1. 13. Majorana Configurations with Discrete
Symmetries
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) ⇔ cos
θ
2
|0 + eiφ
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, where
edges contain a maximum of two vertices. The hyperedges of a
hypergraph can contain any number of vertices, potentially giving
them more applications than graphs.
v3
v4v1
v2
http://en.wikipedia.org/wiki/Hypergraph
2. The Pauli Matrices
Id =
1 0
0 1
, X =
0 1
1 0
, Y =
0 −i
i 0
, Z =
1 0
0 −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 the
form g⊗n, where g ∈ U(2) is a rotation of the Bloch sphere that
permutes the Majorana points.
11. Types of Discrete Symmetries
⊗ A set of Majorana points will have 180 degree rotational
symmetry about the x-axis if and only if the corresponding
state exhibits X⊗n symmetry.
⊗ A set of Majorana points will have 180 degree rotational
symmetry about the y-axis if and only if the corresponding
state exhibits Y ⊗n symmetry.
⊗ A set of Majorana points will have 180 degree rotational
symmetry about the z-axis if and only if the corresponding
state exhibits Z⊗n
symmetry. However, we have proven that
no hypergraph states exhibit Z⊗n
symmetry.
12. Symmetry
⊗ Theorem #1: n=4ℓ
k=3 hypergraph states have Y ⊗n symmetry.
⊗ Conjecture #1: n=2j+1ℓ
k=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−m
k=2j −(m−1) will have X⊗n symmetry.
8. Majorana Points
⊗ Fact: every symmetric n-qubit state |ψ can be written as a
symmetrized 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 points
on 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 that
each hyperedge contains exactly k vertices.
⊗ For a hypergraph to be k-complete, each hyperedge must
contain exactly k vertices and every possible hyperedge of size
k must be contained in that hypergraph. When the
hypergraph has n vertices, the k-complete hypergraph will
have n
k hyperedges of size k. Because of this, we refer to the
k-complete hypergraph on n vertices as the n
k hypergraph.
Abstract
Hypergraph states are a generalization of graph states, which have
proven to be useful in quantum error correction and are resource
states for quantum computation. Quantum entanglement is at the
heart of quantum information; an important related study is that
of local unitary symmetries. In this project, I have studied discrete
symmetries of symmetric hypergraph states (that is, hypergraph
states that are invariant under permutation of qubits). Using
computer aided searches and visualization on the Bloch sphere, we
have found a number of families of states with particular
symmetries.
1. Quantum States
⊗ The qubit, short for quantum bit, is the basic unit of
information in a quantum computer
⊗ Qubits are to bits as quantum computation is to classical
computation.
⊗ The state of a qubit (called a quantum state) is a complex
linear combination of the two basis states, |0 and |1 .
⊗ More familiar to someone with a linear algebra background,
|0 =
1
0
and |1 =
0
1
.
⊗ The quantum state |ψ = α |0 + β |1 is said to be in a
superposition between |0 and |1 .
⊗ However, the vectors are customarily normalized, so α and β
are 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 on
how to entangle the qubits.
⊗ Given a subset S ⊆ {1, 2, . . . , n} of vertices, we write |1S to
denote the computational basis vector that has 1s in positions
given by S and 0s elsewhere.
⊗ Example: For the subset S = {1, 2, 3, 5, 6} of the set of 7
qubits:
|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 of
hyperedges 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, and
Scott 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. G¨uhne, 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 University
October 3−5, 2014
APS Mid−Atlantic MeetingDiscrete Symmetries of Symmetric Hypergraph States
Chase Yetter