For the case of bounded-error quantum protocols, very few lower bounds are currently known (exceptions are inner product [54] and the general “discrep-ancy” bound [108]). In particular, no good lower bounds are known for the disjointness problem. The best known upper bound for this is O(√
n log n) qubits (Section 6.4.2), contrasting with linear classical randomized complexity. Since dis-jointness is a “co-NP-complete” communication problem [14], a good lower bound for this problem would imply lower bounds for all “NP-hard” communication problems. In order to attack this problem, we make an effort to extend the above polynomial-based approach to bounded-error protocols. We consider the approxi-mate rank, denoted ]rank(f )), and show the bound Qcc2(f )≥ (logrank(f ))/2 for] 2-sided bounded-error qubit protocols (again using techniques from [165, 108]).
Unfortunately, lower bounds on ]rank(f ) are much harder to obtain than for rank(f ). If we could prove for the case f (x, y) = g(x∧ y) that rank(f ) roughly] equals the number of monomials gmon(g) of an approximating polynomial for g, then an Ω(√
n) lower bound would follow for disjointness, because we show that disjointness requires at least 2√nmonomials to approximate. Since we prove that the quantities rank(f ) and mon(g) are in fact equal in the exact case, this gives some hope for a similar result ]rank(f )≈ gmon(g) in the approximating case, and hence for resolving the complexity of disjointness. Nevertheless, the bounds that we actually are able to prove for disjointness are disappointingly weak. We end the chapter with a discussion of some of the main open problems for quantum communication complexity.
7.2 Lower Bounds for Exact Protocols
Consider a total function f : {0, 1}n × {0, 1}n → {0, 1}. The communication matrix Mf corresponding to this f is the 2n× 2n Boolean matrix whose (x, y)-entry is f (x, y). We use rank(f ) to denote the rank of Mf (over the reals).
One of the most powerful techniques for lower bounds on classical deterministic communication complexity is the well known log-rank lower bound: Dcc(f ) ≥ log rank(f ). This was first proven by Mehlhorn and Schmidt [121].
As noted in [42, 12], techniques from [165, 108] imply a similar lower bound for quantum protocols with prior entanglement: QccE(f )∈ Ω(log rank(f)). Here we will first prove the log rank(f ) bound for clean quantum protocols and afterwards extend it to general entanglement-enhanced protocols. A clean qubit protocol is a protocol, of the second kind considered in Section 6.3.2, that leaves a clean workspace behind at the end of the protocol: it starts in the state |~0i|0i|~0i (no prior entanglement) and ends with|~0i|f(x, y)i|~0i We use Qccc(f ) for the minimal cost of such clean protocols for f . For simplicity, our proof assumes that the channel is a 1-qubit space. The same proof works if the channel can hold more
qubits. We use the following lemma:
7.2.1. Lemma (Yao [165]; Kremer [108]). The final state of an ℓ-qubit pro-tocol (without prior entanglement) on input (x, y) can be written as
X
i∈{0,1}ℓ
αi(x)βi(y)|Ai(x)i|iℓi|Bi(y)i,
where the αi(x), βi(y) are complex numbers of magnitude ≤ 1, the Ai(x), Bi(y) are unit vectors, and iℓ denotes the last bit of the ℓ-bit string i.
Proof. The proof is by induction on ℓ:
Base step. For ℓ = 0 the lemma is obvious.
Induction step. Suppose after ℓ qubits of communication the state can be
written as X
i∈{0,1}ℓ
αi(x)βi(y)|Ai(x)i|iℓi|Bi(y)i. (7.1) We assume without loss of generality that it is Alice’s turn: she applies Uℓ+1A (x) to her part and the 1-qubit channel. Note that there exist complex numbers αi0(x), αi1(x) and unit vectors Ai0(x), Ai1(x) such that
(Uℓ+1A (x)⊗ I)|Ai(x)i|iℓi|Bi(y)i =
αi0(x)|Ai0(x)i|0i|Bi(y)i + αi1(x)|Ai1(x)i|1i|Bi(y)i.
Thus every element of the superposition (7.1) “splits in two” when we apply Uℓ+1A . Accordingly, we can write the state after Uℓ+1 in the form required by the lemma,
which concludes the proof. 2
7.2.2. Theorem (Buhrman & de Wolf [48]). Qccc(f )≥ log rank(f) + 1.
Proof. Consider a clean ℓ-qubit protocol for f . By Lemma 7.2.1, we can write its final state as X
i∈{0,1}ℓ
αi(x)βi(y)|Ai(x)i|iℓi|Bi(y)i.
The protocol is clean, so the final state is |~0i|f(x, y)i|~0i. Hence all parts of
|Ai(x)i and |Bi(y)i other than |~0i will cancel out, and we can assume without loss of generality that|Ai(x)i = |Bi(y)i = |~0i for all i. Now the amplitude of the
|~0i|1i|~0i-state is simply the sum of the amplitudes αi(x)βi(y) of the i for which iℓ = 1. This sum is either 0 or 1, and equals the acceptance probability P (x, y) of the protocol. Letting α(x) (resp. β(y)) be the dimension-2ℓ−1 vector whose entries are αi(x) (resp. βi(y)) for the i with iℓ = 1, we obtain:
P (x, y) = X
i:iℓ=1
αi(x)βi(y) = α(x)T · β(y).
7.2. Lower Bounds for Exact Protocols 119 Since the protocol is exact, we must have P (x, y) = f (x, y). Hence if we define A as the |X| × d matrix having the α(x) as rows and B as the d × |Y | matrix having the β(y) as columns, then Mf = AB. But now
rank(Mf) = rank(AB)≤ rank(A) ≤ d ≤ 2l−1,
and the theorem follows. 2
We now extend this to the case where Alice and Bob share an unlimited (but finite) number of EPR-pairs at the start of the protocol:
7.2.3. Theorem (Buhrman & de Wolf [48]). Qcc∗E(f ) ≥ log(rank(f )− 1)
2 .
Proof. Let Mf± be the matrix whose (x, y)-entry is (−1)f (x,y); this is the matrix we get if we replace{0, 1} by {+1, −1} in the ordinary communication matrix Mf. Letting J denote the all-1 matrix (which has rank 1), we have Mf± = J− 2Mf, so the ranks of Mf and Mf± differ by at most 1. For m > 0, let f⊕m : Xm× Ym → {0, 1} denote the Boolean function that is the XOR of m independent copies of f , i.e., f⊕m(x1, . . . , xm, y1, . . . , ym) = f (x1, y1)⊕ · · · ⊕ f(xm, ym). Note that Mf±⊕m = (Mf±)⊗m, because the XOR of m Boolean variables in ±-notation is just their product. This implies that rank(Mf±⊕m) = rank(Mf±)m and hence also rank(f⊕m)≥ (rank(f) − 1)m− 1.
Now suppose we have some exact protocol for f that uses ℓ qubits of com-munication and k prior EPR-pairs. We will build a clean qubit protocol without prior entanglement for f⊕m, and then invoke Theorem 7.2.2 to get a lower bound on ℓ. The idea is to establish the prior entanglement once, then to reuse it to cleanly compute f m times, and finally to “uncompute” the entanglement.
First Alice makes k EPR-pairs and sends one half of each pair to Bob (at a cost of k qubits of communication). Now they run the protocol to compute the first instance of f (ℓ qubits of communication). Alice and Bob each copy the answer to a safe place, which we will call their respective ‘answer bits’, and they reverse the protocol (again ℓ qubits of communication). This gives them back the k EPR-pairs and an otherwise clean workspace, which they can reuse. Now they compute the second instance of f , they each XOR the answer into their answer bit (which can be done cleanly), and they reverse the protocol, etc. After all m instances of f have been computed, Alice and Bob both have the answer f⊕m(x, y) left and the k EPR-pairs, which they uncompute using another k qubits of communication (Bob sends his halves of the k EPR-pairs to Alice, who sets them back to |00i).
This gives a clean protocol for f⊕m that uses 2mℓ + 2k qubits and no prior entanglement. By Theorem 7.2.2 we obtain:
2mℓ + 2k ≥ Qccc(f⊕m) ≥ log rank(f⊕m) + 1
≥ log((rank(f) − 1)m− 1) + 1 ≥ m log(rank(f) − 1),
hence
ℓ≥ log(rank(f )− 1)
2 − k
m.
Since this must hold for every m > 0, the theorem follows. 2 We can derive a stronger bound for the Ccc∗E(f )-complexity, which combines classical communication with unlimited prior entanglement:
7.2.4. Theorem (Buhrman & de Wolf [48]). Ccc∗E(f ) ≥ log(rank(f) − 1).
Proof. Let f ∧ f : X2 × Y2 → {0, 1} denote the Boolean function that is the AND of two independent copies of f . Note that Mf ∧f = Mf ⊗ Mf and hence rank(f ∧ f) = rank(f)2. Since a qubit and an EPR-pair can be used to send 2 classical bits via superdense coding (Section 6.2), we can devise a qubit protocol for f ∧ f using Ccc∗E(f ) qubits (compute the two copies of f in parallel using the classical bit protocol). Hence by the previous theorem we obtain Ccc∗E(f )≥ Qcc∗E(f ∧ f) ≥ (log(rank(f ∧ f) − 1))/2 ≥ log(rank(f) − 1). 2 Below we draw some consequences from these log-rank lower bounds. Firstly, the communication matrix MEQn of the equality-problem is the 2n × 2n iden-tity matrix, so rank(EQn) = 2n. This implies Qcc∗E(EQn) ≥ n/2, which is tight up to 1 bit because of superdense coding, and Ccc∗E(EQn) ≥ n (in con-trast, Qcc2(EQn) ∈ Θ(log n) and Ccc∗2(EQn) ∈ O(1)). The disjointness func-tion on n bits is the AND of n disjointnesses on 1 bit (which have rank 2 each), so rank(DISJn) = 2n. The complement of the inner product function has rank(IPn) = 2n. Thus we have the following strong lower bounds, all tight up to 1 bit:3
7.2.5. Corollary (Buhrman & de Wolf [48]).
• Qcc∗E(EQn), Qcc∗E(DISJn), Qcc∗E(IPn)≥ n/2
• Ccc∗E(EQn), Ccc∗E(DISJn), Ccc∗E(IPn)≥ n
Koml´os [107] has shown that the fraction of m×m Boolean matrices that have determinant 0 goes to 0 as m → ∞. Hence almost all 2n× 2n Boolean matrices have full rank 2n, which implies that almost all functions have maximal quantum communication complexity:
7.2.6. Corollary (Buhrman & de Wolf [48]). For almost all total f we have Qcc∗E(f )≥ n/2 and Ccc∗E(f )≥ n.
3The same bounds for IPn are also given in [54]. The bounds for EQn and DISJn are new, and can also be shown to hold for zero-error quantum protocols.
7.3. A Lower Bound Technique via Polynomials 121