Journal Club 15/05
Transversal Injection: Using the Surface Code to
Prepare Non-Pauli Eigenstates
\text{Julien Br\'ehier}
Little prerequisites
- Quantum Error Correction
- Logical operators
- Fault tolerance
- Transversal gates
Little prerequisites
- Quantum Error Correction
- Logical operators
- Fault tolerance
- Transversal gates
Stabilizer codes
\text {For a stabilizer code }\llbracket n,k,d \rrbracket
\forall s \in \mathcal{S},s\bar{\ket{\psi}} = {\color{red}+} \bar{\ket{\psi}}
k = n-rank(\mathcal{S})
\set{\forall s_i,s_j \in \mathcal{S} \subsetneq\mathcal{P}^n,[s_i,s_j]=0}
Kitaev Toric code
Kitaev Toric code
To the surface code
To the surface code
Little prerequisites
- Quantum Error Correction
- Logical operators
- Fault tolerance
- Transversal gates
Encoding information
{\color{blue} \bar{Z}}
{\color{red} \bar{X}}
Encoding information
{\color{blue} \bar{Z}}
Encoding information
{\color{blue} \bar{Z}}
Encoding information
{\color{blue} \bar{Z}}
Little prerequisites
- Quantum Error Correction
- Logical operators
- Fault tolerance
- Transversal gates
Error propagation
To the surface code
To the surface code
\color{blue} Z
To the surface code
\color{blue} Z
To the surface code
\color{blue} Z
To the surface code
\color{blue} Z
\color{blue} Z
Little prerequisites
- Quantum Error Correction
- Logical operators
- Fault tolerance
- Transversal gates
Transversal operator
\color{blue} Z
\color{blue} Z
\color{blue} Z
\color{blue} Z
\color{blue} Z
\color{blue} Z^{\otimes 5} = \bar{Z}
Unfortunately...
Knill-Eastin no go theorem
One then has to think
- Gauge fixed code conversion
- State injection
- Magic state distilation
- Gate teleportation
What else can we do ?
Into the paper .....
Usual initialization
\ket{\psi} = \ket{0}^{\otimes 5}
Usual initialization
\ket{\psi} = \bar{\ket{0}}
Usual initialization
\ket{\psi} = \bar{\ket{0}} = \ket{00000}+\ket{11100}+\ket{00111}+\ket{11011}
Transversal injection
\ket{\chi} = (\alpha\ket{0}+\beta\ket{1})^{\otimes 5}
What's next ?
A treasure hunt !
\ket{\chi}
\ket{\chi}
Z_{1,3,4}
\ket{\chi}
Z_{1,3,4}
Z_{2,3,5}
\ket{\chi}
Z_{1,3,4}
Z_{2,3,5}
X_{1,2,3}
\ket{\chi}
Z_{1,3,4}
Z_{2,3,5}
X_{1,2,3}
X_{3,4,5}
\ket{\chi}
Z_{1,3,4}
Z_{2,3,5}
X_{1,2,3}
X_{3,4,5}
\ket{\Lambda} = \begin{pmatrix}\alpha_L\\\beta_L\end{pmatrix}
Let's look at the steps
What do we need ?
\text{Suppose } syn = (\overbrace{1,0}^{X_i}|\underbrace{0,1}_{Z_i})
Start with
Z_1 \xrightarrow{} \set{0,2,3}
Start with
Z_1 \xrightarrow{} \set{0,2,3}
How could we get 0 ?
Start with
Z_1 \xrightarrow{} \set{0,2,3}
How could we get 0 ?
\emptyset, \set{0,2}, \set{0,3}, \set{2,3}
Start with
Z_1 \xrightarrow{} \set{0,2,3}
How could we get 0 ?
\emptyset, \set{0,2}, \set{0,3}, \set{2,3}
Next :
Z_2 \xrightarrow{} \set{1,2,4}
Next :
Z_2 \xrightarrow{} \set{1,2,4}
How could we get 1 ?
Next :
Z_2 \xrightarrow{} \set{1,2,4}
How could we get 1 ?
Recall we already saw #2
Next :
Z_2 \xrightarrow{} \set{1,2,4}
How could we get 1 ?
Recall we already saw #2
t = z_i \oplus \set{\text{seen qubits}}
Next :
Z_2 \xrightarrow{} \set{1,2,4}
How could we get 1 ?
Recall we already saw #2
t = z_i \oplus \set{\text{seen qubits}}
What do we get ?
Now to the X part
Now to the X part
Just read a table
With trivial syndroms :
With trivial syndroms :
With trivial syndroms :
With trivial syndroms :
\ket{00001} = \alpha^4\beta + \alpha^3\beta^2 - \alpha^2\beta^3 - \alpha\beta^4
What can it achieve ?
What can it achieve ?
What can it achieve ?
- Large distance codes for trivial X syndrome
\mathcal{O}(e^{N-1})
- Although there is a probabilistic aspect to it, depending on the architecture it can save qubits
- Post selection can achieve lower physical error rates