Forrelation Optimally Separates - k Quantum and Classical Query Complexity - Indico

k-Forrelation Optimally Separates
Quantum and Classical Query Complexity
 QIP 2021

 Nikhil Bansal Makrand Sinha

Quantum vs Classical Query Algorithms

 Can query input bits in
 Input x
 Choose bit to query at
 superposition
 n bits random

Quantum Query
 Objective(s) Minimize queries Randomized Query
 Algorithm
 Algorithm
 Quantum Speedup What is the maximal separation? Probability Distribution
 Question [Buhrman-Fortnow-Newman-Röhrig ’03]
 over Decision Trees

 Total functions Partial functions
 c
 Classical queries = (Quantum queries)
 2 1/2 [Simon ’97] [Childs-Cleve-Deotto-Farhi-
 c≤6 [Beals-Buhrman-Cleve-Mosca-de Wolf ’98] O(log n) vs Ω̃(n ) Gutmann-Spielman ’03]

 c≤4 [Aaronson-Ben David-Kothari-Rao-Tal ’21] 1 vs Ω̃(n 1/4) [Beaudrap-Cleve-Watrous ’02]

 c ≥ 5/2 [Aaronson-Ben David-Kothari ’16] 1 vs Ω̃(n 1/2) [Aaronson-Ambainis ’14]

 c ≥ 8/3 − o(1) [Tal ’20] Non-explicit O(1) vs Ω̃(n 2/3−ϵ) [Tal ’20] Non-explicit

Maximal Separation?
 m a l ?
 i Theorem
 o p t 1
 Õ(n 1/2) classical queries
Is this Every ⌈k/2⌉-query quantum algorithm with error − δ can be
 2
 [Aaronson-Ambainis ’14] 1 δ k=2
 simulated with error − with 2k ⋅ n 1−1/k ⋅ δ −2 classical queries
 2 2
 Õ(n 0.999) classical queries
 k = 1000

 What is the task where quantum
 algorithms have the maximal advantage?

 Conjecture k-fold Forrelation problem gives a ⌈k/2⌉ vs Ω̃(n 1−1/k) separation
 [Aaronson-Ambainis ’14]

 Captures the maximal power of • Quantum query algorithms?
 • Quantum circuits
 BQP-complete for promise problems for k ≈ log n
 [Aaronson-Ambainis ’14]

2-Fold Forrelation
 Promise Problem ±
 Input x := (x1, x2) ∈ { 1} n+n

 Rotation
 Distinguish these cases H = Hadamard/Fourier matrix
 Fourier transform
 Each entry

 Can be solved ⟨x1, Hx2⟩ ⟨x1, Hx2⟩ ± 1
 ≤ 0.01 vs ≥ 0.1
 with 1 query n n n

 n
 Almost Orthogonal Far from Orthogonal

 [Aaronson ’10]

 n
 Query Query 1 ⟨x1, Hx2⟩
 amplitude of | 0⟩ = x1(i)Hij x2( j) =
| 0⟩ H
 x2
 H
 x1
 H ∑
 n i,j=1 n

2-Fold Forrelation
 Promise Problem ±
 Input x := (x1, x2) ∈ { 1} n+n

 Rotation
 Distinguish these cases H = Hadamard/Fourier matrix

 Each entry

Can be solved ± 1
 | Forr2(x) | ≤ 0.01 vs Forr2(x) ≥ 0.1
 with 1 query n

 n

 x1(i) Hij

 x2( j) 1 n ⟨x1, Hx2⟩
 ∑
 Forr2(x) = x1(i) ⋅ Hij ⋅ x2( j) =
 n i,j=1 n

k-Fold Forrelation
 Promise Problem ±
 Input x := (x1, …, xk) ∈ { 1}kn

 Rotation
 Distinguish these cases H = Hadamard/Fourier matrix

 Each entry

 Can be solved with ± 1
 | Forrk(x) | ≤ 0.01 vs Forrk(x) ≥ 0.1
 ⌈k/2⌉ queries n

 n

 n
 Query Query
 1
 ∑
| 0⟩ H H H Forrk(x) = x1(i1)Hi1i2 x2(i2)Hi2i3…Hik−1ik xk(ik)
 xk x1 n i1,…,ik=1

 amplitude of | 0⟩

k-Fold Forrelation
 Promise Problem ±
 Input x := (x1, …, xk) ∈ { 1}kn

 Rotation
 Distinguish these cases H = Hadamard/Fourier matrix

 Each entry

Can be solved with ± 1
 | Forrk(x) | ≤ 0.01 vs Forrk(x) ≥ 0.1
 ⌈k/2⌉ queries n

 n

 xk−1(ik−1)
 x1(i1) Hi1i2 Hik−1ik
 n
 xk(ik) Forrk(x) =
 1
 ∑
 x2(i2)
 x1(i1)Hi1i2 x2(i2)Hi2i3…Hik−1ik xk(ik)
 n i1,…,ik=1

Our Results
Theorem 1−1/k
 k-fold Forrelation problem gives a ⌈k/2⌉ vs Ω̃(n ) separation between 500 vs Ω̃(n 0.999)
 −O(k) k = 1000
 quantum and classical query algorithms for advantage δ = 2

 1/2
 Main Contribution: classical lower bound Previous lower bound: Ω̃(n )
 [Aaronson-Ambainis ’14]

 Our proof also works for the non-explicit Rorrelation function introduced by [Tal ’20]

 Replace Hadamard with a Random Orthogonal matrix

 n
 1
 Forrk(x) =
 ∑
 x1(i1)Hi1i2 x2(i2)Hi2i3…Hik−1ik xk(ik)
 n i1,…,ik=1

Our Results
 Theorem 1−1/k
 k-fold Forrelation problem gives a ⌈k/2⌉ vs Ω̃(n ) separation between 500 vs Ω̃(n 0.999)
 −O(k) k = 1000
 quantum and classical query algorithms for advantage δ = 2

 1/2
 Main Contribution: classical lower bound Previous lower bound: Ω̃(n )
 [Aaronson-Ambainis ’14]

 Consequences
 Our proof also works for the non-explicit Rorrelation function introduced by [Tal ’20]

 ‣ Query Complexity of Partial Functions with standard error
 Oϵ(1) vs n 1−ϵ
 separation for error 1/3

 ‣ Query Complexity of Total Functions with standard error

 ∃ total f Classical queries ≥ (Quantum queries) 3−o(1)
 { Explicit

 [SSW ’21] + [Tal ’20] rely on strong properties of random
 ‣ Analogous separations in Communication Complexity by Lifting
 orthogonal matrices that do not hold for Hadamard matrix

Independent Work Analogous results for Rorrelation [Sherstov-Storozhenko-Wu ’21] building on [Tal ’20] Different Techniques

High-level Overview

Quantum vs Classical Query Algorithms
Fact Success probability of any d-query quantum or randomized ̂ ⋅ z where z ∈ {± 1}N and z =
 ∑
 f(z) = f(S)
 ∏
 S S zi
 algorithm is a degree O(d) multilinear polynomial
 S⊆[N] i∈S

 Quantum Query Algorithm Randomized Query Algorithm
 Extremely good at computing dense Can only compute weakly- [Tal ’20]
 polynomials with few queries sparse polynomials [SSW ’21]
 n
 1 L1-norm of coefficients of degree ℓ monomials
 Forrk(x) =
 ∑
 x1(i1)Hi1i2 x2(i2)Hi2i3…Hik−1ik xk(ik)
 n i1,…,ik=1 ≪ number of degree ℓ monomials

 (ℓ) (ℓ)
 ̂ |≤ d d
 for all ℓ ≤ d
 ∑
 | ∂S f(0) | = ∑ | f(S) ≪
 |S|=ℓ |S|=ℓ
 Rest of the talk
 Observation
 Multilinear polynomial p of degree d ≪ n 1−1/k with small [This Work]
 derivatives cannot compute k-Fold Forrelation
 (ℓ)
 d for any ℓ and
 ∑
 | ∂S f(x) | ≤
 2
We only need bound on derivatives of order ≤ k which follow from [Tal ’20] any x ∈ [−1,1]N
 |S|=ℓ

Degree Lower Bounds
 Input (x1, …, xk) ∈ {± 1}kn
 Multilinear polynomial p of degree d ≪ n 1−1/k with small n
 1
 derivatives cannot compute k-Fold Forrelation n ∑
 x1(i1)Hi1i2 x2(i2)Hi2i3…Hik−1ik xk(ik)
 i1,…,ik=1

 Forrk(x)
 Show that such polynomials cannot distinguish
 distributions on 0 vs 1 inputs

 | Forrk(x) | ≤ 0.01 Forrk(x) ≥ 0.1 [p(ℱk)] − [p( )] ≈ 0
 vs

 Uniform Distribution Pseudorandom Distribution ℱk

 Rounding

 e.g. take sign
 Sample from some
Uniform Distribution on {± 1} kn
 distribution k over ℝkn Distribution on {± 1}kn

Degree Lower Bounds via Interpolation
 Input (x1, …, xk) ∈ {± 1}kn
 Multilinear polynomial p of degree d ≪ n 1−1/k with small n
 1
 derivatives cannot compute k-Fold Forrelation n ∑
 x1(i1)Hi1i2 x2(i2)Hi2i3…Hik−1ik xk(ik)
 i1,…,ik=1

 Forrk(x)
 [p(ℱk)] − [p( )] ≈ 0

 Labeled “Smart Path method” by Talagrand
 Many applications in statistical physics,
 probability, convex geometry,…

 Uniform Distribution 
 Pseudorandom Distribution ℱk

 (t) ℱ

 0 Time t 1 Rounding

 e.g. take sign
 Choose smart path to control Sample from some
Uniform Distribution on {± 1}kn
 time derivative of [p( (t))] distribution k over ℝkn Distribution on {± 1}kn

Our Main Technical Contribution
 Multilinear polynomial p of degree d ≪ n 1−1/k with small Choose smart path to control
 derivatives cannot compute k-Fold Forrelation time derivative of [p( (t))]

 (t) ℱ
 [p(ℱk)] − [p( )] ≈ 0
 0 Time t 1
 Lemma For every “time” t
 ℓ(1−1/k) Recall bound on derivatives [Tal ’20]
 k(k−1)

 ∑ ( n)
 1
 ∑
 "Time derivative" ≤ max | ∂S p(x) | ℓth-order ≤ d ℓ/2
 [SSW ’21]
 x∈[−1,1]kn + Our Observation
 ℓ=k |S|=ℓ

 2-Fold
[Raz-Tal ’18]
[Wu ’19]
 ≤ max
 x∈[−1,1]kn
 1
 ∑
 n |S|=2
 | ∂S p(x) | ≤

 2nd order
 d
 n
 Choose d ≈ n 1/2
 { Relies on stochastic calculus tools

 ‣ Gaussian Interpolation
 ‣ Gaussian Integration by Parts
 1 1 d 3/2 d 3
 x∈[−1,1]kn n ∑ ∑
 3-Fold ≤ max | ∂S p(x) | + 2 | ∂S p(x) | ≤ + ‣ Develop new Integration by
 |S|=3
 n |S|=6
 n n 2
 Parts identities for rounding
[This Work]
 2/3
 3rd order 6th order Choose d ≈ n

Proof Ideas

Degree Lower Bounds via Interpolation
 Input (x1, …, xk) ∈ {± 1}kn
 Multilinear polynomial p of degree d ≪ n 1−1/k with small n
 1
 derivatives cannot compute k-Fold Forrelation n ∑
 x1(i1)Hi1i2 x2(i2)Hi2i3…Hik−1ik xk(ik)
 i1,…,ik=1

 Forrk(x)
 [p(ℱk)] − [p( )] ≈ 0

 c a l l
 Small Value

 Uniform Distribution 
 Re Large Value

 Pseudorandom Distribution ℱk

 (t) ℱ

 0 Time t 1 Rounding

 e.g. take sign
 Choose smart path to control Sample from some
Uniform Distribution on {± 1}kn
 time derivative of [p( (t))] distribution k over ℝkn Distribution on {± 1}kn

2-Fold Case: Pseudorandom Distribution
 Pseudorandom Distribution ℱk Input (x1, x2) ∈ {± 1}2n

 x1(i) Hij

 Forrk(x) ≥ 0.1 x2( j)
 t a l k
 Rounding
 i n t his w or k
 N ot s i n our
 e.g. take sign
 h n i que k=2
 Sample from some Tec 1 n
distribution k over ℝkn New Distribution on {± 1}kn Forr2(x) =
 ∑
 x1(i) ⋅ Hij ⋅ x2( j)
 n i,j=1

 Intuition
 X1(i) Hij
 “Covariance Graph” Decision tree or multilinear polynomial
 independent
 X2( j) needs to compute all 2-wise correlations

 independent
 1
 Hij = ±
 n
 [X1(i)X2( j)] = Hij
 1
 Hij2 = 1
 n∑
 [Forr2(X)] =
 (Hn In )
 2n
 In Hn
 X := (X1, X2) Gaussian in ℝ with Covariance Σ = ij

 [Aaronson-Ambainis ’14]

2-Fold Case: The Smart Path
 Uniform Distribution = (1 − t)I + tΣ Pseudorandom Distribution ℱk

 Covariance
 I (t) ℱ Σ
 t a l k s t a l k
 Rounding
 t h is Rounding
 n t h i
 ot i n o t i
 e.g. take signN 0 Time t 1 N
 e.g. take sign
Sample from standard Uniform Distribution Sample from some
 Gaussian over ℝkn distribution k over ℝkn Distribution on {± 1}kn
 on {± 1}kn

 Can be directly handled using Gaussian
 X1(i) Hij Interpolation Formula
 X2( j)
 Multilinear Bound in terms of ∂ij p(x) and final
 1
 polynomial p covariance entries Hij = ±
 n
 1
 ∑
 "Time derivative" ≤ max | ∂ij p(x) |
 ∂ij corresponds to removing xi xj e.g. x1x2 x3⋯xi xj x∈[−1,1]kn n ij

3-Fold Case: Pseudorandom Distribution
 Pseudorandom Distribution ℱk Input (x1, x2, x3) ∈ {± 1}3n

 x1(i1) Hi1i2

 Forrk(x) ≥ 0.1
 t a l k x2(i2)
 Rounding
 n t h is
 N ot i
 e.g. take sign
 Sample from some k=3 Forr3(x) =
 1 n
 ∑
 x1(i)Hij x2( j)Hjl x3(l)
distribution k over ℝkn Distribution on {± 1}kn n i,j,l

 (Hn In )
 2n In Hn
 G, B independent Gaussians in ℝ with covariance Σ=

 G1(i) Hij Hjl
 [Forr3(X)] = 1
 G2( j) B2(l)
 B1( j)

 Intuition
 Decision tree or multilinear polynomial needs
 Take product of these random variables to compute all three-wise correlations now
 X := (G1, G2 ⊙ B1, B2) ∈ ℝ3n Entry-wise product
 [Tal ’20]

3-Fold Case: The Smart Path
 Uniform Distribution Pseudorandom Distribution ℱk

 (t) ℱ
 t a l k
 Rounding t a l k Rounding
 n t his
 i n t his N ot i
 e.g. take signN
 ot 0 Time t 1 e.g. take sign
Sample from standard Uniform Distribution Sample from some
 Gaussian over ℝkn distribution k over ℝkn Distribution on {± 1}kn
 on {± 1}kn

 Interpolate G and B separately z1 = g1, z2 = g2 ⋅ b2, z3 = b3
G1(i) Hij B2(l)
 Hml p(z) = ⋯ + ⋯ z1z2z3 ⋯ + ⋯
 Want bounds in terms of ∂ijℓ p

 =
 G2( j)
 and sixth order derivatives g1g2 ⋅ b2b3
 B1(m)
 Multilinear polynomial p

 Take product of these random variables Other stochastic calculus tools e.g. Gaussian Integration
 X := (G1, G2 ⊙ B1, B2) ∈ ℝ 3n by Parts to relate derivatives after substitution

Summary and Open Problems
Theorem 1−1/k
 k-fold Forrelation problem gives a ⌈k/2⌉ vs Ω̃(n ) separation between Optimal
 quantum and classical query algorithms for advantage δ = 2−O(k) Separation

Relies on stochastic calculus tools
 ‣ Gaussian Interpolation
 ‣ Gaussian Integration by Parts
 ‣ Develop new Integration by Parts identities for rounding

 Open
 Quantum vs Classical Communication Complexity of Total Functions
Problem
 Are these polynomially related?