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

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? 2 n  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 bits from n qubits

5 Holevo’s Theorem ψψ n qubits U b1b1 b2b2 b3b3 bnbn Easy case: b 1 b 2... b n cannot convey more than n bits! Hard case (the general case): ψψ n qubits b1b1 b2b2 b3b3 bnbn U 00 00 00 00 00 m qubits bn+1bn+1 bn+2bn+2 bn+3bn+3 bn+4bn+4 bn+mbn+m (proof omitted here)

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

7 AliceBob Basic communication scenario Resources x 1 x 2  x n Goal: convey n bits from Alice to Bob x 1 x 2  x n

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

9 2. Communication complexity

10 Classical communication complexity f (x,y) x 1 x 2  x n y 1 y 2  y n E.g. equality function: f (x,y) = 1 if x = y, and 0 if x  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 x = y ? x 1 x 2  x n y 1 y 2  y n Probabilistic protocol for Equality (  = 1/ n ): p x ( T ) = x 0 + x 1 T + x 2 T 2 + … + x n  1 T n  1 p y ( T ) = y 0 + y 1 T + y 2 T 2 + … + y n  1 T n  1 Alice: pick random t  {0, 1,…, m  1} send ( t, p x ( t ) mod m ) to Bob (this is only 4 log ( n ) bits) Bob: accept iff p x ( t ) = p y ( t ) mod m (err prob < n / n 2 = 1/ n ) Arithmetic modulo m, for a prime m between n 2 and 2 n 2

12 Quantum communication complexity Qubit communication Prior entanglement f (x,y) x 1 x 2  x n y 1 y 2  y n qubits f (x,y) x 1 x 2  x n y 1 y 2  y n  entangled qubits bits [Y ’93] [CB ’97]

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

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

… n x =x = … 1 y =y = … 0 xy =xy = Alice Bob ii 00 00 bb xy xy  ii 00 00 bb y y Alice xx Communication per x  y -query: 2 ( log n + 3 ) = O ( log n)

16 Appointment scheduling: epilogue Bit communication: Cost: θ( n ) Qubit communication: Cost: O( n 1 / 2 log( n )) Bit communication & prior entanglement: Cost: θ( n 1 / 2 ) Qubit communication & prior entanglement: Cost: θ( n 1 / 2 ) [R ’02] [AA ’03] Cost: θ( n 1 / 2 )

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 [BCW ’98] (It’s a distributed variant of the Deutsch-Jozsa problem … a “constant” vs. “balanced” distinguishing problem)

18 Classical lower bound (*skipped) Theorem: If S  {0,1} n has the property that, for all x, x ′  S, their intersection size is not n/ 4 then  S  < 1.99 n [Frankl and Rödl, 1987] Let some protocol solve restricted equality with k bits comm. ● approximately 2 n /  n input pairs ( x, x ), where Δ ( x ) = n/ 2 Define S = { x : Δ ( x ) = n/ 2 and ( x, x ) yields conv. C } Therefore, 2 n / 2 k  n input pairs ( x, x ) that yield same conv. C ● 2 k conversations of length k For any x, x ′  S, input pair ( x, x ′ ) also yields conversation C Therefore, Δ ( x, x ′)  n/ 2, implying intersection size is not n/ 4 Theorem implies 2 n / 2 k  n n

19 Quantum protocol For each x  {0,1} n, define Protocol: 1.Alice sends  x  to Bob ( log( n ) qubits) 2.Bob measures state in a basis that includes  y  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! Correctness of protocol:

20 Exponential quantum vs. classical separation in bounded-error models O(log n) quantum vs.  (n 1 / 4 / log n) classical communication Output: result of applying M to U   : a log( n ) -qubit state (described classically) M : two-outcome measurement U : unitary operation on log( n ) qubits [R ’99]

21 3. Quantum speed-up is not always possible

22 Inner product IP(x, y) = x 1 y 1 + x 2 y 2 +  + x n y n 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 ) = a 1 x + a 0 mod 2 Given: black box for f Goal: determine a 1 ( a 1 = 0 implies “constant”; a 1 = 1 implies “balanced”) Classically, 2 queries are necessary Quantum mechanically, 1 query is sufficient

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

25 Lower bound for inner product IP(x, y) = x 1 y 1 + x 2 y 2 +  + x n y n mod 2 y1y1ynyny2y2 Alice and Bob’s IP protocol x2x2x1x1xnxn  z  IP(x, y)  Alice and Bob’s IP protocol inverted y1y1y2y2ynynx1x1x2x2xnxn zz Proof:

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

27 4. Simultaneous messages to a third party

28 Equality revisited in simultaneous message model x 1 x 2  x n y 1 y 2  y n f (x,y) Exact protocols: require 2 n bits communication Equality function: f (x,y) = 1 if x = y 0 if x  y

29 Equality revisited in simultaneous message model x 1 x 2  x n y 1 y 2  y n f (x,y) Bounded-error protocols with a shared random key: require only O(1) bits communication Error-correcting code: C( x ) = C( y ) = random k

30 Equality revisited in simultaneous message model x 1 x 2  x n y 1 y 2  y n f (x,y) Bounded-error protocols without a shared key: Classical: θ(n 1 / 2 ) Quantum: θ(log n) [A ’96] [NS ’96] [BCWW ’01]

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

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

33 Quantum protocol for equality in simultaneous message model x 1 x 2  x n y 1 y 2  y n xx yy Orthogonality test xxyy

34 5. One-way communication

35 Hidden matching problem x  {0,1} n matching on {1, 2, …, n } (partition into pairs) Inputs: M =M = [BJK ’04] (i, j, x i  x j ), such that ( i, j )  M Output: 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

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

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

38 6. Nonlocality revisited

39 Communication complexity with distributed outputs b xy a inputs: outputs: (1 bit) 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” b xy a inputs: outputs: ( n bits) ( log n bits) ( n bits) ( log n bits) With classical resources,  ( n ) bits of communication needed for an exact solution With (  00  +  11  ) log n prior entanglement, no communication is needed at all Precondition: either x = y or  ( x, y ) = n / 2 Required postcondition: a = b iff x = y [BCT ’99]

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

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

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 36 th Annual ACM Symposium on Theory of Computing, pages , G. Brassard, “Quantum communication complexity”, Foundations of Physics, 33(11): , R. de Wolf, “Quantum communication and complexity”, Theoretical Computer Science, 287(1): , Available at G. Brassard, R. Cleve, A. Tapp, “Cost of exactly simulating quantum entanglement with classical communication”, Physical Review Letters, 83(9): , H. Buhrman, R. Cleve, W. van Dam, “Quantum entanglement and communication complexity”, SIAM Journal on Computing, H. Buhrman, R. Cleve, A. Wigderson, “Quantum vs. classical communication and computation”, Proceedings of the 30 th Annual ACM Symposium on Theory of Computing, pages 63-68, R. Cleve, H. Buhrman, “Substituting quantum entanglement for communication”, Physical Review A, 56(2): , 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, A. Holevo, “Bounds on the quantity of information transmitted by a quantum communication channel”, Problems of Information Transmission, 9: , B. Kalyanasundaram, G. Schnitger, “The probabilistic communication complexity of set intersection”, Proceedings of 2 nd Annual IEEE Conference on Structure in Complexity Theory, pages 41-47, I. Kremer, Quantum Communication, Master’s thesis, Hebrew University, Computer Science Department, R. Raz, “Exponential separation of quantum and classical communication complexity”, Proceedings of 31 st Annual ACM Symposium on Theory of Computing, pages , A. C.-C. Yao, “Some questions related to distributed computing”, Proceedings of 11 th Annual ACM Symposium on Theory of Computing, pages , A. C.-C. Yao, “Quantum circuit complexity”, Proceedings of 34 th Annual IEEE Symposium on Foundations of Computer Science, pages , 1993.

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47 Table of Contents 1.Communication 2.Communication complexity Equality (P vs D, exponential) [details of probabilistic …] Intersection (Q vs P, polynomial) Special equality (Q vs P, exponential, exact, promise) Raz’s problem (Q vs P, exponential, bounded-error, promise) Inner product (Q just as hard as P, bounded error) 3.Restricted communication: simultaneous messages Equality (Q vs P, exp, bound-err, total, no common randomness) 4.Restricted communication: one round Hidden matching (Q vs P, exponential, bounded error, one-way) 5.Distributed outputs (aka nonlocality/pseudo-telepathy) Special equality revisited Hidden matching revisited