A multi-dimensional cluster state generation system

Abstract

The disclosure regards an optical entanglement system for generating an at least three-dimensional cluster state for measurement-based optical quantum computation, the optical entanglement system having as input a plurality of pairs of quantum states, the system comprising a beam-splitter network, comprising at least twelve beam splitters, and at least eight input modes, wherein each input mode is configured to host a quantum state, a plurality of delay lines, each delay line configured for generating a time delay on one of the quantum states in at least one pair of quantum states, wherein the system is configured for entangling the pairs of quantum states utilizing the beam-splitter network and the induced time delays, thereby generating an at least three-dimensional cluster state.

Description

[0001] A multi-dimensional cluster state generation system

[0002] The present disclosure relates to a novel multi-dimensional cluster state generation system and method directed to measurement-based optical quantum computation.

[0003] Background

[0004] Measurement-based optical quantum computing (MBQC) is a paradigm for quantum computing that relies on manipulating and measuring entangled quantum states of light to perform quantum operations. Unlike traditional circuit-based approaches, MBQC utilizes a cluster state, a highly entangled state of qubits, which serves as a resource for performing computations. In MBQC, quantum information is encoded in the correlations between photons, and computation is achieved by performing measurements on these photons. By changing the measurement bases, different quantum gates can be applied, enabling universal quantum computation. MBQC holds promise for scalable quantum computing due to the robustness of photonic systems and the potential for creating entangled cluster states.

[0005] Nevertheless, MBQC faces certain challenges, such as errors that can be introduced during computation. These errors can arise from various sources, including imperfections in the generation and manipulation of entangled states, photon loss due to imperfect transmission through optical components, and inaccuracies in the measurements performed. These errors can accumulate in the encoded quantum information, and can ultimately compromise the accuracy of the computation. Overcoming these errors typically requires sophisticated error correction techniques, which can be resource-intensive and may limit the scalability of quantum computing systems. Other attempts to overcome errors in MBQC relate to using built-in error correction systems, such as using macronode designs. However, such approaches also hold significant disadvantages, since they do not efficiently provide error correction during quantum computation.

[0006] To address the presence of errors in the computational schemes, two-dimensional cluster states have been developed, enabling single and two-mode Clifford gates. Nevertheless, two-dimensional cluster states cannot incorporate redundancy and error correction efficiently. Therefore, three-dimensional cluster states have also been developed, by involving a combination of a temporal dimension and two spatial dimensions. However, involving spatial dimensions relates to certain challenges, as it leads to a system that has computation scalability tied to hardware scalability.

[0007] Hence, a new method for generating cluster states is needed, that can mitigate errors effectively in measurement-based optical quantum computation, while at the same time can be a building block for a scalable quantum computing unit.

[0008] Summary

[0009] One purpose of the present disclosure is to realize a setup that enables error mitigation in measurement-based optical quantum computation, while at the same time can support a scalable platform. In addition, the present disclosure enables the use of static optical components instead of e.g. variable beam splitters that are used in state of the art solutions, leading to a more efficient and less resource-intensive system. Therefore, the present disclosure relates to an optical entanglement system for generating an at least three-dimensional cluster state for measurement-based optical quantum computation, the optical entanglement system typically having as input a plurality of pairs of quantum states. In the preferred embodiment the system comprises a beamsplitter network, comprising a plurality of beam splitters, preferably at least twelve beam splitters, and a plurality of input modes, preferably at least eight input modes. Each input mode is preferably configured to host a quantum state. The system may comprise a plurality of delay lines, each delay line configured for generating a time delay on one of the quantum states in at least one pair of quantum states. The system is preferably configured for entangling the pairs of quantum states, for example by utilizing the beam-splitter network and preferably the induced time delays, thereby generating an at least three-dimensional cluster state. Advantageously, the system can be configured to utilize only temporal dimensions in order to generate the at least three-dimensional cluster state, by not requiring the use of spatial dimensions. The use of only temporal dimensions can be achieved by utilizing a plurality of delay lines to extend entanglement between macronodes at different times. Further details about this process are found in the sections below. The presently disclosed system can achieve scalability utilizing only temporal dimensions, and such a system can assist in mitigating errors in measurement-based optical quantum computation. As described herein, involving spatial dimensions leads to computation scalability tied to hardware scalability. On the other hand, a computation architecture utilizing only temporal dimensions can achieve scalability that is decoupled from the hardware and enables immediate and resource-efficient scalability.

[0010] The quantum states may be any kind of quantum states, such as Gottesman-Kitaev- Preskill (GKP) Bell pairs or two-mode squeezed states (TMSS), which are also written herein as Einstein-Podolsky-Rosen (EPR) states. These quantum states can be inputted to the input modes of the beam-splitter network, where a plurality of beam splitters are arranged to connect the input modes and entangle the quantum states.

[0011] In an embodiment, each input mode is connected to another input mode by using a beam splitter, forming a pair of entangled modes. Each two sets of entangled modes can be further connected to each other by using two beam splitters, forming a quadruplet state. Further, two such quadruplet states may be entangled by utilizing four beam splitters. Such a process can be continued indefinitely, where the process is having M iterations, where M is an integer number linked to the number of input modes 2Mand to the number of beam splitters 2M'1M.

[0012] For the preferred embodiment described above where M=3, the system relates to 8 input modes and 12 beam splitters. Therefore, by performing three iterations of entanglement using the 12 beam splitters, all quantum states can be connected, forming a macronode for a given time t. Then, using delay lines it is possible to create a multi-dimensional cluster state, by entangling the macronode at a given time t with macronodes at different times. The different times are linked to the delay provided by each delay line, in this case of M=3, a three-dimensional cluster state can be generated, by generating a time delay on a quantum state in three of the four pairs of quantum states. As a result, the delay lines can create a three-dimensional cluster state by extending the entanglement between macronodes of six other different times, resulting in the three axes of a three-dimensional cluster state.

[0013] The above approach is advantageous, as it relates to the generation of a cluster state using a combination of only temporal dimensions (the different times of macronodes) by utilizing a plurality of delay lines. As a result, the presently disclosed approach inherently supports fault-tolerant operations and topological coding due to the nature of the generated cluster states, which is crucial for protecting quantum information and enabling complex algorithms. Further details regarding the generation of the at least three-dimensional cluster state are provided in the detailed description of the present disclosure.

[0014] In a preferred embodiment, the system is configured such that the time delays of the delay lines are multiples of one another, or have identical lengths. Depending on the type of application, different lengths and multiples of lengths may be used. Different lengths may relate to the number of macronodes along the axes of the different dimensions before a periodic boundary is reached. For example, if the length of a delay line is long enough, the delay line may form loops along a certain direction related to a dimension of the cluster state. Specifically, using the delay lines having length nmr, T and nr, where n and m are integer numbers and T may be a fixed time step, it is possible to create a three-dimensional cluster state by extending entanglement between macronodes at time t to macronodes at times t + T, t - T, t + nr, t - nr, t + nmr and t-nmr. This forms the three axes of the three-dimensional cluster state. Such a three-dimensional cluster state is generated by only using temporal dimensions, due to utilizing delay lines having lengths multiples of one another, and not using other means to form spatial dimensions. This is advantageous, as the cluster states can be generated and scaled up without utilizing spatial dimensions, thereby providing a more immediate and resource-efficient scalability compared with scaling the system in spatial dimensions. Hence, utilizing the present disclosure it is possible to scale up the dimensions of the generated cluster states without increasing the number of physical resources used. Such a process can also be described as generating a multidimensional temporally encoded macronode. Further information about this process is included in the detailed description of the present disclosure.

[0015] The present disclosure further relates to a beam splitter system for manipulating quantum states, the system comprising a beam-splitter network, comprising 2Minput modes, wherein each input mode is configured to host a quantum state, and M groups of 2M-1beam splitters, wherein each group of beam splitters is configured to entangle input modes to form pairs of input modes. Each of the beam splitters of the first group of beam splitters can entangle a first input mode with a second input mode, forming a pair of entangled modes, and at least two beam splitters of a subsequent group of beam splitters can entangle a pair of entangled modes with a second pair of entangled modes, forming an extended pair of entangled modes, wherein M is a positive integer number greater than 2. The above process describes the creation of a macronode. For example, for M=3, the beam splitter system coincides with the beam splitter network described in the optical entanglement system described above. By including a plurality of delay lines, it is possible to create an up to 2M-1-dimensional cluster state. The presently disclosed beam splitter network can advantageously be used in the presently disclosed optical entanglement system.

[0016] Furthermore, the present disclosure can become relevant experimentally by involving measurements of the entangled quantum states. The present disclosure therefore further relates to an optical quantum computation system comprising the presently disclosed optical entanglement system, and a plurality of detectors, preferably homodyne detectors, each homodyne detector configured to measure a quantum state, wherein each homodyne detector may be configured to measure at least one quadrature of a quantum state. For example, a homodyne detector may be positioned at the end of an input mode, where the homodyne detector can detect a quantum state.

[0017] In addition, the optical quantum computation system may comprise an error correction scheme. The error correction scheme may be employed using quantum state pairs, such as GKP Bell pairs or EPR states, as inputs, and utilizing the beam splitters and the homodyne detectors it may correct any continuous errors that can occur in the setup. Such continuous errors may be mapped onto discrete errors. A topological error correction scheme may be utilized to correct such discrete errors. Further details regarding the error correction schemes is provided in the detailed description of the present disclosure.

[0018] The present disclosure further relates to a method for generating at least three- dimensional cluster states, comprising the steps: obtaining a plurality of pairs of quantum states, obtaining a beam-splitter network, for example the presently disclosed beam splitter network, preferably comprising at least twelve beam splitters and at least eight input modes, wherein each input mode is configured to host a quantum state, generating a time delay on one of the quantum states in at least one pair of quantum states, and utilizing the generated time delay to obtain an at least three-dimensional cluster state. The presently disclosed method can advantageously be carried out utilizing the presently disclosed optical entanglement system. Description of Drawings

[0019] Various embodiments are described hereinafter with reference to the drawings. The drawings are examples of embodiments and are intended to illustrate some of the features of the presently disclosed multi-dimensional cluster state generation system, and are not limiting to the presently disclosed system and method.

[0020] Fig. 1 shows a schematic of an optical entanglement system.

[0021] Fig. 2 shows a schematic of a macronode entangled to six other macronodes at different times.

[0022] Fig. 3 shows an example of an EPR state or GKP Bell pair generation.

[0023] Fig. 4 shows the steps of a method for generating an at least three-dimensional cluster state.

[0024] Fig. 5 shows an arrangement of data and measurement qubits within a surface code.

[0025] Fig. 6 A, B shows a circuit implementation for even and odd data qubit gates respectively.

[0026] Fig. 7 shows a schematic of an optical entanglement system comprising four delay lines.

[0027] Fig. 8 shows a macronode lattice layout where a macronode is entangled to eight other macronodes.

[0028] Fig. 9 shows an example of an optical entanglement system configured for multiplexing quantum states.

[0029] Fig. 10 shows an example of an optical entanglement system configured for multiplexing quantum states and for cluster state generation.

[0030] Detailed description

[0031] One purpose of the present disclosure is to form an architecture for mitigating errors in measurement-based optical quantum computation, by generating at least three- dimensional cluster states that are encoded temporally. Specifically, the present disclosure in a preferred embodiment relates to an optical entanglement system for generating an at least three-dimensional cluster state for measurement-based optical quantum computation, the optical entanglement system having as input a plurality of pairs of quantum states, the system comprising a beam-splitter network, comprising at least twelve beam splitters, and at least eight input modes, wherein each input mode is configured to host a quantum state. The optical entanglement system further comprises a plurality of delay lines, each delay line configured for generating or introducing a time delay on one of the quantum states in at least one pair of quantum states, wherein the system is configured for entangling the pairs of quantum states utilizing the beamsplitter network and the induced time delays, thereby generating an at least three- dimensional cluster state. In a preferred embodiment, the time delays of the delay lines are multiples of one another, thereby enabling the at least three-dimensional cluster state to be generated utilizing a combination of only temporal dimensions.

[0032] A schematic of an optical entanglement system comprising eight input modes and twelve beam splitters is shown in Fig. 1. This example comprises four sets of quantum states (100) where each quantum state is hosted on an input mode of the eight total input modes (101). The states 1, 8 are the first pair of quantum states, the states 2, 3 are the second pair of quantum states, the states 4, 5 are the third pair of quantum states and the states 6, 7 are the fourth set of quantum states. Twelve beam splitters (102) are utilized to entangle the quantum states and each input mode may be equipped with a homodyne detector (103) to measure the quantum states. Three delay lines can be used, one positioned on the first set of quantum states (105), a second positioned on the second set of quantum states (106) and a third positioned on the fourth set of quantum states (107). The initial step for the generation of a cluster state, is the choice of pairs of quantum states. Such quantum states may be GKP qunaught states, Bell states or any other quantum states.

[0033] In the example shown in Fig. 1, three delay lines are implemented, each positioned on a quantum state of a pair of quantum states. In the schematic shown in Fig. 1 , the first, second and third pair of quantum states are equipped with a delay line. Such a choice is arbitrary, as different quantum states may be equipped with a delay line instead. For example, the second, fourth and sixth quantum states may be equipped with the delay lines instead. Additionally, in order to entangle the quantum states among each other, a series of beam splitters are used. In the example of Fig. 1 , twelve beam splitters are used for that purpose. A first group of beam splitters (108) is utilized to entangle a first quantum state with a second quantum state, forming a pair of entangled modes. For example, in the schematic of Fig. 1 , the quantum states of input modes 1 and 2, are entangled using a beam splitter. Similarly, the quantum states of input modes 3 and 4, as well as 5 and 6 and 7 and 8 are entangled using a beam splitter. Such a process leads to four pairs of entangled modes. Then, the four pairs of entangled modes are further entangled using a second group of beam splitters (109), illustrated with dashed lines. Each beam splitter of the second group of beam splitters entangles a pair of entangled modes to a second pair of entangled modes, forming an extended pair of entangled modes, in this example a quadruplet pair. For example, the pair of entangled modes comprising of quantum states 1 and 2 is entangled to the pair of entangled modes comprising quantum states 3 and 4 via a beam splitter, thereby forming an extended pair of entangled modes, which is a quadruplet pair. The third group of beam splitters (110) may further entangle the extended pairs of entangled modes to another extended pair of entangled modes. Such a process forms a macronode. Using the delay lines having length nmr, rand nr, where n and m are integer numbers and T may be a fixed time step, it is possible to create a three-dimensional cluster state by extending entanglement between macronodes at time t to macronodes at times t + T, t - T, t + nr, t - nr, t + nmr and t-nmr. This forms the three axes of the three-dimensional cluster state. Any other combination of delay lines lengths may also be used, wherein each delay line length is identical to another delay line length or a multiple of another delay line length. The time step T may be adjusted depending on the type of experiment or application. Such an example relates to the case of eight input modes and twelve beam splitters. Employing the delay lines as described above and using certain length on each delay line relates to using a combination of temporal dimensions which are used to generate an at least three-dimensional cluster state. In contrast, state of the art solutions utilize a combination of spatial and temporal dimensions, which is a resourceintensive system. Effectively, the presently disclosed system relates to temporal encoding, enabling the progression of the optical entanglement system to scale to very large size without requiring additional experimental resources, in particular sources of quantum states. Extending the above process, it is possible to apply it on any higher number of input modes, beam splitters and delay lines, as long as certain criteria are met. Further details about the criteria are provided at the next sections of the detailed description.

[0034] Moreover, the plurality of delay lines enables the generation of a multi-dimensional cluster state by utilizing a combination of temporal dimensions. For example, the at least three-dimensional cluster state can be generated by utilizing a combination of only temporal dimensions. As described herein, such a feature is enabled by using delay lines in input lines in order to extend entanglement between macronodes. By using delay lines of certain length enables the formation of a cluster state by using temporal dimensions. By using delay lines of certain length, such as multiples of one another, enables the formation of a cluster state by using temporal dimensions, which is immediately scalable without requiring scalable hardware. The present system is advantageous as it does not rely on using spatial dimensions to generate a multidimensional cluster state. The process of using spatial dimensions for that task is also known as spatial multiplexing. Hence, the system can be configured, such that the at least three-dimensional cluster state is generated without utilizing any spatial dimension.

[0035] In an embodiment, the plurality of delay lines is configured to generate a multidimensional temporally encoded macronode. As described in the paragraph above, using the delay lines having length nmr, r and nr, where n and m are integer numbers and T is a fixed time step, it is possible to create a three-dimensional cluster state by extending entanglement between macronodes at time t to macronodes at times t + T, t - T, t + nr, t - nr, t + nmr and t-nmr. The extension of entanglement between macronodes at the different times is described as a temporally encoded macronode herein. Importantly, these macronodes are defined in time and not in space, enabling the generation of a three-dimensional cluster state by only combining temporal dimensions.

[0036] The beam splitter configuration shown in Fig. 1 may be arranged differently without changing the entanglement structure and the functionality of the system, as the different beam splitters commute. On the other hand, changing the configuration of the input modes may change the entanglement structure, providing flexibility on the system. Specifically, in a configuration with twelve beam splitters and eight input modes, there are 30 different macronode structures that can be achieved by changing the order of input modes. Specific configurations of input modes may be preferred, as they can provide a better application of a surface code to perform topological error correction in the system.

[0037] The present disclosure extends to any further combination of numbers of input modes and beam splitters that satisfy the conditions 2Minput modes and 2M'1M beam splitters, where M is a positive integer number larger than 2. The number M is related to the dimensionality of the cluster state, wherein the maximal dimensionality is 2M-1. The dimensionality is related to the number of delay lines, as the total number of delay lines is equal to the number of dimensions. For example, for M=5, the generated cluster state would be an up to sixteen-dimensional cluster state, and the optical entanglement system would relate to 32 input modes and 80 beam splitters. Extending the process described above, it is possible to realize such a sixteen-dimensional cluster state by incorporating sixteen delay lines.

[0038] The connected macronodes generated by the optical entanglement system described in Fig. 1 can be visualized in Fig. 2. Fig. 2 shows an example of a three-dimensional lattice of connected macronodes, including beam splitters and a two-mode entangled state in the input modes 4 and 5. Data may be encoded in input mode 1 (200) and it can be teleported via the macronode to subsequent macronodes (201). A surface code can be encoded within the plane formed by input modes 1, 2, 3, and 4. The surface code may comprise odd and even data qubits, together with ancillary measure-X and - Z qubits. Examples of surface code and data qubits are provided in the examples section of the detailed description.

[0039] In the present disclosure, a pair of quantum states may be generated by either sending two squeezed states through a beam splitter thereby generating an EPR pair or sending two GKP qunaught states through a beam splitter thereby generating a GKP Bell pair. Such a process can be seen in Fig. 3, and the operations shown (300) are equivalent to the schematic used in Fig. 1 for a pair of quantum states (301). The example shown in Fig. 3, shows the generation of an Einstein-Podolsky-Rosen (EPR) state or of a GKP Bell state comprising a phase rotation of -TT / 2 (304) followed by a beam splitter operation (305) and a final rotation of TT / 2 (306). The optical entanglement system may leverage two-mode entanglement. The GKP Bell pairs (302) may be found in all input modes. Using GKP Bell pairs, it is possible to map continuous errors in the teleportation scheme to discrete errors. These states may be replaced by squeezed vacuum states (303), and therefore it can be advantageous that the system is equipped with a switch. Therefore, the system can be configured, such that each input mode comprises a switch configured to toggle between two states, such as a qunaught state and a squeezed state.

[0040] Advantageously, the system can be configured such that the time delays of the delay lines are multiples of one another, or have identical lengths. Depending on the type of application, different lengths in the time delays may be used, leading to changes in the dimensionality of the cluster state. The delay lines may be implemented in any kind of transmission line suitable for optical signals, such as optical fibers. The delay lines may comprise an extendable optical fiber, which may be chosen depending on the needs of each application. Each delay line may be fixed or variable. A delay line may be any type of an optical delay line. The length of the delay lines may depend on various factors. For example, the longer the length the larger the losses and phase noise that is introduced to the system. In an embodiment, for M=3 and for the case of using three delay lines, it may be beneficial to have lengths of 1, n, and n2respectively, allowing a square surface code.

[0041] The number of components in the optical entanglement system can be generalized. The system can be configured, such that N delay lines are incorporated on at least N pairs of quantum states, each pair of quantum states comprising two quantum states, wherein a plurality of macronodes each comprising at least 2N quantum states are entangled with 2N other macronodes, thereby generating an N-dimensional cluster state, wherein N is a positive integer number.

[0042] Moreover, the optical entanglement system can be configured for multiplexing quantum states. The quantum states are also described herein as input states. Multiplexing of quantum states refers to the process of routing or selecting one of several quantum input states to be used or transmitted through the beam splitter network. An advantage of multiplexing quantum states is that multiple resources can be prepared in parallel, and the system may dynamically choose one or more to use, without increasing noise in the system. Such a feature is enabled due to the ease of scaling the optical entanglement system to accommodate a large number of input optical modes, and owing to its ability to apply identity operations across these modes. For example, as shown in Fig. 9, an embodiment of the system comprises a plurality of input states 900 which can receive a respective quantum state, such as a squeezed vacuum state, or a GKP-encoded qubit. All the sets of quantum states, which in this case are entangled Bell pairs, except for one 901 are replaced with input states. The Bell state that is not replaced comprises an input 903 and an output 904. Then, by selecting appropriate measurement bases at the homodyne detectors 902, one of these input states may be selectively teleported to the designated output mode that is connected via the remaining Bell pair. This teleportation can be performed without increasing the accumulated noise as the number of input states grows.

[0043] In an embodiment, the remaining Bell pair may be substituted with a two-mode squeezed state, thereby simplifying the implementation while maintaining fidelity appropriate for certain Gaussian or near-Gaussian applications.

[0044] Furthermore, the optical entanglement system can be configured such that multiplexing of quantum states can be combined with generation of multi-dimensional cluster states. Such a feature can be advantageous, as it can enable parallel processes without increasing the noise level in the system. Specifically, an advantage of combining multiplexing with cluster state generation in the system is that enables flexible and scalable input-state selection without increasing the accumulated noise. For example, the teleportation-based multiplexing protocol can ensure that only the noise associated with a single teleportation step is introduced, independent of the number of input modes. This enables the system to prepare many candidate input states in parallel and dynamically select the desired one for injection into the cluster state, without affecting the overall fidelity of the computation.

[0045] For example, the multiplexing of quantum states may be combined with cluster state generation to enable state injection into a functional quantum computational cluster, as shown in Fig. 10. For example, one portion of the optical entanglement system may be used to maintain the cluster state's structure 1000 (e.g., by implementing four input modes in a quad-rail configuration), while another portion may be employed to inject auxiliary input states into the cluster 1001, thereby enabling dynamic input routing or magic state injection. This configuration may be particularly useful for measurementbased quantum computing protocols where state injection plays a role in implementing non-Clifford operations.

[0046] The architecture described above may support other forms of input states, such as vacuum states, cat states, or photon-subtracted squeezed states, depending on the application. Similarly, detection schemes may be varied, for example homodyne detectors may be replaced or complemented by heterodyne detectors to facilitate state characterization or enable GKP magic state generation.

[0047] In addition, the measurement bases used at the detectors may be dynamically selectable, allowing for flexible routing and transformation of the quantum states within the system.

[0048] Furthermore, the present disclosure relates to a beam splitter system for manipulating quantum states, the system comprising a beam-splitter network, comprising 2Minput modes, wherein each input mode is configured to host a quantum state, and M groups of 2M-1beam splitters, wherein each group of beam splitters is configured to entangle input modes to form pairs of input modes. Preferably, each of the beam splitters of the first group of beam splitters entangles a first input mode with a second input mode, forming a pair of entangled modes, and at least two beam splitters of a subsequent group of beam splitters entangles a pair of entangled modes with a second pair of entangled modes, forming an extended pair of entangled modes, wherein M is a positive integer number greater than 2. Such a beam splitter system may be used to generate multi-dimensional cluster states or it may also be used to manipulate any kind of quantum states. For example, any application that may require the use of entangled pairs of quantum states may be supposed by such a beam splitter system. In an embodiment where M=3, the beam splitter system comprises eight input modes and three groups of 4 beam splitters. Each of the beam splitters of the first group may entangle a quantum state in an input mode with a second quantum state in a second input mode. Such an example can also be visualized in Fig. 1. For example, the input modes 1 and 2 are entangled by using a beam splitter of the first group of beam splitters (108). Therefore, a total of four pairs of entangled modes can be obtained after utilizing the first group of beam splitters. Then, each of the beam splitters of the second group (109) may entangle a pair of entangled modes to a second pair of entangled modes. For example, as seen in Fig. 1 , the beam splitters (111 , 112) entangle the entangled pair of 1 and 2, to the entangled pair of 3 and 4 input modes, generating an extended pair of entangled modes. Accordingly, the third group of beam splitters (110) may entangle a first extended pair of entangled modes to a second pair of entangled modes. After all twelve beam splitter operations are performed, a macronode can be created. Summarizing the above process, a beam splitter may be used to entangle two input modes forming an entangled mode. Then, two beam splitters may be used to entangle the two entangled modes forming an extended pair of entangled modes. Following the same logic, four beam splitters are needed to entangle the two extended pairs of entangled modes.

[0049] In order to form a cluster state from said macronode, the beam splitter system may comprise a plurality of delay lines, each delay line configured for generating or introducing a time delay on at least one of the quantum states. Utilizing the time delays allows the creation of macronodes of different times, generating a multi-dimensional cluster state. The plurality of delay lines can be configured to generate a multidimensional temporally encoded macronode. For example, using the delay lines having length nmr, rand nr, where n and m are integer numbers and T is a fixed time step, it is possible to create a three-dimensional cluster state by extending entanglement between macronodes at time t to macronodes at times t + T, t - T, t + nr, t - nr, t + nmr and t-nmr.

[0050] As described herein, the beam splitter system may comprise a plurality of homodyne detectors each configured to measure a quantum state. In an embodiment, each of the homodyne detectors may comprise an angle modification system, allowing for the manipulation of the entanglement structure. The angle modification may refer to the adjustment of the phase angle of a local oscillator relative to the measured signal, such as the measured quantum state. By modifying the phase of the local oscillator in a homodyne detector, it is possible to influence the effect of the homodyne detector by altering the entanglement structure and thereby performing a specific computation. The homodyne detectors may be, but are not limited to, optical homodyne detectors.

[0051] The present disclosure further relates to an optical quantum computation system comprising the optical entanglement system according to any one of the embodiments described herein, and a plurality of homodyne detectors, each homodyne detector configured to measure a quantum state, wherein each homodyne detector may be configured to measure at least one quadrature of a quantum state. is configured to conduct gate measurements. In an embodiment, each homodyne detector may be configured to measure a different quadrature for every received quantum state. In addition, each homodyne detector may be configured, such that the at least one quadrature being measured may vary for each time step. The optical quantum computation system may further comprise an error correction scheme.

[0052] The error correction scheme may utilize GKP Bell pairs or qunaught states, said GKP Bell pairs or qunaught states are utilized to correct continuous errors, by mapping said continuous errors to discrete errors.

[0053] Utilizing an at least three-dimensional cluster state allows for the incorporation a topological error correction scheme. The optical quantum computation system can be configured, such that the generated at least three-dimensional cluster state incorporates a topological error correction scheme configured to correct discrete errors. In an embodiment, a surface code may be used as a topological error correction scheme that utilizes a two-dimensional lattice of connected units. The third dimension is required to perform the measurements on the qubits. Extending the dimensions of the cluster state may allow the use of extended topological error correction schemes that may require more than 2 dimensions. An example of a surface code is provided at the example section of the detailed description.

[0054] Moreover, the optical quantum computation system can be configured, such that the pairs of quantum states are Gottesman-Kitaev-Preskill (GKP) Bell pairs. Furthermore, the generated at least three-dimensional cluster state may operate on a plurality of logical qubits, such as Gottesman-Kitaev-Preskill (GKP) qubits.

[0055] In an embodiment, the optical quantum computation system can be configured, such that an approximate GKP magic state can be generated by GKP error correction by utilizing at least one heterodyne detector. Leveraging the GKP error correction inherent in teleportation through the Bell pair, squeezed states can be corrected into |0) or |+) states using homodyne detection, and vacuum states can produce |T) magic states. These corrections can alternatively be performed via Gaussian operations applied to an input GKP Bell pair by adjusting the measurement basis followed by heterodyne detection. Consequently, introducing a heterodyne detector into the system alongside the homodyne detectors can provide the flexibility needed for universal quantum computation including the necessary initialisation.

[0056] In an embodiment, the system comprises at least one beam-splitter and at least two homodyne detectors configured to measure in orthogonal bases. Specifically, at least one detector of the system may comprise at least one beam-splitter and at least two homodyne detectors configured to measure in orthogonal bases. The beam-splitter may be a balanced beam-splitter. A balanced beam-splitter divides the incoming signal to two equal parts, with 50% of the signal transmitted to one direction, and 50% of the signal transmitted to a second direction. As a result, such a configuration can be incorporated in the presently disclosed system. Measuring in the same basis mimics a homodyne detector, while orthogonal measurements enable GKP magic state generation. To isolate a single Bell pair and consequently obtain a magic state, all other modes of the macronode, as well as all macronodes they are connected to, can be measured in the Z-basis. Alternatively, more than one homodyne detector can be replaced by heterodyne detectors, thereby accelerating the generation of magic states. A magic state can for example be a pre-prepared GKP state. In an embodiment, a magic state can be a resource state that is used to execute a non-Clifford gate. For example, for T gates, the magic state can be described as

[0057] A magic state can also be understood as a non-Gaussian, non -stabilizer photonic quantum state that enables universal quantum computing when combined with operations, such as linear optical operations.

[0058] The present disclosure further relates to a method for generating at least three- dimensional cluster states, comprising the steps: obtaining a plurality of pairs of quantum states, obtaining a beam-splitter network, comprising at least twelve beam splitters and at least eight input modes, wherein each input mode is configured to host a quantum state, generating a time delay on a one of the quantum states in at least one pair of quantum states, and utilizing the generated time delay to obtain an at least three-dimensional cluster state. The above steps are depicted in Fig. 4, showing a flowchart of the method steps. At first, the method may relate to obtaining a plurality of pairs of quantum states 400, followed by generating a time delay on a quantum state of at least one pair of quantum states 401. Then, the method relates to obtaining a beamsplitter network 402, and utilizing the generated time delay to obtain an at least three- dimensional cluster state 403. The at least three-dimensional cluster states may be generated by means of the optical quantum computation system according to any one of the embodiments described herein.

[0059] The method may further comprise the steps of: utilizing GKP Bell pairs as pairs of quantum states and utilize the beam-splitter network and homodyne detectors to correct continuous errors and map said continuous errors to discrete errors, and obtaining a topological error correction scheme, configured to correct discrete errors.

[0060] Examples

[0061] The topological error correction scheme described herein may be a surface code. An example of a surface code is given below, assuming a system comprising eight input modes, twelve beam splitters and three delay lines, generating a three-dimensional cluster state. The computational scheme can be centered on the projection of the cluster state onto a Raussendorf-Harrington-Goyal (RHG) lattice via specific homodyne measurements on particular macronodes. The RHG lattice can be a 3-dimensional lattice comprising the 2D surface and one dimension for the computation in which each qubit is connected to a specific subset of neighbouring qubits. The RHG lattice may be utilized for the measurement-based version of the surface code. The surface code is a commonly used topological error correction code, due to its high error threshold and the ability to detect and correct discrete errors arising from teleportation noise with GKP states. In the present disclosure, quantum computation may be achieved by making specific homodyne measurements in a specific basis on the qubits, with the computational output determined by the measurement outcomes.

[0062] A purpose of the computational scheme described by the present disclosure is to teleport a data qubit from mode 1 to T while simultaneously coupling it to ancillary modes 2’, 3’, 6’, and 7’. Such modes are depicted in Fig. 2. These ancillary modes can subsequently play a role in the measure qubits. Each teleportation process may entail a twofold coupling: first, the q quadrature of the data qubit is linked to two ancillary qubits, followed by coupling the p quadrature to the remaining two ancillary qubits. The surface code under consideration places ‘odd’ (500) and ‘even’ (501) data qubits relative to each other as shown in Fig. 5. Even and odd data qubits may differ based on the q and p quadrature coupling: for odd data qubits, the q quadrature is connected to modes 2’ and 3’, and the p quadrature to modes 6’ and 7’ shown in Fig. 2. Conversely, for even data qubits, the p quadrature couples to modes 2’ and 3’, and the q quadrature to modes 6’ and 7’. For example, an even data qubit circuit is shown in Fig. 6A, which is implemented by applying the following measurement basis of the homodyne detection:

[0063] (M s s) = (o, o, o, 7r / 2, o, o, o, TF / 2)

[0064] An even data qubit circuit is shown in Fig. 6B, which is implemented by applying the following measurement basis of the homodyne detection:

[0065] An additional example of an optical entanglement system can be viewed in Fig. 7. In this example, eight input modes, twelve beam splitters and four delay lines are used, as well as four pairs of quantum states. The use of an additional delay line compared to Fig. 1 leads to the generation of a four-dimensional cluster state. Fig. 8 shows the macronode at a time t being entangled to eight other macronodes at different times, leading to a four-dimensional cluster state.

[0066] Items

[0067] 1. An optical entanglement system for generating an at least three-dimensional cluster state for measurement-based optical quantum computation, the optical entanglement system having as input a plurality of pairs of quantum states, the system comprising:

[0068] • a beam-splitter network, comprising o at least twelve beam splitters, and o at least eight input modes, wherein each input mode is configured to host a quantum state,

[0069] • a plurality of delay lines, each delay line configured for generating a time delay on one of the quantum states in at least one pair of quantum states, wherein the system is configured for entangling the plurality of pairs of quantum states utilizing the beam-splitter network and the induced time delays of the plurality of delay lines, thereby generating an at least three- dimensional cluster state.

[0070] 2. The system according to item 1, wherein each input mode comprises a switch configured to toggle between two states, such as a qunaught state and a squeezed state.

[0071] 3. The system according to any one of the preceding items, wherein the at least three-dimensional cluster state is generated by utilizing a combination of only temporal dimensions.

[0072] 4. The system according to any one of the preceding items, wherein the plurality of delay lines is configured to generate a multi-dimensional temporally encoded macronode.

[0073] 5. The system according to any one of the preceding items, wherein the at least three-dimensional cluster state is generated without utilizing any spatial dimension.

[0074] 6. The system according to any one of the preceding items, wherein the time delays of the plurality of delay lines are multiples of one another, or have identical lengths.

[0075] 7. The system according to any one of the preceding items, wherein N delay lines of the plurality of delay lines are incorporated on at least N pairs of the plurality of pairs of quantum states, each pair of the N pairs of quantum states comprising two quantum states, wherein a plurality of macronodes each comprising at least 2N quantum states are entangled with 2N other macronodes, thereby generating an N-dimensional cluster state, wherein N is a positive integer number.

[0076] 8. The system according to any one of the preceding items, wherein the system is configured for multiplexing quantum states.

[0077] 9. The system according to any one of the preceding items, wherein the system is configured such that multiplexing of quantum states is combined with generation of multi-dimensional cluster states.

[0078] 10. A beam splitter system for manipulating quantum states, the system comprising a beam-splitter network, comprising o 2Minput modes, wherein each input mode is configured to host a quantum state, and o M groups of 2M-1beam splitters, wherein each group of beam splitters is configured to entangle input modes to form pairs of input modes, wherein each of the beam splitters of a first group of beam splitters entangles a first input mode with a second input mode, forming a pair of entangled modes, and wherein at least two beam splitters of a subsequent group of beam splitters entangle a pair of entangled modes with a second pair of entangled modes, forming an extended pair of entangled modes, wherein M is a positive integer number greater than 2.

[0079] 11. The beam splitter system according to item 10, further comprising a plurality of delay lines, each delay line configured for generating a time delay on at least one of the quantum states, and wherein the plurality of delay lines is configured to generate a multi-dimensional temporally encoded macronode.

[0080] 12. The beam splitter system according to item 11, further comprising a plurality of homodyne detectors each configured to measure a quantum state.

[0081] 13. The system according to item 12, wherein each of the plurality of homodyne detectors comprises an angle modification system allowing for the manipulation of the entanglement structure.

[0082] 14. The beam splitter system according to any one of the items 10-13, wherein M=3, thereby obtaining a beam splitter system with 8 input modes and 12 beam splitters.

[0083] 15. An optical quantum computation system comprising the optical entanglement system according to any one of the items 1-9, and a plurality of homodyne detectors, each homodyne detector configured to measure a quantum state.

[0084] 16. The optical quantum computation system according to item 15, further comprising an error correction scheme.

[0085] 17. The optical quantum computation system according to item 16, wherein the error correction scheme utilizes GKP Bell pairs, said GKP Bell pairs are utilized to correct continuous errors, by mapping said continuous errors to discrete errors.

[0086] 18. The optical quantum computation system according to item 17, wherein the generated at least three-dimensional cluster state incorporates a topological error correction scheme configured to correct discrete errors.

[0087] 19. The optical quantum computation system according to any one of the items 15-18, wherein each of the plurality of homodyne detectors comprises an angle modification system.

[0088] 20. The optical quantum computation system according to any one of the items 15-19, wherein the plurality of pairs of quantum states are Gottesman- Kitaev-Preskill (GKP) Bell pairs.

[0089] 21. The optical quantum computation system according to any one of the items 15-20, wherein the generated at least three-dimensional cluster state operates on a plurality of logical qubits, such as Gottesman-Kitaev-Preskill (GKP) qubits.

[0090] 22. The optical quantum computation system according to any one of the items 15-21 , the system comprising a plurality of heterodyne detectors.

[0091] 23. The optical quantum computation system according to item 22, wherein an approximate GKP magic state can be generated by GKP error correction by utilizing at least one heterodyne detector of the plurality of heterodyne detectors.

[0092] 24. The optical quantum computation system according to any one of items 15-

[0093] 23, wherein the system comprises at least one beam-splitter and at least two homodyne detectors configured to measure in orthogonal bases.

[0094] 25. The optical quantum computation system according to item 24, wherein the at least one beam-splitter is a balanced beam-splitter.

[0095] 26. A method for generating at least three-dimensional cluster states, comprising the steps

[0096] • obtaining a plurality of pairs of quantum states,

[0097] • obtaining a beam-splitter network, comprising at least twelve beam splitters and at least eight input modes, wherein each input mode is configured to host a quantum state,

[0098] • generating a time delay on one of the quantum states in at least one pair of quantum states, and

[0099] • utilizing the generated time delay to obtain an at least three-dimensional cluster state.

[0100] 27. The method according to item 26, wherein the at least three-dimensional cluster states are generated by means of the system according to any one of the items 15-21.

[0101] 28. The method according to any one of the items 26-27, further comprising the steps

[0102] • utilizing GKP Bell pairs as the plurality of pairs of quantum states and utilize the beam-splitter network and homodyne detectors to correct continuous errors and map said continuous errors to discrete errors, and

[0103] • obtaining a topological error correction scheme, configured to correct discrete errors.

Claims

Claims1. An optical entanglement system for generating an at least three-dimensional cluster state for measurement-based optical quantum computation, the optical entanglement system having as input a plurality of pairs of quantum states, the system comprising:• a beam-splitter network, comprising o at least twelve beam splitters, and o at least eight input modes, wherein each input mode is configured to host a quantum state,• a plurality of delay lines, each delay line configured for generating a time delay on one of the quantum states in at least one pair of quantum states, wherein the time delays of the plurality of delay lines are multiples of one another, wherein the system is configured for entangling the plurality of pairs of quantum states utilizing the beam-splitter network and the induced time delays of the plurality of delay lines, thereby generating an at least three- dimensional cluster state.

2. The system according to claim 1 , wherein each input mode comprises a switch configured to toggle between two states, such as a qunaught state and a squeezed state.

3. The system according to any one of the preceding claims, wherein the at least three-dimensional cluster state is generated by utilizing a combination of only temporal dimensions.

4. The system according to any one of the preceding claims, wherein the plurality of delay lines is configured to generate a multi-dimensional temporally encoded macronode.

5. The system according to any one of the preceding claims, wherein the at least three-dimensional cluster state is generated without utilizing anyspatial dimension.

6. The system according to any one of the preceding claims, wherein the time delays of the plurality of delay lines have identical lengths.

7. The system according to any one of the preceding claims, wherein N delay lines of the plurality of delay lines are incorporated on at least N pairs of quantum states of the plurality of pairs of quantum states, each pair of the N pairs of quantum states comprising two quantum states, wherein a plurality of macronodes each comprising at least 2N quantum states are entangled with 2N other macronodes, thereby generating an N-dimensional cluster state, wherein N is a positive integer number.

8. The system according to any one of the preceding claims, wherein the system is configured for multiplexing quantum states.

9. The system according to any one of the preceding claims, wherein the system is configured such that multiplexing of quantum states is combinedwith generation of multi-dimensional cluster states.

10. A beam splitter system for manipulating quantum states, the system comprising• a plurality of delay lines, each delay line configured for generating a time delay on at least one of the quantum states, wherein the time delays of the plurality of delay lines are multiples of one another, and• a beam-splitter network, comprising o 2Minput modes, wherein each input mode is configured to host a quantum state, and o M groups of 2M-1beam splitters, wherein each group of beam splitters is configured to entangle input modes to form pairs of input modes, wherein each of the beam splitters of a first group of beam splitters entangles a first input mode with a second input mode, forming a pair of entangled modes, and wherein at least two beam splitters of a subsequent group of beam splitters entangle a pair of entangled modes with a second pair of entangled modes, forming an extended pair of entangled modes, wherein M is a positive integer number greater than 2.

11. The beam splitter system according to claim 10, wherein the plurality of delay lines are configured to generate a multi-dimensional temporally encoded macronode.

12. The beam splitter system according to claim 11, further comprising a plurality of homodyne detectors each configured to measure a quantum state.

13. The system according to claim 12, wherein each of the plurality of homodyne detectors comprises an angle modification system allowing for the manipulation of the entanglement structure.

14. The beam splitter system according to any one of the claims 11-13, whereinM=3, thereby obtaining a beam splitter system with 8 input modes and 12beam splitters.

15. An optical quantum computation system comprising the optical entanglement system according to any one of the claims 1-9, and a plurality of homodyne detectors, each homodyne detector configured to measure a quantum state.

16. The optical quantum computation system according to claim 15, further comprising an error correction scheme.

17. The optical quantum computation system according to claim 16, wherein the error correction scheme utilizes GKP Bell pairs, said GKP Bell pairs are utilized to correct continuous errors, by mapping said continuous errors to discrete errors.

18. The optical quantum computation system according to claim 16, wherein the generated at least three-dimensional cluster state incorporates a topological error correction scheme configured to correct discrete errors.

19. The optical quantum computation system according to any one of the claims 15-18, wherein each of the plurality of homodyne detectors comprises an angle modification system.

20. The optical quantum computation system according to any one of the claims 15-19, wherein the plurality of pairs of quantum states are Gottesman- Kitaev-Preskill (GKP) Bell pairs.

21. The optical quantum computation system according to any one of the claims 15-20, wherein the generated at least three-dimensional cluster state operates on a plurality of logical qubits, such as Gottesman-Kitaev-Preskill(GKP) qubits.

22. The optical quantum computation system according to any one of the claims 15-21 , the system comprising a plurality of heterodyne detectors.

23. The optical quantum computation system according to claim 22, wherein an approximate GKP magic state can be generated by GKP error correction by utilizing at least one heterodyne detector of the plurality of heterodyne detectors.

24. The optical quantum computation system according to any one of claims 15- 23, wherein the system comprises at least one beam-splitter and at least two homodyne detectors configured to measure in orthogonal bases.

25. The optical quantum computation system according to claim 24, wherein the at least one beam-splitter is a balanced beam-splitter.

26. A method for generating at least three-dimensional cluster states, comprising the steps• obtaining a plurality of pairs of quantum states,• obtaining a beam-splitter network, comprising at least twelve beam splitters and at least eight input modes, wherein each input mode is configured to host a quantum state,• generating a time delay on one of the quantum states in at least one pair of quantum states of the plurality of pairs of quantum states, wherein each of the time delays are multiples of one another, and• utilizing the generated time delay to obtain an at least three-dimensional cluster state.

27. The method according to claim 26, wherein the at least three-dimensional cluster states are generated by means of the system according to any oneof the claims 15-25.

28. The method according to any one of the claims 26-27, further comprising the steps • utilizing GKP Bell pairs as the plurality of pairs of quantum states and utilize the beam-splitter network and homodyne detectors to correct continuous errors and map said continuous errors to discrete errors, and• obtaining a topological error correction scheme, configured to correct discrete errors.

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