aug 2, 20051 quantum communication complexity richard cleve institute for quantum computing...

Aug 2, 2005 1

Quantum Communication Quantum Communication ComplexityComplexity

Richard Cleve

Institute for Quantum Computing

University of Waterloo

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1. Preliminaries

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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

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How much classical information in How much classical information in nn qubits?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 bits from n qubits

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Holevo’s TheoremHolevo’s Theorem

ψn qubits

Ub1

b2

b3

bn

Easy case:

b1b2 ... bn cannot

convey more than n bits!

Hard case (the general case):

ψn qubits

b1

b2

b3

bn

U00

000

m qubits

bn+1

bn+2

bn+3

bn+4

bn+m

(proof omitted here)

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Entanglement & signalingEntanglement & signaling1100

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21 Example of an entangled state:

No … any operation performed on one qubit has no affect on the state of the other qubit

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”?

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Alice Bob

Basic communication scenarioBasic communication scenario

Resources

x1x2 xn

Goal: convey n bits from Alice to Bob

x1x2 xn

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Basic communication scenarioBasic communication scenarioBit 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]

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2. Communicationcomplexity

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Classical communication complexityClassical communication complexity

f (x,y)

x1x2 xn y1y2 yn

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]

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Classical communication complexityClassical communication complexity

x = y?

x1x2 xn y1y2 yn

Probabilistic protocol for Equality ( = 1/n):

px(T) = x0 + x1T + x2T 2 + … + xn1T

n1

py(T) = y0 + y1T + y2T 2 + … + yn1T

n1

Alice: pick random t {0, 1,…, m1}

send (t, px(t ) mod m) to Bob (this is only 4 log (n) bits)

Bob: accept iff px(t) = py(t) mod m (err prob < n/n2 = 1/n)

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

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Quantum communication complexityQuantum communication complexity

Qubit communication

Prior entanglement

f (x,y)

x1x2 xn y1y2 yn

qubits

f (x,y)

x1x2 xn y1y2 yn

entangled qubits

bits

[Y ’93] [CB ’97]

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Appointment schedulingAppointment scheduling

i (xi = yi = 1)

Classically, (n) bits necessary to succeed with prob. 3/4

For all > 0, O(n1/2 log n) qubits sufficient for error prob. <

0 1 1 0 1 … 01 2 3 4 5 . . . n

1 0 0 1 1 … 11 2 3 4 5 . . . n

x = y =

[KS ’87] [BCW ’98]

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Search problemSearch problem

0 0 0 0 1 0 … 11 2 3 4 5 6 . . . n

x =Given: accessible via queries

i

b xi

i

b

Goal: find i{1, 2, …, n} such that xi = 1

Classically: (n) queries are necessary

Quantum mechanically: O(n1/2) queries are sufficient

i

b xi

i

b

log n

1

x

[G ’96]

Ux

Alternate notation

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0 1 1 0 1 0 … 01 2 3 4 5 6 . . . n

x =

1 0 0 1 1 0 … 1y =

0 0 0 0 1 0 … 0xy =

Alice

Bob

i

00b

xy

i

00b

Bob

y

Bob

y

Alice

x x

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

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Appointment scheduling: epilogueAppointment scheduling: epilogueBit communication:

Cost: θ(n)


Cost: O(n1/2 log(n))


Cost: θ(n1/2)


Cost: θ(n1/2)

[R ’02] [AA ’03]

Cost: θ(n1/2)

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Restricted version of equalityRestricted version of equalityPrecondition (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)

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Classical lower bound (*skipped)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 < 1.99n

[Frankl and Rödl, 1987]

Let some protocol solve restricted equality with k bits comm.

● approximately 2n/n input pairs (x, x), where Δ(x) = n/2

Define S = {x : Δ(x) = n/2 and (x, x) yields conv. C }

Therefore, 2n/2kn input pairs (x, x) that yield same conv. C

● 2k 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 2n/2kn < 1.99n , so k > 0.007n

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Quantum protocolQuantum protocolj

n

j

jxx

1

)1(ψFor 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 yIf (x,y) = n/2 then xy = 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:

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Exponential quantum vs. classical Exponential quantum vs. classical separation in separation in bounded-error modelsbounded-error models

O(log n) quantum vs. (n1/4 / 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]

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3. Quantum speed-up is not always possible

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Inner productInner product

IP(x, y) = x1 y1 + x2 y2 + + xn yn 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]

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Recall Deutsch’s problemRecall Deutsch’s problem

Let f : {0,1} {0,1} be of the form f(x) = a1 x + a0 mod 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

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Bernstein-Vazirani problemBernstein-Vazirani problem(multidimensional Deutsch problem)(multidimensional Deutsch problem)

Let f(x1, x2, …, xn) = a1 x1 + a2 x2 + + an xn + a0 mod 2

Given:

f

b

x1

xn

x2

x2

b f(x1, x2, …, xn)xn

x1

H

H

H

H

H

H

H

H

H

H

1

0

0

0

1

a1

an

a2

Goal: determine a1, a2 , …, an

Classically, n +1 queries are necessary

Quantum mechanically, 1 query is sufficient

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Lower bound for inner productLower bound for inner productIP(x, y) = x1 y1 + x2 y2 + + xn yn mod 2

y1 yny2

Alice and Bob’s IP protocol

x2x1 xn

zIP(x, y)

Alice and Bob’s IP protocol inverted

y1 y2 ynx1 x2 xn

zProof:

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Lower bound for inner productLower bound for inner productIP(x, y) = x1 y1 + x2 y2 + + xn yn 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

x2x1 xn

Alice and Bob’s IP protocol inverted

x1 x2 xnx1 x2 xn

H H H

HHH

0 100

1

H

H

[BV, 1993]

11

11

2

1H

Proof:

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4. Simultaneous messages to a third party

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Equality revisitedEquality revisited in simultaneous message modelin simultaneous message model

x1x2 xn y1y2 yn

f (x,y)

Exact protocols: require 2n bits communication

Equality function:

f (x,y) = 1 if x = y 0 if x y

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x1x2 xn y1y2 yn

f (x,y)

Bounded-error protocols with 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

random k

1 10 1 1 10 11 0 1 1

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x1x2 xn y1y2 yn

f (x,y)Bounded-error protocols without a shared key:

Classical: θ(n1/2)Quantum: θ(log n)[A ’96] [NS ’96] [BCWW ’01]

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Quantum fingerprintsQuantum fingerprintsQuestion 1: how many orthogonal states in k qubits?

Answer: 2k

Answer: 2c2k, for some constant c > 0

Question 3: does this enable k qubits to store c2k bits?

(In other words, log n + O(1) qubits to store n bits?)

Question 2: how many almost orthogonal* states in k qubits?

(* where |xy| ≤ )

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

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Quantum fingerprintsQuantum fingerprints

if x = y, Pr[output = 0] = 1

if x ≠ y, Pr[output = 0] = (1+ 2)/2

Given xy, one can check if x = y or x ≠ y as follows:

Let 000, 001, …, 111 be 2n states on log n + O(1) qubits

such that |xy| ≤ for all x ≠ y

HSWAP

H

x

y

0

Intuition: 0xy +

1yx

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Quantum protocol for equality Quantum protocol for equality in simultaneous message in simultaneous message

modelmodelx1x2 xn y1y2 yn

x y

Orthogonality test

x y

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5. One-way communication

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Hidden matching problemHidden matching problem

x {0,1}nmatching on {1, 2, …, n}(partition into pairs)

Inputs: M =

[BJK ’04]

(i, j, xixj), such that

(i, j) MOutput:

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

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x {0,1}nmatching on {1,2, …, n}Inputs:

Output: (i, j, xixj), (i, j) 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 xixj bits …

Classically, one-way communication is (n) for bounded-error even with a shared classical key (the proof is omitted here)

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x {0,1}nmatching on {1,2, …, n}Inputs: M =

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

Quantum protocol that uses only log n qubits:

Alice sends (log n qubits) to Bob

n

k

kkx

n 1

11

)(

Bob measures in the basis {i j | (i, j ) M }, and

then uses the outcome’s relative phase to deduce xixj

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6. Nonlocality revisited

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Communication complexity Communication complexity with distributed outputswith distributed outputs

b

x y

a

inputs:

outputs:

(1 bit)

(1 bit)

(1 bit)

(1 bit)

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]

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Distributed outputs:Distributed outputs:“spooky Deutsch-“spooky Deutsch- Jozsa”Jozsa”

b

x y

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]

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Distributed-output restricted equalityDistributed-output restricted equality

Bit communication:

Cost: θ(n)


Cost: log n


Cost: zero Cost: zero


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Distributed-output hidden matchingDistributed-output hidden matching

x {0,1}nmatching on {1, 2, …, n}(partition into pairs)

Inputs: 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) M2. (ab)·(ij) = xixj

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Some open problemsSome 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 …)

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Selected references ISelected 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.

http://homepages.cwi.nl/~rdewolf/

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Selected references IISelected 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.

aug 2, 20051 quantum communication complexity richard cleve institute for quantum computing...

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