Multi-way fusion circuits for photonic qubits using hadamard interferometers

Abstract

Multi-way fusion circuits using Hadamard interferometers can perform fusion operations on four or more input qubits, each of which may initially be part of a separately-entangled system of qubits. When the fusion operation succeeds, the four or more input qubits are consumed in a projective entangling measurement that also results in creating a single entangled system that includes the remaining qubits of the initial separately-entangled systems.

Description

PATENT Atorney Docket No. 104456-007110PC-1523316Client Ref. No. PsiQ-642MULTI-WAY FUSION CIRCUITS FOR PHOTONIC QUBITS USING HADAMARD INTERFEROMETERSCROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No.63 / 739,050, filed December 26, 2024, the disclosure of which is incorporated by reference herein.BACKGROUND

[0002] At the most general level, a qubit is a quantum system that can exist in one of two orthogonal states (denoted as |0) and |1) in the conventional bra / ket notation) or in a superposition of the two states (e.g., (|0) + |1)). Multiple qubits can be placed into entangled states, and physical systems of entangled qubits have a variety of applications in quantum computing, quantum communication, and other fields. Thus, techniques for creating entanglement between qubits are desirable.SUMMARY

[0003] Entangled systems of qubits can be created in stages. For instance, small entangled systems of multiple qubits (also referred to as “seed states”), such as Bell states or / / -GHZ states, can be generated, after which seed states can be entangled with each other using “fusion” operations that perform projective entangling measurements on a qubit from each seed state. The fusion operation consumes some or all of the input qubits, while creating entanglement between the remaining qubits of the input seed state.

[0004] Certain embodiments disclosed herein relate to circuits that can perform fusion operations on four or more input qubits, each of which may initially be part of a separately-entangled system of qubits. When the fusion operation succeeds, the four or more input qubits are consumed in a projective entangling measurement that also results in creating a single entangled system that includes the remaining qubits of the initial separately-entangledsystems. In some embodiments, the circuits operate on five or more input qubits and include Hadamard interferometers.

[0005] Some embodiments relate to circuits that operate on an even number of qubits greater than or equal to 4. Such circuits can comprise: a plurality of waveguide pairs to receive a plurality of qubits, wherein each of the qubits is a dual-rail-encoded photonic qubit propagating on one of the waveguide pairs such that the waveguides of the waveguide pair correspond to the rails of the qubits and wherein the total number of qubits in the plurality of qubits is a number (N) that is an even number greater than or equal to 4; a plurality of fourway Hadamard interferometer circuits, each four-way Hadamard interferometer circuit having four input waveguides and four output waveguides, wherein each of the four-way Hadamard interferometer circuits includes a network of beam splitters arranged such that a photon entering on any one of the four input waveguides has an equal probability of exiting on any one of the four output waveguides; the waveguides of the waveguide pairs being coupled to the inputs of the four-way Hadamard interferometer circuits such that each of the four-way Hadamard interferometer circuits receives one rail from each of four different dualrail encoded photonic qubits and the two rails of a given dual-rail encoded photonic qubit are coupled to inputs of two different ones of the four-way Hadamard interferometer circuits; a set of 2N photon detectors, each photon detector coupled to a different one of the output waveguides of the four-way Hadamard interferometer circuits and configured to produce a classical output signal indicating a count of detected photons; and a classical decision logic circuit configured to receive the classical output signals from the set of 2N photon detectors and to determine, based on the received classical output signals, whether an N-way fusion operation on the plurality of qubits succeeded.

[0006] In these and other embodiments, the 2N photon detectors can be configured to operate concurrently to produce the classical output signals for a particular set of received qubits.

[0007] In these and other embodiments, the classical decision logic circuit can be further configured to determine whether the N-way fusion operation succeeded by extracting a detection pattern from the received classical output signals and comparing the detection pattern to a set of success patterns, wherein the N-way fusion operation succeeded if the detection pattern matches one of the success patterns in the set of success patterns.

[0008] For example, the classical decision logic can be further configured such that the detection pattern includes, for each of the four-way Hadamard interferometer circuits, a 4-tuple of counts of detected photons. In some embodiments, the set of success patterns includes only patterns where each 4-tuple corresponds to a total of two detected photons. In some embodiments, the set of success patterns includes only patterns in which, for at least one of the 4-tuples, the two detected photons are detected by two different detectors. For instance, in some embodiments where the number N is equal to 6 and the number of 4-tuples is equal to three, the set of success patterns can consist of: a first subset of success patterns in which, for all three of the 4-tuples, the two detected photons are detected by two different detectors; a second subset of success patterns in which, for one of the three 4-tuples, the two detected photons are detected by the same detector and for the other two 4-tuples, the two detected photons are detected by different detectors such that the other two 4-tuples have either an identical pattern of photon counts or a complementary pattern of photon counts; and a third subset of success patterns in which, for two of the three 4-tuples, the two detected photons are detected by the same detector and for the other 4-tuple, the two detected photons are detected by different detectors.

[0009] In these and other embodiments, each of the four-way Hadamard interferometer circuits can comprise: a first beam splitter coupled between a first waveguide and a second waveguide of the four input waveguides; a second beam splitter coupled between a third waveguide and a fourth waveguide of the four input waveguides; a third beam splitter coupled between a first output waveguide of the first beam splitter and a first output waveguide of the second beam splitter; and a fourth beam splitter coupled between a second output waveguide of the first beam splitter and a second output waveguide of the second beam splitter. Each of the first, second, third, and fourth beam splitters can be, for example, a 50 / 50 beam splitter.

[0010] In these and other embodiments, the N-way fusion operation can correspond to projection onto an N-GHZ state.

[0011] In these and other embodiments, each qubit of the plurality of qubits can initially be part of a respective quantum system of entangled qubits, The N-way fusion operation can consume the plurality of qubits and produce an entangled quantum system from the remaining qubits of the respective quantum systems.

[0012] Some embodiments relate to circuits that operate on an odd number of qubits greater than 4. Such circuits can comprise: a plurality of waveguide pairs to receive a plurality of qubits, wherein each of the qubits is a dual-rail-encoded photonic qubit propagating on one of the waveguide pairs and wherein the total number of qubits in the plurality of qubits is a number (N) that is an odd number greater than 4; a plurality of fourway Hadamard interferometer circuits, each four-way Hadamard interferometer circuit having four input waveguides and four output waveguides, wherein each of the four-way Hadamard interferometer circuits includes a network of beam splitters arranged such that a photon entering on any one of the four input waveguides has an equal probability of exiting on any one of the four output waveguides; a two-way Hadamard interferometer circuit having two input waveguides and two output waveguides, wherein the two-way Hadamard interferometer circuit includes a beam splitter arranged such that a photon entering on either one of the two input waveguides has an equal probability of exiting on any one of the two output waveguides, the waveguides of the waveguide pairs being coupled to the inputs of the four-way Hadamard interferometer circuits and the two-way Hadamard interferometer circuit such that each of the four-way Hadamard interferometer circuits receives one rail from each of four different dual-rail encoded photonic qubits, the two-way Hadamard interferometer circuit receives one rail from each of two different dual-rail encoded photonic qubits, and the two rails of a given dual-rail encoded photonic qubit are coupled to inputs of two different ones of the four- way Hadamard interferometer circuits or the two-way Hadamard interferometer circuit; a set of 2N photon detectors, each photon detector coupled to a different one of the output waveguides of the four-way Hadamard interferometer circuits or the two-way Hadamard interferometer circuit, each of the photon detectors being configured to produce a classical output signal indicating a count of detected photons; and a classical decision logic circuit configured to receive the classical output signals from the set of 2N photon detectors and to determine, based on the received classical output signals, whether an N-way fusion operation on the plurality of qubits succeeded.

[0013] In these and other embodiments, the 2N photon detectors can be configured to operate concurrently to produce the classical output signals for a particular set of received qubits.

[0014] In these and other embodiments, the classical decision logic circuit can be further configured to determine whether the N-way fusion operation succeeded by extracting a detection pattern from the received classical output signals and comparing the detectionpattern to a set of success patterns, wherein the N-way fusion operation succeeded if the detection pattern matches one of the success patterns in the set of success patterns.

[0015] In these and other embodiments, the classical decision logic can be further configured such that the detection pattern includes, for each of the four-way Hadamard interferometer circuits, a 4-tuple of counts of detected photons and, for the two-way Hadamard interferometer circuit, a 2-tuple of counts of detected photons. In some embodiments, the set of success patterns includes only patterns where each 4-tuple corresponds to a total of two detected photons and the 2-tuple corresponds to a total of one detected photon. In some embodiments, the set of success patterns includes only patterns in which for at least one of the 4-tuples, the two detected photons are detected by two different detectors. For instance, in some embodiments where the number N is equal to 7 and the number of 4-tuples is equal to three, the set of success patterns can consist of: a first subset of success patterns in which, for all three of the 4-tuples, the two detected photons are detected by two different detectors; a second subset of success patterns in which, for one of the three 4-tuples, the two detected photons are detected by the same detector and for the other two 4-tuples, the two detected photons are detected by different detectors such that the other two 4-tuples have either an identical pattern of photon counts or a complementary pattern of photon counts; and a third subset of success patterns in which, for two of the three 4-tuples, the two detected photons are detected by the same detector and for the other 4-tuple, the two detected photons are detected by different detectors.

[0016] In these and other embodiments, each of the four-way Hadamard interferometer circuits can comprise: a first beam splitter coupled between a first waveguide and a second waveguide of the four input waveguides; a second beam splitter coupled between a third waveguide and a fourth waveguide of the four input waveguides; a third beam splitter coupled between a first output waveguide of the first beam splitter and a first output waveguide of the second beam splitter; and a fourth beam splitter coupled between a second output waveguide of the first beam splitter and a second output waveguide of the second beam splitter. Each of the first, second, third, and fourth beam splitters can be, for example, a 50 / 50 beam splitter.

[0017] In these and other embodiments, the N-way fusion operation can correspond to projection onto an N-GHZ state.

[0018] In these and other embodiments, each qubit of the plurality of qubits can initially be part of a respective quantum system of entangled qubits. The N-way fusion operation can consume the plurality of qubits and produce an entangled quantum system from the remaining qubits of the respective quantum systems.

[0019] The following detailed description, together with the accompanying drawings, will provide a better understanding of the nature and advantages of the claimed invention.BRIEF DESCRIPTION OF THE DRAWINGS

[0020] FIG. 1 shows two representations of a portion of a pair of waveguides corresponding to a dual-rail-encoded photonic qubit.

[0021] FIG. 2A shows a schematic diagram for coupling of two modes.

[0022] FIG. 2B shows, in schematic form, a physical implementation of mode coupling in a photonic system that can be used in some embodiments.

[0023] FIGs. 3 A and 3B show, in schematic form, examples of physical implementations of a Mach-Zehnder Interferometer (MZI) configuration that can be used in some embodiments.

[0024] FIG. 4A shows another schematic diagram for coupling of two modes.

[0025] FIG. 4B shows, in schematic form, a physical implementation of the mode coupling of FIG. 4A in a photonic system that can be used in some embodiments.

[0026] FIG. 5 shows a four-mode coupling scheme that implements a “spreader,” or “mode-information erasure,” transformation on four modes in accordance with some embodiments.

[0027] FIG. 6 illustrates an example optical device that can implement the four-mode mode-spreading transform shown schematically in FIG. 5 in accordance with some embodiments.

[0028] FIG. 7 shows a circuit diagram for a dual-rail-encoded Bell state generator that can be used in some embodiments.

[0029] FIG. 8A shows a circuit diagram for a dual-rail-encoded type I fusion gate that can be used in some embodiments.

[0030] FIG. 8B shows example results of type I fusion operations using the gate of FIG. 8 A.

[0031] FIG. 9A shows a circuit diagram for a dual-rail-encoded type II fusion gate that can be used in some embodiments.

[0032] FIG. 9B shows an example result of a type II fusion operation using the gate of FIG.9A.

[0033] FIG. 10 illustrates an example of a qubit entangling system in accordance with some embodiments.

[0034] FIG. 11 A shows two representations of a portion of a single waveguide corresponding to a temporally encoded qubit.

[0035] FIG. 1 IB shows an example of an optical circuit that can convert a spatially-encoded qubit to a temporally-encoded qubit.

[0036] FIG. 11C shows an example of an optical circuit that can convert a temporally-encoded qubit to a spatially-encoded qubit.

[0037] FIG. 12 shows a simplified conceptual diagram illustrating operation of a multi-way fusion circuit according to some embodiments.

[0038] FIGs. 13 A and 13B show simplified schematic diagrams of two variations of a type I fusion circuit that can be used in some embodiments.

[0039] FIG. 14 shows a simplified schematic diagram of a type II fusion circuit that can be used in some embodiments.

[0040] FIG. 15A shows a simplified schematic diagram of a three-way fusion circuit according to some embodiments.

[0041] FIG. 15B shows a table listing the success patterns for the circuit of FIG. 15A according to some embodiments.

[0042] FIG. 16 shows a simplified schematic diagram of a balanced four-way fusion circuit according to some embodiments.

[0043] FIG. 17 shows a simplified schematic diagram of an imbalanced four- way fusion circuit according to some embodiments.

[0044] FIG. 18A shows diagrams illustrating the outcomes of type I fusions using the circuits of FIGs. 13 A and 13B for inputs that include two-photon multiphoton states.

[0045] FIG. 18B shows a diagram illustrating the effect of cascading (or concatenating) different variations of a type I fusion circuit according to some embodiments.

[0046] FIG. 19 shows a simplified schematic diagram of an imbalanced four- way fusion circuit according to some embodiments.

[0047] FIGs. 20A and 20B show examples of balanced 5-way fusion circuits according to some embodiments.

[0048] FIGs. 21 A and 21B show examples of balanced 6-way fusion circuits according to some embodiments.

[0049] FIG. 22 shows an example of a balanced 7-way fusion circuit according to some embodiments.

[0050] FIG. 23 shows a simplified schematic diagram of an imbalanced TV-way fusion circuit according to some embodiments.

[0051] FIG. 24 shows a simplified schematic diagram of a Hadamard interferometer circuit that can be used in some embodiments.

[0052] FIG. 25 shows a simplified schematic diagram of a Hadamard-based four-way fusion circuit for dual-rail-encoded qubits according to some embodiments.

[0053] FIG. 26 shows a simplified schematic diagram of a Hadamard-based four-way fusion circuit for temporally encoded qubits according to some embodiments.

[0054] FIG. 27 shows a simplified schematic diagram of a Hadamard-based four-way fusion circuit according to some embodiments.

[0055] FIG. 28A shows a simplified schematic diagram of a Hadamard-based five-way fusion circuit according to some embodiments, and FIG. 28B shows additional details related to the decision logic for the circuit.

[0056] FIG. 29 shows a simplified schematic diagram of a Hadamard-based six-way fusion circuit according to some embodiments.

[0057] FIG. 30 shows a simplified schematic diagram of a Hadamard-based seven-way fusion circuit according to some embodiments.

[0058] FIG. 31 shows a simplified schematic diagram of a Hadamard-based eight-way fusion circuit according to some embodiments.DETAILED DESCRIPTION

[0059] Disclosed herein are examples (also referred to as “embodiments”) of systems and methods for producing entangled states in physical quantum systems, including photonic systems. Such embodiments can be used, for example, in quantum computing as well as in other contexts (e.g., quantum communication) that exploit quantum entanglement. To facilitate understanding of the disclosure, an overview of relevant concepts and terminology is provided in Section 1. With this context established, Sections 2-4 describe examples of multi-way fusion circuits according to various embodiments. Although embodiments are described with specific detail to facilitate understanding, those skilled in the art with access to this disclosure will appreciate that the claimed invention can be practiced without these details.

[0060] Further, embodiments are described herein as creating and operating on systems of qubits, where the quantum state space of a qubit can be modeled as a 2-dimensional vector space. Those skilled in the art with access to this disclosure will understand that techniques described herein can be applied to systems of “qudits,” where a qudit can be any quantum system having a quantum state space that can be modeled as a (complex) / / -dimensional vector space (for any integer / / ), which can be used to encode n bits of information. For the sake of clarity of description, the term “qubit” is used herein, although in some embodiments the system can also employ quantum information carriers that encode information in a manner that is not necessarily associated with a binary bit, such as a qudit.1. Overview of Quantum Computing

[0061] Quantum computing relies on the dynamics of quantum objects, e.g., photons, electrons, atoms, ions, molecules, nanostructures, and the like, which follow the rules of quantum theory. In quantum theory, the quantum state of a quantum object is described by a set of physical properties, the complete set of which is referred to as a mode. In some embodiments, a mode is defined by specifying the value (or distribution of values) of one or more properties of the quantum object. For example, in the case where the quantum object is a photon, modes can be defined by the frequency of the photon, the position in space of the photon (e.g., which waveguide or superposition of waveguides the photon is propagating within), the associated direction of propagation (e.g., the ^-vector for a photon in free space),the polarization state of the photon (e.g., the direction (horizontal or vertical) of the photon’s electric and / or magnetic fields), a time window in which the photon is propagating, the orbital angular momentum state of the photon, and the like.

[0062] For the case of photons propagating in a waveguide, it is convenient to express the state of the photon as one of a set of discrete spatio-temporal modes. For example, the spatial mode kt of the photon is determined according to which one of a finite set of discrete waveguides the photon is propagating in, and the temporal mode tj is determined by which one of a set of discrete time periods (referred to herein as “bins”) the photon is present in. In some photonic implementations, the degree of temporal discretization can be provided by a pulsed laser which is responsible for generating the photons. In examples below, spatial modes will be used primarily to avoid complication of the description. However, one of ordinary skill will appreciate that the systems and methods can apply to any type of mode, e.g., temporal modes, polarization modes, and any other mode or set of modes that serves to specify the quantum state. Further, in the description that follows, embodiments will be described that employ photonic waveguides to define the spatial modes of the photon.However, persons of ordinary skill in the art with access to this disclosure will appreciate that other types of mode, e.g., temporal modes, energy states, and the like, can be used without departing from the scope of the present disclosure. In addition, persons of ordinary skill in the art will be able to implement examples using other types of quantum systems, including but not limited to other types of photonic systems.

[0063] For quantum systems of multiple indistinguishable particles, rather than describing the quantum state of each particle in the system, it is useful to describe the quantum state of the entire many-body system using the formalism of Fock states (sometimes referred to as the occupation number representation). In the Fock state description, the many-body quantum state is specified by how many particles there are in each mode of the system. For example, a multi-mode, two particle Fock state 11001 ! 2,3,4 specifies a two-particle quantum state with one particle in mode 1, zero particles in mode 2, zero particles in mode 3, and one particle in mode 4. Again, as introduced above, a mode can be any property of the quantum object. For the case of a photon, any two modes of the electromagnetic field can be used, e.g., one may design the system to use modes that are related to a degree of freedom that can be manipulated passively with linear optics. For example, polarization, spatial degree of freedom, or angular momentum could be used. The four-mode system represented by the two-particle Fock state 11001 ! 2,3,4 can be physically implemented as four distinctwaveguides with two of the four waveguides having one photon travelling within them. Other examples of a state of such a many-body quantum system include the four-particle Fock state 11111)! 2,3,4 that represents each mode occupied by one particle and the four-particle Fock state 12200 !2, 3,4 that represents modes 1 and 2 respectively occupied by two particles and modes 3 and 4 occupied by zero particles. For modes having zero particles present, the term “vacuum mode” is used. For example, for the four-particle Fock state 12200)-! 2,3,4 modes 3 and 4 are referred to herein as “vacuum modes.” Fock states having a single occupied mode can be represented in shorthand using a subscript to identify the occupied mode. For example, 10010^ 2,3,4 is equivalent to |13).1.1. Qubits

[0064] As used herein, a “qubit” (or quantum bit) is a quantum system with an associated quantum state that can be used to encode information. A quantum state can be used to encode one bit of information if the quantum state space can be modeled as a (complex) two-dimensional vector space, with one dimension in the vector space being mapped to logical value 0 and the other to logical value 1. In contrast to classical bits, a qubit can have a state that is a superposition of logical values 0 and 1. More generally, a “qudit” can be any quantum system having a quantum state space that can be modeled as a (complex) n-dimensional vector space (for any integer / ?), which can be used to encode n bits of information. For the sake of clarity of description, the term “qubit” is used herein, although in some embodiments the system can also employ quantum information carriers that encode information in a manner that is not necessarily associated with a binary bit, such as a qudit. Qubits (or qudits) can be implemented in a variety of quantum systems. Examples of qubits include: polarization states of photons; presence of photons in waveguides; or energy states of molecules, atoms, ions, nuclei, or photons. Other examples include other engineered quantum systems such as flux qubits, phase qubits, or charge qubits (e.g., formed from a superconducting Josephson junction); topological qubits (e.g., Majorana fermions); or spin qubits formed from vacancy centers (e.g., nitrogen vacancies in diamond).

[0065] A qubit can be “dual-rail encoded” such that the logical value of the qubit is encoded by occupation of one of two spatial modes (e.g., waveguides) of the quantum system. For example, the logical 0 and 1 values can be encoded as follows:10)1 = 110)!,2(1)|1>L = |01>l,2(2) where the subscript “L” indicates that the ket represents a logical state (e.g., a qubit value) and, as before, the notationonthe right-hand side of the equations above indicates that there are i particles in a first mode and j particles in a second mode, respectively (e.g., where i and j are integers). In this notation, a two-qubit system having a logical state |0)| 1)L(representing a state of two qubits, the first qubit being in a ‘0’ logical state and the second qubit being in a ‘ 1’ logical state) may be represented using occupancy across four spatial modes by 11001)! 2,3,4 (e.g., in a photonic system, one photon in a first waveguide, zero photons in a second waveguide, zero photons in a third waveguide, and one photon in a fourth waveguide). In some instances throughout this disclosure, the various subscripts are omitted to avoid unnecessary mathematical clutter.A qubit can also be “temporally encoded” such that the logical value of a qubit is encoded based on presence or absence of a photon at a particular location along a waveguide at a particular time. For temporal encoding of qubits, it can be useful to define time bins of fixed duration (e.g., measured using a clock circuit), and two consecutive time bins can be mapped to the logical states of a qubit.1.2. Entangled States

[0066] Many of the advantages of quantum computing relative to “classical” computing (e.g., conventional digital computers using binary logic) stem from the ability to create entangled states of multi-qubit systems. In mathematical terms, a state | / i) of n quantum objects is a separable state if | / >) => and an entangled state is a state that is not separable. One example is a Bell state, which, loosely speaking, is a type of maximally entangled state for a two-qubit system, and qubits in a Bell state may be referred to as a Bell pair. For example, for qubits encoded by single photons in pairs of modes (a dual -rail encoding), examples of Bell states include:>> >> >>>> >> >> >>>> >> >>>> >> >> >>

[0067] More generally, an w-qubit Greenberger-Horne-Zeilinger (GHZ) state (or “ / / -GHZ state”) is an entangled quantum state of n qubits. For a given orthonormal logical basis, an n-GHZ state is a quantum superposition of all qubits being in a first basis state superposed with all qubits being in a second basis state:where the kets above refer to the logical basis. For example, for qubits encoded by single photons in pairs of modes (a dual-rail encoding), a 3-GHZ state can be written:>>> >>>where the kets above refer to photon occupation number in six respective modes (with mode subscripts omitted).1.3. Physical implementations

[0068] Qubits (and operations on qubits) can be implemented using a variety of physical systems. In some examples described herein, qubits are provided in an integrated photonic system employing waveguides, beam splitters, photonic switches, and single photon detectors, and the modes that can be occupied by photons are spatiotemporal modes that correspond to presence of a photon in a waveguide. Modes can be coupled using mode couplers, e.g., optical beam splitters, to implement transformation operations, and measurement operations can be implemented by coupling single-photon detectors to specific waveguides. One of ordinary skill in the art with access to this disclosure will appreciate that modes defined by any appropriate set of degrees of freedom, e.g., polarization modes, temporal modes, and the like, can be used without departing from the scope of the present disclosure. For instance, for modes that only differ in polarization (e.g., horizontal (H) and vertical (V)), a mode coupler can be any optical element that coherently rotates polarization, e.g., a birefringent material such as a waveplate. For other systems such as ion trap systems or neutral atom systems, a mode coupler can be any physical mechanism that can couple two modes, e.g., a pulsed electromagnetic field that is tuned to couple two internal states of the atom / ion.

[0069] In some embodiments of a photonic quantum computing system using dual-rail encoding, a qubit can be implemented using a pair of waveguides. FIG. 1 shows two representations (100, 100') of a portion of a pair of waveguides 102, 104 that can be used to provide a dual-rail-encoded photonic qubit. At 100, a photon 106 is in waveguide 102 and no photon is in waveguide 104 (also referred to as a vacuum mode); in some embodiments, this corresponds to the |0)Lstate of a photonic qubit. At 100', a photon 108 is in waveguide 104, and no photon is in waveguide 102; in some embodiments this corresponds to the 11)Lstate of the photonic qubit. To prepare a photonic qubit in a known logical state, a photon source (not shown) can be coupled to one end of one of the waveguides. The photon source can be operated to emit a single photon into the waveguide to which it is coupled, thereby preparing a photonic qubit in a known state. Photons travel through the waveguides, and by periodically operating the photon source, a quantum system having qubits whose logical states map to different temporal modes of the photonic system can be created in the same pair of waveguides. In addition, by providing multiple pairs of waveguides, a quantum system having qubits whose logical states correspond to different spatiotemporal modes can be created. It should be understood that the waveguides in such a system need not have any particular spatial relationship to each other. For instance, they can be but need not be arranged in parallel.

[0070] Occupied modes can be created by using a photon source to generate a photon that then propagates in the desired waveguide. A photon source can be, for instance, a resonatorbased source that emits photon pairs, also referred to as a heralded single photon source. In one example of such a source, the source is driven by a pump, e.g., a light pulse, that is coupled into a system of optical resonators that, through a nonlinear optical process (e.g., spontaneous four wave mixing (SFWM), spontaneous parametric down-conversion (SPDC), second harmonic generation, or the like), can generate a pair of photons. Many different types of photon sources can be employed. Examples of photon pair sources can include a microring-based spontaneous four wave mixing (SPFW) heralded photon source (HPS). However, the precise type of photon source used is not critical and any type of source, employing any process, such as SPFW, SPDC, or any other process can be used. Other classes of sources can also be employed, such as those that employ atomic and / or artificial atomic systems, e.g., quantum dot sources, color centers in crystals, and the like, and sources can incorporate nonlinear optical materials and / or other materials as desired. In some cases, sources may or may not be coupled to photonic cavities, e.g., as can be the case for artificialatomic systems such as quantum dots coupled to cavities. Other types of photon sources also exist for SPWM and SPDC, such as optomechanical systems and the like.

[0071] In such cases, operation of the photon source may be non-deterministic (also sometimes referred to as “stochastic”) such that a given pump pulse may or may not produce a photon pair. In some embodiments, coherent spatial and / or temporal multiplexing of several non-deterministic sources (referred to herein as “active” multiplexing) can be used to allow the probability of having one mode become occupied during a given cycle to approach 1. One of ordinary skill will appreciate that many different active multiplexing architectures that incorporate spatial and / or temporal multiplexing are possible. For instance, active multiplexing schemes that employ log-tree, generalized Mach-Zehnder interferometers, multimode interferometers, chained sources, chained sources with dump-the-pump schemes, asymmetric multi-crystal single photon sources, or any other type of active multiplexing architecture can be used. In some embodiments, the photon source can employ an active multiplexing scheme with quantum feedback control and the like.

[0072] Measurement operations can be implemented by coupling a waveguide to a singlephoton detector that generates a classical signal (e.g., a digital logic signal) indicating that a photon has been detected by the detector. Any type of photodetector that has sensitivity to single photons can be used. In some embodiments, detection of a photon (e.g., at the output end of a waveguide) indicates an occupied mode while absence of a detected photon can indicate an unoccupied mode.

[0073] Some embodiments described below relate to physical implementations of unitary transform operations that couple modes of a quantum system, which can be understood as transforming the quantum state of the system. For instance, if the initial state of the quantum system (prior to mode coupling) is one in which one mode is occupied with probability 1 and another mode is unoccupied with probability 1 (e.g., a state 110) in the Fock notation introduced above), mode coupling can result in a state in which both modes have a nonzero probability of being occupied, e.g., a state a 110) + a2|01 where In2+ | a212= 1. In some embodiments, operations of this kind can be implemented by using beam splitters to couple modes together and variable phase shifters to apply phase shifts to one or more modes. The amplitudes ai and ai depend on the reflectivity (or transmissivity) of the beam splitters and on any phase shifts that are introduced.

[0074] FIG. 2A shows a schematic diagram 210 (also referred to as a circuit diagram or circuit notation) for coupling of two modes. The modes are drawn as horizontal lines 212, 214, and the mode coupler 216 is indicated by a vertical line that is terminated with nodes (solid dots) to identify the modes being coupled. In the more specific language of linear quantum optics, the mode coupler 216 shown in FIG. 2A represents a 50 / 50 beam splitter that implements a transfer matrix:where T defines the linear map for the photon creation operators on two modes. (In certain contexts, transfer matrix T can be understood as implementing a first-order imaginary Hadamard transform.) By convention the first column of the transfer matrix corresponds to creation operators on the top mode (referred to herein as mode 1, labeled as horizontal line 212), and the second column corresponds to creation operators on the second mode (referred to herein as mode 2, labeled as horizontal line 214), and so on if the system includes more than two modes. More explicitly, the mapping can be written as:al\azt / inputwhere subscripts on the creation operators indicate the mode that is operated on, the subscripts input and output identify the form of the creation operators before and after the beam splitter, respectively and where:>For example, the application of the mode coupler shown in FIG. 2 A leads to the following mappings:at1i.nputazt.„ „ + at „input < outputzoutput / Thus, the action of the mode coupler described by Eq. (9) is to take the input states110), 101), and |11) to>>>111) « y(|20) + |02»

[0075] FIG. 2B shows a physical implementation of a mode coupling that implements the transfer matrix T of Eq. (9) for two photonic modes in accordance with some embodiments. In this example, the mode coupling is implemented using a waveguide beam splitter 200, also sometimes referred to as a directional coupler or mode coupler. Waveguide beam splitter 200 can be realized by bringing two waveguides 202, 204 into close enough proximity that the evanescent field of one waveguide can couple into the other. By adjusting the separation d between waveguides 202, 204 and / or the length I of the coupling region, different couplings between modes can be obtained. In this manner, a waveguide beam splitter 200 can be configured to have a desired transmissivity. For example, the beam splitter can be engineered to have a transmissivity equal to 0.5 (i.e., a 50 / 50 beam splitter for implementing the specific form of the transfer matrix ' / 'introduced above). If other transfer matrices are desired, the reflectivity (or the transmissivity) can be engineered to be greater than 0.6, greater than 0.7, greater than 0.8, or greater than 0.9 without departing from the scope of the present disclosure.

[0076] In addition to mode coupling, some unitary transforms may involve phase shifts applied to one or more modes. In some photonic implementations, variable phase-shifters can be implemented in integrated circuits, providing control over the relative phases of the state of a photon spread over multiple modes. Examples of transfer matrices that define such phase shifts are given by (for applying a +i and ~i phase shift to the second mode, respectively):For silica-on-silicon materials some embodiments implement variable phase-shifters using thermo-optical switches. The thermo-optical switches use resistive elements fabricated on the surface of the chip, that via the thermo-optical effect can provide a change of the refractive index n by raising the temperature of the waveguide by an amount of the order of 10'5K. One of skill in the art with access to the present disclosure will understand that any effect that changes the refractive index of a portion of the waveguide can be used to generate a variable, electrically tunable, phase shift. For example, some embodiments use beam splitters based on any material that supports an electro-optic effect, so-called x2and x3materials such as lithium niobite, BBO, KTP, and the like and even doped semiconductors such as silicon, germanium, and the like.

[0077] Beam-splitters with variable transmissivity and arbitrary phase relationships between output modes can also be achieved by combining directional couplers and variable phase-shifters in a Mach-Zehnder Interferometer (MZI) configuration 300, e.g., as shown in FIG. 3 A. Complete control over the relative phase and amplitude of the two modes 302a, 302b in dual rail encoding can be achieved by varying the phases imparted by phase shifters 306a, 306b, and 306c and the length and proximity of coupling regions 304a and 304b. FIG.3B shows a slightly simpler example of a MZI 310 that allows for a variable transmissivity between modes 302a, 302b by varying the phase imparted by the phase shifter 306. FIGs. 3A and 3B are examples of how one could implement a mode coupler in a physical device, but any type of mode coupler / beam splitter can be used without departing from the scope of the present disclosure.

[0078] In some embodiments, beam splitters and phase shifters can be employed in combination to implement a variety of transfer matrices. For example, FIG. 4A shows, in a schematic form similar to that of FIG. 2A, a mode coupler 400 implementing the following transfer matrix:Thus, mode coupler 400 applies the following mappings:>>The transfer matrix Trof Eq. (15) is related to the transfer matrix T of Eq. (9) by a phase shift on the second mode. This is schematically illustrated in FIG. 4A by the closed node 407 where mode coupler 416 couples to the first mode (line 212) and open node 408 where mode coupler 416 couples to the second mode (line 214). More specifically, Tr= sTs, and, as shown at the right-hand side of FIG. 4 A, mode coupler 416 can be implemented using mode coupler 216 (as described above), with a preceding and following phase shift (denoted by open squares 418a, 418b). Thus, the transfer matrix Trcan be implemented by the physical beam splitter shown in FIG. 4B, where the open triangles represent +i phase shifters.

[0079] Similarly, networks of mode couplers and phase shifters can be used to implement couplings among more than two modes. For example, FIG. 5 shows a four-mode coupling scheme that implements a “spreader,” or “mode-information erasure,” transformation on four modes, i.e., it takes a photon in any one of the input modes and delocalizes the photon amongst each of the four output modes such that the photon has equal probability of being detected in any one of the four output modes. (The well-known Hadamard transformation is one example of a spreader transformation.) As in FIG. 2A, the horizontal lines 512-515 correspond to modes, and the mode coupling is indicated by a vertical line 516 with nodes (dots) to identify the modes being coupled. In this case, four modes are coupled. Circuit notation 502 is an equivalent representation to circuit diagram 504, which is a network of first-order mode couplings. More generally, where a higher-order mode coupling can be implemented as a network of first-order mode couplings, a circuit notation similar to notation 502 (with an appropriate number of modes) may be used.

[0080] FIG. 6 illustrates an example optical device 600 that can implement the four-mode mode-spreading transform shown schematically in FIG. 5 in accordance with some embodiments. Optical device 600 includes a first set of optical waveguides 601, 603 formed in a first layer of material (represented by solid lines in FIG. 6) and a second set of optical waveguides 605, 607 formed in a second layer of material that is distinct and separate fromthe first layer of material (represented by dashed lines in FIG. 6). The second layer of material and the first layer of material are located at different heights on a substrate. One of ordinary skill will appreciate that an interferometer such as that shown in FIG. 6 could be implemented in a single layer if appropriate low loss waveguide crossing were employed.

[0081] At least one optical waveguide 601, 603 of the first set of optical waveguides is coupled with an optical waveguide 605, 607 of the second set of optical waveguides with any type of suitable optical coupler, e.g., the directional couplers described herein (e.g., the optical couplers shown in FIGs. 2B, 3A, 3B). For example, the optical device shown in FIG.6 includes four optical couplers 618, 620, 622, and 624. Each optical coupler can have a coupling region in which two waveguides propagate in parallel. Although the two waveguides are illustrated in FIG. 6 as being offset from each other in the coupling region, the two waveguides may be positioned directly above and below each other in the coupling region without offset. In some embodiments, one or more of the optical couplers 618, 620, 622, and 624 are configured to have a coupling efficiency of approximately 50% between the two waveguides (e.g., a coupling efficiency between 49% and 51%, a coupling efficiency between 49.9% and 50.1%, a coupling efficiency between 49.99% and 50.01%, and a coupling efficiency of 50%, etc.). For example, the length of the two waveguides, the refractive indices of the two waveguides, the widths and heights of the two waveguides, the refractive index of the material located between two waveguides, and the distance between the two waveguides are selected to provide the coupling efficiency of 50% between the two waveguides. This allows the optical coupler to operate like a 50 / 50 beam splitter.

[0082] In addition, the optical device shown in FIG. 6 can include two inter-layer optical couplers 614 and 616. Optical coupler 614 allows transfer of light propagating in a waveguide on the first layer of material to a waveguide on the second layer of material, and optical coupler 616 allows transfer of light propagating in a waveguide on the second layer of material to a waveguide on the first layer of material. The optical couplers 614 and 616 allow optical waveguides located in at least two different layers to be used in a multi-channel optical coupler, which, in turn, enables a compact multi-channel optical coupler.

[0083] Furthermore, the optical device shown in FIG. 6 includes a non-coupling waveguide crossing region 626. In some implementations, the two waveguides (603 and 605 in this example) cross each other without having a parallel coupling region present at the crossing inthe non-coupling waveguide crossing region 626 (e.g., the waveguides can be two straight waveguides that cross each other at a nearly 90-degree angle).

[0084] Those skilled in the art will understand that the foregoing examples are illustrative and that photonic circuits using beam splitters and / or phase shifters can be used to implement many different transfer matrices, including transfer matrices for real and imaginary Hadamard transforms of any order, discrete Fourier transforms, and the like. One class of photonic circuits, referred to herein as “spreader” or “mode-information erasure (MIE)” circuits, has the property that if the input is a single photon localized in one input mode, the circuit delocalizes the photon amongst each of a number of output modes such that the photon has equal probability of being detected in any one of the output modes. Examples of spreader or MIE circuits include circuits implementing Hadamard transfer matrices. (It is to be understood that spreader or MIE circuits may receive an input that is not a single photon localized in one input mode, and the behavior of the circuit in such cases depends on the particular transfer matrix implemented.) In other instances, photonic circuits can implement other transfer matrices, including transfer matrices that, for a single photon in one input mode, provide unequal probability of detecting the photon in different output modes.

[0085] In some embodiments, entangled states of multiple photonic qubits can be created by coupling modes of two (or more) qubits and performing measurements on other modes. By way of example, FIG. 7 shows a circuit diagram for a Bell state generator 700 that can be used in some dual-rail-encoded photonic embodiments. In this example, waveguides (or modes) 732-1 through 732-4 are initially each occupied by a photon (indicated by a wavy line); waveguides (or modes) 732-5 through 732-8 are initially vacuum (unoccupied) modes. (Those skilled in the art will appreciate that other combinations of occupied and unoccupied modes can be used.)

[0086] A first-order mode coupling (e.g., implementing transfer matrix T of Eq. (9)) is performed on pairs of occupied and unoccupied modes as shown by mode couplers 731-1 through 731-4, with each mode coupler 731 having one input waveguide receiving a photon and one input waveguide receiving vacuum. Mode couplers 731 can be, e.g., 50 / 50 beam splitters so that, for example, a photon entering on waveguide 732-1 (or a photon entering on waveguide 732-5) has a 50% probability of emerging on either output of mode coupler 731-1. In some embodiments, mode couplers 731 are implemented as directional couplers.Thereafter, a mode-information erasure coupling (e.g., implementing a four-mode modespreading transform as shown in FIG. 5 or a second-order Hadamard transfer matrix) is performed on one output mode of each mode coupler 731 (in this example, waveguides 733-5 through 733-8 provide inputs to the mode-information erasure coupling), as shown by mode coupler 737. In the following description, mode coupler 737 may also be referred to as a “mode coupler network” or “Hadamard network.” Waveguides 733-5 through 733-8 act as “heralding” modes that are measured and used to determine whether a Bell state was successfully generated on the four output waveguides 733-1 through 733-4. For instance, detectors 738-1 through 738-4 can be coupled to the waveguides 733-5 through 733-8 after second-order mode coupler 737. Each detector 738-1 through 738-4 can output a classical data signal (e.g., a voltage level on a conductor) indicating whether it detected a photon (or the number of photons detected). These outputs can be coupled to classical decision logic circuit 740, which determines whether a Bell state is present on the other four waveguides 733-1 through 733-4. For example, decision logic circuit 740 can be configured such that a Bell state is confirmed (also referred to as “success” of the Bell state generator) if and only if a single photon was detected by each of exactly two of detectors 738-1 through 738-4. In some embodiments, output modes (or waveguides) 733-1 through 733-4 can be mapped to the logical states of two qubits (Qubit 1 and Qubit 2), as indicated in FIG. 7. Specifically, in this example, the logical state of Qubit 1 is based on occupancy of modes 733-1 and 733-2, and the logical state of Qubit 2 is based on occupancy of modes 733-3 and 733-4. It should be noted that generation of a Bell state by Bell state generator 700 is a non-determini stic (or stochastic) process; that is, inputting four photons as shown does not guarantee that a Bell state will be created on modes 733-1 through 733-4. In one implementation, the probability of success is 4 / 32; in another implementation, the success probability is 3 / 16. It should also be noted that there are six detection patterns with one photon in each of two of detectors 738, and that Bell state generator 700 can be expected to produce a Bell state in all six possible arrangements of the four output modes. For a given choice of assignment of modes to dualrail qubits (e.g., as shown in FIG. 7), Bell state generator 700 can produce any of the four two-qubit Bell states defined in Eqs. (3)-(6) above, as well as a “non-qubif ’ maximally entangled state. Different detection patterns at detectors 738 can correspond to different types of Bell states being produced. In some embodiments, based on the particular detection pattern at detectors 738, mode swaps can be selectably applied to modes 733 in order to cast the Bell state into a particular type (e.g., a particular one of the four two-qubit Bell states defined above). In some embodiments, the mode swap can be subsumed into subsequentoperations without the need for active optical switches to implement selectable mode swapping at the output of Bell state generator 700.

[0087] In some embodiments, it is desirable to form quantum systems of multiple entangled qubits (two or more qubits). One technique for forming multi-qubit quantum systems is through the use of an entangling measurement, which is a projective measurement that can be employed to create entanglement between systems of qubits. As used herein, “fusion” (or “a fusion operation” or “fusing”) refers to a projective entangling measurement. A “fusion gate” is a structure that receives two (or more) input qubits, each of which is typically part of a different quantum system. Prior to applying the fusion gate, the different quantum systems need not be entangled with each other. In the case of two input qubits, the fusion gate performs a projective measurement operation on the input qubits that produces either one (“type I fusion”) or zero (“type II fusion”) output qubits in a manner such that the initial two quantum systems are fused into a single quantum system of entangled qubits. Fusion gates are specific examples of a general class of projective entangling measurements and are particularly suited for photonic architectures. Examples of type I and type II fusion gates will now be described.

[0088] FIG. 8A shows a circuit diagram illustrating a type I fusion gate 800 in accordance with some embodiments. The diagram shown in FIG. 8A is schematic with each horizontal line representing a mode of a quantum system, e.g., a photon. In a dual-rail encoding, each pair of modes represents a qubit. In a photonic implementation of the gate the modes in diagrams such as that shown in FIG. 8A can be physically realized using single photons in photonic waveguides. Most generally, a type I fusion gate like that shown in FIG. 8A takes qubit A (physically realized, e.g., by photon modes 843 and 845) and qubit B (physically realized, e.g., by photon modes 847 and 849) as input and outputs a single “fused” qubit that inherits the entanglement with other qubits that were previously entangled with either (or both) of input qubit A or input qubit B.

[0089] For example, FIG. 8B shows the result of type I fusing of two qubits A and B that are each, respectively, a qubit located at the end (i.e., a leaf) of some longer entangled cluster state (only a portion of which is shown). The qubit 857 that remains after the fusion operation inherits the entangling bonds from the original qubits A and B thereby creating a larger linear cluster state. FIG. 8B also shows the result of type I fusing of two qubits A and B that are each, respectively, an internal qubit that belongs to some longer entangled cluster of qubits(only a portion of which is shown). As before, the qubit 859 that remains after fusion inherits the entangling bonds from the original qubits A and B thereby creating a fused quantum system. In this case, the qubit that remains after the fusion operation is entangled with the larger quantum system by way of four other nearest neighbor qubits as shown.

[0090] Returning to the schematic illustration of type I fusion gate 800 shown in FIG. 8A, qubit A is dual-rail encoded by modes 843 and 845, and qubit B is dual-rail encoded by modes 847 and 849. For example, in the case of path-encoded photonic qubits, the logical zero state of qubit A (denoted 10)^) occurs when mode 843 is a photonic waveguide that includes a single photon and mode 845 is a photonic waveguide that includes zero photons (and likewise for qubit B). Thus, type I fusion gate 800 can take as input two dual-rail-encoded photon qubits thereby resulting in a total of four input modes (e.g., modes 843, 845, 847, and 849). To accomplish the fusion operation, a mode coupler (e.g., 50 / 50 beam splitter) 853 is applied between a mode of each of the input qubits, e.g., between mode 843 and mode 849 before performing a detection operation on both modes using photon detectors 855 (which includes two distinct photon detectors coupled to modes 843 and 849 respectively). If desired, one or more mode swap operations can be applied to position the output modes 845 and 845 adjacent to each other. In some embodiments, mode swapping can be accomplished through a physical waveguide crossing as described above or by one or more photonic switches or by any other type of physical mode swap.

[0091] FIG. 8A shows only an example arrangement for a type I fusion gate and one of ordinary skill will appreciate that the position of the mode coupler and the presence of the mode swap region 851 can be altered without departing from the scope of the present disclosure. For example, beam splitter 853 can be applied between modes 845 and 847. Mode swaps are optional and are not necessary if qubits having non-adjacent modes can be dealt with, e.g., by tracking which modes belong to which qubits by storing this information in a classical memory.

[0092] Type I fusion gate 800 is a nondeterministic gate, i.e., the fusion operation succeeds with a certain probability less than 1, and in other cases the quantum state that results is not a larger quantum system that comprises the original quantum systems fused together to form a larger quantum system. More specifically, gate 800 “succeeds,” with probability 50%, when only one photon is detected by detectors 855, and “fails” if zero or two photons are detected by detectors 855. When the gate succeeds, the two quantum systems that qubits A and Bwere a part of become fused into a single larger quantum system with a fused qubit remaining as the qubit that links the two previously unlinked quantum systems (see, e.g., FIG. 8B). However, when the fusion gate fails, it has the effect of removing both qubits from the original quantum systems without generating a larger quantum system.

[0093] FIG. 9A shows a circuit diagram illustrating a type II fusion gate 900 in accordance with some embodiments. Like other diagrams herein, the diagram shown in FIG. 9A is schematic with each horizontal line representing a mode of a quantum system, e.g., a photon. In a dual-rail encoding, each pair of modes represents a qubit. In a photonic implementation of the gate the modes in diagrams such as that shown in FIG. 9A can be physically realized using single photons in photonic waveguides. Most generally, a type II fusion gate such as gate 900 takes qubit A (physically realized, e.g., by photon modes 943 and 945) and qubit B (physically realized, e.g., by photon modes 947 and 949) as input and outputs a quantum state that inherits the entanglement with other qubits that were previously entangled with either (or both) of input qubit A or input qubit B. (For type II fusion, if the input quantum states had a total of N qubits between them, the output quantum state has A - 2 qubits. This is different from type I fusion where input quantum states having a total of N qubits between them leads to an output quantum state having A- 1 qubits.)

[0094] For example, FIG. 9B shows the result of type II fusing of two qubits A and B that are each, respectively, a qubit located at the end (i.e., a leaf) of some longer entangled cluster state (only a portion of which is shown). The resulting quantum system 971 inherits the entangling bonds from qubits A and B thereby creating a larger linear quantum system.

[0095] Returning to the schematic illustration of type II fusion gate 900 shown in FIG. 9A, qubit A is dual-rail encoded by modes 943 and 945, and qubit B is dual-rail encoded by modes 947 and 949. For example, in the case of path encoded photonic qubits, the logical zero state of qubit A (denoted 10)^) occurs when mode 943 is a photonic waveguide that includes a single photon and mode 945 is a photonic waveguide that includes zero photons (and likewise for qubit B). Thus, type II fusion gate 900 takes as input two dual-rail-encoded photon qubits thereby resulting in a total of four input modes (e.g., modes 943, 945, 947, and 949). To accomplish the fusion operation, a first mode coupler (e.g., 50 / 50 beam splitter) 953 is applied between a mode of each of the input qubits, e.g., between mode 943 and mode 949, and a second mode coupler (e.g., 50 / 50 beam splitter) 955 is applied between the other modes of each of the input qubits, e.g., between modes 945 and 947. A detection operation isperformed on all four modes using photon detectors 957(l)-957(4). In some embodiments, mode swap operations (not shown in FIG. 9 A) can be performed to place modes in adjacent positions prior to mode coupling. In some embodiments, mode swapping can be accomplished through a physical waveguide crossing as described above or by one or more photonic switches or by any other type of physical mode swap. Mode swaps are optional and are not necessary if qubits having non-adjacent modes can be dealt with, e.g., by tracking which modes belong to which qubits by storing this information in a classical memory.

[0096] FIG. 9A shows only an example arrangement for the type II fusion gate and one of ordinary skill will appreciate that the positions of the mode couplers and the presence or absence of mode swap regions can be altered without departing from the scope of the present disclosure.

[0097] The type II fusion gate shown in FIG. 9A is a nondeterministic gate, i.e., the fusion operation succeeds with a certain probability less than 1, and in other cases the quantum state that results is not a larger quantum system that comprises the original quantum systems fused together to a larger quantum system. More specifically, the gate “succeeds” in the case where one photon is detected by one of detectors 957(1) and 957(4) and one photon is detected by one of detectors 957(2) and 957(3); in all other cases, the gate “fails.” When the gate succeeds, the two quantum systems that qubits A and B were a part of become fused into a single larger quantum system; unlike type I fusion, no fused qubit remains (compare FIG.8B and FIG. 9B). When the fusion gate fails, it has the effect of removing both qubits from the original quantum systems without generating a larger quantum system.

[0098] FIG. 10 illustrates an example of a qubit entangling system 1001 in accordance with some embodiments. Such a system can be used to generate qubits (e.g., photons) in an entangled state (e.g., a GHZ state, Bell pair, and the like), in accordance with some embodiments. In some embodiments, qubit entangling system 1001 can operate as a resource state generator as described below.

[0099] In an illustrative photonic architecture, qubit entangling system 1001 can include a photon source module 1005 that is optically connected to entangled state generator 1000. Both the photon source module 1005 and the entangled state generator 1000 may be coupled to a classical processing system 1003 such that the classical processing system 1003 can communicate and / or control (e.g., via the classical information channels 1030a-b) the photon source module 1005 and / or the entangled state generator 1000. Photon source module 1005may include a collection of single-photon sources that can provide output photons to entangled state generator 1000 by way of interconnecting waveguides 1032. Entangled state generator 1000 may receive the output photons and convert them to one or more entangled photonic states and then output these entangled photonic states into output waveguides 1040. In some embodiments, output waveguide 1040 can be coupled to some downstream quantum photonic circuit that may use the entangled states, e.g., for performing a quantum computation. For example, the entangled states generated by the entangled state generator 1000 may be used as resource states for one or more interleaving modules as described below.

[0100] In some embodiments, system 1001 may include classical channels 1030 (e.g., classical channels 1030-a through 1030-d) for interconnecting and providing classical information between components. It should be noted that classical channels 1030-a through 1030-d need not all be the same. For example, classical channel 1030-a through 1030-c may comprise a bi-directional communication bus carrying one or more reference signals, e.g., one or more clock signals, one or more control signals, or any other signal that carries classical information, e.g., heralding signals, photon detector readout signals, and the like.

[0101] In some embodiments, qubit entangling system 1001 includes the classical computer system 1003 that communicates with and / or controls the photon source module 1005 and / or the entangled state generator 1000. For example, in some embodiments, classical computer system 1003 can be used to configure one or more circuits, e.g., using a system clock that may be provided to photon sources 1005 and entangled state generator 1000 as well as any downstream quantum photonic circuits used for performing quantum computation. In some embodiments, the quantum photonic circuits can include optical circuits, electrical circuits, or any other types of circuits. In some embodiments, classical computer system 1003 includes memory 1004, one or more processor(s) 1002, a power supply, an input / output (I / O) subsystem, and a communication bus or interconnecting these components. The processor(s) 1002 may execute modules, programs, and / or instructions stored in memory 1004 and thereby perform processing operations.

[0102] In some embodiments, memory 1004 stores one or more programs (e.g., sets of instructions) and / or data structures. For example, in some embodiments, entangled state generator 1000 can attempt to produce an entangled state over successive stages, any one of which may be successful in producing an entangled state. In some embodiments, memory1004 stores one or more programs for determining whether a respective stage was successful and configuring the entangled state generator 1000 accordingly (e.g., by configuring entangled state generator 1000 to switch the photons to an output if the stage was successful, or pass the photons to the next stage of the entangled state generator 1000 if the stage was not yet successful). To that end, in some embodiments, memory 1004 stores detection patterns (described below) from which the classical computing system 1003 may determine whether a stage was successful. In addition, memory 1004 can store settings that are provided to the various configurable components (e.g., switches) described herein that are configured by, e.g., setting one or more phase shifts for the component.

[0103] In some embodiments, some or all of the above-described functions may be implemented with hardware circuits on photon source module 1005 and / or entangled state generator 1000. For example, in some embodiments, photon source module 1005 includes one or more controllers 1007-a (e.g., logic controllers) (e.g., which may comprise field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a “system on a chip” that includes classical processors and memory, or the like). In some embodiments, controller 1007-a determines whether photon source module 1005 was successful (e.g., for a given attempt on a given clock cycle, described below) and outputs a reference signal indicating whether photon source module 1005 was successful. For example, in some embodiments, controller 1007-a outputs a logical high value to classical channel 1030-a and / or classical channel 1030-c when photon source module 1005 is successful and outputs a logical low value to classical channel 1030-a and / or classical channel 1030-c when photon source module 1005 is not successful. In some embodiments, the output of control 1007-a may be used to configure hardware in controller 1007-b.

[0104] Similarly, in some embodiments, entangled state generator 1000 includes one or more controllers 1007-b (e.g., logical controllers) (e.g., which may comprise field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), or the like) that determine whether a respective stage of entangled state generator 1000 has succeeded, perform the switching logic described above, and output a reference signal to classical channels 1030-b and / or 1030-d to inform other components as to whether the entangled state generator 1000 has succeeded.

[0105] In some embodiments, a system clock signal can be provided to photon source module 1005 and entangled state generator 1000 via an external source (not shown) or byclassical computing system 1003 generates via classical channels 1030-a and / or 1030-b. In some embodiments, the system clock signal provided to photon source module 1005 triggers photon source module 1005 to attempt to output one photon per waveguide. In some embodiments, the system clock signal provided to entangled state generator 1000 triggers, or gates, sets of detectors in entangled state generator 1000 to attempt to detect photons. For example, in some embodiments, triggering a set of detectors in entangled state generator 1000 to attempt to detect photons includes gating the set of detectors.

[0106] It should be noted that, in some embodiments, photon source module 1005 and entangled state generator 1000 may have internal clocks. For example, photon source module 1005 may have an internal clock generated and / or used by controller 1007-a and entangled state generator 1000 has an internal clock generated and / or used by controller 1007-b. In some embodiments, the internal clock of photon source module 1005 and / or entangled state generator 1000 is synchronized to an external clock (e.g., the system clock provided by classical computer system 1003) (e.g., through a phase-locked loop). In some embodiments, any of the internal clocks may themselves be used as the system clock, e.g., an internal clock of the photon source may be distributed to other components in the system and used as the master / system clock.

[0107] In some embodiments, photon source module 1005 includes a plurality of probabilistic photon sources that may be spatially and / or temporally multiplexed, i.e., a so-called multiplexed single photon source. In one example of such a source, the source is driven by a pump, e.g., a light pulse, that is coupled into an optical resonator that, through some nonlinear process (e.g., spontaneous four wave mixing, second harmonic generation, and the like) may generate zero, one, or more photons. As used herein, the term “attempt” is used to refer to the act of driving a photon source with some sort of driving signal, e.g., a pump pulse, that may produce output photons non-deterministically (i.e., in response to the driving signal, the probability that the photon source will generate one or more photons may be less than 1). In some embodiments, a respective photon source may be most likely to, on a respective attempt, produce zero photons (e.g., there may be a 90% probability of producing zero photons per attempt to produce a single-photon). The second most likely result for an attempt may be production of a single-photon (e.g., there may be a 9% probability of producing a single-photon per attempt to produce a single-photon). The third most likely result for an attempt may be production of two photons (e.g., there may be an approximately1% probability of producing two photons per attempt to produce a single photon). In some circumstances, there may be less than a 1% probability of producing more than two photons.

[0108] In some embodiments, the apparent efficiency of the photon sources may be increased by using a plurality of single-photon sources and multiplexing the outputs of the plurality of photon sources.

[0109] The precise type of photon source used is not critical and any type of source can be used, employing any photon generating process, such as spontaneous four wave mixing (SPFW), spontaneous parametric down-conversion (SPDC), or any other process. Other classes of sources that do not necessarily require a nonlinear material can also be employed, such as those that employ atomic and / or artificial atomic systems, e.g., quantum dot sources, color centers in crystals, and the like. In some cases, sources may or may be coupled to photonic cavities, e.g., as can be the case for artificial atomic systems such as quantum dots coupled to cavities. Other types of photon sources also exist for SPWM and SPDC, such as optomechanical systems and the like. In some examples the photon sources can emit multiple photons already in an entangled state in which case the entangled state generator 1000 may not be necessary, or alternatively may take the entangled states as input and generate even larger entangled states.

[0110] For the sake of illustration, an example which employs spatial multiplexing of several non-deterministic photon sources is described as an example of a MUX photon source. However, many different spatial MUX architectures are possible without departing from the scope of the present disclosure. Temporal MUXing can also be implemented instead of or in combination with spatial multiplexing. MUX schemes that employ log-tree, generalized Mach-Zehnder interferometers, multimode interferometers, chained sources, chained sources with dump-the-pump schemes, asymmetric multi-crystal single photon sources, or any other type of MUX architecture can be used. In some embodiments, the photon source can employ a MUX scheme with quantum feedback control and the like.[OHl] The foregoing description provides an example of how photonic circuits can be used to implement physical qubits and operations on physical qubits using mode coupling between waveguides. In these examples, a pair of modes can be used to represent each physical qubit. Examples described below can be implemented using similar photonic circuit elements.

[0112] In some embodiments, an entangled system of multiple physical qubits can be mapped to one or more “logical qubits,” and operations associated with a quantumcomputation can be defined as logical operations on logical qubits, which in turn can be mapped to physical operations on physical qubits. In general, the term “qubit,” when used herein without specifying physical or logical qubit, should be understood as referring to a physical qubit.1.4. Temporally Encoded Qubits

[0113] As described above, qubits can be encoded using discrete temporal and / or spatial modes of a photon. One example of spatial encoding is dual-rail encoding as described above with reference to FIG. 1. In some embodiments, temporal encoding can be based on presence or absence of a photon at a particular location along a waveguide at a particular time. FIG.11 A shows two representations (1100, 1100') of a portion of a single waveguide 1102.Photons propagate along waveguide 1102 in the direction indicated by arrow 1110. One section of the waveguide corresponds to a first temporal mode (or time bin) tj, and another section of the waveguide corresponds to a second temporal mode tj+1. In this example, temporal modes tj and t7+1are adjacent temporal modes, meaning that no distinct temporal mode is defined between them. At 1100, a photon is present in temporal mode t7+1and no photon is present in temporal mode t7; in some embodiments, this corresponds to the |0)Lstate of a photonic qubit. At 1100', a photon is present in temporal mode tj and no photon is present in temporal mode t7+1; in some embodiments, this corresponds to the |1)Lstate of the photonic qubit. To prepare a qubit in a known logical state, a photon source (not shown) can be coupled to one end of waveguide 1102. The photon source can be operated to inject a photon into waveguide 1102 at a known time bin, thereby preparing the qubit in a known logical state. Photon sources of the kind described above can be used. It should be understood that different temporally-encoded qubits can be propagating through different sections of the same waveguide at different times.

[0114] In some embodiments, spatially-encoded qubits (e.g., dual-rail encoded qubits as described above with reference to FIG. 1) can be converted to temporally-encoded qubits (e.g., qubits as described above with reference to FIG. 11 A) and vice versa. FIG. 1 IB shows an example of an optical circuit 1120 that can convert a dual-rail-encoded qubit to a temporally-encoded qubit. Optical circuit 1120 includes a 2* 1 multiplexer (mux) 1122 having two input waveguides 1124-0, 1124-1 and an output waveguide 1126. Input path 1124-0 includes a delay line 1128 that adds one time bin of delay. A dual-rail encoded qubit is shown at 1130 using a pair of gray shaded circles to indicate that a photon may be presentin either waveguide 1124-0 (which in this example corresponds to the |0)Lstate of qubit 1130) or waveguide 1124-1 (which corresponds to the |1)Lstate of qubit 1130). The state of qubit 1130 may be a known state or a superposition state in which the photon has a nonzero probability of being in either waveguide 1142-0 or 1124-1. Delay line 1128 can delay a photon in waveguide 1124-0 (if present) by one time bin. A control signal (CTL) can operate 2x1 mux 1122 to couple photons from input waveguide 1124-1 into output waveguide 1126 during a first time bin and to couple photons from input waveguide 1124-0 into output waveguide 1126 during the next time bin. The output of 2x 1 mux 1122 can be a temporally encoded qubit 1130' on output waveguide 1126. It should be noted that temporally-encoded qubit 1130' represents the same quantum state as spatially-encoded qubit 1130, using a pair of temporal modes in the same waveguide rather than a pair of spatial modes in a single time bin.

[0115] FIG. 11C shows an example of an optical circuit 1150 that can convert a temporally-encoded qubit to a dual-rail-encoded qubit. Optical circuit 1150 includes a 1 x2 mux 1152 having an input waveguide 1154 and two output waveguides 1156-0 and 1156-1. Output waveguide 1156-1 includes a delay line 1158 that adds one time bin of delay. A temporally-encoded qubit is shown at 1160 using a pair of shaded circles to indicate that a photon may be present either in a first time bin (corresponding to the 11)Lstate of qubit 1160) or a second time bin (corresponding to the |0)Lstate of qubit 1160). Similarly to FIG. 1 IB, qubit 1160 can be in a known state or in a superposition state in which the photon has a nonzero probability of being in either the first or second time bin. A control signal (CTL) can operate 1 x2 mux 1152 to couple photons from input waveguide 1154 into first output waveguide 1156-1 during a first time bin and to couple photons from input waveguide 1154 into second output waveguide 1156-0 during the next time bin. The result, downstream of delay line 1158, is a dual-rail encoded qubit 1160' occupying a single time bin on output waveguides 1156-0 and 1156-1. It should be noted that spatially-encoded qubit 1160' represents the same quantum state as temporally-encoded qubit 1160, using a pair of spatial modes in a single time bin rather than a pair of temporal modes in a single waveguide.

[0116] In some embodiments, temporal and spatial encodings of qubits may be used in the same system for different purposes. For instance, as described above, operations on one or more dual-rail-encoded qubits can be implemented using linear optical components such as beam splitters and phase shifters. However, for some applications (e.g., storage or longdistance propagation of a quantum state), using a single waveguide to encode the qubit maybe preferable. Accordingly, qubits can be created in either temporal or dual-rail encodings and converted between the two encodings as desired, e.g., using circuits 1120 and 1150 or other similar circuits.2. Multi-way Fusion Circuits Using Type I and Type II Fusions

[0117] According to some embodiments, multi-way fusion circuits (circuits that fuse three or more quantum systems) can be implemented by combining type I and type II fusion circuits of the kind described above with reference to FIGs. 8A-8B and 9A-9B. Such circuits can receive input qubits from each of three or more separately entangled quantum systems, each of which includes multiple qubits. Like the two-way fusion circuits described above, multi-way fusion circuits succeed probabilistically. When the multi-way fusion circuit succeeds, the input qubits are consumed and the remaining qubits become entangled, forming a single entangled quantum system as the output. Classical data is also produced, as described below.

[0118] FIG. 12 shows a simplified conceptual diagram of operation of a multi-way fusion circuit 1200 according to some embodiments. A number (N) of quantum systems 1202-1 through 1202-A are shown at the left. It is assumed that N is at least 3. Each quantum system 1202 includes some number of qubits that are entangled with each other. For purposes of illustration, qubits are shown as circles, and entanglement is indicated by lines connecting qubits. Initially, the different quantum systems 1202-1 through 1202-A are separate from (or not entangled with) each other. From each quantum system 1202-1 through 1202-A, one qubit (denoted herein as Ql-QA, shown as darkened circles in FIG. 12) is selected to be input to multi-way fusion circuit 1200, as indicated by arrows 1210-1 through 1210-AL Multi-way fusion circuit 1200 performs a multi-way fusion operation, which is a projective entangling measurement operation, on input qubits Ql-QA. The fusion operation consumes input qubits Ql-QA. In embodiments described herein, the fusion operation succeeds probabilistically, and multi-way fusion circuit 1200 can produce a classical output signal 1256 indicating whether the fusion operation succeeded. If the fusion operation succeeded, the remaining qubits of entangled systems 1202-1 through 1202-A are “fused” into an entangled quantum system 1230. In embodiments described herein, if the initial quantum systems 1202-1 through 1202-A included a total of M qubits, then the resulting entangled quantum system 1230 includes M — N qubits.

[0119] It should be noted that operation of multi-way fusion circuit 1200 is independent of the particular size (number of qubits) or entanglement pattern (or geometry) of initial quantum systems 1202-1 through 1202-A. Thus, different quantum systems 1202 can include different numbers of qubits and / or different entanglement patterns. When multi-way fusion circuit 1200 succeeds, the resulting entangled quantum system 1230 will have an entanglement pattern that depends on the particular entanglement patterns of initial quantum systems 1202-1 through 1202-7V, the selection of input qubits QI through Q7V, and the particular projective entangling measurement implemented in multi-way fusion circuit 1200. That is, a given multi-way fusion circuit 1200 projects input qubits QI through QA onto a particular A-qubit state (which can be an A-qubit GHZ state or other state), and instances of multi-way fusion circuit 1200 that are configured to project onto different states can produce quantum systems 1230 having different entanglement patterns from the same inputs.

[0120] Further, it should be apparent that quantum systems 1202-1 through 1202-A could be produced using another multi-way fusion circuit, and quantum system 1230 in turn could be an input to another multi-way fusion circuit. Accordingly, multi-way fusion circuits can be chained together to produce entangled quantum systems of arbitrary size and entanglement geometry.

[0121] Example implementations of multi-way fusion circuits for dual-rail-encoded photonic qubits (as described above with reference to FIG. 1) will now be described.2.1.Building Blocks

[0122] To facilitate understanding of the following description, further description of the type I and type II fusion circuits introduced above in Section 1.3 is provided.

[0123] FIGs. 13 A and 13B show simplified schematic diagrams of two variations of a type I fusion circuit 1300, 1300' that can be used in some embodiments. Both variations receive two input qubits (labeled QI and Q2) and produce one output qubit (labeled Qo). Referring first to FIG. 13A, circuit 1300 is an example of a type I fusion circuit of a first variation (also referred to as a “type I vl fusion circuit”). As shown, input qubit QI is encoded on a first pair of waveguides 1302, 1304 such that the logical-0 rail (denoted |0L)) maps to waveguide 1302 and the logical-1 rail (denoted 11L)) maps to waveguide 1304. Similarly, input qubit Q2 is encoded on a second pair of waveguides 1306, 1308 such that the logical-0 rail (denoted 10L)) maps to waveguide 1306 and the logical-1 rail (denoted |1L)) maps to waveguide 1308. As described above, each of input qubits QI and Q2 can be part of a different entangledsystem of multiple qubits. For clarity of illustration, other qubits of the initial entangled systems are not shown. In all circuit diagrams herein, it should be understood that additional waveguides can be provided to propagate the other qubits of the initial entangled systems.

[0124] Waveguides 1304 and 1306, which map to the logical-1 rail of qubit QI and the logical-0 rail of qubit Q2 respectively, include a beam splitter region 1312, which implements a 50 / 50 beam splitter, e.g., in the manner described above with reference to FIG. 2B. To bring waveguides 1304 and 1306 into proximity in region 1312, circuit 1300 can include a mode crossing 1314. Mode crossing 1314 can be implemented, e.g., in the manner described above with reference to FIG. 6. It should be understood that mode crossing 1314 and all other mode crossings described herein can be implemented such that there is no coupling of photons between the crossing waveguides or by using a coupling such as one or more 0 / 100 beam splitters. Downstream of beam splitter region 1312, waveguide 1304 couples to a first photon-counting detector 1321, and waveguide 1306 couples to a second photon-counting detector 1322. In some embodiments, photon-counting detectors can be implemented as photon-number-resolving detectors that can distinguish different numbers of photons incident on the detector. For instance, the detector can output different classical signals indicating 0 incident photons, 1 incident photon, or 2 or more incident photons. Waveguides 1302 and 1308 propagate photons with no other interactions, providing a dual-rail-encoded output qubit Qo having a logical-0 rail (denoted |0L)) that maps to waveguide 1302 and a logical- 1 rail (denoted |1L)) that maps to waveguide 1308. Photon counts from detectors 1321, 1322 can be used by classical decision logic circuitry (not shown in FIG. 13 A) to determine whether the fusion operation succeeded, as described above with reference to FIG. 8A. If the fusion operation succeeded, output qubit Qo is assumed to be in a logically valid state.

[0125] Referring now to FIG. 13B, circuit 1300' is an example of a type I fusion circuit of a second variation (also referred to as a “type I v2 fusion circuit”). As in type I vl fusion circuit 1300, input qubit QI is encoded on a first pair of waveguides 1302', 1304' such that the logical-0 rail (denoted |0L)) maps to waveguide 1302' and the logical-1 rail (denoted |1L)) maps to waveguide 1304'. Similarly, input qubit Q2 is encoded on a second pair of waveguides 1306', 1308' such that the logical-0 rail (denoted |0L)) maps to waveguide 1306' and the logical-1 rail (denoted |1L)) maps to waveguide 1308'. As described above, each of input qubits QI and Q2 can be part of a different entangled system of multiple qubits. For simplicity of illustration, the initial entangled systems are not shown.

[0126] In this case, waveguides 1302' and 1308', which map to the logical-0 rail of qubit QI and the logical-1 rail of qubit Q2 respectively, include a beam splitter region 1312', which implements a 50 / 50 beam splitter, e.g., in the manner described above with reference to FIG.2B. To bring waveguides 1302' and 1308' into proximity in region 1312', circuit 1300' can include a mode crossing 1314', which can be implemented similarly or identically to mode crossing 1314 in circuit 1300 of FIG. 13 A. Downstream of beam splitter region 1312', waveguide 1302' couples to a first photon-counting detector 1321', and waveguide 1308' couples to a second photon-counting detector 1322'. Waveguides 1304' and 1306' propagate photons with no other interactions, providing a dual-rail-encoded output qubit Qo having a logical-0 rail (denoted |0L)) that maps to waveguide 1306' and a logical-1 rail (denoted |1L)) that maps to waveguide 1304'. Photon counts from detectors 1321', 1322' can be used by classical decision logic circuitry to determine whether the fusion operation succeeded, as described above with reference to FIG. 8A. If the fusion operation succeeded, output qubit Qo is assumed to be in a logically valid state.

[0127] Each of type I vl fusion circuit 1300 or type I v2 fusion circuit 1300' can execute a type I fusion operation as described above with reference to FIGs. 8 A and 8B. The difference between type I vl fusion circuit 1300 and type I v2 fusion circuit 1300' lies in which pair of waveguides is used for the measurement and which pair of waveguides maps to the rails of output qubit Qo. Put differently, type I vl fusion circuit 1300 and type I v2 fusion circuit 1300' are equivalent up to a swap of the input qubits. This distinction between variations can provide certain advantages, as will become apparent in connection with examples below. In either variation, it should be understood that a type I fusion operation can be performed on input qubits QI and Q2 that are each part of different entangled systems of qubits and can produce a larger entangled system of qubits, e.g., as illustrated in FIG. 8B. In physical implementations, additional waveguides (not shown) can be provided to propagate the qubits of the input quantum systems that are not acted on by circuit 1300 (or circuit 1300').

[0128] As described above, type I fusion operations are non-deterministic (or stochastic) and succeed with a certain probability less than 1. Success or failure can be determined based on the pattern of detected photons. More specifically, a type I fusion operation in circuit 1300 (or circuit 1300') succeeds when one of detectors 1321 and 1322 (or detectors 1321' and 1322') detects exactly one photon and the other of detectors 1321 and 1322 (or detectors 1321' and 1322') detects zero photons; the operation fails if a total of zero or two photons are detected by detectors 1321 and 1322 (or detectors 1321' and 1322'). When theoperation succeeds, the two separately entangled quantum systems of entangled qubits that qubits QI and Q2 were initially parts of become fused into a single output quantum system of entangled qubits, with qubit Qo included in the output quantum system as the qubit that links the two previously unlinked systems (see, e.g., FIG. 8B). In other words, if QI is part of a system having a first number Ml of entangled qubits and Q2 is part of a system having a second number M2 of entangled qubits, then the result of a successful type I fusion (for either variation vl or v2) is a system having (Ml + M2 - 1) qubits, including Qo but not including QI or Q2. When the fusion operation fails, it has the effect of removing both qubits QI and Q2 from the initial entangled quantum systems without generating entanglement between the initial entangled quantum systems.

[0129] FIG. 14 shows a simplified schematic diagram of a type II fusion circuit 1400 that can be used in some embodiments. Circuit 1400 receives two input qubits (labeled QI and Q2) and produces no output qubits. Input qubit QI is encoded on a first pair of waveguides 1402, 1404 such that the logical-0 rail (denoted |0L)) maps to waveguide 1402 and the logical-1 rail (denoted |1L)) maps to waveguide 1404. Similarly, input qubit Q2 is encoded on a second pair of waveguides 1406, 1408 such that the logical-0 rail (denoted |0L)) maps to waveguide 1406 and the logical-1 rail (denoted |1L)) maps to waveguide 1408. As described above, each of input qubits QI and Q2 can be part of a different entangled system of multiple qubits. For clarity of illustration, the other qubits of the initial entangled systems are not shown.

[0130] Waveguides 1404 and 1406, which map to the logical-1 rail of qubit QI and the logical-0 rail of qubit Q2 respectively, include a first beam splitter region 1412, which implements a 50 / 50 beam splitter, e.g., in the manner described above with reference to FIG.2B. Waveguides 1402 and 1408, which map to the logical-0 rail of qubit QI and the logical-1 rail of qubit Q2 respectively, include a second beam splitter region 1413, which also implements a 50 / 50 beam splitter, e.g., in the manner described above with reference to FIG.2B. To bring waveguides 1404 and 1406 into proximity in region 1412 and to bring waveguides 1402 and 1408 into proximity in region 1413, circuit 1400 can include a mode crossing 1414. Mode crossing 1414 can be implemented, e.g., in the manner described above with reference to FIG. 6. Downstream of beam splitter region 1412, a first pair of photoncounting detectors 1421, 1422 is provided. Waveguide 1404 couples to photon-counting detector 1421, and waveguide 1406 couples to photon-counting detector 1422. Downstream of beam splitter region 1413, a second pair of photon-counting detectors 1423, 1424 isprovided. Waveguide 1402 couples to a third photon-counting detector 1423, and waveguide 1408 couples to a fourth photon-counting detector 1424. Photon counts from detectors 1421-1424 can be used by classical decision logic circuitry (not shown in FIG. 14) to determine whether the fusion operation succeeded, as described above with reference to FIG. 9A.

[0131] Type II fusion circuit 1400 can execute a type II fusion operation as described above with reference to FIGs. 9 A and 9B. It should be understood that a type II fusion operation can be performed on input qubits QI and Q2 that are each part of different entangled systems of qubits and can produce a larger entangled system of qubits, e.g., as illustrated in FIG. 9B. In physical implementations, additional waveguides (not shown) can be provided to propagate the qubits of the input quantum systems that are not acted on by circuit 1400.

[0132] As described above, type II fusion operations are nondeterministic (or stochastic) and succeed with a certain probability less than 1. Success or failure can be determined based on the pattern of detected photons. More specifically, a type II fusion operation in circuit 1400 succeeds when: (1) one of detector pair 1421, 1422 (which are located downstream of the same beam splitter 1412) detects exactly one photon and the other of detector pair 1421, 1422 detects zero photons; and (2) one of detector pair 1423, 1424 (which are located downstream of the same beam splitter 1413) detects exactly one photon and the other of detector pair 1423, 1424 detects zero photons. When the operation succeeds, the two separately entangled quantum systems of entangled qubits that qubits QI and Q2 were initially parts of become fused into a single output quantum system of entangled qubits (see, e.g., FIG. 9B). In other words, if QI is part of a system having a first number Ml of entangled qubits and Q2 is part of a system having a second number M2 of entangled qubits, then the result of a successful type II fusion is a system having (Ml + M2 - 2) qubits. QI and Q2 are both consumed. When the fusion operation fails, it has the effect of removing both qubits QI and Q2 from the initial entangled quantum systems without generating entanglement between the initial entangled quantum systems.2.2.Three-Way Fusion Circuits

[0133] According to some embodiments, a three-way fusion circuit can be constructed by cascading a type I fusion and a type II fusion. FIG. 15A shows a simplified schematic diagram of a three-way fusion circuit 1500 according to some embodiments. Three-way fusion circuit 1500 receives three dual-rail-encoded input qubits QI, Q2, Q3, each of whichcan initially be part of a different entangled system of qubits. (For simplicity of illustration, the remaining qubits of the initial entangled systems are not shown.) Success of three-way fusion circuit 1500 results in the three entangled systems being entangled with each other to form one output entangled system. It should be understood that, although FIG. 15A and other schematic diagrams herein are shown with detectors in a particular arrangement for clarity of illustration, in operation all detectors perform their detection operations within the same detection period (or time bin) and output data from all detectors is concurrently sent to the classical decision logic. For instance, the length of each optical path shown can be selected so that all photons arrive at the detectors simultaneously (meaning within a single detection period or time bin). Further, while type I and type II fusion operations are described herein as discrete operations, it should be understood that operation of a three-way (or more generally an A-way) fusion circuit is not serialized; a single detection pattern across all detectors in the three-way (or TV- way) fusion circuit determines success or failure of the three-way (or TV- way) fusion operation.

[0134] Input qubit QI is encoded on a first pair of waveguides 1502, 1504 such that the logical-0 rail maps to waveguide 1502 and the logical- 1 rail maps to waveguide 1504.Similarly, input qubit Q2 is encoded on a second pair of waveguides 1506, 1508 such that the logical-0 rail maps to waveguide 1506 and the logical- 1 rail maps to waveguide 1508, and input qubit Q3 is encoded on a third pair of waveguides 1510, 1512 such that the logical-0 rail maps to waveguide 1510 and the logical-1 rail maps to waveguide 1512. Input qubits QI and Q2 are input to a type I fusion circuit 1520. In the example shown, type I fusion circuit 1520 is implemented as an instance of type I vl fusion circuit 1300 described above, and waveguides 1502 and 1508 can be mapped to the rails of an intermediate pseudo-qubit Qo as described above with reference to FIG. 13 A, while any photons on waveguides 1504 and 1506 are detected and counted by a first pair of photon-counting detectors 1521, 1522. Qo (and other similar intermediate states) is referred to herein as an “intermediate pseudo-qubit” to denote that the constituent waveguides might or might not be propagating a logically valid qubit state in a given instance; since all detectors in circuit 1500 operate in one time bin to produce a single detection pattern

[0135] Intermediate pseudo-qubit Qo and input qubit Q3 are input to a type II fusion circuit 1530. In the example shown, type II fusion circuit 1530 is implemented as an instance of type II fusion circuit 1400 described above. Photons on waveguides 1502, 1512 are detected by a second pair of photon-counting detectors 1523, 1524, and photons on waveguides 1510,1508 are detected by a third pair of photon-counting detectors 1525, 1526. Components of circuit 1500 can be arranged such that all detectors 1521-1526 operate concurrently (or within the same time bin). Detectors 1521-1526 can output classical logic signals indicating the number of photons detected at each detector onto classical signal paths 1552, 1554.

[0136] Classical decision logic circuit 1550 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 1550 can receive the respective photon count signals from detectors 1521-1526, e.g., via signal paths 1552, 1554, and can determine whether the pattern of photon counts corresponds to a success pattern. As noted above, all signals from detectors 1521-1526 are received concurrently, providing a detection pattern having six elements (one photon count from each detector). Success patterns can correspond to one detector of the first pair of detectors 1521, 1522 detecting one photon (corresponding to successful type I fusion, as described above with reference to FIGs. 8A and 8B) in combination with one detector of the second pair of detectors 1523, 1524 detecting one photon and one detector of the third pair of detectors 1525, 1526 detecting one photon (corresponding to successful type II fusion, as described above with reference to FIGs. 9A and 9B). For instance, classical decision logic circuit 1550 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals are received via signal paths 1552, 1554, classical decision logic circuit 1550 can compare the “readout” pattern of signals received from detectors 1521-1526 to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 1550 can output a classical decision signal on path 1556 indicating whether a match has occurred.

[0137] In some embodiments, classical decision logic circuit 1550 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include QI, Q2, and Q3. Where such heralding signals are provided, classical decision logic circuit 1550 can also determine, based on the received heralding signals, whether the input qubits were in valid logical states. (In examples described herein, a dual-rail-encoded qubit is in a valid logical state if there is exactly one photon propagating in one or the other of the pair of waveguides mapped to the rails; for qubits in entangled states, it is generally not known which of the two waveguides has the photon.) Classical decision logic circuit 1550 can implement the following decision logic: If the received heraldingsignals indicate a valid input state and the readout pattern of photon counts on signal paths 1552, 1554 matches a success pattern, then the decision signal on path 1556 is set to indicate success (e.g., Boolean true); otherwise, the decision signal on path 1556 is set to indicate failure (e.g., Boolean false). It should be understood that heralding signals from upstream classical circuitry can be incorporated in a similar manner into the logic in any of the classical decision logic circuits described herein.

[0138] Success patterns for multi-way fusion circuits such as circuit 1500 can be defined based on whether each of the type I and type II fusions succeeded. For instance, as described above, success of a type I fusion circuit (e.g., circuit 1520) corresponds to one photon in one of detectors 1521, 1522 and zero photons in the other of detectors 1521, 1522. Similarly, success of a type II fusion circuit (e.g., circuit 1530) corresponds to one photon in one detector of each of detector pairs 1523, 1524 and 1525, 1526 and zero photons in the other detector of each pair. Where type I and type II fusion circuits are cascaded (or chained) as shown in FIG. 15 A, the success patterns are the detection patterns where each fusion operation succeeded, which can be expressed as exactly one photon in each pair of detectors coupled to the outputs of the same 50 / 50 beam splitter. If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 1500 is 25% (because type I fusion circuit 1520 and type II fusion circuit 1530 each have a 50% success probability, and circuit 1500 succeeds only if circuits 1520 and 1530 both succeed).

[0139] Further illustrating operation of classical decision logic circuit 1550, FIG. 15B shows a table 1560 listing the success patterns for circuit 1500 according to some embodiments. Success patterns are listed in column 1562. (In the notation of FIG. 15B, detectors D1-D6 correspond to detectors 1521-1526 respectively.) These success patterns correspond to instances where the type I fusion and the type II fusion both succeeded. Each success pattern corresponds to a projective entangling measurement, and column 1564 lists the corresponding projection for each success pattern. In some embodiments, the decision signal on path 1556 can indicate, based on the detection pattern, which projection was detected. In the example of table 1560, there are two projections differing by a sign, and the decision signal can be a three-state signal that indicates failure or, in the case of success, which of the two projections was detected.2.3.Four-Way Fusion Circuits

[0140] According to some embodiments, a four-way fusion circuit can be constructed by cascading (or chaining) multiple type I fusion circuits and a final type II fusion circuit. Such cascades can be “balanced” or “imbalanced.” Examples of each type will now be described.2.3.1. Balanced Four-Way Fusion Circuits

[0141] FIG. 16 shows a simplified schematic diagram of a balanced (or “parallelized”) four-way fusion circuit 1600 according to some embodiments. Four-way fusion circuit 1600 receives four dual-rail-encoded input qubits QI, Q2, Q3, Q4, each of which can be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of four-way fusion circuit 1600 results in the four entangled systems being entangled with each other to form one larger entangled system.

[0142] Input qubit QI is encoded on a first pair of waveguides 1601, 1602 such that the logical-0 rail maps to waveguide 1601 and the logical- 1 rail maps to waveguide 1602.Similarly, input qubit Q2 is encoded on a second pair of waveguides 1603, 1604 such that the logical-0 rail maps to waveguide 1603 and the logical- 1 rail maps to waveguide 1604; input qubit Q3 is encoded on a third pair of waveguides 1605, 1606 such that the logical-0 rail maps to waveguide 1605 and the logical- 1 rail maps to waveguide 1606; and input qubit Q4 is encoded on a fourth pair of waveguides 1607, 1608 such that the logical-0 rail maps to waveguide 1607 and the logical- 1 rail maps to waveguide 1608. Input qubits QI and Q2 are input to a first type I fusion circuit 1621, and input qubits Q3 and Q4 are input to a second type I fusion circuit 1622. In the example shown, each of type I fusion circuits 1621 and 1622 is an instance of type I vl fusion circuit 1300 described above. Thus, waveguides 1601 and 1604 are mapped to the rails of a first intermediate pseudo-qubit Qol as described above with reference to FIG. 13 A, while any photons on waveguides 1602 and 1603 are detected and counted by detectors 1625, 1626. Similarly, waveguides 1605 and 1608 are mapped to the rails of a second intermediate pseudo-qubit Qo2 as described above with reference to FIG. 13 A, while any photons on waveguides 1606 and 1607 are detected and counted by detectors 1627, 1628. Detectors 1625-1628 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 1652.

[0143] Intermediate pseudo-qubits Qol and Qo3 are input to a type II fusion circuit 1630. In the example shown, type II fusion circuit 1630 is implemented as an instance of type II fusion circuit 1400 described above, and photons on waveguides 1601, 1608, 1605, and 1604are detected by detectors 1631, 1632, 1633, 1634 respectively. Detectors 1631-1634 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 1654. Components of circuit 1600 can be arranged such that all detectors 1625-1628 and 1631-1634 operate concurrently (or within the same time bin).

[0144] Classical decision logic circuit 1650 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 1650 can receive the respective photon count signals from detectors 1625-1628 and 1631-1634, e.g., via signal paths 1652, 1654, and can determine whether the pattern of photon counts corresponds to a success pattern. As noted above, all signals from detectors 1625-1628 and 1631-1634 are received concurrently (in the same time bin), providing a detection pattern having eight elements (one photon count from each detector). Success patterns can correspond to conditions where: (1) one (and only one) of detectors 1625 and 1626 detects a single photon; (2) one (and only one) of detectors 1627 and 1628 detects a single photon; (3) one (and only one) of detectors 1631 and 1632 detects a single photon; and (4) one (and only one) of detectors 1633 and 1634 detects a single photon. As described above, classical decision logic circuit 1650 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals are received via signal paths 1652, 1654, classical decision logic circuit 1650 can compare the readout pattern of signals received from detectors 1625-1628 and 1631-1634 to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 1650 can output a classical decision signal 1656 indicating whether a match has occurred. In some embodiments, classical decision logic circuit 1650 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include QI, Q2, Q3, and Q4, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0145] Success patterns for circuit 1600 correspond to patterns indicating that each of type I fusion circuit 1621, type I fusion circuit 1622, and type II fusion circuit 1630 succeeded. As described above, a success pattern for a type I fusion circuit corresponds to one photon in one of the pair of detectors and zero photons in the other, and a success pattern for a type II fusion circuit corresponds to one photon in one of each pair of detectors that are coupled to the same 50 / 50 beam splitter and zero photons in the other detector of each pair. A complete set ofsuccess patterns for circuit 1600 can be derived from these principles. If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 1600 is 12.5% (because type I fusion circuits 1621 and 1622 and type II fusion circuit 1630 each have a 50% success probability, and circuit 1600 succeeds only if all three of circuits 1621, 1622, and 1630 succeed).2.3.2. Imbalanced Four-Way Fusion Circuits

[0146] Circuit 1600 is referred to herein as a “balanced” four- way fusion circuit, reflecting that both inputs to the final type II fusion are intermediate pseudo-qubits. According to some embodiments, it may be useful to construct an “imbalanced” four-way fusion circuit, in which type I fusions are performed in a cascading, or chained, manner, such that the first type I fusion operation is performed on two of the input qubits, and the second type I fusion operation is performed on an intermediate pseudo-qubit output from the first type I fusion operation and a third one of the input qubits. The intermediate pseudo-qubit output from the second type I fusion operation is input to a type II fusion circuit, along with the fourth input qubit. Such circuits are “imbalanced” in the sense that one input to the final type II fusion is one of the input qubits while the other is an intermediate pseudo-qubit.

[0147] FIG. 17 shows a simplified schematic diagram of an imbalanced four- way fusion circuit 1700 according to some embodiments. Four-way fusion circuit 1700 receives four dual-rail-encoded input qubits QI, Q2, Q3, Q4, each of which can initially be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of four-way fusion circuit 1700 results in the four entangled systems being entangled with each other to form one output entangled system.

[0148] Input qubit QI is encoded on a first pair of waveguides 1701, 1702 such that the logical-0 rail maps to waveguide 1701 and the logical- 1 rail maps to waveguide 1702.Similarly, input qubit Q2 is encoded on a second pair of waveguides 1703, 1704 such that the logical-0 rail maps to waveguide 1703 and the logical- 1 rail maps to waveguide 1704; input qubit Q3 is encoded on a third pair of waveguides 1705, 1706 such that the logical-0 rail maps to waveguide 1705 and the logical- 1 rail maps to waveguide 1706; and input qubit Q4 is encoded on a fourth pair of waveguides 1707, 1708 such that the logical-0 rail maps to waveguide 1707 and the logical-1 rail maps to waveguide 1708. Input qubits QI and Q2 areinput to a first type I fusion circuit 1721. In the example shown, type I fusion circuit 1721 is implemented as an instance of type I vl fusion circuit 1300 described above. Thus, waveguides 1701 and 1704 are mapped to the rails of a first intermediate pseudo-qubit Qol as described above with reference to FIG. 13 A, while any photons on waveguides 1702 and 1703 are detected and counted by detectors 1725, 1726. Detectors 1725, 1726 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 1752.

[0149] First intermediate pseudo-qubit Qol and input qubit Q3 are input to a second type I fusion circuit 1722. In the example shown, type I fusion circuit 1722 is implemented as another instance of type I vl fusion circuit 1300 described above. Thus, waveguides 1701 and 1706 are mapped to the rails of a second intermediate pseudo-qubit Qo2 as described above with reference to FIG. 13 A, while any photons on waveguides 1704 and 1705 are detected and counted by detectors 1727, 1728. Detectors 1727, 1728 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 1753.

[0150] Second intermediate pseudo-qubit Qo2 and input qubit Q4 are input to a type II fusion circuit 1730. In the example shown, type II fusion circuit 1730 is implemented as an instance of type II fusion circuit 1400 described above, and photons on waveguides 1701, 1708, 1707, and 1706 are detected by detectors 1731, 1732, 1733, 1734 respectively.Detectors 1731-1734 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 1754. Components of circuit 1700 can be arranged such that all detectors 1725-1728 and 1731-1734 operate concurrently (or within the same time bin).

[0151] Classical decision logic circuit 1750 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 1750 can receive the respective photon count signals from detectors 1725-1728 and 1731-1734, e.g., via signal paths 1752-1754, and can determine whether the pattern of photon counts corresponds to a success pattern. As noted above, all signals from detectors 1725-1728 and 1731-1734 are received concurrently (in the same time bin), providing a detection pattern having eight elements (one photon count from each detector). Success patterns can correspond to conditions where: (1) one (and only one) ofdetectors 1725 and 1726 detects a single photon; (2) one (and only one) of detectors 1727 and 1728 detects a single photon; (3) one (and only one) of detectors 1731 and 1732 detects a single photon; and (4) one (and only one) of detectors 1733 and 1734 detects a single photon. As described above, classical decision logic circuit 1750 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals are received via signal paths 1752-1754, classical decision logic circuit 1750 can compare the readout pattern of signals received from the detectors to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 1750 can output a classical decision signal 1756 indicating whether a match has occurred. In some embodiments, classical decision logic circuit 1750 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include QI, Q2, Q3, and Q4, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0152] Success patterns for circuit 1700 correspond to patterns indicating that each of type I fusion circuit 1721, type I fusion circuit 1722, and type II fusion circuit 1730 succeeded. As described above, a success pattern for a type I fusion circuit corresponds to one photon in one of the pair of detectors and zero photons in the other, and a success pattern for a type II fusion circuit corresponds to one photon in one detector of each pair of detectors coupled to the same 50 / 50 beam splitter and zero in the other detector of each pair. A complete set of success patterns for circuit 1700 can be derived from these principles. If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 1700 is 12.5% (because type I fusion circuits 1721 and 1722 and type II fusion circuit 1730 each have a 50% success probability, and circuit 1700 succeeds only if all three of circuits 1721, 1722, and 1730 succeed). Thus, balanced four-way fusion circuit 1600 and imbalanced fourway fusion circuit 1700 have the same probability of success. However, according to some embodiments, imbalanced four- way fusion circuit 1700 can provide enhanced robustness against errors, particularly where techniques described below are implemented.2.3.3. Multi-way Fusion Circuits with Enhanced Error Robustness

[0153] In examples described herein, success patterns can be defined based on the assumption of an ideal circuit. “Failure patterns” can be defined as any detection pattern that is not a success pattern. In a successful multi-way fusion operation, a circuit such as circuit1500, 1600, or 1700 projects the input qubits onto an / / -GHZ state (e.g., 3-GHZ state for circuit 1500, 4-GHZ state for circuit 1600 or 1700).

[0154] It is expected that in a real circuit occasional errors will occur. Such errors can change the pattern of photon counts received by the classical decision logic and can, in some instances, turn a failure pattern into a success pattern, resulting in a false positive signal output from the classical decision logic. False positives can lead to downstream errors if the resulting state of the qubits from the original quantum system is incorrectly identified. For example, physical errors on the inputs to a multi-way fusion circuit can result in the classical decision logic detecting a success pattern when the input qubits are in fact projected onto a state that is a Pauli rotation away from the expected / / -GHZ state. One type of physical error that can have this effect is distinguishability, where one qubit at the input is distinguishable from the others. Another type is a combination of a “multiphoton” qubit (where the waveguides mapped to a particular qubit propagate a total of two photons instead of one) with photon loss (where the waveguides mapped to a particular qubit propagate a total of zero photons).

[0155] A two-photon multiphoton state consists of a combination of 111), 120), and 102) terms. The wavefunction for such a state can be expressed as:|t / / ) = Pn|ll) + p20|20) + p02|02) , (17) whereIPn l2+ IP20 I2+ IP02 I2= 1 ■ (18) More generally, the density operator for such a state can be expressed as:P = Pn.nl 11X111 + n,2olllX2O| + p11,02|llX02| +wherePll.ll + P20.20 + P02,02 — 1 ■ (20)When one of these terms is input to a type I fusion circuit, the outcome depends on whether the circuit is variation 1 (e.g., circuit 1300 of FIG. 13A) or variation 2 (e.g., circuit 1300') of FIG. 13B.

[0156] FIG. 18A shows diagrams 1801-1806 illustrating behavior of a type I vl fusion circuit (shown in diagrams 1801-1803) and a Type 2 v2 fusion circuit (shown in diagrams 1804-1806) for instances where Q2 is in a logically valid single-photon state (having a combination of 110) and 101) terms) and QI is in a two-photon multiphoton state (having a combination of 111), 120), and 102) terms). Diagrams 1801 and 1804 illustrate the outcomes for a 111) term. In this case, both variations detect a success pattern (indicated by the check mark next to the detectors), and the output qubit is in a logically valid state. Diagrams 1802 and 1805 illustrate the outcomes for a 120) term. In this case, a type I vl fusion circuit (diagram 1802) detects a success pattern although the output qubit is a multiphoton 120) state, while a type I v2 fusion circuit (diagram 1805) detects a failure pattern (indicated by the X next to the detectors). Diagrams 1803 and 1806 illustrate the outcomes for a 102) term. This case is the reverse of the 120) term: a type I v2 fusion circuit (diagram 1806) detects a success pattern although the output qubit is a multiphoton 102) state, while a type I vl fusion circuit (diagram 1803) detects a failure pattern.

[0157] In other words, a type I vl fusion circuit and a type I v2 fusion circuit each detect failure deterministically for one of the “bunched” terms (| 20) or 102), where both photons are in the same mode) and yield a false positive for the other. However, the behavior of the two variations is opposite: variation 1 yields a false positive where variation 2 deterministically fails, and vice versa. According to some embodiments, this property can be exploited to remove two-photon bunched terms in a multi-way fusion circuit by cascading (or chaining) different variations of type I fusion circuits.

[0158] FIG. 18B shows a diagram 1850 illustrating the effect of cascading (or chaining) a type I vl fusion circuit 1852 and a type I v2 fusion circuit 1854 according to some embodiments. Type I vl fusion circuit 1852 fails deterministically for a 02) bunched term, allowing the multiphoton state to be rejected, and type I v2 fusion circuit 1854 fails deterministically for a 120) bunched term, also allowing the multiphoton state to be rejected. It should be noted that, because type I vl fusion circuits and type I v2 fusion circuits are related by a swap of the input qubits, the same effect can be achieved using type I fusion circuits of a single variation. For instance, routing the output of one type I vl fusion circuitinto the second (Q2 in FIG. 13A) input of another type I vl fusion circuit would have the same effect of rejecting any 102) bunched term and any 120) bunched term, as would routing the output of one type I v2 fusion circuit into the first (QI in FIG. 13B) input of another type I v2 fusion circuit.

[0159] A non-bunched two-photon term 111) would not result in a failure pattern in either of type I fusion circuits 1852 or 1854. However, this term can be removed by introducing a basis rotation prior to the final type II fusion. The basis rotation can deterministically convert the 111) term to a superposition of bunched 102) and 120) terms. In some embodiments, the basis rotation can be implemented using 50 / 50 beam splitters. Various basis rotations can be applied, provided that the projection implemented by the fusion operation in the ideal case remains a GHZ projection (and provided that the effect on the 111) term is as described). If a bunched two-photon term combines with a vacuum term (| 00)) as the inputs of a type II fusion circuit, the type II fusion circuit fails deterministically. Thus, a multi-way fusion circuit can be configured to increase the probability that combinations of a multiphoton state at the input and photon loss errors produce a failure pattern.

[0160] FIG. 19 shows a simplified schematic diagram of an imbalanced four- way fusion circuit 1900 according to some embodiments. In many respects, circuit 1900 can be similar to circuit 1700 described above. Like circuit 1700, circuit 1900 receives four dual-rail-encoded input qubits QI, Q2, Q3, Q4, each of which can be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of four- way fusion circuit 1900 results in the four entangled systems being entangled with each other to form one larger entangled system. However, circuit 1900 incorporates enhanced error robustness features that are not shown in circuit 1700.

[0161] Input qubits QI and Q2 are input to a type I vl fusion circuit 1921 that produces a first intermediate pseudo-qubit Qol. Type I vl fusion circuit 1921 can be identical to type I fusion circuit 1721 described above. First intermediate pseudo-qubit Qol and third input qubit Q3 are input to a type I v2 fusion circuit 1922 that produces a second intermediate pseudo-qubit Qo2. As described above with reference to FIG. 18B, this cascade, or chain, of type I fusion circuits deterministically fails if Q4 is a vacuum state (photon loss) and QI is in multiphoton state. If Q2 or Q3 is a multiphoton state and Q4 is a vacuum state, the probability of success is reduced as described below. It should be understood that otherarrangements can be used. For instance, the order of the type I vl and type I v2 fusion circuits can be reversed.

[0162] Second intermediate pseudo-qubit Qo2 passes through a basis rotation circuit 1925, and fourth input qubit Q4 passes through a basis rotation circuit 1926. Basis rotation circuits 1925 and 1926 can each be implemented using a 50 / 50 beam splitter or other passive optical components that apply a desired basis rotation. The basis-rotated qubits are then provided as inputs to a final type II fusion circuit 1930.

[0163] Although a classical decision logic circuit is not shown in FIG. 19, it should be understood that such a circuit can be provided and can operate in the same manner as classical decision logic circuit 1750 described above. The success and failure patterns can be the same as for circuit 1700, and circuit 1900 can have the same success probability as circuit 1700 in the ideal case. However, due to the alternating configuration of the type I fusion circuits and the presence of basis rotation circuits 1925, 1926, circuit 1900 provides enhanced robustness against multiphoton errors. In particular, the combination where QI is a multiphoton state and Q4 is lost fails deterministically. Other combinations of input multiphoton and loss errors may not fail deterministically; however, the success probability may be reduced. For instance, if intermediate pseudo-qubit Qo2 is in a statewhere \b) is one of the bunched terms 102) or 120), basis rotation circuits 1925, 1926 convert Qo2 to a state where the non-bunched 111) term has probability Pi / 2. If pb' < p1', then basis rotation circuits 1925, 1926 can reduce the probability of a false positive when input qubits Q2 and Q3 are multiphoton states. Accordingly, circuit 1900 can produce fewer false positives than circuit 1700.

[0164] In FIG. 19, basis rotation components 1925, 1926 are shown as being located at the inputs to final type II fusion circuit 1930. It should be understood that basis rotation components can be placed elsewhere in the circuit and can improve performance. For instance, optimal placement of basis rotation components may depend on where multiphoton or other errors create issues, which can be determined during optical circuit design.2.4 V-W ay Fusion Circuits

[0165] According to some embodiments, four-way fusion circuits of the kind described above can be generalized to TV-way fusion circuits for any number (N) of qubits (A > 4) by providing cascaded type I fusion circuits to reduce the number of qubits to 2 and a final type II fusion circuit to operate on the remaining two qubits. Either a balanced or imbalanced approach can be used to implement the cascading type I fusion circuits. Examples will now be described.2.4.1. Balanced A-Way Fusion Circuits

[0166] In one approach to a balanced A-way fusion circuit, starting from a final type II fusion, each input can be treated as a pseudo-qubit output of a preceding type I fusion in the cascade, with additional stages with one or more type I fusion circuits added until the total number N of input qubits is accounted for. This approach is referred to herein as a “back-to-fronf ’ approach. Illustrating the back-to-front approach, FIG. 20A shows an example of a balanced 5-way fusion circuit 2000 according to some embodiments. Circuit 2000 receives a set of five dual -rail-encoded input qubits Q1-Q5, each of which can be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of circuit 2000 results in the five entangled systems being entangled with each other to form a single entangled system. It should be understood that, although FIG. 20A and following schematic diagrams herein are shown as including distinct blocks labeled as type I and type II fusion circuits for clarity of illustration, in operation all detectors in all of these blocks perform their detection operations within the same detection period (or time bin) and output data from all detectors is concurrently sent to the classical decision logic. For instance, the length of each optical path shown can be selected so that all photons arrive at the detectors simultaneously (meaning within a single detection period or time bin). Further, while type I and type II fusion operations are described herein as discrete operations, it should be understood that operation of the circuit is not serialized; a single detection pattern across all detectors determines success or failure of the TV-way fusion operation.

[0167] Circuit 2000 is constructed following the back-to-front approach outlined above. At the final stage (s = 3), two input pseudo-qubits are provided to a final type II fusion circuit 2030 that consumes both inputs. Each input to final type II fusion circuit 2030 is the output of a type I fusion circuit 2020-1, 2020-2 at the preceding stage (s = 2), each of which has two inputs. Thus, stage s = 2 consumes four inputs. In the case where N = 5, only one more input qubit is needed, so a preceding stage (s = 1) that includes a single type I fusioncircuit 2021 is provided. If desired, basis rotation circuits 2025, 2026 (which can be similar or identical to basis rotation circuits 1925, 1926 described above) can be provided to apply basis rotation to the input qubits of type II fusion circuit 2030. In this example, the type I fusion circuits in successive stages alternate between variations vl and v2. As described above, this may provide desirable filtering of certain multiphoton states. However, if desired, all type I fusion circuits can be of the same variation. Basis rotations are also optional.

[0168] Classical decision logic circuit 2050 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2050 can receive the respective photon count signals from detectors in each of type I fusion circuits 2021, 2020-1, 2020-2, and type II fusion circuit 2030 and can determine whether the pattern of photon counts corresponds to a success pattern. As noted above, all signals from all detectors are received concurrently (in the same time bin), providing a detection pattern having ten elements (one photon count from each detector). Success patterns can correspond to conditions where each of the type I fusions and the final type II fusion succeeded, which can be defined as described above. As described above, classical decision logic circuit 2050 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern and can determine success or failure by comparing a received pattern to the stored success patterns. In some embodiments, classical decision logic circuit 2050 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI through Q5, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0169] In an alternative approach to a balanced TV-way fusion circuit (referred to herein as a “front-to-back” approach), as many qubits as possible are input to type I fusions at each stage of the cascade. If, at a given stage (s) of the cascade, the number (Ns) of qubits is even, then a number Ns / 2 of type I fusion circuits can be provided at that stage, with each type I fusion circuit receiving two of the Nsqubits and producing one intermediate pseudo-qubit so that the next stage (s + 1) has TVs+1= Ns / 2 qubits. If Nsis odd, then a number (Ns— 1) / 2 of type I fusion circuits can be provided, with one qubit being carried forward to the next stage, so that the next stage (s + 1) has TVs+1= ((TVs-1) / 2) + 1 qubits. The same principle can be applied at each stage until the number of qubits is reduced to 2, and a final type II fusion can be performed on the remaining two qubits. For instance, in balanced 4-way fusion circuit 1600of FIG. 16, for the first stage (s = 1), the number of qubits is Nr= 4, which is even.Accordingly, the first stage includes N± / 2 = 2 type I fusion circuits 1621, 1622. At the second stage (s = 2), the number of qubits is N2= 2, and type II fusion circuit 1630 consumes the final two qubits.

[0170] Further illustrating the front-to-back approach, FIG. 20B shows an example of a balanced 5-way fusion circuit 2051 according to some embodiments. Circuit 2051 receives a set of five dual -rail-encoded input qubits Q1-Q5, each of which can be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of circuit 2051 results in the five entangled systems being entangled with each other to form a single entangled system.

[0171] Circuit 2051 is constructed following the front-to-back approach outlined above. At the first stage (s = 1), the number of qubits is N = 5, which is odd. Accordingly, the first stage includes (Ax— l) / 2 = 2 type I fusion circuits 2060-1, 2060-2. At the second stage (s = 2), the number of qubits is N2= 3. Accordingly, the second stage includes(1V2— l) / 2 = 1 type I fusion circuit 2061. At the third stage (s = 3), a final type II fusion circuit 2070 consumes both qubits. If desired, basis rotation circuits 2065, 2066 (which can be similar or identical to basis rotation circuits 1925, 1926 described above) can be provided to apply basis rotation to the input qubits of type II fusion circuit 2070. In this example, the type I fusion circuits in successive stages alternate between variations vl and v2. As described above, this may provide desirable filtering of certain multiphoton states. However, if desired, all type I fusion circuits can be of the same variation. Basis rotations are also optional.

[0172] Classical decision logic circuit 2080 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2090 can receive the respective photon count signals from detectors in each of type I fusion circuits 2060-1, 2060-2, 2061, and type II fusion circuit 2070 and can determine whether the pattern of photon counts corresponds to a success pattern. As noted above, all signals from all detectors are received concurrently (in the same time bin), providing a detection pattern having ten elements (one photon count from each detector). Success patterns can correspond to conditions where each of the type I fusions and the final type II fusion succeeded, which can be defined as described above. As describedabove, classical decision logic circuit 2080 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern and can determine success or failure by comparing a received pattern to the stored success patterns. In some embodiments, classical decision logic circuit 2080 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI through Q5, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0173] The two approaches illustrated in FIGs. 20A and 20B can produce equivalent circuits in the ideal case, although different constructions may have different performance in the presence of errors.

[0174] FIG. 21 A shows an example of a balanced 6-way fusion circuit 2100 according to some embodiments. Circuit 2100 is constructed according to the back-to-front approach, similarly to circuit 2000 of FIG. 20 A. Circuit 2100 receives a set of six dual -rail-encoded input qubits Q1-Q6, each of which can be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of circuit 2100 results in the six entangled systems being entangled with each other to form one larger entangled system. Circuit 2100 is constructed following the “working-backward” approach outlined above. At the final stage (s = 3), two input pseudo-qubits are provided to a final type II fusion circuit 2130 that consumes both inputs. Each input to final type II fusion circuit 2130 is the output of a type I fusion circuit 2120-1, 2020-2 at the preceding stage (s = 2), each of which has two inputs. Thus, stage s = 2 consumes four inputs. In the case where N = 6, two more input qubit is needed, so a preceding stage (s = 1) that includes two type I fusion circuits 2121-1, 2121-2 is provided. In this example, the inputs to type I fusion circuits 2120-1, 2120-2 at stage s = 2 are balanced, in that each of type I fusion circuits 2120-1, 2120-2 receives one of the input qubits and an intermediate pseudo-qubit from a type I fusion circuit 2121-1 or 2121-2 at a preceding stage. If desired, basis rotation circuits 2125, 2126 (which can be similar or identical to basis rotation circuits 1925, 1926 described above) can be provided to apply basis rotation to the input qubits of type II fusion circuit 2130. In this example, the type I fusion circuits in successive stages alternate between variations vl and v2. As described above, this may provide desirable filtering of certain multiphoton states. However, if desired, all type I fusion circuits can be of the same variation. Basis rotations are also optional.

[0175] Classical decision logic circuit 2150 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2150 can receive the respective photon count signals from detectors in each of type I fusion circuits 2120-1, 2120-2, 2121-1, 2121-2, and type II fusion circuit 2130 and can determine whether the pattern of photon counts corresponds to a success pattern. As noted above, all signals from all detectors are received concurrently (in the same time bin), providing a detection pattern having twelve elements (one photon count from each detector). Success patterns can correspond to conditions where each of the type I fusions and the final type II fusion succeeded, which can be defined as described above. As described above, classical decision logic circuit 2150 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern and can determine success or failure by comparing a received pattern to the stored success patterns. In some embodiments, classical decision logic circuit 2150 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI through Q6, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0176] FIG. 2 IB shows an example of a balanced 6-way fusion circuit 2151 according to some embodiments. Circuit 2151 receives a set of six dual-rail-encoded input qubits Q1-Q6, each of which can be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of circuit 2151 results in the six entangled systems being entangled with each other to form one larger entangled system.

[0177] Circuit 2151 is constructed following the front-to-back approach, similarly to circuit 2051 described above. At the first stage (s = 1), the number of qubits is N = 6, which is even. Accordingly, the first stage includes (Ax) / 2 = 3 type I fusion circuits 2160-1, 2160-2, 2160-3. At the second stage (s = 2), the number of qubits is N2= 3. Accordingly, the second stage includes (JV2— l) / 2 = 1 type I fusion circuit 2161, and the intermediate pseudo-qubit output from type I fusion circuit 2160-3 is propagated to the next stage. At the third stage (s = 3), a final type II fusion circuit 2170 consumes both qubits. If desired, basis rotation circuits 2165, 2166 (which can be similar or identical to basis rotation circuits 1925, 1926 described above) can be provided to apply a basis rotation to the input qubits of type II fusion circuit 2130. In this example, the type I fusion circuits in successive stages alternatebetween variations vl and v2. As described above, this may provide desirable filtering of certain multiphoton states. However, if desired, all type I fusion circuits can be of the same variation. Basis rotations are also optional.

[0178] Classical decision logic circuit 2180 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2180 can receive the respective photon count signals from detectors in each of type I fusion circuits 2160-1, 2160-2, 2160-3, 2161, and type II fusion circuit 2170 and can determine whether the pattern of photon counts corresponds to a success pattern. As noted above, all signals from all detectors are received concurrently (in the same time bin), providing a detection pattern having twelve elements (one photon count from each detector). Success patterns can correspond to conditions where each of the type I fusions and the final type II fusion succeeded, which can be defined as described above. As described above, classical decision logic circuit 2180 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern and can determine success or failure by comparing a received pattern to the stored success patterns. In some embodiments, classical decision logic circuit 2180 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI through Q6, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0179] FIG. 22 shows an example of a balanced 7-way fusion circuit 2200 according to some embodiments. Circuit 2200 receives a set of seven dual-rail-encoded input qubits Ql-Q7, each of which can be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of circuit 2200 results in the seven entangled systems being entangled with each other to form one larger entangled system.

[0180] Circuit 2200 can be constructed using either the front-to-back or back-to-front approach The first stage (s = 1) includes three type I fusion circuits 2220-1, 2220-2, 2220-3, and input qubit Q7 is propagated to the next stage, e second stage includes two type I fusion circuits 2221-1, 2221-2. At the third stage (s = 3), a final type II fusion circuit 2230 consumes the qubits output from type I fusion circuit 2221-1, 2221-2. If desired, basis rotation circuits 2225, 2226 (which can be similar or identical to basis rotation circuits 1925,1926 described above) can be provided to apply basis rotation to the input qubits of type II fusion circuit 2230. In this example, the type I fusion circuits in successive stages alternate between variations vl and v2. As described above, this may provide desirable filtering of certain multiphoton states. Basis rotations are also optional.

[0181] Classical decision logic circuit 2250 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2250 can receive the respective photon count signals from detectors in each of type I fusion circuits 2220-1 through 2220-3, 2221-1, 2221-2, and type II fusion circuit 2230 and can determine whether the pattern of photon counts corresponds to a success pattern. As noted above, all signals from all detectors are received concurrently (in the same time bin), providing a detection pattern having fourteen elements (one photon count from each detector). Success patterns can correspond to conditions where each of the type I fusions and the final type II fusion succeeded, which can be defined as described above. As described above, classical decision logic circuit 2250 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern and can determine success or failure by comparing a received pattern to the stored success patterns. In some embodiments, classical decision logic circuit 2250 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI through Q7, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0182] Circuits 2000, 2050, 2100, 2151, and 2200 illustrate balanced cascading structures that can be extended to any number of stages depending on the number of input qubits. If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for TV-way balanced fusion circuits such as circuits 2000, 2100, or 2200 is 1 / 2W-1(because the type I fusion circuits and the final type II fusion circuit each have a 50% success probability, and the A- way fusion circuit succeeds only if all TV— 1 of the type I and type II fusion circuits succeed). It should be understood that these circuits can be modified. Further, as noted above, although schematic diagrams herein are shown with detectors or fusion circuits in a particular arrangement for clarity of illustration and fusion operations are described sequentially, in operation all detectors in an TV-way fusion circuit perform theirdetection operations within the same detection period (or time bin) and all output data is concurrently sent to the classical decision logic. Accordingly, success or failure of an A-way fusion circuit can be determined based on a single detection pattern across all detectors in the circuit, regardless of the particular value of N. Basis rotation components can be omitted or can be placed elsewhere in the circuit as desired.2.4.2. Imbalanced A-Way Fusion Circuits

[0183] In an imbalanced A-way fusion circuit, a number N — 2 of type I fusion circuits can be coupled in a chain or cascade. The first type I fusion circuit in the chain can receive two of the input qubits, and each subsequent type I fusion circuit can receive a different one of the input qubits and the intermediate pseudo-qubit output from the preceding type I fusion circuit. The output of the last type I fusion circuit and the last remaining input qubit can be input to a final type II fusion circuit. For instance, in imbalanced 4-way fusion circuit 1700 of FIG. 17, the chain of type I fusion circuits includes N — 2 = 2 type I fusion circuits 1721, 1722. The first type I fusion circuit 1721 receives two of the input qubits (QI and Q2), and the second type I fusion circuit 1722 receives a different input qubit (Q3) and the intermediate pseudo-qubit output from first type I fusion circuit 1721. At this point, one input qubit (Q4) remains, and type II fusion circuit 1730 consumes qubit Q4 and the output of the last type I fusion circuit 1722.

[0184] Further illustrating this approach, FIG. 23 shows a simplified schematic diagram of an imbalanced A- way fusion circuit 2300 according to some embodiments. A- way fusion circuit 2300 receives a set of A dual-rail-encoded input qubits QI, Q2, Q3, Q4, ... QA, each of which can be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of A-way fusion circuit 2300 results in the A entangled systems being entangled with each other to form one larger entangled system.

[0185] Circuit 2300 is constructed analogously to circuit 1900. In particular, circuit 2300 includes a cascade of type I fusion circuits 2320-1 through 2320-(A~2). The first type I fusion circuit 2320-1 receives two input qubits QI, Q2. Each other type I fusion circuit 2320-i (2 < i < A — 2) receives intermediate pseudo-qubit Qo(z'-l) from type I fusion circuit 2320-(z-l) and the next input qubit Q(z+1). In this example, the type I fusion circuits 2320-1 through 2320-(A-2) alternate between variations vl or v2: if type I fusion circuit 2320-z is variation vl, then type I fusion circuit 2320-(i+l) is variation v2 and vice versa. Intermediatepseudo-qubit Qo(A~2) from the last type I fusion circuit 2320-(A~2) passes through basis rotation circuit 2325 (which can be similar or identical to basis rotation circuit 1925 of FIG.19), and last input qubit QA passes through basis rotation circuit 2327 (which can be similar or identical to basis rotation circuit 1927 of FIG. 19). A final type II fusion circuit 2330 receives basis-rotated qubits Qo(A~2) and QA and performs type II fusion, similarly to type II fusion circuit 1730 described above

[0186] Classical decision logic circuit 2350 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2350 can receive the respective photon count signals from detectors in each of type I fusion circuits 2320 and type II fusion circuit 2330 and can determine whether the pattern of photon counts corresponds to a success pattern. As noted above, all signals from all detectors are received concurrently (in the same time bin), providing a detection pattern having IN elements (one photon count from each detector). Success patterns can correspond to conditions where each of the type I fusions and the final type II fusion succeeded. As described above, classical decision logic circuit 2350 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern and can determine success or failure by comparing a received pattern to the stored success patterns. In some embodiments, classical decision logic circuit 2350 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI through QA, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0187] If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 2300 is 1 / 2W-1(because type I fusion circuits 2320-1 through 2320-(A~2) and type II fusion circuit 2330 each have a 50% success probability, and circuit 2300 succeeds only if all A-l of the fusion circuits succeed).

[0188] In embodiments where all type I fusions are the same version, if multiphoton and loss errors occur, any combination of a multiphoton state in one of Q2 through Q(A~2) and a loss in QA causes circuit 2300 to fail deterministically. Thus, A — 3 possible multiphotonplus-loss configurations are fully filtered out. Partial filtering (reduced success probability asdescribed above) occurs for a multiphoton state in either QI or Q(TV-1) and a loss in QTV, analogously to the effect described above with reference to circuit 1900. Alternatively, if different type I fusion circuit versions are combined, then any combination of a multiphoton state in one of Q2 through Q(TV~3) and a loss in QA cause circuit 2300 to fail deterministically, and partial filtering occurs for a multiphoton state in either QI or Q(V~2) and a loss in QA. In either case, it is noted that the total number of configurations of multiphoton-plus-loss that can produce Pauli-rotated GHZ states is N2(including instances where multiphoton and loss errors occur on the same qubit; if only instances where the errors occur on different qubits are considered, the total would be N N — 1)). Circuit 2300 can deterministically remove N — 3 of these configurations and partially remove (via reduced success probability) another two. Accordingly, the percentage of Pauli errors filtered out by circuit 2300 increases with increasing N for 3 < N < 6, then starts decreasing.

[0189] It should be understood that circuit 2300 can be modified. Further, as noted above, although schematic diagrams herein are shown with detectors or fusion circuits in a particular arrangement for clarity of illustration and fusion operations are described sequentially, in operation all detectors in an TV-way fusion circuit perform their detection operations within the same detection period (or time bin) and all output data is concurrently sent to the classical decision logic. Accordingly, success or failure of an TV-way fusion circuit can be determined based on a single detection pattern across all detectors in the circuit, regardless of the particular value of N. Basis rotation components can be omitted or can be placed elsewhere in the circuit as desired.2.4.3. Projection onto non-GHZ states

[0190] As described above, the input qubits to a multi-way fusion circuit are assumed to be qubits from separately-entangled systems of qubits, and success of the multi-way fusion circuit results in creating an entangled state among all of the entangled systems that supplied input qubits to the multi-way fusion circuit. The particular entangled state that is created depends on the particular projective measurement. In examples described above, success of the multi-way (or TV-way) fusion circuit corresponds to a projective measurement onto an N-qubit GHZ state.

[0191] Projective measurements onto different TV-qubit states can yield different entanglement patterns or geometries in the resulting entangled system. Accordingly, depending on the particular entanglement geometry that is to be generated, it may bedesirable to project onto a different A-qubit entangled state. According to some embodiments, multi-way fusion circuits of the kind described above can be modified to perform projection onto different A-qubit entangled states by adding additional basis rotation circuits (similar to basis rotation circuits 1925, 1926) on the input paths of one or more of the type I fusion circuits. The particular number, combination, and location of basis rotation circuits can be determined based on the particular projective measurement that is desired.3. N-Way Fusion using Hadamard Interferometer Circuits

[0192] As noted above, A- way fusion can correspond to projection onto an A-qubit state. Such projections can be achieved using a combination of type I and type II fusion circuits, e.g., as described above. In certain cases where the desired projection is onto an / / -GHZ state (for n > 4), an alternative approach uses Hadamard interferometers rather than type I or type II fusion.3.1. 4-Way Fusion Using 4-way Hadamard Interferometer

[0193] FIG. 24 shows a simplified schematic diagram of a Hadamard interferometer circuit 2400 that can be used in some embodiments. Hadamard interferometer circuit 2400 receives four inputs on waveguides 2401-2404 and produces four outputs. Hadamard interferometer circuit 2400 includes four beam splitter regions 2411-2414, each of which implements a 50 / 50 beam splitter, e.g., in the manner described above with reference to FIG. 2B. Beam splitter region 2411 couples waveguides 2401 and 2402 while beam splitter region 2412 couples waveguides 2403 and 2404. Downstream of beam splitter regions 2411 and 2412, beam splitter region 2413 couples waveguides 2401 and 2403 while beam splitter region 2414 couples waveguides 2402 and 2404. To bring waveguides 2401 and 2403 into proximity in region 2413 and to bring waveguides 2402 and 2404 into proximity in region 2414, a mode crossing 2421 can be provided. Mode crossing 2421 can be implemented, e.g., using techniques described above, and Hadamard interferometer circuit 2400 can be implemented in the manner described above with reference to FIG. 6. Circuit symbol 2450 can be used to represent Hadamard interferometer circuit 2400.

[0194] FIG. 25 shows a simplified schematic diagram of a Hadamard-based four-way fusion circuit 2500 according to some embodiments. Four-way fusion circuit 2500 receives four dual-rail-encoded input qubits QI, Q2, Q3, Q4, each of which can initially be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangledsystems are not shown.) Success of four-way fusion circuit 2500 results in the four entangled systems being entangled with each other to form one entangled system.

[0195] Input qubit QI is encoded on a first pair of waveguides 2501, 2502 such that the logical-0 rail maps to waveguide 2501 and the logical- 1 rail maps to waveguide 2502.Similarly, input qubit Q2 is encoded on a second pair of waveguides 2503, 2504 such that the logical-0 rail maps to waveguide 2503 and the logical- 1 rail maps to waveguide 2504; input qubit Q3 is encoded on a third pair of waveguides 2505, 2506 such that the logical-0 rail maps to waveguide 2505 and the logical- 1 rail maps to waveguide 2506; and input qubit Q4 is encoded on a fourth pair of waveguides 2507, 2508 such that the logical-0 rail maps to waveguide 2507 and the logical-1 rail maps to waveguide 2508.

[0196] A first Hadamard interferometer circuit 2511, which can be an instance of circuit 2400, is coupled to waveguides 2501, 2503, 2505, and 2507 (or the logical-0 rails of the four input qubits). A second Hadamard interferometer circuit 2512, which can be another instance of circuit 2400, is coupled to waveguides 2502, 2504, 2506, and 2508 (or the logical-1 rails of the four input qubits). Detectors 2521-2524 are coupled to the outputs of first Hadamard interferometer circuit 2511, and detectors 2525-25828 are coupled to the outputs of second Hadamard interferometer circuit 2512. Detectors 2521-2528 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 2554.

[0197] Classical decision logic circuit 2550 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2550 can receive the respective photon count signals from detectors 2521-2528, e.g., via signal path 2554, and can determine whether the pattern of photon counts corresponds to a success pattern. Success patterns for circuit 2500 correspond to a total of two photons detected by detectors 2521-2524 and a total of two photons detected by detectors 2525-2528, in either: (1) an anti-bunched pattern with two detectors each having one photon (e.g., pattern (1, 1, 0, 0) or (01, 0, 1)) for each group of detectors 2521-2524 and 2525-2528, where the patterns for the two groups are either the same or complementary (e.g., (1, 1, 0, 0) for one group and (0, 0, 1, 1) for the other group, or other patterns where a “1” in one group is “0” in the corresponding position in the other group and vice versa); or (2) an anti -bunched pattern for one group of detectors 2521-2524 or 2525-2528 and a bunchedpattern where both photons are in the same detector (e.g., pattern (2, 0, 0, 0) or (0, 2, 0, 0)) for the other group of detectors 2521-2524 or 2525-2528. As described above, classical decision logic circuit 2550 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals are received via signal path 2554, classical decision logic circuit 2550 can compare the readout pattern of signals received from detectors 2521-2528 to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 2550 can output a classical decision signal 2556 indicating whether a match has occurred. In some embodiments, classical decision logic circuit 2550 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include QI, Q2, Q3, and Q4, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0198] If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 2500 is 9 / 32, or approximately 28%. In a failure pattern in which four photons are detected at the detectors coupled to outputs of one Hadamard interferometer circuit 2511, 2512 and zero photons are detected at the detectors coupled to the output of the other Hadamard interferometer circuit 2511, 2512, the input qubits are measured in the Z basis; in the ideal case, this happens with probability 1 / 8, or 12.5%. Other failure patterns, which happen with probability 19 / 32, or about 59%, correspond to projection of the input qubits onto a non-stabilizer state (in this case not a 4-GHZ state).

[0199] Four-way fusion using Hadamard interferometers may also be well suited for implementations where the qubits are temporally encoded as described above with reference to FIG. 11 A. While temporally-encoded qubits can be converted to spatial encoding (and vice versa), such conversions generally involve active optical switches (e.g., muxes as shown in FIGs. 1 IB and 11 C), which can be lossy, and avoiding an extra active optical switch along the propagation path of the qubits may be desirable.

[0200] FIG. 26 shows a simplified schematic diagram of a Hadamard-based four-way fusion circuit 2600 for temporally encoded qubits according to some embodiments. Fourway fusion circuit 2600 receives four temporally encoded input qubits QI, Q2, Q3, Q4, each of which can initially be part of a different entangled system of qubits. (For simplicity ofillustration, the initial entangled systems are not shown.) Success of four-way fusion circuit 2600 results in the four entangled systems being entangled with each other to form one entangled system.

[0201] Each input qubit is encoded based on presence or absence of a photon in a particular waveguide in one of two time bins, referred to for convenience as tl and t2. Thus, input qubit QI is encoded on a first waveguide 2601 such that the logical-0 state maps to a photon in time bin tl and the logical- 1 state maps to a photon in time bin t2. Similarly, input qubit Q2 is encoded on a second waveguide 2602 such that the logical-0 state maps to a photon in time bin tl and the logical-1 state maps to a photon in time bin t2; input qubit Q3 is encoded on a third waveguide 2603 such that the logical-0 state maps to a photon in time bin tl and the logical-1 state maps to a photon in time bin t2; and input qubit Q4 is encoded on a fourth waveguide 2604 such that the logical-0 state maps to a photon in time bin tl and the logical- 1 state maps to a photon in time bin t2.

[0202] A Hadamard interferometer circuit 2611 (which can be an instance of circuit 2400) is coupled to waveguides 2601-2604. Detectors 2621-2624 are coupled to the outputs of the Hadamard interferometers. Detectors 2621-2624 can output classical logic signals indicating the number of photons detected at each detector during each time bin onto classical signal path 2654.

[0203] Classical decision logic circuit 2650 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, photons in time bin tl arrive at Hadamard interferometer circuit 2611 at the same time (or close enough in time that interference can occur). Likewise, photons in time bin t2 arrive at Hadamard interferometer 2611 at the same time (or close enough in time that interference can occur). Time bins tl and t2 can be separated by any amount of time sufficient to preclude interference between photons in different time bins (and sufficient to allow detectors 2621-2624 to reset between time bins tl and t2). Classical decision logic circuit 2650 can receive the respective photon count signals from detectors 2621-2624 for time bins tl and t2, e.g., via signal path 2654. The pattern of photon counts for time bin tl can be temporarily stored as tl pattern 2661, and the pattern of photon counts for time bin t2 can be temporarily stored as t2 pattern 2662. Based on tl pattern 2661 and t2 pattern 2662, classical decision logic circuit 2650 can determine whether the pattern of photon counts corresponds to a success pattern.Thus, instead of receiving photon patterns from two groups of detectors coupled to two different Hadamard interferometer circuits as in circuit 2500, classical decision logic circuit 2650 receives photon patterns from the same group of detectors for two different time bins. The success patterns for circuit 2600 are analogous to the success patterns for circuit 2500 and correspond to a total of two photons detected by detectors 2621-2624 for each of time bins tl and t2, in either: (1) an anti-bunched pattern with two detectors each having one photon (e.g., pattern (1, 1, 0, 0) or (0 1, 0, 1)) for each of time bins tl and t2, where the two patterns are either the same or complementary (e.g., (1, 1, 0, 0) for one of time bins tl and t2 and (0, 0, 1, 1) for the other of time bins tl and t2, or other patterns where a “1” in one time bin is “0” in the corresponding position in the other time bin and vice versa); or (2) an antibunched pattern for one of time bins tl and t2 and a bunched pattern where both photons are in the same detector (e.g., pattern (2, 0, 0, 0) or (0, 2, 0, 0)) for the other of time bins tl and t2. As described above, classical decision logic circuit 2650 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals for a pair of time bins tl and t2 have been received via signal path 2654, classical decision logic circuit 2650 can compare the readout patterns of signals received as tl pattern 2661 and t2 pattern 2662 to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 2650 can output a classical decision signal 2656 indicating whether a match has occurred. In some embodiments, classical decision logic circuit 2650 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include QI, Q2, Q3, and Q4, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0204] The probability of success for circuit 2600 is the same as for circuit 2500: if all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then the probability of success is approximately 28%. Where temporally encoded qubits are used, circuit 2600 can avoid a conversion from temporal to spatial encoding. 3.2. Expansion to N-Way Fusion

[0205] According to some embodiments, Hadamard-based multi-way fusion circuits can be expanded to instances where the number of quits (N) is greater than four. FIGs. 27-31 show examples of Hadamard-based multi-way fusion circuits for various values of N.

[0206] FIG. 27 shows a simplified schematic diagram of a Hadamard-based four-way fusion circuit 2700 according to some embodiments. Circuit 2700 is similar to circuit 2500 described above, using a different combination of qubit rails as the inputs to each four-way Hadamard interferometer circuits. Four-way fusion circuit 2700 receives four dual-rail-encoded input qubits QI, Q2, Q3, Q4, each of which can initially be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of four-way fusion circuit 2700 results in the four entangled systems being entangled with each other to form one entangled system.

[0207] Input qubit QI is encoded on a first pair of waveguides 2701-0, 2701-1 such that the logical-0 rail maps to waveguide 2701-0 and the logical-1 rail maps to waveguide 2701-1. Similarly, input qubit Q2 is encoded on a second pair of waveguides 2702-0, 2702-1 such that the logical-0 rail maps to waveguide 2702-0 and the logical-1 rail maps to waveguide 2702-1; input qubit Q3 is encoded on a third pair of waveguides 2703-0, 2703-1 such that the logical-0 rail maps to waveguide 2703-0 and the logical-1 rail maps to waveguide 2703-1; and input qubit Q4 is encoded on a fourth pair of waveguides 2704-0, 2704-1 such that the logical-0 rail maps to waveguide 2704-0 and the logical-1 rail maps to waveguide 2704-1.

[0208] A first four- way Hadamard interferometer circuit 2711, which can be an instance of circuit 2400, is coupled to waveguides 2701-0, 2702-1, 2703-0, and 2704-1 (the logical-0 rails of input qubits QI and Q3 and the logical- 1 rails of input qubits Q2 and Q4). A second four- way Hadamard interferometer circuit 2712, which can be another instance of circuit 2400, is coupled to waveguides 2701-1, 2702-0, 2703-1, and 2704-0 (the logical-1 rails of input qubits QI and Q3 and the logical-0 rails of input qubits Q2 and Q4). Detectors 2720-1 through 2720-4 are coupled to the outputs of first four-way Hadamard interferometer circuit 2711, and detectors 2720-5 through 2720-8 are coupled to the outputs of second four- way Hadamard interferometer circuit 2712. Detectors 2720 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 2754.

[0209] Classical decision logic circuit 2750 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2750 can receive the respective photon count signals from the eight detectors 2720, e.g., via signal path 2754, and can determine whether the pattern of photon counts corresponds to a success pattern. As with circuit 2500, success patterns forcircuit 2700 correspond to a total of two photons detected by detectors 2720-1 through 2720-4 and a total of two photons detected by detectors 2720-5 through 2720-8, in either: (1) an anti-bunched pattern with two detectors each having one photon (e.g., pattern (1, 1, 0, 0) or (0 1, 0, 1)) for each group of detectors 2720-1 through 2720-4 and 2720-5 through 2720-8, where the patterns for the two groups are either the same or complementary (e.g., (1, 1, 0, 0) for one group and (0, 0, 1, 1) for the other group, or other patterns where a “1” in one group is “0” in the corresponding position in the other group and vice versa); or (2) an anti-bunched pattern for one group of detectors 2720-1 through 2720-4 or 2720-5 through 2720-8 and a bunched pattern where both photons are in the same detector (e.g., pattern (2, 0, 0, 0) or (0, 2, 0, 0)) for the other group of detectors 2720-1 through 2720-4 or 2720-5 through 2720-8. As described above, classical decision logic circuit 2750 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals are received via signal path 2754, classical decision logic circuit 2750 can compare the readout pattern of signals received from detectors 2720 to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 2750 can output a classical decision signal 2756 indicating whether a match has occurred. In some embodiments, classical decision logic circuit 2750 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI, Q2, Q3, and Q4, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0210] If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 2700 is 9 / 32, or approximately 28%. In a failure pattern in which four photons are detected at the detectors coupled to outputs of one Hadamard interferometer circuit 2711, 2712 and zero photons are detected at the detectors coupled to the output of the other Hadamard interferometer circuit 2711, 2712, the input qubits are measured in the Z basis; in the ideal case, this happens with probability 1 / 8, or 12.5%. Other failure patterns, which happen with probability 19 / 32, or about 59%, correspond to projection of the input qubits onto a non-stabilizer state (in this case not a 4-GHZ state).

[0211] Circuit 2700 illustrates that, for dual-rail-encoded qubits, different combinations of qubit rails can be provided to different Hadamard interferometer circuits without affecting thesuccess probability, as long as each four-way Hadamard interferometer circuit receives rails from four different qubits.

[0212] According to some embodiments, circuit 2700 can be expanded to provide N-way fusion of more than four input qubits. FIG. 28A shows a simplified schematic diagram of a Hadamard-based five-way fusion circuit 2800 according to some embodiments, and FIG. 28B shows additional details related to the decision logic for circuit 2800. Five-way fusion circuit 2800 receives five dual-rail-encoded input qubits QI, Q2, Q3, Q4, Q5 each of which can initially be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of five-way fusion circuit 2800 results in the five entangled systems being entangled with each other to form one entangled system.

[0213] Input qubit QI is encoded on a first pair of waveguides 2801-0, 2801-1 such that the logical-0 rail maps to waveguide 2801-0 and the logical- 1 rail maps to waveguide 2801-1. Similarly, input qubit Q2 is encoded on a second pair of waveguides 2802-0, 2802-1 such that the logical-0 rail maps to waveguide 2802-0 and the logical- 1 rail maps to waveguide 2802-1; input qubit Q3 is encoded on a third pair of waveguides 2803-0, 2803-1 such that the logical-0 rail maps to waveguide 2803-0 and the logical- 1 rail maps to waveguide 2803-1; input qubit Q4 is encoded on a fourth pair of waveguides 2804-0, 2804-1 such that the logical-0 rail maps to waveguide 2804-0 and the logical- 1 rail maps to waveguide 2804-1; and input qubit Q5 is encoded on a fifth pair of waveguides 2805-0, 2805-1 such that the logical-0 rail maps to waveguide 2805-0 and the logical-1 rail maps to waveguide 2805-1.

[0214] A first four- way Hadamard interferometer circuit 2811, which can be an instance of circuit 2400, is coupled to waveguides 2801-0, 2802-1, 2803-0, and 2805-1 (the logical-0 rails of input qubits QI and Q3 and the logical- 1 rails of input qubits Q2 and Q5). A second four-way Hadamard interferometer circuit 2812, which can be another instance of circuit 2400, is coupled to waveguides 2801-1, 2802-0, 2803-1, and 2804-0 (the logical-1 rails of input qubits QI and Q3 and the logical-0 rails of input qubits Q2 and Q4). A two-way Hadamard interferometer circuit 2813, which can be implemented, e.g., using a 50 / 50 beam splitter as described above, is coupled to waveguides 2804-1 and 2805-0 (the logical-1 rail of input qubit Q4 and the logical-0 rail of input qubit Q5).

[0215] Detectors 2820-1 through 2820-4 are coupled to the outputs of first Hadamard interferometer circuit 2811; detectors 2820-5 through 2820-8 are coupled to the outputs of second Hadamard interferometer circuit 2812; and detectors 2820-9 and 2820-10 are coupledto the outputs of two-way Hadamard interferometer circuit 2813. Detectors 2820 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 2854.

[0216] Classical decision logic circuit 2850 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2850 can receive the respective photon count signals from the ten detectors 2820, e.g., via signal path 2854, and can determine whether the pattern of photon counts corresponds to a success pattern. For example, as shown in FIG. 28B, the ten detectors 2820 can be logically grouped into sets. A first set 2861 includes the detectors 2820 whose rails (or waveguides) are connected to interferometer circuit 2811 (detectors 2820-1 through 2820-4 in FIG. 28 A); a second set 2862 includes the detectors whose rails (or waveguides) are connected to interferometer circuit 2812 (detectors 2820-5 through 2820-8 in FIG. 28A); and a third set 2863 includes the detectors 2820 whose rails (or waveguides) are connected to interferometer circuit 2813 (detectors 2820-9 and 2820-10 in FIG. 28A). Each set 2861, 2862, 2863 of detectors provides a detection pattern, which can be represented as an ordered tuple of photon counts from the detectors 2820-z in that set. Success patterns for circuit 2800 can correspond to instances where a total of two photons are detected by each of first set 2861 and second set 2862 and a total of one photon is detected by third set 2863. (In other words, success patterns have the property that cx+ c2+ c3+ c4= 2; c5+ c6+ c7+ c8= 2; and c9+ c10= 1.)

[0217] Additional constraints can also be applied to define success patterns. For instance, two-photon patterns for first set 2861 or second set 2862 can be classified as either bunched or anti-bunched. In a bunched pattern, one detector in the set has two photons while all other detectors in the set have zero photons (e.g., pattern (2, 0, 0, 0) or (00, 2, 0)). In an antibunched pattern, two detectors in the set have one photon each (e.g., pattern (1, 1, 0, 0) or (0 1, 0, 1)). In some embodiments, success patterns can be further defined based on the combination of bunched and anti -bunched patterns across first set 2861 and second set 2862. For example, if both sets 2861, 2862 have an anti -bunched pattern, the patterns for the two sets can be classified as “identical” if both sets have the same anti -bunched pattern (e.g., both sets producing (1, 1,0,0)) or “complementary” if a 0 in any position in one pattern corresponds to a 1 in the other and vice versa (e.g., (1, 1, 0, 0) for one set and (0, 0, 1, 1) for the other set). In some embodiments, only complementary or identical anti-bunched patterns are classifiedas success patterns. For instance, success patterns under this rule include a pattern where the first set has (1, 0, 1, 0) while the second set has (1, 0, 1, 0) and a pattern where the first set has (1, 0, 1, 0) while the second set has (0, 1, 0, 1) but not a pattern where the first set has (1, 0, 1, 0) and the second set has (1, 1, 0, 0).

[0218] As an additional example, a success pattern can be defined as including an antibunched pattern for one of the first or second set 2861, 2862 and a bunched pattern for the other of the first or second set 2861, 2862. That is, if the first set of detectors has an antibunched pattern, then the second set of detectors has a bunched pattern, or vice versa.

[0219] In some embodiments, the set of success patterns for five-way fusion circuit 2800 of FIG. 28 can be defined as including patterns that fit either of the following two categories:

[0220] Category 1: Bunched+ Anti-bunched. One of the first and second sets of detectors has a bunched 4-tuple (both photons in same detector) while the other has an antibunched 4-tuple (two photons in two different detectors). The third set has one photon in one of its two detectors. An example success pattern in this category is:((2, 0, 0, 0), (1, 1, 0, 0), (1, 0)) [Ex. Al] Additional success patterns in this category are derived by applying every combination of one or more of the following operations to the example success pattern: (a) permuting the elements of the first 4-tuple so that the “2” appears in any position; (b) permuting the elements of the second 4-tuple into any order; (c) swapping the first and second 4-tuples; and / or (d) permuting the elements of the final 2-tuple. Examples include:((0, 2, 0, 0), (1, 1, 0, 0), (1, 0)) [Ex. A2] ((0, 2, 0, 0), (1, 1, 0, 0), (0, 1)) [Ex. A3] ((1, 1, 0, 0), (0, 2, 0, 0), (0, 1)) [Ex. A4] ((0, 0, 2, 0), (0, 1, 1, 0), (1, 0)) [Ex. A5] ((0, 1, 1, 0), (0, 0, 2, 0), (1, 0)) [Ex. A6] ((0, 1, 1, 0), (0, 0, 2, 0), (0, 1)) [Ex. A7] In the above, [Ex. A2] corresponds to [Ex. Al] with the first 4-tuple permuted; [Ex. A3] corresponds to [Ex. A2] with the final 2-tuple permuted; [Ex. A4] corresponds to [Ex. A3] with a swap of the 4-tuples; [Ex. A5] corresponds to [Ex. Al] with both 4-tuples permuted;[Ex. A6] corresponds to [Ex. A5] with a swap of the 4-tuples; and [Ex. A7] corresponds to [Ex. A6] with the final 2-tuple permuted. Other success patterns in this category can be derived.

[0221] Category 2: Anti-bunched+ Anti-bunched. Each of the first and second sets of detectors has an anti-bunched 4-tuple, and the patterns in the two 4-tuples are either identical (each position in both 4-tuples has the same value) or complementary (in each position, if one 4-tuple has a 0, the other has a 1 and vice versa). The third set (2 -tuple) has one photon in one of its two detectors. Representative success patterns are:((1, 1, 0, 0), (1, 1, 0, 0), (1, 0)) [Ex. A8] ((1, 1, 0, 0), (0, 0, 1, 1), (1, 0)) [Ex. A9] Here, [Ex. A8] shows identical anti-bunched 4-tuples, and [Ex. A9] shows complementary anti-bunched 4-tuples. Additional success patterns in this category are derived by applying every combination of one or more of the following operations to either of the two representative success patterns: (a) swapping the first and second 4-tuples; (b) applying the same permutation to each of the first and second 4-tuples; and / or (c) permuting the final 2-tuple. Examples include:((0, 0, 1, 1), (1, 1, 0, 0), (1, 0)) [Ex. A10] ((0, 1, 1, 0), (0, 1, 1, 0), (1, 0)) [Ex. All] ((0, 1, 1, 0), (0, 1, 1, 0), (0, 1)) [Ex. A12] Here, [Ex. A10] corresponds to [Ex. A9] with a swap of the 4-tuples; [Ex. All] corresponds to [Ex. A8] with the same permutation of both 4-tuples; and [Ex. A12] corresponds to [Ex. All] with a permutation of the 2-tuple. Other success patterns in this category can be readily derived from these examples.

[0222] As described above, classical decision logic circuit 2850 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals are received via signal path 2854, classical decision logic circuit 2850 can compare the readout pattern of signals received from detectors 2820 to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 2850 can output a classical decision signal 2856 indicating whether a match has occurred. In some embodiments, classical decision logiccircuit 2850 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI, Q2, Q3, Q4, and Q5, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0223] If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 2800 is 9 / 64, or approximately 14%. In a failure pattern in which three photons are detected at the detectors 2820 in one of first or second sets 2861, 2862, one photon is detected at the detectors 2820 in the other of first or second sets 2861, 2862, and zero photons are detected at the detectors 2820 in third set 2863, the output corresponds to measurement of the input qubits in the Z basis. In the case that two photons are detected at the detectors 2820 in one of first or second sets 2861, 2862, one photon is detected at the detectors 2820 in the other of first or second sets 2861, 2862, and two photons are detected at the detectors 2820 in third set 2863, the output corresponds to measurement of the input qubits in the Z basis. In the ideal case, this happens with probability 3 / 8, or 37.5%. Other failure patterns, which happen with probability 31 / 64, or about 48%, correspond to projection of at least a subset of the input qubits onto a non-stabilizer state (in this case not a 5 -GHZ state)

[0224] FIG. 29 shows a simplified schematic diagram of a Hadamard-based six-way fusion circuit 2900 according to some embodiments. Six-way fusion circuit 2900 receives six dual-rail-encoded input qubits QI, Q2, Q3, Q4, Q5, Q6, each of which can initially be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of six-way fusion circuit 2900 results in the six entangled systems being entangled with each other to form one entangled system.

[0225] Input qubit QI is encoded on a first pair of waveguides 2901-0, 2901-1 such that the logical-0 rail maps to waveguide 2901-0 and the logical-1 rail maps to waveguide 2901-1. Similarly, input qubit Q2 is encoded on a second pair of waveguides 2902-0, 2902-1 such that the logical-0 rail maps to waveguide 2902-0 and the logical- 1 rail maps to waveguide 2902-1; input qubit Q3 is encoded on a third pair of waveguides 2903-0, 2903-1 such that the logical-0 rail maps to waveguide 2903-0 and the logical- 1 rail maps to waveguide 2903-1; input qubit Q4 is encoded on a fourth pair of waveguides 2904-0, 2904-1 such that the logical-0 rail maps to waveguide 2904-0 and the logical- 1 rail maps to waveguide 2904-1;input qubit Q5 is encoded on a fifth pair of waveguides 2905-0, 2905-1 such that the logical -0 rail maps to waveguide 2905-0 and the logical-1 rail maps to waveguide 2905-1; and input qubit Q6 is encoded on a sixth pair of waveguides 2906-0, 2906-1 such that the logical-0 rail maps to waveguide 2906-0 and the logical- 1 rail maps to waveguide 2906-1.

[0226] A first four- way Hadamard interferometer circuit 2911, which can be an instance of circuit 2400, is coupled to waveguides 2901-0, 2902-1, 2903-0, and 2906-1 (the logical-0 rails of input qubits QI and Q3 and the logical- 1 rails of input qubits Q2 and Q6). A second four- way Hadamard interferometer circuit 2912, which can be another instance of circuit 2400, is coupled to waveguides 2901-1, 2902-0, 2904-1, and 2905-0 (the logical-1 rails of input qubits QI and Q4 and the logical-0 rails of input qubits Q2 and Q5). A third four- way Hadamard interferometer circuit 2913, which can be another instance of circuit 2400, is coupled to waveguides 2903-1, 2904-0, 2905-1, and 2906-0 (the logical-1 rails of input qubits Q3 and Q5 and the logical-0 rails of input qubits Q4 and Q6).

[0227] Detectors 2920-1 through 2920-4 are coupled to the outputs of first four- way Hadamard interferometer circuit 2911; detectors 2920-5 through 2920-8 are coupled to the outputs of second four- way Hadamard interferometer circuit 2912; and detectors 2920-9 through 2920-12 are coupled to the outputs of third four- way Hadamard interferometer circuit 2913. Detectors 2920 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 2954.

[0228] Classical decision logic circuit 2950 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 2950 can receive the respective photon count signals from the twelve detectors 2920, e.g., via signal path 2954, and can determine whether the pattern of photon counts corresponds to a success pattern. Similarly to circuit 2800, detectors 2920 can be logically grouped into sets, with a first set including the detectors 2920 whose rails (or waveguides) are connected to interferometer circuit 2911 (detectors 2920-1 through 2920-4 in FIG. 29); a second set including the detectors whose rails (or waveguides) are connected to interferometer circuit 2912 (detectors 2920-5 through 2920-8 in FIG. 29); and a third set includes the detectors 2920 whose rails (or waveguides) are connected to interferometer circuit 2913 (detectors 2920-9 through 2920-12 in FIG. 29). Success patterns for circuit 2900can correspond to instances where a total of two photons are detected by each of the three sets.

[0229] As with circuit 2800, two-photon patterns can be classified as bunched or antibunched. For example, a fully anti -bunched success pattern for circuit 2900 can include two of the four detectors in the first set each detecting a single photon, two of the four detectors in the second set each detecting a single photon, and two of the four detectors connected in the third set detecting a single photon. If desired, this can be subject to the further constraint that the tuples in any two sets are either identical or complementary (for instance, two of the three tuples can be identical to each other while the third would be complementary to the others, or all three can be identical). As another example, if one of the sets has a bunched pattern, the other two sets can have anti-bunched arrangements in identical or complementary patterns.

[0230] In some embodiments, the set of success patterns for six-way fusion circuit 2900 of FIG. 29 can be defined as including patterns that fit one of the following three categories:

[0231] Category 1: Fully anti-bunched. Each of the first, second, and third sets of detectors has an anti-bunched 4-tuple. Example success patterns in this category are:((1, 1, 0, 0), (1, 1, 0, 0), (1, 1, 0, 0)) [Ex. Bl] ((0, 1, 1, 0), (0, 1, 1, 0), (0, 1, 1, 0)) [Ex. B2] ((0, 1, 1, 0), (0, 1, 1, 0), (1, 0, 0, 1)) [Ex. B3] ((1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 1, 0)) [Ex. B4] Additional success patterns in this category are derived by permuting the elements of one or more of the 4-tuples. Examples include:((1, 0, 1, 0), (1, 0, 1, 0), (1, 0, 1, 0)) [Ex. B5] ((0, 0, 1, 1), (0, 0, 1, 1), (0, 0, 1, 1)) [Ex. B6] Other success patterns in this category can be readily derived from these examples.

[0232] Category 2: One Bunched+ Two Anti-bunched. One of the three sets of detectors has a bunched 4-tuple (both photons in same detector) while each of the other two has an anti-bunched 4-tuple (two photons in two different detectors). The anti-bunched 4-tuples are subject to the constraint that their patterns are either identical or complementary. Example success patterns in this category are:((2, 0, 0, 0), (1, 1, 0, 0), (1, 1, 0, 0)) [Ex. B7] ((2, 0, 0, 0), (1, 1, 0, 0), (0, 0, 1, 1)) [Ex. B8] In [Ex. B7], the anti-bunched 4-tuples have identical patterns. In [Ex. B8], the anti-bunched 4-tuples have complementary patterns. Additional success patterns in this category are derived by applying every combination of one or more of the following operations to the representative success pattern: (a) permuting the elements of the first 4-tuple so that the “2” appears in any position; (b) permuting the elements of the second 4-tuple into any order and providing a third 4-tuple that is either identical or complementary; and / or (c) rearranging the 4-tuples into any order. Examples include:((0, 0, 2, 0), (1, 1, 0, 0), (1, 1, 0, 0)) [Ex. B9] ((0, 0, 2, 0), (1, 0, 1, 0), (1, 0, 1, 0)) [Ex. B10] ((0, 0, 2, 0), (1, 0, 1, 0), (0, 1, 0, 1)) [Ex. Bl 1] Here, [Ex. B9] is [Ex. B7] with the first 4-tuple permuted; [Ex. B10] is [Ex. B9] with identical permutation of the anti -bunched 4-tuples; and [Ex. Bl 1] is [Ex. B9] with a complementary permutation of the anti-bunched 4-tuples. Other success patterns in this category can be readily derived from these examples.

[0233] Category 3: Two Bunched+ One Anti-bunched. Two of the three sets of detectors have bunched 4-tuples (both photons in same detector) while the third has an antibunched 4-tuple (two photons in two different detectors). An example success pattern in this category is:((2, 0, 0, 0), (2, 0, 0, 0), (1, 1, 0, 0)) [Ex. B12] Additional success patterns in this category are derived by permuting one or more of the 4-tuples and / or rearranging the 4-tuples into any order. Examples include:((2, 0, 0, 0), (1, 1, 0, 0), (0, 0, 2, 0)) [Ex. B13] ((1, 0, 1, 0), (0, 2, 0, 0), (0, 0, 2, 0)) [Ex. B14] Other success patterns in this category can be readily derived from these examples.

[0234] As described above, classical decision logic circuit 2950 can include or have access to a lookup table or other memory structure that stores a representation of each successpattern. When photon-count signals are received via signal path 2954, classical decision logic circuit 2950 can compare the readout pattern of signals received from detectors 2920 to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 2950 can output a classical decision signal 2956 indicating whether a match has occurred. In some embodiments, classical decision logic circuit 2950 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI, Q2, Q3, Q4, Q5, and Q6 and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0235] If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 2900 is considerably higher than that of circuit 2100 (sixway fusion circuit using two-qubit fusions).

[0236] FIG. 30 shows a simplified schematic diagram of a Hadamard-based seven-way fusion circuit 3000 according to some embodiments. Seven-way fusion circuit 3000 receives seven dual-rail-encoded input qubits QI, Q2, Q3, Q4, Q5, Q6, Q7, each of which can initially be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of seven-way fusion circuit 3000 results in the seven entangled systems being entangled with each other to form one entangled system.

[0237] Input qubit QI is encoded on a first pair of waveguides 3001-0, 3001-1 such that the logical-0 rail maps to waveguide 3001-0 and the logical- 1 rail maps to waveguide 3001-1. Similarly, input qubit Q2 is encoded on a second pair of waveguides 3002-0, 3002-1 such that the logical-0 rail maps to waveguide 3002-0 and the logical- 1 rail maps to waveguide 3002-1; input qubit Q3 is encoded on a third pair of waveguides 3003-0, 3003-1 such that the logical-0 rail maps to waveguide 3003-0 and the logical- 1 rail maps to waveguide 3003-1; input qubit Q4 is encoded on a fourth pair of waveguides 3004-0, 3004-1 such that the logical-0 rail maps to waveguide 3004-0 and the logical- 1 rail maps to waveguide 3004-1; input qubit Q5 is encoded on a fifth pair of waveguides 3005-0, 3005-1 such that the logical-0 rail maps to waveguide 3005-0 and the logical-1 rail maps to waveguide 3005-1; input qubit Q6 is encoded on a sixth pair of waveguides 3006-0, 3006-1 such that the logical-0 rail maps to waveguide 3006-0 and the logical- 1 rail maps to waveguide 3006-1; and input qubitQ7 is encoded on a seventh pair of waveguides 3007-0, 3007-1 such that the logical-0 rail maps to waveguide 3007-0 and the logical-1 rail maps to waveguide 3007-1.

[0238] A first four- way Hadamard interferometer circuit 3011, which can be an instance of circuit 2400, is coupled to waveguides 3001-0, 3002-1, 3003-0, and 3007-1 (the logical-0 rails of input qubits QI and Q3 and the logical- 1 rails of input qubits Q2 and Q7). A second four- way Hadamard interferometer circuit 3012, which can be another instance of circuit 2400, is coupled to waveguides 3001-1, 3002-0, 3004-1, and 3005-0 (the logical-1 rails of input qubits QI and Q4 and the logical-0 rails of input qubits Q2 and Q5). A third four- way Hadamard interferometer circuit 3013, which can be another instance of circuit 2400, is coupled to waveguides 3003-1, 3004-0, 3005-1, and 3006-0 (the logical-1 rails of input qubits Q3 and Q5 and the logical-0 rails of input qubits Q4 and Q6). A two-way Hadamard interferometer circuit 3014, which can be implemented, e.g., using a 50 / 50 beam splitter, is coupled to waveguides 3006-1 and 3007-0 (the logical-1 rail of input qubit Q6 and the logical-0 rail of input qubit Q7).

[0239] Detectors 3020-1 through 3020-4 are coupled to the outputs of first four- way Hadamard interferometer circuit 3011; detectors 3020-5 through 3020-8 are coupled to the outputs of second four- way Hadamard interferometer circuit 3012; detectors 3020-9 through 3020-12 are coupled to the outputs of third four- way Hadamard interferometer circuit 3013; and detectors 3020-13 and 3020-14 are coupled to the outputs of two-way Hadamard interferometer circuit 3014. Detectors 3020 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 3054.

[0240] Classical decision logic circuit 3050 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 3050 can receive the respective photon count signals from the fourteen detectors 3020, e.g., via signal path 3054, and can determine whether the pattern of photon counts corresponds to a success pattern. Similarly to circuits described above, detectors 3020 coupled to outputs of the same Hadamard interferometer can be logically grouped into sets (three sets with four detectors each and one set with two detectors), and success patterns for circuit 3000 can correspond to instances where a total of two photons are detected by each of the four sets. In some embodiments, success patterns for circuit 3000 aresimilar to those for circuit 2900 in regard to the sets of four detectors, in combination with the two-detector set having either (1, 0) or (0, 1).

[0241] In some embodiments, the set of success patterns for seven-way fusion circuit 3000 of FIG. 30 can be defined analogously to the definition for six-way fusion circuit 2900 described above. In the seven-way case, each success pattern further includes an additional 2-tuple in either the (0, 1) or (1, 0) state. Thus, the following categories can be defined:

[0242] Category 1: Fully anti-bunched. Each of the first, second, and third sets of detectors has an anti-bunched 4-tuple, and the fourth set of detectors has a 2-tuple in either the (0, 1) or (1, 0) state. Example success patterns in this category include:((1, 1, 0, 0), (1, 1, 0, 0), (1, 1, 0, 0), (1, 0)) [Ex. Cl] ((0, 1, 1, 0), (0, 1, 1, 0), (0, 1, 1, 0), (1, 0)) [Ex. C2] ((0, 1, 1, 0), (0, 1, 1, 0), (1, 0, 0, 1), (0, 1)) [Ex. C3] ((1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 1, 0), (0, 1)) [Ex. C4] ((1, 0, 1, 0), (1, 0, 1, 0), (1, 0, 1, 0)) [Ex. C5] ((0, 0, 1, 1), (0, 0, 1, 1), (0, 0, 1, 1)) [Ex. C6] Other success patterns in this category can be readily derived from these examples.

[0243] Category 2: One Bunched+ Two Anti-bunched. One of the first three sets of detectors has a bunched 4-tuple (both photons in same detector) while each of the other two has an anti-bunched 4-tuple (two photons in two different detectors) and the fourth set of detectors has a 2-tuple in either the (0, 1) or (1, 0) state. The anti -bunched 4-tuples are subject to the constraint that their patterns are either identical or complementary. Example success patterns in this category include:((2, 0, 0, 0), (1, 1, 0, 0), (1, 1, 0, 0), (0,1)) [Ex. C7] ((2, 0, 0, 0), (1, 1, 0, 0), (0, 0, 1, 1), (1, 0)) [Ex. C8] ((0, 0, 2, 0), (1, 1, 0, 0), (1, 1, 0, 0), (1, 0)) [Ex. C9] ((0, 0, 2, 0), (1, 0, 1, 0), (1, 0, 1, 0), (0, 1)) [Ex. CIO] ((0, 0, 2, 0), (1, 0, 1, 0), (0, 1, 0, 1), (1, 0)) [Ex. Cl 1] Other success patterns in this category can be readily derived from these examples.

[0244] Category 3: Two Bunched+ One Anti-bunched. Two of the first three sets of detectors have bunched 4-tuples (both photons in same detector) while the third has an antibunched 4-tuple (two photons in two different detectors) and the fourth set of detectors has a 2 -tuple in either the (0, 1) or (1, 0) state. Example success patterns in this category include:((2, 0, 0, 0), (2, 0, 0, 0), (1, 1, 0, 0), (1, 0)) [Ex. C12] ((2, 0, 0, 0), (1, 1, 0, 0), (0, 0, 2, 0), (1,0)) [Ex. C13] ((1, 0, 1, 0), (0, 2, 0, 0), (0, 0, 2, 0), (0, 1)) [Ex. C14] Other success patterns in this category can be readily derived from these examples.

[0245] As described above, classical decision logic circuit 3050 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals are received via signal path 3054, classical decision logic circuit 3050 can compare the readout pattern of signals received from detectors 3020 to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 3050 can output a classical decision signal 3056 indicating whether a match has occurred. In some embodiments, classical decision logic circuit 3050 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI, Q2, Q3, Q4, Q5, Q6, and Q7, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0246] If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 3000 is considerably higher than that of circuit 2200 (sevenway fusion circuit using two-qubit fusions).

[0247] FIG. 31 shows a simplified schematic diagram of a Hadamard-based eight-way fusion circuit 3100 according to some embodiments. Eight- way fusion circuit 3100 receives eight dual-rail-encoded input qubits QI, Q2, Q3, Q4, Q5, Q6, Q7, Q8, each of which can initially be part of a different entangled system of qubits. (For simplicity of illustration, the initial entangled systems are not shown.) Success of eight- way fusion circuit 3100 results in the eight entangled systems being entangled with each other to form one entangled system.

[0248] Input qubit QI is encoded on a first pair of waveguides 3101-0, 3101-1 such that the logical-0 rail maps to waveguide 3101-0 and the logical- 1 rail maps to waveguide 3101-1. Similarly, input qubit Q2 is encoded on a second pair of waveguides 3102-0, 3102-1 such that the logical-0 rail maps to waveguide 3102-0 and the logical- 1 rail maps to waveguide 3102-1; input qubit Q3 is encoded on a third pair of waveguides 3103-0, 3103-1 such that the logical-0 rail maps to waveguide 3103-0 and the logical- 1 rail maps to waveguide 3103-1; input qubit Q4 is encoded on a fourth pair of waveguides 3104-0, 3104-1 such that the logical-0 rail maps to waveguide 3104-0 and the logical- 1 rail maps to waveguide 3104-1; input qubit Q5 is encoded on a fifth pair of waveguides 3105-0, 3105-1 such that the logical -0 rail maps to waveguide 3105-0 and the logical-1 rail maps to waveguide 3105-1; input qubit Q6 is encoded on a sixth pair of waveguides 3106-0, 3106-1 such that the logical-0 rail maps to waveguide 3106-0 and the logical- 1 rail maps to waveguide 3106-1; input qubit Q7 is encoded on a seventh pair of waveguides 3107-0, 3107-1 such that the logical-0 rail maps to waveguide 3107-0 and the logical-1 rail maps to waveguide 3107-1; and input qubit Q8 is encoded on an eighth pair of waveguides 3108-0, 3108-1 such that the logical-0 rail maps to waveguide 3108-0 and the logical- 1 rail maps to waveguide 3108-1.

[0249] A first four- way Hadamard interferometer circuit 3111, which can be an instance of circuit 2400, is coupled to waveguides 3101-0, 3102-1, 3103-0, and 3108-1 (the logical-0 rails of input qubits QI and Q3 and the logical- 1 rails of input qubits Q2 and Q8). A second four- way Hadamard interferometer circuit 3112, which can be another instance of circuit 2400, is coupled to waveguides 3101-1, 3102-0, 3103-1, and 3104-0 (the logical-1 rails of input qubits QI and Q3 and the logical-0 rails of input qubits Q2 and Q4). A third four- way Hadamard interferometer circuit 3113, which can be another instance of circuit 2400, is coupled to waveguides 3104-1, 3105-0, 3106-1, and 3107-0 (the logical-1 rails of input qubits Q4 and Q6 and the logical-0 rails of input qubits Q5 and Q7). A fourth four-way Hadamard interferometer circuit 3114, which can be another instance of circuit 2400, is coupled to waveguides 3105-1, 3106-0, 3107-1, and 3108-0 (the logical-1 rails of input qubits Q5 and Q7 and the logical-0 rails of input qubits Q6 and Q8).

[0250] Detectors 3120-1 through 3120-4 are coupled to the outputs of first four- way Hadamard interferometer circuit 3111; detectors 3120-5 through 3120-8 are coupled to the outputs of second four- way Hadamard interferometer circuit 3112; detectors 3120-9 through 3120-12 are coupled to the outputs of third four- way Hadamard interferometer circuit 3113; and detectors 3120-13 through 3120-16 are coupled to the outputs of fourth four- wayHadamard interferometer circuit 3114. Detectors 3120 can output classical logic signals indicating the number of photons detected at each detector onto classical signal path 3154.

[0251] Classical decision logic circuit 3150 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 3150 can receive the respective photon count signals from the sixteen detectors 3120, e.g., via signal path 3154, and can determine whether the pattern of photon counts corresponds to a success pattern. Similarly to circuits described above, detectors 3120 coupled to outputs of the same Hadamard interferometer can be logically grouped into sets (four sets with four detectors each). Success patterns for circuit 3100 can correspond to instances where a total of two photons are detected by each of the four sets. In some embodiments, success patterns can also be required to include at least two sets having anti-bunched arrangements, with the other two sets having either bunched or anti-bunched arrangements. If desired, additional constraints can be applied based on whether antibunched sets are identical or complementary.

[0252] In some embodiments, the set of success patterns for eight- way fusion circuit 3100 of FIG. 31 includes any pattern in which at least two of the 4-tuples have an anti -bunched pattern of two photons and zero, one, or two of the 4-tuples have a bunched pattern. Example success patterns include:((0, 0, 1, 1), (0, 2, 0, 0), (0, 2, 0, 0), (0, 1, 0, 1)) [Ex. DI] ((1, 0, 0, 1), (0, 1, 1, 0), (2, 0, 0, 0), (0, 1, 1, 0)) [Ex. D2] ((0, 0, 1, 1), (1, 1, 0, 0), (0, 1, 0, 1), (0, 1, 0, 1)) [Ex. D3] In [Ex. DI], two 4-tuples have a bunched pattern and two 4-tuples have an anti -bunched pattern; in [Ex. D2], one 4-tuple has a bunched pattern and three 4-tuples have an antibunched pattern; and in [Ex. D3], all 4-tuples have an anti-bunched pattern. Other success patterns can be readily derived from these examples.

[0253] As described above, classical decision logic circuit 3150 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals are received via signal path 3154, classical decision logic circuit 3150 can compare the readout pattern of signals received from detectors 3120 to the stored success patterns to determine whether or not a match has occurred. In someembodiments, classical decision logic circuit 3150 can output a classical decision signal 3156 indicating whether a match has occurred. In some embodiments, classical decision logic circuit 3150 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include qubits QI, Q2, Q3, Q4, Q5, Q6, Q7, and Q8, and can incorporate such heralding signals into the decision logic, e.g., as described above.

[0254] If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for circuit 3100 is considerably higher than that of an eight- way fusion circuit using two-qubit fusions (e.g., circuits constructed using principles described in section 2 above).

[0255] It will be appreciated that the circuits described above are illustrative and that variations and modifications are possible. The particular combination of logical rails provided to a given four-way or two-way Hadamard interferometer can be modified, as long as each logical rail passes through exactly one Hadamard interferometer. The pattern illustrated by FIGs. 27-31 can be extended to any number of input qubits.

[0256] The foregoing examples illustrate a general approach for designing N-way fusion circuits using Hadamard interferometers, for a number of qubits N that is greater than or equal to 4. It is assumed that the qubits are dual-rail-encoded photonic qubits so that there are a total of 2N rails. In instances where N is even, 2N is a multiple of 4. Accordingly, a number P = 2N / 4 of four-way Hadamard interferometer circuits can be used, arranged such that each of the four-way Hadamard interferometer circuits receives one rail from each of four different qubits and the two rails of a given qubit are coupled to inputs of two different ones of the four-way Hadamard interferometer circuits. In instances where N is odd, 2N is not a multiple of 4. In this case, a number P = 2(N-l) / 4 of four-way Hadamard interferometer circuits can be used together with a two-way Hadamard interferometer circuit. Each of the four-way Hadamard interferometer circuits receives one rail from each of four different qubits and the two rails of a given qubit are coupled to inputs of two different interferometer circuits; the remaining two rails, which are from two different qubits, are coupled to inputs of the two-way Hadamard interferometer circuit. Provided that there is no instance where both rails of any pair of qubits are input to the same interferometer (or anyinstance where both rails of one qubit are input to the same interferometer), the mapping of qubit rails to inputs of the interferometer circuits can be modified as desired.

[0257] A number 2N of photon-counting detectors are coupled in one-to-one correspondence to the outputs of the Hadamard interferometer circuits. The detectors can be logically grouped into sets, where each set includes the detectors coupled to the outputs of one of the Hadamard interferometer circuits, and each set produces a pattern. Success patterns can be defined based on the combination of photon patterns produced by all of the sets.

[0258] Circuits of this kind exhibit the property that the probability of success increases with the number N of qubits.4. Additional Embodiments

[0259] The foregoing examples are illustrative and can be modified as desired. Although some examples may make reference to use-cases related to quantum computing, it should be apparent from this disclosure that multi-way fusion circuits of the kind described herein can also be used in quantum communication and any other application where production of entangled systems of qubits is desirable. The size of a time bin, the number of spatial and / or temporal modes, and the number of input qubits for a given fusion circuit can be varied as desired. In circuits using cascaded type I and type II fusions, balanced (or parallelized) stages can be combined with chained stages in any manner desired. Mapping of particular logical states of qubits to particular waveguides or time bins is arbitrary and can be modified (e.g., reversed). Various schematic diagrams herein are shown with optical components (e.g., beam splitters) and detectors in a particular arrangement for clarity of illustration, and fusion operations are described sequentially; however, it should be understood that, in operation, all detectors in an 7V-way fusion circuit perform their detection operations within the same detection period (or time bin) and all output data is concurrently sent to the classical decision logic. For instance, optical path lengths can be adjusted so that photons input into any of the input waveguides concurrently also reach a detector concurrently. Accordingly, success or failure of an 7V-way fusion circuit can be determined based on the detection pattern across all IN detectors in the circuit, regardless of the particular value of N. Further, while optical circuit components are shown herein in particular arrangements that illustrate the functionality, the arrangement of components of an optical circuit can be modified without altering the functionality of the circuit.

[0260] Further, embodiments described above include references to specific materials and structures (e.g., optical fibers), but other materials and structures capable of producing, propagating, and operating on photons can be substituted.

[0261] Classical decision logic, as well as control logic to control the switches and other active optical components described herein can be implemented as digital logic circuits with an arrangement of logic gates (AND, OR, NOR, XOR, NAND, NOT, etc.), such as a field programmable gate array (FPGA) or system-on-a-chip (SOC) having a programmable processor and memory, or an on-chip hard-wired circuit, such as an application specific integrated circuit (ASIC). Decision and / or control logic circuits can be implemented on-chip with the waveguides, beam splitters, detectors and / or and other photonic circuit components or off-chip as desired. In some embodiments, photon sources, detectors, and / or other optical circuits can be coupled to an off-chip classical computer system having a processor and a memory, and the off-chip computer system can be programmed to execute some or all of the classical decision logic and / or control logic.

[0262] The following are example embodiments:

[0263] Example 1 : A circuit comprising: a plurality of waveguide pairs to receive a plurality of qubits, wherein each of the qubits is a dual-rail-encoded photonic qubit propagating on one of the waveguide pairs such that the waveguides of the waveguide pair correspond to the rails of the qubits and wherein the total number of qubits in the plurality of qubits is a number (N) that is an even number greater than or equal to 4; a plurality of fourway Hadamard interferometer circuits, each four-way Hadamard interferometer circuit having four input waveguides and four output waveguides, wherein each of the four-way Hadamard interferometer circuits includes a network of beam splitters arranged such that a photon entering on any one of the four input waveguides has an equal probability of exiting on any one of the four output waveguides; the waveguides of the waveguide pairs being coupled to the inputs of the four-way Hadamard interferometer circuits such that each of the four-way Hadamard interferometer circuits receives one rail from each of four different dualrail encoded photonic qubits and the two rails of a given dual-rail encoded photonic qubit are coupled to inputs of two different ones of the four-way Hadamard interferometer circuits; a set of 2N photon detectors, each photon detector coupled to a different one of the output waveguides of the four-way Hadamard interferometer circuits and configured to produce a classical output signal indicating a count of detected photons; and a classical decision logiccircuit configured to receive the classical output signals from the set of 2N photon detectors and to determine, based on the received classical output signals, whether an N-way fusion operation on the plurality of qubits succeeded.

[0264] Example 2: The circuit of Example 1 wherein the 2N photon detectors are configured to operate concurrently to produce the classical output signals for a particular set of received qubits.

[0265] Example 3: The circuit of Example 1 or Example 2 wherein the classical decision logic circuit is further configured to determine whether the N-way fusion operation succeeded by extracting a detection pattern from the received classical output signals and comparing the detection pattern to a set of success patterns, wherein the N-way fusion operation succeeded if the detection pattern matches one of the success patterns in the set of success patterns.

[0266] Example 4: The circuit of Example 3 wherein the classical decision logic is further configured such that the detection pattern includes, for each of the four-way Hadamard interferometer circuits, a 4-tuple of counts of detected photons.

[0267] Example 5: The circuit of Example 4 wherein the set of success patterns includes only patterns where each 4-tuple corresponds to a total of two detected photons.

[0268] Example 6: The circuit of Example 5 wherein the set of success patterns includes only patterns in which, for at least one of the 4-tuples, the two detected photons are detected by two different detectors.

[0269] Example 7: The circuit of Example 5 wherein the number N is equal to 6, the number of 4-tuples is equal to three, and the set of success patterns consists of a first subset of success patterns in which, for all three of the 4-tuples, the two detected photons are detected by two different detectors; a second subset of success patterns in which, for one of the three 4-tuples, the two detected photons are detected by the same detector and for the other two 4-tuples, the two detected photons are detected by different detectors such that the other two 4-tuples have either an identical pattern of photon counts or a complementary pattern of photon counts; and a third subset of success patterns in which, for two of the three 4-tuples, the two detected photons are detected by the same detector and for the other 4-tuple, the two detected photons are detected by different detectors.

[0270] Example 8: The circuit of any preceding Example wherein each of the four-way Hadamard interferometer circuits comprises: a first beam splitter coupled between a firstwaveguide and a second waveguide of the four input waveguides; a second beam splitter coupled between a third waveguide and a fourth waveguide of the four input waveguides; a third beam splitter coupled between a first output waveguide of the first beam splitter and a first output waveguide of the second beam splitter; and a fourth beam splitter coupled between a second output waveguide of the first beam splitter and a second output waveguide of the second beam splitter.

[0271] Example 9: The circuit of Example 8 wherein the first, second, third, and fourth beam splitters are 50 / 50 beam splitters.

[0272] Example 10: The circuit of any preceding Example wherein the N-way fusion operation corresponds to projection onto an N-GHZ state.

[0273] Example 11 : The circuit of any preceding Example wherein each qubit of the plurality of qubits is initially part of a respective quantum system of entangled qubits and wherein the N-way fusion operation consumes the plurality of qubits and produces an entangled quantum system from the remaining qubits of the respective quantum systems.

[0274] Example 12: A circuit comprising: a plurality of waveguide pairs to receive a plurality of qubits, wherein each of the qubits is a dual-rail -encoded photonic qubit propagating on one of the waveguide pairs and wherein the total number of qubits in the plurality of qubits is a number (N) that is an odd number greater than 4; a plurality of fourway Hadamard interferometer circuits, each four-way Hadamard interferometer circuit having four input waveguides and four output waveguides, wherein each of the four-way Hadamard interferometer circuits includes a network of beam splitters arranged such that a photon entering on any one of the four input waveguides has an equal probability of exiting on any one of the four output waveguides; a two-way Hadamard interferometer circuit having two input waveguides and two output waveguides, wherein the two-way Hadamard interferometer circuit includes a beam splitter arranged such that a photon entering on either one of the two input waveguides has an equal probability of exiting on any one of the two output waveguides, the waveguides of the waveguide pairs being coupled to the inputs of the four-way Hadamard interferometer circuits and the two-way Hadamard interferometer circuit such that each of the four-way Hadamard interferometer circuits receives one rail from each of four different dual-rail encoded photonic qubits, the two-way Hadamard interferometer circuit receives one rail from each of two different dual-rail encoded photonic qubits, and the two rails of a given dual-rail encoded photonic qubit are coupled to inputs of two differentones of the four- way Hadamard interferometer circuits or the two-way Hadamard interferometer circuit; a set of 2N photon detectors, each photon detector coupled to a different one of the output waveguides of the four-way Hadamard interferometer circuits or the two-way Hadamard interferometer circuit, each of the photon detectors being configured to produce a classical output signal indicating a count of detected photons; and a classical decision logic circuit configured to receive the classical output signals from the set of 2N photon detectors and to determine, based on the received classical output signals, whether an N-way fusion operation on the plurality of qubits succeeded.

[0275] Example 13: The circuit of Example 12 wherein the 2N photon detectors are configured to operate concurrently to produce the classical output signals for a particular set of received qubits.

[0276] Example 14: The circuit of Example 12 or Example 13 wherein the classical decision logic circuit is further configured to determine whether the N-way fusion operation succeeded by extracting a detection pattern from the received classical output signals and comparing the detection pattern to a set of success patterns, wherein the N-way fusion operation succeeded if the detection pattern matches one of the success patterns in the set of success patterns.

[0277] Example 15: The circuit of Example 14 wherein the classical decision logic is further configured such that the detection pattern includes, for each of the four-way Hadamard interferometer circuits, a 4-tuple of counts of detected photons and, for the two-way Hadamard interferometer circuit, a 2-tuple of counts of detected photons.

[0278] Example 16: The circuit of Example 15 wherein the set of success patterns includes only patterns where each 4-tuple corresponds to a total of two detected photons and the 2-tuple corresponds to a total of one detected photon.

[0279] Example 17: The circuit of Example 16 wherein the set of success patterns includes only patterns in which for at least one of the 4-tuples, the two detected photons are detected by two different detectors.

[0280] Example 18: The circuit of Example 16 wherein the number N is equal to 7, the number of 4-tuples is equal to three, and the set of success patterns consists of: a first subset of success patterns in which, for all three of the 4-tuples, the two detected photons are detected by two different detectors; a second subset of success patterns in which, for one ofthe three 4-tuples, the two detected photons are detected by the same detector and for the other two 4-tuples, the two detected photons are detected by different detectors such that the other two 4-tuples have either an identical pattern of photon counts or a complementary pattern of photon counts; and a third subset of success patterns in which, for two of the three 4-tuples, the two detected photons are detected by the same detector and for the other 4-tuple, the two detected photons are detected by different detectors.

[0281] Example 19. The circuit of any of Examples 12 through 18 wherein each of the four-way Hadamard interferometer circuits comprises: a first beam splitter coupled between a first waveguide and a second waveguide of the four input waveguides; a second beam splitter coupled between a third waveguide and a fourth waveguide of the four input waveguides; a third beam splitter coupled between a first output waveguide of the first beam splitter and a first output waveguide of the second beam splitter; and a fourth beam splitter coupled between a second output waveguide of the first beam splitter and a second output waveguide of the second beam splitter.

[0282] Example 20: The circuit of Example 19 wherein the first, second, third, and fourth beam splitters are 50 / 50 beam splitters.

[0283] Example 21. The circuit of any one of Examples 12 through 20 wherein the N-way fusion operation corresponds to projection onto an N-GHZ state.

[0284] Example 22: The circuit of any one of Examples 12 through 21 wherein each qubit of the plurality of qubits is initially part of a respective quantum system of entangled qubits and wherein the N-way fusion operation consumes the plurality of qubits and produces an entangled quantum system from the remaining qubits of the respective quantum systems.

[0285] It should be understood that all numerical values used herein are for purposes of illustration and may be varied. In some instances, ranges are specified to provide a sense of scale, but numerical values outside a disclosed range are not precluded. Terms such as “synchronized” or “simultaneous” (or “same” or “identical”) should be understood in the engineering rather than the mathematical sense: finite design tolerances can be defined, and events separated by less than the design tolerance may be treated as synchronized or simultaneous. A “time bin” refers to a temporal mode that distinguishes different photonic states in the same waveguide (or spatial mode). The duration of a time bin can be defined based on characteristics of the optical circuits (e.g., there may be some variation in the delay between pumping a photon source and obtaining an output photon from the source), andsuccessive time bins can be separated by arbitrary time periods (e.g., to allow circuit components to recover or change state before receiving the next photon).

[0286] It should also be understood that all diagrams herein are intended as schematic. Unless specifically indicated otherwise, the drawings are not intended to imply any particular physical arrangement of the elements shown therein, or that all elements shown are necessary. Further, Those skilled in the art with access to this disclosure will understand that elements shown in drawings or otherwise described in this disclosure can be modified or omitted and that other elements not shown or described can be added. The terms “upstream” and “downstream” as used herein refer to the direction of photon propagation through an optical circuit (from “upstream” inputs toward “downstream” outputs) and may correspond to any direction in physical space.

[0287] This disclosure provides a description of the claimed invention with reference to specific embodiments. Those skilled in the art with access to this disclosure will appreciate that the embodiments are not exhaustive of the scope of the claimed invention, which extends to all variations, modifications, and equivalents.

Claims

WHAT IS CLAIMED IS:

1. A circuit comprising:a plurality of waveguide pairs to receive a plurality of qubits, wherein each of the qubits is a dual-rail-encoded photonic qubit propagating on one of the waveguide pairs such that the waveguides of the waveguide pair correspond to the rails of the qubits and wherein the total number of qubits in the plurality of qubits is a number (N) that is an even number greater than or equal to 4;a plurality of four-way Hadamard interferometer circuits, each four-way Hadamard interferometer circuit having four input waveguides and four output waveguides, wherein each of the four- way Hadamard interferometer circuits includes a network of beam splitters arranged such that a photon entering on any one of the four input waveguides has an equal probability of exiting on any one of the four output waveguides;the waveguides of the waveguide pairs being coupled to the inputs of the fourway Hadamard interferometer circuits such that each of the four-way Hadamard interferometer circuits receives one rail from each of four different dual-rail encoded photonic qubits and the two rails of a given dual-rail encoded photonic qubit are coupled to inputs of two different ones of the four-way Hadamard interferometer circuits;a set of 2N photon detectors, each photon detector coupled to a different one of the output waveguides of the four-way Hadamard interferometer circuits and configured to produce a classical output signal indicating a count of detected photons; anda classical decision logic circuit configured to receive the classical output signals from the set of 2N photon detectors and to determine, based on the received classical output signals, whether an N-way fusion operation on the plurality of qubits succeeded.

2. The circuit of claim 1 wherein the 2N photon detectors are configured to operate concurrently to produce the classical output signals for a particular set of received qubits.

3. The circuit of claim 1 wherein the classical decision logic circuit is further configured to determine whether the N-way fusion operation succeeded by extracting a detection pattern from the received classical output signals and comparing the detection pattern to a set of success patterns, wherein the N-way fusion operation succeeded if the detection pattern matches one of the success patterns in the set of success patterns.

4. The circuit of claim 3 wherein the classical decision logic is further configured such that the detection pattern includes, for each of the four-way Hadamard interferometer circuits, a 4-tuple of counts of detected photons.

5. The circuit of claim 4 wherein the set of success patterns includes only patterns where each 4-tuple corresponds to a total of two detected photons.

6. The circuit of claim 5 wherein the set of success patterns includes only patterns in which, for at least one of the 4-tuples, the two detected photons are detected by two different detectors.

7. The circuit of claim 5 wherein the number N is equal to 6, the number of 4-tuples is equal to three, and the set of success patterns consists of:a first subset of success patterns in which, for all three of the 4-tuples, the two detected photons are detected by two different detectors;a second subset of success patterns in which, for one of the three 4-tuples, the two detected photons are detected by the same detector and for the other two 4-tuples, the two detected photons are detected by different detectors such that the other two 4-tuples have either an identical pattern of photon counts or a complementary pattern of photon counts; and a third subset of success patterns in which, for two of the three 4-tuples, the two detected photons are detected by the same detector and for the other 4-tuple, the two detected photons are detected by different detectors.

8. The circuit of claim 1 wherein each of the four- way Hadamard interferometer circuits comprises:a first beam splitter coupled between a first waveguide and a second waveguide of the four input waveguides;a second beam splitter coupled between a third waveguide and a fourth waveguide of the four input waveguides;a third beam splitter coupled between a first output waveguide of the first beam splitter and a first output waveguide of the second beam splitter; anda fourth beam splitter coupled between a second output waveguide of the first beam splitter and a second output waveguide of the second beam splitter.

9. The circuit of claim 8 wherein the first, second, third, and fourth beam splitters are 50 / 50 beam splitters.

10. The circuit of claim 1 wherein the N-way fusion operation corresponds to projection onto an N-GHZ state.

11. The circuit of claim 1 wherein each qubit of the plurality of qubits is initially part of a respective quantum system of entangled qubits and wherein the N-way fusion operation consumes the plurality of qubits and produces an entangled quantum system from the remaining qubits of the respective quantum systems.

12. A circuit comprising:a plurality of waveguide pairs to receive a plurality of qubits, wherein each of the qubits is a dual-rail-encoded photonic qubit propagating on one of the waveguide pairs and wherein the total number of qubits in the plurality of qubits is a number (N) that is an odd number greater than 4;a plurality of four-way Hadamard interferometer circuits, each four-way Hadamard interferometer circuit having four input waveguides and four output waveguides, wherein each of the four- way Hadamard interferometer circuits includes a network of beam splitters arranged such that a photon entering on any one of the four input waveguides has an equal probability of exiting on any one of the four output waveguides;a two-way Hadamard interferometer circuit having two input waveguides and two output waveguides, wherein the two-way Hadamard interferometer circuit includes a beam splitter arranged such that a photon entering on either one of the two input waveguides has an equal probability of exiting on any one of the two output waveguides,the waveguides of the waveguide pairs being coupled to the inputs of the fourway Hadamard interferometer circuits and the two-way Hadamard interferometer circuit such that each of the four-way Hadamard interferometer circuits receives one rail from each of four different dual-rail encoded photonic qubits, the two-way Hadamard interferometer circuit receives one rail from each of two different dual-rail encoded photonic qubits, and the two rails of a given dual-rail encoded photonic qubit are coupled to inputs of two different ones of the four- way Hadamard interferometer circuits or the two-way Hadamard interferometer circuit;a set of 2N photon detectors, each photon detector coupled to a different one of the output waveguides of the four-way Hadamard interferometer circuits or the two-way Hadamard interferometer circuit, each of the photon detectors being configured to produce a classical output signal indicating a count of detected photons; anda classical decision logic circuit configured to receive the classical output signals from the set of 2N photon detectors and to determine, based on the received classical output signals, whether an N-way fusion operation on the plurality of qubits succeeded.

13. The circuit of claim 12 wherein the 2N photon detectors are configured to operate concurrently to produce the classical output signals for a particular set of received qubits.

14. The circuit of claim 12 wherein the classical decision logic circuit is further configured to determine whether the N-way fusion operation succeeded by extracting a detection pattern from the received classical output signals and comparing the detection pattern to a set of success patterns, wherein the N-way fusion operation succeeded if the detection pattern matches one of the success patterns in the set of success patterns.

15. The circuit of claim 14 wherein the classical decision logic is further configured such that the detection pattern includes, for each of the four-way Hadamard interferometer circuits, a 4-tuple of counts of detected photons and, for the two-way Hadamard interferometer circuit, a 2-tuple of counts of detected photons.

16. The circuit of claim 15 wherein the set of success patterns includes only patterns where each 4-tuple corresponds to a total of two detected photons and the 2-tuple corresponds to a total of one detected photon.

17. The circuit of claim 16 wherein the set of success patterns includes only patterns in which for at least one of the 4-tuples, the two detected photons are detected by two different detectors.

18. The circuit of claim 16 wherein the number N is equal to 7, the number of 4-tuples is equal to three, and the set of success patterns consists ofa first subset of success patterns in which, for all three of the 4-tuples, the two detected photons are detected by two different detectors;a second subset of success patterns in which, for one of the three 4-tuples, the two detected photons are detected by the same detector and for the other two 4-tuples, the two detected photons are detected by different detectors such that the other two 4-tuples have either an identical pattern of photon counts or a complementary pattern of photon counts; anda third subset of success patterns in which, for two of the three 4-tuples, the two detected photons are detected by the same detector and for the other 4-tuple, the two detected photons are detected by different detectors.

19. The circuit of claim 12 wherein each of the four-way Hadamard interferometer circuits comprises:a first beam splitter coupled between a first waveguide and a second waveguide of the four input waveguides;a second beam splitter coupled between a third waveguide and a fourth waveguide of the four input waveguides;a third beam splitter coupled between a first output waveguide of the first beam splitter and a first output waveguide of the second beam splitter; anda fourth beam splitter coupled between a second output waveguide of the first beam splitter and a second output waveguide of the second beam splitter.

20. The circuit of claim 19 wherein the first, second, third, and fourth beam splitters are 50 / 50 beam splitters.

21. The circuit of claim 12 wherein the N-way fusion operation corresponds to projection onto an N-GHZ state.

22. The circuit of claim 12 wherein each qubit of the plurality of qubits is initially part of a respective quantum system of entangled qubits and wherein the N-way fusion operation consumes the plurality of qubits and produces an entangled quantum system from the remaining qubits of the respective quantum systems.

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