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IEEE Transactions on Computers

Multicasting in Quantum Switching Networks

Manish Kumar Shukla, A. Yavuz Oruç

Published Feb 19, 2010• 9 citations
Publisher

Abstract

In this paper, we present a quantum multicasting network, called a generalized quantum connector (n-GQC), which can be used to multicast quantum information from n inputs to n outputs. This network is recursively constructed using n/2-GQCs and consists of O(n log2n) quantum gates. The key component of the n-GQC is another network, called an n-quantum concentrator (n-QC). This concentrator is also an n × n quantum network, and can route arbitrary quantum states on any m of its inputs to its top m outputs, for any m, 1 ¿ m ¿ n. Its quantum gate complexity is O(n log n). The quantum gate-level depths of n-QC and n-GQC are O(log2n) and O(log3n), respectively. Both n-QC and n-GQC are based on the classical self-routing concentrators and generalized connection networks given by Lee and Oruc [1]. While these networks work for multicasting classical packets, they cannot be used to multicast quantum packets as they employ balancer switches with both forward and backward propagation of packets. We introduce a quantum balancer switch that works using a forward propagation of packets only, thereby facilitating the n-QC and n-GQC designs presented in the paper.

DOI
10.1109/tc.2010.52

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In this paper, we present a quantum multicasting network, called a generalized quantum connector (n-GQC), which can be used to multicast quantum information from n inputs to n outputs. This network is recursively constructed using n/2-GQCs and consists of O(n log2n) quantum gates. The key component of the n-GQC is another network, called an n-quantum concentrator (n-QC). This concentrator is also an n × n quantum network, and can route arbitrary quantum states on any m of its inputs to its top m outputs, for any m, 1 ¿ m ¿ n. Its quantum gate complexity is O(n log n). The quantum gate-level depths of n-QC and n-GQC are O(log2n) and O(log3n), respectively. Both n-QC and n-GQC are based on the classical self-routing concentrators and generalized connection networks given by Lee and Oruc [1]. While these networks work for multicasting classical packets, they cannot be used to multicast quantum packets as they employ balancer switches with both forward and backward propagation of packets. We introduce a quantum balancer switch that works using a forward propagation of packets only, thereby facilitating the n-QC and n-GQC designs presented in the paper.
Quantum switching networks are analogs of classical switching networks in which classical switches are replaced by quantum switches. These networks are used to switch quantum data among a set of quantum sources and receivers. They can also be used to efficiently switch classical data, and help overcome some limitations of classical switching networks by utilizing the unique properties of quantum information systems, such as superposition and parallelism. In this thesis, we design several such networks which can be broadly put in the following three categories: 1. Quantum unicast networks: We give the design of quantum Baseline network (QBN) which is a self-routing and unicast quantum packet switch that uses the Baseline topology. The classical version of the network blocks packets internally even when there are no output contentions and each input packet is addressed to a different output. The QBN uses the principles of quantum superposition and parallelism to overcome such blocking. Also, for assignments that have multiple input packets addressed to an output, this network creates a quantum superposition of all these packets on that output, ensuring that all packets have non-zero probabilities of being observed on that output. 2. Quantum concentrators: We introduce a new network called quantum concentrator, which is a key component of our quantum multicasting network design. This concentrator is also an n × n quantum switching network, to be denoted by n-QC, and which, for any m, 1 ≤ m ≤ n, routes arbitrary quantum states on any m of its inputs to its top m outputs. This network uses O(n log n) quantum gates, and has a gate level depth of O(log 2 n). We also give several variations of this network, the main ones being order-preserving and priority quantum concentrators. 3. Quantum multicast networks: We first design a quantum multicasting network, called a generalized quantum connector (GQC) which can be used to multicast quantum information from n input sources to n outputs. Since general quantum states cannot be copied due to the no-cloning theorem, this network actually multicasts superposed classical information packets, contained in a finite number of qubits at each input. Copying needed for such multicasting is obtained by Wootters and Zurek's quantum copying machines or controlled-not gates. This n-input and n-output (n × n) network, to be denoted by n-GQC, is recursively constructed using n/2-GQCs and uses O(n log 2 n) quantum gates. The time complexity of this network in terms of gate level depth is O(log 3 n). We also give two variations of this network which improve its behavior when routing multicast assignments that have multiple input packets contending for same outputs.
Quantum switching networks are derived from conventional switching networks by replacing the classical switches by quantum switches. We give the quantum circuit design and routing of an n \times m network, called a quantum concentrator that can direct quantum bit packets in arbitrary quantum states from any of its k inputs to some of its k outputs, where 1 \le k \le m \le n. Our designs are based on sparse crossbars which are rectangular grids of 2 \times 2 crosspoints. Sparse crossbar concentrator structures with theoretically minimum crosspoint complexities for any values of n and m are well known but no self-routing algorithms have been reported for such concentrators. We transform two such families of optimal concentrators, called fat-slim and banded sparse crossbars into quantum networks and provide self-routing algorithms for these families of concentrators. In this process, we extend the notion of packet concentration to a quantum network and design self-routing quantum crosspoints from quantum gates. We address issues critical to quantum operation like reversibility and localized self-routing and give a rigorous proof that quantum fat-slim and banded sparse crossbar concentrators are self-routable. The self-routing algorithms described in the paper can be used for both quantum and classical sparse crossbar concentrators by the linearity property of all quantum systems.
We present the design of a self-routing quantum packet switch which improves on a previous design given by the authors [1]. Like the earlier design, this switch too routes packets represented by quantum bits (qubits) but it reduces the routing overhead per packet from O(log N) qubits to O(1) qubits by eliminating the need for an extra "dummy" inputoutput pair and their associated "dummy" packets at each 2 2 internal switch. The quantum self-routing switch creates a superposition of all the maximum size non-blocking subsets of input packets at its outputs, which cannot be achieved by any classical self-routing switching network that is internally blocking. In addition to the network design, we give a method to characterize the output quantum state of the switch using the concepts of frames and balanced matrices.
Presents a switching architecture such that digital data can be switched in the quantum domain. The proposed mechanism supports unicasting as well as multicasting, and is strict-sense nonblocking. In addition, with appropriate interface conversion, this architecture can also be used to switch classical information. This results in a quantum switch that can be used to build classical and quantum information networks. To present this idea, we define the connection digraph which can be used to describe the behavior of a switch at a given time, then we show how a connection digraph can be implemented using elementary quantum gates. Compared with a traditional space or time domain switch, the proposed switching mechanism is much more scalable. Assuming an n/spl times/n quantum switch, the space consumption grows linearly, i.e., O(n), while the time complexity is O(1) for unicasting, and O(log/sub 2/n) for multicasting. Based on these advantages, a high-throughput switching device can be built simply by increasing the number of I/O ports.
Flexible distribution of data in the form of quantum bits (qubits) amongst spatially separated entities is an essential component of envisioned scalable quantum computing architectures. Since qubits cannot be copied, this operation of moving qubits can be relatively costly in terms of resources. Moreover, implementation of quantum gates requires precise and extensive classical control and computation too. Accordingly, we consider the problem of dynamically permuting groups of qubits, i.e., qubit packets using reconfigurable quantum switches in which routing information is calculated classically as a possible way to reduce this cost. We design a 2 x 2 switch based on the controlled-swap quantum gate and show that if switch settings are determined using efficient classical algorithms, then quantum switches can be mapped onto classical non-blocking interconnection switch topologies with low cost by using this switch. Specific quantum switch designs for the optimal Benes network and the planar Spanke-Bcnes network are given.
Large-scale quantum networks with thousands of nodes require scalable network protocols and physical hardware to realize. In this work, we introduce packet switching as a new paradigm for quantum data transmission in both future and near-term quantum networks. We propose a classical-quantum data frame structure and explore methods of frame generation and processing. Further, we present conceptual designs for a quantum reconfigurable optical add-drop multiplexer to realize the proposed transmission scheme. Packet switching allows for a universal design for a next generation Internet where classical and quantum data share the same network protocols and infrastructure. In this new quantum networking paradigm, entanglement distribution, as with quantum key distribution, is an application built on top of the quantum network rather than as a network designed especially for those purposes. For analysis of the network model, we simulate the feasibility of quantum packet switching for some preliminary models of quantum key and entanglement distribution. Finally, we discuss how our model can be integrated with other network models toward a realization of a quantum Internet.