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Quantum Communication Complexity

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### Quantum Communication Complexity

Richard Cleve

Institute for Quantum Computing

University of Waterloo

Aug 2, 2005

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

How much classical information in n qubits?

- 2n1 complex numbers are needed to describe an arbitrary n-qubit pure quantum state:
- 000000 + 001001 + 010010 + + 111111
- 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

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 TheoremEasy case:

Hard case (the general case):

b1b2 ... bncannot convey more than n bits!

(proof omitted here)

qubit

Entanglement & signalingExample 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

Basic communication scenario

Goal: convey n bits from Alice to Bob

x1x2 xn

Alice

Bob

x1x2 xn

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]

complexity

Classical communication complexity

x1x2 xn

y1y2 yn

f (x,y)

E.g. equalityfunction:f (x,y)=1ifx=y,and 0 ifxy

Any deterministic protocol requires n bits communication

Probabilistic protocols can solve with only O(log(n/)) bits communication (error probability )

[Yao ’79]

Classical communication complexity

x1x2 xn

y1y2 yn

x = y?

Probabilistic protocol for Equality ( =1/n):

px(T)= x0+ x1T+ x2T2+ … + xn1Tn1

py(T)= y0+ y1T+ y2T2+ … + yn1Tn1

Arithmetic modulo m, for a prime m between n2 and 2n2

Alice: pick random t {0, 1,…, m1}

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)

x1x2 xn

x1x2 xn

y1y2 yn

y1y2 yn

qubits

f (x,y)

f (x,y)

entangled qubits

bits

Quantum communication complexityQubit communication

Prior entanglement

[Y ’93] [CB ’97]

1 2 3 4 5 . . . n

0 1 1 0 1 … 0

1 0 0 1 1 … 1

Appointment schedulingx=

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]

0 0 0 0 1 0 … 1

x=

i

x

i

i

i

log n

b xi

b xi

b

b

1

Search problemGiven:

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]

xy

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

xy=

0 0 0 0 1 0 … 0

Communication per xy-query:2(logn+ 3) = O(log n)

Qubit communication:

Cost:θ(n)

Bit communication & prior entanglement:

Qubit communication & prior entanglement:

Cost:θ(n1/2)

Cost:θ(n1/2)

Appointment scheduling: epilogueCost:O(n1/2log(n))

Cost:θ(n1/2)

[R ’02] [AA ’03]

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]

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/2kn 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/2kn<1.99n , so k> 0.007n

[Frankl and Rödl, 1987]

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 xy=0, so result is definitely noty

Question: How much communication if error prob. ¼ is ok?

Answer: just 2 bits are sufficient!

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]

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]

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

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

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

zIP(x,y)

IP(x,y) = x1y1 +x2y2 + +xnyn mod 2

Proof:

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)

f (x,y)=1ifx = y

0 ifxy

Equality revisitedin simultaneous message modelx1x2 xn

y1y2 yn

f (x,y)

Exactprotocols: require 2n bits communication

1

1

1

1

1

1

0

1

1

0

1

random k

Equality revisitedin simultaneous message modelx1x2 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

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]

Quantum fingerprints

Question 1: how many orthogonal states in k qubits?

Answer: 2k

Question 2: how many almost orthogonal* states in k qubits? (* where |xy|≤ )

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 examiningx andy

0

H

H

x

S

W

A

P

y

Intuition: 0xy +1yx

Quantum fingerprintsLet 000,001, …,111 be 2n states on log n + O(1) qubits such that |xy|≤ for all x≠y

Given xy, 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

test

Quantum protocol for equality in simultaneous message modelx1x2 xn

y1y2 yn

x

y

x

y

M=

matching on {1, 2, …, n}

(partition into pairs)

x {0,1}n

Inputs:

(i, j,xixj), such that (i, j) M

Output:

Hidden matching problemOnly 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]

Hidden matching problem

M=

matching on {1,2, …, n}

x {0,1}n

Inputs:

Output: (i, j,xixj), (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 xixj bits …

Quantum protocol that uses only log nqubits:

Alice sends (log n qubits) to Bob

Hidden matching problemM=

matching on {1,2, …, n}

x {0,1}n

Inputs:

Output: (i, j,xixj), (i, j) M

Bob measures in the basis {i j|(i, j) M}, and then uses the outcome’s relative phase to deduce xixj

(1 bit)

(1 bit)

(1 bit)

Communication complexity with distributed outputsx

y

inputs:

a

b

outputs:

where a, b, x, y satisfy some relation

E.g. “Bell’s Theorem”

Goal: ab=xy with zero communication

With classical resources, Pr[ab=xy] ≤ 0.75

With 00 + 11 prior entanglement, Pr[ab=xy] = 0.853…

[B ’64] [CHSH ’69]

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]

Qubit communication:

Cost:log n

Cost:θ(n)

Bit communication & prior entanglement:

Qubit communication & prior entanglement:

Cost: zero

Cost: zero

Distributed-output restricted equalityM=

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. (ab)·(ij)= xixj

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]

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 …)

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.

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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