Given x,y ∈ {0,1}ⁿ, Set Disjointness consists in deciding whether x_i = y_i = 1 for some index i ∈ [n]. We study the problem of computing this function in a distributed computing scenario in which the inputs x and y are given to the processors at the two extremities of a path of length d. Each vertex of the path has a quantum processor that can communicate with each of its neighbours by exchanging O(log n) qubits per round. We are interested in the number of rounds required for computing Set Disjointness with constant probability bounded away from 1/2. We call this problem "Set Disjointness on a Line". Set Disjointness on a Line was introduced by Le Gall and Magniez [Le Gall and Magniez, 2018] for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model. However, they were only able to provide a lower bound when the local memory used by the processors on the intermediate vertices of the path is severely limited. More precisely, their bound applies only when the local memory of each intermediate processor consists of O(log n) qubits. In this work, we prove an unconditional lower bound of Ω̃(∛{n d²} + √n) rounds for Set Disjointness on a Line with d + 1 processors. This is the first non-trivial lower bound when there is no restriction on the memory used by the processors. The result gives us a new lower bound of Ω̃ (∛{nδ²} + √n) on the number of rounds required for computing the diameter δ of any n-node network with quantum messages of size O(log n) in the CONGEST model. We draw a connection between the distributed computing scenario above and a new model of query complexity. In this model, an algorithm computing a bi-variate function f (such as Set Disjointness) has access to the inputs x and y through two separate oracles 𝒪_x and 𝒪_y, respectively. The restriction is that the algorithm is required to alternately make d queries to 𝒪_x and d queries to 𝒪_y, with input-independent computation in between queries. The model reflects a "switching delay" of d queries between a "round" of queries to x and the following "round" of queries to y. The technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to this query model. We provide an algorithm for Set Disjointness in this query model with query complexity that matches the round lower bound stated above, up to a polylogarithmic factor. In this sense, the round lower bound we show for Set Disjointness on a Line is optimal.
Given x,y ∈ {0,1}ⁿ, Set Disjointness consists in deciding whether x_i = y_i = 1 for some index i ∈ [n]. We study the problem of computing this function in a distributed computing scenario in which the inputs x and y are given to the processors at the two extremities of a path of length d. Each vertex of the path has a quantum processor that can communicate with each of its neighbours by exchanging O(log n) qubits per round. We are interested in the number of rounds required for computing Set Disjointness with constant probability bounded away from 1/2. We call this problem "Set Disjointness on a Line". Set Disjointness on a Line was introduced by Le Gall and Magniez [Le Gall and Magniez, 2018] for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model. However, they were only able to provide a lower bound when the local memory used by the processors on the intermediate vertices of the path is severely limited. More precisely, their bound applies only when the local memory of each intermediate processor consists of O(log n) qubits. In this work, we prove an unconditional lower bound of Ω̃(∛{n d²} + √n) rounds for Set Disjointness on a Line with d + 1 processors. This is the first non-trivial lower bound when there is no restriction on the memory used by the processors. The result gives us a new lower bound of Ω̃ (∛{nδ²} + √n) on the number of rounds required for computing the diameter δ of any n-node network with quantum messages of size O(log n) in the CONGEST model. We draw a connection between the distributed computing scenario above and a new model of query complexity. In this model, an algorithm computing a bi-variate function f (such as Set Disjointness) has access to the inputs x and y through two separate oracles 𝒪_x and 𝒪_y, respectively. The restriction is that the algorithm is required to alternately make d queries to 𝒪_x and d queries to 𝒪_y, with input-independent computation in between queries. The model reflects a "switching delay" of d queries between a "round" of queries to x and the following "round" of queries to y. The technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to this query model. We provide an algorithm for Set Disjointness in this query model with query complexity that matches the round lower bound stated above, up to a polylogarithmic factor. In this sense, the round lower bound we show for Set Disjointness on a Line is optimal.
Given x,y∈{0,1}n , Set Disjointness consists in deciding whether xi=yi=1 for some index i∈[n] . We study the problem of computing this function in a distributed computing scenario in which the inputs x and y are given to the processors at the two extremities of a path of length d . Each vertex of the path has a quantum processor that can communicate with each of its neighbours by exchanging O(logn) qubits per round. We are interested in the number of rounds required for computing Set Disjointness with constant probability bounded away from 1/2 . We call this problem “Set Disjointness on a Line”. Set Disjointness on a Line was introduced by Le Gall and Magniez [ 14 ] for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model. However, they were only able to provide a lower bound when the local memory used by the processors on the intermediate vertices of the path is severely limited. More precisely, their bound applies only when the local memory of each intermediate processor consists of O(logn) qubits. In this work, we prove an unconditional lower bound of Ω(3nd2+n) rounds for Set Disjointness on a Line with d+1 processors. This is the first non-trivial lower bound when there is no restriction on the memory used by the processors. The result gives us a new lower bound of Ω(3nδ2+n) on the number of rounds required for computing the diameter δ of any n -node network with quantum messages of size O(logn) in the CONGEST model. We draw a connection between the distributed computing scenario above and a new model of query complexity. In this model, an algorithm computing a bi-variate function f (such as Set Disjointness) has access to the inputs x and y through two separate oracles Ox and Oy , respectively. The restriction is that the algorithm is required to alternately make d queries to Ox and d queries to Oy , with input-independent computation in between queries. The model reflects a “switching delay” of d queries between a “round” of queries to x and the following “round” of queries to y . The information-theoretic technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to the number of rounds in this query model. We provide an algorithm for Set Disjointness in this query model with round complexity that matches the round lower bound stated above, up to a polylogarithmic factor. This presents a barrier for obtaining a better round lower bound for Set Disjointness on the Line. At the same time, it hints at the possibility of better communication protocols for the problem.
The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange O(log n) bits per round (here n denotes the number of nodes of the network), it is known that exact computation of the diameter requires Ω(n) rounds, even in networks with constant diameter. In this paper we investigate quantum distributed algorithms for this problem in the quantum CONGEST model, where two adjacent nodes can exchange O(log n) quantum bits per round. Our main result is a O(√D )-round quantum distributed algorithm for exact diameter computation, where D denotes the diameter. This shows a separation between the computational power of quantum and classical algorithms in the CONGEST model. We also show an unconditional lower bound Ω(√ ) on the round complexity of any quantum algorithm computing the diameter, and furthermore show a tight lower bound Ω(√D ) for any distributed quantum algorithm in which each node can use only poly(log n) quantum bits of memory.
The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange O(log n) bits per round (here n denotes the number of nodes of the network), it is known that exact computation of the diameter requires Ω(n) rounds, even in networks with constant diameter. In this paper we investigate quantum distributed algorithms for this problem in the quantum CONGEST model, where two adjacent nodes can exchange O(log n) quantum bits per round. Our main result is a O(√D )-round quantum distributed algorithm for exact diameter computation, where D denotes the diameter. This shows a separation between the computational power of quantum and classical algorithms in the CONGEST model. We also show an unconditional lower bound Ω(√ ) on the round complexity of any quantum algorithm computing the diameter, and furthermore show a tight lower bound Ω(√D ) for any distributed quantum algorithm in which each node can use only poly(log n) quantum bits of memory.
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 the distance 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 [F. L. Gall and F. Magniez, Sublinear-time quantum computation of the diameter in CONGEST networks, in Proc. 2018 ACM Symp. Principles of Distributed Computing (PODC), (2018), pp. 337–346; F. Magniez and A. Nayak, arXiv:2002.11795 ]. We obtain the lower bounds via a reduction from the two-way quantum communication complexity of the set intersection [A. A. Razborov, Izv. Math. 159 (2003)].
This paper considers the triangle finding problem in the CONGEST model of distributed computing. Recent works by Izumi and Le Gall (PODC'17), Chang, Pettie and Zhang (SODA'19) and Chang and Saranurak (PODC'19) have successively reduced the classical round complexity of triangle finding (as well as triangle listing) from the trivial upper bound O(n) to Õ(n^{1/3}), where n denotes the number of vertices in the graph. In this paper we present a quantum distributed algorithm that solves the triangle finding problem in Õ(n^{1/4}) rounds in the CONGEST model. This gives another example of quantum algorithm beating the best known classical algorithms in distributed computing. Our result also exhibits an interesting phenomenon: while in the classical setting the best known upper bounds for the triangle finding and listing problems are identical, in the quantum setting the round complexities of these two problems are now Õ(n^{1/4}) and Θ~(n^{1/3}), respectively. Our result thus shows that triangle finding is easier than triangle listing in the quantum CONGEST model.
We consider the distributed wake-up problem with advice, where nodes are equipped with initial knowledge about the network at large. After the adversary awakens a subset of nodes, an oracle computes a bit string (``the advice'') for each node, and the goal is to wake up all sleeping nodes efficiently. We present the first upper and lower bounds on the message complexity for wake-up in the quantum routing model, introduced by Dufoulon, Magniez, and Pandurangan (PODC 2025). In more detail, we give a distributed advising scheme that, given α bits of advice per node, wakes up all nodes with a message complexity of O(2max{⌊(α−1)/2⌋,0}n3⋅logn) with high probability. Our result breaks the Ω(2αn2) barrier known for the classical port numbering model in sufficiently dense graphs. To complement our algorithm, we give a lower bound on the message complexity for distributed quantum algorithms: By leveraging a lower bound result for the single-bit descriptor problem in the query complexity model, we show that wake-up has a quantum message complexity of Ω(n3/2) without advice, which holds independently of how much time we allow. In the setting where an adversary decides which nodes start the algorithm, most graph problems of interest implicitly require solving wake-up, and thus the same lower bound also holds for other fundamental problems such as single-source broadcast and spanning tree construction.