On the exact analysis of an idealized quantum switch

Abstract

We study an entanglement distribution switch that serves $k$ users in a star topology. The function of the switch is to facilitate end-to-end bipartite entangled state generation for pairs of users. We study a simple variant of this problem, wherein all links connecting the users to the switch are identical, the effects of state decoherence are negligible, and the switch can store an arbitrary number of qubits. We model the system using a discrete-time Markov chain and obtain the capacity of the switch. When the switch operates at capacity, we also present a numerical method for computing the expected number of qubits stored at the switch, which depends on the number of users $k$ and the probability of successful entanglement generation at the link level $p$. We then compare the results of our exact analysis to that of a continuous-time Markov chain model of a quantum switch and argue that the latter is a reasonable approximation to the more realistic model presented in this work.

Introduction

Protocols that exploit quantum communication technology offer two advantages: they can either extend or render feasible the capabilities of their classical counterparts, or they exhibit functionality entirely unachievable through classical means alone. For an example of the former, quantum key distribution (QKD) protocols such as E91 [1] and BBM92 [2] can in principle yield information-theoretic security by using entanglement to generate secure key bits. These raw secret key bits can then be distilled into a one-time pad to encode messages sent between two parties. For an example of the latter, distributed quantum sensing frameworks such as [3] and [4] employ entanglement to overcome the standard quantum limit [5].

While these applications hold a tremendous amount of potential for distributed quantum communication (and even computation, see, e.g., [6]), a substantial challenge is reliable generation of entanglement – an essential component for many of these tasks – especially over a large distance. This is due to the fact that there is an exponential rate-versus-distance decay for quantum state propagation both through terrestrial free-space and optical fiber channels [7], [8]. Quantum repeaters positioned between communicating nodes can overcome this fundamental rate-versus-distance tradeoff [9], [10]. The process of quantum repeater-assisted entanglement generation is illustrated at a high level in Fig. 1 and can be divided into two main steps. In step one, each segment connecting two adjacent nodes attempts to generate an entangled link. Qubits from a successfully-generated entanglement are stored in quantum memories, one in each node (Fig. 1b). Once entangled links are present on all segments, the quantum repeaters perform entanglement swapping [11] on their two locally-held qubits (Fig. 1c). If all swapping operations succeed, this results in an end-to-end entangled link between the communicating parties (Fig. 1d).

In this work, we use the term “quantum switch” instead of “repeater” because in a more complex network than that of Fig. 1, the device will likely be connected to several nodes or users; hence it is reasonable to assume that it will be equipped with entanglement switching logic. Quantum repeaters, switches, and similar devices (e.g., trusted nodes) will serve as building blocks for large-scale quantum networks. It is natural, therefore, to ask questions about their fundamental limits from a mathematical perspective, in order to gain insight into what constitutes efficient operation for such a device, as well as to create a performance comparison basis for future protocols and algorithms that rely on these devices. To this end, we study a quantum switch that serves entangled states to pairs of users in a star topology, with the objective of determining the capacity of the switch, as well as the expected number of stored qubits in memory at the switch when it operates at capacity. We use a discrete-time Markov chain (DTMC) to construct a model that abstracts away various architecture and physical implementation details about the system, e.g., the method used for entanglement generation or how quantum memories are realized.

We focus on the simplest variant of this problem, wherein links connecting users to the switch are identical, there is no quantum state decoherence, and the switch can store arbitrary numbers of qubits. Throughout this paper, we often refer to the number of quantum memories at the switch as its buffer size. An unfortunate property of our DTMC model is that it is difficult to extend to include the aforementioned system characteristics. Prior literature on quantum switch modeling utilizes continuous-time Markov chains (CTMCs) to account for these phenomena. Nevertheless, there is value in studying a quantum switch using a DTMC, as the system is inherently a discrete-time system. Hence, while CTMCs have been shown to be more expressive as a modeling technique, there will undoubtedly be some differences in the resulting performance metrics. To quantify these differences, and determine whether a CTMC model provides a reasonable approximation to the original system, we compare the performance metrics obtained from both models.

Following is a summary of our results:

• the DTMC is stable if and only if the number of users $k\ge 3$;

• the capacity of the switch is given by

  $$C=\frac{qkp}{2},$$

  where $k$ is the number of users or links, $p$ is the probability of successfully generating entanglement at the link level, and $q$ is the probability of a successful swapping operation;

• when the switch operates at capacity (a detailed description of a switching policy that achieves the maximum entanglement switching rate is described in Section 4), the expected number of stored qubits is given by

  $$E\left[Q\right]=\frac{1+\beta }{2(1-\beta )},$$

  where $Q$ is the number of qubits stored at the switch in steady state, across all links, and $\beta$ is in the interval $(0,1)$ and is the unique solution to the following equation¹ when $k\ge 3$:

  $${(\beta p+\overline{p})}^{k-1}(p+\beta \overline{p})-\beta =0;$$

• the expression for the capacity of the switch obtained using the DTMC matches exactly that of the CTMC model found in literature. On the other hand, the CTMC model overestimates the expected number of qubits in memory in steady state, but since the discrepancy is not significant, we conclude that the CTMC model is a reasonable approximation to the behavior of the system considered in this work.

The rest of this paper is organized as follows: in Section 2, we introduce the relevant background for quantum computation and communication. In Section 3 we discuss related work on quantum switch modeling. In Section 4, we formally introduce the DTMC model and state the objectives. The analysis is performed in Section 5. In Section 6, we compare the DTMC model introduced in this work with an existing CTMC model. We conclude in Section 7.