conference poster: discrete symmetries of symmetric hypergraph states
TRANSCRIPT
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