Field of the invention

The invention relates to synchronising events at a plurality of nodes in a quantum network; and, in particular, though not exclusively, to methods and systems for synchronising events at a plurality of nodes in a quantum network, a computer program product enabling a system to perform such methods, and a storage medium storing such computer program product.

Background of the invention

A quantum network comprises a number of quantum nodes that are geographically distant from each other. At the current state of technology, each node is an elaborate experimental setup where one or several qubits reside that can be manipulated using lasers, microwaves and/or other electromagnetic fields. At each node, it is possible to initialize a local qubit to a well-known quantum state; to manipulate the quantum state of a local qubit; and to perform a local qubit readout which collapses its quantum state.

A prerequisite to efficient quantum networking is a reliable method to generate entanglement between two or more qubits that are geographically separated by a considerable distance (typical order of magnitude: 1-100 kilometres). If this can be achieved, practical applications such as quantum key distribution and distributed quantum computing become possible.

In practice, the process of reaching inter-node entanglement comprises two stages: a preparatory stage and an operational stage. During the preparatory stage, each node locally initializes their qubit(s) to a well-known quantum state. In current implementations this involves a process of trial-and-error, and it can take many initialisation attempts to reach a state where the local node is ready to try an entanglement attempt. Therefore, the preparatory stage has a duration that is essentially unpredictable.

During the operational stage, a sequence of manipulations is executed at each node involved in the entanglement process to initiate the entanglement attempt, followed by an application-specific sequence of qubit manipulations. In order to start the operational stage, it is necessary that the preceding preparatory stage has completed on all nodes involved. The underlying physical processes demand that in this operational stage, the operations on different nodes are highly synchronised. The exact requirements depend on the parameters of the qubit, such as its wavelength, but a typical synchronisation requirement is of a level of 1 ns or better.

The timing behaviour and requirements of the preparatory and operational stages poses a problem: at each node, it is not known beforehand how long the preparatory stages of all involved nodes will take; but at the same time, the nodes need to coordinate the start and execution of their operational stages very precisely.

A further problem is posed by the fact that standard computing devices such as microcontrollers do not possess the required timing accuracy to start the operational stage at the determined starting time with sufficient temporal precision. Additionally, the uncertainty in the transmission delay of a signal (jitter) sent between two distant nodes generally increases with the distance and can easily be larger than the allowed timing variation in the synchronised starting time.

Apart from the fact that nodes need to determine a synchronised start time for the operational stage, it is also desirable that the synchronised start time follows the completion of the qubit preparation as quickly as possible, because the locally prepared qubit state at each node degrades. Therefore, a fast method to determine the synchronised start time increases the chance of successful entanglement. Furthermore, a fast method also increases the number of repeated executions (each comprising a preparatory and an operational stage) that can be performed per unit of time, increasing the throughput of the network.

Lago-Rivera et al, ‘Telecom-heralded entanglement between multimode solid- state quantum memories’, Nature 594 (2021), pages 37-40 describe a two-node quantum network where the nodes are placed near to each other (effectively approximately 25 m distant). Entanglements are attempted periodically using synchronised chopper wheels, i.e., rotating shutters with one or more openings, controlled by a single source.

However, this method is not well scalable to larger inter-node distances, as it would become unfeasible to synchronise the chopper wheels at both nodes this way. Moreover, if the period between entanglement attempts is small, there will be a high failure rate due to one of the nodes not having finished its preparation; and if the period between entanglement attempts is large, the method becomes very inefficient.

Hensen et al., ‘Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres’, Nature 526 (2015), pages 682-686 describe a two-node quantum network with a ‘master node’ and a ‘slave node’. The nodes are separated by 1.3 km, which corresponds to a calibrated trigger delay of almost 10 ps with an uncertainty of 10 ns. When the slave node has finished preparing its local qubit, it sends a signal to the master node. When the master node is ready and has received the signal from the slave node, the master node sends a start signal to the slave node. The master node waits for the calibrated trigger delay and then starts the operational stage, the slave node starts as soon as the start signal is received.

This method is relatively inefficient and insufficiently accurate. Moreover, the method does not scale well for larger distances or networks with more than two nodes. There is therefore a need in the art for a method to determine a synchronised starting time between nodes in a quantum network that is reliable, efficient, and scalable both with respect to the distance between nodes and to the number of nodes.

Summary of the invention

It is an aim of embodiments in this disclosure to provide a method to efficiently and reliably determine a synchronised start time of an operational stage in a quantum network.

The inventor has realised that an efficient and reliable synchronisation method can be based on discretising the moments in time that nodes are allowed to transmit communication signals, resulting in a discrete set of allowed transmission times that is shared and synchronised between all nodes in the network. The inventor has furthermore realised that the allowed transmission times are advantageously based on a distribution of transmission times between the nodes, e.g. based on a transmission delay interval characterised by a minimum transmission delay and a maximum transmission delay.

In a first aspect, the disclosure relates to a method for synchronising events at a plurality of nodes in a quantum network. The method comprises receiving or determining transmission delay information about a transmission delay distribution of signals transmitted between a first node comprising a first qubit and a second node comprising a second qubit. The transmission delay information comprises a maximum transmission delay. The method further comprises receiving or determining a synchronised set of allowed transmission times based on the transmission delay distribution, and based on a synchronized clock shared between the nodes. The method further comprises executing, by the first node, a first qubit operation based on a start time. The execution by the first node includes initializing the first qubit and transmitting a first initialisation signal to the second node based on a first allowed transmission time selected from the set of allowed transmission times. The execution by the first node further includes receiving a second initialisation signal from the second node. The second initialisation signal may be indicative of initialisation of the second qubit. The second initialisation signal may be transmitted by the second node based on a second allowed transmission time selected from the set of allowed transmission times. The execution by the first node further includes determining the second allowed transmission time based on the second initialisation signal. For example, the second allowed transmission time may be determined based on a time of reception by the first node of the second initialisation signal or based on a payload of the second initialisation signal. The execution by the first node further includes determining the start time for the first qubit operation based on the first and second allowed transmission times, the transmission delay information and the synchronised clock.

Thus, the method may be considered to comprise a calibration stage, during which the nodes are provided with transmission delay information, and an execution stage, during which the nodes may initialise and manipulate the qubits. Typically, the calibration stage is performed only once for a large number of iterations of the execution stage.

The transmission delay information may comprise a first maximum transmission delay fo.max representative for signals transmitted from the first node to the second node, and a second maximum transmission delay fei.max representative for signals transmitted from the second node to the first node. A maximum transmission delay may represent an amount of time within which substantially all signals can be assumed to have arrived at its destination. The maximum transmission delay may include a safety margin, i.e. , the maximum transmission delay may be larger than the longest expected transmission delay, but is preferably based on and relatively close to actually measured and/or modelled transmission delays.

The transmission delay information may comprise a first transmission delay interval and a first maximum transmission delay indicative of a transmission delay distribution from the first node to the second node, and a second transmission delay interval and a second maximum transmission delay indicative of a transmission delay distribution from the second node to the first node.

For example, a first transmission delay interval Ati2 can be representative for signals transmitted from the first node to the second node, and a second transmission delay interval Afei can be representative for signals transmitted from the second node to the first node. The first transmission delay interval Ati2 may be based on the first maximum transmission delay fo.max and, optionally, a first minimum transmission delay ti2,min. Similarly. The second transmission delay interval Afei may be based on the second maximum transmission delay fei.max and, optionally, a second minimum transmission delay fei.min.

In some embodiments, the transmission delays between the two nodes may be assumed to be symmetric or approximately symmetric, and in that case a single transmission delay distribution may be used to represent transmission delays in both directions. Hence, in these embodiments, only one maximum transmission delay, preferably based on the first and/or second maximum transmission delays, needs to be included in the transmission delay information. Similarly, in these embodiments, only one transmission delay interval, preferably based on the first and/or second transmission delay intervals, needs to be included in the transmission delay information. In some cases, a transmission delay interval between two nodes may stretch from an implied minimum transmission delay (e.g., zero) to the maximum transmission delay between those nodes. In such a case, a single value may be used to determine both a transmission delay interval and a maximum transmission delay. In a more typical case, however, the transmission delay interval will be substantially smaller than the maximum transmission delay.

As was explained above, the method’s execution stage may comprise a preparatory stage and, optionally, an operational stage. During the preparatory stage, each node involved in the synchronised event may initialise its qubit and transmit an initialisation signal to at least the other nodes involved in the synchronised event, the initialisation signal being indicative of completion of the initialisation. For example, when the first node has initialised the first qubit, the first node may transmit the first initialisation signal to the second node (and, optionally, further nodes) at a first allowed transmission time ft, ready. Preferably, the allowed transmission time immediately following the initialisation of the first qubit is selected.

During the preparatory stage, each node involved in the synchronised event listens for an initialisation signal from all other nodes involved in the synchronised event. For example, the first node may listen for a second initialisation signal from the second node, the second signal indicating that the second node has initialised the second qubit. When the second initialisation signal has been received, the first node may determine the second allowed transmission time fe, ready using the synchronised set of allowed transmission times {tready} and, optionally, the transmission delay information.

The point in time when the first node has received the second initialisation signal from the second node may be determined by computing fe, ready + fei.max. In general, the point in time when a node j has received a signal from a node / may be determined by computing t,, ready + ty, max. As used herein, the point in time when a node has received a signal refers, unless otherwise specified, to the point in time that the node has received the signal with a predetermined probability, that is, the point in time where the node is practically guaranteed to have received the signal. This point in time may also be referred to as the latest point in time that the node has potentially received the signal. Generally, the actual reception of the signal is at an earlier point in time.

Thus, based on the determined allowed transmission times t,, ready for all nodes / involved in the synchronised event and based on the transmission delay information for each pair of nodes (/, j) involved in the event (typically the maximum transmission delays ty, _ma x), each node may determine the synchronised start time /start. Since all nodes have the same information, they can independently arrive at the same synchronised start time /start. The synchronised start time /start can be determined such that each node can be sure that the synchronised start time /start is later than the point in time when each node involved in the synchronised event has received the initialisation signal from all other nodes involved in the synchronised event. In particular, the synchronised start time /start can be determined to be later than the moment in time that the first node has received the second initialisation signal and later than the moment in time that the second node has received the first initialisation signal. For example, the synchronised start time /start may be determined based on the first transmission time fi , ready and the first maximum transmission delay fo.max, and on the second allowed transmission time fe, ready and the second maximum transmission delay fei .max; e.g., by selecting the synchronised start time /start such that:

/start — max(/i ,ready + /l2,max, /2, ready + /21 ,max) (1)

In general, the synchronised start time may be determined to be any time later than max(/,, ready + ty, max) for all pairs of nodes (/, _/) involved in the synchronised event.

In an embodiment, the method further comprises an operational stage, during which each node involved in the synchronised event, and in particular the first node, performs the following step: manipulating the first qubit in accordance with the qubit operation, the timing of the manipulation being based on the determined synchronised start time /start and on a signal from the synchronised clock.

As used in this disclosure, determining a value may refer to any method leading to the ability to use that value, and may include computing the value, receiving the value from a different device, or using a predetermined value. For example, the second transmission delay may be computed by the second node, and the computed value may be transmitted to the first node, which may subsequently store the computed value. Thus, both the first node and the second node are understood to have determined the second transmission delay. Similarly, the synchronised set of allowed transmission times can be determined by each node according to a deterministic algorithm, or may be determined centrally and distributed to the nodes.

As used in this disclosure, a node refers to a quantum node in a quantum network, unless the context implies otherwise. In a typical embodiment, each node comprises a qubit and has access to a synchronised clock, preferably a synchronised high- precision clock, for example based on CERN’s White Rabbit design. The synchronised clock should have a timing jitter that is lower than the desired synchronisation accuracy.

As used in this disclosure, a transmission delay is the amount of time between a transmitting device beginning to transmit a signal and a receiving device registering the signal. The transmission delay can be determined by measuring one or more signals using a function generator and a time tagger. The maximum transmission delay may refer to a delay such that at least a predetermined percentage of signals is transmitted within the maximum transmission delay and need not be a maximum in an absolute sense. Similarly, the minimum transmission delay may refer to a delay such that at least a predetermined percentage of signals is transmitted with a transmission delay larger the minimum transmission delay and need not be a minimum in an absolute sense. The percentage of signals with a delay larger than the maximum transmission delay or smaller than the minimum transmission delay is typically substantially zero, e.g. less than 1 in million, less than 1 in 10 ¹² , or even smaller. The minimum and/or maximum transmission delays may be based on a statistical representation of transmission delays, e.g., the maximum transmission delay can be determined as a mean transmission delay plus six times a standard variation of the transmission delay, and the minimum transmission delay can be determined as the mean transmission delay minus six times the standard variation of the transmission delay.

Typically, the calibration stage is performed only once, or periodically (e.g., every minute, every hour, every day, or even every month), while the execution stage (typically comprising a preparatory stage and an operational stage) are repeated a large number of times, typically, tens to hundreds or even millions of times for each calibration stage.

By employing this method, events at different nodes in a quantum network may be synchronised with high precision and efficiency. Analysis of such an approach has shown that for a realistic two-node system, the described method leads to an entanglement attempt rate that is roughly 40% higher than an approach without communication.

In an embodiment, determining the transmission delay information comprises determining a minimum transmission delay between the first and second nodes. In such an embodiment, the synchronised set of allowed transmission times may be based on a transmission delay interval. The transmission delay interval is preferably based on the interval between the minimum transmission delay and the maximum transmission delay.

For example, determining the first transmission delay interval Ati ₂ may comprise determining a minimum transmission delay fi ₂ , _min between the first node and the second node. In such an embodiment, the delay interval Ati ₂ is preferably being based on a difference between the minimum and maximum transmission delays, e.g., Ati ₂ = fi ₂ , _m ax - fi2,min. The minimum transmission delay may e.g. be determined based on measurements, or based on a known relationship with the maximum transmission delay.

Determining the transmission delay interval based on the minimum transmission delay allows for smaller transmission delay intervals and hence for smaller intervals between subsequent allowed transmission times. This in turn can increase the rate with which the manipulation of the qubit may be executed.

In an embodiment, the time interval between two subsequent transmission times in the synchronised set of allowed transmission times is larger than the transmission delay interval, preferably larger than the largest transmission delay interval between all nodes involved in the synchronised event. Preferably, the time interval between subsequent allowed transmission times is a constant.

Thus, the synchronised set of allowed transmission times {/ready} may be defined such that each interval At _N between two subsequent allowed transmission times is larger than the maximum of the first transmission delay interval Ati2 and the second transmission delay interval Afei, e.g. by selecting At _N such that At _N > max(Ati2, Afei).

For example, the synchronised set of allowed transmission times may be defined as fo.N + n Afo, with fo.N being a synchronised reference time, n being an integer number, and At _N being a constant larger than the largest transmission delay interval comprised in the transmission delay information and/or between all nodes involved in the synchronised event. Preferably, At _N is related to the clock rate of the synchronised high- precision clock, e.g. an integer multiple of the clock rate. A constant interval between allowed transmission times results in an easy to compute synchronised set of allowed transmission times.

In an embodiment, determining the second allowed transmission time is based on a reception time when the second initialisation was received and on the transmission delay distribution. For example, determining the second allowed transmission time fe. ready may be based on a second transmission delay f ₂ i, e.g., based on the second maximum transmission delay fei.max or the second minimum transmission delay fei.min. More in particular, the second allowed transmission time may be determined as the allowed transmission time immediately after the time of reception fei , receive of the second initialisation signal minus the second maximum transmission delay, for example by computing: fe, ready ^— Aftrans ' Ceil( (f ₂ 1.receive ^— fcl .max) I Aftrans ) . (2)

Alternatively, the second allowed transmission time may be determined as the allowed transmission time immediately before the time of reception minus the second minimum transmission delay, for example by computing:

^2, ready ⁼ Aftrans ' flOOr( (fel .receive ^— fcl ,min ) I Aftrans ) . (3)

Alternatively, the second allowed transmission time may be determined as the allowed transmission time closest to the time of reception minus the average of the second minimum transmission delay fei.max and the second maximum transmission delay fei.min, for example by computing: fe, ready ^— Aftrans ' TOUnd( (f ₂ 1 .receive ^— (fcl ,min + t21 ,max)/2) I Aftrans ) . (4)

This way, the initialisation signal does not need to comprise a payload comprising an indication of the second allowed transmission time. For example, the initialisation signal can be a binary signal, e.g. a single high or low voltage. In a different embodiment, the second initialisation signal comprises a payload, which may comprise an indication of the second allowed transmission time fe, ready. This can be used if e.g. a packet-based network is used to send the initialisation signals. In such an embodiment, determining the second allowed transmission time (i.e. , the time that the second node sends the second initialisation signal to the first node) may be based on the payload of the second initialisation signal. For example, determining the second allowed transmission time fe, ready may be based on processing the payload of the second initialisation signal. This may reduce the computational burden.

In an embodiment, the synchronised start time is selected from a predetermined set of start times. Preferably, the predetermined set of start times is equally distributed over time. For example, the set of start times may be defined as to, _s + m At _s , with to, _s being a synchronised reference time, m being an integer number, and At _s being a constant. Preferably, At _s is related to the clock rate of the synchronised clock, e.g. an integer multiple of the clock rate. This way, additional uncertainty regarding the start time may be removed.

In an embodiment, execution of the first qubit operation further comprises arming a signal generator based on the start time; generating a start signal in response to the armed signal generator receiving a signal from the synchronised clock; and executing the first qubit operation in response to receiving the start signal.

In such an embodiment, each node may comprise a high-precision signal generator, e.g., a high-precision wavefront generator, which typically initiates all timing- critical operations of a quantum node. Thus, the timing of the manipulation may be based upon reception of a high-precision start signal. The arming of the signal generator is less time-sensitive than the actual starting moment, and the arming window may be selected sufficiently large based on the properties of the microcontroller determining the start time.

In a typical set-up, the quantum event (e.g., entangling) takes place at an intermediate location, usually approximately halfway between the involved nodes. If the propagation time to the intermediate location is not equal for all involved nodes, the timing at the nodes may be adjusted to compensate for this difference in propagation time and to ensure a synchronised event at the intermediate location.

Therefore, in an embodiment, the method may further comprise receiving or determining an intermediate transmission delay difference to an intermediate location between the first node and the second node, the intermediate transmission delay difference being based on a first intermediate transmission delay between the first node and the intermediate location and on a second intermediate transmission delay between the second node and the intermediate location. In such an embodiment, the start time may be based on the determined intermediate transmission delay difference. For example, a first intermediate transmission delay fi,i _M may be defined as the transmission delay between the first node and the intermediate location and, a second intermediate transmission delay f ₂ ,iM may be defined as the transmission delay between the second node and the intermediate location. The intermediate transmission delay difference Afo.iM to an intermediate location IM between the first node and the second node may then be based on or defined as Afo.iM = tzjM - ft ,IM- In such an embodiment, the timing of the manipulation may be additionally based on the determined intermediate transmission delay difference Afo.iM.

This way, the timing at the nodes may be adjusted to ensure a synchronised event at the intermediate location.

In an embodiment, the plurality of nodes comprises exactly two nodes. This is a relatively simple and easy to implement embodiment.

In a different embodiment, the event is synchronised between more than two nodes. In such an embodiment, the method steps described above are performed for each pair of nodes involved in the synchronised event.

In a second aspect, the disclosure relates to a node comprising a qubit and a processor, the node having access to a synchronised high-precision clock, the node being communicatively coupled to one or more other nodes in a quantum network. The processor may be configured to perform method steps comprising receiving or determining transmission delay information about a transmission delay interval between a first node comprising a first qubit and a second node comprising a second qubit, the transmission delay information comprising a maximum transmission delay; and receiving or determining a set of allowed transmission times based on the transmission delay information and based on a synchronised clock shared between the nodes. The method steps may further comprise executing, by the first node, a first qubit operation based on a start time. The execution by the first node may include: initializing the first qubit and transmitting a first initialisation signal to the second node based on a first allowed transmission time selected from the set of allowed transmission times; receiving a second initialisation signal from the second node, the second initialisation signal being indicative of initialisation of the second qubit and the second initialisation signal being transmitted by the second node based on a second allowed transmission time selected from the set of allowed transmission times; determining the second allowed transmission time based on the second initialisation signal; determining the start time for the first qubit operation based on the first and second allowed transmission times, the transmission delay information and the synchronised clock.

In an aspect, the disclosure relates to a quantum network of nodes comprising a plurality of nodes as described above.

One aspect of this disclosure relates to a computer comprising a computer readable storage medium having computer readable program code embodied therewith, and a processor, preferably a microprocessor, coupled to the computer readable storage medium, wherein responsive to executing the computer readable program code, the processor is configured to perform any of the methods described herein.

One aspect of this disclosure relates to a computer program or suite of computer programs comprising at least one software code portion or a computer program product storing at least one software code portion, the software code portion, when run on a computer system, being configured for executing any of the methods described herein.

One aspect of this disclosure relates to a non-transitory computer-readable storage medium storing at least one software code portion, the software code portion, when executed or processed by a computer, is configured to perform any of the methods described herein.

As will be appreciated by one skilled in the art, aspects of the present invention may be embodied as a system, method or computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a "circuit," "module" or "system”. Functions described in this disclosure may be implemented as an algorithm executed by a microprocessor of a computer. Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable medium(s) having computer readable program code embodied, e.g., stored, thereon.

Any combination of one or more computer readable medium(s) may be utilized. The computer readable medium may be a computer readable signal medium or a computer readable storage medium. A computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing. More specific examples (a non- exhaustive list) of the computer readable storage medium would include the following: an electrical connection having one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a computer readable storage medium may be any tangible medium that can contain, or store a program for use by or in connection with an instruction execution system, apparatus, or device.

A computer readable signal medium may include a propagated data signal with computer readable program code embodied therein, for example, in baseband or as part of a carrier wave. Such a propagated signal may take any of a variety of forms, including, but not limited to, electro-magnetic, optical, or any suitable combination thereof. A computer readable signal medium may be any computer readable medium that is not a computer readable storage medium and that can communicate, propagate, or transport a program for use by or in connection with an instruction execution system, apparatus, or device.

Program code embodied on a computer readable medium may be transmitted using any appropriate medium, including but not limited to wireless, wireline, optical fiber, cable, RF, etc., or any suitable combination of the foregoing. Computer program code for carrying out operations for aspects of the present invention may be written in any combination of one or more programming languages, including a functional or an object oriented programming language such as Java(TM), Scala, C++, Python or the like and conventional procedural programming languages, such as the "C" programming language or similar programming languages. The program code may execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer, or entirely on the remote computer, server or virtualized server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider).

Aspects of the present invention are described below with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor, in particular a microprocessor or central processing unit (CPU), or graphics processing unit (GPU), of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer, other programmable data processing apparatus, or other devices create means for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks.

These computer program instructions may also be stored in a computer readable medium that can direct a computer, other programmable data processing apparatus, or other devices to function in a particular manner, such that the instructions stored in the computer readable medium produce an article of manufacture including instructions which implement the function/act specified in the flowchart and/or block diagram block or blocks.

The computer program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other devices to cause a series of operational steps to be performed on the computer, other programmable apparatus or other devices to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks.

The flowchart and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present invention. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustrations, and combinations of blocks in the block diagrams and/or flowchart illustrations, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions.

The invention will be further illustrated with reference to the attached drawings, which schematically will show embodiments according to the invention. It will be understood that the invention is not in any way restricted to these specific embodiments. Identical reference signs refer to identical, or at least similar elements.

Brief of the

Aspects of the invention will be explained in greater detail by reference to exemplary embodiments shown in the drawings, in which:

Fig. 1 depicts a quantum network comprising a plurality of quantum nodes according to an embodiment of the invention;

Fig. 2A and 2B schematically depict methods of determining a synchronised start time according to embodiments of the invention;

Fig. 3A-3C depict various topologies for a quantum network comprising a plurality of quantum nodes according to embodiments of the invention;

Fig. 4 depicts a flow diagram for synchronising events at a plurality of nodes in a quantum network according to an embodiment of the invention; and

Fig. 5 is a block diagram illustrating an exemplary data processing system that may be used for executing methods and software products described in this application. Detailed description

The methods described herein allow for efficient, high-precision synchronisation of events at a plurality of nodes in a quantum network. It is noted that although the depicted examples relate to entanglement, the methods are equally applicable to other systems or other events wherein high-precision synchronisation is required.

Fig. 1 depicts a quantum network comprising a plurality of quantum nodes according to an embodiment of the invention. A quantum network 100 comprises quantum nodes 102,122 that are geographically distant, e.g. being a distance with an order of magnitude of 1-100 km away from each other. Each quantum node comprises one or more qubits 110,130, and equipment to prepare (initialise) and manipulate these qubits using e.g. lasers, microwaves and/or other electromagnetic fields. This equipment is typically controlled by a signal generator 108, e.g., a high-precision wavefront generator. The signal generator can ensure accurate timing of manipulations within a single quantum node.

For each (local) qubit, the respective quantum node can be configured to initialise the qubit to a well-known quantum state; to manipulate the quantum state of the qubit; and to perform readout on the qubit which collapses its quantum state.

It is an aim of quantum nodes in a quantum network to jointly establish quantum events, e.g. entanglement of photons associated with the respective quantum nodes’ qubits. Indeed, a prerequisite of many quantum networking applications is establishment of entanglement between qubits of different quantum nodes. Entanglement is typically achieved at an intermediate location 142, which is generally positioned at a central location between the quantum nodes involved in the entanglement, for instance roughly halfway between two quantum nodes. The intermediate location may comprise further equipment to facilitate a quantum event such as entanglement.

The quantum nodes may further comprise a communication interface 112,132. A quantum node may use the communication interface to communicate with each other node in the network using a classic (i.e. , non-quantum) communication channel 140. For example, each node may have access to a unidirectional communication channel towards each other node, and consequently, each node may also receive unidirectional communication channels form each other node. Alternatively, bidirectional message channels may be used, or a different communication network topology. Several possible network topologies are discussed in more detail below with reference to Fig. 3A-C.

Thus, using these communication channels, each node can send messages to each other node and can receive messages from each other node. Preferably, precautions are taken to ensure that messages that are sent out will always arrive, and that no spurious messages will be injected. Practically speaking, this can be achieved by using a suitable channel encoding known in the art, or by using dedicated signal channels. The quantum nodes also have access to a quantum communication channel 144 for allowing the qubits to interact with each other. As was mentioned above, an interaction may comprise an event, e.g. an entanglement, at the intermediate location. Consequently, the quantum communication channel may comprise a plurality of partial connections of a quantum node with the intermediate location. The classic communication channel and the quantum communication channel can be physically the same channel, e.g. a glass fibre with a time sharing protocol, or they can be physically separate channels. In general, a dedicated quantum communication channel is preferred.

Furthermore, each quantum node comprises, or at least has access to, a synchronised clock 106,126, preferably a synchronised high-precision clock. A set of (mutually) synchronised clocks may be known as a global clock ort a shared clock. Typically, a synchronised clock counts time as a discrete number of ticks since a shared global time t ₀ = 0. For example, the synchronised clock may count with a frequency of 10 MHz, with a timing jitter in the order of 20 ps. The synchronised clocks provide a shared time base to the quantum nodes despite the quantum nodes being geographically distant. Thus, the time of any event in each node, and by extension in the quantum network, has a well-defined value relative to the global clock, that is understood and interpreted the same at all involved nodes.

Proper time distribution mechanisms are known in the art. Moreover, equipment to implement such a global clock infrastructure is commercially available; for example, time-distribution network switches based on CERN’s White Rabbit design may be used. The synchronised clock should have a timing jitter that is lower than the desired synchronisation accuracy.

In the depicted example, the quantum network comprises a first quantum node 102 and a second quantum node 122. Parameters associated with the first quantum node may be indicated with a subscript label “1”, and parameters associated with the second quantum node may be indicated with a subscript label “2”.

A quantum event, such as a process to establish entanglement may comprise a calibration stage and an execution stage. The execution stage may comprise a preparatory stage and an operational stage. Each stage may generally only begin after the previous stage has been completed by all nodes involved in the quantum event. Typically, the calibration stage is performed only once for a large number of iterations of the execution stage.

A transmission delay between a transmitting device and a receiving device may be defined as the amount of time between the transmitting device beginning to transmit a signal and the receiving device registering the signal. The transmission delay may sometimes be referred to as a propagation time. The transmission delays of signals or messages along a classical communication channel exhibit some variation due to, e.g., timing jitter and long-term drift. The possible or probable transmission delays can be determined and characterised for each one-way channel in the system. It is not necessarily the case that the characterisations are symmetrical, that is, the characterisation of transmission delays from a first node to a second node can be different from the characterisation of transmission delays from the second node to the first node. However, these characterisation will often be (very) similar in real-life networks.

During the calibration stage, the quantum nodes determine or receive transmission delay information comprising a transmission delay distribution between each pair of nodes. The transmission delay distribution for a pair of nodes may define a range of possible or probable transmission delays of a signal or message being transmitted from the first node of the pair to the second node of the pair and from the second node to the first node.

The transmission delay distribution may be described using statistical descriptors. In general, it may be assumed that a transmission delay ty between two nodes / and j is variable, but bounded, i.e., fj; _min ty < f^max with fj; _min and t^max denoting, respectively, a minimum and maximum transmission delay. The maximum transmission delay may refer to a delay such that at least a predetermined percentage of signals is registered before the maximum transmission delay and need not be a maximum in the absolute sense. Similarly, the minimum transmission delay may refer to a delay such that at least a predetermined percentage of signals is registered after the minimum transmission delay and need not be a minimum in the absolute sense.

The maximum and/or minimum transmission delays may include a ‘safety margin’, that is, the value for the maximum transmission delay t^max may be larger than the largest transmission delay measured during determination of the transmission delay distribution. This safety margin can be based, for example, on a fixed amount, on a percentage of the largest measured transmission delay, or on the transmission delay interval. Similarly, the minimum transmission delay may be smaller than the smallest transmission delay measured during determination of the transmission delay distribution.

Based on these values, a transmission delay interval At,; between two nodes / and j can be determined as a difference between the maximum transmission delay ty, _ma x and the minimum transmission delay fy.min between those nodes, i.e.,

Thus, the transmission delay interval between two nodes can be a time interval within which a predetermined percentage (typically, substantially all) of signals may be communicated between those two nodes. As each node is assumed to be aware of its own status, the transmission delay from a node to itself may be taken to be zero, i.e.:

For example, in a quantum network with two quantum nodes, there are two non-zero transmission delay intervals: where Ati2 denotes the transmission delay interval for signals from the first node to the second node and Afei denotes the transmission delay interval for signals from the second node to the first node.

Thus, each node may determine and/or receive transmission delay information about the transmission delay distribution. The transmission delay information comprises at least the maximum transmission delay ty, _ma x for each pair of nodes /, j (j # j), or information based on which each maximum transmission delay ty, _ma x can be determined. The transmission delay information furthermore generally comprises the transmission delay interval At,; for each pair of nodes /, j (j # j), or information based on which each transmission delay interval At,j can be determined. In general, the transmission delay information comprises parameters defining a transmission delay distribution for each pair of nodes /, j (j # j). Typical examples of parameters are a set {fy, min, f/j',max} comprising the minimum transmission delay fy.min and the maximum transmission delay ty, _max ; a set {fy.max, At,;} comprising the maximum transmission delay ty, _max and a transmission delay interval At#; or a set {pa, Oij} comprising a mean transmission delay M and a standard deviation oj,.

For example, in a quantum network with two nodes, the transmission delay information may comprise a first transmission delay interval Ati2 and a first maximum transmission delay fo.max indicative of a transmission delay distribution from the first node to the second node, and a second transmission delay interval Afei and a second maximum transmission delay fei.max indicative of a transmission delay distribution from the second node to the first node.

As the minimum transmission delay between two nodes is always at least zero, in principle, a value of fy.min = 0 could be assumed. In that case, only the maximum transmission delay ty, m _ax needs to be known, as the transmission delay interval A^ would be assumed to be equal to the maximum transmission delay, reducing the need for data transfer between the nodes. However, it is generally considerably more efficient to use a more accurate value for tj-.min, as this allows for a smaller value for the transmission delay intervals and hence (as will be explained below) for smaller intervals between subsequent allowed transmission times. A smaller interval between subsequent allowed transmission times relates to an increase in the rate with which the manipulation of the qubit may be executed, and thus network throughput.

It should be noted that the transmission delays, and thus the values for, e.g., the maximum transmission delay ty, _ma x and the transmission delay interval At,;, may depend on the time window over which they have been determined. In particular, the values for ty, _ma x and Ata may vary over time. For example, due to thermal expansion, wires used for communication may be longer in the summer than in the winter, and longer during the day than during the night. Consequently, updated values for ty, m _ax and At; may need to be determined periodically. Frequent updating may lead to more accurate values and typically to smaller transmission delay intervals. If sufficient data are available, measurements may be supplemented by model-based modification to account for, e.g., diurnal and seasonal variation.

Typically, the transmission delay between two nodes is approximately, but not completely symmetric, that is, in general, although, e.g., fo.max “ fei .max it is not true that fo.max = fei .max. The same is true for the other parameters describing the transmission delay distribution. Therefore, the transmission delay distribution may be determined separately for each direction. This may result in a slightly more efficient method. Alternatively, since the transmission delays between two nodes are generally approximately symmetric, the same values may be used to describe the transmission delay distribution in both directions, reducing the number of parameters that needs to be determined and/or shared between the nodes.

When a single set of parameters is used to describe transmissions between two nodes in both directions, care should be taken that the set of parameters correctly describes both connections. For example, the maximum transmission time t _ma x for both directions should be larger than or equal to the largest maximum transmission time for a single direction, which can be achieved by computing, e.g., t _max = rnax(ti ₂ ,max, t21 ,max) , Or tmax ^— fi2,max + 5t, where 5t denotes a ‘safety margin’ that is larger than a presumed maximum difference between ti ₂ , _m ax and fei.max. Similarly, the transmission time interval for both directions should be larger than or equal to the largest transmission time interval for a single direction.

The transmission delay information may be determined, for instance, by the nodes or by a central entity. Determining a transmission delay distribution may comprise determining a statistical description of a number of signal transmissions, e.g. a minimum and maximum transmission delay; a mean transmission delay and a standard deviation; or a combination thereof, such as a minimum and maximum transmission delay where measurements that are more than six standard deviations from the mean transmission delay are discarded. The transmission delay information may be based on measurements, models, or combinations therefore. The transmission delay information may be determined, for example, by doing an all-to-all channel characterization, and distributing this information to all nodes. In general, measurements may be more precise, but especially for very large networks, model-based distributions can be more efficient, as the number of node-to-node channels grows quadratically with the number of nodes.

In some embodiments, the signals exchanged by the nodes using the classical communication channel do not comprise a payload or any other content. In these embodiments, only the timing of the signals is used to convey information. This type of messaging can easily be implemented using a channel, e.g., a physical medium like a wire, that can transport a two-level signal (e.g., a high voltage and a low voltage). Other embodiments may use a signal or message with a payload so that additional information may be shared.

The send time may then be determined by the leading edge of a short pulse emitted by the sender, while the receive time may be determined by the time the pulse’s leading edge is observed by the receiver.

During the calibration stage, the quantum nodes furthermore determine or receive a synchronised set of allowed transmission times {tans} defining a set of discrete moments in time when the nodes are allowed to transmit, e.g., broadcast, an initialisation signal to the other nodes. The initialisation signal, as will be explained in more detail below, indicates that a node has completed preparing its qubit. By discretizing the times at which nodes may transmit the initialisation signal to the other nodes in the system, it is possible to correct for timing variations induced by variation in transmission delays in the classic communication channels.

The synchronised set of allowed transmission times {tans} may be based on the determined transmission delay intervals. In particular, each transmission interval Atans, which is defined as the difference between two consecutive allowed transmission times, should be larger than each of the transmission delay intervals At,; for all nodes /, j in the network, or equivalently, larger than the largest transmission delay interval, i.e.: where N is the number of nodes in the network. So, for example, in a network with two nodes, for all times in {tans}, The synchronised set of allowed transmission times {ftrans} may further be based and on the properties of the synchronised clocks, in particular on the clock period Atciock, where Atciock = 1 / fciock with fciock denoting the clock frequency. For example, for a clock with a clock frequency of 10 MHz, the clock period is Afciock = 1 / (10 MHz) = 100 ns. The synchronised set of allowed transmission times {ftrans} is typically chosen as a regular periodic (one-dimensional) grid, i.e. , the transmission interval Aftrans is generally a constant. A constant transmission interval results in an easy to determine set of allowed transmission times. In such a case, the set of allowed transmission times {ftrans} may be defined as wherein to denotes a synchronised reference time, n a positive integer, and Aftrans denotes a constant transmission interval that is larger than each of the transmission delay intervals At,; for all nodes /, j involved in the event. Preferably, the transmission interval Aftrans is an integer multiple of the clock period Afciock of the synchronised clock.

Preferably, m is the smallest positive integer for which condition (9) holds. However, other embodiments may use a different (larger) value of m, which may be based on the particulars of the used hardware.

If the nodes have access to all transmission delay intervals, they may each determine the set of allowed transmission times {ftrans} themselves. In other embodiments, the set of allowed transmission times {ftrans} may be determined centrally and distributed to the nodes. In that case, the transmission delay information need not comprise the transmission delay intervals.

Thus, at the end of the calibration stage, each node has access to the same set of maximum transmission delays ty, _ma x for all nodes /, j in the network, and to the same set of allowed transmission times {ftrans}. Typically, the calibration stage is performed periodically (e.g., every day or every hour). Typically, the calibration stage is performed only once for a large number of repetitions of the execution stage. The execution stage comprises a preparatory stage during which the qubits are prepared (initialised). The qubits must wait until they are sure that all nodes have finished the preparatory stage, before they can (simultaneously) start the operational stage.

During the preparatory stage, each quantum node may prepare its local qubit. The duration of the preparation is unpredictable, but the quantum node can determine when the local preparation is finished. When the preparation is finished, the quantum node waits for the a time point (preferably the first time point) that is in the set of allowed transmission times {ftrans}, and then sends a signal to all other quantum nodes involved in the event, the signal indicating that the quantum node is ready. The signal indicating that the quantum node is ready may be referred to as a initialisation signal. The time point that a node / transmits its initialisation signal may be referred to as the ready time ft, ready! for example, the first node may transmit its initialisation signal at ft , ready and the second node may transmit its initialisation signal at fe, ready, where ft , ready and t ₂ . ready are times points from the set of allowed transmission times {ftrans}.

As was explained above, the execution stage may comprise a preparatory stage and, optionally, an operational stage. During the preparatory stage, each node involved in the synchronised event may initialise its qubit and transmit an initialisation signal to at least the other nodes involved in the synchronised event, the initialisation signal being indicative of completion of the initialisation. For example, when the first node has initialised the first qubit, the first node may transmit the first initialisation signal to the second node (and, optionally, further nodes) at a first allowed transmission time ft, ready. Preferably, the allowed transmission time immediately following the initialisation of the first qubit is selected.

The initialisation signal can be a ‘bare’ signal, e.g. a single bit not encoding any further information, or a message with a payload. If the initialisation signal comprises a payload, the payload may comprise the ready time of the node transmitting the initialisation signal.

During the entire preparatory stage (that is, both when the quantum node is preparing its local qubit and when it has finished doing so), each quantum node / listens for initialisation signals from the other quantum nodes, indicating that the other quantum nodes have finished preparing their respective qubits. Each quantum node / then determines the ready times tj, ready (j t i) for the other quantum nodes. For example, the first node may listen for an initialisation signal from the second node, the initialisation signal indicating that the second node has prepared the second qubit. When the initialisation signal has been received by the first node, the first node may determine the second ready time fe, ready.

If the initialisation signal comprises a payload, the ready time tj, ready may be determined based on the payload of the respective initialisation signal. Otherwise, the ready time tj, ready of a quantum node j can be determined based on the reception time of the initialisation signal and the transmission delay distributions in the following way.

Each quantum node / may determine a reception time tj,, receive at which the initialisation signal sent by node j was received by node /; e.g. the first quantum node may determine the reception time fei , receive when the initialisation signal transmitted by the second node is registered by the first node. Based on the reception time tj,, receive, the ready time tj, ready of the quantum node j can then be determined using the transmission delay distribution and the set of allowed transmission times {ft _ra ns} and/or the transmission interval Aft _ra ns. For example, the ready time tj, ready may be determined, by each node /, using, e.g.: where round() denotes a function that returns the nearest integer.

For example, in a network with two quantum nodes, the first quantum node may compute and the second quantum node may compute:

The ready time tj, ready may also be determined using alternative computations, such as: where floor() denotes a function returning the largest integer smaller than the function’s argument and ceil() denotes a function returning the smallest integer larger than the function’s argument.

It is possible to perform these calculations using integer arithmetic only, which can be advantageous for a high-speed implementation on a microcontroller. An advantage of using equation (13), based on rounding, to determine the readiness time is that it is robust for transmission delay values that are outside the interval ((/,-. min, tji, max ) , provided they are less than Aftrans/2 away from the average transmission delay (tji, mm + tji, _ma x)/2. However (as will be discussed below in more detail), if the actual transmission tji delay is (much) larger than the assumed maximum transmission delay tji, max, the receiving node / may be too late to start the operational stage at the determined synchronised start time.

Based on the set of readiness times t,. ready for all quantum nodes / and the maximum transmission delays ty.max for all pairs of quantum nodes (/, _/), a synchronised start time /start may be determined. In general, a point in time t,. known that all nodes j are guaranteed to have received a initialisation signal sent by node / can be determined by where /V denotes the number of nodes in the network. Consequently, the point in time t _a ii_known that all nodes / are guaranteed to have received a initialisation signal sent by all other nodes j can be determined by:

This point in time t _a ii_known can be calculated by all quantum nodes once they have received the initialisation signals from all other quantum nodes. Since all nodes have the same information, all quantum nodes will arrive at the same value for Un known.

For example, in a quantum network with two quantum nodes, and

The synchronised start time f _s tart may then be determined based on the time point Un known. For example, the synchronised start time can be the point in time that all nodes are guaranteed to have received all initialisation signals form all other nodes: fstart ^— fall_known- However, to allow for the computation time, the synchronised start time f _s tart is preferably a later moment in time, for example based on a fixed delay: fstart ^— fall_known + fmargim In other embodiments, the start times may be limited to a discretised set of potential start times, similar to the discretised set of allowed transmission times. By using a set of potential start times, additional uncertainty regarding the start time may be removed.

Preferably, the potential start times are equally distributed over time, i.e. , the set of potential start times forms a regular periodical (one-dimensional) grid. Typically, the start times may be based on the clock period of the synchronised clock Atciock, e.g.,

In other embodiments, k may be chosen in larger multiples, e.g., k = 2, 4, 6, ... or k = 8, 16, 24, ... , et cetera. The set of start times may be derived from the clock signal using a frequency divider. The values for k can, e.g., be based on the clock period and on the properties of the electronics used to perform the computations.

All nodes then wait until the time equals the synchronised start time fstart, at which moment they simultaneously start the operational stage. Starting the operational stage may comprise arming the signal generator previous to the synchronised start time, preferably in the interval between the start time immediately preceding the synchronised start time and the (actual) synchronised start time. The armed signal generator may then generate a start signal in response to receiving a signal from the synchronised clock. A first qubit manipulation may then be executed in response to receiving the start signal. Thus, the timing of the manipulation may be based upon reception of a high-precision start signal.

The operational stage may comprise each quantum node manipulating its local qubit in accordance with the qubit operation. The timing of the manipulation can be based on the determined synchronised start time /start and on a signal from the synchronised clock. The operational stage may also comprise a waiting period to allow for travel time differences between each quantum node and an intermediate location where the quantum event, e.g., entangling, may take place.

In an embodiment, the method may further comprise receiving or determining an intermediate transmission delay difference to an intermediate location between the first node and the second node. The intermediate transmission delay difference can be based on a first intermediate transmission delay between the first node and the intermediate location and on a second intermediate transmission delay between the second node and the intermediate location. In such an embodiment, the start time may be based on the determined intermediate transmission delay difference.

For example, a first intermediate transmission delay /ij _M may be defined as the transmission delay between the first node and the intermediate location, and a second intermediate transmission delay / ₂ ,IM may be defined as the transmission delay between the second node and the intermediate location. The intermediate transmission delay difference A/I2,IM to an intermediate location IM between the first node and the second node may then be based on or defined as A/I2,IM = tzjM - fi ,IM. In such an embodiment, the timing of the manipulation may be based, additionally, on the determined intermediate transmission delay difference A/I ₂ ,IM.

This way, the timing at the nodes may be adjusted to ensure a synchronised event at an intermediate location even when the transmission delays to the intermediate location are not equal for all involved nodes.

After the operational stage is concluded, the quantum nodes may begin a new preparatory stage.

Fig. 2A and 2B schematically depict methods of determining a synchronised start time according to embodiments of the invention. In particular, Fig. 2A depicts a plurality of time-related parameters for determining a synchronised starting time /start for a quantum network with two quantum nodes, node 1 and node 2. A time line 202 is subdivided based on counts of the synchronised clock. Because node 1 and node 2 base their time keeping on synchronised clocks, they have the same time base and the same time line may be used for both nodes. The time between two counts of the synchronised clock is the clock period A/ _C iock, which is the reciprocal of the clock frequency. During an calibration stage, transmission delay information has been determined and shared by nodes 1 and 2. In the depicted example, the transmission delay information comprises minimum transmission delays (fo.min and fei.min), maximum transmission delays (fo.max and fei.max), and transmission delay intervals (Ati2 and Afei). Transmission delay interval Ati2 is defined as the difference between the maximum and minimum transmission delays (Ati2 = fi2,max - fi2,min) and denotes the uncertainty in transmission delay for a signal transmitted from node 1 to node 2; similarly transmission delay interval At _2i denotes the uncertainty in transmission delay for a signal transmitted from node 2 to node 1. It is assumed that in between two calibration stages, the transmission delay information does not significantly change, so, for example, the transmission delay interval 208 of a (potential) signal sent by node 1 at t = 36 is the same as the transmission delay interval 204 of the signal sent by node 1 at t = 12.

During the calibration stage, a constant transmission interval A/rans has been selected by nodes 1 and 2, which is larger than the transmission delay intervals in the network. In the depicted example, Ati2 « 3.9 clock periods, At _2i “ 3.2 clock periods, and A/rans = 6 clock periods. A set of allowed transmission times {/trans} is defined by integer multiples of the transmission interval, i.e., in this example, {/trans} = {t = 0, 6, 12, ... }. Furthermore, a set of potential start times has been selected as multiples of eight times the clock period, so {/start} = {t = 0, 8, 16, ... }. Moreover, the nodes have agreed to wait at least four clock periods after the time at which all nodes know that all nodes are aware that all nodes have prepared their qubit (i.e., after / _a ii_known), to allow the nodes sufficient time to finish all computations and to prepare the operational phase.

Based on the transmission delay information, each signal received in a transmission delay interval can be associated with an allowed transmission time, and conversely, each signal sent at an allowed transmission time is (practically) guaranteed to arrive within a known transmission delay interval.

During a preparatory stage, nodes 1 and 2 may prepare their respective qubits. Node 1 may have finished preparing its qubit at a first moment in time 212 at t ~ 8.8. Node 1 waits until the first time point that is an allowed transmission time 214 at t = 12. At that moment, node 1 sends a signal to node 2 indicating that node 1 has finished preparing its qubit. This moment in time may be referred to as the first ready time ft, ready. The signal is received by node 2 at a second moment in time 216 at t ~ 28.3, which may be labelled /21 .receive- As was explained above, based on the transmission delay information, node 2 may determine the corresponding allowed transmission time ft, ready, at which node 1 sent the initialisation signal. Node 2 may do so for example by computing: Based on the first ready time and the transmission delay information, both node 1 and node 2 can compute the latest time by which node 2 could have received the initialisation signal, which may be labelled ft, ready. They can do so by computing:

As nodes 1 and 2 use the same values for the parameters in the computation, they will arrive at the same result.

Similarly, node 2 may finish preparing its qubit at a third moment in time 218 at t = 19.0. Node 2 waits until the first following allowed transmission time 220, labelled fe, ready, at t = 24 and transmits a second initialisation signal to node 1 , which is received by node 1 at a fourth moment in time 222 at t ~ 35.6. Node 1 computes fe. ready in an analogous fashion, and both nodes compute fe, known “ 37.8.

Node 1 and 2 then both compute the time t _a ii_known at which all nodes know that all nodes are aware that all nodes have prepared their qubit, e.g. by computing:

As described above, in the depicted example, the potential start times are multiples of 8 with the condition that the actual start time is at least 4 after t _a ii_known, which in this example may be computed by:

Fig. 2B depicts a plurality of time-related parameters for determining a time ti .known when all nodes are guaranteed to be aware that a first node has prepared its qubit in a quantum network with three quantum nodes, node 1 , node 2, and node 3.

In a quantum network with more than 2 nodes, the transmission delay information comprises a transmission delay distribution for each pair of nodes in the network. For example, in a network with 3 nodes, there are 3 ■ (3-1) = 6 connections. The parameters characterising each connection are distributed to all nodes in the network, so e.g. node 2 also has access to the parameters characterising the connection between nodes 1 and 3.

When e.g. node 1 has finished preparing its local qubit, node 1 transmits, at allowed transmission time ft .ready, a first initialisation signal to node 2 and a second initialisation signal to node 3. When node 2 receives the first initialisation signal, node 2 may compute ti .ready based on (in this example), ti2, receive, fi2,min, and ti2,max in the same way as described above with reference to Fig. 2A. When node 3 receives the second initialisation signal, node 3 may compute ft , ready based on, tn, receive, tn, min, and tn, max in an analogous way. Both nodes subsequently compute ti .known in the same way:

Because both nodes 2 and 3 have computed the same time ti .ready, and both nodes use the same values for tn.max and tn.max, both nodes will arrive at the same value for ti .known.

Fig. 3A-3C depict various topologies for a quantum network comprising a plurality of quantum nodes according to embodiments of the invention. In particular, Fig. 3A depicts a quantum network topology with two quantum nodes 302,304, and two classical communication channels connecting the two quantum nodes. The nodes each determine and, where necessary, share the transmission delay information, the set of allowed transmission times, and the set of potential start times, based on predetermined criteria.

Fig. 3B depicts a quantum network topology with two quantum nodes

312,314, and two classical communication channels connecting the two quantum nodes. The quantum network furthermore comprises a central controller 316 communicatively connected to the quantum nodes. In this embodiment, the quantum nodes share their information (only) with the controller, e.g. times at which test signals were sent and received. Based on this information, the controller determines the transmission delay information, the set of allowed transmission times, and the set of potential start times, and distributes these parameters to the nodes. Mixed forms are also possible, in which, e.g., the nodes determine the transmission delay information and share this with the central controller, while the central controller determines and distributes the set of allowed transmission times and the set of potential start times.

Fig. 3C depicts a quantum network topology with two quantum nodes 322- 330, and twenty (one-way) classical communication channels connecting each pair of quantum nodes. Such a network can be either combined with a decentralised network structure as discussed above with reference to Fig. 3A, or with a centralised or mixed network structure as discussed above with reference to Fig. 3B.

Fig. 4 depicts a flow diagram for synchronising events at a plurality of nodes in a quantum network according to an embodiment of the invention. The method comprises an calibration stage 402, and an execution stage 404. The execution stage comprises a preparatory stage 406 and, optionally, an operational stage 408. The method may be executed by one or more quantum nodes, e.g. quantum nodes as discussed above with reference to Fig. 1.

During the calibration stage, the quantum nodes share information allowing them to successfully set up an entanglement. During the preparatory stage, each quantum node initialises a (local) qubit. Qubit initialisation may involve a trial-and-error process which can take an a priori unknown number of tries, and therefore has an essentially unpredictable duration that is typically different for each of the involved quantum nodes. During the operational stage, each involved quantum node executes a sequence of manipulations to initiate the entanglement attempt. This may be followed by an application-specific sequence of qubit manipulations.

During the operational stage, timing of the sequence of manipulations is critical. This relates both to timing of steps within a single node and to timing of manipulations between nodes, i.e. synchronisation. In a typical use case, the underlying physical processes demand that the manipulations on the involved nodes are synchronised to a level of 1 ns or better during the operational stage. Exact values may depend on the parameters of the system, such as the wavelength of the photons to be entangled.

The calibration stage 402 may comprise steps 410,412. In a step 410, the method comprises receiving or determining transmission delay information about a transmission delay distribution of signals transmitted between a first node comprising a first qubit and a second node comprising a second qubit, the transmission delay information comprising a maximum transmission delay. More in general, each node involved in the event may receive or determine transmission delay information about a transmission delay distribution for each pair of nodes involved in the event. Typically, this information is determined for and distributed to each node in the network. The transmission delay information may comprise at least a maximum transmission delay for each connection between a pair of nodes.

In a step 412, the method comprises receiving or determining a synchronised set of allowed transmission times based on the transmission delay distribution and based on a synchronised clock shared between the nodes. The set of allowed transmission times may be determined by each node individually using a deterministic algorithm, or the set of allowed transmission times may be determined by a central controller and distributed to all nodes involved in the event or, more typically, to all nodes in the network.

The preparatory stage may comprise steps 414-420. In a step 414, the method comprises initializing the first qubit and transmitting a first initialisation signal to the second node based on a first allowed transmission time selected from the synchronised set of allowed transmission times. In a step 416, the method comprises receiving a second initialisation signal from the second node, the second initialisation signal being indicative of initialisation of the second qubit and the second initialisation signal being transmitted by the second node based on a second allowed transmission time selected from the synchronised set of allowed transmission times. The order in which steps 414 and 416 are performed by the first node may depend on the time it takes for the first and second nodes to finish preparing their respective qubits. In a step 418, the method comprises determining the second allowed transmission time based on the second initialisation signal. In a step 420, the method comprises determining the start time for the first qubit operation based on the first and second allowed transmission times, the maximum transmission delay and the synchronised clock.

In a typical embodiment, the preparatory stage is performed by each of the nodes involved in the event with respect to each pair of nodes involved in the event. For example, in a network as shown in Fig. 3A or 3B, steps 414-420 are performed both by node 1 and node 2. When performed by node 1, node 1 is the first node and node 2 is the second node, and when performed by node 2, node 2 is the first node and node 1 is the second node. Similarly in a network as shown in Fig. 3C, steps 414-420 are performed by all nodes 1-5. When performed by node 1, node 1 is the first node and each of nodes 2-5 takes the role of the second node, et cetera.

In some embodiments, the method further comprises operational stage 408 comprising one or more of steps 422-426. In these embodiments, the method may comprise a step 422 comprising arming a signal generator based on the start time. In general, the signal generator can control events in the quantum node with a higher precision than the controller doing the computations and the classical communication with the other nodes involved in the event. Thus, the controller may arm the signal generator with a relatively low temporal accuracy, and the signal generator will start the qubit manipulations with the desired temporal accuracy. To that end, the method may further comprise a step 424 comprising generating a start signal in response to the armed signal generator receiving a signal from the synchronised clock. The method may further comprise a step 426 comprising executing the first qubit operation in response to receiving the start signal. The first qubit operation may comprise manipulating the first qubit in accordance with the qubit operation. Steps 422-426 are generally performed by each of the nodes involved in the event.

After the operational stage is concluded, the quantum nodes may begin a new preparatory stage. In a network with more than two nodes, different selections of nodes can perform different iterations of the operational stage. Depending on the network topology, groups of nodes can even perform the operational stage partially or completely in parallel, possibly after a shared initialisation phase.

By employing this method, events at different nodes in a quantum network may be synchronised with high precision and efficiency. Analysis of such an approach has shown that for a realistic two-node system, the method described in Fig. 4 leads to an entanglement attempt rate that is roughly 40% higher than an approach without communication, i.e. , an approach where the operational stage is started at a fixed time, regardless of whether the nodes have finished preparing their respective qubits. Hence, this method may lead to a roughly 40% higher throughput. Fig. 5 is a block diagram illustrating exemplary data processing systems described in this disclosure. Data processing system 500 may include at least one processor 502 coupled to memory elements 504 through a system bus 506. As such, the data processing system may store program code within memory elements 504. Further, processor 502 may execute the program code accessed from memory elements 504 via system bus 506. In one aspect, data processing system may be implemented as a computer that is suitable for storing and/or executing program code. It should be appreciated, however, that data processing system 500 may be implemented in the form of any system including a processor and memory that is capable of performing the functions described within this specification.

Memory elements 504 may include one or more physical memory devices such as, for example, local memory 508 and one or more bulk storage devices 510. Local memory may refer to random access memory or other non-persistent memory device(s) generally used during actual execution of the program code. A bulk storage device may be implemented as a hard drive or other persistent data storage device. The processing system 500 may also include one or more cache memories (not shown) that provide temporary storage of at least some program code in order to reduce the number of times program code must be retrieved from bulk storage device 510 during execution.

Input/output (I/O) devices depicted as key device 512 and output device 514 optionally can be coupled to the data processing system. Examples of key device may include, but are not limited to, for example, a keyboard, a pointing device such as a mouse, or the like. Examples of output device may include, but are not limited to, for example, a monitor or display, speakers, or the like. Key device and/or output device may be coupled to data processing system either directly or through intervening I/O controllers. A network adapter 516 may also be coupled to data processing system to enable it to become coupled to other systems, computer systems, remote network devices, and/or remote storage devices through intervening private or public networks. The network adapter may comprise a data receiver for receiving data that is transmitted by said systems, devices and/or networks to said data and a data transmitter for transmitting data to said systems, devices and/or networks. Operation modems, cable operation modems, and Ethernet cards are examples of different types of network adapter that may be used with data processing system 500.

As pictured in FIG. 5, memory elements 504 may store an application 518. It should be appreciated that data processing system 500 may further execute an operating system (not shown) that can facilitate execution of the application. Application, being implemented in the form of executable program code, can be executed by data processing system 500, e.g., by processor 502. Responsive to executing application, data processing system may be configured to perform one or more operations to be described herein in further detail. In one aspect, for example, data processing system 500 may represent a client data processing system. In that case, application 518 may represent a client application that, when executed, configures data processing system 500 to perform the various functions described herein with reference to a "client". Examples of a client can include, but are not limited to, a personal computer, a portable computer, a mobile phone, or the like.

In another aspect, data processing system may represent a server. For example, data processing system may represent an (HTTP) server in which case application 518, when executed, may configure data processing system to perform (HTTP) server operations. In another aspect, data processing system may represent a module, unit or function as referred to in this specification.

The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms "a," "an," and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms "comprises" and/or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof.

The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the invention. The embodiment was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments with various modifications as are suited to the particular use contemplated.