[Quantum Information] Quantum Error Correction (QEC)

Quantum Error Correction (QEC)

Notes from RWTH Aachen University course 
“Quantum Information” Summer semester 2020
professor: Müller, Markus

Quantum Error Correction (QEC)

• Problem: the quantum computer cannot be isolated from the environment, therefore, operations can be faulty.
  This leads to errors in quantum gates, initial state preparation, measurements.

• classical: redundancy (by copying)

  • but in quantum cases, by measuring $\alpha|0\rangle+\beta|1\rangle$ will lose the information about $\alpha$ and $\beta$
  • even worse for entangled state $\alpha|000\rangle+\beta|111\rangle$

• Quantum challenges:

  • not only bit flip error, but also phase flip or both
  • errors continuous $U(t)=e^{-iHt}$

• Effective error model: Stochastic independently distributed $X,Y,Z$ errors with probability $p$

• Conceptual setting of a quantum error correcting code
  
  

  

$3$-qubit code

• Idea: distribute one qubit $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ over $3$ physical qubits.
• basis state / codewords $|0\rangle=|000\rangle,|1\rangle=|111\rangle$
  
  
• span codespace
  $$\begin{aligned}|\psi\rangle_L&=\alpha|0\rangle+\beta|1\rangle\\ &=\alpha|000\rangle+\beta|111\rangle\end{aligned}$$

1. Encoding: $C_1NOT_2$ and $C_1NOT_3$ on $|\psi\rangle|0\rangle|0\rangle\rightarrow |\psi\rangle_L$
2. possible occurrence of errors:
   • e.g. bit flip error
     $\alpha|000\rangle+\beta|111\rangle\xrightarrow[]{X_1}\alpha|100\rangle+\beta|011\rangle$
     • by measuring: collapse onto $|011\rangle$ or $|100\rangle$ and lose the information about $\alpha$ and $\beta$
   • Solution indirect measurement via ancilla qubits.
     
     Apply $C_1NOT_3, C_2NOT_3$ on state $|a\rangle|b\rangle|0_C\rangle$ and measuring $|0_C\rangle$
     where $|0_C\rangle$ is an ancilla state.
     If $|ab\rangle=|00\rangle$ or $|ab\rangle=|11\rangle$, $|0_C\rangle$ will not change. $\Rightarrow$ even parity
     If $|ab\rangle=|01\rangle$ or $|ab\rangle=|10\rangle$, $|0_C\rangle$ will change to $|1_C\rangle$. $\Rightarrow$ odd parity
     
   • general consideration:
     
     
     consider $n$-qubit operation $U$ with eigenvalues $\pm 1$
     $$\begin{aligned}|0\rangle|\psi\rangle&\xrightarrow[]{H_1}&1/\sqrt{2}(|0\rangle+|1\rangle)|\psi\rangle\\ &\xrightarrow[]{Controlled-U}&1/\sqrt{2}|0\rangle|\psi\rangle+1/\sqrt{2}|1\rangle U|\psi\rangle\\ &\xrightarrow[]{H_1} & 1/2(|0\rangle+|1\rangle)|\psi\rangle+1/2(|0\rangle-|1\rangle) U|\psi\rangle \\ & &=|0\rangle (1/2)(\mathbb{I}+U)|\psi\rangle+|1\rangle (1/2)(\mathbb{I}-U)|\psi\rangle\end{aligned}$$
     where $(1/2)(\mathbb{I}\pm U)$ is $P_\pm:$ projector operator on the $\pm 1$ eigenspace of $U$
     $\Rightarrow$ realize a measuremnt of $U$
     For $U=Z_1Z_2$
     $\Rightarrow$ This is a contrast to measure $Z_1$ and $Z_2$ individually.
     
   • Two ancilla qubits
     $$\begin{aligned}|\psi\rangle|00\rangle\xrightarrow[]{error}&|\psi'\rangle|00'\rangle|00\rangle_{ancilla} \\ \xrightarrow[]{C_1NOT_4, C_2NOT_4, C_2NOT_5, C_3NOT5}&|\psi'\rangle|00'\rangle Z_1Z_2|0\rangle Z_2Z_3|0\rangle \\ \xrightarrow[]{correction}& |\psi''\rangle|00''\rangle\end{aligned}$$

     $Z_1Z_2$	$Z_2Z_3$	error/correction
     $1$	$1$	$\mathbb{I}$
     $-1$	$1$	$X_1$
     $-1$	$-1$	$X_2$
     $1$	$-1$	$X_3$

   correct only one $X$-error on any of the gate
   If there are two $X$-errors, then QEC fails
   If there are three, then it cannot be detected

• $3$-qubit code is not a complete code
• Robustness $3$-qubit code better than one qubit?
  • error model:
    • bit flip error: $p$
    • no error: $1-p$
  • probability to correct:
    $$p_{correct}=(1-p)^3+3p(1-p)^2$$
    no error on $3$ qubits + the probability of one $X$ error
    $$p_{fail}=3p^2(1-p)+p^3$$
  • For $p<1/2$, $p_{correct}>(1-p)$
    where $(1-p)$ is the probability of single qubit which successes.
    $\Rightarrow$ $3$-qubit is better than single qubit.
• How to increase robustness?
  1. $5$-qubit code
     $p_{correct}^{(5)}\geq p_{correct}^{(3)}$ for small $p$
  2. concatenation
     Encode codes in codes in codes…
     $$p_{fail}^{(n)}=3(p_{fail}^{(n-1)})^2-2(p_{fail}^{(n-1)})^3\leq 3 (p_{fail}^{(n-1)})^2$$
     For $n$ layers of concatenation:
     $$p_{fail}^{(n)}\leq\frac{1}{3}(3p)^{2^n}$$
     $\Rightarrow p_{fail}\rightarrow 0\text{ for }p\leq 1/3$
     
     
     
     
     

     double-exponentially fast in the number of layers $n$

Stabilize Formalism

• Idea: Fix and describe quantum state by set of commuting observations rather than explicit quantum state vector.
• allows us to describe a larger class of multi-qubit states.
• Example:
  • Stabilizer state: $|\Phi^+\rangle=1/\sqrt{2}(|00\rangle+|11\rangle)$
  • Stabilizer: $S_1=Z_1Z_2, S_2=X_1X_2$
  • $S_1S_2=S_2S_1$, they can commute with each other
    i.e. $[S_1,S_2]=0$
  • $|\Phi^+\rangle$ has unique eigenstate $|\psi\rangle$
    $S_1|\psi\rangle=|\psi\rangle\\S_2|\psi\rangle=|\psi\rangle$
  • $S_1, S_2$ generates a $2^2=4$ Abelian group:
    $$\{\mathbb{I},Z_1Z_2,X_1X_2,-Y_1Y_2\}$$
• Generalization of $n$-qubit
  • $n$ generators $S_1,...,S_n$
  • $2^n$ stabilizer operators
  • $S_i|\psi\rangle=|\psi\rangle,\ \forall\ i$
• Stabilizer QEC code:
  • $\{S_i\}$ with $[S_i,S_j] = 0,\ \forall\ i,j$
  • $O_L$: logical operatoers $Z_L, X_L$, with $[O_L,S_i]=0$
  • $\{Z_L,X_L\}=0$
• Quantum hamming bound:
  • How large must a $\mathcal{H}$ space be, so that we can accomadate QEC codes:
    • $n$-qubit, $k$ encoded for up to $t$ errors.
    • if $j$ errors, $j\leq t$
      $$\sum_{j=0}^t{n\choose j}3^j\cdot 2^k\leq 2^n\\\text{ called Quantum Hamming Bound}$$
      ${n\choose j}:j$ errors occur in $n$-qubits
      $3^j:$ $X,Y,Z$ errors
      $2^k:$ error codespace to an $2^k$ dimension orthogonal subspace
    • for 1 logical qubit $k=1$, 1 arbitrary correctable error $t=1$
      $$2(1+3n)\leq2^n$$
      $\Rightarrow n\geq 5$ qubits needed for smallest complete code.
• 9-qubit shor code:
  • $$\begin{aligned}|\psi\rangle_L&=\alpha(|000\rangle+|111\rangle)(|000\rangle+|111\rangle)(|000\rangle+|111\rangle)/\sqrt{8}\\ &+\beta(|000\rangle-|111\rangle)(|000\rangle-|111\rangle)(|000\rangle-|111\rangle)/\sqrt{8}\end{aligned}$$
  • Bit flip errors $X_i$
    can be detected by $\{Z_1Z_2, Z_2Z_3,Z_4Z_5,Z_5Z_6,Z_7Z_8,Z_8Z_9\}$
  • phase flip errors $Z_i$
    can be detected by $\{X_1X_2X_3X_4X_5X_6,X_4X_5X_6X_7X_8X_9\}$
    detect the sign of any of three qubits.
  • after detecting a phase flip in the first block, we don’t know which qubit
    but we can applying $Z_1$-correction
    we get the same correct state.
  • Property:
    • complete code
    • degenerate code: several errors have the same effect on logic state e.g. $Z_1, Z_2$
    • Correction works:  $$\text{attempt + error = identity or stabilizer}$$

Quantum Fault Tolerance

• Problem: Errors can occur before encoding or during encoding
  • uncontrolled error propagation
  • example $(C_1NOT_2)X_1=X_1X_2(C_1NOT_2)$
    $X$ error propagates from a controlled qubit through $CNOT$ gate onto a target qubit
    ( propagate errors from $X_1$ to $X_2$)
    
    
    
    
    
    
    
    
• Fault Tolerant quantum circuit construction: avoid error spread, infection
• $7$-qubit scheme code
  
  
  
  
  
  
• possible, but requires non-local gates, resource overhead

Topological QEC

• idea: quantum information spread out globally, to protect against local errors.
• no concatenation
• increase the size of lattice to increase robustness
• no braiding for gates

Kitaev’s toric code

• $L\times L$ square lattice with physical qubits located on the edges
  
  
  
  
  
• Codespace: $S_Z^{(i)}|\psi\rangle=S_X^{(j)}|\psi\rangle=|\psi\rangle$
• Periodic boundry conditions: lattice is embedded on the surface of a torus. $\Rightarrow$ Toric code
  
  
  
  
  
• number of qubits and operatoins:
  • $L^2$ plaquette $\Rightarrow$ $2L^2$ physics qubits
  • $L^2$ plaquette $\Rightarrow$ $L^2$ $Z$-stabalizer
  • $L^2$ plaquette $\Rightarrow$ $L^2$ $X$-stabalizer
• not all stabilizers are independent:
  $$\prod_i S_Z^{(i)}=\mathbb{I}\Rightarrow\prod_i^{L^2-1} S_Z^{(i)}=S_Z^{(L^2)}$$
  • only $L^2-1$ independent $Z(X)$-stabalizers.
  • $k=$ number of logic qubits$=(\#$physical qubits$)-(\#$independent stabalizer constrains$)=2$ (encoded qubits)

  Topological property: independent of size, lattice structure

• minimum-weight perfect matching (MWPM)
• logic quantum computations: transversal gates
• Transversality: logic operations = bitwise physical operations (local operations)
  $$H^{\otimes n}=H_1\otimes H_2\otimes...\otimes H_n$$
  if one $H_i$ is wrong, only effect one qubit $\Rightarrow$ Fault Tolerant

Readings

Nielsen and Chuang

    10 Quantum error-correction

Quantum Error Correction for Beginners