Quantum Error Correction and the Floquet Codes

Julien Bréhier

University of Bonn March 14, 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

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

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

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

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

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

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

PHFC Example

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

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

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
Test experimentally