Anyons and Topological Quantum Computation

Presented by: Jonah Sobel (Dated: Thursday, 3/18/21)

I. INTRODUCTION

When restricted to two spatial dimensions, particles behave with fewer restrictions than normal fermions and bosons in three spatial dimensions. Emergent phenomena known as quasiparticles can occur in strange circumstances, in which a microscopically complex system behaves as if it contained weakly interacting particles in a vacuum. One such example is known as an anyon, a quasiparticle that emerges in two spatial dimensions. Anyons are of particular interest as they have been theoretically shown to solve a major problem in the implementation of quantum computing. A quantum computer created from anyons is known as a topological quantum computer, and while researchers have yet to provide experimental (and undisputed) evidence of non-abelian anyons, recent promising findings offer hope that topological quantum computation may be realized soon enough. Anyons were first theorized in 1977 by Leinaas and Myrheim [1] as a two dimensional ex- ception to the boson-fermion particle divide. Frank Wilczek was the first to explore the properties of anyons in depth and published two papers on anyons in 1982 [2]. Then, in 1985 it was predicted that the particles that exist in fractional quantum hall effect states would be anyons, and later confirmed by Wilczek. Moore, Read, and Wen showed that non-abelian anyons could be realized in fractional quantum hall effect states [3]. Then, Alexei Kitaev showed that these non-abelian anyons could be used to build a topological quantum computer. More recently, two independent teams of scientists, one in the US and one in France, announced experimental evidence of Abelian anyons in 2020 [4, 5].

II. WHAT IS AN ANYON?

Bosons and fermions are distinguished from one another by their spin numbers. Bosons possess integer spins, whereas fermions possess half-integer spins. This leads to important symmetry properties: under interchange of identical particles, the wave function of bosons 2 is symmetric:

|ψ1ψ2i = |ψ2ψ1i whereas that of fermions is anti-symmetric:

|ψ1ψ2i = −|ψ2ψ1i

In two dimensions, anyons obey fractional statistics. As described by Nayak and Simon [6], when two identical particles each make a counterclockwise half-revolution about each other, the particles return to their original wave function with an adopted phase of eiθ:

iθ |ψ1ψ2i = e |ψ2ψ1i

Similarly, when they make a clockwise half-revolution, their wavefunction adopts a phase of e−iθ. Directions of clockwise and counterclockwise are only meaningful and clearly deﬁned in two spatial dimensions. This is why this scheme would not make sense in three or more dimensions. A unique consequence of these statistics is that when anyons are exchanged twice in the same direction, their wavefunction picks up a phase of e2iθ. This is also commonly written with θ = 2πs, where s is the spin quantum number:

2s |ψ1ψ2i = (−1) |ψ2ψ1i

In 1988, JürgFröhlich demonstrated that, according to the spin-statistics theorem for particle exchange, it was possible for the particle exchange to be non-abelian. This is achieved when the system exhibits degeneracy (so that multiple distinct states of the system have the same configuration of particles). According to Trebst, exchanging the particles contributes not just a phase change, but sends the system into a different state with the same particle configuration. This means that particle exchange corresponds to a linear transformation on the subspace of degenerate states [7]. Without degeneracy, the subspace is one-dimensional, and so all linear transformations commute (because they are essentially just multiplications by a phase). With degeneracy, this subspace has higher dimension and so these linear transformations will not always commute.

III. TOPOLOGICAL QUANTUM COMPUTING

One of the biggest problems that quantum computing faces is that of decoherence and stability. A quantum system is said to be coherent as long as there is a definite phase 3 relation between states (this is necessary for quantum computing). Every system is loosely coupled with the state of its environment, and so decoherence can be understood to be the loss of information from a quantum system to its surrounding environment. If a system were perfectly isolated from any external stimuli, it would, in theory, maintain coherence indefi- nitely. Decoherence cannot be undone, as it is effectively non-unitary. Therefore, it would be impossible to measure or manipulate a state that is isolated as such, and so decoherence seems to be a necessary component of this scheme. However, decoherence poses a major challenge to achieving quantum supremacy as many important quantum algorithms require much longer coherence times than are currently achievable. This is why, today, there is a large effort focused on minimizing (or at least controlling) decoherence.

A traditional quantum computer relies on trapped quantum particles – these particles are easily disturbed, and small cumulative perturbations quickly cause these states to deco- here. A topological quantum computer instead uses quantum braids, which are much more stable and resistant to decoherence. The same perturbations do not aﬀect the topological properties of the quantum braids. Like fermions, anyons cannot occupy the same state, so their world lines cannot intersect or merge. This allows their paths in 3 (2 spatial + 1 time) dimensional spacetime to form stable braids. When braided, the quantum state of the anyonic system is transformed only according to the topological class of the trajectories of the anyons. So, small errors introduced in the trajectories will not aﬀect the information stored in the state of the system.

A common example of this process uses Fibonacci anyons, as detailed by Trebst [7]. Fibonacci anyons are the simplest non-abelian anyons. In a system of anyons, the ground state of the system (or vacuum) is denoted by 1. This trivial particle is its own anti-particle. In order to have a non-trivial system of anyons, we need at least one more particle type. The Fibonacci anyonic system is the simplest such system, and so it has one additional nontrivial anyon of type τ. These anyons are called Fibonacci anyons, and they are also their own anti-particles. Anyons are combined in a process similar to the addition of quantum spins to form a new total spin – this is called the fusion of anyons. We may also think of fusion as a measurement, because the resulting anyons may be of several diﬀerent types, 4

each with its own probability. According to Trebst, given an anyon X, if the fusion of X with any other anyon produces an anyon of the same type, then X is said to be Abelian. The trivial anyon of type 1 is Abelian, because 1 ⊗ x = x for any type x. For our Fibonacci system of anyons, the fusion rules are as follows:

1. 1 ⊗ 1 = 1

2. 1 ⊗ τ = τ ⊗ 1 = τ

3. τ ⊗ τ = 1 ⊕ τ

where ⊗ is the same tensor product as for 1/2-spin particles, and ⊕ is the direct sum and represents the two possible fusion channels [6]. When extended to three anyons:

τ ⊗ τ ⊗ τ = τ ⊗ (1 ⊕ τ) = τ ⊗ 1 ⊕ τ ⊗ τ = τ ⊕ 1 ⊕ τ = 1 ⊕ 2τ

The fusion of three anyons yields a ﬁnal state with total charge τ in 2 ways or 1 in one way. So, we use 3 states to deﬁne our basis. According to Trebst, these fusion paths represent an orthonormal basis of the degenerate ground-state manifold. In the Fibonacci 1 system, when two τ anyons fuse, the probability of seeing 1 is p0 = ϕ2 and the probability √ 1 1+ 5 of seeing τ is p1 = ϕ , where ϕ = 2 (the Golden Ratio). When two anyons are exchanged in the plane, there are two possible exchanges which are not topologically equivalent. The convention is to move the right-hand anyon in front of and across the left-hand anyon; this is called right-handed braiding. The inverse of this is called left-handed braiding. This is depicted in the following image:

FIG. 1. Here, each line represents the world line of an anyon. Time ﬂows upwards. The image on the left depicts right-handed braiding and the image on the right depicts left-handed braiding (unfortunately formatted diagram). [7] 5

Now, Bonesteel discusses how to encode these anyons as superpositions of 0 or 1 [8]. First, we limit our basis to a 2-D Hilbert space. So, we only consider 2 states with total charge τ. Here, we group the two leftmost anyons into the control group and call the rightmost anyon the noncomputational anyon. We let |0i be when the control group has a total charge equal to 1, and |1i when the control group has a total charge equal to τ.

Basis states are |(•, •)0i and |(•, •)1i, where (•, •)a describes two anyons with total q-spin a. Q-spin is a q-deformed spin quantum number [8]. When a third anyon is added, the Hilbert space is 3 dimensional and is spanned by |((•, •)0, •)1i, |((•, •)1, •)1i, and |((•, •)0, •)0i. For quantum computing, take triplets of anyons with total q-spin 1, and let the logical qubit

states be |0Li = |((•, •)0, •)1i and |1Li = |((•, •)1, •)1i. Therefore, the state |((•, •)0, •)0i with total q-spin 0 is noncomputational. These qubits are measured by determining the q-spin of the two leftmost anyons. However, error may be introduced through transitions to noncomputational states, called leakage errors.

To swap the two computational anyons, apply the gate σ1:

  e−i4π/5 0 0    −i2π/5  σ1 =  0 −e 0  (1)   0 0 −e−i2π/5

To swap the rightmost computational anyon with the noncomputational anyon, apply the √ 5−1 gate σ2, where τ = 2 is the inverse golden ratio:

 √  −τe−iπ/5 −i τe−iπ/10 0  √   −iπ/10  σ2 = −i τe −τ 0  (2)   0 0 −e−i2π/5

The upper 2 × 2 section acts on the computational anyons and are used to perform qubit rotations whereas the lower right element acts as a phase on the noncomputational qubit. 6

FIG. 2. (a) Here are examples of a simple 3-anyon braid. (b) A general 3-anyon braid and its matrix representation.

A. Gates

Adiabatically braiding these anyons results in unitary transformation [8]. Trebst notes that these braid operators are a result of two subclasses of operators: R and F matrices [7]. The R matrix is the topological phase imparted onto anyons during the braid. As they wind, the anyons pick up a phase due to the Ahronov-Bohm eﬀect. The F matrix is a result of the physical rotation of the anyon. As they braid, the control group is always the two leftmost anyons, so braiding changes which two anyons are in the control group. Therefore, this changes the basis. We evaluate our system by fusing together the anyons in the control group ﬁrst, so exchanging these anyons will rotate the system. The order of the anyons in the control group matters because they are non-abelian and an exchange will transform the system. Trebst derives the basis for the Fibonacci system to be:

{|111i, |ττ1i, |1ττi, |τ1τi, |τττi}

In this representation, the F −matrix is given by   1    1      F =  1       ϕ−1 ϕ−1/2   ϕ−1/2 −ϕ−1 7

And the R−matrix is given by

R = diag(e4πi/5, e−3πi/5, e−3πi/5, e4πi/5, e−3πi/5)

Finally, the braid matrix is given by   e4πi/5    e−3πi/5    −1   B = FRF =  e−3πi/5       ϕ−1e−4πi/5 −ϕ−1/2e−2πi/5   −ϕ−1/2e−2πi/5 −ϕ−1

These are the explicit matrix representations of the basis transformation F and the braid matrix B for the fusion of two Fibonacci anyons. With these, Trebst derives matrix representations of Hamiltonians that describe the interactions between Fibonacci anyons. Later in their paper, they also derive these representations for the fusion of three anyons [7].

To find braids that act as single qubit gates, Bonesteel performed a brute force search of 3-anyon braids with up to 46 exchanges [8]. This yielded braids approximating the target gate to a distance within ≈ 1 − 2 × 1−−3. The distance between two matrices U, V is defined as = ||U − V || where ||O|| is the operator norm of O. The operator norm of O is the square-root of the highest eigenvalue of O†O. According to a theorem by Solovay and Kitaev, given a set of gates generated by infinite braids, which is sufficiently dense in the space of all gates, braids approximating arbitrary single-qubit gates to any required accuracy may be found efficiently (with the length of the braid growing like ∼ | log |c with c ≈ 4). This problem is much more complicated for generating two-qubit gates, but Bonesteel found constructions for controlled rotation gates that just require finding a finite number of 3- anyon braids. These can be systematically improved with Solovay-Kitaev [8].

Bonesteel provides their method as follows: weave the control pair of anyons from the control qubit through the anyons forming the target qubit. Weaving means that the target anyons remain fixed in place while the control pair is moved around them. If the q-spin of the control pair is 0, then this results in the identity. If the q-spin of the control pair is 1, then this results in a transition. Then, weave the control pair through only two target 8 anyons at a time. The only nontrivial case is when the control pair has q-spin of 1, and is equivalent to a single anyon, this reduces the problem of constructing two-qubit gates to the problem of finding a finite number of 3-anyon braids [8].

FIG. 3. (a) An example of weaving with 3 anyons. (b) This shows the implementation of a controlled rotation using 6 anyons, where the bottom two blue lines act as the weaving control pair, and the top two green lines act as the target pair. [8]

In fact, Bonesteel demonstrates the implementation of a controlled-NOT gate in the following ﬁgure:

FIG. 4. In (c), we see the implementation of a CNOT gate for topological quantum computation. [8]

These braids can be used to formalize how braids act on our Hilbert space and construct universal quantum gates [8].

Freedman, Larsen and Wang showed that “the topological modular functor from Witten- ChernSimons theory is universal for quantum computation in the sense a quantum circuit 9 computation can be eﬃciently approximated by an intertwining action of a braid on the functor’s state space” [9]. They demonstrate that such a computational model is polynomi- ally equivalent to the quantum circuit model. Therefore a topological quantum computer has equivalent computational properties to a conventional quantum computer [9]. Note that error does exist in the topological quantum computing model – error may be introduced as thermal ﬂuctuations, which are random stray anyons that interfere with the braiding process. This is thought to be solved (or at least remedied) by increasing the separation distance between anyons [10].

IV. WHERE WE ARE NOW AND WHERE WE ARE HEADED

In April of 2020, Abelian anyons were detected at the Ecole´ normale supérieurein Paris, and researchers published their results in Science. As predicted, they managed to “experi- mentally demonstrate Abelian fractional statistics at filling factor ν = 1/3 by measuring the current correlations resulting from the collsion between anyons at a beam-splitter... [and] we extract ϕ = π/3 in agreement with predictions” [5]. This was achieved with the fractional quantum hall effect implemented by applying a strong B−field to a two dimensional electron gas (here they used Gallium Arsenide and Aluminum Gallium Arsenide), but the full extent of their experiment is beyond the scope of this brief review. Then again in July of 2020, researchers at Purdue University separately announced their detection of Abelian anyons. They used a different approach (involving interferometry) but also looked at the ν = 1/3 fractional quantum hall state. The phase that they generated with anyonic braiding was ϕ = 2π/3 [4]. Inconclusive experimental evidence of non-abelian anyons was announced in October of 2013 [11], but these results are currently contested [12] (it is proposed that these researchers ac- tually just observed Coulomb effects rather than evidence of non-abelian braiding). Moving forward, there is much excitement surrounding the study of the fractional quantum hall effect state of ν = 5/2, as it is widely believed that this is the most promising place to observe non-abelian anyons [13]. If (and when) non-abelian anyons are conclusively observed, the prospect of constructing and implementing a topological quantum computer will 10 be one very large step closer to reality, and quantum supremacy will loom on the horizon.

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