Conference Poster: Discrete Symmetries of Symmetric Hypergraph States

13. Majorana Configurations with DiscreteSymmetries

(

6

3

) (

510

255

) (

511

256

) (

511

128

)

(

511

64

) (

511

32

) (

256

65

) (

384

65

)

14. Future Outlook

A paper on this material is currently in preparation. We believe we

have proofs for the conjectures. It would be ideal to prove that the

only hypergraph states with X⊗n and Y⊗n symmetry are elements

of the families in the conjectures on panel 12.

6. The Bloch Sphere

|0〉

|ψ〉

φ

θ

|1〉

(θ, φ) ⇔ cosθ

2|0〉+ e iφ sin

θ

2|1〉 = |ψ〉

spherical coordinates ⇔ vector in C2

point on the sphere ⇔ state of a qubit

3. Hypergraphs

The hypergraph is a way to visually represent collections of sets.The more well-known graph contains vertices and edges, whereedges contain a maximum of two vertices. The hyperedges of ahypergraph can contain any number of vertices, potentially givingthem more applications than graphs.

v3

v4v1

v2

http://en.wikipedia.org/wiki/Hypergraph

2. The Pauli Matrices

Id =

[

1 00 1

]

, X =

[

0 11 0

]

, Y =

[

0 −i

i 0

]

, Z =

[

1 00 −1

]

(Denoted σ0, σ1, σ2, σ3, respectively)

10. Discrete Symmetries of Symmetric States

Theorem [1]

Any discrete LU symmetry of an n-qubit symmetric state is of theform g⊗n, where g ∈ U(2) is a rotation of the Bloch sphere thatpermutes the Majorana points.

11. Types of Discrete Symmetries

⊗ A set of Majorana points will have 180 degree rotationalsymmetry about the x-axis if and only if the correspondingstate exhibits X⊗n symmetry.

⊗ A set of Majorana points will have 180 degree rotationalsymmetry about the y-axis if and only if the correspondingstate exhibits Y⊗n symmetry.

⊗ A set of Majorana points will have 180 degree rotationalsymmetry about the z-axis if and only if the correspondingstate exhibits Z⊗n symmetry. However, we have proven thatno hypergraph states exhibit Z⊗n symmetry.

12. Symmetry

⊗ Theorem #1:(

n=4ℓk=3

)

hypergraph states have Y⊗n symmetry.

⊗ Conjecture #1:(

n=2j+1ℓk=2j+1

)

hypergraph states have Y⊗n

symmetry.

⊗ Theorem #2:(

n=2j+1−2

k=2j−1

)

hypergraph states have X⊗n

symmetry.

⊗ Conjecture #2:(

n=2j+1ℓ−mk=2j−(m−1)

)

will have X⊗n symmetry.

8. Majorana Points

⊗ Fact: every symmetric n-qubit state |ψ〉 can be written as asymmetrized product of n 1-qubit states.

⊗ |ψ〉 = α∑

π∈Sn

∣

∣ψπ(1)

⟩ ∣

∣ψπ(2)

⟩

...∣

∣ψπ(n)

⟩

(where α is a normalization factor and Sn is the group of permutations of {1, 2, ..., n})

⊗ These 1-qubit states, |ψ1〉 , |ψ2〉 , ..., |ψn〉, thought of as pointson the Bloch sphere, are called the Majorana points for |ψ〉.

9. Algorithm to find the Majorana Points of a

symmetric |ψ〉

Given symmetric |ψ〉

1. Find coefficients d0, d1, ..., dn such that |ψ〉 =

n∑

k=0

dk

∣

∣

∣D

(k)n

⟩

where∣

∣

∣D

(k)n

⟩

=1

√

(

n

k

)

∑

wt(I )=k

|I 〉 is the weight k Dicke state.

2. Construct the Majorana polynomial

p(z) =

n∑

k=0

(−1)k

√

(

n

k

)

dkzk

3. Find the roots of the Majorana polynomial, say λ1, λ2, ..., λn(not necessarily distinct).

4. Take the inverse stereographic projection of λ∗k, 1 ≤ k ≤ n.

These are the Majorana points.

4. k-uniformity

⊗ When one says that a hypergraph is k-uniform, it means thateach hyperedge contains exactly k vertices.

⊗ For a hypergraph to be k-complete, each hyperedge mustcontain exactly k vertices and every possible hyperedge of sizek must be contained in that hypergraph. When thehypergraph has n vertices, the k-complete hypergraph willhave

(

n

k

)

hyperedges of size k . Because of this, we refer to thek-complete hypergraph on n vertices as the

(

n

k

)

hypergraph.

Abstract

Hypergraph states are a generalization of graph states, which haveproven to be useful in quantum error correction and are resourcestates for quantum computation. Quantum entanglement is at theheart of quantum information; an important related study is thatof local unitary symmetries. In this project, I have studied discretesymmetries of symmetric hypergraph states (that is, hypergraphstates that are invariant under permutation of qubits). Usingcomputer aided searches and visualization on the Bloch sphere, wehave found a number of families of states with particularsymmetries.

1. Quantum States

⊗ The qubit, short for quantum bit, is the basic unit ofinformation in a quantum computer

⊗ Qubits are to bits as quantum computation is to classicalcomputation.

⊗ The state of a qubit (called a quantum state) is a complexlinear combination of the two basis states, |0〉 and |1〉.

⊗ More familiar to someone with a linear algebra background,

|0〉 =

[

10

]

and |1〉 =

[

01

]

.

⊗ The quantum state |ψ〉 = α |0〉+ β |1〉 is said to be in asuperposition between |0〉 and |1〉.

⊗ However, the vectors are customarily normalized, so α and βare restricted to the following condition: |α|2 + |β|2 = 1

5. Hypergraph States

Here is how hypergraph states are constructed from hypergraphs.Each vertex is a qubit and each hyperedge gives instructions onhow to entangle the qubits.

⊗ Given a subset S ⊆ {1, 2, . . . , n} of vertices, we write |1S〉 todenote the computational basis vector that has 1s in positionsgiven by S and 0s elsewhere.

⊗ Example: For the subset S = {1, 2, 3, 5, 6} of the set of 7qubits:

|1S〉 = |1110110〉

⊗ The formula for the hypergraph state is the following:

|ψ〉 =∑

S⊂{1,...n}

(−1)#{e∈E : e⊆S} |1S〉

⊗ Example: For the hypergraph in panel 3, the number ofhyperedges contained in S = {1, 2, 3, 5, 6} is 3. So the sign of|1S〉 is (−1)3 = −1.

7. Stereographic Projection

P′

P

Points on the Bloch Sphere −→ C2

Acknowledgments. This work was supported by National Science

Foundation grant #PHY-1211594. I thank my research advisors

Dr. David W. Lyons and Dr. Scott N. Walck.

Lebanon Valley College Mathematical Physics Research Group

http://quantum.lvc.edu/mathphys

References

[1] Curt D. Cenci, David W. Lyons, Laura M. Snyder, andScott N. Walck.Symmetric states: local unitary equivalence via stabilizers.Quantum Information and Computation, 10:1029–1041,November 2010.arXiv:1007.3920v1 [quant-ph].

[2] M. Rossi, M. Huber, D. Bruß, and C. Macchiavello.Quantum hypergraph states.New Journal of Physics, 15(11):113022, 2013.

[3] O. Guhne, M. Cuquet, F. E. S. Steinhoff, T. Moroder,M. Rossi, D. Bruß, B. Kraus, and C. Macchiavello.Entanglement and nonclassical properties of hypergraph states.2014.arXiv:1404.6492 [quant-ph].

Lebanon Valley College

Pennsylvania State UniversityOctober 3−5, 2014

APS Mid−Atlantic MeetingDiscrete Symmetries of Symmetric Hypergraph States

Chase Yetter