Quantum Chebyshevs Inequality and Applications Yassine Hamoudi, - PowerPoint PPT Presentation

Quantum Chebyshev’s Inequality and Applications Yassine Hamoudi, Frédéric Magniez IRIF , Université Paris Diderot, CNRS CQT 2019 arXiv: 1807.06456

Buffon’s needle A needle dropped randomly on a floor with equally spaced parallel lines will cross one of the lines with probability 2/ π . Bu ff on, G., Essai d'arithmétique morale , 1777. � 2

Monte Carlo algorithms: Use repeated random sampling and statistical analysis to estimate parameters of interest � 3

Monte Carlo algorithms: Use repeated random sampling and statistical analysis to estimate parameters of interest Empirical mean: 1/ Repeat the experiment n times: n i.i.d. samples x 1 , …, x n ~ X 2/ Output: (x 1 +…+ x n )/n � 3

Monte Carlo algorithms: Use repeated random sampling and statistical analysis to estimate parameters of interest Empirical mean: 1/ Repeat the experiment n times: n i.i.d. samples x 1 , …, x n ~ X 2/ Output: (x 1 +…+ x n )/n Law of large numbers: x 1 + . . . + x n n →∞ E ( X ) n � 3

˜ μ = x 1 + . . . + x n Empirical mean: x 1 , . . . , x n ∼ X with n How fast does it converge to E(X) ? � 4

˜ μ = x 1 + . . . + x n Empirical mean: x 1 , . . . , x n ∼ X with n How fast does it converge to E(X) ? Chebyshev’s Inequality: multiplicative error 0 < ε < 1 | ˜ Objective: with high probability ( finite) μ − E ( X ) | ≤ ϵ E ( X ) E ( X ), Var ( X ) ≠ 0 � 4

˜ μ = x 1 + . . . + x n Empirical mean: x 1 , . . . , x n ∼ X with n How fast does it converge to E(X) ? Chebyshev’s Inequality: multiplicative error 0 < ε < 1 | ˜ Objective: with high probability ( finite) μ − E ( X ) | ≤ ϵ E ( X ) E ( X ), Var ( X ) ≠ 0 Number of samples needed: O ( ϵ 2 E ( X ) 2 ) E ( X 2 ) ϵ 2 E ( X ) 2 ) = O ( E ( X 2 ) O ( ϵ 2 ( E ( X ) 2 − 1 )) Var ( X ) 1 (in fact ) � 4

˜ μ = x 1 + . . . + x n Empirical mean: x 1 , . . . , x n ∼ X with n How fast does it converge to E(X) ? Chebyshev’s Inequality: multiplicative error 0 < ε < 1 | ˜ Objective: with high probability ( finite) μ − E ( X ) | ≤ ϵ E ( X ) E ( X ), Var ( X ) ≠ 0 Relative second moment Number of samples needed: O ( ϵ 2 E ( X ) 2 ) E ( X 2 ) ϵ 2 E ( X ) 2 ) = O ( E ( X 2 ) O ( ϵ 2 ( E ( X ) 2 − 1 )) Var ( X ) 1 (in fact ) � 4

˜ μ = x 1 + . . . + x n Empirical mean: x 1 , . . . , x n ∼ X with n How fast does it converge to E(X) ? Chebyshev’s Inequality: multiplicative error 0 < ε < 1 | ˜ Objective: with high probability ( finite) μ − E ( X ) | ≤ ϵ E ( X ) E ( X ), Var ( X ) ≠ 0 Relative second moment Number of samples needed: O ( ϵ 2 E ( X ) 2 ) E ( X 2 ) ϵ 2 E ( X ) 2 ) = O ( E ( X 2 ) O ( ϵ 2 ( E ( X ) 2 − 1 )) Var ( X ) 1 (in fact ) n = Ω ( ϵ 2 ) Δ 2 Δ 2 ≥ E ( X 2 ) In practice: given an upper-bound , take samples E ( X ) 2 � 4

Example: edge counting Problem: approximate the number m of edges in an n -vertex graph G � 5

Example: edge counting Problem: approximate the number m of edges in an n -vertex graph G Estimator X := 1. Sample a vertex v ∈ V uniformly at random 2. Sample a neighbor w of v uniformly at random 3. If deg(v) < deg(w) (or deg(v) = deg(w) and v < lex w) Output n*deg(v) Else Output 0 � 5

Example: edge counting Problem: approximate the number m of edges in an n -vertex graph G Estimator X := 1. Sample a vertex v ∈ V uniformly at random 2. Sample a neighbor w of v uniformly at random 3. If deg(v) < deg(w) (or deg(v) = deg(w) and v < lex w) Output n*deg(v) Else Output 0 Lemma: E(X) = m and E(X 2 )/E(X) 2 ≤ O( √ n ). (when m ≥ Ω (n)) [Goldreich, Ron’08] [Seshadhri’15] � 5

Example: edge counting Problem: approximate the number m of edges in an n -vertex graph G Estimator X := 1. Sample a vertex v ∈ V uniformly at random 2. Sample a neighbor w of v uniformly at random 3. If deg(v) < deg(w) (or deg(v) = deg(w) and v < lex w) Output n*deg(v) Else Output 0 Lemma: E(X) = m and E(X 2 )/E(X) 2 ≤ O( √ n ). (when m ≥ Ω (n)) [Goldreich, Ron’08] [Seshadhri’15] Consequence: O( √ n/ ε 2 ) samples to approximate m with error ε . � 5

Other applications Counting with Markov chain Monte Carlo methods: Counting vs. sampling [Jerrum, Sinclair’96] [ Š tefankovi č et al.’09], Volume of convex bodies [Dyer, Frieze'91], Permanent [Jerrum, Sinclair, Vigoda’04] Data stream model: Frequency moments, Collision probability [Alon, Matias, Szegedy’99] [Monemizadeh, Woodru ff ’] [Andoni et al.’11] [Crouch et al.’16] Testing properties of distributions: Closeness [Goldreich, Ron’11] [Batu et al.’13] [Chan et al.’14] , Conditional independence [Canonne et al.’18] Estimating graph parameters: Number of connected components, Minimum spanning tree weight [Chazelle, Rubinfeld, Trevisan’05] , Average distance [Goldreich, Ron’08] , Number of triangles [Eden et al. 17] etc. � 6

Random variable X over sample space Ω ⊂ R + Classical sample: one value x ∈ Ω , sampled with probability p x � 7

Random variable X over sample space Ω ⊂ R + Classical sample: one value x ∈ Ω , sampled with probability p x Quantum sample: one (controlled-) execution of a quantum sampler or , where S − 1 S X X S X | 0 ⟩ = ∑ p x | ψ x ⟩ | x ⟩ x ∈Ω with ψ x = arbitrary unit vector � 7

Random variable X over sample space Ω ⊂ R + Classical sample: one value x ∈ Ω , sampled with probability p x Quantum sample: one (controlled-) execution of a quantum sampler or , where S − 1 S X X S X | 0 ⟩ = ∑ p x | ψ x ⟩ | x ⟩ x ∈Ω with ψ x = arbitrary unit vector Question: can we estimate E(X) with less samples in the quantum setting? � 7

Previous Works

The Amplitude Estimation algorithm [Brassard et al.’11] [Brassard et al.’11] [Wocjan et al.’09] S X | 0 ⟩ = ∑ Given one can obtain (with 1 ancillary qubit + controlled rotation): p x | ψ x ⟩ | x ⟩ x ∈Ω � 9

The Amplitude Estimation algorithm [Brassard et al.’11] [Brassard et al.’11] [Wocjan et al.’09] S X | 0 ⟩ = ∑ Given one can obtain (with 1 ancillary qubit + controlled rotation): p x | ψ x ⟩ | x ⟩ x ∈Ω p x | ψ x ⟩ | x ⟩ ( | 1 ⟩ ) S Y | 0 ⟩ = ∑ 1 − x x | 0 ⟩ + M Ω M Ω x ∈Ω where M Ω = max{ x ∈ Ω } 9 �

The Amplitude Estimation algorithm [Brassard et al.’11] [Brassard et al.’11] [Wocjan et al.’09] S X | 0 ⟩ = ∑ Given one can obtain (with 1 ancillary qubit + controlled rotation): p x | ψ x ⟩ | x ⟩ x ∈Ω p x | ψ x ⟩ | x ⟩ ( | 1 ⟩ ) S Y | 0 ⟩ = ∑ 1 − x x | 0 ⟩ + M Ω M Ω x ∈Ω 1 − E ( X ) E ( X ) = | φ 0 ⟩ | 0 ⟩ + | φ 1 ⟩ | 1 ⟩ M Ω M Ω and | φ 0 ⟩ , | φ 1 ⟩ are some unit vectors. where M Ω = max{ x ∈ Ω } 9 �

The Amplitude Estimation algorithm [Brassard et al.’11] [Brassard et al.’11] [Wocjan et al.’09] S X | 0 ⟩ = ∑ Given one can obtain (with 1 ancillary qubit + controlled rotation): p x | ψ x ⟩ | x ⟩ x ∈Ω p x | ψ x ⟩ | x ⟩ ( | 1 ⟩ ) S Y | 0 ⟩ = ∑ 1 − x x | 0 ⟩ + M Ω M Ω x ∈Ω 1 − E ( X ) E ( X ) = | φ 0 ⟩ | 0 ⟩ + | φ 1 ⟩ | 1 ⟩ M Ω M Ω and | φ 0 ⟩ , | φ 1 ⟩ are some unit vectors. where M Ω = max{ x ∈ Ω } Observation: The Grover's operator G = S − 1 Y ( I − 2 | 0 ⟩⟨ 0 | ) S Y ( I − 2 I ⊗ | 1 ⟩⟨ 1 | ) has eigenvalues , where . e ±2 i θ θ = sin − 1 ( E ( X )/ M Ω ) 9 �

The Amplitude Estimation algorithm [Brassard et al.’11] [Brassard et al.’11] [Wocjan et al.’09] S X | 0 ⟩ = ∑ Given one can obtain (with 1 ancillary qubit + controlled rotation): p x | ψ x ⟩ | x ⟩ x ∈Ω p x | ψ x ⟩ | x ⟩ ( | 1 ⟩ ) S Y | 0 ⟩ = ∑ 1 − x x | 0 ⟩ + M Ω M Ω x ∈Ω 1 − E ( X ) E ( X ) = | φ 0 ⟩ | 0 ⟩ + | φ 1 ⟩ | 1 ⟩ M Ω M Ω and | φ 0 ⟩ , | φ 1 ⟩ are some unit vectors. where M Ω = max{ x ∈ Ω } Observation: The Grover's operator G = S − 1 Y ( I − 2 | 0 ⟩⟨ 0 | ) S Y ( I − 2 I ⊗ | 1 ⟩⟨ 1 | ) has eigenvalues , where . e ±2 i θ θ = sin − 1 ( E ( X )/ M Ω ) Algorithm: 1/ Apply Phase Estimation on G for steps t ≥ Ω ( M Ω /( ϵ E ( X ))) | ˜ ˜ to get an estimate s.t. . θ − | θ || ≤ 1/ t θ μ = M Ω ⋅ sin 2 ( ˜ ˜ 2/ Output as an estimate to E(X). θ ) 9 �

The Amplitude Estimation algorithm [Brassard et al.’02] [Brassard et al.’11] [Wocjan et al.’09] S X | 0 ⟩ = ∑ Given one can obtain (with 1 ancillary qubit + controlled rotation): p x | ψ x ⟩ | x ⟩ x ∈Ω p x | ψ x ⟩ | x ⟩ ( | 1 ⟩ ) S Y | 0 ⟩ = ∑ 1 − x x | 0 ⟩ + M Ω M Ω x ∈Ω 1 − E ( X ) E ( X ) = | φ 0 ⟩ | 0 ⟩ + | φ 1 ⟩ | 1 ⟩ M Ω M Ω and | φ 0 ⟩ , | φ 1 ⟩ are some unit vectors. where M Ω = max{ x ∈ Ω } O ( E ( X ) ) quantum samples to obtain | ˜ M Ω Result: μ − E ( X ) | ≤ ϵ E ( X ) ϵ 10 �