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Effcient quantum protocols for XOR functions

This paper explores efficient quantum communication protocols for computing XOR functions between two parties, Alice and Bob, who share no common input. We analyze the communication complexity involved and delve into the fundamental limits established by the Log-rank Conjecture, focusing particularly on lower bounds. Key connections to Fourier analysis are discussed, highlighting significant functions like Equality and Hamming Distance. This research not only confirms quantum implications for low-degree XOR functions but also opens avenues for addressing open problems in communication complexity.

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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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