quantum query complexity and the adversary bound part ii
play

Quantum query complexity and the adversary bound Part II: Learning - PowerPoint PPT Presentation

Jul 29, 2023 •190 likes •639 views

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 ∅ ∅ ∅

Recommend

Quantum query complexity and the adversary bound Part I: The adversary bound Alexander Belov

Quantum query complexity and the adversary bound Part I: The adversary bound Alexander Belov

Quantum query complexity and the adversary bound Part I: The adversary bound Alexander Belov University of Latvia 22nd EWSCS, 5-10 March 2017, Palmse 1 / 36 Introduction Settings Why? Sub-question 1 Sub-question 2 Sub-question 3 Short

536 views • 36 slides
The Multiplicative Quantum Adversary Robert palek Quantum query complexity Quantum query

The Multiplicative Quantum Adversary Robert palek Quantum query complexity Quantum query

The Multiplicative Quantum Adversary Robert palek Quantum query complexity Quantum query complexity Given a function f: {0,1} n {0,1} m Quantum query complexity not necessarily Boolean output Given a function f: {0,1} n

1.43k views • 115 slides
Adversary Lower Bound for the k -sum Problem Robert Alexander Belov Spalek University of

Adversary Lower Bound for the k -sum Problem Robert Alexander Belov Spalek University of

Adversary Lower Bound for the k -sum Problem Robert Alexander Belov Spalek University of Latvia Google, Inc. January 22, 2013 QIP 2013, Beijing, China This work has been supported by the European Social Fund within the project Support

721 views • 45 slides
How Quantum Computing This Lower Bound Is . . . Can Help With (Continuous) How Quantum . . .

How Quantum Computing This Lower Bound Is . . . Can Help With (Continuous) How Quantum . . .

Known Advantages of . . . Need to Consider . . . We Consider Only . . . Non-Quantum Lower . . . How Quantum Computing This Lower Bound Is . . . Can Help With (Continuous) How Quantum . . . Auxiliary Quantum . . . Optimization Main

285 views • 24 slides
Quantum Hamiltonian Complexity Itai Arad Centre of Quantum Technologies National University of

Quantum Hamiltonian Complexity Itai Arad Centre of Quantum Technologies National University of

Quantum Hamiltonian Complexity Itai Arad Centre of Quantum Technologies National University of Singapore QIP 2015 1 18 / Quantum Hamiltonian Complexity condensed complexity QHC matter theory physics Local Hamiltonians 2 18 / Local

733 views • 37 slides
Complexity of cutting planes and branch-and-bound in mixed-integer optimization Amitabh Basu

Complexity of cutting planes and branch-and-bound in mixed-integer optimization Amitabh Basu

Complexity of cutting planes and branch-and-bound in mixed-integer optimization Amitabh Basu Michele Conforti Marco Di Summa Hongyi Jiang March 10, 2020 Abstract We investigate the theoretical complexity of branch-and-bound (BB)

368 views • 19 slides
All Quantum Adversaries Are Equivalent Robert palek joint work with Mario Szegedy Quantum

All Quantum Adversaries Are Equivalent Robert palek joint work with Mario Szegedy Quantum

All Quantum Adversaries Are Equivalent Robert palek joint work with Mario Szegedy Quantum query complexity Want to compute Boolean function f Input queried by oracle calls O x | i , b , z = | i , b x i , z Allow arbitrary

195 views • 18 slides
Quantum Lecture 7 The Holevo bound Typical sequences and subspaces Compression Mikael

Quantum Lecture 7 The Holevo bound Typical sequences and subspaces Compression Mikael

Quantum Lecture 7 The Holevo bound Typical sequences and subspaces Compression Mikael Skoglund, Quantum Info 1/15 The Holevo Bound Assume a discrete random variable X X with pmf p ( x ) is embedded on a set of states x , as

372 views • 8 slides
On the Query Complexity of Real Functionals Hugo Fre, Walid Gomaa, Mathieu Hoyrup Hugo

On the Query Complexity of Real Functionals Hugo Fre, Walid Gomaa, Mathieu Hoyrup Hugo

Outline Introduction Complexity of Norms Query Complexity One Oracle Access On the Query Complexity of Real Functionals Hugo Fre, Walid Gomaa, Mathieu Hoyrup Hugo Fre, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real

936 views • 28 slides
Active Adversary Lecture 7 CCA Security MAC Active Adversary An active adversary can inject

Active Adversary Lecture 7 CCA Security MAC Active Adversary An active adversary can inject

Active Adversary Lecture 7 CCA Security MAC Active Adversary An active adversary can inject messages into the channel Eve can send ciphertexts to Bob and get them decrypted Chosen Ciphertext Attack (CCA) If Bob decrypts all ciphertexts for Eve,

352 views • 14 slides
On Characterizing the Relationship between Lower Bound Methods in Communication Complexity

On Characterizing the Relationship between Lower Bound Methods in Communication Complexity

On Characterizing the Relationship between Lower Bound Methods in Communication Complexity Jiahui Liu, Prayaag Venkat Columbia University, University of Maryland College Park Thursday, August 10 Introduction: Communication Complexity

480 views • 29 slides
Quantum Weirdness Part 6 Quantum Weirdness in Materials Quantum Cryptography Quantum

Quantum Weirdness Part 6 Quantum Weirdness in Materials Quantum Cryptography Quantum

Quantum Weirdness Part 6 Quantum Weirdness in Materials Quantum Cryptography Quantum Teleportation Quantum Snake Oil Quantum Weirdness in Materials Why Materials Behave as they do Combining Atoms Into Molecules Molecular Orbital Theory

1.17k views • 72 slides
Research on Quantum Computational Complexity and Quantum Cryptography at ERATOQCI Project, JST

Research on Quantum Computational Complexity and Quantum Cryptography at ERATOQCI Project, JST

Research on Quantum Computational Complexity and Quantum Cryptography at ERATOQCI Project, JST Hiroshi IMAI ERATO QCI Project, JST Dept. Computer Science, Univ. Tokyo 21st Century COE Security Program and RDI, Chuo University 2003

356 views • 19 slides
Quantum Computability and Complexity and the Limits of Quantum Computation Eric Benjamin, Kenny

Quantum Computability and Complexity and the Limits of Quantum Computation Eric Benjamin, Kenny

Quantum Computability and Complexity and the Limits of Quantum Computation Eric Benjamin, Kenny Huang, Amir Kamil, Jimmy Kittiyachavalit University of California, Berkeley December 7, 2003 This paper will present an overview of the current

101 views • 9 slides
Quantum Weirdness Part 4 Interpretations Of Quantum Mechanics Quantum Tunnelling

Quantum Weirdness Part 4 Interpretations Of Quantum Mechanics Quantum Tunnelling

Quantum Weirdness Part 4 Interpretations Of Quantum Mechanics Quantum Tunnelling Interpretations of Quantum Mechanics What does it mean? The Copenhagen Interpretation Niels Bohr 1885-1962 Danish Physicist A wavefunction describes the

852 views • 58 slides
Circuit Complexity Circuit model aims to offer unconditional lower bound results. Computational

Circuit Complexity Circuit model aims to offer unconditional lower bound results. Computational

Circuit Complexity Circuit model aims to offer unconditional lower bound results. Computational Complexity, by Fu Yuxi Circuit Complexity 1 / 58 Synopsis 1. Boolean Circuit Model 2. Uniform Circuit 3. P / poly 4. Karp-Lipton Theorem 5. NC and

913 views • 59 slides
Quadratically Tight Relations for Randomized Query Complexity Rahul Jain Hartmut Klauck Srijita

Quadratically Tight Relations for Randomized Query Complexity Rahul Jain Hartmut Klauck Srijita

Quadratically Tight Relations for Randomized Query Complexity Quadratically Tight Relations for Randomized Query Complexity Rahul Jain Hartmut Klauck Srijita Kundu Troy Lee Miklos Santha Swagato Sanyal Jevg enijs Vihrovs Centre for

550 views • 26 slides
Quantum Oracle Classification The Case of Group Structure Mark

Quantum Oracle Classification The Case of Group Structure Mark

Quantum Oracle Classification The Case of Group Structure Mark Zhandry Princeton University Query Complexity x O:X Y O(x) Info about O Examples: Pre-image of given

932 views • 58 slides
Outline The Adversary 1 A Cracking Example! 2 The Adversary 1/44 Whos our adversary?

Outline The Adversary 1 A Cracking Example! 2 The Adversary 1/44 Whos our adversary?

Outline The Adversary 1 A Cracking Example! 2 The Adversary 1/44 Whos our adversary? What does a typical program look like? The Adversary 2/44 Whos our adversary? What does a typical program look like? What valuables does the

1.14k views • 64 slides
Active Adversary Lecture 7 CCA Security MAC Active Adversary Active Adversary An active

Active Adversary Lecture 7 CCA Security MAC Active Adversary Active Adversary An active

Active Adversary Lecture 7 CCA Security MAC Active Adversary Active Adversary An active adversary can inject messages into the channel Active Adversary An active adversary can inject messages into the channel Eve can send ciphertexts to Bob

2.08k views • 160 slides
A linear bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

A linear bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

A linear bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces Dominique Attali Jean-Daniel Boissonnat Laboratoire LIS PRISME-INRIA SM2002 2 Introduction Applications : mesh generation medial

1.05k views • 47 slides
Complexity of Well-Quasi-Orderings and Well-Structured Transition Systems Part IV: Complexity of

Complexity of Well-Quasi-Orderings and Well-Structured Transition Systems Part IV: Complexity of

Complexity of Well-Quasi-Orderings and Well-Structured Transition Systems Part IV: Complexity of WSTS Verification Philippe Schnoebelen LSV, CNRS & ENS Cachan + Oxford 1-year visitor Oxford Dept. Comp. Sci, Mar. 9th, 2012 Part IV.a: Upper

774 views • 53 slides
MAC. SKE in Practice. Lecture 5 Active Adversary Active Adversary An active adversary can

MAC. SKE in Practice. Lecture 5 Active Adversary Active Adversary An active adversary can

MAC. SKE in Practice. Lecture 5 Active Adversary Active Adversary An active adversary can inject messages into the channel Active Adversary An active adversary can inject messages into the channel Eve can send ciphertexts to Bob and get them

2.08k views • 155 slides
Strong Direct Sum for Randomized Query Complexity Eric Blais Joshua Brody University of Waterloo

Strong Direct Sum for Randomized Query Complexity Eric Blais Joshua Brody University of Waterloo

Strong Direct Sum for Randomized Query Complexity Eric Blais Joshua Brody University of Waterloo Swarthmore College Conference on Computational Complexity New Brunswick, New Jersey July 18, 2019 Outline Introduction Strong Direct

593 views • 35 slides

More recommend