DECOHERENCE PATTERNS OF TOPOLOGICAL QUBITS

DECOHERENCE PATTERNS OF TOPOLOGICAL QUBITS

Feng-Li Lin (National Taiwan Normal Univ)

Based on 1603.05136 (JHEP) with Pei-Hua Liu(NTNU)

& 1406.6249 (NJP) with Shi-Hao Ho(NCTS), Sung-Po Chao(NCTS) & Chung-Hsien Chou(NCKU)

Talk @ NTU for Chiral Matter & Topology--- 2018/12/06

Outline

1. Motivation

2. Majorana zero modes & Topological qubit

3. Reduced dynamics

4. Robust topological qubits

5. Accelerating topological qubit

6. Conclusion

I. Motivation:Topological Order in Open

System

Motivation

nMost physical qubits are not robust against quantum decoherence, i.e., leaking its quantum information into the environment.

nThe robust qubits play key roles for reliable quantum computations. Looking for robust qubit is a urgent task.

nThe fundamental way of making robust qubit not by fine tuning is to implement it based on physical principle.

nOne way is to make the qubit topological, i.e., protected by the topological order.

II. Majorana zero mode & Topological qubit

Majorana zero modes (MZMs)

nThe Dirac fermion localizing on a topological defect (such as monopole, string or domain wall) results in MZMs. See Jackiw-Rebbi or Jackiw-Rebbi.

nIn some sense, a MZM is half of the Dirac fermion, and squares itself to unity. Two different MZMs anti-commute with each other.

Kitaev’s chain: 1D superconductor

nThe Z_2 1D p-wave superconductor is a model of TSc.

nThis can be seen by introducing the fractional Majorana fields

nThe edge Majorana modes are robust and protected by Z_2.

nTwo phases: one has no dangling Majorana modes, one does.

Topological Qubit

•

We can pair up two spatially separated Majorana zero modes to form a qubit. We call it topological qubit.

•

However, this is not really a qubit due to Z2 parity:

•

Thus, we cannot form a state like as

long as Z2 parity is preserved.

Way out

nOne way-out is to use 4 MZMs to form one topological qubit by just restricting to the parity even states, e.g., a logical qubit as

nAnother way-out is to couple the MZMs to the environment so that the MZMs are now open system, which can no

longer preserve Z2 parity.

nThe 2^nd way is what we will consider in this talk. So, 2 MZMs make one topological qubit.

Fermionic or Bosonic environments

nIn this work, we consider two ways of coupling the Kitaev’s chains to the environments:

1. The fermionic channel:

These interactions breaks the parity symmetry.

2. The bosonic channel:

These interactions preserve the parity symmetry. We will see this will cause different decoherence patterns from the fermionic ones.

MZMs as Open System

•

Hamiltonian:

•

Environment operator, e.g.

•

They are assumed to Ohmic-type:

•

They obey the locality constraint:

O^a = _{a a}^† O^ab = _{a b}^†

III. Reduced dynamics

RDM in Interaction picture

⇢₀ = ⇢_M,0 ⌦ |0i^eh0|

• In general, it is difficult to obtain the closed form of RDM.

• It usually needs approximation. A common one is the Markov approximation which leads to Lindblad master equation for RDM.

Derive exact RDM for MZMs (1)

•

Use Clifford algebra of MZMs, we get

•

Note

bosonic time ordering for O operators!!

Derive exact RDM for MZMs (2)

•

Use the above and the locality constraint, we get

•

No odd parity terms like

•

These correlators are bosonic under time

ordering, and the closed form can be obtained by

Wick contraction.

Exact RDM for 2 MZMs

c.f.

IV. Robust topological qubits

Characterizing decoherence

nFor qubit systems, the most unambiguous way to

characterize quantum decoherence is to check if the reduced density matrix becomes a pointer state or not.

nA particular pointer state is the Gibbs state, i..e, thermalization.

Purity and Concurrence

nIf the state of the qubits does not reduce to a pointer state, it means the decoherence is incomplete, and could be

further purified. Then we can characterize the purity by

nFor two-qubit state, we can characterize the quantum entanglement between the two qubits by concurrence:

nZero concurrence implies no entanglement, but the state could be still quantum.

are the square roots of eigenvalues, in decreasing order, of

Single qubit: Memory in strongly correlated environment

For initial state described by Recall that

Fermionic environments:

Bosonic environments:

Diagonal elements in bosonic environments protected by (fermion) parity so that the qubit state decohere completely but does not thermalize for sub-Ohmic environment. This is not case for the fermionic one.

Special features

nThere is no retarded Green function appearing in the final form of the reduced dynamics. This is related to the fact that the Majorana modes are dissipationless, i.e.,

generating no heat.

nThe symmetric Green function appearing above is the

Majorana-dressed one as discussed. It control the overall time dependence.

nTurn out that this time factor for Ohmic-like spectrum has a closed form, and has a critical point at Q=1.

Time dependence factor --- critical at Q=1

Define the single qubit by With this we may choose as the single qubit operators. The time dependence of reduced density matrix is determined by

Fermionic:

Bosonic:

Here

Effective gap-ness

n The quantum information of the probe is carried away by the collective excitations of the environment, which is

specified by the spectral density.

nThe Ohmic-like spectrum has no gap at low energy, and one would expect the complete decoherence.

nHowever, the super-Ohmic spectrum suppress more the low energy modes than the higher energy ones.

nAdding the topological nature of the Majorana modes, we see an effective gap emerging for super-Ohmic cases.

Two Qubits: Special case for uniform environments

Before studying the reduced dynamics for more general initial states:

Let us first consider a simple case: choose the initial state as , i.e.,

Fermionic:

Bosonic:

Again, diagonal elements in bosonic environments protected by (fermion) parity.

Two qubits in fermionic environments I

Two qubits in bosonic environments II

The red lines all turn into the pointer states but the blue lines do not.

Parity-violating bosonic environments

n In the above, we mainly consider the parity- preserving bosonic environments, i.e.,

n For the parity-violating ones, i.e.,

for sub-Ohmic environment

Only for particular set of initial states, it will be thermalized.

C.f. Spin-Boson model

ref. S.T. Wu PRA89p034301

V. Accelerating topological qubit

MZMs in motion

•

We can generalize the above to the MZMs in motion, either boost or in acceleration.

Frame of MZMs (M-frame) vs Frame of Environment (E-frame)

Local frame change

•

The overall RDM formalism is the same as for the static case.

•

Only thing to take care is the change of the local time frame according to the relative motions.

Given local times for MZMs

Influence functionals in local frames

•

Influence functional in M-frame:

•

Influence functional in E-frame:

Worldline of constant a!

Transition Rate

• Treating the MZMs as kind of Unruh-DeWit detector, then the transition amplitude is

• The full transition probability is

• In the current setup, it is

P_i!f := X

m

|A^(m)_i!f|² = lim

t!1hf|X

m

hm|U(t)|0i|iihi|h0|U^†(t)|mi|fi = lim

t!1hf|Tr^E⇢^D(t)|fi .

P_0!1 = lim

t!1

1

2(1 e^I1(t)+I2(t))

Decoherence à zero transition!

Thermalization

• We find that even for robust topological qubit will be thermalized due to Unruh effect.

M

T E

E M

T

Boosted MZMs

Overtaking/Decoherence inertial impedance

Anti-Unruh phenomenon (1)

Anti-Unruh phenomenon (2)

Decoherence inertial impedance

•

From the above, we see a highly nontrivial non- equilibrium phenomenon, we call it decoherence inertial impedance.

•

There is initial resistance for the system to

against the change caused by the external forces such as acceleration.

•

That is, the large acceleration cause less decoherence.

•

The anti-Unruh and decoherence inertial

impedance/overtaking imply each other.

Information backflow: coupling modulation

Information backflow:

Incoherent motions

Different

Initial L’s Different a2’s

Circular motions

• We see the similar overtaking, anti-Unruh and information back flow for the circular motions of MZMs.

overtaking backflow

VI. Conclusions

Conlusions

• Our works are the first systematic study of the topological order in open system.

• This is an interesting interplay between topological order and (relativistic) quantum information.

• By the locality constraint, the reduced dynamics can be solved exactly.

• We find the robust topological qubits in the super-Ohmic environment.

• By setting MZMs in motion, we find the universal thermalization, anti-Unruh, decoherence inertial impedance and information backflow.