Quantum Communication Complexity

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Quantum Communication Complexity

• 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 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
• Holevos Theorem 1973 implies cannot extract
  more than n bits from n qubits

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Holevos Theorem
Easy case
Hard case (the general case)
b1b2 ... bn cannot convey more than n bits!
(proof omitted here)
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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
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Basic communication scenario
Goal convey n bits from Alice to Bob
x1x2 ? xn
Alice
Bob
x1x2 ? xn
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Basic communication scenario
H 73 BW 92
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2. Communication complexity
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Classical communication complexity
x1x2 ? xn
y1y2 ? yn
f (x,y)
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 complexity
x1x2 ? xn
y1y2 ? yn
x y?
Probabilistic protocol for Equality (? 1/n)
px(T) x0 x1T x2T 2 xn?1T n?1 py(T)
y0 y1T y2T 2 yn?1T n?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 4
log (n) bits)
Bob accept iff px(t) py(t) mod m (err prob
lt n/n2 1/n)
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Quantum communication complexity
Qubit communication
Prior entanglement
Y 93 CB 97
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Appointment scheduling
x
y
i (xi yi 1)
Classically, ?(n) bits necessary to succeed with
prob. ? 3/4
For all ? gt 0, O(n1/2 log n) qubits sufficient
for error prob. lt ?
KS 87 BCW 98
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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
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1 2 3 4 5 6 . . . n
0 1 1 0 1 0 0
x
Alice
1 0 0 1 1 0 1
y
Bob
0 0 0 0 1 0 0
x?y
?
Communication per x?y-query 2(log n 3) O(log
n)
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Appointment scheduling epilogue
Cost O(n1/2 log(n))
Cost ?(n1/2)
R 02 AA 03
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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
(Its a distributed variant of the Deutsch-Jozsa
problem a constant vs. balanced
distinguishing problem)
BCW 98
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Classical lower bound (skipped)
Theorem If S ? 0,1n has the property that,
for all x, x' ? S, their intersection size is
not n/4 then ?S? lt 1.99n
Let some protocol solve restricted equality with
k bits comm.
? 2k conversations of length k
? approximately 2n/?n input pairs (x, x),
where ?(x) n/2
Therefore, 2n/2k?n input pairs (x, x) that
yield same conv. C
Define S x ?(x) n/2 and (x, x) yields
conv. C
For any x, x' ? S, input pair (x, x' ) also
yields conversation C
Therefore, ?(x, x') ? n/2, implying
intersection size is not n/4
Theorem implies 2n/2k?n lt 1.99n , so k gt 0.007n
Frankl and Rödl, 1987
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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 Bobs 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!
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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
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3. Quantum speed-up is not always possible
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Inner 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 Deutschs 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 problem(multidimensional
Deutsch problem)
Let f(x1, x2, , xn) a1 x1 a2 x2 ? an xn
a0 mod 2
Given
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 product
IP(x, y) x1 y1 x2 y2 ? xn yn mod 2
Proof
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Lower bound for inner product
IP(x, y) x1 y1 x2 y2 ? xn yn mod 2
?0?
?1?
?0?
?0?
?x2?
?x1?
?xn?
Proof
Alice and Bobs IP protocol
Alice and Bobs 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 Holevos
Theorem)
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4. Simultaneous messages to a third party
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Equality revisited in
simultaneous message model
x1x2 ? xn
y1y2 ? yn
f (x,y)
Exact protocols require 2n bits communication
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Equality revisited in
simultaneous message model
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
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Equality revisited in
simultaneous message model
x1x2 ? xn
y1y2 ? yn
f (x,y)
Bounded-error protocols without a shared key
Classical ?(n1/2)
Quantum ?(log n)
A 96 NS 96 BCWW 01
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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 gt 0
Question 3 does this enable k qubits to store
c2k bits? (In other words, log n O(1) qubits to
store n bits?)
Answer no recall Holevos 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 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, Proutput 0 1
if x ? y, Proutput 0 (1 ?2)/2
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Quantum protocol for equality
in simultaneous message model
x1x2 ? xn
y1y2 ? yn
??x?
??y?
??x?
??y?
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5. One-way communication
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Hidden matching problem
Only one-way communication (Alice to Bob) is
permitted
Quantum protocol can be exponentially more
efficient than any classical protocoleven with a
shared key
BJK 04
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Hidden matching problem
matching on 1,2, , n
M
x ? 0,1n
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 Alices message Bob can repeat
his side of the protocol using several
edge-disjoint matchings, which yields information
about several xi?xj bits
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Hidden matching problem
matching on 1,2, , n
M
x ? 0,1n
Inputs
Output (i, j, xi?xj), (i, j ) ? M
Bob measures in the basis ?i? ? ? j? (i, j ) ?
M , and then uses the outcomes relative phase
to deduce xi?xj
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6. Nonlocality revisited
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Communication complexity with distributed outputs
x
y
inputs
b
a
outputs
where a, b, x, y satisfy some relation
E.g. Bells Theorem Goal a?b x?y with
zero communication
With classical resources, Pra?b x?y 0.75
With ?00? ?11? prior entanglement, Pra?b
x?y 0.853
B 64 CHSH 69
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Distributed outputsspooky Deutsch- Jozsa
x
y
inputs
(n bits)
(n bits)
b
a
outputs
(log n bits)
(log n 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?)log n prior entanglement, no
communication is needed at all
BCT 99
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Distributed-output restricted equality
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Distributed-output hidden matching
(b, i, j), such that 1. (i, j) ? M 2.
(a?b)(i?j) xi?xj
Outputs a ? 0,1log n
With prior entanglement, no communication
necessary without prior entanglement, one-way
communication is ?(?n), even to achieve success
probability ¾
B 04
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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 )

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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.

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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, Masters
  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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THE END
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Table of Contents

• Communication
• Communication complexity
• Equality (P vs D, exponential) details of
  probabilistic
• Intersection (Q vs P, polynomial)
• Special equality (Q vs P, exponential, exact,
  promise)
• Razs problem (Q vs P, exponential,
  bounded-error, promise)
• Inner product (Q just as hard as P, bounded
  error)
• Restricted communication simultaneous messages
• Equality (Q vs P, exp, bound-err, total, no
  common randomness)
• Restricted communication one round
• Hidden matching (Q vs P, exponential, bounded
  error, one-way)
• Distributed outputs (aka nonlocality/pseudo-telepa
  thy)
• Special equality revisited
• Hidden matching revisited