Quantum Control Theory and Graphs

Transcription

1 Quantum Control Theory and Graphs Simone Severini 1 UCL August 23, Joint work with Chris Godsil, Jamie Smith (UWaterloo), and Steve Kirkland (Hamilton Institute); Leslie Hogben, Michael Young, Domenico D Alessandro (Iowa State University), Daniel Burgarth (Imperial College); Supported by the Royal Society. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

2 Real-world motivations There are (at least) two important real-world motivations for a quantum approach to control theory and graphs: Technology Nature Study information transfer in engineered nanodevices Study energy transport in natural biosystems Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

3 Introduction We look at graphs as models of physical networks: The vertices correspond to physical objects (specifically to quantum mechanical particles); The edges correspond to physical interactions. Each vertex/particle enjoys properties that specify its state (imagine, e.g., a complex vector associated to each vertex; sometimes in Combinatorics this is called a vector labeling). In this talk, we are interested in two directions: Controlling the state of a system: preparing the network (i.e., the state of each vertex/particle) in an arbitrary global state. (As a special case,) Transfering information between vertices: "moving" the state of a particle into another particle. In the study of this topic, the Physics background is minimal, but the Mathematics turns out to be fairly rich. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

4 The Physics background Fact Let G be a graph on n vertices (possibly with self-loops). We associate the standard basis vector i {1,..., n} in C n to a vertex i. Let A be the adjacency matrix of G. A (coherent) quantum dynamics on G is governed by the equation e iat i = U(t)i = v t, where U(t) := e iat with t R >0 ; v t is the state of the system at time t. The interface between quantum control theory and graphs is (also) concerned with the following general problem: Problem ("The problem") Given graphs {G 1, G 2,.., G k }. Study the collection of matrices U 1 (t 1 ), U 2 (t 2 ),..., U k (t k ) (t 1, t 2,..., t k R 0 ) and the subgroup of the unitary group that they generate under matrix multiplication. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

5 Controllable graphs Let G be a graph on n vertices and let z R n. Let W z := ( z Az... A n 1 z ) an n n matrix with entries in Z 0. When z is the characteristic vector of some set S V, the matrix W S is called a walk matrix of G with respect to S. The pair (G, S) is said to be controllable if the matrix W S is invertible (i.e., det(w S ) = 0). A graph G is said be controllable if (G, 1) is controllable. Theorem A pair (G, S) is controllable if and only if the unitary matrices U A (s) = e iat and U S (t) = e izzt t (t, t R 0 ) generate a dense subgroup of the unitary group U(n) (n 2). a a "if": C. Godsil, SS, Phys. Rev. A 81, (2010). arxiv: v3 [quant-ph]; "only if": D. Burgarth, D. D Alessandro, L. Hogben, SS, M. Young, IEEE Trans. Auto. Contr. (to appear). arxiv: v1 [quant-ph]. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

6 Controllable graphs: examples Indeed, controllability occurs together with the ability of constructing with reasonable accuracy any unitary matrix of the appropriate dimension (which corresponds to preparing the physical network in an arbitrary global state). One can verify by exhaustive search that there are no controllable graphs on n 5 vertices. The following are all controllable (connected, nonisomorphic) graphs on 6 vertices: Regular graphs are not controllable. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

7 Controllable graphs: open problems Problem How diffi cult is to determine the smallest S such that (G, S) is a controllable pair? Problem Asymptotically almost surely is every graph controllable? Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

8 Zero-forcing Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

9 Zero-forcing Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

10 Zero-forcing Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

11 Zero-forcing Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

12 Zero-forcing Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

13 Zero-forcing Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

14 Zero-forcing Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

15 Zero-forcing Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

16 Zero-forcing The set of the initially colored vertices is called a zero-forcing set. Theorem If S is a zero-forcing set then the pair (G, S) is controllable. a a D. Burgarth, V. Giovannetti, Phys. Rev. Lett. 99, , (2007). arxiv: v1 [quant-ph]. S. Fallat and L. Hogben, LAA, 426: , Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

17 Perfect state transfer There is perfect state transfer (PST) in a graph G between vertex i and vertex j if there is t R >0 such that j T U(t)i = 1. Theorem (1) Every hypecube has PST between antipodal vertices for t = π/2; (2) An n-path has PST only if n = 2, 3; (3) The Cayley graphs of Z k 2 have PST if and only if the Cayely elements do not sum up to the identity. ((4) For Z n we have a complete classification.) a a M. Christandl, N. Datta, A. Ekert, A. Landahl Phys. Rev. Lett. 92, (2004); Godsil Bernasconi, Godsil, SS, Phys. Rev. A (2008). arxiv: v1 [quant-ph]; X. Zhang, C. Godsil LAA, 435(10) 2011; C Godsil. arxiv: v2 [math.co]. N. Saxena, SS, I. Shparlinski, Int. J. of Quantum Inf. 5 (2007). arxiv:quant-ph/ v1. M. Bašić. arxiv: v1 [cs.dm] A graph G is periodic if there is t R >0 such that i T U(t)i = 1 for every i. Theorem A connected regular graph is periodic if and only if its eigenvalues are integers. a a C. Godsil, arxiv: v2 [math.co]. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

18 Pretty good state transfer There is pretty good state transfer (PGST) in G between vertex i and vertex j if for every ɛ > 0 there is t R >0 such that j T U(t)i > 1 ɛ. Theorem There is PGST in the n-path if and only if n = p 1 or 2p 1, where p is a prime, or if n = 2 m 1, for every m 1. a a C. Godsil, S. Kirkland, SS, J. Smith, Phys. Rev. Lett. 109, (2012). arxiv: v2 [quant-ph]; see also L. Vinet, A. Zhedanov, for other examples of PGST. arxiv: v2 [quant-ph] Fact "In principle" we have a method for primality testing based on a physical dynamics. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

19 Average mixing Define a uni-stochastic matrix M(t) := U(t) U( t), where " " denotes Schur product. The average mixing matrix is 1 T M(G ) := lim M(t)dt = T T 0 r E 2 r, where each E r is an idempotent in the spectral decomposition of the adjacency matrix. (E.g., for the n-cycle with n odd M(C n ) = n 1 J + 1 n 2 n I, where J is the all-ones matrix.) Theorem The average mixing matrix of a graph is rational. a a C. Godsil. arxiv: v3 [math.co]. A graph is uniform mixing at time t if U(t) is flat, i.e., all entries of (t) have the same absolute value. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

20 Open problems Problem Given suffi cient combinatorial conditions for PST/PGST. Problem Determine which graphs are uniform mixing. Problem Construct a theory of state transfer under perturbations. (This would correspond to study more realistic "noisy" systems.) Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

21 Conclusions We have seen some notions and results related to quantum control and graphs. The area suggests many open problems that is worth considering. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22

22 Quantum Physics of Information Workshop, Aug Where? Rm 601 Pao Yue-Kong Library, Minhang Campus, Shanghai Jiao Tong University Who? Charlier Bennett (IBM Watson Research Center), Quanhua Xu (Université Franche-Comté and Wuhan University), Mingsheng Ying (Tsinghua University), Giannicola Scarpa (CWI, Amsterdam), Fernando Brandao (ETH, Zurich), Andreas Winter (University of Bristol and National University of Singapore), Shunlong Luo (Chinese Academy of Sciences), David Poulin (University of Sherbrooke), Stephanie Wehner (National University of Singapore), Renato Renner (ETH, Zurich), Guihua Zeng (SJTU), Richard Jozsa (University of Cambridge) Free registration; please Sandy Nie to confirm attendance. Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, / 22