Symmetry in quantum walks on graphs

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1 NSF Workshop on Quantum Information Processing and Nanoscale Systems, September 2007 Symmetry in quantum walks on graphs Hari Krovi and Todd Brun Communication Sciences Institute Ming Hsieh Department of Electrical Engineering USC

2 The structure of the walk is Quantum walks Û = Ŝ(Î Ĉ). Ŝ is the shift matrix and Ĉ is the coin matrix. Ŝ = v i v(i), i v, i, where v(i) is the vertex connected to v along direction i. For the square, Ŝ = 00, 1 10, , 2 01, , 1 11, , 2 00, , 1 00, , 2 11, , 1 01, , 2 10, 2 Ŝ encodes the structure of the graph much like the adjacency matrix.

3 Quantum walks1 C can be any unitary matrix in general. Usually matrices some structure are considered. The matrices we use are the Grover matrix and the discrete Fourier transform matrix. Ĝ = 2 Ψ Ψ I = ˆD = 1 d 2 d d... d 2 2 d d d d d... d 1, where Ψ = 1 d i i ω ω 2... ω d , where ω = exp(2πi/d). 1 ω d 1 ω 2(d 1)... ω (d 1)(d 1)

4 Quantum hitting time To define the hitting time for quantum walks analogous to classical walks, we need the following ingredients. Measure at every step to see if the particle is found at the final vertex. Projective measurement with operators ˆP f and ˆQ f = Î ˆP f, where ˆP f = x f x f Îc. The walk is a repeated application of MÛ. p(t) = Tr{ ˆP f Û( ˆQ f Û) t 1 ρ 0 (Û ˆQf ) t 1 Û ˆPf } Define superoperators Nρ = ˆQ f ÛρÛ ˆQf and Yρ = ˆP f ÛρÛ ˆPf p(t) = Tr{YN t 1 ρ 0 } The hitting time is again τ(v i, v f ) = t=0 tp v i (t). This can be summed like a geometric series to give: τ = Tr{Y(I N) 2 ρ 0 }

5 Graph reduction 2n basis states 0, R, 1, L, 1, R, 2, L, 2, R,..., n 1, R, n, L.

6 Automorphism groups Direction preserving automorphisms of the n = 2 hypercube.

7 Automorphism groups1 Automorphisms that require an interchange of directions of the n = 2 hypercube.

8 Symmetries A unitary matrix ˆR is a symmetry of Û if ˆRÛ ˆR = Û. Automorphisms have a representation as a permutation matrix on the Hilbert space of Û. Direction preserving ˆP v Îc and others ˆP v ˆP c. Every automorphism is a symmetry of the shift (Ŝ) matrix. Direction-preserving automorphisms are always symmetries of Û. ( ˆP v Îc){Ŝ(Î Ĉ)}( ˆP v Îc) = {( ˆP v Îc)Ŝ( ˆP v Îc) }(Î Ĉ) Other automorphisms ˆP v ˆP c need not always be symmetries. {( ˆP v ˆP c )Ŝ( ˆP v ˆP c ) }(Î ˆP c Ĉ ˆP c )

9 Representation theory-brief overview Every finite group has a unitary representation on a given vector space V. Every representation decomposes into irreducible representations (irreps). These irreps partition V and can be identified with subspaces in V such that V = W 1 W k. The number of times each irrep occurs in a given representation can be determined using characters. The regular representation consists of all the irreps with each irrep occurring multiple times. The group of direction-preserving automorphisms of a Cayley graph form a regular representation on the Hilbert space of the vertices H v.

10 Symmetry and degeneracy Unitary operator Û defined on H has a group of symmetries whose representation is σ(g). Every irrep of σ(g) lies inside an eigenspace of Û. If an irrep in σ has a dimension d, then there must be an eigenspace of Û with degeneracy of at least d. Discrete walks on Cayley graphs Group of direction preserving automorphisms G give a regular representation on H v. Every irrep of G occurs in the regular representation. Representation on H v H c is ˆP v Îc. This is not a regular representation but it turns out that every irrep occurs in this representation too.

11 Symmetry and degeneracy1 If there is an irrep with dim(irrep) > dim(coin), then there is a degenerate eigenspace of Û with dim > dim(coin). For Cayley graphs on the symmetric group Γ(S n, Y ), Y is a set of n 1 transpositions, this happens for n 5. Continuous walks on Cayley graphs Degree of freedom at the final vertex is 1, since there is no coin. If dim(eigenspace of Û) > 1, then we have infinite hitting times. If dim(irrep) > 1, then we will have infinite hitting times. For any Cayley graph defined on any non-abelian group, this happens.

12 Quotient graphs Infinite hitting times and exponential speed-up are both related. Walk can be confined to a subspace. In some cases this confinement can be related to the symmetry of the graph. The walk in such a case is actually on a different graph.

13 Quotient graphs1 Using some subgroup H of the automorphism group, we can find the quotient graph. The walk will remain in this graph if the initial state respects the symmetries of the group H.

14 Quotient graphs2

15 Quotient graphs3