Review：Majorana噪声模型

Aasen, D., Hell, M., Mishmash, R. V., Higginbotham, A., Danon, J., Leijnse, M., Jespersen, T. S., Folk, J. A., Marcus, C. M., Flensberg, K., & Alicea, J. (2016). Milestones toward Majorana-based quantum computing. Physical Review X, 6(3), 1–28. https://doi.org/10.1103/PhysRevX.6.031016

Knapp, C., Beverland, M., Pikulin, D. I., & Karzig, T. (2018). Modeling noise and error correction for Majorana-based quantum computing. Quantum, 2. https://doi.org/10.22331/q-2018-09-03-88

1. Stochastic Majorana noise models

Naively: with same for each step. However, such a model will on average make the state spend an equal amount of time in each parity: even and odd. In physical systems, there is a finite gap between two different parity, making the naive model unrealistic.

Introduce the concept of MZM islands.

4 types of errors:

• Quasiparticle event: application of a single MAjorana operator: $|\psi \rangleM \to \gamma{j,a}|\psi \rangle_M$.
• Pair-wise dephasing event: application of a pair of Majorana operators belonging to the same island: $|\psi \rangleM \to \gamma{j,a}\gamma_{j,b}|\psi \rangle_M$.
• Correlated event: application of Majorana operators from multiple islands involved in the same measurement.
• Measurement bit-flip: flipping of the classical bit storing the outcome of a 2k-MZM parity measurement: e.g. .

For a Majorana circuit, the possible noise can be described as follow:

Majorana circuit noise (MC): in a given step for k-island measurement (k = 0 for an idle island), implement the following sequence:

• 1. For each island that begins the time step with odd parity, apply a quasiparticle event with probability .
• 1. For the set of islands involved in the same k-island mea- surement, do the following:
  • (a) For each island in the set, apply one single-island noise event: either a quasiparticle event with $p{qp}^{(k)}p{pair}^{(k)}$.
  • (b) Apply a correlated event to the set: either an odd correlated event with probability $p{cor,odd}^{(k)}p{cor,even}^{(k)}k\leq 1$, do nothing.
• 1. Apply the measurement projector perfectly, then flip the classical bit storing the measurement outcome with probability . For an idle island, do nothing.

MC is defined by the parameter set ${p{odd}^{(k)},p{qp}^{(k)},p{pair}^{(k)},p{cor,odd}^{(k)},p{cor,even}^{(k)},p{mst}^{(k)}}$.

2. Physical System

Take the j-th tetron (a 4-MZM island) and its Hamiltonian can be written as:

$$H = E_C(\hat n_s - n_g)^2 + \sum_k E_k\hat n_{\Delta,k} + \sum_{a \neq b} \delta E_{ab}i\gamma_{j,a}\gamma_{j,b}\tag{2.1}$$

• The first term is the charging energy. is the number of fermions on the island when the island is not grounded (i.e. Coulomb-blockaded).
• The second term is the BCS Hamiltonian written in terms of the quasiparticle number operator $\hat n{\Delta,k}\hat n{\Delta,k}$ counts the above-gap quasiparticles.
• The last term is the MZM hybridization energy. $\delta E{ab}\gamma{j,a}\gamma{j,b}\delta E{a b} \propto e^{-L_{a b} / \xi}$.
• The simple model does not include the overlap of MZMs from different islands.

The eigen states of the Hamiltonian can be written as:

$$\left|n_{s},\left\{n_{\Delta, k}\right\}_{\Delta}\right\rangle\left|i \gamma_{j, 1} \gamma_{j, 2}, i \gamma_{j, 3} \gamma_{j, 4}\right\rangle\tag{2.2}$$

The first part only contains the non-topological part, and the second part denotes the (almost) degenerate MZM space. How we pair up the MZMs (1,2 and 3,4) is chosen randomly.

We can set . The gs is:

$$|g\rangle |\psi_i\rangle_M = |0,\{0\}_\Delta \rangle |\psi_i\rangle_M\tag{2.3}$$

where denotes the odd and even parity.

Fig1: Lowest states for the H when $EC = 2\DeltaE_C = \Delta/2|g\rangle\delta E{ab}|e{C,\pm}\rangle|e\Delta \rangle$.

Two “lowest order” error processes:

• Thermally excited quasi-particles, when a Cooper pair breaks into two electrons, one occupies one of the non-local fermionic states formed by the MZMs while the other occupies a state in the continuum above the superconducting gap, denoted by ,
• Extrinsic quasi-particle poisoning, when a quasiparticle tunnels onto or off of the superconducting island, thereby changing the to- tal fermion parity and charge of the island, denoted by .

[Fig2] This triangle schematically illustrates how the ground state change into two different error states, with the error operation and the corresponding lifetime.

3. A Physical Noise Model

We can model the noise by a Hamiltonian which divide the noise into x and x parts:

$$H(t) = h_z (t) \sigma_z + h_x(t) \sigma_x\tag{3.1}$$

The coupling may have 2 physical contributes:

• Overlap between MZMs across the same island, splitting .
• Residual charging energy, splitting exponentially small in , with a prefactor .

These two mechanisms weakly lifts the topological degeneracy between and .

The coupling arises from the MZMs residing opposite the valve, ans the splitting is exponentially small in . Here is the width of the seperation between islands, and the is the MZM decay length into the barrier.

In this model the noise depends on the parameters and . We notice that there is no local noise sources that can lead to a splitting without exponential suppress.

3.1 Dephasing time

3.2 Relaxation time