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Quantum Communication Complexity

Quantum Communication Complexity. Richard Cleve Institute for Quantum Computing University of Waterloo. 1. Preliminaries. How does quantum information affect the communication costs of information processing tasks?. Potential applications Context in which to explore interesting

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Quantum Communication Complexity

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  1. Quantum Communication Complexity Richard Cleve Institute for Quantum Computing University of Waterloo Aug 2, 2005

  2. 1. Preliminaries

  3. How does quantum information affect the communication costs of information processing tasks? • Potential applications • Context in which to explore interesting • properties of quantum information • Interplay with quantum algorithms, • nonlocality, and information theory

  4. How much classical information in n qubits? • 2n1 complex numbers are needed to describe an arbitrary n-qubit pure quantum state: • 000000 + 001001 + 010010 +  + 111111 • Does this mean that an exponential amount of classical information is somehow stored in n qubits? • No … • Holevo’s Theorem [1973] implies: cannot extract more than n bitsfrom n qubits

  5. b1 b1 U U ψ b2 ψ b2 b3 b3 n qubits n qubits bn bn 0 bn+1 0 bn+2 m qubits 0 bn+3 0 bn+4 0 bn+m Holevo’s Theorem Easy case: Hard case (the general case): b1b2 ... bncannot convey more than n bits! (proof omitted here)

  6. qubit qubit Entanglement & signaling Example of an entangled state: Can be used to perform some intriguing feats, such as teleportation, superdense coding, and “pseudo-telepathy” Can entangled states be used to “signal instantaneously”? No … any operation performed on one qubit has no affect on the state of the other qubit

  7. Resources Basic communication scenario Goal: convey n bits from Alice to Bob x1x2  xn Alice Bob x1x2  xn

  8. Bit communication: Qubit communication: Cost:n Cost:n Bit communication & prior entanglement: Qubit communication & prior entanglement: Cost:n Cost:n/2superdense coding Basic communication scenario [H ’73] [BW ’92]

  9. 2. Communication complexity

  10. Classical communication complexity x1x2  xn y1y2  yn f (x,y) E.g. equalityfunction:f (x,y)=1ifx=y,and 0 ifxy Any deterministic protocol requires n bits communication Probabilistic protocols can solve with only O(log(n/)) bits communication (error probability ) [Yao ’79]

  11. Classical communication complexity x1x2  xn y1y2  yn x = y? Probabilistic protocol for Equality ( =1/n): px(T)= x0+ x1T+ x2T2+ … + xn1Tn1 py(T)= y0+ y1T+ y2T2+ … + yn1Tn1 Arithmetic modulo m, for a prime m between n2 and 2n2 Alice: pick random t {0, 1,…, m1} send (t, px(t) mod m)to Bob (this is only 4log(n) bits) Bob: accept iff px(t) =py(t) mod m(err prob <n/n2= 1/n)

  12. x1x2  xn x1x2  xn y1y2  yn y1y2  yn qubits f (x,y) f (x,y)   entangled qubits bits Quantum communication complexity Qubit communication Prior entanglement [Y ’93] [CB ’97]

  13. 1 2 3 4 5 . . . n 1 2 3 4 5 . . . n 0 1 1 0 1 … 0 1 0 0 1 1 … 1 Appointment scheduling x= y= i (xi=yi=1) Classically, (n)bits necessary to succeed with prob.  3/4 For all >0, O(n1/2logn)qubits sufficient for error prob. < [KS ’87] [BCW ’98]

  14. 1 2 3 4 5 6 . . . n 0 0 0 0 1 0 … 1 x= i x i i i log n b  xi b  xi b b 1 Search problem Given: accessible via queries Ux Alternate notation Goal: find i{1, 2, …, n} such that xi=1 Classically:(n)queries are necessary Quantum mechanically:O(n1/2) queries are sufficient [G ’96]

  15. xy y x x y i i 0 0 0 0 b b Bob Alice Bob 1 2 3 4 5 6 . . . n x= 0 1 1 0 1 0 … 0 Alice y= 1 0 0 1 1 0 … 1 Bob xy= 0 0 0 0 1 0 … 0  Communication per xy-query:2(logn+ 3) = O(log n)

  16. Bit communication: Qubit communication: Cost:θ(n) Bit communication & prior entanglement: Qubit communication & prior entanglement: Cost:θ(n1/2) Cost:θ(n1/2) Appointment scheduling: epilogue Cost:O(n1/2log(n)) Cost:θ(n1/2) [R ’02] [AA ’03]

  17. Restricted version of equality Precondition (i.e. promise): either x = y or (x,y) =n/2 Hamming distance Classically, (n) bits communication are still necessary for an exact solution Quantum mechanically, O(log n) qubits communication are sufficient for an exact solution (It’s a distributed variant of the Deutsch-Jozsa problem … a “constant” vs. “balanced” distinguishing problem) [BCW ’98]

  18. Classical lower bound (*skipped) Theorem: If S {0,1}n has the property that, for all x, x′S, their intersection size is notn/4 then S < 1.99n Let some protocol solve restricted equality with k bits comm. ● 2k conversations of length k ● approximately2n/n input pairs (x, x), where Δ(x)=n/2 Therefore, 2n/2kn input pairs (x, x) that yield same conv. C Define S= {x : Δ(x)=n/2 and (x, x) yields conv. C } For any x, x′S, input pair (x, x′)also yields conversation C Therefore, Δ(x, x′)n/2,implying intersection size is notn/4 Theorem implies 2n/2kn<1.99n , so k> 0.007n [Frankl and Rödl, 1987]

  19. For each x {0,1}n, define Quantum protocol • Protocol: • Alice sends x to Bob (log(n) qubits) • Bob measures state in a basis that includes y Correctness of protocol: If x = y then Bob’s result is definitely y If (x,y) =n/2 then xy=0, so result is definitely noty Question: How much communication if error prob. ¼ is ok? Answer: just 2 bits are sufficient!

  20. Exponential quantum vs. classical separation in bounded-error models : a log(n)-qubit state (described classically) M: two-outcome measurement U: unitary operation on log(n) qubits Output: result of applying M to U O(log n) quantum vs. (n1/4 /log n) classical communication [R ’99]

  21. 3. Quantum speed-up is not always possible

  22. Inner product IP(x,y)=x1y1+x2y2+ +xnyn mod 2 Classically, (n) bits of communication are required, even for bounded-error protocols Quantum protocols also require (n) communication [KY ’95] [CNDT ’98] [NS ’02]

  23. Recall Deutsch’s problem Let f:{0,1}  {0,1} be of the form f(x) =a1x+a0mod 2 Given: black box for f Goal: determine a1 (a1 = 0 implies “constant”; a1 =1 implies “balanced”) Classically, 2 queries are necessary Quantum mechanically, 1 query is sufficient

  24. a1 0 x1 x1 H H f a2 0 x2 x2 H H     H H xn 0 an xn H H 1 1 b b  f(x1, x2, …, xn) H H Bernstein-Vazirani problem(multidimensional Deutsch problem) Let f(x1, x2, …, xn) =a1x1+a2x2+ +anxn +a0mod 2 Given: Goal: determine a1, a2,…, an Classically, n+1 queries are necessary Quantum mechanically, 1 query is sufficient

  25. Lower bound for inner product x1 x2 xn y1 y2 yn z Alice and Bob’s IP protocol Alice and Bob’s IP protocol inverted x1 x2 xn y1 y2 yn zIP(x,y) IP(x,y) = x1y1 +x2y2 + +xnyn mod 2 Proof:

  26. Lower bound for inner product H H H H H H H H IP(x,y)=x1y1+x2y2+ +xnyn mod 2 0 0 0 1 x1 x2 xn Proof: Alice and Bob’s IP protocol Alice and Bob’s IP protocol inverted x1 x2 xn x1 x2 xn 1 [BV, 1993] Since n bits are conveyed from Alice to Bob, n qubits communication necessary (by Holevo’s Theorem)

  27. 4. Simultaneous messages to a third party

  28. Equality function: f (x,y)=1ifx = y 0 ifxy Equality revisitedin simultaneous message model x1x2  xn y1y2  yn f (x,y) Exactprotocols: require 2n bits communication

  29. 0 1 1 1 1 1 1 0 1 1 0 1 random k Equality revisitedin simultaneous message model x1x2  xn y1y2  yn f (x,y) Bounded-error protocolswith a shared random key:require only O(1) bits communication Error-correcting code: C(x) =0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 0 C(y) =0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0

  30. Equality revisitedin simultaneous message model x1x2  xn y1y2  yn f (x,y) Bounded-error protocolswithout a shared key: Classical: θ(n1/2) Quantum:θ(logn) [A ’96] [NS ’96] [BCWW ’01]

  31. Quantum fingerprints Question 1: how many orthogonal states in k qubits? Answer: 2k Question 2: how many almost orthogonal* states in k qubits? (* where |xy|≤ ) Answer: 2c2k, for some constant c> 0 Question 3:does this enable k qubits to store c2kbits? (In other words, log n+O(1) qubits to store n bits?) Answer:no … recall Holevo’s Theorem However, it does enable one to check ifx=yorx≠yby only examiningx andy

  32. 0 H H x S W A P y Intuition: 0xy +1yx Quantum fingerprints Let 000,001, …,111 be 2n states on log n + O(1) qubits such that |xy|≤  for all x≠y Given xy, one can check if x=y or x≠y as follows: if x=y, Pr[output= 0] = 1 if x≠y, Pr[output= 0] = (1+2)/2

  33. Orthogonality test Quantum protocol for equality in simultaneous message model x1x2  xn y1y2  yn x y x y

  34. 5. One-way communication

  35. M= matching on {1, 2, …, n} (partition into pairs) x {0,1}n Inputs: (i, j,xixj), such that (i, j) M Output: Hidden matching problem Only one-way communication (Alice to Bob) is permitted Quantum protocol can be exponentially more efficient than any classical protocol—even with a shared key [BJK ’04]

  36. Hidden matching problem M= matching on {1,2, …, n} x {0,1}n Inputs: Output: (i, j,xixj), (i, j) M Classically, one-way communication is (n) for bounded-error even with a shared classical key (the proof is omitted here) Intuition: With Alice’s message Bob can repeat his side of the protocol using several edge-disjoint matchings, which yields information about several xixj bits …

  37. Quantum protocol that uses only log nqubits: Alice sends (log n qubits) to Bob Hidden matching problem M= matching on {1,2, …, n} x {0,1}n Inputs: Output: (i, j,xixj), (i, j) M Bob measures in the basis {i  j|(i, j) M}, and then uses the outcome’s relative phase to deduce xixj

  38. 6. Nonlocality revisited

  39. (1 bit) (1 bit) (1 bit) (1 bit) Communication complexity with distributed outputs x y inputs: a b outputs: where a, b, x, y satisfy some relation E.g. “Bell’s Theorem” Goal: ab=xy with zero communication With classical resources, Pr[ab=xy] ≤ 0.75 With 00 + 11 prior entanglement, Pr[ab=xy] = 0.853… [B ’64] [CHSH ’69]

  40. Distributed outputs:“spooky Deutsch-Jozsa” x y inputs: (n bits) (n bits) a b outputs: (logn bits) (logn bits) Precondition: either x =y or (x,y) =n/2 Required postcondition: a =b iff x =y With classical resources, (n) bits of communication needed for an exact solution With (00 +11)logn prior entanglement, no communication is needed at all [BCT ’99]

  41. Bit communication: Qubit communication: Cost:log n Cost:θ(n) Bit communication & prior entanglement: Qubit communication & prior entanglement: Cost: zero Cost: zero Distributed-output restricted equality

  42. M= matching on {1, 2, …, n} (partition into pairs) x {0,1}n Inputs: Distributed-output hidden matching (b, i, j), such that 1.(i, j)M 2. (ab)·(ij)= xixj Outputs: a  {0,1}logn With prior entanglement, no communication necessary; without prior entanglement, one-way communication is (n), even to achieve success probability ¾ [B ’04]

  43. Some open problems • Develop some “Killer Apps” • Exponential separation between one-round quantum and multi-round classical? • Are the qubit communication and the prior entanglement models equivalent? • The distributed-output scenario can be viewed as a two-prover interactive proof system, raising questions about their expressive power in a quantum world (may come up on Thursday …)

  44. Selected references I • Z. Bar-Yossef, T.S. Jayram, I. Kerenidis, “Exponential separation of quantum and classical one-way communication complexity”, Proceedings of 36th Annual ACM Symposium on Theory of Computing, pages 128-137, 2004. • G. Brassard, “Quantum communication complexity”, Foundations of Physics, 33(11): 1593-1616, 2003. • R. de Wolf, “Quantum communication and complexity”, Theoretical Computer Science, 287(1): 337-353, 2002. Available at http://homepages.cwi.nl/~rdewolf/ • G. Brassard, R. Cleve, A. Tapp, “Cost of exactly simulating quantum entanglement with classical communication”, Physical Review Letters, 83(9): 1874-1877, 1999. • H. Buhrman, R. Cleve, W. van Dam, “Quantum entanglement and communication complexity”, SIAM Journal on Computing, 2000. • H. Buhrman, R. Cleve, A. Wigderson, “Quantum vs. classical communication and computation”, Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 63-68, 1998. • R. Cleve, H. Buhrman, “Substituting quantum entanglement for communication”, Physical Review A, 56(2): 1201-1204, 1997.

  45. Selected references II • R. Cleve, W. van Dam, P. Høyer, A. Tapp, “Quantum entanglement and the communication complexity of the inner product function”, Lecture Notes in Computer Science, 1509: 61-74, 1999. • A. Holevo, “Bounds on the quantity of information transmitted by a quantum communication channel”, Problems of Information Transmission, 9: 177-183, 1973. • B. Kalyanasundaram, G. Schnitger, “The probabilistic communication complexity of set intersection”, Proceedings of 2nd Annual IEEE Conference on Structure in Complexity Theory, pages 41-47, 1987. • I. Kremer, Quantum Communication, Master’s thesis, Hebrew University, Computer Science Department, 1995. • R. Raz, “Exponential separation of quantum and classical communication complexity”, Proceedings of 31st Annual ACM Symposium on Theory of Computing, pages 358-367, 1999. • A. C.-C. Yao, “Some questions related to distributed computing”, Proceedings of 11th Annual ACM Symposium on Theory of Computing, pages 209-213, 1979. • A. C.-C. Yao, “Quantum circuit complexity”, Proceedings of 34th Annual IEEE Symposium on Foundations of Computer Science, pages 352-361, 1993.

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