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Quantum Communication Complexity

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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

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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