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 outline Introduction Basic Adversary Spectral Adversary Dual Adversary Composition 2 / 36
� � � Settings Introduction A computational problem: Settings f : [ q ] n ⊇ D → { 0 , 1 } Why? Sub-question 1 Sub-question 2 Sub-question 3 A computational device (deterministic, randomised, or quantum) : Short outline Basic Adversary Device output Spectral Adversary Dual Adversary Composition x 1 x 2 x 3 . . . x n Query complexity: number of queries to the input string � (worst-case) required to solve the problem. Both upper and lower bounds. � 3 / 36
Why? Introduction Settings Why? Sub-question 1 Sub-question 2 Why quantum query complexity? Sub-question 3 Short outline Basic Adversary Spectral Adversary Dual Adversary � Why quantum query complexity? Composition � Why quantum query complexity? � Why quantum query complexity? 4 / 36
Sub-question 1 Introduction Why query complexity? Settings Why? Reason 1: For some problems, this is the right model. Sub-question 1 Sub-question 2 Sub-question 3 Short outline Basic Adversary Spectral Adversary In hypothesis testing, we are interested in reducing the number � Dual Adversary of experiments. Experiments are expensive, computation is not Composition so. In property testing, complexity is traditionally measured in the � number of samples. 5 / 36
Sub-question 1 Introduction Why query complexity? Settings Why? Reason 1: For some problems, this is the right model. Sub-question 1 Sub-question 2 Reason 2: Because we can. Sub-question 3 Short outline Basic Adversary Spectral Adversary Dual Adversary We are interested in time complexity usually, but in most cases � Composition we can only prove lower bounds by proving lower bound on query complexity. In cryptography, proofs in the random oracle model are, � essentially, query complexity lower bounds. 6 / 36
Sub-question 1 Introduction Why query complexity? Settings Why? Reason 1: For some problems, this is the right model. Sub-question 1 Sub-question 2 Reason 2: Because we can. Sub-question 3 Reason 3: Query algorithm can give insights into the problem. Short outline Basic Adversary Spectral Adversary Dual Adversary Composition Trying to minimise query complexity of an algorithms result in � our better understanding of algorithms and the problem. One might hope to implement query-efficient algorithm � time-efficiently. 7 / 36
Sub-question 1 Introduction Why query complexity? Settings Why? Reason 1: For some problems, this is the right model. Sub-question 1 Sub-question 2 Reason 2: Because we can. Sub-question 3 Reason 3: Query algorithm can give insights into the problem. Short outline Reason 4: Query complexity gives impossibility results. Basic Adversary Spectral Adversary Dual Adversary Composition We can prove that black-box approaches are doomed. � We can exclude a lower bound via query complexity. � 8 / 36
Sub-question 2 Introduction Why quantum query complexity? Settings Why? Sub-question 1 Sub-question 2 Sub-question 3 Short outline Basic Adversary Spectral Adversary Dual Adversary Composition 9 / 36
Sub-question 3 Introduction Why quantum query complexity? Settings Why? Reason 1: You do not need to understand quantum computation to Sub-question 1 Sub-question 2 study quantum query complexity. Sub-question 3 Short outline Basic Adversary Spectral Adversary Dual Adversary Composition Quantum query complexity is tightly characterised by a � semi-definite optimisation problem: the adversary bound. 10 / 36
Sub-question 3 Introduction Why quantum query complexity? Settings Why? Reason 1: You do not need to understand quantum computation to Sub-question 1 Sub-question 2 study quantum query complexity. Sub-question 3 Reason 2: In some sense, we understand quantum query Short outline complexity better than randomised query complexity. Basic Adversary Spectral Adversary Dual Adversary Composition Randomised query complexity is a collection of separated � results. Quantum query complexity start resembling a theory. � 11 / 36
Sub-question 3 Introduction Why quantum query complexity? Settings Why? Reason 1: You do not need to understand quantum computation to Sub-question 1 Sub-question 2 study quantum query complexity. Sub-question 3 Reason 2: In some sense, we understand quantum query Short outline complexity better than randomised query complexity. Basic Adversary Reason 3: Quantum query complexity provides upper bound on Spectral Adversary Dual Adversary approximate polynomial degree. Composition Approximating polynomials are used in various fields (e.g., � learning theory) One can get an approximating polynomial by constructing a � quantum query algorithm. 12 / 36
Short outline Introduction Settings Why? Sub-question 1 We are mostly considering total functions Sub-question 2 Sub-question 3 f : [ q ] n → { 0 , 1 } . Short outline Basic Adversary Spectral Adversary Dual Adversary Composition For such functions, one can only get a polynomial improvement � (at most 6-th power) We will mostly consider sub-quadratic improvements. � 13 / 36
Introduction Basic Adversary Main idea Quantum Version k -threshold function Spectral Adversary Basic Adversary Dual Adversary Composition 14 / 36
Main idea Introduction Basic Adversary Distinguishing inputs Main idea Quantum Version k -threshold function Spectral Adversary OR function: Dual Adversary Composition ······ 100 ... 00 ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ 010 ... 00 ◗◗◗◗◗◗◗◗◗◗◗◗◗ 001 ... 00 000 ... 10 000 ... 01 ❉❉❉❉❉❉❉❉ ❥ q ❥ ❥ q ❥ q ❥ ❥ q ❥ q ❥ ❥ q ❥ q ❥ ❥ q ❥ ❥ q ❥ q ❥ 000 ... 00 No randomised algorithm can distinguish all these pairs of inputs in o ( n ) queries. Hence, randomised query complexity is Ω( n ) . 15 / 36
Quantum Version Introduction Basic Adversary ······ 100 ... 00 ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ 010 ... 00 ◗◗◗◗◗◗◗◗◗◗◗◗◗ 001 ... 00 000 ... 10 000 ... 01 ❉❉❉❉❉❉❉❉ ❥ q ❥ Main idea q ❥ ❥ q ❥ ❥ q ❥ q ❥ ❥ Quantum Version q ❥ ❥ q ❥ q ❥ ❥ q ❥ k -threshold function q ❥ 000 ... 00 Spectral Adversary Dual Adversary Composition Select: X ⊆ f − 1 (1) Y ⊆ f − 1 (0) Relation ∼ between X and Y 16 / 36
Quantum Version Introduction Basic Adversary ······ 100 ... 00 ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ 010 ... 00 ◗◗◗◗◗◗◗◗◗◗◗◗◗ 001 ... 00 000 ... 10 000 ... 01 ❉❉❉❉❉❉❉❉ ❥ q ❥ Main idea q ❥ ❥ q ❥ ❥ q ❥ q ❥ ❥ Quantum Version q ❥ ❥ q ❥ q ❥ ❥ q ❥ k -threshold function q ❥ 000 ... 00 Spectral Adversary Dual Adversary Calculate: Composition m : minimum, over x ∈ X , of the number of y with x ∼ y : 1 m ′ : minimum, over y ∈ Y , of the number of x with x ∼ y : n ℓ x,j : number of y ∈ Y such that x ∼ y and x j � = y j : 1 ℓ ′ y,j : number of x ∈ X such that x ∼ y and x j � = y j : 1 ℓ max : maximum of ℓ x,j ℓ ′ y,j over x ∼ y and j such that x j � = y j : 1 �� � = Ω( √ n ) mm ′ Q ( f ) = Ω ℓ max 17 / 36
k -threshold function Introduction Basic Adversary Main idea k -threshold function: Quantum Version k -threshold function f ( x ) = 1 if there are at least k ones in the input. Spectral Adversary Dual Adversary X ⊆ f − 1 (1) : x of Hamming weight k Composition Y ⊆ f − 1 (0) : y of Hamming weight k − 1 ∼ : x ∼ y if x and y differ in 1 position x : 111 . . . 11100 . . . 00 y : 111 . . . 11000 . . . 00 18 / 36
k -threshold function Introduction Basic Adversary x : 111 . . . 11100 . . . 00 Main idea y : 111 . . . 11000 . . . 00 Quantum Version k -threshold function m : minimum, over x ∈ X , of number y with x ∼ y : k Spectral Adversary m ′ : minimum, over y ∈ Y , of number x with x ∼ y : n − k + 1 Dual Adversary ℓ x,j : number of y ∈ Y , such that x ∼ y and x j � = y j : 1 Composition ℓ ′ y,j : number of x ∈ X , such that x ∼ y and x j � = y j : 1 ℓ max : maximum of ℓ x,j ℓ ′ y,j over x ∼ y and j such that x j � = y j : 1 �� � mm ′ �� � Q ( k -threshold ) = Ω = Ω k ( n − k + 1) ℓ max This is tight! 19 / 36
Introduction Basic Adversary Spectral Adversary Matrix Representation Example Theorem Proof Spectral Adversary Spectral Adversary Flavours Dual Adversary Composition 20 / 36
Matrix Representation Introduction We had a relation x ∼ y Basic Adversary Spectral Adversary 000 001 010 100 Matrix Representation ♦ ♦ ⑧ ♦ ♦ ⑧ ♦ ♦ Example ⑧ ⑧ ♦ ♦ ⑧ ⑧ ♦ ♦ ♦ ♦ ⑧ ⑧ ♦ ♦ ⑧ ⑧ Theorem ♦ ♦ ⑧ ♦ ♦ ⑧ ♦ ♦ ⑧ ⑧ ♦ ♦ ⑧ ⑧ ♦ ♦ Proof 011 101 110 111 Spectral Adversary Flavours Dual Adversary Composition Represent as a 01-matrix: 011 101 110 111 000 0 0 0 0 Γ = 001 1 1 0 0 010 1 0 1 0 100 0 1 1 0 21 / 36
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