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Effcient quantum protocols for XOR functions

Effcient quantum protocols for XOR functions. Shengyu Zhang The Chinese University of Hong Kong. Communication complexity. Two parties, Alice and Bob, jointly compute a function on input . known only to Alice and only to Bob.

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Effcient quantum protocols for XOR functions

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  1. Effcient quantum protocols for XOR functions ShengyuZhang The Chinese University of Hong Kong

  2. Communication complexity • Two parties, Alice and Bob, jointly compute a function on input . • known only to Alice and only to Bob. • Communication complexity*1: how many bits are needed to be exchanged? *1. A. Yao. STOC, 1979.

  3. Computation modes • Deterministic: Players run determ. protocol. --- • Randomized: Players have access to random bits; small error probability allowed. --- • Quantum: Players send quantum messages. • Share resource? (Superscript.) • : share entanglement. • : share nothing • Error? (Subscript) • : bounded-error. • : zero-error, fixed length.

  4. Lower bounds • Not only interesting on its own, but also important because of numerous applications. • to prove lower bounds. • Question: How to lower bound communication complexity itself? • Communication matrix

  5. Log-rank conjecture Log Rank Conjecture*2 • Rank lower bound*1 • Conj: The rank lower bound is polynomially tight. • combinatorial measure linear algebra measure. • 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. Lovász, Saks. FOCS, 1988. *3. Lovász. Book Chapter, 1990. *4. Valiant. Info. Proc. Lett., 2004. *5. Ambainis, Schulman, Ta-Shma, Vazirani, Wigderson, SICOMP 2003.

  6. Log-rank conjecture: quantum version Log Rank Conjecture • Rank lower bound • Quantum: rank lower bound *1 *1. Buhrman, de Wolf. CCC, 2001.

  7. Log-rank conjecture for XOR functions • Since Log-rank conjecture appears too hard in its full generality,… • let’s try some special class of functions. • XOR functions: . --- • The linear composition of and . • Include important functions such as Equality, Hamming Distance, Gap Hamming Distance. • Interesting connections to Fourier analysis of functions on .

  8. Digression: Fourier analysis • can be written as • , and characters are orthogonal • : Fourier coefficients of • Parseval: If , then . • Two important measures: • --- Spectral norm. • --- Fourier sparsity. • Cauchy-Schwartz: for

  9. Log-rank Conj. For XOR functions • Interesting connections to Fourier analysis: • 1. . • Log-rank Conj: • Thm.*1 • 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.

  10. Quantum • 2. *1 • This paper: , where . • Recall classical: • Confirms quantum Log-rank Conjecture for low-degree XOR functions. • 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.

  11. About quantum protocol • Much simpler. • comes very naturally. • Inherently quantum. • Not from quantizing any classical protocol.

  12. Goal: compute where Add phase Fourier:

  13. Goal: compute where Add phase Decoding + Fourier • One more issue: Only Alice knows ! Bob doesn’t. • It’s unaffordable to send . • Obs: . Measure Measure A random and . Recall our target:. What’s the difference? The derivative: . Good: . Bad: . (That’s where the factor of comes from.)

  14. Goal: compute where Add phase Decoding + Fourier • One more issue: Only Alice knows ! Bob doesn’t. • It’s unaffordable to send . • Obs: . Measure Measure A random and . Recall our target:. What’s the difference? The derivative: . Good: . Bad: . (That’s where the factor of comes from.)

  15. Goal: compute where Add phase Decoding + Fourier • One more issue: Only Alice knows ! Bob doesn’t. • It’s unaffordable to send . • Obs: . • Thus in round 2, Alice and Bob can just encode the entire . Measure Measure A random and . Recall our target:. What’s the difference? The derivative: . Good: . Bad: . (That’s where the factor of comes from.)

  16. Goal: compute where Add phase Decoding + Fourier Measure Measure A random and . Compute . At last, , a constant function. Cost: . Used trivial bound:

  17. Open problems • Get rid of the factor ! • What can we say about additive structure of for Boolean functions ? Say, ?

  18. Thanks

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