Towards a Distributed Quantum Computing Ecosystem (Invited Paper)

Home , Quantum machine, Quantum teleportation

Daniele Cuomo Marcello Calefﬁ Angela Sara Cacciapuoti University of Naples Federico II University of Naples Federico II University of Naples Federico II [email protected] marcello.calefﬁ@unina.it [email protected]

Abstract—The Quantum Internet, by enabling quantum com- Sycamore, by sampling from the output distribution of 53- munications among remote quantum nodes, is a network capable qubits random quantum circuits [11]. By neglecting some of supporting functionalities with no direct counterpart in the performance-enhancing techniques as pointed out by IBM classical world. Indeed, with the network and communications functionalities provided by the Quantum Internet, remote quan- [12], [13], Google estimated that “a state-of-the-art supercom- tum devices can communicate and cooperate for solving chal- puter would require approximately 10,000 years to perform the lenging computational tasks by adopting a distributed computing equivalent task” that required just 200 seconds on Sycamore. approach. The aim of this paper is to provide the reader with an By ignoring the noise effects and by coarsely oversimpli- overview about the main challenges and open problems arising fying, the computing power of a quantum computer scales with the design of a Distributed Quantum Computing ecosystem. For this, we provide a survey, following a bottom-up approach, exponentially with the number of qubits that can be embedded from a communications engineering perspective. We start by and interconnected within [2]. And one of the reasons lays introducing the Quantum Internet as the fundamental underlying in a principle of quantum mechanics known as superposition infrastructure of the Distributed Quantum Computing ecosystem. principle. Specifically, a classical bit encodes one of two Then we go further, by elaborating on a high-level system mutually exclusive states - usually denoted as 0 and 1 - being abstraction of the Distributed Quantum Computing ecosystem. Such an abstraction is described through a set of logical layers. in only one state at a certain time. Conversely, a qubit can be Thereby, we clarify dependencies among the aforementioned in an extra mode - called superposition i.e., it can be in a layers and, at the same time, a road-map emerges. combination of the two basic states [2], [3]. Hence – thanks to the superposition principle – a qubit I.INTRODUCTION offers richer opportunities for carrying information and com- Nowadays, a tremendous amount of heterogeneous players puting, since it can do more than just flipping between 0 and entered the quantum race, ranging from tech giants - such as 1. And this quantum advantage grows exponentially with the IBM and Google in fierce competition to build a commercial number of qubits. In fact, while n classical bits are only in quantum computer - to states and governments, with massive one of the 2n possible states at any given moment, an n-qubit public funds to be distributed over the next years [1], [2], [3], register can be in a superposition of all the 2n possible states. [4]. To give a flavor of the above, let us consider one of In 2017, the European Commission launched a e1-billion the killer applications of the quantum computing: chemical flagship program to support the quantum research for ten years reaction simulation [14]. As highlighted in [15], the amount of starting from 2018, and a first e132-million tranche is being information needed to fully describe the energy configurations provided during the following three years [5]. In 2018, the of a relatively simple molecule such as caffeine is astoundingly United States of America launched the National Quantum large: 1048 bits. For comparison, the number of atoms in Initiative funded with $1.2-billion over ten years and China the Earth is estimated between 1049 and 1050 bits. Hence, is keeping up, investing billions to commercialize quantum describing the energy configuration of caffeine at one single technologies [6]. instant needs roughly a number of bits comparable to 1 to These huge efforts are justified by the disruptive potential 10 per cent of all the atoms on the planet. But this energy of a quantum computer, beyond anything classical computers

arXiv:2002.11808v2 [quant-ph] 28 Mar 2020 configuration description becomes suddenly feasible with a could ever achieve. Indeed, by exploiting the rules of quantum quantum processor embedding roughly 160 noiseless qubits, mechanics, a quantum computer can tackle classes of problems thanks to the superposition principle. that choke conventional machines. These problems include Unfortunately, qubits are very fragile and easily modified by chemical reaction simulations, optimization in manufacturing interactions with the outside world, via a noise process known and supply chains, financial modeling, machine learning and as decoherence [16], [3]. Indeed, decoherence is not the only enhanced security [7], [8], [9]. Hence, the quantum computing source of errors in quantum computing. Errors practically arise has the potential to completely change markets and industries. with any operation on a quantum state. However, isolating At the end of 2019, Google achieved the so-called quan- the qubits from the surrounding is not the solution, since tum supremacy1 with a 54-qubits quantum processor, named the qubits must be manipulated to fulfill the communication and computing needs, such as reading/writing operations. 1The term was coined by J. Preskill in 2012 [10] to describe the moment when a programmable quantum device would solve a problem that cannot be Moreover, the challenges for controlling and preserving the solved by classical computers, regardless of the usefulness of the problem. quantum information embedded in a single qubit get harder as the number of qubits within a single device increases, due to coupling effects. In this regard, Quantum Error Cor- rection (QEC) represents a fundamental tool for protecting quantum information from noise and faulty operations [17], [18]. However, QEC operates by spreading the information of one logical qubit into several physical qubits. Hence, solving problems of practical interest, such as integer factorization – which constitutes one of the most widely adopted algorithms for securing communications over the current Internet – or molecule design may require millions of physical qubits [6], [7]. Hence, on one hand researchers worldwide are leveraging on the advancement of different technologies for qubit imple- mentation superconducting circuits, ion traps, quantum dots, and diamond vacancies among the others and innovative QEC techniques to scale the number of qubits beyond two-digits. On the other hand, the Quantum Internet, i.e., a network enabling Fig. 1. Distributed Quantum Computing speed-up. The volume of cubes quantum communications among remote quantum nodes, has represents the ideal quantum computing power, i.e., in absence of noise been recently proposed as the key strategy to significantly scale and errors. As evident by comparing the power available at isolated devices up the number of qubits [19], [20], [1], [21], [22]. versus the power achievable through clustered devices, the interconnection of quantum processors via the Quantum Internet provides an exponential In fact, the availability of such a network and the adoption computing speed-up with respect to isolated devices. of a distributed computing paradigm allows us to regard the Quantum Internet jointly as a virtual quantum computer with a number of qubits that scales linearly with the number of network2 able to transmit qubits and to distribute entangled interconnected devices. quantum states3 among remote quantum devices [3], [1], [2], In this light, the aim of this paper is to provide the [19], [20], [21], [23]. reader with an overview about the main challenges and open In fact,the availability of the corresponding underlying problems arising with the design of a Distributed Quantum network infrastructure and the adoption of the distributed Computing ecosystem. quantum computing paradigm [25] allows us to regard the We start in Sec. II by introducing the Quantum Internet – Quantum Internet – jointly – as a virtual quantum computer as the fundamental underlying communication infrastructure with a number of qubits that scales linearly with the number of a Distributed Quantum Computing ecosystem – as well as of interconnected devices. Hence, the Quantum Internet may some of its unique key applications. Then we go further in enable an exponential speed-up [26], [25], [1] of the quan- Sec. III, by conceptualizing a high-level system abstraction tum computing power with just a linear amount of physical of the Distributed Quantum Computing ecosystem from a resources, represented by the interconnected quantum proces- communication engineering perspective. Such an abstraction sors. Indeed, by comparing the computing power achievable is described through a set of (logical) layers, with the higher with quantum devices working independently versus working depending on the functionalities provided by the lower ones. as a unique quantum cluster, the gap comes out - as depicted Thereby, we clarify dependencies among the aforementioned in Fig. 1 [1]. layers. Since, each layer of the ecosystem has some related Specifically, increasing the number of isolated devices lays open challenges, within Sec. IV we survey such challenges to a linear speed-up, with a double growth in computational and open problems. Finally, in Sec. V we conclude the paper power by doubling the number of devices. Conversely, increas- with some perspectives. ing the number of clustered devices provides an exponential growth, with a significant advantage clearly visible with just two interconnected devices. For instance, a single 10-qubit II.THE QUANTUM INTERNET processor can represent 210 states thanks to the superposition principle , hence two isolated 10-qubit processors can repre- As mentioned in Sec. I, one promising approach to address sent 211 states. But if we interconnect the two processors, the challenges arising with large-scale quantum processor the resulting virtual device can represent up to 218 states realization is to mimic modern high-performance computing infrastructures – where thousands of processors, memories 2 We refer the reader to [23], [24] for a discussion about the differences and storage units are inter-connected via a communication underlying the notion of “Quantum Internet” versus “quantum network”. network, and the computational tasks are solved by adopting 3The deepest difference between classical and quantum mechanics lays in a distributed approach. the concept of quantum entanglement, a sort of correlation with no counterpart in the classical world. For an in-depth introduction to quantum entanglement To this aim, it is mandatory to design and deploy the we refer the reader to the classical book [17] , whereas a concise description Quantum Internet, which formally defines a global quantum can be found in [3]. [1], depending on the number of qubits devoted to fulfill the Distributed communication needs of the clustered processors as discussed Distributed Quantum Quantum Algorithm in Sec. IV-B. Compiler Before analyzing with further details in Sec. III the resulting Virtual Distributed Quantum Computing ecosystem from a communi- Quantum cation engineering perspective, it is worthwhile to note that Processor the availability of the Quantum Internet infrastructure en- Local ables unparalleled capabilities not restricted to the distributed Operations computing [21], [2]. Specifically, applications such as blind Quantum computing, secure communications and noiseless communi- Processor cations have already been theorized or even experimentally verified , as recently overviewed by an IETF Quantum Internet Draft [23]. Blind quantum computing [27], [28] refers to a server-client architecture where clients can send sensitive data to server, Quantum which elaborates inputs without knowing their values. This Link Remote functionality allows to achieve a twofold goal: preserving data Classical Operations confidentiality as well as solving tasks that are intractable for Link the client – that can be a classical computer – but tractable for the server, which implements the quantum paradigm. Fig. 2. A high-level system abstraction of the distributed quantum computing Secure communications in the quantum field refers to the ecosystem. The lowest layer provides the communication/network functionalities and consists of quantum processors interconnected with both classical and class of communication protocols that exploit quantum me- quantum links. Thanks to the underlying communication infrastructure, both chanics in order to get benefits unattainable using classical local and remote qubit operations can be executed. Hence, from a computing Internet. For instance, in the field of quantum cryptography, perspective, the two lowest levels concur to build a virtual quantum processor with a number of qubits that scales with the number of inter-connected researchers study strategies for sharing keys among parties physical quantum processors. The virtual processor acts as an interface for in total secrecy [29], [30], [31]. Whilst quantum byzantine the distributed quantum compiler, which mapsa quantum algorithm intoa agreement, a protocol used by multiple entities to distributively sequence of local and remote operations, so that the available computing resources are optimized with respect to both the hardware and the network agree on a common decision, allows to achieve the consensus constraints. in a constant number of rounds, whereas classical protocols scale polynomially with the number of processors [32]. Finally, the Quantum Internet provides the underlying in- Distributed Quantum Computing ecosystem. For the sake of frastructure to achieve transmission rates exceeding the fun- clarity, we restrict our attention to a network composed by two damental limits of conventional (quantum) Shannon theory quantum processors, directly inter-connected. Nevertheless, [33], [34]. Specifically, by exploiting the capability of quantum the discussion in the following can be easily extended to more particles to propagate simultaneously among multiple space- complex network topologies, provided that end-to-end routing time trajectories, quantum superpositions of noisy channels and network functionalities such as [36], [37], [38], [39], [40], can behave as perfect noiseless quantum communication chan- [41] are available. nels, even if no quantum information can be sent throughout Starting from bottom, in Fig. 2 we have the communi- either of the noisy component channels individually [35]. cation infrastructure underlying the Quantum Internet: spatially remote quantum devices able to communicate quantum III.DISTRIBUTED QUANTUM COMPUTING ECOSYSTEM information by means of a synergy of both classical and The overall aim of classical distributed computing is to quantum communication resources. Indeed, as we overview in deal with hard computational problems by splitting out the Sec. IV-B, the transmission of quantum information generally computational tasks among several classical devices, in order requires the transmission of classical information as well, to lighten the loads on single devices. hence the availability of a classical network infrastructure – As mentioned in Sec. II, with the network infrastructure pro- such as the classical Internet – is required [2], [3], [21]. This vided by the Quantum Internet, this paradigm can be extended constraint has been highlighted by interconnecting the two to quantum computing as well: remote quantum devices can quantum processors with both a classical and a quantum link. communicate and cooperate for solving computational tasks By exploiting the communication functionalities provided by adopting a distributed computing approach [26], [25]. Since by the lowest level, both local and remote qubit operations can an entirely new paradigm – characterized by unconventional be executed. Specifically, the local operations – i.e., operations phenomena ruled out by the quantum mechanics [1], such as between qubits stored within the same quantum processor– no-cloning and entanglement – is involved, a new ecosystem can be executed by exploiting the physical (or logical, whether needs to be engineered. the quantum device should natively implement QEC func- The infographic in Fig. 2 is a stack depicting dependencies tionalities [42]) controls and readouts functionalities provided among a possible set of layers that together provide the by the device. Conversely, the remote operations – i.e., the q1 q2 q3 q4 q5 q6 q7 q8 components of the proposed ecosystem and considering open challenges related with.

q0 q15 q14 q13 q12 q11 q10 q9 IV. OPEN CHALLENGES AHEAD The aim of this section is to describe and to discuss some of Fig. 3. Coupling map of the ibmqx3 architecture: the nodes represent the the open problems related with the proposed Distributed Quan- qubits while the edges represent the possibility to have interactions between two qubits, i.e., to implement the CNOT operation. As an instance, a CNOT tum Computing ecosystem. For this, some layers are individu- operation can be directly executed between qubits q1 and q2 but not between ally discussed. Speciﬁcally, in Sec. IV-A quantum processors qubits q1 and q3 . are considered, paying particular attention to drawbacks induced by local operations involving multiple qubits. Sec. IV-B REVERSEDCNOT SWAP introduces the interconnection between remote quantum pro-

|ψq1 i H H H H cessors, discussing the quantum teleportation as a mean to transfer quantum information between interconnected devices. |ψ i q2 H H H H After that, in Sec. IV-C we discuss the gate teleportation as a CNOT strategy to perform remote operations. Sec. IV-D describes the |ψq i 3 layer responsible for abstracting and optimizing the execution Fig. 4. A CNOT operation between non-adjacent qubits can be implemented of the quantum algorithms, based on the characteristics of the through a sequence of swapping operations, with each swap consisting of three underlying system. Finally, in Sec. IV-D we discuss some of CNOT (with the in-between CNOT being reverse, i.e., with target and control the current standardization efforts. qubits swapped) between adjacent qubits [46]. Thus, the circuit performs a CNOT between q1 and q3, leaving q2 unaltered. Note that H denotes the A. Quantum Processor Hadamard gate mapping a basis state into an even superposition of the basis states [47], and that the quantum state stored within qubit qi is denoted– The most basic element of a distributed quantum computing by adopting the standard bra-ket notation for describing quantum states– as ecosystem is identifiable with the local quantum computing |ψ i. qi device. Here the qubits are connected according to some directed and connected graph – namely, the coupling map – that accounts for the hardware limitations resulting from operations between qubits stored at different quantum devices controlling and preserving the quantum information from – pose further constraints, as discussed in Sec. IV. decoherence and noise. As an example, Fig. 3 depicts the Thanks to the abstraction provided by the two layers re- coupling map of the ibmqx3 quantum processor [46], with siding at the very bottom, a virtual quantum processor is nodes representing qubits while edges represent the possibility obtained, where remote qubits are interconnected through to have interactions between two qubits, i.e., to implement one virtual connections made possible by the remote operations. of the fundamental quantum operations: the CNOT operation4. Clearly, remote operations are likely to be characterized by In fact, there exists a universal quantum gate set5 with CNOT delays and error rates notably larger than those characterizing as the only operator belonging the set that involves more than local operations. Hence, local operations should be preferred one qubit. Thus, we can focus on problems related to the CNOT over remote ones as much as possible, even though remote operation keeping the discourse general. operations are unavoidable whenever the number of qubits It is immediate to observe that only nodes – i.e., qubits – required to perform the computational task exceeds the number linked by an edge can directly interact [49]. Nevertheless, of qubits available at a single device. Hence, an optimization for an algorithm designer it is crucial being able to define a must be performed by the distributed quantum compiler, so circuit without restrictions on interactions. Indeed, a circuit that the different operations required by the quantum algorithm programming model can easily abstract from this restriction as well as the input to the algorithm itself are properly [17], resulting in a fully connected graph at the cost of an allocated among the qubits of the different devices. overhead due to indirect execution of the desired operation. Finally, at the very top we have the quantum algorithm, For instance, let us consider to carry out a CNOT between the which is completely independent and unaware of the phys- two non-adjacent qubits q and q of Fig. 3. By performing ical/logical constraints imposed by both the hardware and 1 3 two state swapping operations for each edge belonging to the network particulars, thanks to the abstraction provided by the shortest-path from node q towards node q , the overall result underlying levels. We can think at this module as a minimal 1 3 will be equivalent to a CNOT between q and q , keeping other service for defining quantum algorithms, but also, in a wider 1 3 states unchanged. Fig. 4 clarifies the process above. perspective, as a platform where interesting functionalities are The overhead induced by the swapping operations explains available - allowing, for instance, quantum machine learning how important is the topological organization of devices and [43] or quantum optimization algorithms [44], [45]. the circuit design as well. However, several complications have been omitted so far. In- deed, communication protocols intrinsically imply overhead, 4The CNOT operation involves two qubits, referred to as control qubit and i.e., computing resources need to be dedicated, in order to deal target qubit. It works as follows [47]: if the control qubit is 1, then the target qubit value is flipped. Otherwise, nothing happens. with transmission processes and errors correction. Therefore, 5 A universal gate set is any set of gates which any operation possible on it’s worth going further towards this discussion, expanding a quantum computer can be reduced to [48]. ION TRAP SOURCE

SUPERCONDUCTING QUBITS |ψi H QUANTUM DOTS

|Φ+i

SATELLITE-FREE X Z |ψi SPACE CHANNEL

GROUND-FREE DESTINATION SPACE CHANNEL Fig. 6. Quantum teleportation circuit. First two wires belong to source, OPTICAL FIBER CHANNEL whereas the bottom wire belongs to destination. A generic state |ψi is initially stored at the source . Once the teleportation process is fulﬁlled, the original state is available at the destination, regardless of its value. |Φ+i represents an EPR pair, that is a couple of maximally entangled qubits [50]. The result of the measurement process at the source is sent to the destination via a classical link . The carried classical bits are thus used for determining whether gates X and Z – corresponding to a bit- and a phase-ﬂip, respectively [47] – must be performed or not, in order to recover the original state |ψi from the EPR pair member available at the destination.

Fig. 5. Pictorial representation of a matter-flying transducer, which is needed resources is classical: two bits must be transmitted from the to convert matter qubits – i.e., qubits for information processing/storing – into source to the destination. The other resource is quantum: an flying qubits – i.e., qubits for information transmission – and vice versa. entangled pair of qubits must be generated and shared between the source and the destination. B. Quantum Link In the context of the distributed quantum computing ecosys- As mentioned in Sec. III, the Quantum Internet requires tem, quantum teleportation constitutes the foundation of a both classical and quantum links. In this perspective, a distinc- communication paradigm known as teledata [54], which gen- tion between matter and flying qubits – i.e., between qubits eralizes the concept of moving state among qubits to remote for information processing/storing and qubits for information devices. transmission – must be made [2]. To provide a concrete example of the teledata concept, As regards to the matter qubits, several candidate technolo- a further distinction must be made between communication gies are available, each one with its pros and cons [25]. Con- qubits and data qubits. Specifically, within each quantum versely, as regards to the flying qubits, there exists a general device, a subset of matter qubits is reserved for fulfilling the consensus about the adoption of photons as qubit substrate communication needs to generate entanglement. These qubits [2]. However, heterogeneity arises by considering the different are called communication qubits [23], to distinguish them physical channels the photons propagate through, ranging from from the remaining matter qubits within the device devoted free-space optical channels (either ground or satellite free- to processing/storage, referred to as data qubits. space) to optical fibers. Thus, a transducer for matter-flying As an example, consider two ibmqx2 architectures [55] conversion is needed as depicted in Fig. 5 and discussed with interconnected via quantum teleportation as depicted in Fig. 7. further details in [2]. And communication models need to take + The c0, c1 pair is in the state |Φ i - that is the standard notation into account sucha technological heterogeneity, with the aim to denote a couple of maximally entangled qubits [17], also of providing a black box for upper protocol layers with one known as EPR pair [50]. Any kind of interaction between common logic. remote devices involves communication qubits, but not all the Furthermore, quantum mechanics does not allow an un- data qubits are connected with them. As already explained known qubit to be copied or even simply observed/measured. in Sec. IV-A, interactions between non-adjacent qubits are As a consequence, the communication techniques utilized feasible but they imply an overhead. A solution would be to interconnect spatially remote quantum devices cannot be adding more qubits to the communication set, but it means directly borrowed from classical communications. In this con- sacrifice further valuable resources for processing and storage. text, quantum teleportation is widely accepted as one of the For this reason, the selection of the communication qubit set most promising quantum communication technique between is a crucial task within the distributed quantum computing remote quantum nodes [51], [52], [3]. Quantum teleportation ecosystem and it implies a carefully evaluation of the trade- has been experimentally verified [53] and it requires, as off between the number of data and communication qubits. depicted in Fig. 6, a pair of parallel resources6. One of these Next section shows how to exploit teleportation in order to 6 For an in-depth discussion about the quantum teleportation process, we not only send information but also perform joint operations refer the readers to [3]. between remote qubits. q0 q4 SOURCE |ψi Z

|Φ+i q2 q1 c0 c1 q5 q7

|Φ+i

q3 q6 H

Fig. 7. Coupling map representing physical architecture of two quantum |φi X devices ibmqx2 interconnected via the quantum teleportation paradigm. Nodes labeled with c and c denote communication qubits, and the dotted 0 1 DESTINATION line indicates that they are in the entangled state |Φ+i, i.e., they form an EPR pair. Conversely, qi with i ∈ {0, 1, 2, 3} denote data qubits – i.e., qubits Fig. 8. Quantum telegate circuit implementing a CNOT operation between available for computing tasks. remote qubits. Specifically, a CNOT operation between qubits placed at different devices – say qubits q0 and q4 in Fig. 7– is performed through CNOTs between each qubit and the member of an EPR pair stored at the C. Teleporting Gates same device, followed by single-qubit gates and measurements. Note that |ψi and |φi denote the generic initial states stored at q0 and q4, respectively, + Distributed quantum computation requires the capability to whereas |Φ i denotes the EPR pair stored by the communications qubits – say qubits c0 and c1 in Fig. 7. perform quantum operations on qubits belonging to remote quantum devices. As mentioned in the previous section, one possible solution algorithm designers may benefit from an abstraction which is to resort to the teledata concept, by moving the quantum hides the complications due to physical features. information from a quantum device to another via the telepor- However, as mentioned in Sec. III, within the Virtual Quan- tation process, through an entangled pair. tum Processor remote qubits belonging to remote quantum However, an entangled pair allows one to implement also a devices are interconnected through virtual connections made so-called teleporting gate, or telegate [56]. From a theoretical possible by the remote operations as described in Secs. IV-B perspective, we have already observed that providing the CNOT and IV-C. Unfortunately, remote operations are likely to be operation - together with other single qubit gates - is enough to characterized by delays and error rates larger than local perform any kind of quantum algorithm. Therefore, returning operations. The reason underlying this statement is that each to stack dependencies of Fig. 2, we can conceptualize a protocol step needed for realizing remote operations – from the service that provides a set of remote operations based on entanglement distribution to the gate operations – is affected teleported gates. Such service will directly interact with the by decoherence [3], i.e., by a quantum-specific noise process. physical system, exploiting the entanglement generation and Decoherence affects local operations as well. However, since distribution functionality [3]. the effects of such a noise become stronger as function of Specifically, by considering the topology depicted in Fig. 7, the time [3], [2], remote operations, by involving further it is possible to implement the CNOT as the joint operation away parties, are more vulnerable. It follows that, given a between qubits belonging to spatially remote devices. Indeed, particular quantum circuit describing a quantum algorithm, by exploiting the availability of two communication qubits the Distributed Quantum Compiler is required to optimize the – c0 and c1 – shared between the two remote devices and circuit8 so that the number of remote operations is minimized storing an EPR pair, it is possible to perform a remote CNOT as much as possible to limit the decoherence effects. 7 operation with, for example, q0 as control qubit and q4 as Furthermore, the compiler should be able to optimize the target qubit. Fig. 8 shows the corresponding circuit, consisting circuit so that it can be executed regardless of the underlying of local CNOT operations and single-qubit operations and network topology. Indeed, it may be the case that two remote measurements. quantum devices are not directly connected – i.e., no communication qubits are shared between the two devices – even D. Distributed Quantum Compiler though some remote operations between qubits stored at these Concepts discussed so far provide some fundamental under- devices are required. Hence, the compiler should optimize lying communication functionalities enabling the distributed the corresponding quantum circuit so that the entanglement quantum computing paradigm. swapping operations9 among remote devices are minimized. Indeed, the definition of a quantum algorithm can be very Finally, as discussed in Sec. IV-B, there exists a trade-off abstract. In general, it goes through the definition of a quantum circuit – where a computation is a sequence of quantum gates 8 Note that a circuit optimization is needed also in case of a single quantum processor [46], [58] to reduce the overhead arising with the swap operations on a register – as those depicted in Figs. 6 and 8. And discussed in Sec. IV-A. 9 Entanglement swapping is a technique used to entangle distant nodes – 7 For a more comprehensive presentation of the telegate, we refer the reader without physically sending an entangled qubit through the entire distance – to [57]. by swapping the entanglement generated at intermediate nodes [52]. between data and communication qubits. The larger is the [5] E. Cartlidge, “Europe’s e1-billion quantum flagship announces grants.” number of communication qubits in a device, the higher is the Science, vol. 362, no. 6414, p. 512, 2018. [6] E. Gibney, “The quantum gold rush.” Nature, vol. 574, no. 7776, pp. rate of remote operations achievable at the price of reducing 22–24, 2019. the number of qubits devoted to computation. [7] K. Bourzac, “4 tough chemistry problems that quantum computers will solve.” IEEE Spectrum, vol. 54, no. 11, pp. 7–9, 2017. E. What’s Next [8] M. Schuld, I. Sinayskiy, and F. Petruccione, “An introduction to quantum machine learning.” Contemporary Physics, vol. 56, no. 2, pp. 172–185, Considering modern available technologies, it seems reason- 2015. able to envision that we will see a first attempt of interconnec- [9] D. Gottesman, H.-K. Lo, N. Lutkenhaus, and J. Preskill, “Security tion among quantum computers located nearby. Likely, one of of quantum key distribution with imperfect devices.” in International Symposium on Information Theory, 2004, p. 136. the main tech companies currently providing cloud access to [10] J. Preskill, “Quantum computing and the entanglement frontier.” in 25th isolated quantum computers – such as IBM [59], HIQ [60], Solvay Conference on Physics, October 2011. Amazon Braket [61] or Azure Quantum [62] – will scale [11] F. Arute, K. Arya, R. Babbush et al., “Quantum supremacy using a the available quantum computing power by interconnecting programmable superconducting processor.” Nature, vol. 574, no. 7779, pp. 505–510, 2019. few quantum computers located few meters away within a [12] E. Pednault, J. Gunnels, D. Maslov, and J. Gambetta, “On quantum quantum farm [2], [4]. After that, consider that some of these supremacy.” IBM Research Blog, vol. 21, 2019. companies will have quantum computers distributed over the [13] E. Pednault, J. A. Gunnels, G. Nannicini et al., “Leveraging secondary storage to simulate deep 54-qubit sycamore circuits.” arXiv preprint, p. world. Thus, it would not be surprising to see interconnection arXiv:1910.09534, 2019. among quantum computers or even among quantum clusters [14] Y. Cao, J. Romero, J. P. Olson et al., “Quantum chemistry in the age of miles away from each other. quantum computing.” Chemical reviews, vol. 119, no. 19, pp. 10 856– However, interconnecting quantum devices implies the need 10 915, 2019. [15] G. Drouin, “IBM Qs Dr.Robert Sutor explains the state of the quantum for a communication standard. In other words, remote devices computing industry.” May 2018. – likely to have technological differences – will need to [16] A. S. Cacciapuoti and M. Caleffi, “Toward the quantum internet: agree on network protocols in order to exchange interpretable A directional-dependent noise model for quantum signal processing.” in IEEE International Conference on Acoustics, Speech and Signal information. Thus, the Quantum Internet will need a logical Processing, May 2019, pp. 7978–7982. architecture as well as the classical Internet does – i.e., with [17] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum the TCP/IP Internet protocol suite being the standard de- Information. Cambridge University Press, 2010. [18] Z. Babar, P. Botsinis, D. Alanis, S. X. Ng, and L. Hanzo, “The Road facto. Research in this direction has already started, within From Classical to Quantum Codes: A Hashing Bound Approaching the Internet Engineering Task Force (IETF), where researchers Design Procedure.” IEEE Access, vol. 3, pp. 146–176, 2015. are trying to conceptualize the Quantum Internet as a service- [19] H. J. Kimble, “The quantum internet.” Nature, vol. 453, no. 7198, pp. oriented platform [24], [23]. 1023–1030, June 2008. [20] S. Pirandola and S. L. Braunstein, “Physics: Unite to Build a Quantum Internet.” Nature, vol. 532, no. 7598, pp. 169–171, Apr. 2016. ONCLUSIONS V. C [21] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: a vision for With this paper, we have presented a layered ecosystem that the road ahead.” Science, vol. 362, no. 6412, 2018. [22] D. Awschalom, K. K. Berggren, H. Bernien et al., “Development of outlines a possible key strategy towards large-scale quantum Quantum InterConnects for Next-Generation Information Technologies.” processor design based on the distributed quantum computing p. arXiv:1912.06642, 2019. paradigm. Within the envisioned ecosystem, the lowest layers [23] W. Kozlowski, S. Wehner, R. V. Meter, B. Rijsman, A. S. Cacciapuoti, and M. Caleffi, “Architectural Principles for a Quantum Internet.” integrate the Quantum Internet as the fundamental underlying Internet Engineering Task Force, Internet-Draft draft-irtf-qirg-principles- infrastructure providing networking and communication func- 03, Mar. 2020, work in Progress. tionalities among remote quantum devices. Conversely, the up- [24] C. Wang, A. Rahman, and R. Li, “Applications and Use Cases for per layers are responsible for mapping the quantum algorithm the Quantum Internet.” Internet Engineering Task Force, Internet- Draft draft-wang-qirg-quantum-internet-use-cases-04, Mar. 2020, work onto the underlying physical infrastructure by optimizing the in Progress. available computational resources as well as by accounting for [25] R. V. Meter and S. J. Devitt, “The path to scalable distributed quantum the constraints induced by the hardware/network configuration. computing.” Computer, vol. 49, no. 9, pp. 31–42, Sept 2016. [26] A. Yimsiriwattana and S. J. Lomonaco Jr, “Distributed quantum computing: A distributed Shor algorithm.” in Quantum Information and EFERENCES R Computation II, vol. 5436. International Society for Optics and [1] M. Caleffi, A. S. Cacciapuoti, and G. Bianchi, “Quantum internet: Photonics, 2004, pp. 360–372. From communication to distributed computing!” in Proceedings of [27] A. Broadbent, J. Fitzsimons, and E. Kashefi, “Universal Blind Quantum the 5th ACM International Conference on Nanoscale Computing and Computation.” in 50th Annual IEEE Symposium on Foundations of Communication, 2018, invited paper. Computer Science, Oct. 2009, pp. 517–526. [2] A. S. Cacciapuoti, M. Caleffi, F. Tafuri, F. S. Cataliotti, S. Gherar- [28] D. Aharonov, M. Ben-Or, and E. Eban, “Interactive proofs for quantum dini, and G. Bianchi, “Quantum Internet: Networking Challenges in computations.” in Proceedings of Innovations in Computer Science. Distributed Quantum Computing.” IEEE Network, vol. 34, no. 1, pp. Tsinghua University Press, 2010. 137–143, January 2020. [29] G. Xu, K. Xiao, Z. Li, N. X., and R. M., “Controlled Secure Direct [3] A. S. Cacciapuoti, M. Caleffi, R. Van Meter, and L. Hanzo, “When Communication Protocol via the Three-Qubit Partially Entangled Set of Entanglement Meets Classical Communications: Quantum Teleportation States.” Computers, Materials & Continua, vol. 58, no. 3, pp. 809–827, for the Quantum Internet.” IEEE Transactions on Communications, pp. 2019. 1–1, 2020, invited paper. [30] X. Chen, Y. Wang, G. Xu, and Y. Yang, “Quantum network communi- [4] M. Caleffi, D. Chandra, D. Cuomo, S. Hasaanpour, and A. S. Caccia- cation with a novel discrete-time quantum walk.” IEEE Access, vol. 7, puoti, “The Rise of the Quantum Internet.” IEEE Computer, 2020. pp. 13 634–13 642, 2019. [31] K. Sutradhar and H. Om, “Efficient quantum secret sharing without a [58] D. Ferrari and M. Amoretti, “Efficient and effective quantum compiling trusted player.” Quantum Information Processing, vol. 19, no. 2, p. 73, for entanglement-based machine learning on ibm q devices.” Interna- 2020. tional Journal of Quantum Information, vol. 16, no. 08, p. 1840006, [32] M. Ben-Or and A. Hassidim, “Fast quantum Byzantine agreement.” in 2018. Proceedings of the thirty-seventh annual ACM symposium on Theory of [59] IBM Quantum Experience. [Online]. Available: https:// computing. Citeseer, 2005, pp. 481–485. quantum-computing.ibm.com [33] M. Caleffi and A. S. Cacciapuoti, “Quantum Switch for the Quantum [60] Quantum Computing Software HiQ. [Online]. Available: https: Internet: Noiseless Communications through Noisy Channels.” IEEE //hiq.huaweicloud.com/ Journal on Selected Areas in Communications, pp. 1–1, 2020. [61] Amazon Braket. [Online]. Available: https://aws.amazon.com/braket/ [34] A. S. Cacciapuoti and M. Caleffi, “Capacity bounds for quantum [62] Azure Quantum. [Online]. Available: https://azure.microsoft.com/it-it/ communications through quantum trajectories.” arXiv preprint, p. services/quantum/ arXiv:1912.08575, 2019. [35] L. Gyongyosi, S. Imre, and H. V. Nguyen, “A Survey on Quantum Channel Capacities.” IEEE Communications Surveys Tutorials, vol. 20, no. 2, pp. 1149–1205, 2018. [36] M. Caleffi, “Optimal routing for quantum networks.” IEEE Access, vol. 5, pp. 22 299–22 312, 2017. [37] L. Gyongyosi and S. Imre, “Multilayer optimization for the quantum internet.” Scientific reports, vol. 8, no. 1, p. 12690, 2018. [38] A. Pirker and W. Dur,¨ “A quantum network stack and protocols for reliable entanglement-based networks.” New Journal of Physics, vol. 21, no. 3, p. 033003, 2019. [39] L. Gyongyosi and S. Imre, “Topology adaption for the quantum internet.” Quantum Information Processing, vol. 17, no. 11, p. 295, 2018. [40] K. Chakraborty, F. Rozpedek, A. Dahlberg, and S. Wehner, “Distributed routing in a quantum internet.” arXiv preprint, p. arXiv:1907.11630, 2019. [41] S. Shi and C. Qian, “Modeling and designing routing protocols in quantum networks.” arXiv preprint, p. arXiv:1909.09329, 2019. [42] J. M. Gambetta, J. M. Chow, and M. Steffen, “Building logical qubits in a superconducting quantum computing system.” npj Quantum Infor- mation, vol. 3, no. 1, p. 2, 2017. [43] F. Tacchino, C. Macchiavello, D. Gerace, and D. Bajoni, “An artificial neuron implemented on an actual quantum processor.” npj Quantum Information, vol. 5, no. 1, pp. 1–8, 2019. [44] G. Verdon, J. M. Arrazola, K. Brdler, and N. Killoran, “A Quantum Approximate Optimization Algorithm for continuous problems.” p. arXiv:1902.00409, 2019. [45] E. Farhi and A. W. Harrow, “Quantum Supremacy through the Quantum Approximate Optimization Algorithm.” p. arXiv:1602.07674, 2016. [46] A. Zulehner, A. Paler, and R. Wille, “An efficient methodology for mapping quantum circuits to the IBM QX architectures.” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 38, no. 7, pp. 1226–1236, July 2019. [47] E. G. Rieffel and W. H. Polak, Quantum computing: A gentle introduction. MIT Press, 2011. [48] N. Raychev, “Universal quantum operators.” International Journal of Scientific and Engineering Research, vol. 6, no. 6, pp. 1369–1371, 2015. [49] N. M. Linke, D. Maslov, M. Roetteler et al., “Experimental comparison of two quantum computing architectures.” Proceedings of the National Academy of Sciences, vol. 114, no. 13, pp. 3305–3310, 2017. [50] A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Physical Review, vol. 47, pp. 777–780, May 1935. [51] C. H. Bennett, G. Brassard, C. Crepeau´ et al., “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels.” Physical Review Letters, vol. 70, pp. 1895–1899, Mar. 1993. [52] R. Van Meter, “Quantum networking and internetworking.” IEEE Net- work, vol. 26, no. 4, pp. 59–64, 2012. [53] J.-G. Ren, P. Xu, H.-L. Yong et al., “Ground-to-satellite quantum teleportation.” Nature, vol. 549, no. 7670, p. 70, 2017. [54] R. Van Meter, K. Nemoto, W. J. Munro, and K. M. Itoh, “Distributed Arithmetic on a Quantum Multicomputer.” in 33rd International Sym- posium on Computer Architecture, June 2006, pp. 354–365. [55] Qiskit. Beckend information. [Online]. Available: https://github.com/ qiskit/ibmq-device-information [56] R. D. V. Meter III, “Architecture of a quantum multicomputer optimized for Shor’s factoring algorithm.” arXiv preprint, pp. quant–ph/0 607 065, 2006. [57] K. S. Chou, J. Z. Blumoff, C. S. Wang et al., “Deterministic teleportation of a quantum gate between two logical qubits.” Nature, vol. 561, no. 7723, pp. 368–373, 2018.