Quantum Error Correction and the Floquet Codes

Julien Bréhier

University of Bonn 20.05.2025

Journey Through Quantum Error Correction

  • Introduction
  • The Toric Code
  • The Toric Honeycomb Floquet Code (THFC)
  • Cutting Slices
  • Transversal Operators
  • Lattice Surgery (LS)
  • Prospects

Important notions

  • Stabilizer codes
  • Distance of a code
  • 'Good codes'
  • Logical operators
  • Minimum Weight Perfect Matching
  • Ancilla measurements
  • Error Correcting
  • Fault tolerance
  • Transversal operators
  • Lattice Surgery

Stabilizer codes

Stabilizer codes

\text {For a stabilizer code }\llbracket n,k,d \rrbracket

Stabilizer codes

\text {For a stabilizer code }\llbracket n,k,d \rrbracket
\set{\forall s_i,s_j \in \mathcal{S} \subsetneq\mathcal{P}^n,[s_i,s_j]=0}

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}}
\set{\forall s_i,s_j \in \mathcal{S} \subsetneq\mathcal{P}^n,[s_i,s_j]=0}

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 3-Qubit Code

The 3-Qubit Code

https://arxiv.org/pdf/0905.2794

The 3-Qubit Code

https://arxiv.org/pdf/0905.2794

s = \langle Z_1Z_2,Z_1Z_3\rangle

The 3-Qubit Code

https://arxiv.org/pdf/0905.2794

s = \langle Z_1Z_2,Z_1Z_3\rangle
|\bar{0}\rangle = |000\rangle \\ |\bar{1}\rangle = |111\rangle

The Steane Code

https://arthurpesah.me/blog/2023-03-16-stabilizer-formalism-2/

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/

The Search for Good Codes

The Search for Good Codes

You might remember :

The Search for Good Codes

` \text {For a stabilizer code }\llbracket n,k,d \rrbracket ,

You might remember :

Journey Through Quantum Error Correction

  • Introduction
  • The Toric Code
  • The Toric Honeycomb Floquet Code (THFC)
  • Cutting Slices
  • Transversal Operators
  • Lattice Surgery (LS)
  • Prospects

Hamiltonian

A_v = \prod_{i \in v} \sigma^x_i, \,\,\\ B_p = \prod_{i \in p} \sigma^z_i

Logical Operators

https://errorcorrectionzoo.org/c/toric

Minimum Weight Perfect Matching

https://arthurpesah.me/blog/2023-05-13-surface-code/

Journey Through Quantum Error Correction

  • Introduction
  • The Toric Code
  • The Toric Honeycomb Floquet Code (THFC)
  • Cutting Slices
  • Transversal Operators
  • Lattice Surgery (LS)
  • Prospects

Measurement Method Through Ancilla

Other Measurements

Kitaev Spin Model

THC Yields No Qubit

e = 3p
q = 2p
s = (p - 1) + 2 = p + 1
|\mathcal{G}| = 3p - 1
g = \frac{|\mathcal{G}| - |\mathcal{S}|}{2} = \frac{3p - 1 - (p + 1)}{2} = p - 1
k + g + s = n \quad \Longleftrightarrow \quad k + (p - 1) + (p + 1) = 2p

Floquet-ness

Toric Code ?

Two Protected Operators

Journey Through Quantum Error Correction

  • Introduction
  • The Toric Code
  • The Toric Honeycomb Floquet Code (THFC)
  • Cutting Slices
  • Transversal Operators
  • Lattice Surgery (LS)
  • Prospects

Setting boundaries

https://quantumcomputing.stackexchange.com/questions/4395/how-is-computation-done-in-a-2d-surface-code-array

The Surface Code

The Planar Honeycomb Floquet Code (PHFC)

Careful with the edges

X_a
Y_a
Y_b
Z_a
Z_b
X_b

Notion of threshold

Journey Through Quantum Error Correction

  • Introduction
  • The Toric Code
  • The Toric Honeycomb Floquet Code (THFC)
  • Cutting Slices
  • Transversal Operators
  • Lattice Surgery (LS)
  • Prospects

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

Journey Through Quantum Error Correction

  • Introduction
  • The Toric Code
  • The Toric Honeycomb Floquet Code (THFC)
  • Cutting Slices
  • Transversal Operators
  • Lattice Surgery (LS)
  • Prospects

Idea of Fault Tolerance

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}

Surface Code Example

S = \sqrt{Z}

Surface Code Example

S = \sqrt{Z}
\bar{S} = S^{\otimes 9}

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*}

In PHFC

Journey Through Quantum Error Correction

  • Introduction
  • The Toric Code
  • The Toric Honeycomb Floquet Code (THFC)
  • Cutting Slices
  • Transversal Operators
  • Lattice Surgery (LS)
  • Prospects

Important notions

  • Stabilizer codes
  • Distance of a code
  • 'Good codes'
  • Logical operators
  • Minimum Weight Perfect Matching
  • Ancilla measurements
  • Error Correcting
  • Fault tolerance
  • Transversal operators
  • Lattice Surgery

Future ideas

  • Publish the results comparing the two techniques
  • Compare different implementations of the planar codes (there exist a CSS version)
  • Test experimentally perhaps

Copy of Quantum Error Correction and the Floquet Codes

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