1. Unbounded-Error Classical and Quantum Communication Complexity
Kazuo Iwama (Kyoto U.)
Harumichi Nishimura (Osaka Pref. U.)
Rudy Raymond (IBM Research, TRL)
Shigeru Yamashita (NAIST)
December 21, 2007, QIP2008, New Delhi

2. Two Messages in This Talk
(Result) Relationship between unbounded error classical and quantum communication complexity
(Technique) Connection from/to arrangement of points and hyperplanes to/from unbounded error one-way communication complexity (probably an easier way to analyze communication protocols)

3. Communication Complexity (CC)
Input x
communication
(bits or qubits)
Input y
Output f(x,y)
CC of f :=the minimum amount of communication to compute f
・ Introduced by Yao (1979)
・ Application to lower bound of complexity theory
Quantum CC, initiated by Yao (1993)

4. Restrictions or Variants
What to compute?
(Total/Partial) Boolean Function or Relations
What kind of interactions?
One-way, Two-way, Simultaneous Message Passing (SMP)
What kind of additional resources?
Public coins or Entanglement
What kind of probability required to evaluate?
Exact,
(Two-sided) Bounded-error (>2/3 for yes-instance and <1/3 for no-instance),
One-sided Bounded-error (>1/2 and 0)
Unbounded-error (>1/2 and <1/2),
Nondeterministic (>0 and 0),
etc.

5. Unbounded-Error CC
The strongest model but still hard for many functions
Introduced by Paturi and Simon (1986)
One-way CC is at most two-way CC plus one [Paturi-Simon'86].
Equality: 2-bit, Index function (or Random Access Coding): log(n) bits, but Inner Product: > n/2 [Forster’01]. In fact, almost all functions require linear bounds [Alon-Frank-Rodl ’85]
Our Results: characterization of CC by the smallest dim. of arrangement kf
One-way CC (Alice → Bob)
Two-way CC (Alice and Bob both know f(x,y))
SMP CC
Unbounded quantum/classical one-way/two-way/SMP CCs
are equivalent up to constant multiplicative factor 4.
In one-way case, quantum CC is
exactly the half of classical CC.

6. Arrangement (1) Arrangements Points and Hyperplanes
realizes f(x, y) if
px is above hy ⇒ f(x,y) = 0
px is below hy ⇒　f(x,y) = 1
kf := the minimum of dimensions k of arrangements that realize f

7. Arrangement (2) hyperplanes h1 h1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 h1
000
001
010
011
101
100
111
110 h2 h3
3 bit RAC
000
001
010
011
101
100
111
110 Y X p1
000
001
010
011
100
101
110
111 p2 p3 p4
points p5 p6 p7 p8
The above arrangement realizes f(x,y) and no arrangments on R2 exist, so kf=3.

8. One-way CC: Finding good probability distributions
One-way CC from Alice to Bob (for RAC)
m bits
Alice has
n bits x
Bob has i in [n]
Encoding states:
Probability vectors of dim. 2m
Decoding:
Vectors of dim.
2m to output 1 on

9. Arrangement and One-way CC with classical bits (1)
Instead of adjusting probability distributions, think in arrangements
Arrangement realizing f(x,y) in Rk
Probability distribution of Alice's message r(x)=(r1,r2,..,rk) ⇔ point p(x) in Rk
Probability distribution of Bob's guess on receiving messages q(y) = (q1,q2,..,qk) ⇔ hyperplane h(y) in Rk
For any point p(x) and hyperplane h(y)
or point p(x) above the plane h(y)
or point p(x) below the plane h(y)

10. Arrangement and One-way CC with classical bits (2)
Unbounded-error CC:
The smallest dimension of arrangement realizing f : kf
Previous Result by Paturi&Simon’86:
Our result:

11. Quantum One-way CC: Finding good quantum states and measurements
One-way CC from Alice to Bob (for RAC)
m qubits
Alice has
n bits x
Bob has i in [n]
Encoding states
Binary POVMs

12. Arrangement and One-way CC with quantum bits (1)
Instead of considering quantum states and POVMs, think in arrangements
Arrangement realizing f(x,y) in Rk
Quantum state by Alice ⇔point p(x) in Rk
POVM by Bob on receiving Alice's message ⇔　hyperplane h(y) in Rk
For any point p(x) and hyperplane h(y)
or point p(x) above the plane h(y)
or point p(x) below the plane h(y)

13. Quantum States ⇔ Points
n-qubit state:
Let N=2n and k=N2-1. For any r in Rk, there exists an n-qubit state of such that
Bloch vector
For k = 3, quantum states lie in the Bloch ball. The topology is more complicated
for higher k. See [Kimura&Kossakowski’04]
Small ball of radius 1/(N-1)
Large ball of radius 1
Bloch vectors of n-qubit states

14. POVMs ⇔ Hyperplanes
Let N=2n and k=N2-1. For any hyperplane h in Rk, there exist a POVM {E0,I-E0} with positives such that:
Bounding eigenvalues of summation of SU(N) generators by techniques in [Kimura&Kossakowski’04]
Notices:
The dimension of n qubits space is quadratically bigger than that of n bits!

15. Arrangement and One-way CC with quantum bits (2)
Our result: Unbounded-error quantum CC is also characterized by kf
As a result, quantum CC is always half of classical one.
Arrangement
Bloch vector of quantum states
shrink
and shift
Small ball
Large ball

16. Arrangement and Two-way CC (1)
Yao-Kremer : final quantum state for computing f(x,y)
Success probability:
Arrangement in 2 (2n-1) – 2 (n-1) –dimensional real space

17. Arrangement and Two-way Communication Complexity (2)
Results: (the cost of two-way is about that of one-way)
=Q1(f)
[Paturi-Simon 86]

18. Summary Arrangement is a useful tool for unbounded error quantum CC.
Its geometric view has already existed (Bloch vectors)!
Tight bounds in terms of minimal dimension
Quantum CC is almost (exactly in one-way case!) the half of classical CC
What about margin of arrangement?
Show tighter results on weakly unbounded-error CC
Quantum CC is at least 1/3 of classical CC (c.f. [Klauck'01])
Unfortunately, we don't know how to apply the technique to the bounded-error model...
Exp. separation between quantum and classical for partial Boolean functions [Gavinsky-Kempe-Kerenidis-Raz-de Wolf'07], ??? for total ones