Relaxation of individual edge majoranas can occur due to (i) local symmetry-breaking terms, which hybridize majoranas on the same edge, (ii) tunneling through the bulk and hybridizing with majoranas across the other end, or (iii) hybridizing with the bulk. (b) Relaxation of operators O = γ 1, γ 2, γ 1 γ 2 γ N γ N − 1 is plotted. The three draids shown at t = 2 T 0, 4 T 0, … may be combined into a single draid of γ 2, γ 7. (a) n f = 1 scheme for creating Z 2 × Z 2 SPT phase. Here the fastest drive period is T 0 = 0.01, the slowest drive period is 2 6 n f − 1 T 0, and the effective Hamiltonian describes dynamics stroboscopically at period of T F = 2 6 n f T 0. The results exhibit strong suppression of the Hamiltonian as the number of fractal layers is increased. U err is also plotted and denotes the Frobenius-norm error in the unitarity of the time-evolution matrix this error provides an effective lower bound to the relaxation amplitudes defined above as it occurs simply due to numerical inaccuracy in computing the time-evolution matrix at exponentially long times. In particular, we plot S γ ( t ) = max i j 1 − ⟨ γ i ( t ) γ j ( t ) γ j γ i ⟩, which approaches 1 when the (fastest relaxing) majorana pair operator γ i γ j relaxes. We show only the majorana pair, which relaxes the fastest in time for a given realization of the protocol, denoted by the number of fractal layers, n f = 1, 2, 3. (b) Relaxation of coherence of all majorana pairs in time, probed for a system of eight majoranas, and averaged over 100 disorder realizations. Note that the draids at time 4 T, for instance, may be simplified to a single draid between majoranas 1, 4. (a) Protocol for cancellation of a generic Hamiltonian of majoranas. We show that such driving can lead to significantly more decoupled majorana fermions, and be used for the purposes of engineering a variety of Hamiltonians. An appropriate choice of quantum dots, when periodically modulated in time, can realize double braids between arbitrary two majoranas housed in the quantum wires. We propose an architecture of quantum dots coupled to topological quantum wires to realize such periodic double braiding. Significantly, these double braids do not require physically braiding majorana fermions, which is experimentally rather demanding, or making measurements, which is inherently stochastic. Rapid double braiding can utilize this phase to reduce the spatial overlap between majorana fermions that are braided with those that are not. In this work, we note and exploit a peculiar property of majorana fermions that distinguishes them from usual fermions-braiding them twice bestows upon them a robust π phase. Unfortunately, a number of experimental difficulties make the realization of tightly localized, sufficiently isolated majoranas, challenging. When spatially separated, a group of majorana fermions forms a quantum register that is nearly free from decoherence, and which can be simply manipulated by braiding majorana fermions around one another. Majorana fermions are particles that represent an important building block of future quantum computers. We propose an architecture that implements draids between distant majorana modes within a quantum register using a setup with multiple quantum dots and also discuss measurement-based ways of implementing the same. The robustness of this protocol can be shown to parallel the topological robustness of physically braided majoranas. In current experimental setups, this could lead to suppression of this coupling by a few orders of magnitude. For instance, we show that draids can be performed by periodically modulating the coupling between a quantum dot and a topological superconducting wire to dynamically suppress the hybridization of majoranas in the quantum wire. Remarkably, draids can be implemented without having to physically braid majoranas or performing projective measurements. Nontrivial topological models can be obtained by selectively applying draids to some of the overlapping (imperfect) majoranas. Suppressing all couplings can drastically reduce residual majorana dynamics, producing a more robust computational subspace. Rapid draiding can be used to dynamically suppress some or all intermajorana couplings. The protocols rely on double braids- draids-which flip the signs of both majoranas, as one is taken all the way around the other. We propose and analyze a family of periodic braiding protocols in systems with multiple localized Majorana modes ( majoranas) for the purposes of Hamiltonian engineering.
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