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Page 1: Effcient  quantum protocols for XOR functions

Effcient quantum protocols for XOR functions

Shengyu Zhang

The Chinese University of Hong Kong

Page 2: Effcient  quantum protocols for XOR functions

Communication complexity

β€’ Two parties, Alice and Bob, jointly compute a function on input . – known only to Alice and only to Bob.

β€’ Communication complexity*1: how many bits are needed to be exchanged?

π‘₯ 𝑦

*1. A. Yao. STOC, 1979.

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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.– Share resource? (Superscript.)

β€’ : share entanglement. β€’ : share nothing

– Error? (Subscript)β€’ : bounded-error. β€’ : zero-error, fixed length.

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Lower boundsβ€’ Not only interesting on its own, but also important

because of numerous applications.– to prove lower bounds.

β€’ Question: How to lower bound communication complexity itself?

β€’ Communication matrix

π‘₯β†’ 𝐹 (π‘₯ , 𝑦 )

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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.– 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.

≀ log𝑂 (1 )π‘Ÿπ‘Žπ‘›π‘˜ (𝑀𝐹 )Log Rank Conjecture*2

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Log-rank conjecture: quantum versionβ€’ Rank lower bound

β€’ Quantum: rank lower bound *1

*1. Buhrman, de Wolf. CCC, 2001.

≀ log𝑂 (1 )π‘Ÿπ‘Žπ‘›π‘˜ (𝑀𝐹 )Log Rank Conjecture

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Log-rank conjecture for XOR functions

β€’ Since Log-rank conjecture appears too hard in its full generality,…

β€’ let’s try some special class of functions.β€’ XOR functions: . ---– The linear composition of and .– Include important functions such as Equality,

Hamming Distance, Gap Hamming Distance.β€’ Interesting connections to Fourier analysis of

functions on .

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Digression: Fourier analysis

β€’ can be written as

– , and characters are orthogonal – : Fourier coefficients of – Parseval: If , then .

β€’ Two important measures: – --- Spectral norm.– --- Fourier sparsity.

β€’ Cauchy-Schwartz: for

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Log-rank Conj. For XOR functions

β€’ Interesting connections to Fourier analysis:β€’ 1. . β€’ Log-rank Conj: β€’ Thm.*1

β€’ Thm.*1 .– : degree of as polynomial over .– Fact*2. .

*1. Tsang, Wong, Xie, Zhang, FOCS, 2013.*2. Bernasconi and Codenotti. IEEE Transactions on Computers, 1999.

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Quantum

β€’ 2. *1

β€’ This paper: , where .– Recall classical: – Confirms quantum Log-rank Conjecture for low-

degree XOR functions.β€’ This talk: A simpler case .

– . *2

*1. Lee and Shraibman. Foundations and Trends in Theoretical Computer Science, 2009.*2. Buhrman and de Wolf. CCC, 2001.

Ξ© ( log π‘Ÿπ‘Žπ‘›π‘˜πœ– (𝑀 𝑓 βˆ˜βŠ• ))β‰₯

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About quantum protocol

β€’ Much simpler.

β€’ comes very naturally.

β€’ Inherently quantum.– Not from quantizing any classical protocol.

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|πœ“ ⟩=βˆ‘π›Όβˆˆπ΄

οΏ½Μ‚οΏ½ (𝛼 ) πœ’ 𝛼 (π‘₯ )|𝐸 (𝛼 ) ⟩ |πœ“ ⟩

|πœ“ β€² ⟩=βˆ‘π›Όβˆˆ 𝐴

οΏ½Μ‚οΏ½ (𝛼 ) πœ’ 𝛼 (π‘₯ ) πœ’ 𝛼(𝑦 )|𝐸 (𝛼 ) ⟩Add phase

|πœ“ β€² ⟩

βˆ‘π›Όβˆˆπ΄

𝑓 (𝛼 ) πœ’π›Ό (π‘₯+ 𝑦 )|𝛼 ⟩|𝐸 (𝛼 ) ⟩ β†’|𝛼 ⟩

Fourier:

|0 ⟩ +|1 ⟩√2

βŠ—

βˆ‘π‘‘

|0 ⟩+ 𝑓 (π‘₯+𝑦+𝑑)|1 ⟩√2

|𝑑 ⟩

Goal: compute where

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|πœ“ ⟩=ΒΏ+⟩ βˆ‘π›Όβˆˆπ΄

𝑓 (𝛼 ) πœ’π›Ό (π‘₯ )|𝐸 (𝛼 ) ⟩ |πœ“ ⟩Add phase |πœ“ β€² ⟩

Decoding + Fourier

βˆ‘π‘‘

|0 ⟩+ 𝑓 (π‘₯+𝑦+𝑑)|1 ⟩√2

|𝑑 ⟩

Measure Measure

A random and .Recall our target: . What’s the difference? The derivative: . Good: .Bad: .(That’s where the factor of comes from.)

β€’ One more issue: Only Alice knows ! Bob doesn’t.

β€’ It’s unaffordable to send .β€’ Obs: .

Goal: compute where

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|πœ“ ⟩=ΒΏ+⟩ βˆ‘π›Όβˆˆπ΄

𝑓 (𝛼 ) πœ’π›Ό (π‘₯ )|𝐸 (𝛼 ) ⟩ |πœ“ ⟩Add phase |πœ“ β€² ⟩

Decoding + Fourier

βˆ‘π‘‘

|0 ⟩+ 𝑓 (π‘₯+𝑦+𝑑)|1 ⟩√2

|𝑑 ⟩

Measure Measure

A random and .Recall our target: . What’s the difference? The derivative: . Good: .Bad: .(That’s where the factor of comes from.)

β€’ One more issue: Only Alice knows ! Bob doesn’t.

β€’ It’s unaffordable to send .β€’ Obs: .

𝐴

{0,1 }𝑛

Goal: compute where

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|πœ“ ⟩=ΒΏ+⟩ βˆ‘π›Όβˆˆπ΄

𝑓 (𝛼 ) πœ’π›Ό (π‘₯ )|𝐸 (𝛼 ) ⟩ |πœ“ ⟩Add phase |πœ“ β€² ⟩

Decoding + Fourier

βˆ‘π‘‘

|0 ⟩+ 𝑓 (π‘₯+𝑦+𝑑)|1 ⟩√2

|𝑑 ⟩

Measure Measure

A random and .Recall our target: . What’s the difference? The derivative: . Good: .Bad: .(That’s where the factor of comes from.)

β€’ One more issue: Only Alice knows ! Bob doesn’t.

β€’ It’s unaffordable to send .β€’ Obs: .

β€’ Thus in round 2, Alice and Bob can just encode the entire .

Goal: compute where

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|πœ“ ⟩=ΒΏ+⟩ βˆ‘π›Όβˆˆπ΄

𝑓 (𝛼 ) πœ’π›Ό (π‘₯ )|𝐸 (𝛼 ) ⟩ |πœ“ ⟩Add phase |πœ“ β€² ⟩

Decoding + Fourier

βˆ‘π‘‘

|0 ⟩+ 𝑓 (π‘₯+𝑦+𝑑)|1 ⟩√2

|𝑑 ⟩

Measure Measure

A random and .Compute .

At last, , a constant function.Cost: .

Used trivial bound:

Goal: compute where

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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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