qubits.** a, **We braid the non-Abelian anyons to entangle three qubits encoded in eight D3Vs. **b, **Quantum state tomography allows for reconstructing the density matrix, which can be represented in a 3D bar plot and is found to be consistent with the desired highly entangled GHZ-state.

## Conclusion

Our experiments show the first observation of non-Abelian exchange statistics, and that braiding of the D3Vs can be used to perform quantum computations. With future additions, including error correction during the braiding procedure, this could be a major step towards topological quantum computation, a long-sought method to endow qubits with intrinsic resilience against fluctuations and noise that would otherwise cause errors in computations.

## Acknowledgements

*We would like to thank Katie McCormick, our Quantum Science Communicator, for helping to write this blog post.*

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