The topological quantum computation has been a Holy Grail of quantum computing. A magical solution to (many) of our noisy problems and yet, to the moment of writing this tutorial it has not yet been found. Previously we briefly discussed what are those topological quantum states and why people are interested with them. We also mentioned few words about crabs and sweet times of playing Heroes of Might and Magic III and Pokémon Silver.

If you have read the previous tutorial you have a brief idea that those topological states are represented by pairings of modes. If state is a configuration of pairings then processing such state should consist of altering the pairings. This is what people refer to as braiding (although it does have a more strict, group-theoretical definition). In 1D, you could braid edges of pairs $\alpha$ and $\beta$ using the braiding operator.

\begin{aligned} U_{\alpha,\beta} = \exp\{\frac{\pi}{4} \gamma_\alpha \gamma_\beta\} \end{aligned}

If you know QuTip and have the Majorana operators from the the previous tutorial implementing this operator becomes really easy

def Uij(L, g1, i, g2, j, start=0, Opers=None):
ga = g1(L, i, start=start, Opers=Opers)
gb = g2(L, j, start=start, Opers=Opers)
return ((np.pi/4.)*ga*gb).expm()


Previously we have achieved a topological phase transition by performing adiabatic changes of parameters of the Kitaev Hamiltonian operator describing one-dimensional topological semiconductor. There also exists a unitary description of such topological phase transition, after all, it is a sequence of braids that makes one of the fermions delocalized on the far edges of the 1DTS.

What we were doing last time consisted of modifying the values of onsite energy and enabling couplings between the sites. The uncoupled sites with onsite energy are regular fermions described with regular fermion operators $a_n$ and $a_n^\dagger$. If we consider a sequence of same-species local braids

\begin{aligned} U_c = \prod_{j=1}^{L-1} U_{\gamma_{(j,l)},\gamma_{(j+1,l)}} \end{aligned}

we would find that applying it to the regular fermions turns them into quasifermions

$U_c a_n U_c^\dagger = \begin{cases} -f_{(n,l)(n+1,l)}, & \text{if}\ n

which implies that same-species braids applied locally are equivalent to the adiabatic manipulations of $\mu$ and $w$ parameters turning $H^{(N)}_j$ into $H^{(T)}_j$ regimes. A more detailed explanation of that can be found in the Section II.E of our paper (Narozniak et al., 2021).

From that we know that such phase transition is a sequence of braids acting on neighboring sites exchanging local Majorana edge modes indicated by the same species. This moves pairing further and further. For the system of length $L=3$ if we start with filled state

\begin{aligned} \left \vert 111 \right>_N &= \prod_{j=1}^{L} a^\dagger_j \left \vert 000 \right>_N \\ &= \left \vert \psi_0 \right> \end{aligned}

and apply the following sequence of local braids

\begin{aligned} \left \vert 111 \right>_T &= U_{\gamma_{(1,l)},\gamma_{(2,l)}} U_{\gamma_{(2,l)},\gamma_{(3,l)}} \left \vert 111 \right>_N \\ &= \left \vert 0_L \right> \end{aligned}

we obtain the highly delocalized pairing on the edges just as last time! We could simulate this effect in Python with QuTip as follows

vacuum = tensor([basis(2, 0) for j in range(L)])

filled = vacuum
for n in range(L):

psi0L = filled
for i, n in enumerate(reversed(range(L-1))):
U = Uij(L, gl, n, gl, n+1, Opers=Opers)
psi = U*psi0L


Just looking at the operators does not provide us with much of intuition, however such topological phase transition implemented using a sequence of local braids becomes more clear once we draw a braiding diagram

and see the final state indeed possesses the Majorana Zero Modes on its edges. Numerically the results also match our previous tutorial.

They might look too perfect, but if you execute the adiabatic code yourself and shorten the adiabatic sweep time you will witness noisiness in the plots which is not there in case of sequence of unitary braids.

The interesting thing here is that we do not really need fermions to perform braids. In the Majorana picture the difference between presence and absense is simply different ordering of modes, which means that we could braid fermions with lacks of fermions or simply with vacuum. Let us try it now by starting with a almost completely filled state, leaving one empty lattice site at the end.

\begin{aligned} \left \vert 110 \right>_N &= \prod_{j=1}^{L-1} a^\dagger_j \left \vert 000 \right>_N \\ &= \left \vert \psi_0 \right>\end{aligned}

We apply exactly same unitary sequence of braids on this almost completely filled state

\begin{aligned} \left \vert 110 \right>_T &= U_{\gamma_{(1,l)},\gamma_{(2,l)}} U_{\gamma_{(2,l)},\gamma_{(3,l)}} \left \vert 110 \right>_N \\ &= \left \vert 1_L \right> \end{aligned}

and we receive another topological regime state of exactly same energy. This is allowed because the ground state of topological regime Hamiltonian is degenerate (remember why we call them zero modes?). We also verify it with Python.

psi1L = a_(L, L-1, Opers=Opers)*filled
for i, n in enumerate(reversed(range(L-1))):
U = Uij(L, gl, n, gl, n+1, Opers=Opers)
psi1L = U*psi1L


Visually braiding the edge of the vacuum looks exactly the same. At the final state the highly delocalized fermion is a lack of fermion.

Numerically exactly opposite pairing order becomes detectable.

In this sense the delocalized fermion could be labeled as logical zero state $\left \vert 0_L \right>$ and delocalized vacuum would then be the logical one state $\left \vert 1_L \right>$. An arbitrary topological logical state would be then

\begin{aligned} \left \vert \psi_L \right> &= \alpha \left \vert 1_L \right> + \beta \left \vert 1_L \right> \\ \left \vert \alpha \right \vert^2 + \left \vert \beta \right \vert^2 &= 1 \end{aligned}

and next time we might define some topological qubits based on those states and also some quantum gates to process them.

Today we have simulated the topological phase transitions by applying sequences of unitary braids on topological states. The full source code of this simulation is published under MIT licence on GitHub. If you find errors please tweet me and let me know.

1. Narozniak, M., Dartiailh, M. C., Dowling, J. P., Shabani, J., & Byrnes, T. (2021). Quantum gates for Majoranas zero modes in topological superconductors in one-dimensional geometry. Phys. Rev. B, 103(20), 205429. https://doi.org/10.1103/PhysRevB.103.205429