The Polynomial Method Strikes Back...The π-distinctness problem This generalizes element...
Transcript of The Polynomial Method Strikes Back...The π-distinctness problem This generalizes element...
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:
π₯1 π₯2 π₯3 π₯πβ―
ππ₯ππ₯
Why query complexity?Complexity theoretic motivation
Algorithmic motivation
Other applications
Quantum query complexity
Quantum query complexity: Minimum number of uses of ππ₯ in a quantum
circuit that for every input π₯, outputs π(π₯) with error β€ 1/3.π π
Example: .
Then π ORπ = π ANDπ = Ξ π [Grover96, Bennett-Bernstein-Brassard-Vazirani97]
Classically, we need Ξ π queries for both problems.
ππ₯π0 ππ₯π1 ππ
Lower bounds on quantum query complexityPositive-weights adversary method Negative-weights adversary method
Polynomial method
Approximate degree
Approximate degree: Minimum degree of a polynomial π(π₯1, β¦ , π₯π) with real
coefficients such that βπ₯ β 0,1 π, π π₯ β π π₯ β€ 1/3. ΰ·ͺdeg(π)
Theorem ([Beals-Buhrman-Cleve-Mosca-de Wolf01]): For any π,
π π β₯1
2ΰ·ͺdeg(π)
ΰ·ͺdeg ORπ = ΰ·ͺdeg = Ξ π π ORπ = π = Ξ π
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,β¦]
[Minsky-Papert69, Beigel93, Sherstov08]
[Beigel94, Bouland-Chen-Holden-Thaler-Vasudevan16]
[Bogdanov-Ishai-Viola-Williamson16]
The π-distinctness problem
This generalizes element distinctness, which is 2-distinctness.
Upper bounds
[Ambainis07]
[Belovs12]
Lower bounds
[Aaronson-Shi04]
π-distinctness: Given π numbers in π = {1,β¦ , π }, does any number appear β₯π times?
Our result: π Distπ = .
π-junta testing
Upper bounds
[AtΔ±cΔ±-Servedio07]
[Ambainis-Belovs-Regev-deWolf16]
Lower bounds
[AtΔ±cΔ±-Servedio07]
[Ambainis-Belovs-Regev-deWolf16]
π-junta testing: Given the truth table of a Boolean function, decide if
(YES) the function depends on at most π variables, or
(NO) the function is far (at least πΏπ 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 π π β« !
Surjectivity: Given π numbers in π (π = Ξ(π)), does every π β [π ] appear in the list?
Our result: and a new proof of .
Summary of results
Surjectivity upper bound
π SURJ = ΰ·¨π(π Ξ€3 4)
Surjectivity lower bound
π SURJ = ΰ·©Ξ©(π Ξ€3 4)
π-distinctness
π Distπ = ΰ·©Ξ©(π34 β
12π)
Image size testing
π IST = ΰ·©Ξ©( π)
π-junta testing
π Juntaπ = ΰ·©Ξ©( π)
Statistical distance
π SDU = ΰ·©Ξ©( π)
Shannon entropy
π Entropy = ΰ·©Ξ©( π)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 π1, π2, and π3 represent the acceptance probability of
algorithms (that output 0 or 1) π΄1, π΄2, and π΄3.
Algorithm: If π΄1 outputs 1, then output π΄2, else output π΄3.
Polynomial: π1 π₯ π2 π₯ + 1 β π1 π₯ π3(π₯).
Example: Implementing an if-then-else statement
Key idea: This is well defined even if ππ β [0,1]
Γ
=
Overview of the upper boundIdea 2: Polynomials are not algorithms
Overview of lower bounds
π
π π π π
ANDπ β ORπ
Open for 10+ years! Proof uses the method of dual polynomials.
PS: See Adam Boulandβs talk at 11:15 for an alternate proof using quantum arguments.
Lower bound [Sherstov13, Bun-Thaler13]: ΰ·ͺdeg ANDπ β ORπ = Ξ© ππ .
Proof 1: Use robust quantum search [HΓΈyer-Mosca-de Wolf03]
Proof 2: βπ, π, ΰ·ͺdeg = π ΰ·ͺdeg π ΰ·ͺdeg π [Sherstov13]
Upper bound: ΰ·ͺdeg ANDπ β ORπ = π ππ .
π Γ π input bits
Dual polynomialsApproximate degree can be expressed as a linear program
ΰ·ͺdeg π β€ π iff there exists a polynomial π of degree π, i.e., π = Οπ: π β€π πΌπ π₯π, s.t.,
βπ₯ β 0,1 π, π π₯ β π π₯ β€ 1/3
ΰ·ͺdeg π > π iff there exists π: 0,1 π β β,
1. Οπ₯ |π π₯ | = 1 (1) π is β1 normalized
2. If deg π β€ π then Οπ₯π π₯ π π₯ = 0 (2) π has pure high degree π
3. Οπ₯π π₯ β1 π(π₯) > 1/3. (3) π is well correlated with π
Lower bound for ANDπ β ORπ
Proof strategy for ΰ·ͺdeg ANDπ β ORπ = Ξ© ππ :
1. Start with πAND and πOR witnessing ΰ·ͺdeg = Ξ© π and ΰ·ͺdeg = Ξ© π
2. Combine these into π witnessing ΰ·ͺdeg ANDπ β ORπ = Ξ© ππ using the technique
of dual block composition.
ΰ·ͺdeg π > π iff there exists π: 0,1 π β β,
1. Οπ₯ |π π₯ | = 1 (1) π is β1 normalized
2. If deg π β€ π then Οπ₯π π₯ π π₯ = 0 (2) π has pure high degree π
3. Οπ₯π π₯ β1 π(π₯) > 1/3. (3) π is well correlated with π
Dual block composition for π β π
Composed dual automatically satisfies (1) and (2).
[Sherstov13, Bun-Thaler13] show that property (3) is also satisfied for ANDπ β ORπ.
ππβπ = 2π ππ sgn ππ π₯1 , β¦ , sgn ππ π₯π Οπ=1π ππ π₯π [Shi-Zhu09, Lee09, Sherstov13]
ΰ·ͺdeg π > π iff there exists π: 0,1 π β β,
1. Οπ₯ |π π₯ | = 1 (1) π is β1 normalized
2. If deg π β€ π then Οπ₯π π₯ π π₯ = 0 (2) π has pure high degree π
3. Οπ₯π π₯ β1 π(π₯) > 1/3. (3) π is well correlated with π
ΰ·ͺdeg SURJ = ΰ·©Ξ© π3/4
Reduction to a composed function
SURJ reduces to ANDπ β ORπ function, restricted
to inputs with Hamming weight β€ π.
We denote this function ANDπ β ORπβ€π.
β ΰ·ͺdeg SURJ = ΰ·¨π ΰ·ͺdeg ANDπ β ORπβ€π
SURJ π₯1, β¦ , π₯π = α₯
πβ π
α§
πβ π
(π₯π = π? )
Surjectivity: Given π numbers in π (π = Ξ(π)), does every π β [π ] appear in the list?
Converse [Ambainis05, Bun-Thaler17]: ΰ·ͺdeg SURJ = ΰ·©Ξ© ΰ·ͺdeg ANDπ β ORπβ€π
ANDπ β ORπ β ANDπ β ORπβ€π
ΰ·ͺdeg ANDπ β ORπ = Ξ π π = Ξ(π)
ΰ·ͺdeg ANDπ β ORπβ€π = ΰ·©Ξ ΰ·ͺdeg SURJ
Progress so far towards ΰ·ͺdeg SURJ = ΰ·©Ξ© π3/4
1. We saw that ΰ·ͺdeg SURJ = ΰ·©Ξ ΰ·ͺdeg ANDπ β ORπβ€π .
2. We saw using dual block composition that
ΰ·ͺdeg ANDπ β ORπ = Ξ© π π = Ξ©(π), when π = Ξ π .
Does the constructed dual also work for ANDπ β ORπβ€π? No.
ΰ·ͺdeg πβ€π» > π iff there exists π,
1. Οπ₯ |π π₯ | = 1 (1) π is β1 normalized
2. If deg π β€ π then Οπ₯π π₯ π π₯ = 0 (2) π has pure high degree π
3. Οπ₯π π₯ β1 π(π₯) > 1/3. (3) π is well correlated with π
4. π π₯ = 0 if π₯ > π» (4) π is only supported on the promise
Dual witness for ΰ·ͺdeg ANDπ β ORπβ€π
Fix 1: Use a dual witness πOR for ORπ that only certifies ΰ·ͺdeg = Ξ© π1/4 and satisfies
a βdual decay conditionβ, i.e., πOR π₯ is exponentially small for π₯ β« π1/4. Thus the
composed dual has degree Ξ© π π1/4 = Ξ©(π3/4) and almost satisfies condition (4).
Fix 2: Although condition (4) is only βalmost satisfiedβ in our dual witness, we can
postprocess the dual to have it be exactly satisfied [Razborov-Sherstov10].
ΰ·ͺdeg πβ€π» > π iff there exists π,
1. Οπ₯ |π π₯ | = 1 (1) π is β1 normalized
2. If deg π β€ π then Οπ₯π π₯ π π₯ = 0 (2) π has pure high degree π
3. Οπ₯π π₯ β1 π(π₯) > 1/3. (3) π is well correlated with π
4. π π₯ = 0 if π₯ > π» (4) π 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 Ξ© π .
Key property we exploit:
Any polynomial like this must
still have degree Ξ© π !
Open problems
[Ambainis07, Belovs-Ε palek13]
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