Analytical Profile of Coleus Forskohlii | Forskolin .pptx
[01] Quantum Error Correction for Beginners
1. Quantum Error Correction
for Beginners
section 1-9
2020/04/17 WebEx
National Institute of Informatics
Shin Nishio
2. Paper
[1] Simon J. Devitt, Kae Nemoto, William J. Munro
Quantum Error Correction for Beginners
(Submitted on 18 May 2009, last revised 21 Jun 2013 (this version, v4))
arXiv[quant-ph] 0905.2794
Hands-on: implementation based on [1], written in Qiskit by Shin
https://github.com/parton-quark/QEC-for-Beginners
Abstract
QEC and fault-tolerant quantum computation
• Theoretical aspect of QIS
• Significant development since 1995
• An introduction for researchers other than QEC
2
9. 9
A. Coherent quantum errors
B. Environmental decoherence
C. loss, leakage measurement and initialization
e.g. Rotation around the X-axis
| ⟩φ !"#$% = %
&
'
𝑒&∈)*| ⟩0 = cos(𝑁𝜖) | ⟩0 + 𝑖 sin(𝑁𝜖) | ⟩1
Undesired gate op
10. 10
A. Coherent quantum errors
B. Environmental decoherence
C. loss, leakage measurement and initialization
e.g. Rotation around the X-axis
| ⟩φ !"#$% = %
&
'
𝑒&∈)*| ⟩0 = cos(𝑁𝜖) | ⟩0 + 𝑖 sin(𝑁𝜖) | ⟩1
Undesired gate op
Probability
𝑃 | ⟩0 = cos!
(𝑁𝜖) ≈ 1 − 𝑁𝜖 !
𝑃 | ⟩1 = sin!
(𝑁𝜖) ≈ 𝑁𝜖 !
𝑃"##$# ≈ 𝑁𝜖 !
11. 11
A. Coherent quantum errors
B. Environmental decoherence
C. loss, leakage measurement and initialization
Assumption (For simplicity)
• two level quantum system ! ⟩𝑒4 , | ⟩𝑒5
• Environment is initialized in the state | ⟩𝐸 = | ⟩𝑒4
• Environment couples to the qubit during the wait
• controlled flip when | ⟩1
• nothing when | ⟩0
Second algorithm
14. 14
A. Coherent quantum errors
B. Environmental decoherence
C. loss, leakage measurement and initialization
Qubit loss: tracing out
𝑇𝑟&(𝜌)
where 𝑖 is the index of the lost qubit
Initialization
• incoherent: initial state
𝜌& = 1 − 𝑝- ⟩|0 ⟨ |0 + 𝑝- ⟩|1 ⟨ |1
where 𝑝- is the probability of initialization
• coherent
⟩| 𝜑 = 𝛼 ⟩|0 + 𝛽 ⟩|1
where 𝑎 ) + 𝛽 ) = 1 and 𝛽 ) ≪ 1
15. 4. The three-qubit code
15
⟩| 𝜑
⟩|0
⟩|0
3-qubit code
• does not represent full quantum code
• cannot simultaneously correct for both
bit and phase flips
• encodes a single logical qubit into three
physical qubits
• # of errors that can be corrected: t
𝑡 =
𝑑 − 1
2
18. 18
• no error correction scheme will, in general, fully restore
a corrupted state to the original logical state.
• The 3-qubit code can correct one bit-flip error
• others are relaxing these assumptions
19. 5. The nine-qubit code
19
9-qubit code by Shor
• uses 3-qubit code
• can correct
• 1 bit-flip & 1 phase-flip
• one of each
→sufficient to correct for an arbitrary single qubit error
} on one of the nine qubits
• bit-flip and phase-flip on the same qubit
→ Independently corrected
21. 21
Correction
• X errors(for each 3-qubit block): same as 3-qubit code
• Z errors
1. Change the basis
1st block
2nd block
3rd block
4. Change the basis
2. Check the parity(1st & 2nd ) 3. Check the parity(2nd & 3rd )
22. 22
Conclusion
• Quantum Errors
• coherent, environmental decoherence, loss, and so on
• the 3-qubit code
• only for bit-flip, uses syndrome
• the 9-qubit code
• combination of Z-error correction & X-error correction