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

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)

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)

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.

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.

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

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

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

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)

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:

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

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)

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

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!

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

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

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

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