the polynomial method strikes back...the ๐-distinctness problem this generalizes element...
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The Polynomial Method Strikes Back: Tight Quantum Query Bounds via Dual Polynomials
Robin KothariMicrosoft Research
arXiv:1710.09079
Justin ThalerGeorgetown University
Mark BunPrinceton University
Query complexity
Goal:
๐ฅ1 ๐ฅ2 ๐ฅ3 ๐ฅ๐โฏ
๐๐ฅ๐๐ฅ
Why query complexity?Complexity theoretic motivation
Algorithmic motivation
Other applications
Quantum query complexity
Quantum query complexity: Minimum number of uses of ๐๐ฅ in a quantum
circuit that for every input ๐ฅ, outputs ๐(๐ฅ) with error โค 1/3.๐ ๐
Example: .
Then ๐ OR๐ = ๐ AND๐ = ฮ ๐ [Grover96, Bennett-Bernstein-Brassard-Vazirani97]
Classically, we need ฮ ๐ queries for both problems.
๐๐ฅ๐0 ๐๐ฅ๐1 ๐๐
Lower bounds on quantum query complexityPositive-weights adversary method Negative-weights adversary method
Polynomial method
Approximate degree
Approximate degree: Minimum degree of a polynomial ๐(๐ฅ1, โฆ , ๐ฅ๐) with real
coefficients such that โ๐ฅ โ 0,1 ๐, ๐ ๐ฅ โ ๐ ๐ฅ โค 1/3. เทชdeg(๐)
Theorem ([Beals-Buhrman-Cleve-Mosca-de Wolf01]): For any ๐,
๐ ๐ โฅ1
2เทชdeg(๐)
เทชdeg OR๐ = เทชdeg = ฮ ๐ ๐ OR๐ = ๐ = ฮ ๐
Examples:
Other applications of approximate degreeUpper bounds
[Klivans-Servedio04, Klivans-Servedio06, Kalai-Klivans-Mansour-Servedio08]
[Kahn-Linial-Samorodnitsky96, Sherstov09]
[Thaler-Ullman-Vadhan12, Chandrasekaran-Thaler-Ullman-Wan14]
[Tal14, Tal17]
Lower bounds
[Sherstov07, Shi-Zhu07, Chattopadhyay-Ada08, Lee-Shraibman08,โฆ]
[Minsky-Papert69, Beigel93, Sherstov08]
[Beigel94, Bouland-Chen-Holden-Thaler-Vasudevan16]
[Bogdanov-Ishai-Viola-Williamson16]
The ๐-distinctness problem
This generalizes element distinctness, which is 2-distinctness.
Upper bounds
[Ambainis07]
[Belovs12]
Lower bounds
[Aaronson-Shi04]
๐-distinctness: Given ๐ numbers in ๐ = {1,โฆ , ๐ }, does any number appear โฅ๐ times?
Our result: ๐ Dist๐ = .
๐-junta testing
Upper bounds
[Atฤฑcฤฑ-Servedio07]
[Ambainis-Belovs-Regev-deWolf16]
Lower bounds
[Atฤฑcฤฑ-Servedio07]
[Ambainis-Belovs-Regev-deWolf16]
๐-junta testing: Given the truth table of a Boolean function, decide if
(YES) the function depends on at most ๐ variables, or
(NO) the function is far (at least ๐ฟ๐ in Hamming distance) from having this property.
Our result: .
Summary of results
Surjectivity
Quantum query complexity
[Beame-Machmouchi12, Sherstov15]
Approximate degree
[Aaronson-Shi04, Ambainis05, Bun-Thaler17]
[Sherstov18]
SURJ is the first natural function to have ๐ ๐ โซ !
Surjectivity: Given ๐ numbers in ๐ (๐ = ฮ(๐)), does every ๐ โ [๐ ] appear in the list?
Our result: and a new proof of .
Summary of results
Surjectivity upper bound
๐ SURJ = เทจ๐(๐ ฮค3 4)
Surjectivity lower bound
๐ SURJ = เทฉฮฉ(๐ ฮค3 4)
๐-distinctness
๐ Dist๐ = เทฉฮฉ(๐34 โ
12๐)
Image size testing
๐ IST = เทฉฮฉ( ๐)
๐-junta testing
๐ Junta๐ = เทฉฮฉ( ๐)
Statistical distance
๐ SDU = เทฉฮฉ( ๐)
Shannon entropy
๐ Entropy = เทฉฮฉ( ๐)Reduction
Intuition and ideas
Overview of the upper bound
Polynomials are algorithms
Polynomials are not algorithms
Overview of the upper boundIdea 1: Polynomials are algorithms
Imagine that polynomials ๐1, ๐2, and ๐3 represent the acceptance probability of
algorithms (that output 0 or 1) ๐ด1, ๐ด2, and ๐ด3.
Algorithm: If ๐ด1 outputs 1, then output ๐ด2, else output ๐ด3.
Polynomial: ๐1 ๐ฅ ๐2 ๐ฅ + 1 โ ๐1 ๐ฅ ๐3(๐ฅ).
Example: Implementing an if-then-else statement
Key idea: This is well defined even if ๐๐ โ [0,1]
ร
=
Overview of the upper boundIdea 2: Polynomials are not algorithms
Overview of lower bounds
๐
๐ ๐ ๐ ๐
AND๐ โ OR๐
Open for 10+ years! Proof uses the method of dual polynomials.
PS: See Adam Boulandโs talk at 11:15 for an alternate proof using quantum arguments.
Lower bound [Sherstov13, Bun-Thaler13]: เทชdeg AND๐ โ OR๐ = ฮฉ ๐๐ .
Proof 1: Use robust quantum search [Hรธyer-Mosca-de Wolf03]
Proof 2: โ๐, ๐, เทชdeg = ๐ เทชdeg ๐ เทชdeg ๐ [Sherstov13]
Upper bound: เทชdeg AND๐ โ OR๐ = ๐ ๐๐ .
๐ ร ๐ input bits
Dual polynomialsApproximate degree can be expressed as a linear program
เทชdeg ๐ โค ๐ iff there exists a polynomial ๐ of degree ๐, i.e., ๐ = ฯ๐: ๐ โค๐ ๐ผ๐ ๐ฅ๐, s.t.,
โ๐ฅ โ 0,1 ๐, ๐ ๐ฅ โ ๐ ๐ฅ โค 1/3
เทชdeg ๐ > ๐ iff there exists ๐: 0,1 ๐ โ โ,
1. ฯ๐ฅ |๐ ๐ฅ | = 1 (1) ๐ is โ1 normalized
2. If deg ๐ โค ๐ then ฯ๐ฅ๐ ๐ฅ ๐ ๐ฅ = 0 (2) ๐ has pure high degree ๐
3. ฯ๐ฅ๐ ๐ฅ โ1 ๐(๐ฅ) > 1/3. (3) ๐ is well correlated with ๐
Lower bound for AND๐ โ OR๐
Proof strategy for เทชdeg AND๐ โ OR๐ = ฮฉ ๐๐ :
1. Start with ๐AND and ๐OR witnessing เทชdeg = ฮฉ ๐ and เทชdeg = ฮฉ ๐
2. Combine these into ๐ witnessing เทชdeg AND๐ โ OR๐ = ฮฉ ๐๐ using the technique
of dual block composition.
เทชdeg ๐ > ๐ iff there exists ๐: 0,1 ๐ โ โ,
1. ฯ๐ฅ |๐ ๐ฅ | = 1 (1) ๐ is โ1 normalized
2. If deg ๐ โค ๐ then ฯ๐ฅ๐ ๐ฅ ๐ ๐ฅ = 0 (2) ๐ has pure high degree ๐
3. ฯ๐ฅ๐ ๐ฅ โ1 ๐(๐ฅ) > 1/3. (3) ๐ is well correlated with ๐
Dual block composition for ๐ โ ๐
Composed dual automatically satisfies (1) and (2).
[Sherstov13, Bun-Thaler13] show that property (3) is also satisfied for AND๐ โ OR๐.
๐๐โ๐ = 2๐ ๐๐ sgn ๐๐ ๐ฅ1 , โฆ , sgn ๐๐ ๐ฅ๐ ฯ๐=1๐ ๐๐ ๐ฅ๐ [Shi-Zhu09, Lee09, Sherstov13]
เทชdeg ๐ > ๐ iff there exists ๐: 0,1 ๐ โ โ,
1. ฯ๐ฅ |๐ ๐ฅ | = 1 (1) ๐ is โ1 normalized
2. If deg ๐ โค ๐ then ฯ๐ฅ๐ ๐ฅ ๐ ๐ฅ = 0 (2) ๐ has pure high degree ๐
3. ฯ๐ฅ๐ ๐ฅ โ1 ๐(๐ฅ) > 1/3. (3) ๐ is well correlated with ๐
เทชdeg SURJ = เทฉฮฉ ๐3/4
Reduction to a composed function
SURJ reduces to AND๐ โ OR๐ function, restricted
to inputs with Hamming weight โค ๐.
We denote this function AND๐ โ OR๐โค๐.
โ เทชdeg SURJ = เทจ๐ เทชdeg AND๐ โ OR๐โค๐
SURJ ๐ฅ1, โฆ , ๐ฅ๐ = แฅ
๐โ ๐
แง
๐โ ๐
(๐ฅ๐ = ๐? )
Surjectivity: Given ๐ numbers in ๐ (๐ = ฮ(๐)), does every ๐ โ [๐ ] appear in the list?
Converse [Ambainis05, Bun-Thaler17]: เทชdeg SURJ = เทฉฮฉ เทชdeg AND๐ โ OR๐โค๐
AND๐ โ OR๐ โ AND๐ โ OR๐โค๐
เทชdeg AND๐ โ OR๐ = ฮ ๐ ๐ = ฮ(๐)
เทชdeg AND๐ โ OR๐โค๐ = เทฉฮ เทชdeg SURJ
Progress so far towards เทชdeg SURJ = เทฉฮฉ ๐3/4
1. We saw that เทชdeg SURJ = เทฉฮ เทชdeg AND๐ โ OR๐โค๐ .
2. We saw using dual block composition that
เทชdeg AND๐ โ OR๐ = ฮฉ ๐ ๐ = ฮฉ(๐), when ๐ = ฮ ๐ .
Does the constructed dual also work for AND๐ โ OR๐โค๐? No.
เทชdeg ๐โค๐ป > ๐ iff there exists ๐,
1. ฯ๐ฅ |๐ ๐ฅ | = 1 (1) ๐ is โ1 normalized
2. If deg ๐ โค ๐ then ฯ๐ฅ๐ ๐ฅ ๐ ๐ฅ = 0 (2) ๐ has pure high degree ๐
3. ฯ๐ฅ๐ ๐ฅ โ1 ๐(๐ฅ) > 1/3. (3) ๐ is well correlated with ๐
4. ๐ ๐ฅ = 0 if ๐ฅ > ๐ป (4) ๐ is only supported on the promise
Dual witness for เทชdeg AND๐ โ OR๐โค๐
Fix 1: Use a dual witness ๐OR for OR๐ that only certifies เทชdeg = ฮฉ ๐1/4 and satisfies
a โdual decay conditionโ, i.e., ๐OR ๐ฅ is exponentially small for ๐ฅ โซ ๐1/4. Thus the
composed dual has degree ฮฉ ๐ ๐1/4 = ฮฉ(๐3/4) and almost satisfies condition (4).
Fix 2: Although condition (4) is only โalmost satisfiedโ in our dual witness, we can
postprocess the dual to have it be exactly satisfied [Razborov-Sherstov10].
เทชdeg ๐โค๐ป > ๐ iff there exists ๐,
1. ฯ๐ฅ |๐ ๐ฅ | = 1 (1) ๐ is โ1 normalized
2. If deg ๐ โค ๐ then ฯ๐ฅ๐ ๐ฅ ๐ ๐ฅ = 0 (2) ๐ has pure high degree ๐
3. ฯ๐ฅ๐ ๐ฅ โ1 ๐(๐ฅ) > 1/3. (3) ๐ is well correlated with ๐
4. ๐ ๐ฅ = 0 if ๐ฅ > ๐ป (4) ๐ is only supported on the promise
Looking back at the lower boundsHow did we resolve questions that have resisted attack by the adversary method?
What is the key new ingredient in these lower bounds?
Lower bound for OR:
Any polynomial like this must
have degree ฮฉ ๐ .
Key property we exploit:
Any polynomial like this must
still have degree ฮฉ ๐ !
Open problems
[Ambainis07, Belovs-ล palek13]
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