This document discusses quantum error correction for beginners. It introduces key concepts like the 4-qubit error detecting code, stabilizer formalism, and the 7-qubit Steane code for error correction. It also covers how to digitize quantum errors using ancilla qubits and parity measurements to determine the error syndrome. Environmental noise is modeled using the Lindblad formalism and error correction circuits are presented.

1. Quantum Error Correction
for Beginners(2)
section 1-9
2020/04/24 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

3. Paper
3
Xavier
Shin (1)
Shin (2)

4. 6. Quantum Error Detection
4
• Post-selected quantum computation
• encode ancilla qubits with error detecting circuits
• faster & requires fewer qubits than active QEC
• the 4-qubit code
• most simple QED
• encoding 2 qubits state into 4 qubit
• 2 qubits ancilla
• no information about where the error occured
• measurement ancilla qubits
• bit flip in the 4 qubit → ⟩|10
• phase flip in the 4 qubit → ⟩|01

5. 7. Stabilizer formalism
5
• The quantum state can be described by using operators 𝑂
and eigenvalue 𝜆.
• 𝑂, 𝜆 = 𝑋, 1 ∶
⟩|# $ ⟩|%
&
• 𝑂, 𝜆 = 𝑌, −1 ∶
⟩|# '( ⟩|%
&
• 𝑂, 𝜆 = 𝑍% 𝑍& + 𝑋% 𝑋&, 2 ∶
⟩|## $ ⟩|%%
&
𝑂, 𝜆 = 𝑍% + 𝑍&, 0 : ⟩|01 or ⟩|10
𝑂, 𝜆 = 𝑍% + 𝑍&, +𝑍% 𝑍&, 3 = 𝑍% + 𝑍&, 2 : ⟩|00

6. 6
Stabilizer formalism
• Introduced by Daniel Gottesman[2]
• The quantum state can be described by using operators 𝑂
and eigenvalue 𝜆.
• Let {𝑆𝑖} be 𝑛 chosen operators from the Pauli group.
• the Pauli group 𝒫 = ±1, ±𝑖 × 𝐼, 𝑋, 𝑌, 𝑍 ⊗*
• 𝑂 = ∑(+%
*
𝑆(
• Use 𝑆( with maximum eigenvalue as Quantum State
[2] Daniel Gottesman, Stabilizer Codes and Quantum Error Correction, arXiv:quant-ph/9705052 (1997)
GHZ state
⟩| 𝐺𝐻𝑍 =
⟩|000 + ⟩|111
2
𝑆! = 𝐾"
+ 𝐾#
+ 𝐾$
𝐾"
= 𝜎% ⊗ 𝜎% ⊗ 𝜎% = XXX
𝐾#
= 𝜎& ⊗ 𝜎& ⊗ 𝜎' = ZZI
𝐾$
= 𝜎' ⊗ 𝜎& ⊗ 𝜎( = IZZ

7. 7
• Stabilizer state ⟩|Ψ
𝐾 ⟩|Ψ = ⟩|Ψ for ∀𝐾 ∈ 𝑆(stabilizer operator)
e.g. σ, ⟩|0 = ⟩|0 : ⟩|0 is stabilized by the operator σ,
• Stabilizer group
for 𝑎 ∈ 𝐴, the set of element of G which keep 𝑎 unchanged
𝑆- 𝑎 = 𝑔 ∈ 𝐺 𝑔𝑎 = 𝑎}
example of stabilizer state
• 𝐾 = 𝑋𝑋, 𝑍𝑍 Bell state
⟩|## $ ⟩|%%
&
• 𝐾 = 𝑍𝑍𝐼, 𝐼𝑍𝑍, 𝑋𝑋𝑋 GHZ state
⟩|### $ ⟩|%%%
&

8. 8. QEC with stabilizer code
8
• The 7-qubit Steane code[3]
• 𝑛, 𝑘, 𝑑 = [[7,1,3]], 𝑡 =
.'%
&
= 1
[3] Steane, Andrew M. "Error correcting codes in quantum theory." Physical Review Letters 77.5 (1996): 793.
Physical Qubits
Logical qubits
distance
Encoding
fix the encoded data into two codewords
̅𝑍 = 𝑍𝑍𝑍𝑍𝑍𝑍𝑍 = 𝑍⊗"
where ̅𝑍 ⟩|0 # = ⟩|0 #, ̅𝑍 ⟩|1 # = ⟩−|1 #

9. 9
A. State preparation
circuit for prepare the [[7,1,3]]
logical state ⟩|0 #
parity measurement(operator measurement)
⟩| 𝜓 $ =
⟩| 𝜓 % + 𝑈 ⟩| 𝜓 % ⟩|0 + ⟩| 𝜓 % − 𝑈 ⟩| 𝜓 % ⟩|1
2
anc ⟩|0
⟩| 𝜓 $ = ⟩| 𝜓 % + 𝑈 ⟩| 𝜓 %
anc ⟩|1
⟩| 𝜓 $ = ⟩| 𝜓 % − 𝑈 ⟩| 𝜓 %This image is from [1]
This image is from [1]

10. 10
B. Error Correction
This image is from [1]

11. 9. Digitization of quantum errors
11
Digitization of quantum noise
→ examine the stability of QIP, calculate thresholds for QEC
A. Systematic gate errors
assumption:
• N-qubit unitary operator 𝑈9 is applied inaccurately
• resultant operation: 𝒰9
𝒰) ⟩| 𝜓 * = 𝑈+ 𝑈) ⟩| 𝜓 * = /
,
𝛼, 𝐸, 23𝜓# *
coherent error op perfectly applied
where 𝐸& ∈ 𝒫'
[QEC] append ancilla blocks =>𝐴(
)
, =>𝐴(
*
for {𝑋, 𝑍} correction

12. 12
↓measure ancilla
data blocks: 𝐸& =>𝜓+
#
with 𝛼&
,
correction: 𝐸&
-

13. 13
B. Environmental decoherence
The Lindblad formalism: Environmental effects
Hamiltonian
coherent, dynamical incoherent
assumption
• not undergoing any coherent evolution
Solve 𝜕.
1. Dephasing
𝐿/ = 𝑍
2. Spontaneous emission/absorption
(same rate)
𝐿, = ⟩|0 ⟨ |1
𝐿0 = ⟩|1 ⟨ |0
𝜌 𝑡 = 1 − 𝑝 𝑡 𝜌 0 + 𝑝1 𝑡 𝑋𝜌 0 𝑋 + 𝑝2 𝑡 𝑌𝜌 0 𝑌 + 𝑝3 𝑡 𝑍𝜌 0 𝑍

14. 14
𝜌 𝑡 = 1 − 𝑝 𝑡 𝜌 0 + 𝑝% 𝑡 𝑋𝜌 0 𝑋 + 𝑝- 𝑡 𝑌𝜌 0 𝑌 + 𝑝& 𝑡 𝑍𝜌 0 𝑍
Encoded data block
no error
Probabilities(after measure anc)
• no error: 1 − 𝑝(𝑡)
• a single X,Y or Z error: 𝑝L 𝑡 , 𝑝M 𝑡 , 𝑝,(𝑡)
where

15. 15
Conclusion
• Stabilizer formalism
• represent Quantum state with fewer parameter
• easy to determine the operations
• Parity measurement
• project an arbitrary input state to a ±1 eigenstate of U
• Digitization of Quantum Error
• correction with simple 𝑍, 𝑋 gate is useful for correct
continuous errors