Mapping graph state orbits under local complementation

Presented by Zhi Han

Simon Fraser University & Aalto University

slides.com/zhihan/lc

Overview

  • Rubik's cube
  • Group Theory and Cayley graphs
  • Local complementation forms a group
    • Graph states
    • Local complementation

43 quintillion permutations!

Speedcubers

World Cube Association

http://cube20.org/qtm/

This is in the paper

Is this an efficient way of solving the cube?

How can we use group theory to speed up solving the cube, if we don't compute every state/action?

How do speedcubers actually solve the cube?

https://jperm.net/3x3/cfop

How to find God's algorithm?

CFOP discovered using group theory and brute force?

brute force?

Group theory

Example of Groups

Ref. [2]

Ref. [2]

Cayley graph. A cayley graph always defines a group.

Isomorphic

Ref. [2]

Can always go from a Cayley diagram (states) to a diagram of actions.

R^3BR^2B^3 = R

Can do algebra, etc...

Ref. [2]

R^3BR^2B^3 = R

Can do algebra, etc...

Mulitplication Table

Ref. [2]

Definition of a Group

  1. Closure: Any two composed actions must belong to the group.

  2. Identity: There exists an action that does nothing.

  3. Inverses: The inverse action must be part of the group.

  4. Associativity: Rearranging the parentheses when composing actions does not affect the result.

  1. Closure: \(g_1 g_2 \in G \quad \forall g_1 ,g_2 \in G\)

  2. Identity: \( e \in G \quad eg = ge = g\)

  3. Inverses: \(g \in G \implies g^{-1} \in G\)

  4. Associativity: \(( g_1 g_2) g_3 = g_1 (g_2 g_3)\)

Ref. [2]

Ref. [2]

Ref. [2]

Group theory summary

  • Group specified by a Cayley diagram.
  • Groups \(G\) are the actions that act on a set \(S\) states.
  • Cayley diagrams are not very scalable. 
  • Bonus slides below: Orbit - Stabilizer Theorem

Orbit stabilizer theorem

The actual paper

The actual paper

State \(S\): Graph states

Group \(G\): Local complementation on qubit \(i\)

|G\rangle=\prod_{(i, j) \in E} {C_Z}_{i j}|+\rangle^{\otimes|V|}

A graph state is specified by a graph \(G = (V, E)\). \(V\) is the qubit initalized to \(|+\rangle\) and for each edge apply the \(C_Z \) operation.

\(|+\rangle\)

\(|+\rangle\)

\(|+\rangle\)

\(|+\rangle\)

C_Z = |00\rangle \langle 00|+|01\rangle \langle 01|+|10\rangle \langle 10|-|11\rangle \langle 11|

Graph states

C_Z
C_Z
C_Z
C_Z

Cluster states

Image [1]

Why do we care?

https://pennylane.ai/qml/demos/tutorial_mbqc.html

Graph state

Measuring graph state

Constructing graph state

entanglement in graph states are considered a resource in measurement based quantum computers

Why do we care?

https://pennylane.ai/qml/demos/tutorial_mbqc.html

entanglement in graph states are considered a resource in measurement based quantum computers

knowing how to transform a graph state without consuming entanglement is very useful

knowing how to transform a graph state without consuming entanglement is very useful

entanglement in graph states are considered a resource in measurement based quantum computers

Local complementation doesn't change the entanglement

Why do we care?

Local complementation doesn't change the entanglement

Why do we care?

Local complementation doesn't change the entanglement

  • Classical Communication (CC)
  • Local Operations (LO)
U_\alpha^{\mathrm{LC}}=\sqrt{-i X_\alpha} \bigotimes_{i \in N_G(\alpha)} \sqrt{i Z_i}

LOCC = entanglement invariant operations

knowing how to transform a graph state without consuming entanglement is very useful

entanglement in graph states are considered a resource in measurement based quantum computers

Local complementation doesn't change the entanglement

What other graph state operations doesn't change entanglement?

  1. LC = Local complementation 

  2. LPM = (Local Pauli) measurements

  3. CC = Classical communication

What other graph state operations doesn't change entanglement?

LPM = (Local Pauli) measurements

Why do we care?

Image [1]

  1. LC = Local complementation 
  2. LPM = (Local Pauli) measurements
  3. CC = Classical communication

Vertex minor problem: NP-complete

Local complementation

U^{LC}_α|G\rangle = |LC_α(G)\rangle

That is, in the neighbourhood of α, it removes edges if they are present, and adds any edges are missing (see Fig. 1a).

\(L_3\)

\(L_3\)

\(C_3\)

\(L_3\): Full Cayley graph for class 3

\(C_3\): \(L_3\) under isomorphism graphs

\(L_3\)

\(C_3\)

We denote orbits \(C_i\) when isomorphic graphs are considered equal (unlabelled graph states), and \(L_i\) otherwise (labelled graph states).

\(L_4\)

\(C_4\)

[3]. They only compute up to entanglement class 148, or 8 qubits. Exact scaling is not known but seems doubly exponential

Cayley graph of \(L_{10}\)

\(L_{10}\)

Cayley graph of \(L_{10}\)

Adjacency Matrix of \(L_{10}\). 

Note that number of group actions doesn't equal the set

Cayley graph of \(L_{10}\)

Distance Matrix of \(L_{10}\)

Computing the correlations for each orbit

\(E_S\) Schmidt measure

\(d\): distance between graph states in the orbit

\(\chi\) chromatic index

rwd: rank index

|e|: minimum number edges

\(E_S\) Schmidt measure

\(d\): distance between graph states in the orbit

\(\chi\) chromatic index

rwd: rank index

|e|: minimum number edges

Summary

  • Local complementation is a operation that preserves entanglement.
  • Knowing the Cayley graph for Local Complementation on graph states might help us develop a God's algorithm for error correcting quantum states.
  • This paper computes the Cayley graphs for up to 9 qubits. 

[1] J. C. Adcock, S. Morley-Short, A. Dahlberg, and J. W. Silverstone, Mapping Graph State Orbits under Local Complementation, Quantum 4, 305 (2020).

 

[2] http://www.math.clemson.edu/~macaule/classes/m20_math4120/index.html

 

[3] Cabello, A., Danielsen, L. E., Lopez-Tarrida, A. J. & Portillo, J. R. Optimal preparation of graph states. Physical Review A 83, 042314 (2011).

 

[4] https://pennylane.ai/qml/demos/tutorial_mbqc.html

References

[1] https://www.semanticscholar.org/paper/Quantum-Computing-with-Cluster-States-Gelo-Tabia/9a42a72224e9199298e0cd37eb11739f216c86ad

Images