World’s First Lattice of Non-Abelian Anions – Google AI Blog

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Posted by Trond Andersen and Yuri Lensky, Researchers, Google Quantum AI Team

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

While this sounds like common sense, it only applies to our familiar three-dimensional world. The researchers predicted that for a special type of particle, called an anion, that is limited to moving only in a two-dimensional (2D) plane, quantum mechanics allows for something entirely different. Anions are indistinguishable from each other, and some non-abelian anions have a special property that causes observable differences in the shared quantum state during exchange, making it possible to tell when they have been exchanged, despite being completely indistinguishable from each other. While the researchers were successful in detecting their relatives, abelian anions, whose change during exchange is more subtle and impossible to detect directly, realizing “non-abelian exchange behavior” has proven more difficult due to challenges both with the control that with the sensing.

In “Non-Abelian braiding of graph vertices in a superconducting processor”, published in Nature, we report for the first time the observation of this non-Abelian braiding behavior. Non-Abelian anons could open up a new avenue for quantum computing, where quantum operations are achieved by swapping particles over one another like strings are swapped over one another to create braids. Implementing this new switching behavior on our superconducting quantum processor could be an alternative route to so-called topological quantum computing, which benefits from being robust against ambient noise.

Exchange statistics and non-abelian anions

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

Non-Abelian anion exchange can be visualized in a similar way, where strings are made by extending particle positions in the time dimension to form “world lines”. Imagine plotting the positions of two particles with respect to time. If the particles remained stationary, the graph would simply consist of two parallel lines, representing their constant positions. But if we swap the positions of the particles, the world lines wrap around each other. Swap them a second time and you’ve tied a knot.

While somewhat difficult to visualize, kinks in four dimensions (three spatial dimensions plus one time dimension) can still be unraveled easily. They are trivial: like a shoelace, just pull one end and it unwinds. But when the particles are confined to two spatial dimensions, the kinks are in full three dimensions, and as we know from our daily 3D lives, they can’t always be untied easily. The entanglement of non-Abelian anion world lines can be used as quantum computing operations to transform the state of particles.

A key aspect of non-Abelian anions is “degeneracy”: the complete state of several separate anions is not fully specified by local information, allowing the same configuration of anions to represent superpositions of different quantum states. Coiling of non-abelian anyons around each other can change the encoded state.

How to create a non abelian anyon

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

To produce non-Abelian anions, we need to control degeneracy (i.e. the number of wave functions that causes all stabilizer measurements to be +1). Since a stabilizer measurement returns two possible values, each stabilizer halves the degeneracy of the system, and with enough stabilizers, only one wave function satisfies the criterion. So, an easy way to increase degeneration is to bond two stabilizers together. In doing so, we remove an edge in the stabilizer grid, resulting in two points where only three edges intersect. These points, referred to as “degree 3 vertices” (D3V), are predicted to be non-Abelian anions.

To weave the D3Vs, we have to move them, which means we have to stretch and squash the stabilizers into new shapes. We accomplish this by implementing two-qubit gates between the anions and their neighbors (middle and right panels shown below).

Non-abelian anyons in stabilizer codes. a: Example of a knot made by intertwining two anyones worldlines. 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 a sublattice in the regular surface code. This constraint is broken when D3V (yellow triangles) are introduced. c: Process for forming and moving D3Vs (predicted to be non-abelian anions). We start with the surface code, where each square corresponds to a joint measurement of the four qubits at 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 anions. We move the D3Vs by applying two-qubit gates between nearby sites (right panel).

Now that we have a way to make and move non-abelian anions, we need to test their anion behavior. For this we examine three characteristics that one would expect from non-abelian anions:

The “Melting Rules” – What happens when non-Abelian anions collide with each other? Exchange Stats: What happens when they are intertwined around each other? Topological quantum computing primitives: Can we encode qubits in non-abelian anions and use entanglement to perform two-qubit entanglement operations? The fusion rules of non abelian anions

We investigate fusion rules by studying how a pair of D3Vs interact with the bishop-like platelet violations introduced above. Specifically, we create a pair and fly one around a D3V by gated a single qubit.

While the rules of bishops in chess dictate that plate violations can never meet, the dislocation in the chessboard lattice allows them to break this rule, meet its partner, and annihilate themselves with it. License plate violations are now gone! But it brings the non-Abelian Anyons back into contact with each other, and the Anyons suddenly transform into the violations of the missing platelet. As strange as this behavior may seem, it is a manifestation of exactly the fusion rules we expect these entities to obey. This establishes confidence that D3Vs are, in fact, non-abelian anions.

Demonstration of anion fusion rules (starting from panel I, bottom left). We form and separate two D3Vs (yellow triangles), then form two adjacent tile violations (red squares) and pass one between the D3Vs. The D3Vs deformation of the “chessboard” changes the bishop’s rules of plate violations. While previously lying on adjacent squares, they are now able to move along the same diagonals and collide (as shown by the red lines). When they collide, they destroy each other. The D3Vs are brought together and surprisingly transform into the missing violations of the adjacent red plaque. Observation of non-Abelian exchange statistics

Now that we’ve established the fusion rules, we want to see the real smoking gun of non-abelian anions: the non-abelian exchange stats. We create two pairs of non-abelian anion, then weave them by wrapping one of each pair around each other (shown below). When we merge the two pairs again, two pairs of platelet violations appear. The simple act of wrapping anions around each other changed the observables of our system. In other words, if you closed your eyes while the non-abelian anonis were being swapped, you would still be able to tell that they had been swapped once your eyes were opened. This is the hallmark of non-Abelian statistics.

Twist the non-abelian anions. We make two pairs of D3Vs (panel II), then bring one from each pair around each other (III-XI). When merging the two pairs again in panel XII, two pairs of license plate violations appear! Twisting the non-Abelian anions changed the system observables from panel I to panel XII; a direct manifestation of non-Abelian trading statistics. Topological quantum computing

Finally, having established their fusion rules and exchange statistics, we demonstrate how we can use these particles in quantum computations. Non-Abelian anions can be used to encode information, represented by logical qubits, which should be distinct from the actual physical qubits used in the experiment. It can be shown that the number of logical qubits encoded in N D3V is N/2–1, so we use N=8 D3V to encode three logical qubits and perform entanglement to interlace them. By studying the resulting state, we find that the entangling actually led to the formation of the well-known quantum entangled state called the Greenberger-Horne-Zeilinger (GHZ) state.

Using non-abelian anions as logical qubits. a, We intertwine the non-abelian anions to entangle three qubits encoded in eight D3Vs. b, State quantum tomography reconstructs the density matrix, which can be represented in a 3D bar graph and is consistent with the desired highly entangled GHZ state. Conclusion

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

Thanks

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

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