CNot gates in the Honeycomb Floquet Code
Julien Bréhier
Aachen GM 30/09/2025
Holidays
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}
The Steane Code
\begin{align*}
& IIIXXXX \\
& IXXIIXX \\
& XIXIXIX \\
& IIIZZZZ \\
& IZZIIZZ \\
& ZIZIZIZ
\end{align*}
https://arthurpesah.me/blog/2023-03-16-stabilizer-formalism-2/
The Steane Code
\begin{align*}
\bar{X} & = XXXXXXX \\
\bar{Z} & = ZZZZZZZ.
\end{align*}
https://arthurpesah.me/blog/2023-03-16-stabilizer-formalism-2/
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}
Subsystem 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}}
\set{\forall s_i,s_j \in \mathcal{S} \subsetneq\mathcal{P}^n,[s_i,s_j]=0}
Subsystem 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}}
\set{\forall s_i,s_j \in \mathcal{S} \subsetneq\mathcal{P}^n,[s_i,s_j]=0}
\text{We measure }c \in \mathcal{C}\text{ such that}\\
\forall s\in \mathcal{S}, \exist \{c_i\}_{i\in I}\in\mathcal{C}, \prod_I c_i = s
Subsystem 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}}
\set{\forall s_i,s_j \in \mathcal{S} \subsetneq\mathcal{P}^n,[s_i,s_j]=0}
\text{We measure }c \in \mathcal{C}\text{ such that}\\
\forall s\in \mathcal{S}, \exist \{c_i\}_{i\in I}\in\mathcal{C}, \prod_I c_i = s
\text{and (preferably) }\forall (c,s) \in \mathcal{(C,S)}, [c,s]=0
Measurement Method Through Ancilla
Floquet-ness
Other Measurements
Two Protected Operators
The Planar Honeycomb Floquet Code (PHFC)
Careful with the edges
X_a
Y_a
Y_b
Z_a
Z_b
X_b
Threshold
Lattice surgery
What is it ?
CNot gate
CNot gate
Init\ket{\bar{0}}_A \xrightarrow{} \mathcal{M} (\bar{X}_A\bar{X}_T)\\
\xrightarrow{} \mathcal{M}(\bar{Z}_C\bar{Z}_A)\xrightarrow{} \mathcal{M}(\bar{X}_A)
Init\ket{\bar{+}}_A \xrightarrow{} \mathcal{M}(\bar{Z}_C\bar{Z}_A) \\
\xrightarrow{} \mathcal{M} (\bar{X}_A\bar{X}_T)\xrightarrow{} \mathcal{M}(\bar{Z}_A)
Space time diagram
Transversal gate
The Steane Code
\begin{align*}
& IIIXXXX \\
& IXXIIXX \\
& XIXIXIX \\
& IIIZZZZ \\
& IZZIIZZ \\
& ZIZIZIZ
\end{align*}
\bar{U} = U^{\otimes n}
The Steane Code
\begin{align*}
& IIIXXXX \\
& IXXIIXX \\
& XIXIXIX \\
& IIIZZZZ \\
& IZZIIZZ \\
& ZIZIZIZ
\end{align*}
\begin{align*}
\bar{X} & = XXXXXXX \\
\bar{Z} & = ZZZZZZZ.
\end{align*}
\bar{U} = U^{\otimes n}
PHFC Example
https://quantumcomputing.stackexchange.com/questions/24269/what-is-formally-a-transversal-operator
PHFC Example
https://quantumcomputing.stackexchange.com/questions/24269/what-is-formally-a-transversal-operator
In PHFC
In PHFC
In PHFC
Z_a | {\color{red}X_b}
(X_bZ_b)Y_bX_aY_a(Z_a|X_b)
Z_a | {\color{red}X_b}
(X_bZ_b)Y_bX_aY_a(Z_a|X_b)
\mathcal{G}^C = r^C(-7)r^C(-6)r^C(-2)r^C(-1)\\
\mathcal{G}^T = r^T(-7)r^T(-6)r^T(-2)r^T(-1)
Z_a | {\color{red}X_b}
Z_a | {\color{red}X_b}
(X_bZ_b)Y_bX_aY_a(Z_a|X_b)
\begin{align*}
\mathcal{G}^C &= r^C(-7)r^C(-6)r^C(-2)r^C(-1)\\
& \times {\color{red} r^T(-7)r^T(-6)r^T(-2)} \\
\mathcal{G}^T &= r^T(-7)r^T(-6)r^T(-2)r^T(-1)
\end{align*}
Z_a | X_b {\color{blue}Z_b}
Z_a | X_b {\color{blue}Z_b}
(Z_a)|X_b (Z_b)
Z_a | X_b {\color{blue}Z_b}
\begin{align*}
\mathcal{G}^C &= r^C(-3)r^C(-1)\\
\mathcal{G}^T &= r^T(-3)r^T(-1)
\end{align*}
(Z_a)|X_b (Z_b)
Z_a | X_b {\color{blue}Z_b}
\begin{align*}
\mathcal{G}^C &= r^C(-3)r^C(-1)\\
\mathcal{G}^T &= r^T(-3)r^T(-1)\\
&\times {\color{blue}r^C(-3)}
\end{align*}
(Z_a)|X_b (Z_b)
Z_a | X_b Z_b{\color{green}Y_b}
Z_a | X_b Z_b{\color{green}Y_b}
(Y_aZ_a)|X_b (Z_bY_b)
Z_a | X_b Z_b{\color{green}Y_b}
\begin{align*}
\mathcal{R}^C &= r^C(-5)r^C(-4)r^C(-2)r^C(-1)\\
\mathcal{R}^T &= r^T(-5)r^T(-4)r^T(-2)r^T(-1)
\end{align*}
(Y_aZ_a)|X_b (Z_bY_b)
Z_a | X_b Z_b{\color{green}Y_b}
\begin{align*}
\mathcal{R}^C &= r^C(-5)r^C(-4)r^C(-2)r^C(-1)\\
& \times {\color{red}r^T(-5)r^T(-4)}\\
\mathcal{R}^T &= r^T(-5)r^T(-4)r^T(-2)r^T(-1)
\end{align*}
(Y_aZ_a)|X_b (Z_bY_b)
Z_a | X_b Z_bY_b{\color{red}X_a}
Z_a | X_b Z_bY_b{\color{red}X_a}
(X_aY_a)Z_a|X_bZ_b(Y_bX_a)
Z_a | X_b Z_bY_b{\color{red}X_a}
\begin{align*}
\mathcal{B}^C &= r^C(-7)r^C(-6)r^C(-2)r^C(-1)\\
\mathcal{B}^T &= r^T(-7)r^T(-6)r^T(-2)r^T(-1)
\end{align*}
(X_aY_a)Z_a|X_bZ_b(Y_bX_a)
Z_a | X_b Z_bY_b{\color{red}X_a}
\begin{align*}
\mathcal{B}^C &= r^C(-7)r^C(-6)r^C(-2)r^C(-1)\\
\mathcal{B}^T &= r^T(-7)r^T(-6)r^T(-2)r^T(-1)\\
& \times {\color{blue}r^C(-7)r^C(-6)}
\end{align*}
(X_aY_a)Z_a|X_bZ_b(Y_bX_a)
Z_a | X_b Z_bY_bX_a{\color{green}Y_a}
Z_a | X_b Z_bY_bX_a{\color{green}Y_a}
(Y_b)X_a(Y_a)
Z_a | X_b Z_bY_bX_a{\color{green}Y_a}
(Y_b)X_a(Y_a)
\begin{align*}
\mathcal{B}^C &= r^C(-3)r^C(-1)\\
\mathcal{B}^T &= r^T(-3)r^T(-1)
\end{align*}
Z_a | X_b Z_bY_bX_a{\color{green}Y_a}
(Y_b)X_a(Y_a)
\begin{align*}
\mathcal{B}^C &= r^C(-3)r^C(-1)\\
\mathcal{B}^T &= r^T(-3)r^T(-1)
\end{align*}
Z_a | X_b Z_bY_bX_aY_a{\color{blue}Z_a}
Z_a | X_b Z_bY_bX_aY_a{\color{blue}Z_a}
(Z_bY_b)X_a(Y_aZ_a)
Z_a | X_b Z_bY_bX_aY_a{\color{blue}Z_a}
(Z_bY_b)X_a(Y_aZ_a)
\begin{align*}
\mathcal{R}^C &= r^C(-5)r^C(-4)r^C(-2)r^C(-1)\\
\mathcal{R}^T &= r^T(-5)r^T(-4)r^T(-2)r^T(-1)
\end{align*}
Z_a | X_b Z_bY_bX_aY_a{\color{blue}Z_a}
(Z_bY_b)X_a(Y_aZ_a)
\begin{align*}
\mathcal{R}^C &= r^C(-5)r^C(-4)r^C(-2)r^C(-1)\\
\mathcal{R}^T &= r^T(-5)r^T(-4)r^T(-2)r^T(-1)
\end{align*}
Z_a | X_b Z_bY_bX_aY_aZ_a{\color{red}X_b}
Z_a | X_b Z_bY_bX_aY_aZ_a{\color{red}X_b}
(X_b Z_b)Y_bX_aY_a(Z_aX_b)
Z_a | X_b Z_bY_bX_aY_aZ_a{\color{red}X_b}
(X_b Z_b)Y_bX_aY_a(Z_aX_b)
\begin{align*}
\mathcal{G}^C &= r^C(-7)r^C(-6)r^C(-2)r^C(-1)\\
\mathcal{G}^T &= r^T(-7)r^T(-6)r^T(-2)r^T(-1)
\end{align*}
Z_a | X_b Z_bY_bX_aY_aZ_a{\color{red}X_b}
(X_b Z_b)Y_bX_aY_a(Z_aX_b)
\begin{align*}
\mathcal{G}^C &= r^C(-7)r^C(-6)r^C(-2)r^C(-1)\\
\mathcal{G}^T &= r^T(-7)r^T(-6)r^T(-2)r^T(-1)
\end{align*}
So is our problem solved ?
In PHFC
MWPM (Hypergraph)
Syndrome surgery
MWPM (Hypergraph)
Syndrome surgery
MWPM (Hypergraph)
Syndrome surgery
MWPM (Hypergraph)
Syndrome surgery
MWPM (Hypergraph)
Syndrome surgery
How to :
- Split the diagram in two before the Cnot gates
- Correct up to the Cnot gates using MWPM
- Copy the tentative error onto the post block
- Match corresponding hexes to see the double flips for the first 4 rounds post Cnot
Conclusion
Conclusion
- Built the PHFC from the literature
Conclusion
- Built the PHFC from the literature
- Implemented the LS protocols for XX and ZZ measurements
Conclusion
- Built the PHFC from the literature
- Implemented the LS protocols for XX and ZZ measurements
- Fused the protocols to make a CNot (cf. Threshold plot)
Conclusion
- Built the PHFC from the literature
- Implemented the LS protocols for XX and ZZ measurements
- Fused the protocols to make a CNot (cf. Threshold plot)
- Created the traversal gate circuits with proper stabilizers
Quantum Error Correction and the Floquet Codes
By Julien Bréhier
