Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

1 Introduction

Yao [Yao79] introduced the classical model of communication complexity, and also subsequently introduced its quantum analogue [Yao93]. Communication complexity has important applications in several disciplines, such as lower bounds on circuits, data structures, streaming algorithms, and many other areas (see, for example, [KN97, RY20] and the references therein). The basic model of communication complexity involves two parties, usually called Alice and Bob, who wish to jointly compute $F (x, y)$ for a known function $F : {0, 1}^{n} \times {0, 1}^{n} \to {0, 1}$ , where Alice holds $x \in {0, 1}^{n}$ and Bob holds $y \in {0, 1}^{n}$ . The parties use a communication protocol agreed upon in advance to compute $F (x, y)$ . They are individually computationally unbounded and the cost is the amount of communication between the parties on the worst-case input.

Consider the $n$ -bit Equality function, denoted ${E Q}_{n} : {0, 1}^{n} \times {0, 1}^{n} \to {0, 1}$ (or simply $E Q$ when $n$ is clear from context), and defined as ${E Q}_{n} (x, y) = 1$ iff $x = y$ . This is arguably the simplest and most basic problem in communication complexity. It is well known that its deterministic communication complexity equals $n$ , which is maximal. However, Yao [Yao79] already showed that if we allow some small constant error probability, then the communication complexity becomes much smaller. In this paper we pin down the small-error communication complexity of Equality in various communication models. Our bounds are essentially optimal both in terms of $n$ and in terms of the error. While our optimal upper bounds only give small improvements over known bounds, Equality is such a fundamental communication problem that we feel it is worthwhile to pin down its complexity as precisely as possible and to find protocols that are as efficient as possible.

1.1 Prior work

Given a function $F : {0, 1}^{n} \times {0, 1}^{n} \to {0, 1}$ , define the $2^{n} \times 2^{n}$ communication matrix of $F$ , denoted $M_{F}$ , by $M_{F} (x, y) = F (x, y)$ . Define the $ε$ -approximate rank of a matrix $M$ , denoted ${r k}_{ε} (M)$ , to be the minimum number of rank-1 matrices needed such that their sum is $ε$ -close to $M$ entrywise (equivalently, ${r k}_{ε} (M)$ is the minimum rank among all matrices that are $ε$ -close to $M$ entrywise). If the rank-1 matrices are additionally constrained to be entrywise nonnegative, then the resulting measure is called the $ε$ -approximate nonnegative-rank of $M$ , denoted ${r k}_{ε}^{+} (M)$ . By definition, ${r k}_{ε}^{+} (M_{F}) \geq {r k}_{ε} (M_{F})$ . Denote $ε$ -error randomized communication complexity by $R_{ε}^{p r i} (\cdot)$ when the players have access to private randomness, and $R_{ε}^{p u b} (F)$ when the players have access to public randomness (i.e., shared coin flips). Let $Q_{ε}^{p r i} (\cdot)$ denote $ε$ -error quantum communication complexity, assuming private randomness. In all quantum communication models under consideration in this paper, Alice and Bob do not have access to pre-shared entanglement.

Krause [Kra96] showed the following lower bound on the randomized communication complexity of a Boolean function in terms of the approximate nonnegative-rank of its communication matrix.

Theorem 1.1 ([Kra96]).

Let $F : {0, 1}^{n} \times {0, 1}^{n} \to {0, 1}$ be a Boolean function and $ε > 0$ . Then,

R_{ε}^{p r i} (F) \geq log {r k}_{ε}^{+} (M_{F}) .

Analogous to this, the following lower bound is known on the quantum communication complexity of a Boolean function, due to Nielsen [Nie98] and Buhrman and de Wolf [BW01].

Theorem 1.2 ([Nie98, Bw01]).

Let $F : {0, 1}^{n} \times {0, 1}^{n} \to {0, 1}$ be a Boolean function and let $ε > 0$ . Then,

Q_{ε}^{p r i} (F) \geq \frac{1}{2} log {r k}_{ε} (M_{F}) .

A similar proof as that of [BW01] can be used to show that the quantum communication complexity of a Boolean function is bounded below by the logarithm of its approximate psd-rank, which we define below. Let $M$ be a matrix with nonnegative real entries. A rank- $d$ psd-factorization of $M$ consists of a set of $d \times d$ complex¹¹1Often this definition is restricted to real matrices. This can change the psd-rank by a constant factor, but no more than that [LWW17, Section 3.3]. psd matrices $A_{i}$ (one for each row of $M$ ) and $B_{j}$ (one for each column of $M$ ), such that for all $i, j$ we have $Mij=\rm tr(AiBj)$ . The psd-rank of $M$ , denoted ${r k}^{p s d} (M)$ , is the minimal $d$ for which $M$ has such a psd factorization. This notion has gained a lot of interest in areas such as semidefinite optimization, communication complexity, and others. See Fawzi et al. [FGP $^{+}$ 15] for an excellent survey. The $ε$ -approximate psd-rank of $M$ , which we denote by ${r k}_{ε}^{p s d} (M)$ , is the minimum psd-rank among all matrices that are $ε$ -close to $M$ entrywise.

Theorem 1.3.

Let $F : {0, 1}^{n} \times {0, 1}^{n} \to {0, 1}$ be a Boolean function and let $ε > 0$ . Then,

Q_{ε}^{p r i} (F) \geq log {r k}_{ε}^{p s d} (M_{F}) + 1.

For completeness, we prove this in Appendix A. It is easy to show that ${r k}_{ε}^{p s d} (M_{F}) \leq {r k}_{ε}^{+} (M_{F})$ . Alon [Alo09] showed the following bounds on the approximate rank of the Identity matrix.

Theorem 1.4 ([Alo09]).

There exists a positive constant $c$ such that the following holds for all integers $n > 0$ and $1 / 2^{n / 2} \leq ε \leq 1 / 4$ . Let $I$ denote the $2^{n} \times 2^{n}$ Identity matrix. Then,

{r k}_{ε} (I) \geq \frac{c n}{ε^{2} log (\frac{1}{ε})} .

Note that the $2^{n} \times 2^{n}$ Identity matrix is the communication matrix of the $n$ -bit Equality function. Theorems 1.1 and 1.4 thus imply that for $1 / 2^{n / 2} \leq ε \leq 1 / 4$ ,

R_{ε}^{p r i} ({E Q}_{n}) \geq log (\frac{n}{} ε^{2}) - log log (\frac{1}{ε}) - O (1) .

(1)

Newman [New91] proved the following theorem that shows that public-coin protocols can be converted to private-coin protocols with an additive error, with a small additive cost. For the following form, see for example, [KN97, Claim 3.14].

Theorem 1.5 (cf. [Kn97, Claim 3.14]).

Let $F : {0, 1}^{n} \times {0, 1}^{n}$ be a Boolean function. For every $δ > 0$ and every $ε > 0$ ,

R_{ε + δ}^{p r i} (F) \leq R_{ε}^{p u b} (F) + log (\frac{n}{δ^{2}}) + O (1) .

Relabeling variables, Theorem 1.5 is equivalent to

R_{ε (1 + δ)}^{p r i} (F) \leq R_{ε}^{p u b} (F) + log (\frac{n}{ε^{2} δ^{2}}) + O (1) .

1.2 Our results

In this section we list our results, first those for randomized communication complexity, and then those for quantum communication complexity.

1.2.1 Randomized communication complexity

We give an improved version of Newman’s theorem (Theorem 1.5), which allows us to convert a public-coin protocol to a private-coin one with an optimal dependence on the error. Our proof follows along similar lines as that of Newman’s. Our key deviation is that we use a multiplicative form of the Chernoff bound, where previously an additive version was used.

Theorem 1.6.

Let $F : {0, 1}^{n} \times {0, 1}^{n} \to {0, 1}$ be a Boolean function. For all $ε \in [0, 1 / 2)$ and all $δ \in (0, 1]$ ,

R_{ε (1 + δ)}^{p r i} (F) \leq R_{ε}^{p u b} (F) + log (\frac{n}{ε}) + log (\frac{6}{δ^{2}}) .

To compare Theorem 1.5 and Theorem 1.6, consider the $(1 / n)$ -error private-coin randomized communication complexity of ${E Q}_{n}$ . The $ε$ -error public-coin communication complexity of ${E Q}_{n}$ is at most $log (1 / ε)$ (and this can be shown to be tight up to an additive constant). Thus, Theorem 1.5 can at best give an upper bound of

R_{1 / n}^{p r i} ({E Q}_{n}) \leq log n + log (n^{3}) + O (1) = 4 log n + O (1) .

In contrast, Equation (1) implies $R_{1 / n}^{p r i} ({E Q}_{n}) \geq 3 log n - log log n - O (1)$ . On the other hand, Theorem 1.6 implies a tight upper bound (up to the additive $log log n + O (1)$ term) of $3 log n + O (1)$ on the $(1 / n)$ -error private-coin communication complexity of ${E Q}_{n}$ , by converting the $log (1 / ε)$ -cost public-coin protocol for ${E Q}_{n}$ to a private-coin protocol.

Theorem 1.7.

For all positive integers $n > 0$ and for all $ε \in [0, 1 / 2)$ ,

R_{ε}^{p r i} (E Q)

\leq log (\frac{n}{ε^{2}}) + 4.

This shows Alon’s theorem (Theorem 1.4) is tight up to the $O (log (1 / ε))$ factor, not only for approximate rank, but also for communication complexity. Theorem 1.7 and Theorem 1.1 also imply that the approximate-rank lower bound in Theorem 1.4 is nearly tight even restricting to nonnegative approximations to the Identity matrix.

Corollary 1.8.

Let $n > 0$ be an integer, and let $I$ denote the $2^{n} \times 2^{n}$ Identity matrix. Then for all $ε \in [0, 1 / 2)$ ,

{r k}_{ε}^{+} (I) \leq \frac{16 n}{ε^{2}} .

To compare the performance of Theorem 1.5 with that of Theorem 1.6 in a more general setting, we consider the natural problem of converting a public-coin protocol to a private-coin protocol while allowing the error to double. Setting $δ = ε$ in Theorem 1.5 and relabeling parameters, we obtain

R_{ε}^{p r i} (F) \leq R_{ε / 2}^{p u b} (F) + log (\frac{n}{ε^{2}}) + O (1) .

However, Theorem 1.6 yields the following improved dependence on $ε$ by setting $δ = 1$ and relabeling parameters.

R_{ε}^{p r i} (F) \leq R_{ε / 2}^{p u b} (F) + log (\frac{n}{ε}) + 4.

1.2.2 Quantum communication complexity

Prior to this work, the best known lower bound on the $ε$ -error quantum communication complexity of Equality was $Ω (log (n / ε))$ [BW01, Proposition 3], with a constant hidden in the $Ω (\cdot)$ that is less than $1 / 2$ . Theorem 1.2 and Theorem 1.4 imply that

Q_{ε}^{p r i} ({E Q}_{n}) \geq log (\frac{\sqrt{n}}{ε}) - log log (\frac{1}{ε}) - O (1) .

(2)

In terms of upper bounds, we exhibit a one-way quantum communication upper bound with an optimal dependence on $ε$ , that uses only pure-state messages (and hence does not use even private randomness). In particular, by choosing $ε$ to be an arbitrary small polynomial in the input size, this implies that the factor of $1 / 2$ in Theorem 1.2 cannot be improved when $F = {E Q}_{n}$ . Let $Q_{ε}^{p u r e, \to} (F)$ be the $ε$ -error quantum communication complexity of $F$ , where the protocols are one-way and Alice is only allowed to send a pure state to Bob. We show the following.

Theorem 1.9.

For all positive integers $n > 0$ and for all $ε \in [0, 1 / 2)$ ,

Q_{ε}^{p u r e, \to} ({E Q}_{n}) \leq log (\frac{n}{ε}) + 4.

The proof uses the probabilistic method to analyze random linear codes. Nayak [Nay99] already used the same upper bound technique to show an upper bound on the bounded-error one-way quantum communication complexity of ${E Q}_{n}$ . They did not explicitly derive this error-dependence, but it follows immediately from their construction by plugging in codes with length $O (n / ε)$ and relative distance $1 / 2 - \sqrt{ε}$ in [Nay99, pp.16–17]. We also show that this is nearly tight:

Theorem 1.10.

There exists an absolute constant $c$ such that the following holds. For all positive integers $n > 0$ and for all $ε \in [1 / 2^{n}, 1 / 4]$ ,

Q_{ε}^{p u r e, \to} ({E Q}_{n}) \geq log (\frac{n}{ε}) - log log (\frac{1}{ε}) - c .

While the pure-state protocol of Theorem 1.9 has optimal dependence on $ε$ (up to the $log log (1 / ε)$ term), it does not match the $n$ -dependence of the lower bound of Equation (2); in fact, one-way pure-state protocols cannot match this (Theorem 1.10). However, if we allow one-way mixed-state messages, then we can give a better upper bound and close the gap:

Theorem 1.11.

For all positive integers $n > 0$ and for all $ε \in [0, 1 / 2)$ ,

Q_{ε}^{p r i} ({E Q}_{n}) \leq log (\frac{\sqrt{n}}{ε}) + 3.

The proof is again probabilistic, using known concentration properties of overlaps of random projectors to allow us to show the existence of appropriate mixed-state messages for Alice and appropriate measurements for Bob. Theorems 1.3 and 1.11 also imply upper bounds on the $ε$ -approximate psd-rank of the Identity matrix.

Corollary 1.12.

Let $n > 0$ be an integer, and let $I$ denote the $2^{n} \times 2^{n}$ Identity matrix. Then for all $ε \in [0, 1 / 2)$ ,

{r k}_{ε}^{p s d} (I) \leq \frac{4 \sqrt{n}}{ε} .

As noted by Lee, Wei and de Wolf [LWW17, Theorem 17], Alon’s approximate rank lower bound (Theorem 1.4) almost immediately gives a lower bound of ${r k}_{ε}^{p s d} (I) = Ω (\frac{\sqrt{n}}{ε \sqrt{log (1 / ε)}})$ . This shows that our upper bound in Corollary 1.12 is tight up to a multiplicative $O (\sqrt{log (1 / ε)})$ factor.

2 Preliminaries

All logarithms in this paper are taken to base 2. We use $exp (x)$ to denote $e^{x}$ , where $e$ denotes Euler’s number. For strings $x, y \in {0, 1}^{n}$ , define their Hamming distance by $d (x, y) := | {i \in [n] : x_{i} \neq y_{i}} |$ . For an event $X$ , let $I (X) \in {0, 1}$ denote the indicator of $X$ , which is 1 iff $X$ occurs.

Definition 2.1 (Linear code).

For integers $N \geq n$ , a linear code is a linear function $C : {0, 1}^{n} \to {0, 1}^{N}$ .

One may view a linear code as an $N \times n$ matrix $M$ over $F_{2}$ ; an input $x \in {0, 1}^{n}$ is mapped to $N$ -bit codeword $M x$ (where the matrix product is taken over $F_{2}$ ). Choosing a random linear code corresponds to choosing an $M$ with uniformly random binary entries.

We use the following well-known multiplicative form of the Chernoff bound [MU05, Theorem 4.4].

Lemma 2.2.

Let $Z_{1}, \dots, Z_{n}$

be independent random variables taking values in

{0, 1}

. Let

Z = \sum_{i = 1}^{n} Z_{i}

, and let

μ = E [Z]

. Then for all

δ \in [0, 1]

	$Pr [Z \geq (1 + δ) μ]$	$\leq exp (- δ^{2} μ / 3),$
	$Pr [Z \leq (1 - δ) μ]$	$\leq exp (- δ^{2} μ / 2) .$

We refer the reader to [KN97, RY20] for the basics of classical communication complexity, and to [Wol02] for an introduction to quantum communication complexity.

3 An improved form of Newman’s theorem

Proof of Theorem 1.6.

Let $Π$ be a public-coin protocol that computes $F$ with error $ε$ . Assume without loss of generality that all the random coins are tossed at the beginning of the protocol. That is, for every $x, y \in {0, 1}^{n}$ ,

Pr r [Π (x, y, r) \neq F (x, y)] \leq ε .

(3)

Set

B = \frac{6 n}{δ^{2} ε}

(4)

and independently choose random strings $r_{1}, \dots, r_{B}$ according to the same distribution as used by $Π$ . For two strings $x, y \in {0, 1}^{n}$ and an index $j \in [B]$ , let $I_{j, x, y}$ denote the indicator event of $r_{j}$ being a “bad random string” for $x, y$ :

I_{j, x, y} := {\begin{matrix} 1 & Π (x, y, r_{j}) \neq f (x, y) 0 & otherwise. \end{matrix}

(5)

Fix two arbitrary strings $x, y \in {0, 1}^{n}$ . Equation (3) implies ${Pr}_{r_{1}, \dots, r_{B}, j \in [B]} [I_{j, x, y} = 1] \leq ε$ . By linearity of expectation and our choice of $B$ in Equation (4),

E_{r_{1}, \dots, r_{B}} ⎡ ⎣ \sum j \in [B] I_{j, x, y} ⎤ ⎦ \leq B ε = \frac{6 n}{δ^{2}} .

We now give an upper bound on ${Pr}_{r_{1}, \dots, r_{B}} [\sum_{j \in [B]} I_{j, x, y} \geq B ε (1 + δ)]$ . Assume without loss of generality that ${Pr}_{r_{1}, \dots, r_{B}, j \in [B]} [I_{j, x, y} = 1] = ε$ , and hence $E_{r_{1}, \dots, r_{B}} [\sum_{j \in [B]} I_{j, x, y}] = B ε$ (since the desired probability could only be smaller otherwise). By a Chernoff bound (Lemma 2.2),

Pr r_{1}, \dots, r_{B} ⎡ ⎣ \sum j \in [B] I_{j, x, y} \geq B ε (1 + δ) ⎤ ⎦ \leq exp (- \frac{δ^{2} \cdot 6 n}{3 δ^{2}}) = exp (- 2 n) < 2^{- 2 n} .

By a union bound over all $x, y \in {0, 1}^{n}$ ,

	$Pr r_{1}, \dots, r_{B} ⎡ ⎣ \sum j \in [B] I_{j, x, y} \geq B ε (1 + δ) for some x, y \in {0, 1}^{n} ⎤ ⎦$	$\leq \sum x, y \in {0, 1}^{n} Pr r_{1}, \dots, r_{B} ⎡ ⎣ \sum j \in [B] I_{j, x, y} \geq B ε (1 + δ) ⎤ ⎦$
		$< 2^{2 n} \cdot 2^{- 2 n} = 1.$

Hence there exists a choice of $r_{1}, \dots, r_{B}$ such that the following holds for all $x, y \in {0, 1}^{n}$ :

\sum j \in [B] I_{j, x, y} < B ε (1 + δ) .

(6)

Fixing this choice of $r_{1}, \dots, r_{B}$ , Protocol 1 gives a private-coin protocol for $F$ .

Alice samples $j \in [B]$ uniformly at random, and sends it to Bob.
Alice and Bob perform the public-coin protocol $Π$ assuming $r_{j}$ was the public random string.

Protocol 1 A private-coin protocol for

F

To show the correctness of this protocol, our choice of $B$ (Equation (4)) and Equations (5) and (6) imply that for all $x, y \in {0, 1}^{n}$ ,

Pr j \in [B] [Π (x, y, r_{j}) \neq f (x, y)]

< \frac{B ε (1 + δ)}{B} = ε (1 + δ) .

Hence the protocol has error probability less than $ε (1 + δ)$ . The cost of the first step of the protocol is $log B$ , and the cost of the second step is at most $R_{ε}^{p u b} (F)$ . Thus, we have,

R_{ε (1 + δ)}^{p r i} (F) \leq R_{ε}^{p u b} (F) + log B = R_{ε}^{p u b} (F) + log \frac{6 n}{δ^{2} ε} = R_{ε}^{p u b} (F) + log \frac{n}{ε} + log \frac{6}{δ^{2}} .

Note that if $Π$ was a one-way protocol, then Protocol 1 is a one-way private-coin protocol. ∎

4 Communication complexity upper bounds

In this section we show randomized and quantum communication upper bounds for Equality.

4.1 Randomized upper bound

As an application of Theorem 1.6, we recover an optimal small-error private-coin communication complexity upper bound for ${E Q}_{n}$ from a naive public-coin protocol of cost $log (2 / ε)$ and error $ε / 2$ :

R_{ε}^{p r i} ({E Q}_{n}) \leq log (\frac{2}{} ε) + log (\frac{n}{ε}) + 3 = log (\frac{n}{ε^{2}}) + 4.

(7)

This proves Theorem 1.7. Newman’s theorem (Theorem 1.5) would only give an upper bound of

In particular, for $ε = 1 / n$ we improve the upper bound from $4 log n + O (1)$ to $3 log n + O (1)$ , which turns out to be essentially optimal.

4.2 Quantum upper bound with only pure states

We require the following property of random linear codes.

Claim 4.1.

Let $n$ be a positive integer and let $δ > 0$ . Let $x \neq y \in {0, 1}^{n}$ be two arbitrary but fixed strings. Let $N = 4 n / δ^{2}$ . Let $C : {0, 1}^{n} \to {0, 1}^{N}$ be a random linear code. Then

Pr C [\frac{d (C (x), C (y))}{N} \notin [\frac{1}{2} - δ, \frac{1}{2} + δ]] < 2^{- 2 n} .

Proof of Claim 4.1.

For each $i \in [N]$ , the random variable $Z_{i} := I [C (x)_{i} = C (y)_{i}]$ equals 1 with probability $1 / 2$ and $0$ with probability $1 / 2$ . Further, $Z_{i}$ and $Z_{j}$ are independent for all $i \neq j \in [N]$ . Define $Z = \sum_{i = 1}^{N} Z_{i} = d (C (x), C (y))$ . We have $E [Z] = N / 2$ . By a Chernoff bound (Lemma 2.2),

Pr C [∣ ∣ ∣ \frac{d (C (x), C (y))}{N} - \frac{1}{2} ∣ ∣ ∣ \geq δ] = Pr C [∣ ∣ ∣ Z - \frac{N}{2} ∣ ∣ ∣ \geq 2 δ \cdot \frac{N}{2}] \leq 2 exp (- 4 δ^{2} N / 6) < 2^{- 2 n},

where the last inequality holds by our choice of $N$ . ∎

By a union bound over all $x, y \in {0, 1}^{n}$ , Claim 4.1 implies the following corollary.

Corollary 4.2.

Let $n$ be a positive integer, let $δ > 0$ and let $N = 4 n / δ^{2}$ . Then there exists a linear code $C : {0, 1}^{n} \to {0, 1}^{N}$ such that for all $x \neq y \in {0, 1}^{n}$ ,

\frac{d (C (x), C (y))}{N} \in [\frac{1}{2} - δ, \frac{1}{2} + δ] .

We now prove Theorem 1.9.

Proof of Theorem 1.9.

Set $δ = \sqrt{ε} / 2$ . Let $N = 4 n / δ^{2} = 16 n / ε$ and let $C : {0, 1}^{n} \to {0, 1}^{16 n / ε}$ be the code obtained from Corollary 4.2. The following is a protocol for ${E Q}_{n}$ .

Alice, on input $x \in {0, 1}^{n}$ prepares state $| ϕ_{x} ⟩ := \frac{1}{\sqrt{N}} \sum_{i \in [N]} (- 1)^{C (x)_{i}} | i ⟩$ , and sends Bob $| ϕ_{x} ⟩$ .
Define $| ϕ_{y} ⟩ := \frac{1}{\sqrt{N}} \sum_{i \in [N]} (- 1)^{C (y)_{i}} | i ⟩$ . Bob measures with respect to the projectors $| ϕ_{y} ⟩ ⟨ ϕ_{y} |$ and $I - | ϕ_{y} ⟩ ⟨ ϕ_{y} |$ , and outputs 1 on observing the first measurement outcome, and 0 otherwise.

This protocol succeeds with probability 1 when $x = y$ . The only error arises when $x \neq y$ and Bob observes the first measurement outcome. Thus, the error probability of this protocol equals

	$max x \neq y \in {0, 1}^{n} \| ⟨ ϕ_{x} \| ϕ_{y} ⟩ \|^{2}$	$= max x \neq y \in {0, 1}^{n} {⎛ ⎝ \frac{1}{N} \sum i \in [N] (- 1)^{C (x)_{i} + C (y)_{i}} ⎞ ⎠}^{2}$
		$= max x \neq y \in {0, 1}^{n} {(1 - \frac{2 d (C (x), C (y))}{N})}^{2}$
		$\leq 4 δ^{2} = ε,$

where the last inequality follows from Corollary 4.2 and the last equality follows from our choice of $δ$ . The number of qubits sent from Alice to Bob is $log N = log (16 n / ε) = log (n / ε) + 4$ . ∎

We show in Section 5 that the protocol in the previous proof is nearly optimal if one restricts to one-way communication with only pure states.

4.3 Quantum upper bound with mixed states

In the last section we gave a $log (n / ε) + O (1)$ quantum upper bound on the $ε$ -error communication complexity of ${E Q}_{n}$ , where Alice was only allowed to send a pure state to Bob. In this section we show that allowing Alice to send a mixed state to Bob gives a communication upper bound with a better (in fact optimal) $n$ -dependence. Our protocol is based on concentration properties of overlaps of random projectors.

Consider two rank- $r$ projectors $P$ and $Q$ acting on $C^{d}$ . The largest possible inner product $\rm tr(PQ)$ between them is $r$ , which occurs iff $P = Q$ . However, when one or both of the projectors are Haar-random, then we expect their inner product to be much smaller, namely only $r^{2} / d$ . This is because if we take the spectral decompositions $P = \sum_{i = 1}^{r} | u_{i} ⟩ ⟨ u_{i} |$ and $Q = \sum_{j = 1}^{r} | v_{j} ⟩ ⟨ v_{j} |$ , then

\rm tr(PQ)=r∑i,j=1|⟨ui,vj⟩|2,

and the expected squared inner product between a random $d$

-dimensional unit vector

u_{i}

and any fixed unit vector

v_{j}

, is

1 / d

. Hayden, Leung and Winter [HLW06, Lemma III.5] showed that this inner product is very tightly concentrated around its expectation.

Claim 4.3 ([Hlw06, Lemma III.5]).

Let $P$ and $Q$ be rank- $r$ projectors on $C^{d}$ , where $P$ is random²²2More precisely, $P$ is a projection onto a uniformly chosen $r$ -dimensional subspace from all $r$ -dimensional subspaces of $C^{d}$ . We do not elaborate more on this here since it is not relevant for us. and $Q$ is fixed. Let $δ \in [0, 1]$ . Then

Pr[\rm tr(PQ)≥(1+δ)r2d]≤exp(−r2δ25)<2−r2δ2/5.

The following corollary then follows by setting parameters suitably.

Corollary 4.4.

For every integer $n > 0$ and all $ε \in [0, 1 / 2)$ , there exists a set ${P_{x} : x \in {0, 1}^{n}}$ of $2^{n}$ rank- $r$ projectors on $C^{d}$ , with $r = \sqrt{10 n}$ and $d = 2 r / ε$ , such that $\rm tr(PxPy)<εr$ for all $x \neq y \in {0, 1}^{n}$ .

Proof.

Fix $δ = 1$ and choose rank- $r$ projectors ${P_{x} : x \in {0, 1}^{n}}$ independently and uniformly at random. Claim 4.3 and our choice of parameters implies that for all $x \neq y \in {0, 1}^{n}$ ,

Pr[\rm tr(PxPy)≥2r2d]=Pr[% \rm tr(PxPy)≥εr]<2−r2δ2/5=2−2n.

The corollary now follows by applying a union bound over all distinct $x, y \in {0, 1}^{n}$ . ∎

We now prove Theorem 1.11.

Proof.

Let ${P_{x} : x \in {0, 1}^{n}}$ be projectors on $C^{d}$ as guaranteed by Corollary 4.4, each of rank $r = \sqrt{10 n}$ , with $d = 2 \sqrt{10 n} / ε$ . The following is our protocol for ${E Q}_{n}$ .

Alice, on input $x \in {0, 1}^{n}$ , sends the $log d$ -qubit mixed state $ρ_{x} := P_{x} / r$ to Bob.
Bob, on input $y \in {0, 1}^{n}$ , measures w.r.t. projectors $P_{y}, I - P_{y}$ , and outputs 1 on observing the first measurement outcome, and 0 otherwise.

Protocol 2 A mixed-state protocol

Π

for

F

To see the correctness of this protocol, first observe that the protocol outputs the correct answer with probability 1 if $x = y$ , because $\rm tr(Pxρx)=\rm tr(Px)/r=1$ . If $x \neq y$ , then the error probability is the probability of Bob observing the first measurement outcome, which is

Pr[Π(x,y)≠EQn(x,y)]=\rm tr(Pyρx)=\rm tr(PyPx)/r<ε,

from Corollary 4.4. The cost is $log d = log (2 \sqrt{10 n} / ε) \leq log (\sqrt{n} / ε) + 3$ qubits of communication. ∎

5 Quantum one-way lower bound

In this section we prove lower bounds on the one-way quantum communication complexity of any function whose communication matrix has a large number of distinct rows. As a consequence we obtain our lower bound for ${E Q}_{n}$ of Theorem 1.10.

Let $F : {0, 1}^{n} \times {0, 1}^{n} \to {0, 1}$ be a Boolean function. We consider the model where communication is one-way, and Alice is only allowed to send a pure state to Bob. Suppose there exists a protocol of cost $log d$ that computes $F$ to error $ε$ . Any such protocol looks like the following.

Alice, on input $x \in {0, 1}^{n}$ , sends a message $| ϕ_{x} ⟩$ to Bob, where $| ϕ_{x} ⟩$ is a unit vector in $C^{d}$ .
Bob, on input $y$ , measures with respect to projectors $P_{y}, I - P_{y}$ .

The acceptance probability of the protocol is $∥ P_{y} | ϕ_{x} ⟩ ∥^{2}$ . Thus, we have

∥ P_{y} | ϕ_{x} ⟩ ∥^{2} \geq 1 - ε, ∥ (I - P_{y}) | ϕ_{x} ⟩ ∥^{2} \leq ε for all x, y \in F^{- 1} (1),

(8)

and

∥ P_{y} | ϕ_{x} ⟩ ∥^{2} \leq ε, ∥ (I - P_{y}) | ϕ_{x} ⟩ ∥^{2} \geq 1 - ε for all x, y \in F^{- 1} (0) .

(9)

Claim 5.1.

Let $F : {0, 1}^{n} \times {0, 1}^{n} \to {0, 1}$ be a Boolean function with $N$ distinct rows in $M_{F}$ . Let $X \subseteq {0, 1}^{n}$ be an arbitrary subset of size $N$ that indexes distinct rows in $M_{F}$ . For a one-way quantum communication protocol as above that computes $F$ to error $ε \leq 1 / 2$ , we have

2 - 2 \sqrt{ε (1 - ε)} \leq ∥ | ϕ_{x_{1}} ⟩ - | ϕ_{x_{2}} ⟩ ∥^{2} \leq 2 + 4 \sqrt{ε}

for all distinct $x_{1}, x_{2} \in X$ .

Proof.

Fix any two distinct $x_{1}, x_{2} \in X$ , and let $| ϕ_{x_{1}} ⟩, | ϕ_{x_{2}} ⟩ \in C^{d}$ be the messages sent by Alice on inputs $x_{1}, x_{2}$ , respectively. Recall that $∥ | ϕ_{x_{1}} ⟩ ∥ = ∥ | ϕ_{x_{2}} ⟩ ∥ = 1$ . Because of the assumption that the rows of $M_{F}$ indexed by $X$ are all distinct, there is a $y \in {0, 1}^{n}$ such that $F (x_{1}, y) \neq F (x_{2}, y)$ . Without loss of generality assume $F (x_{1}, y) = 1$ and $F (x_{2}, y) = 0$ . Write

	$\| ϕ_{x_{1}} ⟩$	$= P_{y} \| ϕ_{x_{1}} ⟩ + (I - P_{y}) \| ϕ_{x_{1}} ⟩,$
	$\| ϕ_{x_{2}} ⟩$	$= P_{y} \| ϕ_{x_{2}} ⟩ + (I - P_{y}) \| ϕ_{x_{2}} ⟩ .$

Thus,

$∥ \| ϕ_{x_{1}} ⟩ - \| ϕ_{x_{2}} ⟩ ∥^{2}$	$= ∥ P_{y} (\| ϕ_{x_{1}} ⟩ - \| ϕ_{x_{2}} ⟩) ∥^{2} + ∥ (I - P_{y}) (\| ϕ_{x_{1}} ⟩ - \| ϕ_{x_{2}} ⟩) ∥^{2}$	since $P_{y}$ and $I - P_{y}$ are orthogonal projectors
	$\geq (∥ P_{y} \| ϕ_{x_{1}} ⟩ ∥ - ∥ P_{y} \| ϕ_{x_{2}} ⟩) ∥)^{2} + (∥ (I - P_{y}) \| ϕ_{x_{1}} ⟩ ∥ - ∥ (I - P_{y}) \| ϕ_{x_{2}} ⟩ ∥)^{2}$	by the triangle inequality
	$\geq 2 (\sqrt{1 - ε} - \sqrt{ε})^{2}$	by Equations (8) and (9), and since $F (x_{1}, y) = 1$ and $F (x_{2}, y) = 0$
	$= 2 - 2 \sqrt{ε (1 - ε)} .$

For the upper bound, first define $p := ∥ P_{y} | ϕ_{x_{1}} ⟩ ∥^{2} \geq 1 - ε$ , and $q := ∥ (I - P_{y}) | ϕ_{x_{2}} ⟩ ∥^{2} \geq 1 - ε$ .

$∥ \| ϕ_{x_{1}} ⟩ - \| ϕ_{x_{2}} ⟩ ∥^{2}$	$= ∥ P_{y} (\| ϕ_{x_{1}} ⟩ - \| ϕ_{x_{2}} ⟩) ∥^{2} + ∥ (I - P_{y}) (\| ϕ_{x_{1}} ⟩ - \| ϕ_{x_{2}} ⟩) ∥^{2}$
	$\leq (∥ P_{y} \| ϕ_{x_{1}} ⟩ ∥ + ∥ P_{y} \| ϕ_{x_{2}} ⟩ ∥)^{2} + (∥ (I - P_{y}) \| ϕ_{x_{1}} ⟩ ∥ + ∥ (I - P_{y}) \| ϕ_{x_{2}} ⟩ ∥)^{2}$	by the triangle inequality
	$= (\sqrt{p} + \sqrt{1 - q})^{2} + (\sqrt{1 - p} + \sqrt{q})^{2}$
	$= 2 + 2 \sqrt{p (1 - q)} + 2 \sqrt{(1 - p) q} \leq 2 + 4 \sqrt{ε} .$

∎

We now state our main result of this section.

Theorem 5.2.

There exists an absolute constant $c$ such that the following holds. Let $F : {0, 1}^{n} \times {0, 1}^{n}$ be a Boolean function with $N$ distinct rows in $M_{F}$ . Then for all $ε \in [1 / N, 1 / 4]$ ,

Q_{ε}^{p u r e, \to} (F) \geq log (\frac{log N}{ε}) - log log (\frac{1}{ε}) - c .

Proof.

Let $X \subseteq {0, 1}^{n}$ be an arbitrary set of $N$ elements that index distinct rows in $M_{F}$ . Consider a protocol of cost $log d$ , as described in the beginning of this section, that computes $F$ to error $ε$ . Claim 5.1 implies existence of vectors $| ϕ_{x} ⟩ \in C^{d}$ for all $x \in X$ , such that

2 - 2 \sqrt{ε (1 - ε)} \leq ∥ | ϕ_{x_{1}} ⟩ - | ϕ_{x_{2}} ⟩ ∥^{2} \leq 2 + 4 \sqrt{ε}

(10)

for all distinct $x_{1}, x_{2} \in X$ . For each $x \in X$ , define a real vector $| ϕ_{x}^{R} ⟩ \in R^{2 d}$ by

| ϕ_{x}^{R} ⟩ = \sum j \in [d] | j ⟩ (R (| ϕ_{x} ⟩_{j}) | 0 ⟩ + C (| ϕ_{x} ⟩_{j}) | 1 ⟩),

where $R (| ϕ_{x} ⟩_{j})$ and $C (| ϕ_{x} ⟩_{j})$ denote the real and complex components of the $j$ ’th coordinate of $| ϕ_{x} ⟩$ , respectively. Note that each $| ϕ_{x}^{R} ⟩$ is a unit vector, since the $| ϕ_{x} ⟩$ are unit vectors. For all distinct $x_{1}, x_{2} \in X$ , we have

	$\| ϕ_{x_{1}} ⟩ - \| ϕ_{x_{2}} ⟩$	$= \sum j \in [d] \| j ⟩ (R (\| ϕ_{x_{1}} ⟩_{j} - \| ϕ_{x_{2}} ⟩_{j}) + i \cdot C (\| ϕ_{x_{1}} ⟩_{j} - \| ϕ_{x_{2}} ⟩_{j})),$
	$\| ϕ_{x_{1}}^{R} ⟩ - \| ϕ_{x_{2}}^{R} ⟩$	$= \sum j \in [d] \| j ⟩ ((R (\| ϕ_{x_{1}} ⟩_{j} - \| ϕ_{x_{2}} ⟩_{j}) \| 0 ⟩) + (C (\| ϕ_{x_{1}} ⟩_{j} - \| ϕ_{x_{2}} ⟩_{j}) \| 1 ⟩)) .$

Hence, Equation (10) implies

∥ | ϕ_{x_{1}}^{R} ⟩ - | ϕ_{x_{2}}^{R} ⟩ ∥^{2} = ∥ | ϕ_{x_{1}} ⟩ - | ϕ_{x_{2}} ⟩ ∥^{2} \in [2 - 2 \sqrt{ε (1 - ε)}, 2 + 4 \sqrt{ε}]

(11)

for all distinct $x_{1}, x_{2} \in X$ . Since ${∥ v - w ∥}^{2} = {∥ v ∥}^{2} + {∥ w ∥}^{2} - 2 ⟨ v, w ⟩$ for real vectors $v, w$ , we obtain

| ⟨ ϕ_{x_{1}}^{R} | ϕ_{x_{2}}^{R} ⟩ | \leq 2 \sqrt{ε}

for all distinct $x_{1}, x_{2} \in X$ . Now consider the $N \times N$ matrix $M$ whose rows and columns are indexed by strings in $X$ , defined by

M_{x, y} = ⟨ ϕ_{x}^{R} | ϕ

	$\| ϕ_{x_{1}} ⟩$	$= P_{y} \| ϕ_{x_{1}} ⟩ + (I - P_{y}) \| ϕ_{x_{1}} ⟩,$
	$\| ϕ_{x_{2}} ⟩$	$= P_{y} \| ϕ_{x_{2}} ⟩ + (I - P_{y}) \| ϕ_{x_{2}} ⟩ .$

	$\| ϕ_{x_{1}} ⟩ - \| ϕ_{x_{2}} ⟩$	$= \sum j \in [d] \| j ⟩ (R (\| ϕ_{x_{1}} ⟩_{j} - \| ϕ_{x_{2}} ⟩_{j}) + i \cdot C (\| ϕ_{x_{1}} ⟩_{j} - \| ϕ_{x_{2}} ⟩_{j})),$
	$\| ϕ_{x_{1}}^{R} ⟩ - \| ϕ_{x_{2}}^{R} ⟩$	$= \sum j \in [d] \| j ⟩ ((R (\| ϕ_{x_{1}} ⟩_{j} - \| ϕ_{x_{2}} ⟩_{j}) \| 0 ⟩) + (C (\| ϕ_{x_{1}} ⟩_{j} - \| ϕ_{x_{2}} ⟩_{j}) \| 1 ⟩)) .$

Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

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This week in AI

1 Introduction

1.1 Prior work

Theorem 1.1 ([Kra96]).

Theorem 1.2 ([Nie98, Bw01]).

Theorem 1.3.

Theorem 1.4 ([Alo09]).

Theorem 1.5 (cf. [Kn97, Claim 3.14]).

1.2 Our results

1.2.1 Randomized communication complexity

Theorem 1.6.

Theorem 1.7.

Corollary 1.8.

1.2.2 Quantum communication complexity

Theorem 1.9.

Theorem 1.10.

Theorem 1.11.

Corollary 1.12.

2 Preliminaries

Definition 2.1 (Linear code).

Lemma 2.2.

3 An improved form of Newman’s theorem

Proof of Theorem 1.6.

4 Communication complexity upper bounds

4.1 Randomized upper bound

4.2 Quantum upper bound with only pure states

Claim 4.1.

Proof of Claim 4.1.

Corollary 4.2.

Proof of Theorem 1.9.

4.3 Quantum upper bound with mixed states

Claim 4.3 ([Hlw06, Lemma III.5]).

Corollary 4.4.

Proof.

Proof.

5 Quantum one-way lower bound

Claim 5.1.

Proof.

Theorem 5.2.

Proof.