Adiabatic braiding of Majorana zero modes can be used for topologically protected quantum information processing. While extremely robust to many environmental perturbations, these systems are vulnerable to noise with high-frequency components. Ironically, slower processes needed for adiabaticity allow more noise-induced excitations to accumulate, resulting in an antiadiabatic behavior that limits the precision of Majorana gates if some noise is present. In a recent publication (2017 Phys. Rev. B 96 075158), fast optimal protocols were proposed as a shortcut for implementing the same unitary operation as the adiabatic braiding in a low-energy effective model. These shortcuts sacrifice topological protection in the absence of noise but provide performance gains and remarkable robustness to noise due to the shorter evolution time. Nevertheless, gates optimized for vanishing noise are suboptimal in the presence of noise. If we know the noise strength beforehand, can we design protocols optimized for the existing unavoidable noise, which will effectively correct the noise-induced errors? We address this question in the present paper, focusing on the same low-energy effective model. We find such optimal protocols using simulated-annealing Monte Carlo simulations. The numerically found pulse shapes, which we fully characterize, are in agreement with Pontryagin's minimum principle, which allows us to arbitrarily improve the approximate numerical results (due to discretization and imperfect convergence) and obtain numerically exact optimal protocols. The protocols are bang–bang (sequence of sudden quenches) for vanishing noise, but contain continuous segments in the presence of multiplicative noise due to the nonlinearity of the master equation governing the evolution. We find that such noise-optimized protocols completely eliminate the above-mentioned antiadiabatic behavior. The final error corresponding to these optimal protocols monotonically decreases with the total time (in three different regimes). A liner fit to 1/ τ indicates extrapolation of the cost function to finite value in the limit. However, quadratic and cubic fits are more suggestive of the cost function extrapolating to zero in the limit of infinite time. Our results set the precision limit of the device as a function of the noise strength and total time.
Adiabatic braiding of Majorana zero modes can be used for topologically protected quantum information processing. While extremely robust to many environmental perturbations, these systems are vulnerable to noise with high-frequency components. Ironically, slower processes needed for adiabaticity allow more noise-induced excitations to accumulate, resulting in an antiadiabatic behavior that limits the precision of Majorana gates if some noise is present. In a recent publication (2017 Phys. Rev. B 96 075158), fast optimal protocols were proposed as a shortcut for implementing the same unitary operation as the adiabatic braiding in a low-energy effective model. These shortcuts sacrifice topological protection in the absence of noise but provide performance gains and remarkable robustness to noise due to the shorter evolution time. Nevertheless, gates optimized for vanishing noise are suboptimal in the presence of noise. If we know the noise strength beforehand, can we design protocols optimized for the existing unavoidable noise, which will effectively correct the noise-induced errors? We address this question in the present paper, focusing on the same low-energy effective model. We find such optimal protocols using simulated-annealing Monte Carlo simulations. The numerically found pulse shapes, which we fully characterize, are in agreement with Pontryagin's minimum principle, which allows us to arbitrarily improve the approximate numerical results (due to discretization and imperfect convergence) and obtain numerically exact optimal protocols. The protocols are bang–bang (sequence of sudden quenches) for vanishing noise, but contain continuous segments in the presence of multiplicative noise due to the nonlinearity of the master equation governing the evolution. We find that such noise-optimized protocols completely eliminate the above-mentioned antiadiabatic behavior. The final error corresponding to these optimal protocols monotonically decreases with the total time (in three different regimes). A liner fit to 1/ τ indicates extrapolation of the cost function to finite value in the limit. However, quadratic and cubic fits are more suggestive of the cost function extrapolating to zero in the limit of infinite time. Our results set the precision limit of the device as a function of the noise strength and total time.
In topological quantum computing, unitary operations on qubits are performed by adiabatic braiding of non-Abelian quasiparticles, such as Majorana zero modes, and are protected from local environmental perturbations. In the adiabatic regime, with timescales set by the inverse gap of the system, the errors can be made arbitrarily small by performing the process more slowly. To enhance the performance of quantum information processing with Majorana zero modes, we apply the theory of optimal control to the diabatic dynamics of Majorana-based qubits. While we sacrifice complete topological protection, we impose constraints on the optimal protocol to take advantage of the nonlocal nature of topological information and increase the robustness of our gates. By using the Pontryagin's maximum principle, we show that robust equivalent gates to perfect adiabatic braiding can be implemented in finite times through optimal pulses. In our implementation, modifications to the device Hamiltonian are avoided. Focusing on thermally isolated systems, we study the effects of calibration errors and external white and 1/f (pink) noise on Majorana-based gates. While a noise-induced antiadiabatic behavior, where a slower process creates more diabatic excitations, prohibits indefinite enhancement of the robustness of the adiabatic scheme, our fast optimal protocols exhibit remarkable stability to noise and have the potential to significantly enhance the practical performance of Majorana-based information processing.
In topological quantum computing, unitary operations on qubits are performed by adiabatic braiding of nonAbelian quasiparticles and are protected from local environmental perturbations. Due to the condition of adiabaticity, a perfectly accurate topological quantum computer appears to be an infinitely slow one. By using the Pontryagin’s maximum principle, here we show that such topologically protected quantum gates can be implemented exactly in finite times through optimal diabatic pulses. In our implementation, modifications to the device Hamiltonian are avoided. We study the effects of calibration errors and external noise and show that our optimally fast pulse shapes are also remarkably robust. Using Monte Carlo simulations, we furthermore find more robust, continuous protocols with similar performance.
Majorana zero modes have emerged as one of the most promising platforms for topological quantum computation, since their non-Abelian braiding statistics allow quantum information to be encoded nonlocally and manipulated through braiding operations that are, in principle, protected against local perturbations. In practice, however, a braid is only as robust as its physical implementation: finite-time operation, residual couplings, and environmental noise can all convert local excitations into logical errors during the exchange process. Here, we address this question through a microscopic comparison of two representative braiding architectures, a nanowire trijunction and a quantum-dot-assisted setup, simulating the full time-dependent Bogoliubov--de Gennes dynamics under both noiseless and noisy conditions. We show that the dot-assisted architecture consistently achieves a lower error over a shorter timescale than the trijunction, owing to its more localized exchange mechanism. This advantage persists in the presence of noise, and a spatially resolved analysis further reveals that, in the dot-assisted geometry, fast noise localized on the dot produces a smaller error than equivalent noise on the wires, whereas slow, quasi-static noise on the dot becomes the dominant limitation. Taken together, these findings link the different error contributions directly to device geometry, pointing to concrete design principles for noise-resilient Majorana-based quantum gates.
Majorana zero modes (MZMs) appear as edge states in topological superconductors and are envisioned as promising candidates for application towards fault-tolerant quantum computation. Quantum information may be stored in a degenerate ground state subspace constructed from MZMs while quantum gates may be performed by braiding MZMs and exploiting their non-Abelian properties. While the topological protection of MZMs ensures that they are robust against decoherence, their dynamical manipulation can nonetheless induce undesirable transitions to excited states, manifesting as a “diabatic error.”This thesis studies the diabatic error accumulated as MZMs are transported across a superconducting wire, which is an essential building block for physical braiding schemes. The transport protocol considered relies on appropriately tuning sections (“piano keys”) of a wire between distinct topological phases via electric gates to control local chemical potentials. First, the diabatic error is examined for transport which uses multiple piano keys. The error is calculated through numerical simulation of the transport protocol and is complemented by approximate analytical expressions calculated using an effective, low-energy model. It is found that an optimal number of keys can be used to reduce the error. Furthermore, this thesis explores transport protocols performed on wires which feature disorder on the chemical potential and noise in the piano key tuning. The rise in the diabatic error due to these additional effects with respect to parameters-of-interest including the disorder strength, disorder correlation length, noise amplitude, and noise frequency cut-offs is characterized using numerical methods. For disorder, it is found that the error’s behaviour is principally controlled by the statistics of the minimum bulk energy gap. Spatially correlated disorder is also studied which presents non-monotonic behaviour for the error. Finally, for noise, it is found that optical transitions can cause a substantial increase in the error as the frequency modes within the noise approach the minimum energy gap.
Braiding Majorana zero modes does not by itself provide a universal set of quantum gates. Additional operations can be supplied through ancillary states, whose preparation need not have the same protection as braiding. Here, we consider purification of the non-Gaussian ancilla used for an entangling gate in Bravyi's scheme. We keep his elementary braid-and-fusion comparison and change how its outputs are combined. An additional layer compares the product of the three stabilizer signs. The first accepted errors giving an incorrect output then require three faulty input blocks rather than two. As a result, the preparation error decreases cubically instead of quadratically, and the asymptotic raw-copy overhead is reduced. We also construct a preliminary sequence in which a processed block is paired with a fresh copy from the same noisy source. For independent inputs supported in the even-parity sector and ideal purification operations, this sequence establishes a sufficient distillability fidelity of 59.95 percent after free Gaussian twirling. At a common source fidelity, it reduces the expected cost of preparing an input for the recurrence by more than seven orders of magnitude relative to the balanced construction studied here. Finally, we examine imperfect purification operations in an effective syndrome-noise model. The additional layer leaves the leading operation-error coefficient unchanged, although a nonzero error floor remains.
Braiding Majorana zero-modes around each other is a promising route towards topological quantum computing. Yet, two competing maximes emerge when implementing Majorana braiding in real systems: On the one hand, perfect braiding should be conducted adiabatically slowly to avoid non-topological errors. On the other hand, braiding must be conducted fast such that unavoidable decoherence effects introduced by the environment are negligible. This competition results in an intermediate time scale for Majorana braiding that is optimal, but generally not error-free. Here, we calculate this intermediate time scale for a T-junction of short one-dimensional topological superconductors coupled to a bosonic bath that generates fluctuations in the local electric potential, which stem from, e.g., environmental photons or phonons of the substrate. We thereby obtain boundaries for the speed of Majorana braiding with a predetermined gate fidelity. Our results emphasize the general susceptibility of Majorana-based information storage in finite-size systems and can serve as a guide for determining the optimal braiding times in future experiments.