Scalable quantum control and non-Abelian anyon creation in the Kitaev honeycomb model

The Kitaev honeycomb model is a system allowing for experimentally realizable quantum computation with topological protection of quantum information. Practical implementation of quantum information processing typically relies on adiabatic, i.e., slow dynamics. Here we show that the restriction to adiabatic dynamics can be overcome with optimal control theory, enabled by an extension of the fermionization of the Kitaev honeycomb model to the time-dependent case. Moreover, we present a quantum control method that is applicable to large lattice models due to subexponential scaling.

Figure 1

A honeycomb lattice showing the three kinds of interactions between neighboring spins which are on the vertices of hexagonal plaquettes. Red, blue, and green links indicate ${\sigma }^x{\sigma }^x$, ${\sigma }^y{\sigma }^y$, and ${\sigma }^z{\sigma }^z$ interactions, respectively. The hexagonal plaquette operators correspond to many conserved quantities in the system. A generic plaquette whose spin sites are numbered 1 to 6 is highlighted in gray.

Figure 2

Logarithmic comparison of infidelity between initial and adiabatic target states for linear control pulses and optimized control pulses, at various timescales, for the simplest six-qubit lattice. 100 time steps were used in both cases. We can see more clearly the dramatic improvement in fidelity at approximately $T=0.8$. The minimum infidelity reached by optimized pulses, on the order of ${10}^{-9}$ is reached at time $T\approx 0.9$, many orders of magnitude less than the time to reach this infidelity with the linear ramp, which is at $T\approx 1350$.

Figure 3

Six-qubit lattice: Heisenberg infidelities based on differing targets (circles) are compared with their corresponding state fidelities (squares). These are compared overall with state fidelities achieved from linear ramp pulses (triangles). Different colors represent different values of $T_{\text{ad}}$, which are the timescales of 100 (red, e.g., upper squares), 200 (orange, middle squares), and 300 (green, lower squares) that define the target unitary $W_{\text{ad}}$. All optimized pulses are comprised of 200 time steps. There is a significant improvement at times between $T=2$ and $T=4$. Thereafter, there is effectively perfect fidelity with the presence of numerical noise. It confirms that both Heisenberg and state infidelities improve markedly at the same time $T$ and this is an improvement on the fidelities achieved with a linear ramp pulse.

Figure 4

Ten-qubit lattice: Heisenberg infidelities between optimized unitaries and an optimized target are shown (orange), as compared with infidelities between the target and a unitary defined by a linear ramp control (blue). Each optimized pulse consists of 200 time steps. The drop time is $T\approx 10$, later than for the six-qubit lattice.

Figure 5

Thirty-qubit lattice with open boundary conditions and pulses with 400 time steps. Heisenberg infidelities between optimized unitaries and an optimized target are shown. The spectral gap condition requires that $T_{\text{ad}}\gg 30$ calculated analogously to the six- and ten-qubit lattices. The results show that $T_d$ is within the order of magnitude of 30 rather than many orders of magnitude larger as required by adiabaticity, highlighting the improvement in timescale achieved.

Figure 6

Drop time (above) and minimum required time steps (below) compared with lattice size by number of qubits. These quantities are defined as the earliest time $T$ and minimum number of time steps in an optimized piecewise-constant pulse, respectively, where Heisenberg infidelity drops below ${10}^{-6}$. They are compared with quadratic growth curves (blue) to show subexponential growth.

Figure 7

Simple sets of $y$ (above) and $z$ (below) link flips that preserve the vortex sector while being gauge inequivalent to the trivial link sector.

Figure 8

Topological interpretation of the set of $y$ and $z$ flips that preserve vortex sector but are gauge inequivalent to the trivial link sector which topologically amounts to no loop at all.

Figure 9

Clearly $[Z_6,W_p]=[Z_6,Z_1{X}_2{Y}_3{Z}_4{X}_5{Y}_6]\ne 0$ but $[Z_6{Z}_2,W_p]=[Z_6{Z}_2,Z_1{X}_2{Y}_3{Z}_4{X}_5{Y}_6]=0$.

Figure 10

A lattice where the gray spin sites are acted on with $D_j$ operators and link eigenvalues on yellow links are flipped. It is not possible to flip only a single line of $z$ links using ${\mathrm{\Gamma }}_k$ operators alone.