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Researchers are fabricating quantum processors powerful enough to execute small instances of quantum algorithms. Scalability concerns are motivating distributed-memory multicomputer architectures, and experimental efforts have demonstrated some of the building blocks for such a design. Numberous systems are emerging with the goal of enabling local and distributed quantum computing.

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... Distributed architectures are envisioned as a long-term solution to provide practical applications of quantum computing [1], [2], [3]. They offer a physical substrate to scale up horizontally computing resources, rather than relying on vertical scale-up, coming from single hardware advancement. ...

... Starting from ≺, we can define a constraint to add to formulation (3). Namely, ∀i ∈ [k], ∀e ∈ δ − (P T i ) the following holds: ...

... The two-qubits assumption is general and can be extended to multi-qubits protocols.3 This step implies communication. ...

Practical distributed quantum computing requires the development of efficient compilers, able to make quantum circuits compatible with some given hardware constraints. This problem is known to be tough, even for local computing. Here, we address it on distributed architectures. As generally assumed in this scenario, telegates represent the fundamental remote (inter-processor) operations. Each telegate consists of several tasks: i) entanglement generation and distribution, ii) local operations, and iii) classical communications. Entanglement generations and distribution is an expensive resource, as it is time-consuming and fault-prone. To mitigate its impact, we model an optimization problem that combines running-time minimization with the usage of that resource. Specifically, we provide a parametric ILP formulation, where the parameter denotes a time horizon (or time availability); the objective function count the number of used resources. To minimize the time, a binary search solves the subject ILP by iterating over the parameter. Ultimately, to enhance the solution space, we extend the formulation, by introducing a predicate that manipulates the circuit given in input and parallelizes telegates' tasks.

... Among applications supported by quantum networks, distributed quantum computing is of particular interest as it leverages the power of interconnected quantum computers to create a virtual quantum machine with processing capabilities that surpass its physical constituents alone [3]- [5]. Distributed quantum computing becomes even more interesting in the noise intermediate scale quantum machines (NISQ) scenario where there is a clear tradeoff between the size of quantum computers, in terms of number of qubits, and the fidelity of quantum operations, given the fact that physical separation directly reduces cross talk among qubits [6]. When the quantum network scenario is considered, the complexity of distributed quantum computing extends in at least two dimensions. ...

... The propagation of quantum control information between neighbors in the network is embedded in the formalism through the walker's shift operator. The propagation in the case of non-neighboring nodes needs both coin and shift operators as defined in (5) and (6). Note that generic computation on the data qubits can be performed with (7) and (10) given states in the superposition (15), although it is not possible to evolve the walker to that superposition from |A, c A without shift operators. ...

... where S f is defined in (6). The coin operator definition is also time-independent, although it depends on the minimum path p chosen. ...

Quantum networks are complex systems formed by the interaction among quantum processors through quantum channels. Analogous to classical computer networks, quantum networks allow for the distribution of quantum computation among quantum computers. In this work, we describe a quantum walk protocol to perform distributed quantum computing in a quantum network. The protocol uses a quantum walk as a quantum control signal to perform distributed quantum operations. We consider a generalization of the discrete-time coined quantum walk model that accounts for the interaction between a quantum walker system in the network graph with quantum registers inside the network nodes. The protocol logically captures distributed quantum computing, abstracting hardware implementation and the transmission of quantum information through channels. Control signal transmission is mapped to the propagation of the walker system across the network, while interactions between the control layer and the quantum registers are embedded into the application of coin operators. We demonstrate how to use the quantum walker system to perform a distributed CNOT operation, which shows the universality of the protocol for distributed quantum computing. Furthermore, we apply the protocol to the task of entanglement distribution in a quantum network.

... Distributed quantum computing usually refers to the act of performing a quantum computational task on two or more distinct quantum processors and combining the results to produce a complete output [18,20,[283][284][285][286]. Consider a quantum computational task that is divisible into several subroutines. ...

... Distributed quantum computing is also applicable at the nanoscale, specifically in designing quantum chips. As the architecture of a single quantum processor approaches its limits, combining multiple processors with classical and quantum communication links could provide a promising solution [285]. In IBM's recent development roadmap, it is considered that getting beyond single-chip processors is the key to overcoming the scalability problem [287,288]. ...

The quantum internet is envisioned as the ultimate stage of the quantum revolution, which surpasses its classical counterpart in various aspects, such as the efficiency of data transmission, the security of network services, and the capability of information processing. Given its disruptive impact on the national security and the digital economy, a global race to build scalable quantum networks has already begun. With the joint effort of national governments, industrial participants and research institutes, the development of quantum networks has advanced rapidly in recent years, bringing the first primitive quantum networks within reach. In this work, we aim to provide an up-to-date review of the field of quantum networks from both theoretical and experimental perspectives, contributing to a better understanding of the building blocks required for the establishment of a global quantum internet. We also introduce a newly developed quantum network toolkit to facilitate the exploration and evaluation of innovative ideas. Particularly, it provides dual quantum computing engines, supporting simulations in both the quantum circuit and measurement-based models. It also includes a compilation scheme for mapping quantum network protocols onto quantum circuits, enabling their emulations on real-world quantum hardware devices. We showcase the power of this toolkit with several featured demonstrations, including a simulation of the Micius quantum satellite experiment, a testing of a four-layer quantum network architecture with resource management, and a quantum emulation of the CHSH game. We hope this work can give a better understanding of the state-of-the-art development of quantum networks and provide the necessary tools to make further contributions along the way.

... In this treatise, we aim for answering the remaining open questions on how much advantage we can glean from the indefinite causal order of quantum channels to improve the capacities of both entanglement-assisted classical and quantum communication. More specifically, within the Quantum Internet framework, multiple quantum devices are interconnected via pre-shared entanglement for facilitating various applications that require the exchange of classical and quantum information amongst the quantum devices, including quantum communications [25], [26], quantum cryptography [27], quantum sensing [28], [29], distributed quantum computation [30], [31], blind quantum • We derive the general formulation of entanglementassisted classical communication capacity over quantum trajectory for various scenarios involving quantum Pauli channels. • We determine the operating region where entanglementassisted communication over quantum trajectory obtains capacity gain against classical trajectory. ...

... Observe from (30) and (31), the following inequality always holds: C E,Q ≥ C E,C . The equality holds when ...

The unique and often-weird properties of quantum mechanics allow an information carrier to propagate through multiple trajectories of quantum channels simultaneously. This ultimately leads us to quantum trajectories with an indefinite causal order of quantum channels. It has been shown that indefinite causal order enables the violation of bottleneck capacity, which bounds the amount of the transferable classical and quantum information through a classical trajectory with a well-defined causal order of quantum channels. In this treatise, we investigate this beneficial property in the realm of both entanglement-assisted classical and quantum communications. To this aim, we derive closed-form capacity expressions of entanglement-assisted classical and quantum communication for arbitrary quantum Pauli channels over classical and quantum trajectories. We show that by exploiting the indefinite causal order of quantum channels, we obtain capacity gains over classical trajectory as well as the violation of bottleneck capacity for various practical scenarios. Furthermore, we determine the operating region where entanglement-assisted communication over quantum trajectory obtains capacity gain against classical trajectory and where the entanglement-assisted communication over quantum trajectory violates the bottleneck capacity.

... In this treatise, we aim for answering the remaining open questions on how much advantage we can glean from the indefinite causal order of quantum channels to improve the capacities of both entanglement-assisted classical and quantum communication. More specifically, within the Quantum Internet framework, multiple quantum devices are interconnected via pre-shared entanglement for facilitating various applications that require the exchange of classical and quantum information amongst the quantum devices, including quantum communications [25], [26], quantum cryptography [27], quantum sensing [28], [29], distributed quantum computation [30], [31], blind quantum computation [32], [33], quantum-secure direct-communication (QSDC) [34]- [37], and quantum-secure secret-sharing [38]. Therefore, the pre-shared entanglement can be viewed as the primary consumable resources for enabling entanglement-assisted classical and quantum communications within the Quantum Internet framework [31], [39]- [42]. ...

... Observe from (30) and (31), the following inequality always holds: C E,Q ≥ C E,C . The equality holds Remark. ...

The unique and often-weird properties of quantum mechanics allow an information carrier to propagate through multiple trajectories of quantum channels simultaneously. This ultimately leads us to quantum trajectories with an indefinite causal order of quantum channels. It has been shown that indefinite causal order enables the violation of bottleneck capacity, which bounds the amount of the transferable classical and quantum information through a classical trajectory with a well-defined causal order of quantum channels. In this treatise, we investigate this beneficial property in the realm of both entanglement-assisted classical and quantum communications. To this aim, we derive closed-form capacity expressions of entanglement-assisted classical and quantum communication for arbitrary quantum Pauli channels over classical and quantum trajectories. We show that by exploiting the indefinite causal order of quantum channels, we obtain capacity gains over classical trajectory as well as the violation of bottleneck capacity for various practical scenarios. Furthermore, we determine the operating region where entanglement-assisted communication over quantum trajectory obtains capacity gain against classical trajectory and where the entanglement-assisted communication over quantum trajectory violates the bottleneck capacity.

... Quantum repeaters allow for two end-points to perform entanglement swapping, allowing them to establish Bell pairs even if the repeater is controlled by an adversary. This provides a tremendously strong security guarantee, and also allows for the distillation of Bell pairs between parties which may be used for other applications including, and beyond, QKD (e.g., distributed quantum computing [8][9][10] and distributed quantum sensing [11][12][13][14]). Much current research is devoted to this future Quantum Internet, in particular to its behavior and performance in terms of maximizing entanglement distribution rates [15][16][17][18][19][20][21][22][23][24][25], with [22][23][24][25] particularly focusing on routing within a Quantum Internet. ...

In this paper, we consider quantum key distribution (QKD) in a quantum network with both quantum repeaters and a small number of trusted nodes. In contrast to current QKD networks with only trusted nodes and the true Quantum Internet with only quantum repeaters, such networks represent a middle ground, serving as near-future QKD networks. In this setting, QKD can be efficiently and practically deployed, while providing insights for the future true Quantum Internet. To significantly improve the key generation efficiency in such networks, we develop a new dynamic routing strategy that makes routing decisions based on the current network state, as well as evaluate various classical/quantum post-processing techniques. Using simulations, we show that our dynamic routing strategy can improve the key rate between two users significantly in settings with asymmetric trusted node placement. The post-processing techniques can also increase key rates in high noise scenarios. Furthermore, combining the dynamic routing strategy with the post-processing techniques can further improve the overall performance of the QKD network.

... Quantum networks bring new opportunities for secure communication [1,2], distributed sensing [3][4][5], and distributed quantum computing [6,7]. The ability to faithfully transmit quantum information over long distances is a key prerequisite for the construction of large-scale quantum networks. ...

The generation of multiple entangled qubit pairs between distributed nodes is a prerequisite for a future quantum Internet. To achieve a practicable generation rate, standard protocols based on photonic qubits require multiple long-term quantum memories, which remains a significant experimental challenge. In this paper, we propose a novel protocol based on 2m-dimensional time-bin photonic qudits that allows for the simultaneous generation of multiple (m) entangled pairs between two distributed qubit registers and we outline a specific implementation of the protocol based on cavity-mediated spin-photon interactions. By adopting the qudit protocol, the required qubit memory time is independent of the transmission loss between the nodes, in contrast to standard qubit approaches. As such, our protocol can significantly boost the performance of near-term quantum networks.

... The long-term goal in the exploration of quantum computation and communication is to conceive the perfectly secure Quantum Internet [9], which is an emerging concept in the landscape of quantum engineering, as portrayed in the stylized illustration of Fig. 2. The concept is reminiscent of that of the classical Internet, interconnecting multiple quantum nodes in the quantum network. The Quantum Internet will facilitate the perfectly secure exchange of quantum information, whilst supporting a plethora of other compelling applications such as distributed quantum computation, blind quantum computation, quantum-secure secret-sharing, and many more [10], [11]. For example, multiple inter-connected quantum computers can jointly act as a distributed quantum computer and can perform more advanced computational tasks than a single quantum computer. ...

Moore’s Law has prevailed since 1965, predicting that the integration density of chips will be doubled approximately every 18 months or so, which has resulted in nanoscale in- tegration associated with 7 nm technologies at the time of writing. At this scale however we are about to enter the transitory range between classical and quantum physics. Based on the brilliant proposition by Feynman a new breed of information bearers was born, where the quantum bits are mapped for example to the spin of an electron. As a benefit, the alluring properties of the nano-scale quantum world have opened up a whole spate of opportunities in signal processing and communications, as discussed in this easy-reading discourse requiring no background in quantum physics.

... The great challenge for quantum communication 1 is how to overcome loss 2 , the dominant source of noise through free space and telecom fibres. Many applications [3][4][5][6][7] , including quantum key distribution (QKD) 8,9 (i.e., the task of sharing a secret random key between two distant parties), suffer from an exponential ratedistance scaling 10,11 . Determining the most efficient protocols for distributing quantum information, entanglement, and secure keys is of vital importance to realise the full capability of the quantum internet 12 . ...

The field of quantum communications promises the faithful distribution of quantum information, quantum entanglement, and absolutely secret keys, however, the highest rates of these tasks are fundamentally limited by the transmission distance between quantum repeaters. The ultimate end-to-end rates of quantum communication networks are known to be achievable by an optimal entanglement distillation protocol followed by teleportation. In this work, we give a practical design for this achievability. Our ultimate design is an iterative approach, where each purification step operates on shared entangled states and detects loss errors at the highest rates allowed by physics. As a simpler design, we show that the first round of iterations can purify completely at high rates. We propose an experimental implementation using linear optics and photon-number measurements which is robust to inefficient operations and measurements, showcasing its near-term potential for real-world practical applications.

... The primary purpose of quantum communication [1,2] is a transfer of quantum states between remote parties. Such transmission is of vital importance for a variety of applications such as secure transfer of classical messages using quantum key distribution (QKD) [3,4], quantum metrology [5][6][7] and distributed computations [8,9] to name few. The QKD is one of the most mature quantum technologies nowadays, but it's direct communication distance is fundamentally limited by losses [10]. ...

We present a scheme of quantum repeater that uses entangled multimode coherent states which are obtained by electro-optic modulation of symmetric and antisymmetric Schr\"odinger cat states. In this method subcarrier modes of the phase modulated states generated by the remote parties are sent to a symmetric beam splitter at the central node. The entangled coherent states are heraldedly prepared by photon counting measurements at the output channels of the beam splitter. We study how the effects of decoherence in the quantum channel affect statistics of photocounts and corresponding fidelity. We show how the proposed scheme can be useful for extending range of quantum key distribution with sub carrier wave encoding by exploiting quantum teleportation with the generated entanglement.

... Quantum networks bring new opportunities for secure communication [1,2], distributed sensing [3][4][5], and distributed quantum computing [6,7]. The ability to faithfully transmit quantum information over long distances is a key prerequisite for the construction of large-scale quantum networks. ...

Generating multiple entangled qubit pairs between distributed nodes is a prerequisite for a future quantum internet. To achieve a practicable generation rate, standard protocols based on photonic qubits require multiple long-term quantum memories, which remains a significant experimental challenge. In this paper, we propose a novel protocol based on $2^m$-dimensional time-bin photonic qudits that allow for the simultaneous generation of multiple ($m$) entangled pairs between two distributed qubit registers and outline a specific implementation of the protocol based on cavity-mediated spin-photon interactions. By adopting the qudit protocol, the required qubit memory time is independent of the transmission loss between the nodes in contrast to standard qubit approaches. As such, our protocol can significantly boost the performance of near-term quantum networks.

... However, one of the main goals of quantum communication is to provide a worldwide connectivity via complex networks -similar to the current internet -with an ultimate security based on the laws of physics rather than computational complexity [3]. The Quantum Internet vision is moving towards interconnecting seamlessly multiple nodes and enabling applications beyond QKD such as blind and distributed quantum computing [4]. For such applications to reach their maximum capabilities, a seamless medium allowing the coexistence of high power classical channels with quantum channels would be beneficial for the integration of quantum technologies with the current optical infrastructure. ...

The feasibility of coexisting a quantum channel with carrier-grade classical optical channels over Hollow Core Nested Antiresonant Nodeless Fibre (HC-NANF) is experimentally explored for the first time in terms of achievable quantum bit error rate (QBER), secret key rate (SKR) as well as classical signal bit error rates (BER). A coexistence transmission of 1.6 Tbps is achieved for the classical channels simultaneously with a quantum channel over a 2 km-long HC-NANF with a total coexistence power of 0 dBm. To find the best and worst wavelength position for the classical channels, we simulated different classical channels bands with different spacing between the quantum and classical channels considering the crosstalk generated from both Raman scattering and four-wave-mixing (FWM) on the quantum channel. Following our simulation, we numerically estimate the best (Raman spectrum dip) and worst locations (Raman spectrum peak) of the classical channel with respect to its impact on the performance on the quantum channel in terms of SKR and QBER. We further implemented a testbed to experimentally test both single-mode fibre (SMF) and HC-NANF in the best and worst-case scenarios. In the best-case scenario, the spacing between quantum and classical is 200 GHz (1.6 nm) with 50 GHz (0.4 nm) spacing between each classical channel. The SKR was preserved without any noticeable changes when coexisting the quantum channel with eight classical channels at 0 dBm total coexistence power in HC-NANF compared to a significant drop of 73% when using SMF at
$-$
24 dBm total coexistence power which is 250 times lower than the power used in HC-NANF. In the worst-case scenario using the same powers, and with 1 THz (8 nm) spacing between quantum and classical channels, the SKR dropped 10% using the HC-NANF, whereas in the SMF the SKR plummeted to zero.

... An emerging focus has been drawn to architecting the quantum internet [2,3]: a global network built using quantum repeaters [4,5] and satellites [6] to distribute entanglement at high rates to multiple distant users ondemand [7][8][9]. There are several well-known applications of shared entanglement, a new information currency: distributed quantum computing [10], communications with physics-based security [11], provably-secure access to quantum computers on the cloud [12], and entanglement-enhanced distributed sensing [13][14][15][16]. In this paper, we design a system for another impactful application of shared entanglement: substantially improving classical communication rates in certain regimes. ...

Pre-shared entanglement can significantly boost communication rates in the high thermal noise and low-brightness transmitter regime. In this regime, for a lossy-bosonic channel with additive thermal noise, the ratio between the entanglement-assisted capacity and the Holevo capacity - the maximum reliable-communications rate permitted by quantum mechanics without any pre-shared entanglement - scales as $\log(1/{\bar N}_{\rm S})$, where the mean transmitted photon number per mode, ${\bar N}_{\rm S} \ll 1$. Thus, pre-shared entanglement, e.g., distributed by the quantum internet or a satellite-assisted quantum link, promises to significantly improve low-power radio-frequency communications. In this paper, we propose a pair of structured quantum transceiver designs that leverage continuous-variable pre-shared entanglement generated, e.g., from a down-conversion source, binary phase modulation, and non-Gaussian joint detection over a code word block, to achieve this scaling law of capacity enhancement. Further, we describe a modification to the aforesaid receiver using a front-end that uses sum-frequency generation sandwiched with dynamically-programmable in-line two-mode squeezers, and a receiver back-end that takes full advantage of the output of the receiver's front-end by employing a non-destructive multimode vacuum-or-not measurement to achieve the entanglement-assisted classical communications capacity.

... Applications on NISQ devices [20] are typically hindered by finite coherence times and insufficient gate precision, as well as by the overall small number of available qubits. A crucial step in surpassing the size limitation of current-generation quantum computers will be the transition to distributed quantum computers [21,22]. Similar to the limitations of monolithic NISQ devices, however, near-term interconnect hardware that is required for the facilitation of quantum gates across distributed quantum processing units (QPUs, c.f. Fig. 1) is also facing challenges. ...

We study a variation of the Trotter-Suzuki decomposition, in which a Hamiltonian exponential is approximated by an ordered product of two-qubit operator exponentials such that the Trotter step size is enhanced for a small number of terms. Such decomposition directly reflects hardware constraints of distributed quantum computers, where operations on monolithic quantum devices are fast compared to entanglement distribution across separate nodes using interconnects. We simulate non-equilibrium dynamics of transverse-field Ising and XY spin chain models and investigate the impact of locally increased Trotter step sizes that are associated with an increasingly sparse use of the quantum interconnect. We find that the overall quality of the approximation depends smoothly on the local sparsity and that the proliferation of local errors is slow. As a consequence, we show that fast local operations on monolithic devices can be leveraged to obtain an overall improved result fidelity even on distributed quantum computers where the use of interconnects is costly.

... In fact, the different mechanism of quantum mechanics introduce new challenge in the network design due to the network's needs to cater quantum phenomenon and to abide by the law of quantum mechanics [26]. Distributed quantum computation has been discussed and analyzed by few, such as [27], and some of its form has been proposed to be used to run Shor's algorithm, such as in [28], [29]. Additionally, the initiative to establish quantum internet, mentioned as the ultimate phase in the quantum revolution [26], hope to achieve a network that can interconnects remote quantum devices through quantum links in synergy with the classical ones. ...

As the development of quantum computing hardware is on the rise, its potential application to various research areas has been investigated, including to machine learning. Recently, there have been several initiatives to expand the work to quantum federated learning (QFL). However, challenges arise due to the fact that quantum computation poses different characteristics from classical computation, giving an even more challenge for a federated setting. In this paper, we present a high-level overview of the current state of research in QFL. Furthermore, we also describe in brief about quantum computation and discuss its present limitations in relation to QFL development. Additionally, possible approaches to deploy QFL are explored. Lastly, remarks and challenges of QFL are also presented.

... 6. No geometrical constraint: As in photonic systems [57,[66][67][68][69] and distributed architectures [70][71][72], we assume that two-qubit gates are applicable to any pair of physical qubits without geometrical constraint. Classical computation can also be performed without such geometrical constraint. ...

Scalable realization of quantum computation to attain substantial speedups over classical computation requires fault tolerance. Conventionally, fault-tolerant quantum computation (FTQC) demands excessive space overhead of physical qubits per logical qubit. A more recent protocol to achieve constant-space-overhead FTQC using quantum low-density parity-check (LDPC) codes thus attracts considerable attention but suffers from another drawback: it incurs polynomially long time overhead. We construct an alternative avenue of FTQC to achieve constant space overhead and only quasi-polylogarithmic time overhead simultaneously via techniques for using concatenations of different quantum Hamming codes rather than quantum LDPC codes. Our protocol achieves FTQC even if a decoder has nonzero runtime, unlike the existing constant-space-overhead protocol. These results establish a foundation for realizing FTQC within feasibly bounded space overhead yet negligibly tiny slowdown.

... One of the great promises of quantum technology is the development of quantum networks, which will allow global distribution of entanglement for tasks such as distributed quantum computing [1,2], distributed quantum sensing [3,4] and quantum-secure communication [5][6][7][8]. To leverage the full potential of quantum networks we require protocols that draw an efficiency advantage from genuine multi-partite entanglement as opposed to strictly pair-wise correlations such as Bell states. ...

One of the great promises of quantum technology is the development of quantum networks, which will allow global distribution of entanglement for tasks such as distributed quantum computing, distributed quantum sensing and quantum-secure communication. To leverage the full potential of quantum networks we require protocols that draw an efficiency advantage from genuine multi-partite entanglement as opposed to strictly pair-wise correlations such as Bell states. Multi-user entanglement such as Greenberger-Horne-Zeilinger (GHZ) states have already found application in quantum conference key agreement, quantum secret sharing and quantum communication complexity problems. However, a true network advantage has not yet been achieved. In this work we create a six-photon graph-state network from which we derive either a four-user GHZ state for direct quantum conference key agreement or the required amount of Bell pairs for the equivalent pair-wise protocol. We show that the GHZ-state protocol has a more than two-fold rate advantage by only consuming half the amount of network resources per secure conference key bit.

... The emergence of terrestrial quantum networks in large metropolitan areas demonstrates an increasing maturity of quantum technologies. A networked infrastructure enables increased capabilities for distributed applications in delegated quantum computing [1,2], quantum communications [3,4], and quantum sensing [5]. However, extending these applications over global scales is currently not possible owing to exponential losses in optical fibres. ...

In satellite-based quantum key distribution (QKD), the number of secret bits that can be generated in a single satellite pass over the ground station is severely restricted by the pass duration and the free-space optical channel loss. High channel loss may decrease the signal-to-noise ratio due to background noise, reduce the number of generated raw key bits, and increase the quantum bit error rate (QBER), all of which have detrimental effects on the output secret key length. Under finite-size security analysis, higher QBER increases the minimum raw key length necessary for non-zero key length extraction due to less efficient reconciliation and post-processing overheads. We show that recent developments in finite key analysis allow three different small-satellite-based QKD projects CQT-Sat, UK-QUARC-ROKS, and QEYSSat to produce secret keys even under very high loss conditions, improving on estimates based on older finite key bounds.

... The great challenge for quantum communication [1] is how to overcome loss [2], the dominant source of noise through free space and telecom fibers. Many applications [3][4][5][6][7], including quantum key distribution (QKD) [8,9] (i.e., the task of sharing a secret random key between two distant parties), suffer from an exponential rate-distance scaling [10]. Determining the most efficient protocols for distributing quantum information, entanglement, and secure keys is of vital importance to realise the full capability of the quantum internet [11]. ...

The field of quantum communications promises the faithful distribution of quantum information, quantum entanglement, and absolutely secret keys. The highest rates of these tasks are fundamentally limited by the transmission distance between quantum repeaters. Previous protocols are unable to saturate the ultimate rates, which motivates our paper. We use entanglement purification or equivalently, quantum error detection, to remove the effects of the bosonic pure-loss channel, and prove the highest rates of our protocol saturate the ultimate limits. By simplifying our protocol, we show that states can be purified completely in a single attempt at rates which are close to the ultimate limits. We propose a linear-optics implementation which is robust to inefficient operations and measurements, showcasing its near-term potential for real-world practical applications.

... However, one of the main goals of quantum communication is to provide a worldwide connectivity via complex networkssimilar to the current internet -with an ultimate security based on the laws of physics rather than computational complexity [3]. The Quantum Internet vision is moving towards interconnecting seamlessly multiple nodes and enabling applications beyond QKD such as blind and distributed quantum computing [4]. For such applications to reach their maximum capabilities, a seamless medium allowing the coexistence of high power classical channels with quantum channels would be beneficial for the integration of quantum technologies with the current optical infrastructure. ...

The feasibility of coexisting a quantum channel with carrier-grade classical optical channels over Hollow Core Nested Antiresonant Nodeless Fibre (HC-NANF) is experimentally explored for the first time in terms of achievable quantum bit error rate (QBER), secret key rate (SKR) as well as classical signal bit error rates (BER). A coexistence transmission of 1.6 Tbps is achieved for the classical channels simultaneously with a quantum channel over a 2 km-long HC-NANF with a total coexistence power of 0 dBm. To find the best and worst wavelength position for the classical channels, we simulated different classical channels bands with different spacing between the quantum and classical channels considering the crosstalk generated from both Raman scattering and four-wave-mixing (FWM) on the quantum channel. Following our simulation, we numerically estimate the best (Raman spectrum dip) and worst locations (Raman spectrum peak) of the classical channel with respect to its impact on the performance on the quantum channel in terms of SKR and QBER. We further implemented a testbed to experimentally test both single mode fibre (SMF) and HC-NANF in the best and worst-case scenarios. In the best-case scenario, the spacing between quantum and classical is 200 GHz (1.6 nm) with 50 GHz (0.4 nm) spacing between each classical channel. The SKR was preserved without any noticeable changes when coexisting the quantum channel with eight classical channels at 0 dBm total coexistence power in HC-NANF compared to a significant drop of 73% when using SMF at -24 dBm total coexistence power which is 250 times lower than the power used in HC-NANF. In the worst-case scenario using the same powers, and with 1 THz (8 nm) spacing between quantum and classical channels, the SKR dropped 10% using the HC-NANF, whereas in the SMF the SKR plummeted to zero.

... Collective effects via the interaction between quantum emitters and reservoirs have attracted much attention as a consequence of quantum mechanics over the years and are now an indispensable part of modern technology. Especially, extensive efforts have been made to understand collective dynamics of atoms and traveling photons, which leads to novel opportunities for quantum networks and quantum information processing [1][2][3]. The recent excitement of the field is due largely to the remarkable progress of the waveguide quantum electrodynamics (wQED) technology [4][5][6][7]; it enables atoms couple directly to a waveguide, which differs strongly from the cavity-QED approaches of confining photons in all spatial directions. ...

We investigate the stimulated emission of superradiant atoms coupled to a waveguide induced by a coherent-state photon pulse. We provide an analytical result when a short $\pi$ pulse is incident, which shows that the atoms emit photons coherently into the output pulse, which remains a coherent state in the short pulse limit. An incident pulse is amplified in phase-preserving manner, where noise is added almost entirely in the phase direction in phase space. This property improves the ratio of intensity signal to noise after the amplification for sufficiently short pulses. This is a unique feature different from general phase-preserving linear amplifiers, where the signal-to-noise ratio deteriorates in the amplification process. We also discuss the dependence of the photon-emission probability on pulse parameters, such as the pulse area and the duration.

... Although quantum memories may alleviate this requirement by several orders of magnitude [4], tens of thousands of qubits are still needed. To mitigate the severe demands concerning the integration of qubits, distributed quantum computing is suggested as an alternative [5][6][7]. Similar to a distributed classical computer, a distributed quantum computer is realized by linking small quantum computers using classical and quantum communication channels. ...

Quantum communication between remote superconducting systems is being studied intensively to increase the number of integrated superconducting qubits and to realize a distributed quantum computer. Since optical photons must be used for communication outside a dilution refrigerator, the direct conversion of microwave photons to optical photons has been widely investigated. However, the direct conversion approach suffers from added photon noise, heating due to a strong optical pump, and the requirement for large cooperativity. Instead, for quantum communication between superconducting qubits, we propose an entanglement distribution scheme using a solid-state spin quantum memory that works as an interface for both microwave and optical photons. The quantum memory enables quantum communication without significant heating inside the refrigerator, in contrast to schemes using high-power optical pumps. Moreover, introducing the quantum memory naturally makes it possible to herald entanglement and parallelization using multiple memories.

... In order to move qubits between any two parties over long distances, quantum state teleportation is preferred to quantum communication, whose fidelity decreases exponentially with the channel length, due to loss [27]. Quantum teleportation requires end-to-end entanglement generation, i.e., probably the most important general-purpose service in the future Quantum Internet [28][29][30][31][32][33]. In Figure 6, an example of quantum network stack architecture, inspired by the TCP/IP one, is presented [34,35]. ...

Private set intersection is an important problem with implications in many areas, ranging from remote diagnostics to private contact discovery. In this work, we consider the case of two-party PSI in the honest-but-curious setting. We propose a protocol that solves the server-aided PSI problem using delegated blind quantum computing. More specifically, the proposed protocol allows Alice and Bob (who do not have any quantum computational resources or quantum memory) to interact with Steve (who has a quantum computer) in order for Alice and Bob to obtain set intersection such that privacy is preserved. In particular, Steve learns nothing about the clients' input, output, or desired computation. The proposed protocol is correct, secure and blind against a malicious server, and characterized by a quantum communication complexity that is linear in the input size.

... According to [1] the quantum applications involve algorithms ranging from quantum chemistry to astrophysics to machine learning-relevant matrix operations. The quantum approaches suite to applications that involves sets of discrete states modeled by the superposition of a number of qubits. ...

... With each computational step, entanglement is consumed by the subsequent measurement of qubits in a certain direction. Cluster state provides MQC with multiple advantages over the circuit-based model including decomposing large computational space into steps by re-utilizing the cluster [143], concatenation of gate simulations [144], distributed [145][146][147], loss-tolerant [148,149], and fault-tolerant [150,151] quantum computation. Apart from MQC, cluster state remains an efficient resource for quantum repeaters, as discussed previously. ...

Color centers in wide band gap semiconductors are prominent candidates for solid state quantum technologies due to their attractive properties - optical interfacing, long coherence, spin-photon and spin-spin entanglement, and scalability. Silicon carbide color centers integrated into photonic devices span a wide range of applications in quantum information processing in a material platform with advanced availability and processing capabilities. Recent progress in emitter generation and characterization, nanofabrication, device design and quantum optical studies have amplified the scientific interest in this platform. We discuss the promising directions in the development of quantum networking, simulation and computing hardware using silicon carbide integrated photonics.

... Quantum networking has mainly developed over the last two decades as relayed point-to-point quantum communication links [1], with primary application on quantum security and quantum key distribution (QKD) [2]. However, to address the Quantum Internet vision towards interconnecting seamlessly multiple quantum nodes and enable applications beyond QKD such as blind and distributed quantum computing [3], fully dynamic quantum networks [4] need to be deployed overcoming the relayed quantum links approach and allowing the co-existence of classical and quantum channels over the same fibre. Entanglement distribution has recently proved a very powerful technology for dynamic quantum networking, and demonstrations of static [5,6] and active [7] entanglement distribution concepts has opened the road for the deployment of fully functional quantum networks that allow simultaneous multi-point quantum nodes interconnection. ...

We experimentally prove the feasibility of wavelength multiplexing bright 100 Gbps classical communication with multiple single-photon level entanglement channels over SMF-28e fibre in a quantum network. This minimises the resources needed for quantum networks.

... Thus A and B can share an EPR pair. 1 A delicate step of a repeater-based quantum channel is that the Pauli correction for teleportation can be deferred. For example, suppose that a qubit is to be sent from node A to B via a router R. Originally, A sends R the binary measurement outcome m 1 and then R does a Pauli correction according to m 1 . ...

A quantum network, which involves multiple parties pinging each other with quantum messages, could revolutionize communication, computing and basic sciences. The future internet will be a global system of various packet switching quantum and classical networks and we call it quantum internet. To build a quantum internet, unified protocols that support the distribution of quantum messages within it are necessary. Intuitively one would extend classical internet protocols to handle quantum messages. However, classical network mechanisms, especially those related to error control and reliable connection, implicitly assume that information can be duplicated, which is not true in the quantum world due to the no-cloning theorem and monogamy of entanglement. In this paper, we investigate and propose protocols for packet quantum network intercommunication. To handle the packet loss problem in transport, we propose a quantum retransmission protocol based on the recursive use of a quantum secret sharing scheme. Other internet protocols are also discussed. In particular, the creation of the logical process-to-process connections is accomplished by a quantum version of the three-way handshake protocol.

... It depends upon the technology used for qubits. Qubits realized with superconducting circuits, exhibits 100 micro-seconds of decoherence time [317], and a much larger decoherence time has been reported with trapped ions [318]. Reliability of a teleportation system is measured with quantum fidelity which is considered as the fundamental figure of merit, and larger the imperfections introduced by decoherence, the lower is the fidelity. ...

The advanced notebooks, mobile phones, and internet applications in today’s world that we use are all entrenched in classical communication bits of zeros and ones. Classical internet has laid its foundation originating from the amalgamation of mathematics and Claude Shannon’s theory of information. However, today’s internet technology is a playground for eavesdroppers. This poses a serious challenge to various applications that rely on classical internet technology, and it has motivated the researchers to switch to new technologies that are fundamentally more secure. By exploring the quantum effects, researchers paved the way into quantum networks that provide security, privacy, and range of capabilities such as quantum computation, communication, and metrology. The realization of Quantum Internet (QI) requires quantum communication between various remote nodes through quantum channels guarded by quantum cryptographic protocols. Such networks rely upon quantum bits (qubits) that can simultaneously take the value of zeros and ones. Due to the extraordinary properties of qubits such as superposition, entanglement, and teleportation, it gives an edge to quantum networks over traditional networks in many ways. At the same time, transmitting qubits over long distances is a formidable task and extensive research is going on satellite-based quantum communication, which will deliver breakthroughs for physically realizing QI in near future. In this paper, QI functionalities, technologies, applications and open challenges have been extensively surveyed to help readers gain a basic understanding of the infrastructure required for the development of the global QI.

... On the other hand, we are aware that the performed evaluation is not exhaustive and further work is necessary to fully characterize the proposed approach, for example in designing a reasonable method for tuning the α parameter in Eq. 1. In particular, we plan to extend the evaluation to circuits and devices with more qubits, and to distributed quantum computing architectures as well [20,21,22]. As a final remark, it is worth noting that it is possible to further mitigate the effect of noise by using an ensemble of diverse mappings (EDM) approach, as suggested by Tannu and Qureshi [23]. ...

Quantum compilation is the problem of translating an input quantum circuit into the most efficient equivalent of itself, taking into account the characteristics of the device that will execute the computation. Compilation strategies are composed of sequential passes that perform placement, routing and optimization tasks. Noise-adaptive compilers do take the noise statistics of the device into account, for some or all passes. The noise statics can be obtained from calibration data, and updated after each device calibration. In this paper, we propose a novel noise-adaptive compilation strategy that is computationally efficient. The proposed strategy assumes that the quantum device coupling map uses a heavy-hexagon lattice. Moreover, we present the application-motivated benchmarking of the proposed noise-adaptive compilation strategy, compared with some of the most advanced state-of-art approaches. The presented results seem to indicate that our compilation strategy is particularly effective for deep circuits and for square circuits.

... However, ultimately the goal of quantum communication is to provide a worldwide connectivity via complex networkssimilar to the current internet -with an ultimate security based on the on the laws of physics rather than computational complexity [3]. The Quantum Internet vision is moving towards interconnecting seamlessly multiple nodes and enable applications beyond QKD such as blind and distributed quantum computing [4]. For such applications to reach their maximum capabilities, fully dynamic quantum networks [5] need to be deployed overcoming the relayed point-to-point quantum links approach and allowing the coexistence of classical and quantum channels over the same fibre in optical networks with the ability to switch between different nodes providing multihops connectivity. ...

We demonstrate for the first time a trusted-node-free fully-meshed metro network with dynamic discrete-variable quantum key distribution DV-QKD networking capabilities across four optical network nodes. A QKD-aware centralised SDN controller is utilised to provide dynamicity in switching and rerouting. The feasibility of coexisting a quantum channel with carrier-grade classical optical channels over field-deployed and laboratory-based fibres is experimentally explored in terms of achievable quantum bit error rate, secret key rate as well as classical signal bit error rate. Moreover, coexistence analysis over multi-hops configuration using different switching scenarios is also presented. The secret key rate dropped 43 % when coexisting one classical channel with 150 GHz spacing from the quantum channel for multiple links. This is due to the noise leakage from the Raman scattering into the 100 GHz bandwidth of the internal filter of the Bob DV-QKD unit. When coexisting four classical channels with 150 GHz spacing between quantum and the nearest classical channel, the quantum channel deteriorates faster due to the combination of Raman noise, other nonlinearities and high aggregated launch power causing the QBER value to exceed the threshold of 6 % leading the SKR to reach a value of zero bps at a launch power of 7 dB per channel. Furthermore, the coexistence of a quantum channel and six classical channels through a field-deployed fibre Test Network is examined.

... We further note that other recent demonstrations in the TFLN platform include densely integrated and re-configurable linear optical circuit elements [106,107], highly efficient electro-optic modulators [108,109], and integrated single photon detectors [110]. Thus, alongside source engineering, this platform has many of the components needed to implement an on-chip linear optical quantum computer [111,112]. ...

This article reviews recent progress in quasi-phasematched X⁽²⁾nonlinear nanophotonics, with a particular focus on dispersion-engineered nonlinear interactions. Throughout this article, we establish design rules for the bandwidth and interaction lengths of various nonlinear processes, and provide examples for how these processes can be engineered in nanophotonic devices. In particular, we apply these rules towards the design of sources of non-classical light and show that dispersion-engineered devices can outperform their conventional counterparts. Examples include ultra-broadband optical parametric amplification as a resource for measurement-based quantum computation, dispersion-engineered spontaneous parametric downconversion as a source of separable biphotons, and synchronously pumped nonlinear resonators as a potential route towards single-photon nonlinearities.

... Q uantum communication can be used to connect distant quantum devices into a quantum network. At short distances, networking quantum devices provides a path towards a scalable distributed quantum computer 1 . At larger distances, interconnected quantum networks allow for communication tasks between distant users on a quantum internet. ...

In order to bring quantum networks into the real world, we would like to determine the requirements of quantum network protocols including the underlying quantum hardware. Because detailed architecture proposals are generally too complex for mathematical analysis, it is natural to employ numerical simulation. Here we introduce NetSquid, the NETwork Simulator for QUantum Information using Discrete events, a discrete-event based platform for simulating all aspects of quantum networks and modular quantum computing systems, ranging from the physical layer and its control plane up to the application level. We study several use cases to showcase NetSquid’s power, including detailed physical layer simulations of repeater chains based on nitrogen vacancy centres in diamond as well as atomic ensembles. We also study the control plane of a quantum switch beyond its analytically known regime, and showcase NetSquid’s ability to investigate large networks by simulating entanglement distribution over a chain of up to one thousand nodes.

We investigate the stimulated emission of superradiant atoms coupled to a waveguide induced by a coherent-state photon pulse. We provide an analytical result when a short π pulse is incident, which shows that the atoms emit photons coherently into the output pulse, which remains a coherent state in the short pulse limit. An incident pulse is amplified in a phase-preserving manner, where noise is added almost entirely in the phase direction in phase space. This property improves the ratio of the intensity signal to noise after the amplification for sufficiently short pulses. This is a unique feature different from general phase-preserving linear amplifiers, where the signal-to-noise ratio deteriorates in the amplification process. We also discuss the dependence of the photon-emission probability on pulse parameters, such as the pulse area and the duration.

In quantum networks an important goal is to reduce resource requirements for the transport and communication of quantum information. Quantum network coding presents a way of doing this by distributing entangled states over a network that would ordinarily exhibit contention. In this work, we study measurement-based quantum network coding (MQNC), which is a protocol particularly suitable for noisy intermediate-scale quantum devices. MQNC has previously been studied experimentally on a superconducting processor, however the resulting states did not have a usable degree of entanglement. We adapt MQNC to the newer superconducting processor ibm_cairo and obtain a much improved degree of entanglement, enabling us to demonstrate successful teleportation of quantum information. The teleportation is shown to occur with fidelity higher than could be achieved via classical means, made possible by considering qubits from a polar cap of the Bloch Sphere. We also present a generalization of MQNC with a simple mapping onto the heavy-hex processor layout and a direct mapping onto a proposed logical error-corrected layout. Our work provides some useful techniques for testing and successfully carrying out quantum network coding.

This chapter discusses how to build a quantum network. Basic concepts of quantum communication networks and the quantum Internet are introduced, followed by quantum teleportation and relay concepts. Then entanglement distribution, a key technique that enables quantum networking by employing quantum teleportation concepts, is described, starting with a brief description of how to generate entangled states. Entanglement swapping, Bell state measurements, and the Hong–Ou–Mendel effect are described, followed by continuous-variable (CV) quantum teleportation and then quantum network coding basics. The section after that is devoted to the engineering of entangled states by photon addition and subtraction. After describing photon subtraction and addition concepts, we discuss how to apply them to build a hybrid CV–discrete-variable (DV) quantum network. Hybrid quantum networks are described in terms of hybrid DV–CV entangled states as well as state teleportation and entanglement swapping through entangling measurements. The generation of entangled macroscopic light states is then discussed, given that macroscopic light states are much more tolerant of channel loss than discrete-variable states. The section ends with noiseless amplification as a possible approach to extend transmission distances between quantum nodes. The focus of the chapter then moves to cluster state-based quantum networking. The cluster states concept is introduced first, followed by cluster state processing. The same section describes cluster state-based quantum networking as both quantum Internet and distributed quantum computing enabling technology. The section that follows describes surface code-based and quantum low-density parity-check code-based quantum networking concepts. The final section relates to entanglement-assisted communication networks.

In this chapter, we provide an overview of quantum communications (QuComs), quantum error correction (QEC), quantum networks, and quantum sensing (QuSen) and discuss their relevance. We then discuss the basic principles of QuComs, QEC, QuNets, and QuSen. The last section of the chapter is devoted to the organization of the book, with a detailed description of each chapter.

We demonstrate for the first time a four-node trusted-node-free metro network configuration with dynamic discrete-variable quantum key distribution DV-QKD networking capabilities across four optical network nodes. The network allows the dynamic deployment of any QKD link between two nodes of the network, while a QKD-aware centralised software-defined networking (SDN) controller is utilised to provide dynamicity in switching and rerouting. The feasibility of coexisting a quantum channel with carrier-grade classical optical channels where both the quantum and classical channels are in the C-band over field-deployed metropolitan networks and laboratory-based fibres (<10 km) is experimentally explored in terms of achievable quantum bit error rate, secret key rate as well as classical signal bit error rate. Moreover, coexistence analysis over multi-hops configuration using different switching scenarios is also presented. The secret key rate dropped 43% when coexisting one classical channel with 150 GHz spacing from the quantum channel for multiple links. This is due to the noise leakage from the Raman scattering into the 100 GHz bandwidth of the internal filter of the Bob DV-QKD unit. When coexisting four classical channels with 150 GHz spacing between quantum and the nearest classical channel, the quantum channel deteriorates faster due to the combination of Raman noise, other nonlinearities and high aggregated launch power causing the QBER value to exceed the threshold of 6% leading the SKR to reach a value of zero bps at a launch power of 7 dB per channel. Furthermore, the coexistence of a quantum channel and six classical channels through a field-deployed fibre test network is examined.

Color centers in wide bandgap semiconductors are prominent candidates for solid-state quantum technologies due to their attractive properties including optical interfacing, long coherence times, and spin–photon and spin–spin entanglement, as well as the potential for scalability. Silicon carbide color centers integrated into photonic devices span a wide range of applications in quantum information processing in a material platform with quantum-grade wafer availability and advanced processing capabilities. Recent progress in emitter generation and characterization, nanofabrication, device design, and quantum optical studies has amplified the scientific interest in this platform. We provide a conceptual and quantitative analysis of the role of silicon carbide integrated photonics in three key application areas: quantum networking, simulation, and computing.

The high-fidelity storage, distribution, and processing of quantum information prefers qubits with different physical properties. Thus hybrid quantum gates interfacing different types of qubits are essential for the realization of complex quantum network structures. A Rydberg-atom-based physical quantum controlled-Z (CZ) gate is proposed to hybridly process the polarization-encoded single-photon optical qubit and the “Schrödinger cat” microwave qubit. The degradation of the fidelity under the influence of various noise channels, such as microwave cavity loss, spontaneous emission of atom states, and nonadiabaticity effect, etc., has been analyzed through detailed theoretical analysis by deriving the input-output relation of qubit fields. The feasibility and the challenges of the protocol within current technology are also discussed by analyzing the possible experimental parameter settings. Our work opens prospects for future large-scale quantum architecture based on hybrid quantum modules.

Quantum communications capacity using direct transmission over length-L optical fiber scales as R∼e−αL, where α is the fiber's loss coefficient. The rate achieved using a linear chain of quantum repeaters equipped with quantum memories, probabilistic Bell state measurements (BSMs), and switches used for spatial multiplexing, but no quantum error correction, was shown to surpass the direct-transmission capacity. However, this rate still decays exponentially with the end-to-end distance, viz., R∼e−sαL, with s<1. We show that the introduction of temporal multiplexing—i.e., the ability to perform BSMs among qubits at a repeater node that were successfully entangled with qubits at distinct neighboring nodes at different time steps—leads to a subexponential rate-vs-distance scaling, i.e., R∼e−tαL, which is not attainable with just spatial or spectral multiplexing. We evaluate analytical upper and lower bounds to this rate and obtain the exact rate by numerically optimizing the time-multiplexing block length and the number of repeater nodes. We further demonstrate that incorporating losses in the optical switches used to implement time multiplexing degrades the rate-vs-distance performance, eventually falling back to exponential scaling for very lossy switches. We also examine models for quantum memory decoherence and describe optimal regimes of operation to preserve the desired boost from temporal multiplexing. Quantum memory decoherence is seen to be more detrimental to the repeater's performance over switching losses.

We consider the task of faithfully simulating a distributed quantum measurement, wherein we provide a protocol for the three parties, Alice, Bob and Charlie, to simulate a repeated action of a distributed quantum measurement using a pair of non-product approximating measurements by Alice and Bob, followed by a stochastic mapping at Charlie. The objective of the protocol is to utilize minimum resources, in terms of classical bits needed by Alice and Bob to communicate their measurement outcomes to Charlie, and the common randomness shared among the three parties, while faithfully simulating independent repeated instances of the original measurement. To achieve this, we develop a mutual covering lemma and a technique for random binning of distributed quantum measurements, and, in turn, characterize a set of sufficient communication and common randomness rates required for asymptotic simulatability in terms of single-letter quantum information quantities. In the special case, where the Charlie’s action is restricted to a deterministic mapping, we develop a one-shot performance characterization of the distributed faithful simulation problem. Furthermore, using these results we address a distributed quantum rate-distortion problem, where we characterize the achievable rate distortion region through a single-letter inner bound. Finally, via a technique of single-letterization of multi-letter quantum information quantities, we provide an outer bound for the rate-distortion region.

By harnessing unique quantum mechanical phenomena, such as superposition and entanglement, quantum computers offer the possibility to drastically outperform classical computers for certain classes of problems. The realization of this potential, however, presents a substantial challenge, because noise and imperfections associated with the materials used to fabricate devices can obscure the delicate quantum mechanical effects that enable quantum computing. Hence, progress in synthesis, characterization, and modeling of materials for quantum computing have driven many exciting advances in recent years and will become increasingly important in the years to come. As progressively more complex, multi-qubit systems come online, and as significant government and industrial investment drives research forward, new challenges and opportunities for materials science continue to emerge. The articles in this issue survey the current state of materials science progress and obstacles for some leading quantum computing platforms; opportunities for deeper involvement by materials scientists abound. Ultimate realization of the full potential of quantum computers will require a multidisciplinary effort spanning many traditional areas of expertise.Graphic abstract

Spin-based, quantum-photonics promise to realize distributed quantum computing and quantum networks. The performance depends on an efficient entanglement distribution where cavity quantum electrodynamics could boost the efficiency. The central challenge is the development of compact devices with large spin-photon coupling rates and a high operation bandwidth. Photonic crystal cavities comprise strong field confinement but require highly accurate positioning of atomic systems in mode field maxima. Negatively charged silicon-vacancy centers in diamond emerged as promising atom-like systems. Spectral stability and access to long-lived, nuclear-spin memories enabled elementary demonstrations of quantum network nodes, including memory-enhanced quantum communication. In a hybrid approach, we deterministically place SiV-containing nanodiamonds inside one hole of one-dimensional, freestanding, Si3N4-based photonic crystal cavities and coherently couple individual optical transitions to cavity modes. We optimize light–matter coupling utilizing two-mode composition, waveguiding, Purcell-enhancement, and cavity-resonance tuning. The resulting photon flux increases by 14 compared to free space. Corresponding lifetime-shortening below 460 ps puts potential operation bandwidth beyond GHz rates.

We propose a novel scheme of quantum parallel teleportation for arbitrary unknown multi-qudit states under the principal of flow split. With the assistance of the central server and the pre-shared entanglement distributed between the intermediate nodes in each path, any arbitrary unknown multi-qudit state can be successfully restored by the final receiver via parallel transmission through various paths. Due to parallel teleportation with distinct paths, this quantum network teleportation can resist network congestion. In addition, only two-qudit measurements are required by the intermediate nodes and the sender node and only one-qudit unitary operation is required by the destination node which reduces the technical requirements of network nodes and makes our protocol more practical and flexible. The analysis also demonstrates that our propose scheme can achieve a good performance in terms of communication cost and time delay. Finally, the unknown states of multiple qudits with hybrid dimensions could be teleported via quantum channels with different dimensions.

The negatively-charged nitrogen-vacancy colour centre in diamond has long been identified as a platform for quantum computation. However, despite beautiful proof of concept experiments, a pathway to true scalability has proven elusive. Now a group from Oxford and Grenoble-Alpes have shown coupling between nitrogen-vacancy centres and open Fabry-Perot cavities in a way that proves a clear route to scalable quantum computing (Johnson et al 2015 New J. Phys.
17 122003). And all at the relatively balmy temperature of 77 K.

Quantum computers are designed to outperform standard computers by running
quantum algorithms. Areas in which quantum algorithms can be applied include
cryptography, search and optimisation, simulation of quantum systems, and
solving large systems of linear equations. Here we briefly survey known quantum
algorithms, with an emphasis on a broad overview of their applications rather
than their technical details. We include a discussion of recent developments
and near-term applications of quantum algorithms.

The exceptionally long quantum coherence times of phosphorus donor nuclear spin qubits in silicon, coupled with the proven scalability of silicon-based nano-electronics, make them attractive candidates for large-scale quantum computing. However, the high threshold of topological quantum error correction can only be captured in a two-dimensional array of qubits operating synchronously and in parallel-posing formidable fabrication and control challenges. We present an architecture that addresses these problems through a novel shared-control paradigm that is particularly suited to the natural uniformity of the phosphorus donor nuclear spin qubit states and electronic confinement. The architecture comprises a two-dimensional lattice of donor qubits sandwiched between two vertically separated control layers forming a mutually perpendicular crisscross gate array. Shared-control lines facilitate loading/unloading of single electrons to specific donors, thereby activating multiple qubits in parallel across the array on which the required operations for surface code quantum error correction are carried out by global spin control. The complexities of independent qubit control, wave function engineering, and ad hoc quantum interconnects are explicitly avoided. With many of the basic elements of fabrication and control based on demonstrated techniques and with simulated quantum operation below the surface code error threshold, the architecture represents a new pathway for large-scale quantum information processing in silicon and potentially in other qubit systems where uniformity can be exploited.

Recent progress in quantum information has led to the start of several large
national and industrial efforts to build a quantum computer. Researchers are
now working to overcome many scientific and technological challenges. The
program's biggest obstacle, a potential showstopper for the entire effort, is
the need for high-fidelity qubit operations in a scalable architecture. This
challenge arises from the fundamental fragility of quantum information, which
can only be overcome with quantum error correction. In a fault-tolerant quantum
computer the qubits and their logic interactions must have errors below a
threshold: scaling up with more and more qubits then brings the net error
probability down to appropriate levels ~ $10^{-18}$ needed for running complex
algorithms. Reducing error requires solving problems in physics, control,
materials and fabrication, which differ for every implementation. I explain
here the common key driver for continued improvement - the metrology of qubit
errors.

We provide a current perspective on the rapidly developing field of Majorana
zero modes in solid state systems. We emphasize the theoretical prediction,
experimental realization, and potential use of Majorana zero modes in future
information processing devices through braiding-based topological quantum
computation. Well-separated Majorana zero modes should manifest non-Abelian
braiding statistics suitable for unitary gate operations for topological
quantum computation. Recent experimental work, following earlier theoretical
predictions, has shown specific signatures consistent with the existence of
Majorana modes localized at the ends of semiconductor nanowires in the presence
of superconducting proximity effect. We discuss the experimental findings and
their theoretical analyses, and provide a perspective on the extent to which
the observations indicate the existence of anyonic Majorana zero modes in solid
state systems. We also discuss fractional quantum Hall systems (the 5/2 state)
in this context. We describe proposed schemes for carrying out braiding with
Majorana zero modes as well as the necessary steps for implementing topological
quantum computation.

This article provides an introduction to surface code quantum computing. We
first estimate the size and speed of a surface code quantum computer. We then
introduce the concept of the stabilizer, using two qubits, and extend this
concept to stabilizers acting on a two-dimensional array of physical qubits, on
which we implement the surface code. We next describe how logical qubits are
formed in the surface code array and give numerical estimates of their
fault-tolerance. We outline how logical qubits are physically moved on the
array, how qubit braid transformations are constructed, and how a braid between
two logical qubits is equivalent to a controlled-NOT. We then describe the
single-qubit Hadamard, S and T operators, completing the set of required gates
for a universal quantum computer. We conclude by briefly discussing physical
implementations of the surface code. We include a number of appendices in which
we provide supplementary information to the main text.

This review describes recent groundbreaking results in Si, Si/SiGe and
dopant-based quantum dots, and it highlights the remarkable advances in
Si-based quantum physics that have occurred in the past few years. This
progress has been possible thanks to materials development for both Si quantum
devices, and thanks to the physical understanding of quantum effects in
silicon. Recent critical steps include the isolation of single electrons, the
observation of spin blockade and single-shot read-out of individual electron
spins in both dopants and gated quantum dots in Si. Each of these results has
come with physics that was not anticipated from previous work in other material
systems. These advances underline the significant progress towards the
realization of spin quantum bits in a material with a long spin coherence time,
crucial for quantum computation and spintronics.

The quantum multicomputer consists of a large number of small nodes and a qubus interconnect for creating entangled state between the nodes. The primary metric chosen is the performance of such a system on Shor's algorithm for factoring large numbers: specifically, the quantum modular exponentiation step that is the computational bottleneck. This dissertation introduces a number of optimizations for the modular exponentiation. My algorithms reduce the latency, or circuit depth, to complete the modular exponentiation of an n-bit number from O(n^3) to O(n log^2 n) or O(n^2 log n), depending on architecture. Calculations show that these algorithms are one million times and thirteen thousand times faster, when factoring a 6,000-bit number, depending on architecture. Extending to the quantum multicomputer, five different qubus interconnect topologies are considered, and two forms of carry-ripple adder are found to be the fastest for a wide range of performance parameters. The links in the quantum multicomputer are serial; parallel links would provide only very modest improvements in system reliability and performance. Two levels of the Steane [[23,1,7]] error correction code will adequately protect our data for factoring a 1,024-bit number even when the qubit teleportation failure rate is one percent.

The first generation of quantum computers are on the horizon, fabricated from quantum hardware platforms that may soon be able to tackle certain tasks that cannot be performed or modelled with conventional computers. These quantum devices will not likely be universal or fully programmable, but special-purpose processors whose hardware will be tightly co-designed with particular target applications. Trapped atomic ions are a leading platform for first generation quantum computers, but are also fundamentally scalable to more powerful general purpose devices in future generations. This is because trapped ion qubits are atomic clock standards that can be made identical to a part in 10^15, and their quantum circuit connectivity can be reconfigured through the use of external fields, without modifying the arrangement or architecture of the qubits themselves. In this article we show how a modular quantum computer of any size can be engineered from ion crystals, and how the wiring between ion trap qubits can be tailored to a variety of applications and quantum computing protocols.

The recent development of chip-scale integrated quantum photonic circuits has radically changed the way in which quantum optic experiments are performed, and provides a means to deliver complex and compact quantum photonic technologies for applications in quantum communications, sensing, and computation. Silicon photonics is a promising material system for the delivery of a fully integrated and large-scale quantum photonic technology platform, where all key components could be monolithically integrated into single quantum devices. In this chapter, we provide an overview of the field silicon quantum photonics, presenting the latest developments in the generation, manipulation, and detection of quantum states of light key on-chip functions that form the basic building blocks of future quantum information processing and communication technologies.

We are going to need a quantum Internet, and to build it, we need quantum internetworking technology. This book is my contribution to both the technical and social work of getting there. It is based on my experiences during 15 years of work on classical computing systems and networks, followed by a decade of research on quantum computing systems and networks. Quantum information, including both quantum computing and quantum communication, is poised to have a large and sustained impact on the fields of theoretical and experimental quantum physics, theoretical computer science (or informatics) and ultimately the information technology industry. One important subfield is quantum networking, especially using quantum repeaters, which are the focus of this tome. Quantum signals are weak and very fragile, and, in general, cannot be copied or amplified. Engineering quantum communication sessions that can reliably exchange data over long distances, in topologically complex networks built on heterogeneous technologies and managed by many independent organizations, requires an extraordinarily broad range of expertise, which few individuals anywhere have in toto. Over the next 300 or so pages, we will attempt to lay a common foundation on which each person can erect his or her contribution.

The accelerated development of quantum technology has reached a pivotal
point. Early in 2014, several results were published demonstrating that several
experimental technologies are now accurate enough to satisfy the requirements
of fault-tolerant, error corrected quantum computation. While there are many
technological and experimental issues that still need to be solved, the ability
of experimental systems to now have error rates low enough to satisfy the
fault-tolerant threshold for several error correction models is a tremendous
milestone. Consequently, it is now a good time for the computer science and
classical engineering community to examine the {\em classical} problems
associated with compiling quantum algorithms and implementing them on future
quantum hardware. In this paper, we will review the basic operational rules of
a topological quantum computing architecture and outline one of the most
important classical problems that need to be solved; the decoding of error
correction data for a large-scale quantum computer. We will endeavour to
present these problems independently from the underlying physics as much of
this work can be effectively solved by non-experts in quantum information or
quantum mechanics.

Quantum computers promise to exceed the computational efficiency of
ordinary classical machines because quantum algorithms allow the
execution of certain tasks in fewer steps. But practical implementation
of these machines poses a formidable challenge. Here I present a scheme
for implementing a quantum-mechanical computer. Information is encoded
onto the nuclear spins of donor atoms in doped silicon electronic
devices. Logical operations on individual spins are performed using
externally applied electric fields, and spin measurements are made using
currents of spin-polarized electrons. The realization of such a computer
is dependent on future refinements of conventional silicon electronics.

Over the past several decades, quantum information science has emerged to seek answers to the question: can we gain some advantage by storing, transmitting and processing information encoded in systems that exhibit unique quantum properties? Today it is understood that the answer is yes, and many research groups around the world are working towards the highly ambitious technological goal of building a quantum computer, which would dramatically improve computational power for particular tasks. A number of physical systems, spanning much of modern physics, are being developed for quantum computation. However, it remains unclear which technology, if any, will ultimately prove successful. Here we describe the latest developments for each of the leading approaches and explain the major challenges for the future.

Quantum computers promise to increase greatly the efficiency of solving problems such as factoring large integers, combinatorial optimization and quantum physics simulation. One of the greatest challenges now is to implement the basic quantum-computational elements in a physical system and to demonstrate that they can be reliably and scalably controlled. One of the earliest proposals for quantum computation is based on implementing a quantum bit with two optical modes containing one photon. The proposal is appealing because of the ease with which photon interference can be observed. Until now, it suffered from the requirement for non-linear couplings between optical modes containing few photons. Here we show that efficient quantum computation is possible using only beam splitters, phase shifters, single photon sources and photo-detectors. Our methods exploit feedback from photo-detectors and are robust against errors from photon loss and detector inefficiency. The basic elements are accessible to experimental investigation with current technology.

Silicon Photonics III: Systems and Applications

- D Bonneau
- J W Silverstone
- M G Thompson
- L Pavesi
- D J Lockwood

Silicon Photonics III: Systems and Applications

- bonneau