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Graph Spectra and Continuous Quantum Walks Gabriel Coutinho, Chris Godsil September 1, 2020 Preface If A is the adjacency matrix of a graph X, we define a transition matrix U(t) by U(t) = exp(itA). Physicists say that U(t) determines a continuous quantum walk. Most ques- tions they ask about these matrices concern the squared absolute values of the entries of U(t) (because these may be determined by measurement, and the entries themselves cannot be). The matrices U(t) are unitary and U(t) = U(−t), so we may say that the physicists are concerned with ques- tions about the entries of the Schur product M(t) = U(t) ◦ U(t) = U(t) ◦ U(−t). Since U(t) is unitary, the matrix M(t) is doubly stochastic. Hence each column determines a probability density on V (X) and there are two extreme cases: (a) A row of M(t) has one nonzero entry (necessarily equal to 1). (b) All entries in a row of U(t) are equal to 1/|V (X)|. In case (a), there are vertices a and b of X such that |U(t)a,b| = 1. If a =6 b, we have perfect state transfer from a to b at time t, otherwise we say that X is periodic at a at time t. Physicists are particularly interested in perfect state transfer. In case (b), if the entries of the a-row of M(t) is constant, we have local uniform mixing at a at time t. (Somewhat surprisingly these two concepts are connected—in a number of cases, perfect state transfer and local uniform mixing occur on the same graph.) The quest for graphs admitting perfect state transfer or uniform mixing has unveiled new results in spectral graph theory, and rich connections to other fields of mathematics. This also brought us to explore more topics in iii quantum information and their interplay with combinatorics. This book is, in part, a progress report on these questions. For us, the biggest surprise is the extent to which tools from algebraic graph theory prove useful. So we treat this in somewhat more detail than is strictly necessary. Some of it is standard, some is old stuff repackaged, and some is new material (e.g., controllability, strongly cospectral vertices) that has been developed to deal with quantum walks. But combinatorics is not everything: we also meet with Lie groups, various flavours of number theory, and almost periodic functions. (And so a second surprise is the number of different mathematical areas entangled with our topic.) we do not treat discrete quantum walks. We do not treat quantum algo- rithms or quantum computation, nor do we deal with questions about com- plexity, error correction, non-local games, and the quantum circuit model. We discuss a little of the related physics. We have focussed on questions that are both mathematically interesting and have some physical signifi- cance, since this overlap is often a sign of a fruitful outcome. We have had useful comments on these notes from many people, includ- ing Dave Witte Morris, Tino Tamon, Sasha Jurišic and members of his seminar, Alexis Hunt, David Feder, Henry Liu, Harmony Zhan, Nicholas Lai, Xiaohong Zhang. This is a preliminary version intended to be used a class notes for a grad course at UFMG in 2020. We are still working on it, and several chapters lack references, exercises, and revision. We wel- come any comments to [email protected] or [email protected]. Contents Preface iii Contents v I Introduction 1 1 Continuous Quantum Walks 3 1.1 Basics . 4 1.2 Products . 6 1.3 State Transfer and Mixing . 7 1.4 Symmetry and Periodicity . 8 1.5 Spectral Decomposition for Adjacency Matrices . 9 1.6 Using Spectral Decomposition . 11 1.7 Complete Graphs . 12 1.8 Bipartite Graphs . 13 1.9 Uniform Mixing on Bipartite Graphs and Hadamard matrices 15 1.10 Vertex-Transitive Graphs . 16 1.11 Pretty Good State Transfer . 18 1.12 Oriented Graphs . 20 1.13 Universal State Transfer . 21 1.14 Path Plots . 23 1.15 One More Plot . 27 2 Physics 29 2.1 Quantum Systems . 29 2.2 Density Matrices and POVMs . 31 2.3 Qubits and 2 × 2 Hermitians . 34 v Contents 2.4 Entanglement . 36 2.5 Walks: Adjacency Matrix . 37 2.6 Walks: More Matrices . 39 2.7 Orbits of Density Matrices . 40 II Spectra of Graphs 45 3 Equitable Partitions 47 3.1 Partitions and Projections . 47 3.2 Walks and orthogonal matrices . 50 3.3 Automorphisms . 52 3.4 Quotients of the d-cube and of complete bipartite graphs . 54 4 Spectrum and Walks 57 4.1 Formal power series . 58 4.2 Walk generating function . 59 4.3 Closed Walks at a Vertex . 60 4.4 Walks between two vertices . 62 4.5 The Jacobi trick . 64 4.6 All Walks . 65 4.7 1-Sums and 2-Sums . 68 4.8 Reduced Walks . 70 4.9 Hermitian matrices . 71 4.10 No transfer at all on some signed graphs . 72 5 Walk Modules 75 5.1 Walk Modules . 75 5.2 Dual degree and covering radius . 76 5.3 Controllable pairs, symmetries, rational functions . 78 5.4 Controllable subsets . 80 5.5 Controllable Graphs . 82 5.6 Isomorphism . 83 5.7 Isomorphism of Controllable Pairs . 86 5.8 Laplacians . 87 5.9 Control Theory . 88 6 Cospectral Vertices 93 vi Contents 6.1 Walk modules of cospectral vertices . 93 6.2 Characterizing Cospectral Vertices . 94 6.3 Walk-Regular Graphs . 95 6.4 State Transfer on Walk-Regular Graphs . 97 6.5 Parallel Vertices . 98 6.6 Strongly Cospectral Vertices . 99 6.7 Examples of Strongly Cospectral Vertices . 101 6.8 Characterizing Strongly Cospectral Vertices . 103 6.9 Matrix Algebras . 105 IIIState Transfer and Periodicity 109 7 State transfer 111 7.1 Three Inequalities . 112 7.2 Near Enough: Cospectrality . 114 7.3 Ratio Condition . 117 7.4 Using Integrality . 122 7.5 Perfect state transfer . 123 7.6 Bipartite and Regular Graphs . 126 7.7 No Control . 127 8 Recurrence and approximations 131 8.1 Periods . 131 8.2 Tori . 133 8.3 Dense Subgroups of U(n) . 137 8.4 Periods of Functions . 139 8.5 Almost Periodic . 140 8.6 Means . 141 8.7 Zeros of Transition Functions . 142 9 Real State Transfer 145 9.1 Real State Transfer . 146 9.2 Pretty Good State Transfer . 148 9.3 Algebras . 149 9.4 Timing . 150 9.5 Periodicity and Eigenvalues . 151 9.6 Algebraic States . 153 vii Contents 9.7 Fractional Revival . 155 9.8 Symmetry . 156 9.9 Commutativity . 158 9.10 Cospectrality . 163 9.11 Characterizing Fractional Revival . 164 10 Paths and Trees 169 10.1 Recurrences for Paths . 169 10.2 Eigenvalues and Eigenvectors . 171 10.3 Line Graphs and Laplacians . 175 10.4 Inverses . 177 10.5 Eigenthings for Laplacians of Paths . 178 10.6 No Transfer on Paths . 180 10.7 PST on Laplacians . 180 10.8 Twins . 182 10.9 No Laplacian Perfect State Transfer on Trees . 183 11 Pretty Good State Transfer 185 11.1 PGST on Paths . 185 11.2 Phases and Pretty Good State Transfer . 186 11.3 No PGST on Paths . 188 11.4 Path Laplacians . 190 11.5 Double Stars . 193 12 Joins and Products 197 12.1 Eigenvalues and Eigenvectors of Joins . 197 12.2 Spectral Idempotents for Joins . 199 12.3 The Transition Matrix of a Join . 201 12.4 Joins with K2 and K2 . 202 12.5 Irrational Periods and Phases . 205 12.6 The Direct Product . 206 12.7 Double Covers and Switching Graphs . 208 12.8 Laplacians and Complements . 210 IVAlgebraic Connections 213 13 Average Mixing 215 viii Contents 13.1 Average Mixing . 215 13.2 Properties of the Average Mixing Matrix . 217 13.3 Uniform Average Mixing? . 219 13.4 An Ordering . 220 13.5 Strongly Cospectral Vertices . 221 13.6 Integrality . 222 13.7 Near Enough Implies Strongly Cospectral . 224 13.8 Average Mixing on Paths . 225 13.9 Path Laplacians . 226 13.10 Cycles . 228 13.11 Pseudocyclic Schemes . 230 13.12 Average Mixing on Pseudocyclic Graphs . 232 13.13 Average Mixing on Oriented Graphs . 233 13.14 Average States . 234 14 Orthogonal Polynomials 237 14.1 Examples . 238 14.2 The Three-Term Recurrence . ..
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