Quantum Control Theory and Graphs

Simone Severini1

UCL

August 23, 2012

1Joint work with Chris Godsil, Jamie Smith (UWaterloo), and Steve Kirkland (HamiltonInstitute); 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, 2012 1 / 22

Real-world motivations

There are (at least) two important real-world motivations for a quantumapproach to control theory and graphs:

Technology Nature

Study information transfer Study energy transportin engineered nanodevices in natural biosystems

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Introduction

We look at graphs as models of physical networks:I The vertices correspond to physical objects (specifically to quantummechanical particles);

I The edges correspond to physical interactions.

Each vertex/particle enjoys properties that specify its state (imagine, e.g., acomplex vector associated to each vertex; sometimes in Combinatorics this iscalled a vector labeling).In this talk, we are interested in two directions:

I Controlling the state of a system: preparing the network (i.e., the state ofeach vertex/particle) in an arbitrary global state.

I (As a special case,) Transfering information between vertices: "moving" thestate of a particle into another particle.

In the study of this topic, the Physics’background is minimal, but theMathematics turns out to be fairly rich.

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The Physics backgroundLet G be a graph on n vertices (possibly with self-loops).We associate the standard basis vector i ∈ {1, ...,n} in Cn to a vertex i.Let A be the adjacency matrix of G .

FactA (coherent) quantum dynamics on G is governed by the equation

e−iAt i = U(t)i = vt ,

where U(t) := e−iAt with t ∈ R>0; vt is the state of the system at time t.

The interface between quantum control theory and graphs is (also) concernedwith the following general problem:

Problem ("The problem")

Given graphs {G1,G2, ..,Gk}.Study the collection of matrices 〈U1(t1),U2(t2), ...,Uk (tk )〉 (t1, t2, ..., tk ∈ R≥0)and the subgroup of the unitary group that they generate under matrixmultiplication.

Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, 2012 4 / 22

Controllable graphs

Let G be a graph on n vertices and let z ∈ Rn .

Let Wz :=(z Az . . . An−1z

)an n× n matrix with entries in Z≥0.

When z is the characteristic vector of some set S ⊆ V , the matrix WS iscalled a walk matrix of G with respect to S .The pair (G , S) is said to be controllable if the matrix WS is invertible (i.e.,det(WS ) 6= 0). A graph G is said be controllable if (G , 1) is controllable.

TheoremA pair (G ,S) is controllable if and only if the unitary matrices UA(s) = e−iAt andUS (t) = e−izz

T t ′ (t, t ′ ∈ R≥0) generate a dense subgroup of the unitary groupU(n) (n ≥ 2).a

a"if": C. Godsil, SS, Phys. Rev. A 81, 052316 (2010). arXiv:0910.5397v3 [quant-ph]; "onlyif": D. Burgarth, D. D’Alessandro, L. Hogben, SS, M. Young, IEEE Trans. Auto. Contr. (toappear). arXiv:1111.1475v1 [quant-ph].

Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, 2012 5 / 22

Controllable graphs: examples

Indeed, controllability occurs together with the ability of constructing withreasonable accuracy any unitary matrix of the appropriate dimension (whichcorresponds to preparing the physical network in an arbitrary global state).

One can verify by exhaustive search that there are no controllable graphs onn ≤ 5 vertices.The following are all controllable (connected, nonisomorphic) graphs on 6vertices:

Regular graphs are not controllable.

Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, 2012 6 / 22

Controllable graphs: open problems

ProblemHow diffi cult is to determine the smallest S such that (G , S) is a controllable pair?

ProblemAsymptotically almost surely is every graph controllable?

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Zero-forcing

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Zero-forcing

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Zero-forcing

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Zero-forcing

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Zero-forcing

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Zero-forcing

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Zero-forcing

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Zero-forcing

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Zero-forcing

The set of the initially colored vertices is called a zero-forcing set.

TheoremIf S is a zero-forcing set then the pair (G , S) is controllable.a

aD. Burgarth, V. Giovannetti, Phys. Rev. Lett. 99, 100501, (2007). arXiv:0704.3027v1[quant-ph]. S. Fallat and L. Hogben, LAA, 426: 558582, 2007.

Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, 2012 16 / 22

Perfect state transferThere is perfect state transfer (PST) in a graph G between vertex i andvertex j if there is t ∈ R>0 such that |jTU(t)i| = 1.

Theorem(1) Every hypecube has PST between antipodal vertices for t = π/2; (2) Ann-path has PST only if n = 2, 3; (3) The Cayley graphs of Zk

2 have PST if andonly if the Cayely elements do not sum up to the identity. ((4) For Zn we have acomplete classification.) a

aM. Christandl, N. Datta, A. Ekert, A. Landahl Phys. Rev. Lett. 92, 187902 (2004); GodsilBernasconi, Godsil, SS, Phys. Rev. A 78. 052320 (2008). arXiv:0808.0510v1 [quant-ph]; X.Zhang, C. Godsil LAA, 435(10) 2011; C Godsil. arXiv:1102.4898v2 [math.CO]. N. Saxena, SS, I.Shparlinski, Int. J. of Quantum Inf. 5 (2007). arXiv:quant-ph/0703236v1. M. Bašic.arXiv:1104.1825v1 [cs.DM]

A graph G is periodic if there is t ∈ R>0 such that |iTU(t)i| = 1 for every i .

TheoremA connected regular graph is periodic if and only if its eigenvalues are integers.a

aC. Godsil, arXiv:0806.2074v2 [math.CO].

Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, 2012 17 / 22

Pretty good state transfer

There is pretty good state transfer (PGST) in G between vertex i andvertex j if for every ε > 0 there is t ∈ R>0 such that |jTU(t)i| > 1− ε.

TheoremThere is PGST in the n-path if and only if n = p − 1 or 2p − 1, where p is aprime, or if n = 2m − 1, for every m ≥ 1.a

aC. Godsil, S. Kirkland, SS, J. Smith, Phys. Rev. Lett. 109, 050502 (2012).arXiv:1201.4822v2 [quant-ph]; see also L. Vinet, A. Zhedanov, for other examples of PGST.arXiv:1205.4680v2 [quant-ph]

Fact"In principle" we have a method for primality testing based on a physicaldynamics.

Simone Severini (UCL) Shanghai Conference on Algebraic Combinatorics 2012 August 23, 2012 18 / 22

Average mixing

Define a uni-stochastic matrix M(t) := U(t) ◦ U(−t), where "◦" denotesSchur product.

The average mixing matrix is

M(G ) := limT→∞

1T

∫ T0M(t)dt = ∑

rE ◦2r ,

where each Er is an idempotent in the spectral decomposition of theadjacency matrix. (E.g., for the n-cycle with n odd M(Cn) = n−1

n2 J +1n I ,

where J is the all-ones matrix.)

TheoremThe average mixing matrix of a graph is rational.a

aC. Godsil. arXiv:1103.2578v3 [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, 2012 19 / 22

Open problems

ProblemGiven suffi cient combinatorial conditions for PST/PGST.

ProblemDetermine which graphs are uniform mixing.

ProblemConstruct a theory of state transfer under perturbations. (This would correspondto study more realistic "noisy" systems.)

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Conclusions

We have seen some notions and results related to quantum control andgraphs.The area suggests many open problems that is worth considering.

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Quantum Physics of Information Workshop, 27-28 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), FernandoBrandao (ETH, Zurich), Andreas Winter (University of Bristol and NationalUniversity of Singapore), Shunlong Luo (Chinese Academy of Sciences),David Poulin (University of Sherbrooke), Stephanie Wehner (NationalUniversity of Singapore), Renato Renner (ETH, Zurich), Guihua Zeng(SJTU), Richard Jozsa (University of Cambridge)

Free registration; please email Sandy Nie (yingyunie@sjtu.edu.cn) toconfirm attendance.

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