Effcient quantum protocols for XOR functions
Shengyu Zhang
The Chinese University of Hong Kong
Communication complexity
β’ Two parties, Alice and Bob, jointly compute a function on input . β known only to Alice and only to Bob.
β’ Communication complexity*1: how many bits are needed to be exchanged?
π₯ π¦
*1. A. Yao. STOC, 1979.
Computation modes
β’ Deterministic: Players run determ. protocol. ---β’ Randomized: Players have access to random bits;
small error probability allowed. --- β’ Quantum: Players send quantum messages.β Share resource? (Superscript.)
β’ : share entanglement. β’ : share nothing
β Error? (Subscript)β’ : bounded-error. β’ : zero-error, fixed length.
4
Lower boundsβ’ Not only interesting on its own, but also important
because of numerous applications.β to prove lower bounds.
β’ Question: How to lower bound communication complexity itself?
β’ Communication matrix
π₯β πΉ (π₯ , π¦ )
5
Log-rank conjectureβ’ Rank lower bound*1
β’ Conj: The rank lower bound is polynomially tight.β’ combinatorial measure linear algebra measure.β’ Equivalent to a bunch of other conjectures.β related to graph theory*2; nonnegative rank*3, Boolean
roots of polynomials*4, quantum sampling complexity*5.
*1. Melhorn, Schmidt. STOC, 1982. *2. LovΓ‘sz, Saks. FOCS, 1988.*3. LovΓ‘sz. Book Chapter, 1990.
*4. Valiant. Info. Proc. Lett., 2004.*5. Ambainis, Schulman, Ta-Shma, Vazirani, Wigderson, SICOMP 2003.
β€ logπ (1 )ππππ (ππΉ )Log Rank Conjecture*2
6
Log-rank conjecture: quantum versionβ’ Rank lower bound
β’ Quantum: rank lower bound *1
*1. Buhrman, de Wolf. CCC, 2001.
β€ logπ (1 )ππππ (ππΉ )Log Rank Conjecture
Log-rank conjecture for XOR functions
β’ Since Log-rank conjecture appears too hard in its full generality,β¦
β’ letβs try some special class of functions.β’ XOR functions: . ---β The linear composition of and .β Include important functions such as Equality,
Hamming Distance, Gap Hamming Distance.β’ Interesting connections to Fourier analysis of
functions on .
Digression: Fourier analysis
β’ can be written as
β , and characters are orthogonal β : Fourier coefficients of β Parseval: If , then .
β’ Two important measures: β --- Spectral norm.β --- Fourier sparsity.
β’ Cauchy-Schwartz: for
Log-rank Conj. For XOR functions
β’ Interesting connections to Fourier analysis:β’ 1. . β’ Log-rank Conj: β’ Thm.*1
β’ Thm.*1 .β : degree of as polynomial over .β Fact*2. .
*1. Tsang, Wong, Xie, Zhang, FOCS, 2013.*2. Bernasconi and Codenotti. IEEE Transactions on Computers, 1999.
Quantum
β’ 2. *1
β’ This paper: , where .β Recall classical: β Confirms quantum Log-rank Conjecture for low-
degree XOR functions.β’ This talk: A simpler case .
β . *2
*1. Lee and Shraibman. Foundations and Trends in Theoretical Computer Science, 2009.*2. Buhrman and de Wolf. CCC, 2001.
Ξ© ( log πππππ (π π ββ ))β₯
About quantum protocol
β’ Much simpler.
β’ comes very naturally.
β’ Inherently quantum.β Not from quantizing any classical protocol.
|π β©=βπΌβπ΄
οΏ½ΜοΏ½ (πΌ ) π πΌ (π₯ )|πΈ (πΌ ) β© |π β©
|π β² β©=βπΌβ π΄
οΏ½ΜοΏ½ (πΌ ) π πΌ (π₯ ) π πΌ(π¦ )|πΈ (πΌ ) β©Add phase
|π β² β©
βπΌβπ΄
π (πΌ ) ππΌ (π₯+ π¦ )|πΌ β©|πΈ (πΌ ) β© β|πΌ β©
Fourier:
|0 β© +|1 β©β2
β
βπ‘
|0 β©+ π (π₯+π¦+π‘)|1 β©β2
|π‘ β©
Goal: compute where
|π β©=ΒΏ+β© βπΌβπ΄
π (πΌ ) ππΌ (π₯ )|πΈ (πΌ ) β© |π β©Add phase |π β² β©
Decoding + Fourier
βπ‘
|0 β©+ π (π₯+π¦+π‘)|1 β©β2
|π‘ β©
Measure Measure
A random and .Recall our target: . Whatβs the difference? The derivative: . Good: .Bad: .(Thatβs where the factor of comes from.)
β’ One more issue: Only Alice knows ! Bob doesnβt.
β’ Itβs unaffordable to send .β’ Obs: .
Goal: compute where
|π β©=ΒΏ+β© βπΌβπ΄
π (πΌ ) ππΌ (π₯ )|πΈ (πΌ ) β© |π β©Add phase |π β² β©
Decoding + Fourier
βπ‘
|0 β©+ π (π₯+π¦+π‘)|1 β©β2
|π‘ β©
Measure Measure
A random and .Recall our target: . Whatβs the difference? The derivative: . Good: .Bad: .(Thatβs where the factor of comes from.)
β’ One more issue: Only Alice knows ! Bob doesnβt.
β’ Itβs unaffordable to send .β’ Obs: .
π΄
{0,1 }π
Goal: compute where
|π β©=ΒΏ+β© βπΌβπ΄
π (πΌ ) ππΌ (π₯ )|πΈ (πΌ ) β© |π β©Add phase |π β² β©
Decoding + Fourier
βπ‘
|0 β©+ π (π₯+π¦+π‘)|1 β©β2
|π‘ β©
Measure Measure
A random and .Recall our target: . Whatβs the difference? The derivative: . Good: .Bad: .(Thatβs where the factor of comes from.)
β’ One more issue: Only Alice knows ! Bob doesnβt.
β’ Itβs unaffordable to send .β’ Obs: .
β’ Thus in round 2, Alice and Bob can just encode the entire .
Goal: compute where
|π β©=ΒΏ+β© βπΌβπ΄
π (πΌ ) ππΌ (π₯ )|πΈ (πΌ ) β© |π β©Add phase |π β² β©
Decoding + Fourier
βπ‘
|0 β©+ π (π₯+π¦+π‘)|1 β©β2
|π‘ β©
Measure Measure
A random and .Compute .
At last, , a constant function.Cost: .
Used trivial bound:
Goal: compute where
Open problems
β’ Get rid of the factor !
β’ What can we say about additive structure of for Boolean functions ? Say, ?
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