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Quantum query complexity and the adversary bound Part II: Learning graphs Alexander Belov University of Latvia 22nd EWSCS, 5-10 March 2017, Palmse 1 / 35 Learning graphs Dual Adversary Certificate Structure Examples Idea Construction

  1. Quantum query complexity and the adversary bound Part II: Learning graphs Alexander Belov University of Latvia 22nd EWSCS, 5-10 March 2017, Palmse 1 / 35

  2. Learning graphs Dual Adversary Certificate Structure Examples Idea Construction Feasibility Learning graphs Objective value Summary OR function Symmetry Element Distinctness Triangle Detection 2 / 35

  3. Dual Adversary Learning graphs Recall the dual adversary bound Dual Adversary Certificate Structure Examples � max X j [ [ z, z ] ] Idea minimise z Construction j ∈ [ n ] Feasibility Objective value � X j [ [ x, y ] ] = 1 whenever f ( x ) � = f ( y ) ; subject to Summary j : x j � = y j OR function Symmetry X j is a p.s.d. D × D matrix for all j ∈ [ n ] , Element Distinctness Triangle Detection 3 / 35

  4. Dual Adversary Learning graphs Recall the dual adversary bound Dual Adversary Certificate Structure Examples � max X j [ [ z, z ] ] Idea minimise z Construction j ∈ [ n ] Feasibility Objective value � X j [ [ x, y ] ] = 1 whenever f ( x ) � = f ( y ) ; subject to Summary j : x j � = y j OR function Symmetry X j is a p.s.d. D × D matrix for all j ∈ [ n ] , Element Distinctness Triangle Detection How do we ensure the feasibility condition? In general this is difficult, � but there is a way for functions with short certificates. � 3 / 35

  5. Certificate Structure Learning graphs Dual Adversary Certificate Structure Examples Idea Function Construction f : [ q ] n ⊇ D → { 0 , 1 } Feasibility Objective value For x ∈ f − 1 (1) , write out: Summary OR function Symmetry � � M x = S ⊆ [ n ] | x S is enough to deduce f ( x ) = 1 . Element Distinctness Triangle Detection The set of all M x is the certificate structure of f . (Interested in inclusion-wise minimal M x only.) 4 / 35

  6. Example Learning graphs Element Distinctness Problem Function f : [ q ] n → { 0 , 1 } : there are two equal elements. Dual Adversary Certificate Structure Examples 1147 1417 1471 Idea ���� ���� ���� ❏ ���� ���� ���� ���� ❏ ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ❏ Construction ✴✴ ✴✴ ✴✴ ❏ ❏ ❏ t t t ❏ ❏ ❏ t t t ✎ ✎ ✎ Feasibility ❏ ❏ ❏ t t t ❏ ❏ ❏ t t t ✎ ✎ ✎ t t t ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� Objective value ▲ ❏ ▲ ❏ ▲ ❏ ❇ ❇ ❇ ❁ ❁ ❁ ✮ ���� ♣♣♣♣♣♣ ▲ ❏ ✴ ✮ ���� ♣♣♣♣♣♣ ▲ ❏ ✴ ✮ ���� ♣♣♣♣♣♣ ▲ ❏ ✴ ���� rrrrr ���� rrrrr ���� rrrrr t t t ❇ ❁ ❇ ❁ ❇ ❁ ✎✎ ���� ✮ ▲ ❏ ✎✎ ���� ✮ ▲ ❏ ✎✎ ���� ✮ ▲ ❏ t ✕✕ ���� ✴ t ✕✕ ���� ✴ t ✕✕ ���� ✴ ❁ ❏ ❁ ❏ ❁ ❏ t ✜✜ ▲ t ▲ ✜✜ t ✜✜ ▲ ❇ ❇ ❇ ❏ ❏ ❏ t ▲ t ▲ t ▲ Summary ���� t ���� t ���� t ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ▲ ▲ ▲ ❏ ❏ ❏ ❇ ❇ ❇ ❁ ❁ ❁ ✴ ✴ ✴ ���� ♣♣♣♣♣♣ ❏ ���� rrrrr ▲ ✮ ���� ♣♣♣♣♣♣ ❏ ���� rrrrr ▲ ✮ ���� ♣♣♣♣♣♣ ❏ ���� rrrrr ▲ ✮ t t t OR function ❁ ❁ ❁ ❏ ▲ ❇ ❏ ▲ ❇ ❏ ▲ ❇ ✮ t t ✮ ✮ t ✴ ✴ ✴ ❏ ✎✎ ❏ ✎✎ ❏ ✎✎ ✕✕ ✜✜ ▲ ❁ ✕✕ ✜✜ ▲ ❁ ✕✕ ✜✜ ▲ ❁ ❇ t ❇ t ❇ t ❏ ❏ ❏ ▲ t ▲ t ▲ t ���� t ���� t ���� t ���� ���� ���� ���� ✴ ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ✴ ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ✴ ���� ���� ���� ���� ���� ���� ���� ���� 1 ❏ 2 3 4 1 ❏ 2 3 4 1 ❏ 2 3 4 Symmetry ❏ ✴ ❏ ✴ ❏ ✴ t t t ❏ ❏ ❏ t t t ❏ ✎✎ ❏ ✎✎ ❏ ✎✎ t t t ❏ ❏ ❏ t t t t t t ���� ���� ���� Element Distinctness ∅ ∅ ∅ Triangle Detection 4117 4171 4711 ���� ���� ���� ❏ ���� ���� ���� ���� ❏ ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ❏ ���� ✴✴ ✴✴ ✴✴ ❏ ❏ ❏ t t t ❏ ❏ ❏ t t t ✎ ✎ ✎ ❏ ❏ ❏ t t t ❏ ❏ ❏ t t t ✎ ✎ ✎ t t t ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ▲ ❏ ▲ ❏ ▲ ❏ ❇ ❁ ❇ ❁ ❇ ❁ ✮ ���� ♣♣♣♣♣♣ ✴ ✮ ���� ♣♣♣♣♣♣ ✴ ✮ ���� ♣♣♣♣♣♣ ✴ ���� rrrrr ▲ ❏ ���� rrrrr ▲ ❏ ���� rrrrr ▲ ❏ t t t ❇ ❁ ❇ ❁ ❇ ❁ ✎✎ ���� ✮ ▲ ❏ ✎✎ ���� ✮ ▲ ❏ ✎✎ ���� ✮ ▲ ❏ t ✕✕ ���� t ✕✕ ���� t ✕✕ ���� ✴ ✴ ✴ ❏ ❏ ❏ t ❁ ✜✜ ▲ t ❁ ✜✜ ▲ t ❁ ✜✜ ▲ ❇ ❇ ❇ ❏ ❏ ❏ ▲ ▲ ▲ t t t ���� t ���� t ���� t ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ▲ ▲ ▲ ❏ ❏ ❏ ❇ ❇ ❇ ❁ ❁ ❁ ✴ ✴ ✴ ���� ♣♣♣♣♣♣ ❏ ▲ ✮ ���� ♣♣♣♣♣♣ ❏ ▲ ✮ ���� ♣♣♣♣♣♣ ❏ ▲ ✮ ���� rrrrr t ���� rrrrr t ���� rrrrr t ❁ ❁ ❁ ❏ ❇ ❏ ❇ ❏ ❇ ▲ ✮ ▲ ✮ ▲ ✮ ✴ t ✴ t ✴ t ✎✎ ✎✎ ✎✎ ✕✕ ❏ ▲ ❁ ✕✕ ❏ ▲ ❁ ✕✕ ❏ ▲ ❁ ✜✜ ❇ t ✜✜ ❇ t ✜✜ ❇ t ❏ ❏ ❏ ▲ t ▲ t ▲ t ���� t ���� t ���� t ���� ���� ���� ���� ���� ���� ✴ ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ✴ ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ✴ ���� ���� ���� ���� ���� ���� ���� ���� 1 ❏ 2 3 4 1 ❏ 2 3 4 1 ❏ 2 3 4 ✴ ✴ ✴ ❏ ❏ ❏ t t t ❏ ❏ ❏ t t t ❏ ✎✎ ❏ ✎✎ ❏ ✎✎ t t t ❏ ❏ ❏ t t t t t t ���� ���� ���� 5 / 35 ∅ ∅ ∅

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