Which Quantum Walk on Graphs?

Transcription

1 Which Quantum Walk on Graphs? Christino Tamon Department of Computer Science Clarkson University August 13, 2015

2 Outline I Walks on graphs (random and quantum) I State transfer on graphs (motivation) I Known results and open problems

3 Algebraic graph theory: graphs are matrices The path P 3 w u v Its adjacency and degree matrices: A = D =

4 Walk on graphs u w v A = The matrix power A t encodes the following: A t uv = #t-step walks ending at u (starting from v) u ; v cospectral: A t uu = A t vv for all t.

5 Discrete-time random walk u w v P := AD 1 = = = The matrix power P t encodes: P t uv = P[at u after t steps (starting from v)] P is not symmetric, but p D 1 P p D is.

6 Continuous-time random walk u w v L := I e Lt = 1X k =0 p D 1 P p D ( 1) k t k k! L k The matrix exponential (e L ) t encodes: (e Lt ) uv = P[at u at time t (starting from v)] Heat kernel: Chung & Yau (1999)

7 \Quantum" walk u w v e iat = 1X k =0 ( it) k k! A k The matrix exponential (e ia ) t encodes: (e iat ) uv = amplitude at u at time t (starting from v) Recover classical interpretation by squaring: j(e iat ) uv j 2 = P[at u if measured at time t (starting from v)]

8 \Quantum" walk u w v U (t) = exp 0 i t C A More explicitly U (t) = p cos(pp p p p 2t) i 2 sin( p 2t) cos( p 2t) p 3 1 i 2 p sin( 2t) 2 p cos( 2t) p i 2 sin( p 2t) 5 cos( 2t) 1 i 2 sin( 2t) 1 + cos( 2t) U (t) is symmetric. U (=p 2)0;2 = 1, negative certainty.

9 Have you heard about... a quantum particle who walked into a discrete graph: I it can be in many places at once (superposition) I it may cancel itself out (interference) C. Addams, NY Times \Is it weird in here or is it just me?" - Steven Wright

10 Quantum walk (Farhi & Gutmann 1998) A continuous-time quantum walk on a graph G is given by U (t) := e imt where M is a Hermitian matrix related to G. Based on Schrodinger's equation i~ 0 (t) = H (t) =) (t) = e iht (0) with Hamiltonian H.

11 Motivation Why quantum walks? I paradigm to design quantum algorithms: Grover (1996), Szegedy (2004), Farhi-Goldstone-Gutmann (2008) I universal model for quantum computation: Childs (2009) I information transfer in quantum networks: Bose (2003), Christandl et al. (2004) I beautiful interplay with algebraic combinatorics Godsil (2011)

12 Spectral decomposition Theorem If M is Hermitian with eigenvalues f 1 ; : : : ; dg, there are Hermitian matrices fe 1 ; : : : ; E d g so M = dx r =1 re r where E r E s = rse r and P d r =1 E r = I. As an application: 1X ( it) k k! M k k =0 {z } innite = e imt = dx e i r t E r r =1 {z } nite

13 Quantum walk on K 2 u v Since we have A = e ita = e it 2 " # 0 1 = " # e it " # " # 1 1 = 1 1 " # " cos t i sin t i sin t cos t #

14 Graph Laplacians u w v Combinatorial, signless and normalized (San Diego): L := I L := D A = Q := D + A = p D 1 A p D 1 = p p 2 1 p p

15 Regular graphs Laplacian quantum walk on a k-regular graph: e itl = e it(ki A) ; since D = ki = e itki e ita ; because A,D commute = e itk e ita ; using e I = e I These walks are equivalent (up to phase factors and time reversal): j(e itl ) uv j = j(e ita ) uv j

16 Motivation: quantum spin network Bose (2003), Christandl et al. (2004): u w v

17 Motivation: quantum spin network Bose (2003), Christandl et al. (2004): u w v

18 Motivation: quantum spin network Bose (2003), Christandl et al. (2004): u w v

19 Quantum spin networks In a n-vertex graph G, assign to each vertex u of G a unit-norm vector u 2 C 2 (a quantum particle): u = a 0 j0i + a 1 j1i where fj0i; j1ig is an orthonormal basis for C 2. The state of all n particles is a unit-norm vector in (C 2 ) n : X = a z jz 1 i : : : jz n i z 2f0;1g n

20 Hamiltonian Consider a Hamiltonian acting on (C 2 ) n : H := X (u ;v )2E (X u X v + Y u Y v + Z u Z v ) where 2 f0; 1g and the Pauli matrices are " # " # i X = ; Y = ; Z = 1 0 i 0 " # Notation: view u 2 [n] Standard to omit. X u = I : : : I X {z} uth I : : : I

21 Information transfer Let the system do a unitary walk (t) = exp( ih t) (0) Goal: there is a time so for any unit-norm 2 C 2 (0) = j0i : : : j0i () = j0i j0i : : : It suces to take = j1i.

22 Back to quantum walk Project H onto its invariant subspace W spanned by fje u ig n u=1 where e u 2 f0; 1g n has Hamming weight one. We get a n n matrix The goal becomes ~H 8 >< >: L if = +1 A if = 0 Q if = 1 je n i = exp( i ~H t)je 1 i

23 Example: P 3 H = XXI + IXX + YYI + IYY + (ZZI + IZZ ) Let W = spanfj100i; j010i; j001ig. Restricted to W, we get the blue submatrix (adjacency matrix of P 3 ). Pauli Y is similar.

24 State transfer (Bose 2003, Christandl et al. 2004) A graph G has perfect state transfer (relative to M ) between vertices u and v at time t if for some 2 C. exp( imt)e u = e v Choices for M : adjacency matrix, or one of the Laplacians or anything Hermitian?

25 Rest of talk State Transfer in Quantum Walk relative to Combinatorial Laplacian I Tight conditions for perfect state transfer I Families of graphs (with or without) I Questions

26 Three equalities (Godsil) Say G has Laplacian perfect state transfer between u and v:

27 Three equalities (Godsil) Say G has Laplacian perfect state transfer between u and v: X 1 = j(e ilt ) uv j = j e i t r (E r ) uv j; by spectral theorem X r j(e r ) uv j; e i t r = sgn((e r ) uv ) r q q(e r ) uu (E r ) vv ; E r e u / E r e v X r s X r (E r ) uu s X r (E r ) vv ; (E r ) uu = (E r ) vv = 1 since P r E r = I

28 Three equalities (Godsil) Say G has Laplacian perfect state transfer between u and v: X 1 = j(e ilt ) uv j = j e i t r (E r ) uv j; by spectral theorem X r j(e r ) uv j; e i t r = sgn((e r ) uv ) r q q(e r ) uu (E r ) vv ; E r e u / E r e v X r s X r (E r ) uu s X r (E r ) vv ; (E r ) uu = (E r ) vv = 1 since P r E r = I Conditions: integrality and strong cospectrality e i r t = sgn((e r ) uv ); E r e u = E r e v

29 Conditions Let supp(u) := fr : E r e u 6= 0g and u ;v := fr 2 supp(u) : E r e u = E r e v g Thm (see Coutinho's thesis) G has Laplacian perfect state transfer between u and v at time if and only if all of the following hold: I E r e u = E r e v. I supp(u) Z. I r 2 + u ;v i r = gcd(supp(u)) is even. Moreover, is an odd multiple of = gcd(supp(u)).

30 Cubes u v Laplacian perfect state transfer occurs on K 2 at time t = =2. " # " # cos(t) e il(k2)t = e it i sin(t)! 0 1 t = =2 i sin(t) cos(t) 1 0 Thm (Christandl et al. 2004) The n-cube Q n has Laplacian perfect state transfer at time =2. proof: The quantum walk on G H is given by e il(g H )t = e il(g)t e il(h )t Therefore e il(q n )t = (e il(k2)t ) n.

31 Perpetual example: P 3 True/False: There is perfect state transfer relative to L in P 3. u w v

32 Perpetual example: P 3 True/False: There is perfect state transfer relative to L in P 3. u w v Thm (Coutinho & Liu) No tree has Laplacian perfect state transfer except for K 2. (P 3 is a tree) Thm (w. Alvir, Dever, Lovitz, Myer, Xu, Zhan) K 2 + G has Laplacian perfect state transfer i jgj 2 (mod 4). (P 3 = K 2 + K 1 )

33 Odd Trees Thm (Coutinho & Liu) No odd order G with an odd number of spanning trees has Laplacian perfect state transfer. Proof: By Matrix Tree Theorem: Y r 6=0 r = jgj (#spanning trees): The number of even integer eigenvalues r for which E r e u 6= 0 must be at least two for perfect state transfer to occur. A bit more is true: (Ghorbani) If G has odd order and #spanning trees 6 0 (mod 4), then L has no nonzero even integer eigenvalue.

34 Double cones Lemma: If G has Laplacian perfect state transfer at time t, so does G provided jgjt 2 2Z. Proof: Let n = jgj. Note L G = ni J L G. Thus exp( itl G ) = e itn e itj exp(itl G ) But, e itj = (e itn 1)J =n + I. Thm (w. Alvir, Dever, Lovitz, Myer, Xu, Zhan) K 2 + H has Laplacian perfect state transfer i jh j 2 (mod 4). Proof: K 2 [ H = K 2 + H and K 2 has perfect state transfer at time t = =2. Need 2 + jh j 2 4Z. Take quotient for only if.

35 Coronas True/False: There is perfect state transfer relative to L in P 4. u a b v (Frucht & Harary) The corona G H is formed by taking one copy of G and jgj copies of H. Then connect the j th vertex of G to each vertex in the j th copy of H. Note: P 4 = K 2 K 1. Thm (w. Ackelsberg, Brehm, Chan, Mundinger) No corona G H has Laplacian perfect state transfer whenever jgj 2 and jh j 1.

36 Why Coronas 3K 2 K 1 (spiky cocktail party graph) Thm (w. Ackelsberg, Brehm, Chan, Mundinger) nk 2 K 1 has Laplacian pretty good state transfer for all n. Coutinho, Godsil, Guo, & Vanhove (2015): the only strongly regular graph with perfect state transfer is nk 2 for n even.

37 Some questions True/False: There is Laplacian perfect state transfer on a graph with odd order.

38 Some questions True/False: There is Laplacian perfect state transfer on a graph with odd order. True/False: There is Laplacian perfect state transfer on a graph at time =odd.

39 Some References S. Bose. \Quantum Communication through an Unmodulated Spin Chain," Phys. Rev. Lett. 91:207901, M. Christandl, N. Datta, A. Ekert, A. Landahl. \Perfect state transfer in quantum spin networks," Phys. Rev. Lett. 92:187902, G. Coutinho, H. Liu. \No Laplacian Perfect State Transfer on Trees," arxiv.org/ E. Farhi, S. Gutmann. \Quantum computation and decision trees," Phys. Rev. A 58:915, C. Godsil. \State Transfer on Graphs," Discrete Mathematics 312(1): , 2011.