GHZ state analyser with auxiliary photons

Abstract

An apparatus is disclosed. The apparatus comprises an optical circuit. The optical circuit comprises an arrangement of a first interferometer, one or more second interferometers, and one or more detectors. The apparatus is configured to probabilistically project a multi-qubit input state onto a multi-qubit GHZ state. A method for entangling a plurality of multi-qubit photonic states using such an apparatus is also disclosed herein.

Description

GHZ state analyser with auxiliary photons Technical Field

[0001] The present disclosure relates to methods and apparatuses for performing measurements of photonic qubit states. In particular, but not exclusively, the present disclosure relates to an apparatuses for performing an entangling measurement, and methods for entangling a plurality of multi -qubit photonic states. Background

[0002] Various physical systems have been considered for quantum information processing. Many architectures rely on embodying information in matter-based systems such as ions or spin states in quantum dots. An alternative is linear optical quantum computing, in which information is encoded in electromagnetic field modes. Linear optical quantum computing uses linear optical elements such as beamsplitters and phase shifters to manipulate quantum information encoded in electromagnetic field modes. Photon detectors are used to process and read out information. Photonic platforms provide many advantages including the ability to facilitate quicker gate operations compared to the decoherence time of quantum information, fast read-out measurements, and efficient qubit transfer. Furthermore, photonic systems can largely operate at room temperature.

[0003] There are however some challenges in using linear optics to process quantum information, in particular (i) the lack of deterministic entangling operations and (ii) the impact of photon loss (due, for example, to absorption of leakage), and often these two challenges exacerbate one another. For example, one may be able to improve the theoretical probability of a successful entangling operation by increasing the number of optical elements involved, but each additional optical element may introduce further photon loss, and therefore information loss, into the system. These challenges increase the difficulty of reliably constructing large, entangled states, such as the highly entangled cluster states that are typically required for measurement-based quantum computing. Besides measurement-based quantum computing, these challenges also apply to building quantum networks that rely on the distribution of shared entangled states for applications such as quantum key distribution and entanglement enhanced quantum sensing. Summary

[0004] According to an aspect of the present disclosure, an apparatus is provided. The apparatus comprises an optical circuit. The optical circuit comprises a first interferometer, one or more second interferometers, and a detector arrangement. The first interferometer is arranged to receive, as 2 ^^ input optical modes, a plurality of ^^ dual-rail encoded photonic qubits, each qubit encoded as probability amplitudes corresponding to the photon occupation of two orthogonal optical modes, wherein ^^ is an integer greater than two. The first interferometer is further arranged to interfere the ^^ qubits such that (i) a beamsplitter interaction is performed on the first mode of the first qubit and the second mode of the ^^^^ℎqubit, and (ii) a beamsplitter interaction is performed on the second mode of the ^^^^ℎqubit and the first mode of the ( ^^ + 1)^^ℎqubit for all ^^ between 1 and ^^− 1. The first interferometer is further arranged to output a plurality of ^^ interfered qubits as 2 ^^ output optical modes towards the detector arrangement or the one or more second interferometers. The second interferometer is arranged to receive, as two primary input optical modes, an interfered qubit from the first interferometer. The second interferometer is further arranged to receive, in a first auxiliary input optical mode and a second auxiliary inputoptical mode, two auxiliary single photons. The second interferometer is further arranged to interfere the primary and auxiliary input optical modes such that (i) a first beamsplitter interaction is performed on the two auxiliary input optical modes, (ii) a second beamsplitter interaction is performed on a first primary input optical mode of the received interfered qubit and the first auxiliary input optical mode, and (iii) a third beamsplitter interaction is performed on a second primary input optical mode of the received interfered qubit and the second auxiliary input optical mode. The second interferometer is further arranged to output four output optical modes towards the detector arrangement. The detector arrangement comprises one or more photodetectors, the detector arrangement configured to measure a number of photons in each of the output optical modes received from the first interferometer or the one or more second interferometers.

[0005] The apparatus may be referred to as an assisted ^^-GHZ state analyser. Advantageously, the apparatus enables a user to entangle multiple multi-qubit photonic steps in one step. The user further advantageously enables a user to generate large, entangled states from two-qubit resource states (e.g. Bell states) – for example, one is able to generate an ^^-qubit GHZ state from ^^ Bell states in one go. Moreover, by utilising auxiliary single photons, the probability of successfully measuring a ^^-qubit GHZ state is improved.

[0006] The apparatus may further comprise control circuitry for indicating, based on the measured number of photons in each of the output optical modes, that a ^^-qubit GHZ state measurement has been performed on the ^^ qubits.

[0007] The number ^^ of second interferometers may be one ( ^^ = 1). In some examples, the number ^^ of second interferometers may be greater than one, for example the number ^^ of second interferometers may be equal to the number ^^ of qubits received by the first interferometer ( ^^ = ^^). Multiple second interferometers enable a greater probability of successfully measuring a ^^-qubit GHZ state. Furthermore, the use of several multiple second interferometers improve the photon loss tolerance of the apparatus.

[0008] The input optical modes and output optical modes may be spatial modes.

[0009] The first interferometer may comprise a spatial interferometer having 2 ^^ input ports for inputting the ^^ qubits to the interferometer and 2 ^^ output ports for outputting the ^^ interfered qubits. The first interferometer may further comprise a plurality of waveguides arranged to pass through the first interferometer to connect the 2 ^^ input ports to the 2 ^^ output ports. The plurality of waveguides may be arranged to provide coupling locations between pairs of the waveguides, wherein a waveguide coupler is arranged at each of at least a subset of the coupling locations such that at each of those coupling locations the two optical modes carried by the two respective waveguides are capable of coupling with each other in a beamsplitter interaction.

[0010] A second interferometer may comprise a spatial interferometer having four input ports for inputting the interfered qubit and the two auxiliary single photons to the second interferometer and four output ports for outputting the output optical modes towards the detector arrangement. The second interferometer may comprise a plurality of waveguides arranged to pass through the second interferometer to connect the four input ports with the four output ports; wherein the plurality of waveguides are arranged to p rovide coupling locations between pairs of the waveguides, wherein a waveguide coupler is arranged at each of at least a subset of the coupling locations such that at each of those coupling locations the two optical modes carried by the two respective waveguides are capable of coupling with each other in a beamsplitter interaction.

[0011] One or more of the first interferometer or second interferometer(s) may be provided on a photonic integrated circuit.

[0012] The input optical modes and output optical modes may be polarization modes. The input optical modes and output optical modes may be temporal modes. The input optical modes and output optical modes may be spectral modes. By utilising degrees of freedom beyond spatial degrees of freedom, the number of linear optical elements and detectors may be advantageously reduced.

[0013] The apparatus may further comprise one or more single photon sources configured to provide auxiliary single photons to the number of second interferometers.

[0014] The first interferometer and the one or more second interferometers may be provided as regions of a third interferometer.

[0015] Each beamsplitter interaction may be a 50 / 50 beamsplitter interaction.

[0016] According to an aspect of the present disclosure, a method is provided for entangling a plurality of ^^ photonic multiqubit states, wherein ^^ is an integer greater than two. The method comprises performing a ^^-qubit GHZ state measurement on a set of qubits, the set comprising one qubit from each of the ^^ photonic multiqubit states, wherein each qubit is encoded as probability amplitudes corresponding to the photon occupation of two orthogonal optical modes. Performing the ^^-qubit state measurement comprises providing each qubit of the set of qubits to a first interferometer, the first interferometer configured to (i) perform a beamsplitter interaction on the first mode of the first qubit and the second mode of the ^^^^ℎqubit, (ii) perform a beamsplitter interaction on the second mode of the ^^^^ℎqubit and the first mode of the ( ^^ + 1)^^ℎqubit for all ^^ between 1 and ^^− 1, and (iii) output a plurality of ^^ interfered qubits as 2 ^^ output optical modes. Performing the ^^-qubit state measurement further comprises providing an interfered qubit and a pair of auxiliary single photons to a corresponding second interferometer, the second interferometer configured to (i) perform a first beamsplitter interaction on two auxiliary input optical modes comprising the two auxiliary single photons, (ii) perform a second beamsplitter interaction on a first optical mode of the interfered qubit and the first auxiliary input optical mode, and (iii) perform a third beamsplitter interaction on a second optical mode of the interfered qubit and the second auxiliary input optical mode. Performing the ^^-qubit state measurement further comprises measuring a number of photons in each output optical mode output from the first interferometer or second interferometer.

[0017] Advantageously, the method enables a user to entangle multiple multi-qubit photonic states in one step. This is beneficial in several circumstances. For example, the method enables a generation of heralded, entangled resource states for fault-tolerant quantum computation and communication. In comparison to other known methods for generating heralded, entangled resource states, which typically involve performing Bell state measurements, the methods described herein lead to a significant increase in success probability, and a reduction in the linear optical circuit depth, multiplexing, and overall physical resources required for generating those heralded, entangled resource states.

[0018] The method further advantageously enables a user to generate large, entangled states from two -qubit resource states (e.g. Bell states) – for example, one is able to generate an ^^-qubit GHZ state from ^^ Bell states in one go.

[0019] The method may further comprise, based on the measured photon occupation of the output modes, performing corrective operations on at least one unmeasured photonic qubit of the entangled state.

[0020] At least one photonic multiqubit state may comprise a Bell state.

[0021] At least one photonic multiqubit state may comprise a 3-qubit GHZ state.

[0022] Many modifications and other embodiments set out herein will come to mind to a person skilled in the art in light of the teachings presented herein. Therefore, it will be understood that the disclosure herein is not to be limited to the specific embodiments disclosed herein. Moreover, although the description provided herein provides example embodiments in the context of certain example combinations of elements, steps and / or functions, it will be appreciated that different combinations of elements, steps and / or functions may be provided by alternative embodiments without departing from the spirit or scope of the disclosure.

[0023] Many modifications and other embodiments set out herein will come to mind to a person skilled in the art in light of the teachings presented herein. Therefore, it will be understood that the disclosure herein is not to be limited to the specific embodiments disclosed herein. Moreover, although the description provided herein provides example embodiments in the context of certain example combinations of elements, steps and / or functions, it will be appreciated that different combinations of elements, steps and / or functions may be provided by alternative embodiments without departing from the spirit or scope of the disclosure. Brief Description of the Figures

[0024] Illustrative embodiments of the present disclosure will now be described by way of example only, with reference to the accompanying figures.

[0025] Fig.1 shows an optical mode representation of an apparatus according to an example .

[0026] Fig.2 shows a table of possible measurement outcomes that may be output from the apparatus of Fig.1 and the input state that can be inferred from those outcomes.

[0027] Fig.3 shows an optical mode representation of an apparatus according to an example .

[0028] Fig.4 shows a graph of the success probability of an assisted ^^-qubit GHZ measurement in the presence of photon loss.

[0029] Fig.5 shows a graph of the success probability of an assisted ^^-qubit GHZ measurement in the presence of photon loss.

[0030] Fig.6 shows a block diagram of an apparatus for performing an assisted ^^-qubit GHZ measurement according to an example.

[0031] Fig.7 shows a flowchart of a method for entangling a plurality of multiqubit states.

[0032] Throughout the description and the drawings, like reference numerals refer to like parts. Detailed Description

[0033] Embodiments of the disclosure are described with reference to the accompanying drawings. However, it should be appreciated that the disclosure is not limited to the embodiments, and all changes and / or equivalents or replacements thereto also belong to the scope of the disclosure. The same or similar reference denotations may be used to refer to the same or similar elements throughout the specification and the drawings.

[0034] As used herein, the terms “have”, “may have”, “include”, or “may include” a feature (e.g. a number, function, operation, or a component such as a part) indicate the existence of the feature and do not exclude the existence of other features. Throughout the description and claims of this specification, the words “comprise” and “contain” and variations of them mean “including but not limited to”, and they are not intended to (and do not) exclude other components, integers or steps. Throughout the description and claims of this specification, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article isused, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.

[0035] As used herein, the terms “A or B”, “at least one of A and / or B”, or “one or more of A and / or B” may include all possible combinations of A and B. For example, “A or B”, “at least one of A or B”, “at least one of A and B” may indicate all of (1) including at least one A, (2) including at least one B, or (3) including at least one A and at least one B.

[0036] As used herein, the terms “first” and “second” may modify various components regardless of importance and do not limit the components. These terms are only used to distinguish one component from another. For example, reference to a first component and a second component may indicate different components from each other regardless of the order or importance of the components.

[0037] It will be understood that when an element (e.g. a first element) is referred to as being (physically, operatively or communicatively) “coupled with / to”, or “connected with / to” another element (e.g. a second element), it can be coupled with / to the other element directly or via a third element. In contrast, it will be understood that when an element (e.g. a first element) is referred to as being “directly coupled with / to” or “directly connected with / to” another element (e.g. a second element), no element (e.g. a third element) intervenes between the element and the other element.

[0038] The terms as used herein are provided merely to describe some embodiments thereof, but not to limit the scope of other embodiments of the disclosure. It is to be understood that the singular forms “a”, “an”, and “the” include plural references unless the context clearly dictates otherwise. All terms including technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the embodiments of the disclosure belong. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealised or overly formal sense unless expressly so defined herein.

[0039] Just as a classical bit has a state – a computational basis state 0 or a computational basis state 1 – a quantum bit (a “qubit”) also has a state. A qubit may be in either of the computational basis states, in Dirac notation written as |0^^^and |1^^^respectively, or may be in a linear combination – a superposition - of those states e.g. |Ψ^= ^^|0^^^+ ^^|1^^^. Abstractly, a qubit can be understood as a normalized vector in a two-dimensional Hilbert space. For the purposes of this document, such a qubit will often be referred to as a logical qubit, and the subscript ^^ is used throughout this specification to denote a logical state. A measurement of the qubit in the computational basis will typically project the qubit onto either the 0 state or the 1 state with a probability dependent on the real or complex valued parameters ^^ and ^^. A logical qubit may be realized using one or more physical qubits. Physical qubits are understood to mean physical quantum systems, the quantum properties of which can be interpreted as qubit states.

[0040] Many physical systems can be used to realize a physical qubit. Dynamically, quantum objects such as photons, electrons, ions and atoms obey the laws of quantum mechanics. In a physical quantum system, the quantum degrees of freedom within that physical quantum system are often referred to as modes. For example, a photonic system may be described by the spatial modes of the photonic system (which paths or superpositions of paths the photons within the photonic system take), the temporal modes of the photonic system (which time bins or superpositions of time bins the photons within the photonic system are in), the polarization modes of thephotonic system (for example, whether the photons of the photonic system are horizontally polarized, vertically polarized, or some superposition of the two), the frequency modes of the photonic system (which frequencies or superpositions of frequencies the photons within the photonic system are in) or any intersection of these modes (for example, the quantum state of photons in a waveguide is often described in terms of spatiotemporal modes).

[0041] Photonic qubit states may be “dual-rail encoded”, which means that the logical value of a qubit can be expressed in terms of which of two modes a photon resides in. That is, the logical state of the qubit may be considered to be a computational basis state |0^^^if the photon is in a first mode, while the logical state of the qubit may be considered to be a computational basis state |1^^^if the photon is in a second mode orthogonal to the first mode. The logical state of the qubit may be described by a superposition state if the photon is in a superposition of the first and second modes. Without loss of generality, in the dual-rail encoding, the logical states |0^^^and |1^^^of a qubit can be expressed as: |0^^^= |11, 02^^^ ^^(EQ.1) |1^^^= |01, 12^^^ ^^(EQ.2) where the subscripts 1 and 2 refer to first and second orthogonal modes respectively, and the corresponding 0 or 1 value represents the number of photons in that mode. The subscript “DR” denotes a dual-rail encoded photonic state.

[0042] A beamsplitter is a four-port optical device, taking two optical modes as input and outputting two optical modes as output. In an ideal beamsplitter, the mode coupling is linear. Both input modes are partially transmitted and partially reflected by the beamsplitter. In an ideal beamsplitter, the total number of photons incident upon the beamsplitter at a point in time is conserved such that the number of photons output from the output ports is the same as the number of photons input on the input ports, but the photons are distributed between the two output modes in proportion to a transmission coefficient (or a reflection coefficient). The effect of a 50 / 50 beamsplitter on the dual-rail encoded computational basis states (EQ.1 and EQ.2) is, without loss of generality, taken to be: ^^ ^^|11,02^^^ ^^→ |11′,02′^^^ ^^+ |01′, 12′^^^ ^^(EQ.3) ^^ ^^|01,12^^^ ^^→ |11′,02′^^^ ^^− |01′, 12′^^^ ^^(EQ.4) up to normalization, where 1’ and 2’ indicate the first and second output modes of the beamsplitter. If two single photons are incident on a lossless 50 / 50 beamsplitter at the same time, one in each of two input modes of the beamsplitter, then due to interference the output modes become entangled. The output state can be expressed as a superposition of two configurations – one in which both photons are deflected to a first output mode and one in which both photons are deflected to the second output mode. The probability of the two photons being found in a particular one of the output modes is 50%. That is, up to normalisation, the effect of the 50 / 50 beamsplitter on two input modes carrying a single photon each is: ^^ ^^|11,12^^^ ^^→ |21′,02′^^^ ^^+ |01′, 22′^^^ ^^(EQ.5)

[0043] This effect is known as “photon bunching” and occurs even with imperfect 50 / 50 beamsplitters.

[0044] The term “beamsplitter interaction” as used herein is understood to mean a mode coupling interaction that couples two input modes to two output modes in a same or similar manner to the way in which a beamsplitterwould. A beamsplitter interaction (as used herein) accordingly may be imparted by a beamsplitter or by some other mechanism.

[0045] Two examples of entangled quantum states are the canonical ^^-qubit GHZ states (where ^^ is an integer) defined (up to normalization) as: | ^^ ^^ ^^ ^^ ±^^^= |0^^^^^^± |1^^^^^^(EQ.6) where ⊗ indicates the Kronecker product. If one were to be provided with a large supply of ^^-qubit GHZ states of the form shown in (EQ.6) and measure the state of each of the ^^ qubits, then one would expect all ^^ qubits to be in logical state 0 around 50% of the time, and to be in logical state 1 around 50% of the time. States that are equivalent to the canonical ^^-qubit GHZ states of (EQ.6) up to local (single qubit) Clifford rotations are also known as ^^-qubit GHZ states: any ^^-qubit GHZ state may be transformed to a canonical ^^-qubit GHZ state via a sequence of one or more local operations. For example, the state (1⁄√2 )(|0,0,1^^^± |1,1,0^^^) is also an example of a 3-GHZ state.

[0046] As used herein, to perform a “ ^^-qubit GHZ measurement” is understood to mean projecting a multiqubit state onto an ^^-qubit GHZ state. Several examples of apparatuses capable of performing a ^^-qubit GHZ measurement are described herein.

[0047] Fig.1 shows a schematic diagram of an apparatus configured to perform a ^^-qubit GHZ measurement according to an example. The apparatus may be referred to as an unassisted ^^-qubit GHZ state analyser. More particularly, Fig.1 depicts an optical mode representation of an optical circuit 100 of the apparatus that is suitable for performing a ^^-qubit GHZ measurement according to an example. The optical circuit 100 may be referred to as a ^^-GHZ state analyser circuit. In the optical circuit 100 shown in Fig.1, the number of qubits on which the measurement is performed is ^^, which may be any integer greater than two. Each horizontal line in Fig. 1 represents an optical mode. The optical modes may be, for example, spatial modes, temporal modes, polarisation modes or spectral modes, or may be some intersection or combination of those modes, for example spatiotemporal modes. Each vertical line represents a beamsplitter interaction 104.

[0048] Each of the ^^ dual-rail encoded photonic qubits, ^^1 to ^^ ^^, is encoded as probability amplitudes corresponding to the photon occupation of a pair of orthogonal optical modes. For example, if a photon is present in first optical mode 106-1A and no photon is present in second optical mode 106-1B then the qubit ^^1 is in a computational basis state |0^^^; if a photon is present in the second optical mode 106-1B and no photon is present in first optical mode 106-1A then the qubit ^^1 is in a computational basis state |1^^^; if a photon is in a superposition of being in the first mode 106-1A and the second mode 106-1B, then the qubit ^^1 is in a superposition state. The ^^ qubits, ^^1 to ^^ ^^, are provided to an interferometer 102 in which a plurality of beamsplitter interactions 104 are performed. A 50 / 50 beamsplitter interaction 104 is performed between the first mode 106-1A of the first qubit ^^1 and the second mode 106-NB of the ^^^^ℎqubit ^^ ^^. A 50 / 50 beamsplitter interaction 104 is also performed on the second mode 106- ^^B of the ^^^^ℎqubit and the first mode 106-( ^^ + 1)A of the ( ^^ + 1)^^ℎqubit for all ^^ between 1 and ^^− 1, wherein ^^ is an integer. As the beamsplitter interactions 104 on Fig.1 are performed on different modes, the order of the beamsplitter interactions 104 may be swapped or they may be performed at the same time. After the beamsplitter interactions 104, measurements of the number of photons of each output mode are performed, as indicated by the measurement symbols 108. In this example, themeasurements are photon number resolving measurements capable of determining a number of photons in each output mode.

[0049] The measurements 108 collectively may be made by an arrangement of one or more photon number resolving (PNR) photodetectors, which may be referred to as PNR detectors. A PNR detector may be capable of resolving the number of photons in an output mode. A PNR detector may comprise, for example, a superconducting nanowire detector that generates an output signal, the intensity of which is dependent on the number of photons that strike the detector. A PNR detector may comprise a transition edge sensor. In some examples the measurements 108 collectively may be made by one or more pseudo-threshold detectors capable of distinguishing between 0, 1, or more than one photon in a given output mode.

[0050] Consider, for example, the three-qubit ( ^^ = 3) logical input product state: |0^^^^3= |11 ^^ ,01 ^^^|12 ^^ , 02 ^^^|13 ^^ , 03 ^^^ (EQ. 7)in which the subscripts ^^ ^^, ^^ ^^, ( ^^ = 1,2,3), on the right-hand side indicate a corresponding input mode 106- ^^A, 106- ^^B. At the output of the interferometer 102, the resulting state is given (up to normalisation) by:in which the subscripts ^^ ^^’, ^^ ^^’, ( ^^ = 1,2,3), on the right-hand side indicate a corresponding output mode. Next consider, the effect of the interferometer 102 on the complementary logical input state: |1^^^^3= |01 ^^ ,11 ^^^|02 ^^ , 12 ^^^|03 ^^ , 13 ^^^(EQ. 8)which becomes (up to normalisation):

[0051] A property of the two input states of (EQ.7) and (EQ.8) is that any pair of input optical modes 106 that undergo a beamsplitter interaction in the interferometer 102 has only one photon, and these are the only input states for which this property holds. If one considers any other input state to the interferometer 102, at least one pair of input optical modes that undergo a beamsplitter interaction 104 in the interferometer 102 receives two photons and the beamsplitter interaction will result in a state in which both of the photons are in the same mode. The input states given in (EQ.7) and (EQ.8) are the only two input states that have any overlap in their detection patterns. Accordingly, if the measurements 108 yield certain outcomes (e.g. patterns of photon numbers) , then this corresponds to a measurement of a canonical 3-qubit GHZ state. This property holds if the apparatus is extended to ^^ > 3 qubits.

[0052] The left-hand column of Fig. 2 shows the measurement outcomes (photon numbers) that may be measured by the photon number resolving measurements in response to a three -qubit dual-rail encoded state formed of three photons being provided to an optical circuit such as that depicted in Fig.1 (with ^^ equal to three). For example, “210000” indicates that a measurement yielded two photons in output mode 106 -1A’, a measurement yielded one photon in output mode 106-1B’ and so on. The right-hand column of Fig.2 indicates the input state that can be inferred to have been measured by the optical circuit, based on measuring a corresponding outcome indicated in the left-hand column. For example, if a measurement outcome yields a single photon in modes 106-1A’, 106-1B’ and 106-2B’, then one can infer that the logical input state that was measured by the apparatus was a 3-qubit GHZ state and in particular the canonical 3GHZ+ state (see EQ.6). As indicated in the table of Fig.2, the optical circuit 100 performs an equal-weight projection on to the canonical 3GHZ+ and 3GHZ- states and each of the separable states indicated in the table, and each inferred state is associated with four different detection patterns.

[0053] Given a maximally mixed ^^-qubit input state, the optical circuit 100 of Fig. 1 may be capable of successfully projecting the input state onto a canonical ^^-qubit GHZ state with success probability:and is capable of distinguishing between the two canonical GHZ states.

[0054] The skilled person will appreciate that the optical circuit 100 may be varied in many ways while still performing the same function. For example, in the description above for all qubits ^^1 to ^^ ^^ a logical 0 is encoded by a photon in a first mode 106- ^^A and no photon in a second mode 106- ^^B, and a logical 1 is encoded by photon in a second mode 106- ^^B and no photon in a first mode 106- ^^A; however, by flipping the encoding for one or more of the qubits, the optical circuit 100 may project onto a ^^-qubit GHZ state other than a canonical ^^-qubit GHZ state, for example (if ^^ ^^ ^^ 3), the state (1⁄√2 )(|0,1,0^^^± |1,0,1^^^).

[0055] The optical modes of optical circuit 100 may comprise spatial modes, e.g. each input mode 106- ^^A, 106- ^^B may represent a path into the apparatus or optical circuit, and each output mode 106- ^^ ^^′, 106- ^^ ^^′ may represent a path out of the interferometer 102 towards a photodetector aligned with that output path. In some examples, the apparatus may be implemented in bulk optics and the beamsplitter interaction may be implemented by a bulk optic beamsplitter arranged to couple two input paths to two output paths. In some examples, the interferometer 102, and optionally the detectors for performing the measurements 108, may be provided on a photonic integrated circuit. In a photonic integrated circuit, the beamsplitter interactions may be implemented using, for example, a directional waveguide coupler. The skilled person will appreciate that any other element capable of suitably coupling two input paths to two output paths may be used to impart a beamsplitter interaction on spatial modes.

[0056] The optical modes of optical circuit 100 may comprise temporal modes, e.g. each input mode 106- ^^A, 106- ^^B and output mode 106- ^^A’, 106- ^^B’ may represent a time bin (unit of time or time window) in which a photon may be found. A beamsplitter interaction may be imparted by a temporal mode coupling device. The skilled person would appreciate that any suitable temporal mode coupling device might be used.

[0057] In a first example of a temporal mode coupling device, a temporal mode coupling device may comprise a reconfigurable beamsplitter and a delay line (e.g. optical fibre). The reconfigurable beamsplitter may have two input ports and two output ports and be controllable to interfere photons received at substantially the same time (that is, received such that the wave packets of those photons overlap) through two input ports based on one ormore control signals received from control logic. In other words, the reconfigurable beamsplitter may have a tuneable effective transmission coefficient (or conversely, a tuneable effective reflection coefficient). The delay line may couple a first output port of the reconfigurable beamsplitter to a first input port of the reconfigurable beamsplitter and may have a length designed to delay light by an amount of time equal to the duration of a time bin. Accordingly, the reconfigurable beamsplitter may be controlled to selectively interfere an optical mode received from the delay line with an optical mode received from the second input port. To perform a beamsplitter interaction between for example two photons in two time bins, the photons may be sent sequentially to the second input port of the reconfigurable beamsplitter, the reconfigurable beamsplitter may be controlled to route the first photon to the first output port and into the delay line, and then controlled to interfere the first photon and the second photon. The reconfigurable beamsplitter may be implemented in any suitable way, for example as a Mach- Zehnder interferometer. For example, a Mach-Zehnder interferometer may comprise two 50:50 beamsplitters with a reconfigurable thermo-optic phase shifter on one internal path between the two 50:50 beamsplitters. Optionally, a further phase shifter may be arranged on an external path (input or output) of a Mach-Zehnder interferometer. By tuning the phase shifter(s), the effective transmission coefficient of the Mach -Zehnder interferometer is reconfigurable.

[0058] In a second example of a temporal mode coupling device, a temporal mode coupling device may comprise, for example, a quantum memory device comprising an atomic system having discrete energy states. The atomic system may comprise a single type of neutral atoms or ions or a plurality of different types of neutral atoms and / or ions. The atomic system may be an atomic ensemble comprising a plurality of atoms, such as a gas of atoms, or ions, such as a rare earth ion locked in a host medium. Such an atomic system may use energy levels associated with electrons in inner or outer shells of atoms, ions, or crystal defects. Examples of suitable atomic ensembles include vapours containing Rubidium. Other energy levels of atomic system may be used including Rydberg – type atomic systems. Additionally, or alternatively the atomic system may comprise a single neutral atom or ion, for example single 87Rb atoms in a magneto-optical-trap (MOT). The quantum memory device may be used to controllably place, for example, a photon in the first temporal mode into a superposition of being stored in the memory and being not stored in the memory, and then to controllably release any photon (or quantum state thereof) stored in the memory at the time the second temporal mode passes through, in order to interfere temporal modes in a similar way as a reconfigurable beamsplitter can be used to interfere modes. Additionally, an example of a suitable quantum memory device being used in such a way is described in international patent application number PCT / GB2021 / 052447 filed on 21 September 2021 in the name of ORCA Computing Limited, the content of which is incorporated herein by reference.

[0059] The optical modes of optical circuit 100 may comprise spectral modes e.g. each input mode 106- ^^A, 106- ^^B may represent a particular frequency or frequency window. The beamsplitter interactions may be imparted using, for example, a frequency mixer or a coupled ring resonator system.

[0060] The optical modes of optical circuit 100 may comprise polarization modes. For example, a beamsplitter interaction may be implemented using a wave plate.

[0061] Fig.3 shows a schematic diagram of an apparatus configured to perform a ^^-qubit GHZ measurement state measurement according to an example. Mor particularly, Fig.1 depicts an optical mode representation of an optical circuit 300 of the apparatus that is suitable for performing a ^^-qubit GHZ measurement according to an example. The optical circuit 300 is another example of a ^^-GHZ state analyser circuit. As the circuit 300 utilisesauxiliary modes, it may also be referred to as an assisted ^^-GHZ state analyser circuit. The apparatus may be referred to as an assisted ^^-qubit GHZ state analyser. In the optical circuit 300 shown in Fig.3, the number of qubits on which the measurement is performed is ^^, which may be any integer greater than two. Each horizontal line in Fig.3 represents an optical mode, and each vertical line represents a beamsplitter interaction. Similar to the optical circuit 100 of Fig.1, the optical modes of optical circuit 300 may be, for example, spatial modes, temporal modes, polarisation modes or spectral modes, or may be some intersection or combination of those modes, for example spatiotemporal modes. Similar to the optical circuit 100 of Fig.1, the beamsplitter interactions may be implemented using any suitable mode coupling elements.

[0062] The optical circuit 300 of Fig.3 comprises a first interferometer 102 which is functionally equivalent to that depicted in Fig.1, and a second interferometer 302. The first interferometer 102 is arranged to receive, as 2 ^^ input optical modes (106-1A, 106-1B,…,106-NA, 106-NB), a plurality of ^^ dual-rail encoded photonic qubits, ^^1 to ^^ ^^, each qubit encoded as probability amplitudes corresponding to the photon occupation of two orthogonal optical modes 106- ^^A, 106- ^^B ( ^^ =. The first interferometer 102 is further arranged to interfere the ^^ qubits such that a beamsplitter interaction 104 is performed on the first mode 106-1A of a first qubit ^^1 and the second mode 106-NB of the ^^^^ℎqubit ^^ ^^, and such that a beamsplitter interaction is performed on the second mode 106- ^^B of the ^^^^ℎqubit and the first mode of 106-( ^^ + 1)A of the ( ^^ + 1)^^ℎqubit for all ^^ between and including 1 and ^^− 1. The first interferometer 102 is further arranged to output ^^ interfered qubits as 2 ^^ output optical modes (one such interfered qubit is identified by numeral 304 in the figure). An interfered qubit comprises a pair of optical modes output from the first interferometer that together may constitute a dual-rail encoded qubit.

[0063] The second interferometer 302 is arranged to receive, directly or indirectly from the first interferometer 102, an interfered qubit 304 as two primary input modes. The second interferometer 302 is further arranged to receive, in a first auxiliary input optical mode 306-1A and a second auxiliary input optical mode 306-1B, a pair of single photons from one or more photon sources 308. The second interferometer 302 is further arranged to interfere the primary input modes and auxiliary modes such that a first beamsplitter interaction 104 is performed on the two auxiliary input modes, a second beamsplitter interaction is performed on a first primary input optical mode of the received interfered qubit 304 and the first auxiliary input optical mode 306 -1A, and a third beamsplitter interaction 104 is performed on a second primary input optical mode of the received interfered qubit 304 and the second auxiliary input optical mode 306-1B. The second interferometer 302 is arranged to output four output optical modes (106-1A’, 106-1B’, 306-1A’, 306-1B’) for measurement 108. The second and third beamsplitter interactions of the second interferometer 302 act upon different modes and so the order of the second and third beamsplitter interactions may be swapped, or the beamsplitter interactions may occur at the same time.

[0064] As with the optical circuit 100 described above, in the optical circuit 300 each measurement 108 is a photon number resolving measurement capable of measuring the number of photons in measured output modes.

[0065] The photon source(s) 308 may comprise one or more single-photon sources. A single-photon source is a photon source that probabilistically or deterministically outputs one photon per attempt. A deterministic single- photon source may comprise, for example, a quantum dot photon source. A probabilistic single-photon source may comprise a heralded single-photon source. A heralded single-photon source may rely upon a probabilistic non-linear process such as spontaneous four-wave mixing (SFWM) or spontaneous parametric down conversion (SPDC), in which a non-linear photonic element is pumped by an optical source (e.g. a laser) and probabilistically produces pairs of entangled photons, the detection of one heralding the generation of the other. Suitable materialsfrom which non-linear photonic elements may be formed include but are not limited to periodically-poled lithium niobate (PPLN) or potassium titanyl phosphate (KTP). Furthermore, while two photon sources 308 are depicted in Fig.3, the skilled person will appreciate that more or fewer photon sources may be utilised depending on the architecture of the apparatus in which the optical circuit 300 is implemented. For example, if modes 306-1A and 306-1B are temporal modes, then a single photon source 308 may be utilised to produce a single photon in each of two temporal modes (time bins).

[0066] Advantageously, the optical circuit 300 may increase the probability of successfully projecting an input state onto a ^^-qubit GHZ state, even in the presence of photon loss.

[0067] The skilled person will appreciate that many variations of the optical circuit 300 may be made while still performing a similar function. For example, in the description above for all qubits ^^1 to ^^ ^^ a logical 0 is encoded by a photon in a first mode 106- ^^A and no photon in a second mode 106- ^^B, and a logical 1 is encoded by photon in a second mode 106- ^^B and no photon in a first mode 106- ^^A; however, by flipping the encoding for one or more of the qubits, the optical circuit 300 may project onto a ^^-qubit GHZ state other than a canonical ^^-qubit GHZ state. As another example, while the second interferometer 302 is shown interfering the auxiliary modes 306-1A and 306-1B with the optical modes of the first qubit ^^1 in Fig.3, this is for illustrative purposes only; the second interferometer may be arranged to interfere the paired optical modes of any of the qubits Q1 to QN with the pair of auxiliary modes 306-1A, 306-1B.

[0068] The optical circuit 300 may include one or more single-mode phase shift operations, for example to shift a phase of a first mode relative to a second mode. In some examples, the second interferometer 302 may include one or more phase shift operations without detracting from the function of the apparatus (e.g. without inhibiting the apparatus from performing a ^^-GHZ measurement). For example, it may not matter whether the two auxiliary modes 306 of the second interferometer 302 are in a state (1⁄√2 )(|0,2^^^ ^^+ |2,0^^^ ^^) or (1⁄√2 )(|0,2^^^ ^^− |2,0^^^ ^^) after the first beamsplitter interaction. In some examples, phase shifts may be implemented by the same elements that implement the beamsplitter interactions (e.g. by a reconfigurable beamsplitter).

[0069] While the optical circuit 300 depicted in Fig.3 includes one second interferometer 302 that couples a single interfered qubit 304 (corresponding to qubit ^^1) to a pair of auxiliary modes 306-1A, 306-1B, further second interferometers 302 may be included that couple further interfered qubits 304 output from the first interferometer 102 with further pairs of auxiliary modes. For example, a further copy of second interferometer 304 may couple an interfered qubit 304 (corresponding to the ^^^^ℎqubit ^^ ^^) to a pair of auxiliary modes 306- ^^A, 306- ^^B.

[0070] While the first interferometer 102 and second interferometer 302 of the optical circuit 300 are shown as distinct interferometers, this is for illustrative purposes only. The first interferometer and second interferometer(s) may be distinct from one another, for example provided on distinct yet coupled photonic inte grated circuits. Alternatively, the first interferometer 102 and one or more second interferometers may be comprised in a third interferometer e.g. the first interferometer 102 and one or more second interferometers are interferometer regions of a composite interferometer.

[0071] In some examples, a second interferometer 302 may be applied to every other interfered qubit output from the first interferometer 102, for example a second interferometer may be applied to all odd numbered interfered qubits (e.g. interfered qubits corresponding to input qubit ^^1, ^^3,…) or to all even numbered interfered qubits (e.g. interfered qubits corresponding to input qubit ^^2, ^^4,…). For example, consider an assisted 4-GHZstate analyser circuit ( ^^ = 4) in which input qubits ^^1, ^^2, ^^3, and ^^4 are interfered on a first interferometer 102 and then odd numbered interfered qubits (or, alternatively, even numbered interfered qubits) are provided to respective second interferometers 302 where they are interfered with respective pairs of auxiliary modes. In this example, the probability of successfully measuring a 4-GHZ state is 25 / 128, which is greater than the probability of successfully measuring a 4-GHZ state using an unassisted GHZ state analyser (EQ.9) which is 1 / 8.

[0072] In some examples, a second interferometer 302 may be applied to every interfered qubit output from the first interferometer 102. For example, consider an assisted 4-GHZ state analyser circuit ( ^^ = 4) in which input qubits ^^1, ^^2, ^^3, and ^^4 are interfered on a first interferometer 102 and then all interfered qubits are provided to respective second interferometers 302 where they are interfered with respective pairs of auxiliary modes. In this example, the probability of successfully measuring a 4-GHZ state is 17 / 64, which is greater than the probability of successfully measuring a 4-GHZ state using an unassisted GHZ state analyser (EQ.9) which is 1 / 8. This improvement of performance is noticeable also when the optical circuit is applied to a greater number of photons. In the absence of photon loss, the success probability of an assisted ^^-GHZ state analyser circuit in which the number of second interferometers 302 is equal to the number of qubits ^^ is believed to be:

[0073] When an assisted ^^-GHZ state analyser circuit includes multiple second interferometers 302, there may be some redundancy in the measurement outcomes that indicate a successful ^^-GHZ measurement, and this provides some tolerance to photon loss. Roughly speaking, some successful measurement outcomes may remain as such if one or more of the second interferometers were removed, and for those measurement outcomes it may not matter is corresponding auxiliary photons are lost. Using a simple model in which all qubit photons and auxiliary photons experience a lossy channel with a loss rate of 1 − ^^, the probability of a successful 3-GHZ measurement from an assisted 3-GHZ analyser circuit with three second interferometers 302 is :

[0074] Similarly, the probability of a successful 4-GHZ measurement from an assisted 4-GHZ analyser circuit with four second interferometers 302 is:

[0075] Fig.4 shows a plot of the probability of successfully performing a 3-GHZ state measurement in the presence of photon loss using an unassisted 3-GHZ state analyser circuit e.g. optical circuit 100 of Fig.1 (solid curve), using an assisted 3-GHZ state analyser circuit with one second interferometer 302 e.g. optical circuit 300 of Fig.3 (dashed curve), and using an assisted 3-GHZ state analyser circuit with three second interferometers 302 (dotted curve).

[0076] Fig.5 shows a plot of the probability of successfully performing a 4-GHZ state measurement in the presence of photon loss using an unassisted 4-GHZ state analyser circuit e.g. optical circuit 100 of Fig.1 (solid curve), using an assisted 4-GHZ state analyser circuit with one second interferometer 302 e.g. optical circuit 300of Fig.3 (dashed curve), and using an assisted 4-GHZ state analyser circuit with four second interferometers 302 (dotted curve).

[0077] Fig.6 illustrates an apparatus 600 according to an example. The apparatus 600 provides an example implementation of the optical circuit 300 described above in relation to Fig.3. The skilled person will appreciate that other architectures would also be suitable. Apparatus 600 comprises an optical circuit. The optical circuit includes a spatial interferometer 610 comprising a first interferometer 680 and a second interferometer 690. More particularly, the first interferometer 680 and the second interferometer 690 are shown as first and second interferometer regions of a composite interferometer 610. The apparatus 600 further comprises a detector arrangement 650. The skilled person will appreciate that while the spatial interferometer 610 and detector arrangement 650 are illustrated as separate entities in the figure, this is for illustration purposes only. The example apparatus 600 shown in Fig.6 is suitable for probabilistically performing an assisted 4-qubit GHZ measurement (that is, ^^ = 4), and the detection outcome indicates whether or not the measurement has been successful. The skilled person would appreciate that the apparatus 600 may be further adapted to perform an ^^-qubit GHZ measurement for an arbitrary number of qubits ^^.

[0078] The interferometer 610 of Fig.6 is arranged to receive, as 2 ^^ (in this example, eight) input optical modes, a plurality of ^^ (in this example, four) dual-rail encoded photonic qubits, each photonic qubit encoded as probability amplitudes corresponding to the photon occupation of two optical modes. In this example, the input optical modes are spatial optical modes. The interferometer 610 accordingly comprises 2 ^^ + 2 input ports 620, labelled 620-1A, 620-1B, 620-2A,…, 620-5B in the figure. The first received qubit is defined by the photon occupation of the two optical paths immediately coupled to input ports 620-1A and 620-1B; the second received qubit is defined by the photon occupation of the two optical paths immediately coupled to input ports 620-2A and 620-2B; the third received qubit is defined by the photon occupation of the two optical paths immediately coupled to input ports 620-3A and 620-3B; and the fourth qubit is defined by the photon occupation of the two optical paths immediately coupled to input ports 620-4A and 620-4B. Two further auxiliary input ports 620-5A and 620- 5B are configured to receive a single photon from a photon source (not shown in Fig.6). For example, the two auxiliary input ports may be coupled using e.g. optical fibre to respective single-photon sources.

[0079] The interferometer 610 further comprises 2 ^^ + 2 output ports 630, labelled 630-1A, 630-1B,…,630-5B in the figure, for outputting the 2 ^^ + 2 output modes towards the detector arrangement 150. The interferometer 610 comprises a plurality of optical paths, shown as solid lines in Fig.6, that are arranged to pass through the interferometer to connect the 2 ^^ + 2 input ports 620 to the 2 ^^ + 2 output ports 630. The plurality of optical paths are arranged so as to provide coupling locations between pairs of the plurality of optical paths. A 50 / 50 beamsplitter interaction, denoted by a beamsplitter 640 in the figure, occurs at each of the coupling locations such that at each coupling location the two optical modes carried by the two respective optical paths couple with one another in a beamsplitter interaction.

[0080] The interferometer 610 may be designed and manufactured in any suitable and desired way.

[0081] As a first example, the interferometer 610 may comprise a photonic integrated circuit. The integrated circuit may be implemented in silicon nitride (Si3N4) or any other suitable material. The interferometer may comprise a plurality of etched waveguides to form the optical paths between the input ports and the output ports. The 50 / 50 beamsplitter interactions may be implemented by directional coupling, in which in a small coupling region two waveguides are situated close enough to one another that the evanescent fields between the twowaveguides couple, the length of the coupling region and the separation of the waveguides chosen to provide a transmission coefficient of 50%. Crossing points at which no beamsplitter interaction is performed, for example such that the optical path from input port 620-1A can cross the optical paths from input ports 620-1B, 620-2A and 620-2B without interference, may be implemented by providing one or more of said optical paths at different depths within the substrate material of the integrated circuit. As an alternative, depending on how the detectors are deployed, the interferometer on the integrated circuit may be of a planar design, such that there are no crossing points at which no beamsplitter interaction is performed. As another alternative, while the apparatus is shown with input ports 620 on one side of the apparatus and detectors on the other, different geometries and form factors are equally applicable: for example, the apparatus may be configured in a circular or other geometry in which all detectors 660 are arranged in the interior of the device while all input ports 620 are arranged around the exterior of the apparatus.

[0082] Beamsplitter interactions in a photonic integrated circuit may be implemented using, for example, an on- chip Mach-Zehnder interferometer (MZI). For example, each MZI may comprise two coupled directional couplers with a phase shifter provided on one internal optical path therebetween. The phase shifter may comprise an electro- optic modulator (EOM). By applying a suitable control signal to the EOM, the phase imparted by the phase shifter may be adjusted, enabling mitigation of manufacturing defects in the directional couplers.

[0083] In a second example, the interferometer 610 may be implemented in bulk optics, with optical paths provided in free space or optical fibre, and beamsplitter interactions provided by beamsplitters. Optical fibre may advantageously enable reliable “long-distance” interactions such as that between the first input port 620-1A and eighth input port 620-4B (in contrast the nearest-neighbouring waveguide interactions between e.g. second input port 620-1B and third input port 620-2A may be thought of as “short distance” in comparison.

[0084] The detector arrangement 650 comprises 2 ^^ + 2 photodetectors which in Fig.6 are labelled 660-1, to 660-10 and are each aligned with a respective output port 630. The photodetectors comprise photon number resolving (PNR) detectors, capable of determining how many photons are received. For example, the detectors may comprise superconducting nanowire detectors that generate an output signal intensity proportional to the (discrete) number of photons that strike a detector. For example, the PNR detectors may comprise transition edge sensors (TESs). The choice of detector may depend on the form factor of the apparatus.

[0085] The skilled person will appreciate that the interferometer and detectors may be connected in any of a number of ways. For example, the detectors may be implemented on-chip with the interferometer. For example, the output ports 630 may be coupled to the detectors with optical fibres.

[0086] The apparatus 600 further comprises control logic 670. The control logic 170 is configured to receive detection signals from the photodetectors 660, for example electrical signals indicative of a number of photons incident on each detector at a particular point in time or within a particular time window. The control logic 670 is further configured to determine whether or not an ^^-qubit GHZ measurement has successfully been performed. In other words, the control logic 670 is further configured to determine whether or not the detection signals from the photodetectors 660 imply that the photon configuration input into the input ports 610-1A to 620-4B of the interferometer 610 correspond to an ^^-qubit GHZ (in this example 4-qubit GHZ) state. The control logic 170 is further configured to, in the event that a ^^-qubit GHZ measurement is successfully performed, provide an indication that the ^^-qubit GHZ measurement was successfully performed. In some examples, the control logic 670 may generate an output signal (electrical, audible, visual or otherwise) only when the detection signals implythat the photon configuration input into the interferometer correspond to an ^^-qubit GHZ state. In other examples, the control logic may generate an output signal for each point in time / time window in which detection signals are collected, and that output signal may contain information indicating whether or not an ^^-qubit GHZ measurement was performed successfully at that point in time / within that time window.

[0087] The control logic 670 may be implemented in a hardware controller. In some examples, the controller may be a general or dedicated processor, such as a central processing unit (CPU) or a graphics processing unit (GPU). In other examples, the controller may be implemented in a dedicated, application-specific processing unit. For example, the controller may comprise an application-specific integrated circuit (ASIC) or an application- specific standard product (ASSP) or another domain-specific architecture (DSA). Alternatively, the controller may be implemented in adaptive computing hardware (that is, hardware comprising configurable hardware blocks / configurable logic blocks) that has been configured to perform the required functions, for example in a configured field programmable gate array (FPGA).

[0088] The arrangement of the coupling locations and beamsplitters 640 is such that the first interferometer 680 is configured to interfere the ^^ dual-rail encoded photonic qubits such that (i) a beamsplitter interaction is performed on the second mode of the ^^^^ℎqubit and the first mode of the ( ^^ + 1)^^ℎqubit for all ^^ between 1 and ^^− 1, wherein ^^ is an integer, and (ii) a beamsplitter interaction is performed on the first mode of the first qubit and the second mode of the ^^^^ℎqubit. The arrangement of the coupling locations and beamsplitters 640 is such that the second interferometer 690 is configured to interfere an interfered qubit output from the first interferometer and a pair of single photons.

[0089] The skilled person will appreciate that the dual-rail encoding can be adapted such that the apparatus 600 probabilistically projects a 4-qubit input state onto a 4-GHZ state that is not a canonical 4-GHZ state. For example, the by flipping the encoding for one or more of the qubits, the apparatus 600 may probabilistically project a 4- qubit input state onto a 4-qubit GHZ state other than a canonical 4-qubit GHZ state.

[0090] The skilled person will appreciate that the apparatus 600 may be varied in many ways. For example, other implementations of the apparatus 600 may utilise other configurations of optical paths, crossing points and beamsplitter interactions without detracting from the function of the apparatus. While each qubit is defined by the presence / absence of photons in neighbouring input ports, in other examples a qubit may be defined by a presence / absence of photons in non-neighbouring input ports, which may enable for a reduction in crossing points. In some architectures, for example when the interferometer 610 is implemented in a planar photonic integrated circuit, there is a chance of photon loss at crossing points, and so advantageously by reducing the number of crossing points, the accuracy of the apparatus may be improved.

[0091] In other examples, the apparatus 600 may include further second interferometer regions 690 to couple further interfered qubits with pairs of auxiliary modes. For example, an apparatus may have 2( ^^ + ^^) input ports (where ^^ is the number of second interferometer regions), 2( ^^ + ^^) output ports, and a detector arrangement with 2( ^^ + ^^) photodetectors. A first interferometer region 680 may interfere the optical modes of the ^^ qubits and output ^^ interfered qubits. Each second interferometer region 690 of the ^^ second interferometer regions may be configured to: receive, as two primary input optical modes, an interfered qubit from the first interferometer; receive, in a respective pair of auxiliary optical modes, a corresponding pair of single photons; interfere the primary and auxiliary optical modes; and output four output optical modes.

[0092] In the discussions above, measurement apparatuses have been described that are capable of performing an ^^-qubit GHZ measurement. A useful application for performing a ^^-qubit GHZ measurement is in entangling a plurality of ^^ photonic multiqubit states.

[0093] Fig.7 shows a flowchart of a method 700 for entangling a plurality of ^^ photonic multiqubit states, wherein ^^ is an integer. The method comprises performing an ^^-qubit GHZ measurement on a set of photonic qubits, the set comprising one photonic qubit from each of the ^^ photonic multiqubit states. Each qubit of the set of qubits is dual rail encoded as probability amplitudes corresponding to the photon occupation of two orthogonal optical modes. For example, with reference to Fig.6, each of the qubits may be dual-rail encoded in the input ports to which the photonic qubit is provided.

[0094] At 710, the method comprises providing each qubit of the set of qubits to a first interferometer, the first interferometer configured to (i) perform a beamsplitter interaction on the first mode of the first qubit and the second mode of thequbit; and (ii) perform a beamsplitter interaction on the second mode of the ^^^^ℎqubit and the first mode of the ( ^^ + 1)^^ℎqubit for all ^^ between 1 and ^^− 1, wherein ^^ is an integer. For example, with reference to the apparatus 600 of Fig.6, beamsplitter interactions are performed between the first mode of the first qubit and the second mode of the fourth qubit, between the second mode of the first qubit and the first mode of the second qubit, between the second mode of the second qubit and first mode of the third qubit, and between the second mode of the third qubit and the first mode of the fourth qubit .

[0095] At 720, the method comprises providing an interfered qubit and a pair of auxiliary single photons to a second interferometer, the second interferometer configured to: perform a first beamsplitter interaction on two auxiliary input optical modes comprising the two auxiliary single photons; perform a second beamsplitter interaction on a first optical mode of the interfered qubit and the first auxiliary input optical mode; and perform a third beamsplitter interaction on a second optical mode of the interfered qubit and the second auxiliary input optical mode. For example, with reference to Fig.6 second interferometer region 690 is configured to receive an interfered qubit from the first interferometer, to receive a pair of auxiliary single photons from input ports 620- 5A and 620-5B, to interfere the interfered qubit and the auxiliary single photons, and to output four modes via output ports 630-4A to 630-5B.

[0096] At 730, the method comprises measuring a number of photons in each output mode from the first interferometer or second interferometer to determine whether a ^^-qubit GHZ measurement has been performed. For example, with reference to the apparatus 600, the photodetectors 660 are arranged to measure the number of photons received from each output port 630-1A to 630-5B. From the measurement outcome, the control logic 670 is able to confirm that a projection of the output state of the interferometer 610 on to a 4-GHZ state has been performed. By projecting the output modes onto an ^^-qubit GHZ state, the remaining qubits of the photonic multiqubit photonic states are entangled.

[0097] An example of a use case for linear-optics based measurement apparatuses and methods such as those described herein is in the generation of large, heralded entangled states, which find use in, for example, measurement-based quantum computing.

[0098] In a typical approach to generating large, heralded entangled states, a large number of low-level few- qubit photonic entangled resource states (for example, photonic 3-qubit GHZ states) are generated. Bell state measurements, in which an input state is probabilistically projected onto a two-qubit Bell state, are then used to entangle pairs of those few-qubit resource states to generate slightly larger entangled states – for example, a Bellstate measurement can be used to probabilistically entangle two 3-qubit GHZ states to create a 4-qubit GHZ state. A figure of merit often used to evaluate linear optical Bell state measurements is the success probability – the probability that a maximally mixed input state is projected onto a known Bell state. For linear optical schemes that do not take advantage of auxiliary qubits, the optimal success probability of an individual Bell state measurement is known to be 1 / 2 (one half). Successive Bell state measurements may then be used to build larger entangled states. For example, an ^^-qubit GHZ state is typically created by combining two smaller GHZ states using a Bell state measurement. With Bell state measurements, the generation of an ^^-qubit GHZ state can accordingly be broken down into 3-qubit GHZ states and a binary tree of Bell state measurements, the binary tree having a depth of [log2( ^^ − 2)]. This means that the probability of successfully generating a ^^-qubit GHZ state [log ^ sing Bell state measurements is given by 2(^−2 ] u−22). Moreover, using only Bell state measurements, the minimal resource states that are required must contain at least three entangled qubits, for example, a 3 -qubit GHZ state, which may also only be probabilistically generated. The use of Bell state measurements to entangle resource states means that the initial low-level resource states cannot themselves be Bell states as the low-level resource states must comprise at least three qubits.

[0099] In contrast, the apparatuses and methods described herein can be used to generate larger resource states from more basic two-qubit entangled states such as Bell states. Bell states are a viable and easier-to-generate candidate for the low-level resource states than three-qubit GHZ states. For example, consider a number ^^ of Bell states. Labelling each pair with the superscript ( ^^), where ^^ is an integer between 1 and ^^, and labelling the qubits in each pair with A and B, the initial state of the system of ^^ Bell pairs can be written as: (EQ.13)A subset of the qubits, in this example comprising one qubit of each Bell state (for example those labelled A), are then dual rail encoded and provided to an assisted N-GHZ state analyser apparatus such as apparatus 600 of Fig. 6. Upon successfully performing an ^^-qubit GHZ measurement on those qubits, the remaining qubits (labelled B) are placed into (unnormalized) state:As indicated in (EQ.14) the remaining qubits of the multiqubit photonic states are projected onto an entangled state which in this example is one of the two canonical ^^-qubit GHZ states that the apparatus 600 is capable of measuring.

[0100] The skilled person will appreciate that the method 700 of FIG.8 may be varied or extended in a number of ways. For example, the method may further comprise performing one or more corrective operations on one or more of the qubits of the resulting entangled state. For example, if control logic 670 indicates that a projection onto the canonical NGHZ- state has been performed, and so the resulting entangled state is in the NGHZ- state then a Pauli-Z operation may be performed on any qubit of that resulting state may be performed, for exampleusing an optical phase shifter on one mode of a dual rail encoding of that qubit, to transform that resulting state to a NGHZ+ state.

[0101] In some examples, the method 700 may further comprise providing a further interfered qubit and a further pair of auxiliary single photons to a corresponding further second interferometer; and measuring a number of photons in each output optical mode from the further second interferometer.

[0102] As indicated above, the methods and apparatuses described herein accordingly enable the generation of large, entangled states such as ^^-qubit GHZ states using more basic resource states than 3-GHZ states, for example 2-qubit Bell states.

[0103] Variations of the described embodiments are envisaged.

[0104] All of the features disclosed in this specification (including any accompanying claims, abstract and drawings, and / or all of the steps of any method or process so disclosed, may be combined in any combination, except combinations where at least some of such features and / or steps are mutually exclusive.

[0105] Each feature disclosed in this specification (including any accompanying claims, abstract and drawings), may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise. Thus, unless expressly stated otherwise, each feature disclosed is one example only of a generic series of equivalent or similar features.

[0106] The disclosure is not restricted to the details of any foregoing embodiments. The disclosure extends to any novel one, or any novel combination, of the features disclosed in this specification (including any accompanying claims, abstract and drawings), or to any novel one, or any novel combination, of the steps of any method or process so disclosed. The claims should not be construed to cover merely the foregoing embodiments, but also any embodiments which fall within the scope of the claims.

Claims

Claims 1. An apparatus comprising: an optical circuit comprising: a first interferometer; one or more second interferometers; and a detector arrangement; wherein the first interferometer is arranged to: receive, as 2 ^^ input optical modes, a plurality of ^^ dual-rail encoded photonic qubits, each qubit encoded as probability amplitudes corresponding to the photon occupation of two orthogonal optical modes, wherein ^^ is an integer greater than two; interfere the ^^ qubits such that: a beamsplitter interaction is performed on the first mode of the first qubit and the second mode of the ^^^^ℎqubit; a beamsplitter interaction is performed on the second mode of the ^^^^ℎqubit and the first mode of the ( ^^ + 1)^^ℎqubit for all ^^ between 1 and ^^−1; and output a plurality of ^^ interfered qubits as 2 ^^ output optical modes towards the detector arrangement or the one or more second interferometers; wherein each second interferometer is arranged to: receive, as two primary input optical modes, an interfered qubit from the first interferometer; receive, in a first auxiliary input optical mode and a second auxiliary input optical mode, two auxiliary single photons; interfere the primary and auxiliary input optical modes such that: a first beamsplitter interaction is performed on the two auxiliary input optical modes; a second beamsplitter interaction is performed on a first primary input optical mode of the received interfered qubit and the first auxiliary input optical mode; and a third beamsplitter interaction is performed on a second primary input optical mode of the received interfered qubit and the second auxiliary input optical mode; and output four output optical modes towards the detector arrangement; and wherein the detector arrangement comprises one or more photodetectors, the detector arrangement configured to measure a number of photons in each of the output optical modes received from the first interferometer or the one or more second interferometers.

2. An apparatus according to claim 1, further comprising control circuitry for indicating, based on the measured number of photons in each of the output optical modes, that a ^^-qubit GHZ state measurement has been performed on the ^^ qubits.

3. An apparatus according to claim 1 or claim 2, wherein the number of second interferometers is one.

4. An apparatus according to claim 1 or claim 2, wherein the number of second interferometers is equal to the number ^^ of qubits received by the first interferometer.

5. An apparatus according to any preceding claim, wherein the input optical modes and output optical modes are spatial modes.

6. An apparatus according to claim 5, wherein the first interferometer is a spatial interferometer having 2 ^^ input ports for inputting the ^^ qubits to the interferometer and 2 ^^ output ports for outputting the ^^ interfered qubits.

7. An apparatus according to claim 5 or claim 6, wherein the first interferometer comprises a plurality of waveguides arranged to pass through the first interferometer to couple the 2 ^^ input ports to the 2 ^^ output ports; wherein the plurality of waveguides is arranged to provide coupling locations between pairs of the waveguides, wherein a waveguide coupler is arranged at each of at least a subset of the coupling locations such that at each of those coupling locations the two optical modes carried by the two respective waveguides are capable of coupling with each other in a beamsplitter interaction.

8. An apparatus according to any of claims 5 to 8, wherein the second interferometer is a spatial interferometer having four input ports for inputting the interfered qubit and the two auxiliary single photons to the second interferometer and four output ports for outputting the output optical modes towards the detector arrangement.

9. An apparatus according to claim 8, wherein the second interferometer comprises a plurality of waveguides arranged to pass through the second interferometer to couple the four input ports with the four output ports; wherein the plurality of waveguides are arranged to provide coupling locations between pairs of the waveguides, wherein a waveguide coupler is arranged at each of at least a subset of the coupling locations such that at each of those coupling locations the two optical modes carried by the two respective waveguides are capable of coupling with each other in a beamsplitter interaction.

10. An apparatus according to any preceding claim, wherein at least one of the first interferometer or one or more of the second interferometers are provided on a photonic integrated circuit.

11. An apparatus according to any of claims 1 to 4, wherein the input optical modes and output optical modes are polarization modes.

12. An apparatus according to any of claims 1 to 4, wherein the input optical modes and output optical modes are temporal modes.

13. An apparatus according to any of claims 1 to 4, wherein the input optical modes and output optical modes are spectral modes.

14. An apparatus according to any preceding claim, wherein the apparatus further comprises one or more photon sources configured to provide auxiliary single photons to the number of second interferometers.

15. An apparatus according to any preceding claim, wherein the first interferometer and the one or more second interferometers are provided as regions of a third interferometer.

16. An apparatus according to any preceding claim, wherein each beamsplitter interaction is a 50 / 50 beamsplitter interaction.

17. An apparatus according to any preceding claim, wherein ^^ is an integer greater than three.

18. A method for entangling a plurality of ^^ photonic multiqubit states, wherein ^^ is an integer greater than two, the method comprising performing a ^^-qubit GHZ state measurement on a set of photonic qubits, the set comprising one photonic qubit from each of the ^^ photonic multiqubit states, wherein each qubit is encoded as probability amplitudes corresponding to the photon occupation of two orthogonal optical modes, wherein performing the ^^-qubit state measurement comprises: providing each qubit of the set of qubits to a first interferometer, the first interferometer configured to: perform a beamsplitter interaction on the first mode of the first qubit and the second mode of the ^^^^ℎqubit; and perform a beamsplitter interaction on the second mode of the ^^^^ℎqubit and the first mode of the ( ^^ + 1)^^ℎqubit for all ^^ between 1 and ^^− 1; and output a plurality of ^^ interfered qubits as 2 ^^ output optical modes; providing an interfered qubit and a pair of auxiliary single photons to a corresponding second interferometer, the second interferometer configured to: perform a first beamsplitter interaction on two auxiliary input optical modes comprising the two auxiliary single photons; perform a second beamsplitter interaction on a first optical mode of the interfered qubit and the first auxiliary input optical mode; and perform a third beamsplitter interaction on a second optical mode of the interfered qubit and the second auxiliary input optical mode; and measuring a number of photons in each output optical mode output from the first interferometer or second interferometer.

19. The method of claim 18, further comprising: providing a further interfered qubit and a further pair of auxiliary single photons to a corresponding further second interferometer; andmeasuring a number of photons in each output optical mode output from the further second interferometer.

20. The method of claim 18 or 19, further comprising, based on the measured photon occupation of the output modes, performing corrective operations on at least one unmeasured photonic qubit of the entangled state.

21. The method of any of claims 18 to 20, wherein at least one photonic multiqubit state comprises a Bell state.

22. The method of any of claims 18 to 21, wherein at least one photonic multiqubit state comprises a 3-qubit GHZ state.

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