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〉 + e i φ sin θ 2 |1〉 = |ψ 〉 spherical coordinates ⇔ vector in C 2 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. v 3 v 4 v 1 v 2 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 σ 0 ,σ 1 ,σ 2 ,σ 3 , 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=2 j +1 ℓ k =2 j +1 ) hypergraph states have Y ⊗n symmetry. ⊗ Theorem #2: ( n=2 j +1 -2 k =2 j -1 ) hypergraph states have X ⊗n symmetry. ⊗ Conjecture #2: ( n=2 j +1 ℓ-m k =2 j -(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 ∈S n ψ π (1) ψ π (2) ... ψ π (n) (where α is a normalization factor and S n is the group of permutations of {1, 2, ..., n}) ⊗ These 1-qubit states, |ψ 1 〉 , |ψ 2 〉 ,..., |ψ n 〉, 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 d 0 , d 1 ,..., d n such that |ψ 〉 = n k =0 d k 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 d k z k 3. Find the roots of the Majorana polynomial, say λ 1 ,λ 2 ,...,λ n (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 |1 S 〉 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: |1 S 〉 = |1110110〉 ⊗ The formula for the hypergraph state is the following: |ψ 〉 = S ⊂{1,...n} (−1) #{e ∈E : e ⊆S } |1 S 〉 ⊗ 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 |1 S 〉 is (−1) 3 = −1. 7. Stereographic Projection P ′ P Points on the Bloch Sphere −→ C 2 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 Meeting Discrete Symmetries of Symmetric Hypergraph States Chase Yetter
