Distributed Quantum computers abide from the exit complexity of the knowledge. The exit complexity is the accrue of the nodal information needed to clarify the total egress system with deference to a distinguished exit node. The core objective of this paper is to compile an arrogant methodology for assessing the exit complexity of the knowledge in distributed quantum computers. The proposed methodology is based on contouring the knowledge using the unlabeled binary trees, hence building an benchmarked and a computer based model. The proposed methodology dramatizes knowledge autocratically calculates the exit complexity. The methodology consists of several amphitheaters, starting with detecting the baron aspect of the tree of others entitled express knowledge and then measure the volume of information and the complexity of behavior destining from the bargain of information. Then calculate egress resulting from episodes that do not lead to the withdrawal of the information. In the end is calculated total egress complexity and then appraised total exit complexity of the system. Given the complexity of the operations within the Distributed Computing Quantity, this research addresses effective transactions that could affect the three-dimensional behavior of knowledge. The results materialized that the best affair where total exit complexity as minimal as possible is a picture of a binary tree is entitled at the rate of positive and negative cardinal points medium value. It could be argued that these cardinal points should not amass the upper bound apex or minimum.
Distributed Quantum computers abide from the exit complexity of the knowledge. The exit complexity is the accrue of the nodal information needed to clarify the total egress system with deference to a distinguished exit node. The core objective of this paper is to compile an arrogant methodology for assessing the exit complexity of the knowledge in distributed quantum computers. The proposed methodology is based on contouring the knowledge using the unlabeled binary trees, hence building an benchmarked and a computer based model. The proposed methodology dramatizes knowledge autocratically calculates the exit complexity. The methodology consists of several amphitheaters, starting with detecting the baron aspect of the tree of others entitled express knowledge and then measure the volume of information and the complexity of behavior destining from the bargain of information. Then calculate egress resulting from episodes that do not lead to the withdrawal of the information. In the end is calculated total egress complexity and then appraised total exit complexity of the system. Given the complexity of the operations within the Distributed Computing Quantity, this research addresses effective transactions that could affect the three-dimensional behavior of knowledge. The results materialized that the best affair where total exit complexity as minimal as possible is a picture of a binary tree is entitled at the rate of positive and negative cardinal points medium value. It could be argued that these cardinal points should not amass the upper bound apex or minimum.
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.
The focus of this paper is on {\em quantum distributed} computation, where we investigate whether quantum communication can help in {\em speeding up} distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication {\em does not} help in substantially speeding up distributed algorithms for these problems compared to the classical setting. In order to obtain this result, we extend the technique of Das Sarma et al. [SICOMP 2012] to obtain a uniform approach to prove non-trivial lower bounds for quantum distributed algorithms for several graph optimization (both exact and approximate versions) as well as verification problems, some of which are new even in the classical setting, e.g. tight randomized lower bounds for Hamiltonian cycle and spanning tree verification, answering an open problem of Das Sarma et al., and a lower bound in terms of the weight aspect ratio, matching the upper bounds of Elkin [STOC 2004]. Our approach introduces the {\em Server model} and {\em Quantum Simulation Theorem} which together provide a connection between distributed algorithms and communication complexity. The Server model is the standard two-party communication complexity model augmented with additional power; yet, most of the hardness in the two-party model is carried over to this new model. The Quantum Simulation Theorem carries this hardness further to quantum distributed computing. Our techniques, except the proof of the hardness in the Server model, require very little knowledge in quantum computing, and this can help overcoming a usual impediment in proving bounds on quantum distributed algorithms.
We study lower bounds for quantum distributed computing, where a set of nodes (representing quantum computers) interconnected by an underlying network consisting of (bandwidth-restricted) links, communicate using quantum communication. Nodes have unlimited computational power and may share an unlimited number of entangled qubits. Our main contribution is a simple uniform technique for proving lower bounds for quantum distributed algorithms. In particular, we identify a new quantum communication model called the Server model which provides a connection between distributed algorithms and communication complexity: it is strong enough to capture the hardness of several distributed computing problems, while weak enough that several hard problems in two-party communication complexity remain hard; to this end, we identify a set of communication complexity techniques that can be carried over to the Server model, namely techniques based on nonlocal games. We show that these techniques serve as a fundamental tool in proving lower bounds in quantum distributed computing. The new techniques help us to prove several non-trivial quantum distributed lower bounds (which are the first-known quantum bounds for fundamental global problems such as minimum spanning tree, shortest paths etc.), some of which are new even in the classical setting. First, it allows us to show that all previous classical lower bounds of more than twenty verification and optimization graph problems in [Das Sarma et al., STOC 11] also hold in the quantum setting. Many of these bounds are tight, implying a large class of problems that do not gain an advantage from quantum effects. Our results also imply the following new results in the classical communication models: (1) the first randomized lower bounds for Hamiltonian cycle and spanning tree verification problems in both the distributed computing and the communication complexity model, answering the open problem of Das Sarma et al. and subsuming many bounds in [Babai, Frankl, and Simon, FOCS 96], and (2) the first lower bound that is tight for all weight aspect ratios, matching previous upper bounds of [Elkin, STOC 04]. Submitted for a Regular Presentation. The full version of this paper can be found as [15] at http://arxiv.org/abs/1207.5211. ∗Department of Computer Science, Ben-Gurion University, Beer-Sheva, 84105, Israel. E-mail: elkinm@cs.bgu.ac.il. †Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Centre for Quantum Technologies, National University of Singapore, Singapore 117543. E-mail: hklauck@gmail.com. Research at the Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation. ‡Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371. Work partially done while at University of Vienna, Austria. E-mail: danupon@gmail.com. §Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Department of Computer Science, Brown University, Providence, RI 02912, USA. E-mail: gopalpandurangan@gmail.com. Supported in part by the following research grants: Nanyang Technological University grant M58110000, Singapore MOE Academic Research Fund (AcRF) Tier 2 grant MOE2010-T2-2-082, and a grant from the US-Israeli Binational Science Foundation (BSF).
We investigate how much quantum distributed algorithms can outperform classical distributed algorithms with respect to the message complexity (the overall amount of communication used by the algorithm). Recently, Dufoulon, Magniez and Pandurangan (PODC 2025) have shown a polynomial quantum advantage for several tasks such as leader election and agreement. In this paper, we show an exponential quantum advantage for a fundamental task: routing information between two specified nodes of a network. We prove that for the family of ``welded trees" introduced in the seminal work by Childs, Cleve, Deotto, Farhi, Gutmann and Spielman (STOC 2003), there exists a quantum distributed algorithm that transfers messages from the entrance of the graph to the exit with message complexity exponentially smaller than any classical algorithm. Our quantum algorithm is based on the recent "succinct" implementation of quantum walks over the welded trees by Li, Li and Luo (SODA 2024). Our classical lower bound is obtained by ``lifting'' the lower bound from Childs, Cleve, Deotto, Farhi, Gutmann and Spielman (STOC 2003) from query complexity to message complexity.
We investigate how much quantum distributed algorithms can outperform classical distributed algorithms with respect to the message complexity (the overall amount of communication used by the algorithm). Recently, Dufoulon, Magniez and Pandurangan (PODC 2025) have shown a polynomial quantum advantage for several tasks such as leader election and agreement. In this paper, we show an exponential quantum advantage for a fundamental task: routing information between two specified nodes of a network. We prove that for the family of “welded trees” introduced in the seminal work by Childs, Cleve, Deotto, Farhi, Gutmann and Spielman (STOC 2003), there exists a quantum distributed algorithm that transfers messages from the entrance of the graph to the exit with message complexity exponentially smaller than any classical algorithm. Our quantum algorithm is based on the recent “succinct” implementation of quantum walks over the welded trees by Li, Li and Luo (SODA 2024). Our classical lower bound is obtained by “lifting” the lower bound from Childs, Cleve, Deotto, Farhi, Gutmann and Spielman (STOC 2003) from query complexity to message complexity. A full version of this paper can be found on arXiv [21].
We consider the following question in query complexity: Given a classical query algorithm in the form of a decision tree, when does there exist a quantum query algorithm with a speed-up (i.e., that makes fewer queries) over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up in the number of queries. In particular, our results give a bounded-error quantum query algorithm of cost O(√s) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size s. This recovers an O(√n) algorithm for the Search problem, for example. Lin and Lin [Theory of Computing'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor. Their upper bounds are in terms of a quantity which we call the “guessing complexity” of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Information and Computation'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any measure studied in classical complexity theory. We also show a polynomial separation between rank and its natural randomized analog for the complete binary AND-OR tree. Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to edges of the tree. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are strictly stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes in their construction could yield better upper bounds.