Inconsistency in fault tolerant quantum error correction scheme

1
Inconsistency in fault tolerant quantum
error correction scheme!!

• 3 ingredients of fault tolerant quantum error
  correction scheme
• (1) Fast gate operation ultra-short time
• (2) Pure ancilla supplies cold temperature
• (3) Markovian noise
• However, (3) Markov approximation is valid only
  in high temperature, long time

2
Inconsistency in fault tolerant quantum
error correction scheme!!

• So we need to consider non-Markovian noise
• New threshod
• Pessimistic result for physical implementation

3
Toward better and realistic QEC

• (1) Quantum Error Correcting Codes (QECC)
• pro no need for the detail
• con at least 5 qubits to correct 1 error,
  too much resource
• (2) Decoherence Free Subspace (DFS)
• encoding qubits in a safe space to avoid
  decoherence
• pro need to know interation
• con 2 physical qubits at least, normally
  does not exist
• (3) Dynamical Decoupling (DD) Active Error
  Correction
• remove decoherence by external pulses
• protecting qubits from errors before they
  occur
• - can work even for a single qubit
• -work best for non-Markovian noise

4
Dynamical Decoupling of General Environment
(Shiokawa-Hu, Quantum Information Processing
2007)

• Bang-bang control (Viola-Lloyd 1998)
• oscillator bath
• Oscillator based
• qubits
• QBM Hamiltonian

5
(No Transcript)
6
BB with Pure dephasing model

• In the presence of decoupling pulses, at
• To avoid decoherence due to resonance of
  decoupling pulses

7
Our Enemy General Environment

• (1) Ohmic (a1)
• (2) sub-Ohmic (alt1)
• bath has long range time correlations
• (3) super-Ohmic (agt1)
• bath has ultra-short
• range time
• correlation

8
1/f noise in quantum computer architechture

• often attributable to (but not limited to)
  charge fluctuations in electrodes providing
  control voltages
• trapped ions (Turchette 00)
• quantum dots (Burkard 99, Levy 01)
• doped silicon (Kane 98, Vrijen 00)
• electrons on helium (Platzman 99)
• superconducting qubits (Nakamura 02).

9
Successful Decoupling ( I )

• Strong ignore Hs during the pulse
• Instantaneous rotation (switching) of interaction

10

• Leakage ellimination
• We split N level spin operator

11

• Leakage ellimination (II)
• The decoupling operator

12
Successful Decoupling ( II )

• Fast ignore bath Hamiltonian
• necessary and sufficient?

• Yes for Ohmic
• Not sufficient for super-Ohmic
• Not necessary for 1/f noise
• For a generic environment, even at T 0,
• bath characteristic time is not
• and depend sensitively on the bath

13
Problem of pure dephasing

• Pure dephasing model predicts unpleasant
  features
• To avoid decoherence due to resonance of
  decoupling pulses
• (1) However, this is still too fast to implement
• (2) Resonance is not observed in experiments

14
DD in ESBM

• We want to study the model with realistic
  features that agrees with experiments
• Oscillator based qubit model has attractive
  features exactly solvable with arbitrary pulses
  while retaining realistic features such as
  multilevel structure, leakage, etc
• We study DD on the qubit obtained by
• dynamical level reduction (ESBM)

15

• First, we apply decoupling pulses on QBM
• and study fluctuations of oscillator
• variables.
• If the decoupling is successful,
• fluctuations induced from environment will be
  elliminated and recover their intrinsic pure
  state values.

16
(No Transcript)
17
DD effect on ESBMTS and B. L. Hu,
quant-ph/0507177 (2005).

• Plots of the logarithm of the decay factor at
  T0, 1 GHz.
• 100GHz, 0.1GHz, N 100
  for Ohmic,
• 100GHz, 0.5GHz, N 30
  for 1/f,
• 30GHz, 0.01GHz, N 100
  for super-Ohmic case.

18
DD on ESBM predicts correct (nonresonant)
behavior
nonresonant
resonant -vanishng coherence
Pure dephasing
Our model from QBM

• (a) Coherence at t 0.25ns (b) 1st excited
  state population at t 0.15ns
• T0, 1 GHz, 100GHz,
• 0.125GHz (Ohmic), 0.2GHz
  (1/f), 0.005GHz (super-Ohmic).

19
We studied dynamical decouplin gin general
environment

• Pulse parameter p
• (1) Ohmic environement p0.3 lt 1
• fast - difficult to implement
• (2) super-Ohmic environement p0.1 ltlt 1
• ultra-fast - impossible!!
• (3) 1/f noise
  p0.8 1
• slow - easy to implement

20
Strong continuous field QZE

• For strong and constant pulse limit,
• Eigenstate of the total Hamiltonian H
• and the total evolution
• Upto O(1), environment is decoupled from the
  system
• - Similar mechanism to quantum Zeno effect

21
(No Transcript)
22
Summary

• Proposed new DD scheme for physical qubits
• obtained by taking lower two levels of
  oscillator potential
• DD works well for environment with long range
  correlation
• slow pulses are enough to eliminate 1/f
  noise.
• Ideal methods to deal with non-Markovian
  noise
• Qubit decoherence based on ESBM predicts correct
  non-
• resonant behavior, agrees with experiments
• DD does not work for super-Ohmic environment.
• Pure state UCR can be recovered by our DD scheme
• Decay can be suppressed in the strong continous
  field.
• This case is similar to the frequent
  measurement due to QZE and the suppression has a
  different physical origin.