Complexity of Eccentricities and All-Pairs Shortest Paths in the Quantum CONGEST Model

Computing the distance parameters of a network, including the diameter, radius, eccentricities and the all-pairs shortest paths (APSP) is a central problem in distributed computing. This paper investigates he dtistance parameters in the quantum CONGEST models and establishes almost linear lower bounds on eccentricities and APSP, which match the classical upper bounds. Our results imply that there is not quantum speedup for these two problems. In contrast with the diameter and radius, exchanging quantum messages is able to save the communication when the networks have low diameters [Le Gall and Magniez, PODC 2018]. We obtain the lower bounds via a reduction from the two-way quantum communication complexity of the set intersection [Razborov, Izvestiya Mathematics 2003].


Introduction
In distributed computing, the graph G = (V, E) represents the topology of the network, each node represents a processor with an unique ID, and messages can be transferred directly through the edges connecting two nodes. In the classical CONGEST model, the computation proceeds with round-based synchrony and each node can send one O(log n)-bit message to each adjacent node per round, where n denotes the number of nodes. More specifically, protocols in the model are executed round by round. In each round, each node can send/receive a message of O(log n) bits to/from each of its neighbours, i.e., each pair of neighbours can exchange a message of O(log n) bits. After that, all nodes implement local computation and enter the next round simultaneously. As for the quantum version defined in [EKNP14], the only difference is that each node can send qubits instead of classical bits. The round complexity of a distributed problem f in the CONGEST model (both classical and quantum version) denotes the smallest number of communication rounds needed to compute f .
The diameter of a graph, denoted by D, is defined as the maximum distance between any two nodes. It is a fundamental parameter of the network since in at least D rounds, every two nodes are possible to know each other. The eccentricity with respect to some node v is defined as the maximum distances from v to any other node. So the diameter is the maximum eccentricities over all nodes. The radius of a graph is defined to be the minimum eccentricities over all nodes. Diameters, radius and eccentricities are three basic distance parameters in distributed network.
We study the distance computation in the quantum CONGEST model and focus on investigating whether introducing quantum communication can speed up distributed computation compared to the classical setting.
Related works. Elkin, Klauck, Nanongkai and Pandurangan [EKNP14] proved that quantum communication does not offer significant advantages over classical communication for many fundamental problems in distributed computing, including minimum spanning tree (MST), minimum cut, s-source distance, shortest path tree (SPT) and shortest s-t paths.
However, based on distributed Grover's search [Gro96], Le Gall and Magniez [GM18] proposed aÕ( √ nD)-round quantum algorithm to exactly compute the diameter in the CONGEST model when the graph has a low diameter, where D is the diameter of the underlying network graph, exhibiting a quantum advantage in the CONGEST model, as the classical round complexity is Ω (n). They also proved a tight lower boundΩ( √ nD) for any distributed quantum algorithms if each node can use only poly(log n) quantum bits. An unconditional quantum lower boundΩ( √ n) for diameters was also proved in [GM18], which was later improved to Ω( 3 √ nD 2 + √ n) by Magniez and Nayak [MN20].
In the classical CONGEST model, there is a series of studies on distance computation. Earlier, Frischknecht, Holzer and Wattenhofer [FHW12] showed that computing the diameter of an unweighted graph in CONGEST model requiresΩ(n) rounds, which is tight up to a logarithmic factors since even the All-Pairs Shortest Path problem(APSP) on an unweighted graph can be resolved in O(n) rounds [PRT12,HW12]. Abboud, Censor-Hillel and Khoury [ACK16] later gave a same lower bound ofΩ(n) on diameters even in sparse networks. As a result, the round complexities of computing diameter, radius eccentricities and APSP in the classical CONGEST model are all iñ Θ(n) regime.
Theorem 1.1. Any quantum protocol computing all-pairs shortest paths in the CONGEST model requires Ω( n log n ) rounds.
Theorem 1.2. Any quantum protocol computing the exact eccentricities of all nodes in the CON-GEST model requires Ω( n log 2 n ) rounds.
Remark 1.4. The lower bounds on eccentricities and APSP we establish still hold even if the networks have constant diameters. Thus there is no quantum advantages even for the networks with constant diameters. However, for such networks, it is known that the quantum round complexities of the diameter and radius are quadratically smaller than the classical round complexities in the CONGEST model [FHW12,GM18,MN20].
Our lower bounds are tight up to a logarithmic factor. Thus quantum computation cannot speedup the eccentricities and APSP computation substantially. Our results and related works are collected in Table 1

Classical and quantum CONGEST model
In the classical CONGEST model, the communication network can be seen as a graph G = (V, E). Usually we assume that there are n nodes and m edges in G, and the nodes are assigned with unique identifiers. Each node represents a processor with unlimited computational power, i.e., the consumption of any local computation in a single processor is ignored. Each edge connecting two nodes represents a communication channel with O(log n) bits of bandwidth.
In the quantum CONGEST model, adjacent nodes are allowed to exchange quantum bits (qubits), i.e., the classical channels are replaced by quantum channels with the same bandwidth O(log n). And naturally each node can locally do some quantum computation. Nodes may own qubits which are entangled with the qubits owned by other nodes. In this paper, we assume that initially distinct nodes do not share entanglement. But they can, for example, locally create a pair of entangled qubits, and send one to other nodes.
For both classical and quantum CONGEST models, the algorithm is implemented round by round in a synchronous manner. In each round, every node sends/receives a message of O(log n) bits to/from each neighbour, then implements local computation. At the end of the algorithm, every node has its own output (maybe empty). The round complexity of an algorithm is defined to be the number of communication rounds in the process in the worst case. And the round complexity of a problem is the least round complexity of any algorithm solving this problem.

Graph notation and problem definition
For a graph G = (V, E), the distance between two nodes u and v is the length of the shortest path between u and v. We use d(u, v) to denote the distance between u and v. The eccentricity with respect to some node u is denoted by The diameter of a graph G, denoted by D, is the maximum eccentricities over all nodes, i.e., the maximum distance between any two nodes; while the radius of a graph G, denoted by R, is the minimum eccentricities over all nodes: Any node with the minimum eccentricity is called a center of the graph, so the radius is the eccentricity of any center node.
Before describing problems of distance computation in the distributed setting, recall the definition of approximation for optimization problems.
Given a maximization problem P and an instance x, OP T (x) denotes the value of the optimal solution of the instance x, and SOL A (x) denotes the solution that the algorithm A obtains for instance x. For any ρ ≥ 1, we say that the algorithm A computes a ρ-(multiplicative) approximation Similarly for a minimization problem P and ρ ≥ 1, we say that an algorithm A computes a ρ-approximation Definition 2.1 (Diameter/radius). Given a network with underlying graph G = (V, E), the diameter computation, which is a maximization problem, requires that all nodes should have the same output, which is the exact value (or an approximation) of the diameter of G. Similarly the radius computation, which is a minimization problem, requires nodes to output the same value, which is the exact value (or an approximation) of the radius of G.
Definition 2.2 (Eccentricities). Given a network with the underlying graph G = (V, E), the eccentricities computation requires each node u to output its eccentricity, or an approximation.
Remarks. If an algorithm A has node u outputsê(u), we say that, for ρ ≥ 1, A computes a ρ-approximation of the eccentricity if Definition 2.3 (APSP). Given a network with the underlying graph G = (V, E), the APSP computation requires each node u to output the distance between u and v for each node v.

Two-party communication complexity
A main tool to obtain lower bounds in the CONGEST model is via reductions to two-way communication complexity and apply the various existing methods proving lower bounds on communication complexity. Communication complexity was first introduced by Yao in [Yao79]. Consider two players, usually called Alice and Bob, and assume that Alice and Bob receives inputs x, y ∈ {0, 1} k , respectively. The players want to compute a Boolean function f : {0, 1} k × {0, 1} k → {0, 1} by exchanging messages. At the end of the protocol, both Alice and Bob output f (x, y). The minimum communication required for the players to compute the function is the communication complexity of f .
In [Yao93], Yao introduced the model of quantum communication complexity, where the players are allowed to communicate with qubits. The communication cost of a quantum protocol is the maximum (over all inputs) number of qubits that the players exchange. The quantum communication complexity of a function f is the minimum communication cost of any quantum protocol that computes f with probability at least 2/3. Definition 2.4 (Disjointness). For any integer k ≥ 1, the disjointness function DISJ k : It is well known that the (randomized) classical communication complexity of the disjointness function is Θ(k) [Raz92], while its quantum communication complexity is Θ( Raz03]. Braverman, Garg, Ko, Mao and Touchette further proved an almost tight lower bound on the bounded-round quantum communication complexity of disjointness.
The classical lower bound of the disjointness has been widely applied to obtain numerous tight lower bounds in the classical CONGEST model. However, its quantum lower bound seems to not always have enough capability to capture the hardness in the quantum CONGEST model. We further introduce the intersection function and its quantum communication complexity.

Warm up
Based on Grover's search for the maximum eccentricities over all nodes, Le Gall and Magniez [GM18] proposed a distributed quantum algorithm of computing the exact diameter withinÕ( √ nD) rounds in the CONGEST model, where D is the diameter of the network. Similarly, apply Grover's search for the minimum eccentricities over all nodes, we can obtain anÕ( √ nD)-round distributed algorithm to compute the exact radius of the network.
The diameter is the maximum eccentricities over all nodes and the radius is the minimum eccentricities over all nodes. These two parameters have many similar properties. Before we introduce the lower bound of radius computation, we should know about how to prove the lower bound of the diameter computation.
Let G d be a line network, which consists of d + 1 nodes, denoted by A 0 , . . . , A d . Here A 0 and A d are the two end points and A 1 , . . . , A d−1 are the intermediate nodes. Each edge of G d is a quantum channel of bandwidth B qubits. The nodes A 0 and A d receive k-bit inputs x, y ∈ {0, 1} k , respectively. The disjointness on a line L k,d is a computational problem over the network G d that requires to compute the DISJ k (x, y). We say an algorithm solves L k,d with probability p, if there exists a node outputs the right answer with probability at least p. The following lemma relates the round complexity of L k,d to the two-party communication complexity of DISJ k (x, y). Le Gall and Magniez [GM18] showed that any quantum distributed algorithm that computing the diameter of the network requiresΩ( nD/s) rounds if each node uses at most s qubits of memory where D is the diameter of the network via a reduction from disjointness on a line L k,d In [ACK16], the authors also exhibited a reduction from disjointness to diameter computation [ACK16]. Thus Lemma 2.5 implies that computing the diameter needΩ(  Based on the approach of proving the lower bound of computing the diameter, we can directly obtain the the following theorem. Proof. First of all, there exists a graph G reducing DISJ n (x, y) to radius computation [ACK16], where the graph G has Θ(n) nodes. The graph G has two parts, one is simulated by Alice (denoted by G a ) and the other is simulated by Bob (denoted by G b ), which has Θ(log n) edges connecting G a and G b . Assume that there exists a r-round quantum distributed algorithm A that computes the radius, we can use the algorithm to construct a protocol that computes the DISJ n (x, y), where x ∈ {0, 1} n and y ∈ {0, 1} n are Alice's input and Bob's input, respectively.
The protocol works as follows. Alice and Bob can jointly simulate the graph G: Alice simulates G a (which depends on x), while Bob simulates G b (which depends on y). To simulate the r-round quantum distributed algorithm A, Alice and Bob need to exchange messages corresponding to the communication occurring along the Θ(log n) edges between G a and G b . Because there are Θ(log n) channels(edges) and the bandwidth of each channel is O(log n), one round of communication in algorithm A can be replaced with two messages of O(log 2 n) qubits. Finally, Alice and Bob can compute the radius of G, thus compute DISJ n (x, y). From lemma 2.5, we conclude that r log 2 n = Ω( n r + r), which implies r =Ω( √ n).
Replacing each edges connecting G a and G b with a path of length d, we obtain an instance for radius computation, which is reduced from L n,d with d = Θ(D).
We consider the following two cases.
• There is no upper bound on the memory of each node. Assume that there exists an r-round quantum distributed algorithm that computes the radius, we can obtain an r-round protocol for L k,d . By Lemma 3.2, we obtain an lower boundΩ( 3 √ nD 2 ) as desired.
1. To construct an instance G = (V, E), the vertex set V are partitioned into two disjoint sets V a and V b ; 2. Edge set E includes several edges that are independent of x and y, such as edges between V a and V b ; 3. A set of edges between nodes in V a (resp. V b ) is created according to x (resp. y), denoted by E a (resp. E b ).

Lower bounds
We briefly outline the reductions from two-party communication complexity to the computation in the CONGEST model. And thus the lower bound on communication complexity implies the lower bound in the CONGEST model. Given a distributed problem P, we briefly describe the reduction from communication complexity to P. For a Boolean function f : {0, 1} k × {0, 1} k → {0, 1}, let x, y ∈ {0, 1} k be the input of f . The reduction is via a construction of instance G = (V, E) of P, which consists of three steps shown in Figure 1.
If f (x, y) can be inferred from the solution of P on G, then the lower bound on the complexity of P can be obtained from the following argument. 1) Assume that there is a r-round distributed algorithm A solving P; 2) the player Alice (resp. Bob) knows x (resp. y) and simulate the execution of A on G for each node in V a (resp. V b ); 3) Alice and Bob obtain the solution of P on G and compute f (x, y). The communication between Alice and Bob is exactly the same with the communication between nodes in V a and V b during running A on G. Suppose there are s edges between nodes in V a and V b . Recall that edges have bandwidth O(log n) in CONGEST model. The communication complexity of f is O(rs · log n), which should be beyond any lower bound of the communication complexity of f .

APSP
We follow the above framework to prove a nearly tight lower bound on the round complexity of APSP in the quantum CONGEST model, i.e., to prove theorem 1.1 via a reduction from the intersection functions in two-party communication setting.
Let n ≥ 5 be an integer. Set s = (n − 1)/2 , k = s(s − 1)/2 and ρ ≤ k. Alice and Bob are given an instance (x, y) ∈ {0, 1} k × {0, 1} k of the intersection function. Alice and Bob who receive binary string x, y of length k respectively, are supposed to compute INT k,ρ (x, y). We establish the reduction by first constructing an instance G = (V, E) of APSP. Let |V | = n. In the first step of Figure 1 If n is odd, we include all (a 0 , a i ) and (a i , b i ) in E. The rest of edges are constructed according to the inputs x, y ∈ {0, 1} k . Notice that k = s(s − 1)/2. There are k pairs of distinct indices in [s]. Each pair (i, j) with 1 ≤ i < j ≤ s is associated with an unique index p ∈ [k], and the existence of the edge (a i , a j ) (resp. (b i , b j )) is determined by x p (resp. y p ). More precisely, suppose these pairs (i, j) are sorted in lexicographic order, and (i p , j p ) denotes the p th pair. For p ∈ [k], if x p = 0, Alice adds an edge between a ip and a jp into E. If y p = 0, Bob adds an edge (b ip , b jp ). Formally, we have An example for n = 8 (and thus k = 3), x = (0, 1, 0) and y = (1, 1, 0) can be found in figure 2. Proof. We use the graphical notation u → v to denote an edge (u, v) traversed in a path. There is always a path of the form (a i → a 0 → a j → b j ), so d(a i , b j ) ≤ 3. If x p = y p = 1, a i and b j share no common neighbour, and thus d(a i , b j ) = 3. Otherwise they share a common neighbour a j if x p = 0, or b i if y p = 0, and thus d(a i , b j ) = 2.
So the number of (i, j) such that 1 ≤ i < j ≤ s and d(a i , b j ) = 3 is the same with the number of p ∈ [k] with x p = y p = 1, or the size of the intersection of x and y. |x ∩ y| can be inferred from the solution of APSP, and so does INT k,ρ (x, y) for any ρ. We are now ready to prove the quantum lower bound of APSP.
Proof of Theorem 1.1. Let A be an r-round quantum protocol which computes APSP. For n ≥ 5, we set parameters s = (n − 1)/2 and k = s(s − 1)/2. Alice and Bob, respectively receiving x and y, compute INT k,ρ (x, y) by exchanging quantum messages. They construct the instance G = (V a V b , E) described above. We will show how Alice simulates A on the nodes of V a . Bob does the same for V b .
At the t th round, for each message sent to a node in V b by some node in V a while executing the t th round of A, Alice sends the same message to Bob along with the ID of the receiver. For each message received from Bob, Alice "virtually" allocate it to the corresponding node. Communication inside V a and local computation on nodes of V a can be done without communicating with Bob.
After simulating A, Alice knows d(a i , b j ) for i, j ∈ [s]. She counts the number of (i, j) such that 1 ≤ i < j ≤ s and d(a i , b j ) = 3 to obtain |x ∩ y| and decide INT k,ρ (x, y) for any ρ. The quantum communication complexity between Alice and Bob is basically the same with the number of qubits passing over edges between V a and V b while simulating A. The set of edges between V a and V b is {(a i , b i ) : i ∈ [s]}, which is of size s. Recall that A is of r rounds and edges have bandwidth O(log n) qubits. For any ρ, the quantum communication complexity of INT k,ρ is O(rs · log n). By setting ρ = k/2 , Lemma 2.7 gives a lower bound Ω(k). Therefore, r = Ω( k s·log n ) = Ω( n log n ) as s = Θ(n) and k = Θ(n 2 ).

Eccentricities
To prove the quantum lower bounds of computing the exact and approximate eccentricities shown in Theorem 1.2 and Theorem 1.3, we again establish a reduction from the intersection function, and construct an underlying network graph following the idea of [ACK16].
We include some additional nodes a 1 , · · · , a n−n in V so that |V | = n. For convenience, let B(p, i) denote the i th bit in binary expression of integer p − 1. We describe the edges independent of x and y. For each p ∈ [k], edges between a p and nodes in i∈[s] {a 0 i , a 1 i } are added according to binary expression of integer p − 1, i.e., a p is connected to a B(p,i) i for each i ∈ [s]. Moreover, a p and a 1 , a 1 and a 2 , a 2 and a 3 are all connected. Edges between nodes of V b are linked in a similar way except that b p is connected to b p for each p ∈ [k]. In addition, a 1 is connected to a i for each i ∈ [n − n ]. As for edges between V a and V b , an edge linking a 3 and b 3 are added.
See also Figure 3 for better understanding. The set of red lines is an example of edges added according to x and y. . . . . . .
Since s = log 2 (k − 1) + 1, for i ∈ [s], one can always find integers x, y ∈ {0, · · · , k − 1} such that the i th bit in binary expression of x (y) is 0 (1). And thus indices q 0 = x + 1 ∈ [k], q 1 = y + 1 ∈ [k] satisfying B(q 0 , i) = 0 and B(q 1 , i) = 1. There are paths (a p → a 1 → a q 0 → a 0 Proof. It is equivalent to prove d(a p , b p ) < 5 if and only if x p = y p = 1 because b p is the only neighbour of b p and d(a p , b p ) = d(a p , b p ) + 1. If x p = y p = 1, there is a path (a p → a 3 → b 3 → b p ) and thus d(a p , b p ) ≤ 3. We now only need to prove the necessity. For q ∈ [k] \ {p}, obviously d(a p , a q ) = d(b p , b q ) = 2. Furthermore, d(a p , b q ) = 3 because there is a path of length 3 between a p and b q , and they share no neighbour. d(a q , b p ) = 3 by symmetry.
We use the graphical notation u ; v to denote any shortest path from u to v. If edge (a 3 , b 3 ) is excluded from E, the shortest path between a p and b p will be of the form (a p ; a q ; b p ) or (a p ; b q ; b p ) for some q ∈ A corollary of Proposition 4.2 and Proposition 4.3 tells that for any p ∈ [k], a p has eccentricity less than 6, i.e., e(a p ) < 6, if x p = y p = 1, and e(a p ) = 6 otherwise. So the number of p ∈ [k] such that e(a p ) < 6 is the same with the number of p ∈ [k] with x p = y p = 1. And for any ρ, INT k,ρ (x, y) can be inferred by computing eccentricities. We are ready to prove the quantum lower bound of computing exact eccentricities.
Proof of Theorem 1.2. Let A be any r-round quantum protocol which computes the exact eccentricity for each node. By the same argument mentioned in the proof of Theorem 1.1, Alice and Bob simulate A on the instance G = (V a V b , E a E b E c ) described above. After that, Alice knows e(a p ) for p ∈ [k]. By counting the number of p ∈ [k] satisfying e(a p ) < 6, she can obtain |x ∩ y|. Recall that k = Θ(n) and the set of edges between nodes simulated by Alice and those simulated by Bob, which is E c , is of size Θ(s) = Θ(log k) = Θ(log n). The quantum communication complexity of INT k,ρ is O(r · log 2 n) for any ρ. Combining Lemma 2.7, r = Ω( k log 2 n ) = Ω( n log 2 n ).

Approximate eccentricities
The above argument can be extended to prove a more general quantum lower bound of approximating eccentricities by introducing a parameter and replacing some edges of instance G = (V, E) with paths of length . More specifically, except for edges between V a and V b , i.e., edges in E c , and edges between a 1 and additional nodes a 1 , · · · , a n−n , other edges (including the edges added according to x and y) are all replaced with paths containing − 1 intermediate nodes. There are 3k + 2ks + 4 + |x| + |y| edges needed to be replaced. Since the replacement may change the number of nodes, we redefine some parameters. Let n ≥ 31 − 8 be an integer where ≥ 1 is an undetermined parameter. We set k to be the maximum integer satisfying 3k + 4( log 2 (k − 1) + 1) + 6 + ( − 1)(3k + 2k( log 2 (k − 1) + 1) + 4 + 2k) ≤ n, and set s = log 2 (k − 1) + 1, n = 3k + 4s + 6 + ( − 1)(3k + 2ks + 4 + |x| + |y|). We again include n − n additional nodes to make sure |V | = n. The instance G is constructed in the same way except that intermediate nodes are added. By following the argument in the proof of Proposition 4.2, for p ∈ [k], we have: d(a p , b q ) ≤ 3 + 1; d(a p , b 1 ) ≤ 3 + 1.
Thus, e(a p ) = 3 + 1 if x p = y p = 1 and e(a p ) = 5 + 1 otherwise. We omit intermediate nodes because the eccentricity of a p cannot be achieved by any intermediate nodes. The ratio of the eccentricity of a p in the case x p = 0 or y p = 0 and the one in the case x p = y p = 1 is 5 +1 3 +1 = 5 3 − ε when is sufficiently large. Thus, an algorithm of eccentricities with an approximation ratio better than 5 3 will induce a quantum communication protocol for the intersection function with inputs x and y.
Proof of Theorem 1.3. For any constant 0 < ε < 2 3 set = 2 9ε , let A be any r-round quantum protocol which computes a ( 5

Summary and Open Problems
In this paper, we give nearly tight bounds on the complexity of the eccentricities and all pairs shortest paths in the quantum CONGEST model and prove that there is no quantum speedup for these two problems. In contrast, the computation of diameter and radius have lower round complexity in the quantum CONGEST model than classical CONGEST model when the diameter is small [GM18]. It is interesting that quantum communication plays different role for different distance parameters.
For now we only consider the round complexity of the distance parameters of the unweighted network in the quantum CONGEST models. Whether there is quantum speedup for diameter computation in weighted network is still an open problem.