Quantum entanglement generation in photonic circuits with error robustness

Abstract

Photonic circuits can generate entangled "seed" states of two or more photonic qubits, such as Bell states, 3-GHZ states, 4-GHZ states, and 4-Line states. A set of input waveguides to receive photons is coupled to via a network of linear optical components to a set of output waveguides including a first subset of output waveguides and a second subset of output waveguides. The network of linear optical components can implement a unitary transform operation that probabilistically produces either a success output state or a failure output state. Photon detectors coupled to the second subset of output waveguides can provide photon counts to a classical decision logic circuit that determines whether the output state is the success output state or the failure output state. The unitary transform can be selected and output waveguides can be assigned to the subsets in a manner that provides robustness against errors in the circuit.

Description

QUANTUM ENTANGUEMENT GENERATION IN PHOTONIC CIRCUITS WITH ERROR ROBUSTNESSCROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of U.S. Provisional Application No. 63 / 542,061, filed October 2, 2023, 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] Certain embodiments described herein relate to photonic circuits that can generate entangled states of two or more photonic qubits. Such states are referred to herein as “seed states.” Examples include Bell states (a maximally entangled state of two qubits) and 3-GHZ states (a maximally entangled state of three qubits). In some embodiments, a photonic seed state generation circuit can include a set of input waveguides to receive photons and a set of output waveguides including a first subset of output waveguides and a second subset of output waveguides. (It should be understood that the output waveguides need not have any particular physical arrangement.) A network of linear optical components can couple the input waveguides to the output waveguides. The network of linear optical components can implement a unitary transform operation that probabilistically produces, in response to an input event that includes a photon being input to each of the input waveguides, an output state that is either: a success output state in which the first subset of output waveguides propagates one of a set of target entangled qubit states while the second subset of output waveguides propagates a corresponding one of a set of success patterns of photons; or a failure outputstate in which the first subset of output waveguides propagates one of a set of invalid qubit states while the second subset of output waveguides propagates a corresponding one of a set of failure patterns of photons. Each of the output waveguides in the second subset of output waveguides can be coupled to a photon detector that is configured to produce a classical output signal indicating a number of photons detected. A classical decision logic circuit can be coupled to the photon detectors and configured to receive a readout pattern comprising the respective classical output signals from the photon detectors, to determine whether each of the input waveguides received a photon, to compare the readout pattern to the set of success patterns, and to output a classical decision signal that indicates success in the event that each of the input waveguides received a photon and the readout pattern matches one of the success patterns and indicates failure otherwise. With appropriate assignment of the output waveguides to the first and second subsets, the success patterns and the failure patterns can be defined such that a false indication of success occurs only if multiple errors occur. For example, the unitary transform operation can satisfies one or both of the following criteria: (a) a coordinate distance between any one of the success patterns and any one of the failure patterns is at least 2; or (b) for any failure pattern with a coordinate distance less than 2 from one of the success patterns, the first subset of waveguides propagates an invalid qubit state in which none of the qubits is in a valid logical state.

[0004] 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

[0005] FIG. 1 shows two representations of a portion of a pair of waveguides corresponding to a dual-rail-encoded photonic qubit.

[0006] FIG. 2A shows a schematic diagram for coupling of two modes.

[0007] FIG. 2B shows, in schematic form, a physical implementation of mode coupling in a photonic system that can be used in some embodiments.

[0008] 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.

[0009] FIG. 4A shows another schematic diagram for coupling of two modes.

[0010] 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.

[0011] 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.

[0012] 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.

[0013] FIG. 7 shows a circuit diagram for a dual-rail-encoded Bell state generator that can be used in some embodiments.

[0014] FIG. 8A shows a circuit diagram for a dual-rail-encoded type I fusion gate that can be used in some embodiments.

[0015] FIG. 8B shows example results of type I fusion operations using the gate of FIG. 8A.

[0016] FIG. 9A shows a circuit diagram for a dual-rail-encoded type II fusion gate that can be used in some embodiments.

[0017] FIG. 9B shows an example result of a type II fusion operation using the gate of FIG. 9A.

[0018] FIG. 10 illustrates an example of a qubit entangling system in accordance with some embodiments.

[0019] FIG. 11 A shows two representations of a portion of a single waveguide corresponding to a temporally encoded qubit.

[0020] FIG. 1 IB shows an example of an optical circuit that can convert a spatially- encoded qubit to a temporally-encoded qubit.

[0021] FIG. 11C shows an example of an optical circuit that can convert a temporally- encoded qubit to a spatially-encoded qubit.

[0022] FIG. 12 shows a simplified circuit schematic of a seed state generator circuit according to some embodiments.

[0023] FIG. 13 shows a high-level schematic diagram of a photonic Bell state generator circuit according to some embodiments.

[0024] FIGs. 14A and 14B show schematic diagrams of implementations of the optical circuit of FIG. 13 according to various embodiments.

[0025] FIG. 15 shows a high-level schematic diagram of a photonic Bell state generator circuit according to some embodiments.

[0026] FIG. 16 shows a schematic diagram of an implementation of the circuit of FIG. 15 according to some embodiments.

[0027] FIG. 17 shows a high-level schematic diagram of a photonic Bell state generator circuit for temporally encoded qubits according to some embodiments.

[0028] FIG. 18 shows a high-level schematic diagram of a photonic Bell state generator circuit for temporally encoded qubits according to some embodiments.

[0029] FIG. 19 shows a high-level schematic diagram of a photonic 3 -GHZ state generator circuit according to some embodiments.

[0030] FIG. 20 shows a schematic diagram of an implementation of the circuit of FIG. 19 according to some embodiments.

[0031] FIG. 21 shows a high-level schematic diagram of a photonic 3 -GHZ state generator circuit with temporally encoded qubits according to some embodiments.

[0032] FIG. 22 shows a simplified schematic diagram of a spatial Hadamard-4 (“H4”) 3- GHZ state generator circuit with time-bin encoded outputs according to some embodiments.

[0033] FIG. 23 shows a simplified schematic diagram of a hybrid (spatial -temporal) H4 3- GHZ state generator circuit according to some embodiments.

[0034] FIG. 24 shows a simplified schematic diagram of a temporal 3 -GHZ state generator circuit according to some embodiments.

[0035] FIG. 25 illustrates a simplified schematic diagram of a spatial 4*3-DFT 4-GHZ generator circuit according to some embodiments.

[0036] FIG. 26 shows a simplified schematic diagram of a hybrid 4x3 DFT 4-GHZ generator circuit according to some embodiments.

[0037] FIG. 27 shows a simplified schematic diagram of a temporal 4*3-DFT 4-GHZ generator circuit according to some embodiments.

[0038] FIG. 28 A shows a simplified schematic diagram of a spatial 4x3 DFT 4-Line generator circuit according to some embodiments.

[0039] FIG. 28B shows a circuit diagram of a four-dimensional DFT coupler according to some embodiments.

[0040] FIG. 29 shows a simplified schematic diagram of a hybrid 4x3 DFT 4-Line generator circuit according to some embodiments.

[0041] FIG. 30 shows a simplified schematic diagram of a temporal 4x3-DFT 4-line generator circuit according to some embodiments.

[0042] FIG. 31 shows a simplified schematic diagram of a circuit that implements an expanded version of the spatial 4x3-DFT 4-GHZ generator circuit of FIG. 25 according to some embodiments.DETAILED DESCRIPTION

[0043] 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, Section 2 describes examples of entanglement generators 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.

[0044] 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 ri), which can be used to encode n bits of information. For thesake 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

[0045] 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.

[0046] 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.

[0047] 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 | 1001)1 2 3 4specifies 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)42,3,4 can be physically implemented as four distinct waveguides 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 1111 l)x,2,3,4that represents each mode occupied by one particle and the four- particle Fock state |2200)1 2 3 4that 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)42 3 4modes 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, |0010)1 2 3 4is equivalent to |13).1.1. Qubits

[0048] 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 ri), 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 encodeinformation 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).

[0049] 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:|0)L= | 10)lj2(1)|1>L = |01>2(2) where the subscript “L” indicates that the ket represents a logical state (e.g., a qubit value) and, as before, the notation \ij)1>2on the 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

[0050] 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 | / >) of n quantumobjects is a separable state if \i ) = | )<8) ... ® |i / in), 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:

[0051] More generally, an / / -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

[0052] 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 modecouplers, 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.

[0053] 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.

[0054] 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 iscoupled 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 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.

[0055] 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.

[0056] 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.

[0057] 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 ofbeing occupied, e.g., a state a 110) += 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.

[0058] 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:where 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:Thus, the action of the mode coupler described by Eq. (9) is to take the input states 110), 101), and 111) to

[0059] 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 T introduced above). If other transfer matrices are desired, thereflectivity (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.

[0060] 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 a 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.

[0061] 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, butany type of mode coupler / beam splitter can be used without departing from the scope of the present disclosure.

[0062] 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. 2 A, 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.

[0063] 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.

[0064] 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 from the 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.

[0065] 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.

[0066] 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.

[0067] 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 in the 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).

[0068] 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.

[0069] 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.)

[0070] 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 mode spreading 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-deterministic (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 alsobe 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 subsequent operations without the need for active optical switches to implement selectable mode swapping at the output of Bell state generator 700.

[0071] 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.

[0072] 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 (physicallyrealized, 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.

[0073] 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.

[0074] 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.

[0075] 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.

[0076] 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 B were 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.

[0077] 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.)

[0078] 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.

[0079] 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 bymodes 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 is performed 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.

[0080] 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.

[0081] 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.

[0082] 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 anentangled 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.

[0083] 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 1005 may 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.

[0084] 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.

[0085] 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 opticalcircuits, 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.

[0086] 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, memory 1004 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.

[0087] 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.

[0088] 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.

[0089] 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 by classical 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.

[0090] 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.

[0091] 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” isused 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 approximately 1% 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.

[0092] 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.

[0093] 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.

[0094] 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.

[0095] 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.

[0096] In some embodiments, an entangled system of multiple physical qubits can be mapped to one or more “logical qubits,” and operations associated with a quantum computation 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

[0097] 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 t7 +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 tj; 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 tj+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 beunderstood that different temporally-encoded qubits can be propagating through different sections of the same waveguide at different times.

[0098] 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 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 present in 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 2x 1 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.

[0099] 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 1154into 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.

[0100] 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 may be 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. Seed State Generation

[0101] For quantum computing, quantum communication, and other applications, it can be useful to produce entangled systems of a few qubits, such as Bell states, 3 -GHZ states, or larger / / -GHZ states. Such systems are referred to herein as “seed states.” In linear optical implementations, seed states can be generated for spatially-encoded qubits (e.g., as shown in FIG. 1) or temporally-encoded qubits (e.g., as shown in FIG. 11 A) by propagating a set of input photons through a network of beam splitters and phase shifters that implements an appropriate unitary transform matrix. For instance, one photon can be input to each input waveguide of the network. (Alternatively, for some networks, some input waveguides can receive photons while other input waveguides receive vacuum, or no photons.) The transform can probabilistically (or stochastically, or non-deterministically) produce any one of several possible output states on the output waveguides of the network. Some, though generally not all, of these output states can include photons in a desired seed state propagating on a first subset of the output waveguides (referred to herein as “propagation outputs”). To determine whether the seed state was produced in a given instance, the remaining output waveguides of the network (referred to herein as “heralding outputs”) can be coupled to detection circuitry, which can include photon-counting detectors that indicate the number of photons detected. In accordance with the unitary transform matrix, certain patterns of photons on the heralding outputs occur when a desired seed state is produced on the propagation outputs, while other patterns of photons on the heralding occur when the stateon the propagation outputs is not a desired seed state. As used herein, a “success pattern” refers to a pattern of photons on the heralding outputs of a seed state generator circuit that, in an ideal circuit, occurs when the seed state is produced on the propagation outputs, and a “failure pattern” refers to a detection pattern in the heralding outputs of a seed state generator circuit that, in an ideal circuit, occurs when a state other than the seed state is present on the propagation outputs; states other than the seed state can include states with one or more invalid qubit states on the propagation outputs. (An “ideal circuit” in this context is a circuit with no photon loss, no detector inefficiency, and no photonic or electronic noise.) It should be understood that the success patterns and failure patterns for a given seed state and unitary transform matrix (and the probabilities of occurrence) can be computed using matrix algebra and / or simulations.

[0102] In this context, Bell state generator 700 of FIG. 7 (described in detail above) can be understood as a seed state generator circuit. A unitary transform, implemented by a network of linear optical components that includes mode couplers 731-1 through 731-4 and mode coupler 737, can generate a Bell state on the four propagation outputs (labeled as the logical states of Qubit 1 and Qubit 2). Detectors 738-1 through 738-4 are coupled to the four heralding outputs of the network. Classical decision logic circuit 740 can receive the photon counts from detectors 738-1 through 738-4 and determine whether the counts correspond to a success pattern or a failure pattern. In this example, four photons are input on input waveguides 732-1 through 732-4 while four other input waveguides 732-5 through 732-8 receive vacuum (no photons). The success patterns have exactly one photon in each of two of detectors 738-1 through 738-4, in any combination, while any pattern with more or fewer than two total photons, or with two photons in the same one of detectors 738-1 through 738-4 is a failure pattern. A detection pattern can be expressed as a vector, with each component representing photon count in one of the detectors. In this example, success patterns can include (1, 1, 0, 0), (1, 0, 1, 0), (0, 1, 1, 0), and other permutations of two zeroes and two ones. Failure patterns can include (2, 0, 0, 0), (2, 0, 0, 2), (0, 1, 0, 0), and other patterns that do not include two zeroes and two ones.

[0103] Bell state generator 700 is susceptible to detector errors. For instance, photoncounting detectors may occasionally generate dark counts (or noise), where a photon is counted even though no photon is actually present. A dark count can convert a failure pattern such as (0, 1, 0, 0) into a success pattern such as (1, 1, 0, 0). This can result in classical decision logic 740 producing a false positive, i.e., indicating success when a Bell state is notactually present on the propagation outputs. False positives may lead to downstream errors. It should be understood that false negatives are also possible; however, false negatives are less likely to produce downstream errors.

[0104] Certain embodiments described herein relate to seed state generators with increased robustness against false positives resulting from detector noise or inefficiency, which can be achieved, e.g., by increasing the coordinate distance between success patterns and failure patterns. As used herein, “coordinate distance” refers generally to a measure of the number of changes required to convert one pattern of photons in the detectors into another. Examples of specific coordinate-distance metrics include Manhattan distance (or taxicab distance), Hamming distance, edit distance, and the like.

[0105] FIG. 12 shows a conceptual schematic diagram of a generalized seed state generator circuit 1200 according to some embodiments. Seed state generator circuit 1200 includes a number (N) of input paths (e.g., waveguides) 1201-1 through 1202-N, each of which can receive an input photon from a photon source (not shown). The photon sources can be, for example, a set of heralded single photon sources as described above. Each heralded single photon source can produce a pair of photons, one of which is delivered to one of input paths 1202-1 through 1202-N while the other is delivered to a detector (not shown) that produces a heralding signal indicating whether a photon was detected. As described below, circuit 1200 can use the heralding signals from the photon sources to determine whether the input state is valid in a given instance of operation.

[0106] Input paths 1202-1 through 1202-N are coupled as inputs to a network 1210 of linear optical components (e.g., beam splitters and phase shifters) that implements an N*N unitary transform. As described above, the unitary transform can be defined with reference to a unitary matrix, the coefficients (or elements) of which can be implemented using beam splitters and phase shifters. The particular implementation depends on the transform; examples of transforms and implementations are described below. Network 1210 has N output paths, which are divided (logically and / or physically) into a number (M) of heralding outputs 1216-1 through 1216-M and a number (N-M) of propagation outputs 1214-1 through 1214-(N-M). In some embodiments, N-M can be a multiple of two, and photons on propagation outputs 1214-1 through 1214-(N~M) can be interpreted as an entangled system of (N-M) / 2 dual-rail-encoded qubits.

[0107] Heralding outputs 1216-1 through 1216-M are coupled to a set of M photoncounting detectors 1220-1 through 1220-M. In some embodiments, a mode information erasure circuit 1218 can be disposed along the optical path of heralding outputs 1216-1 through 1216-M, downstream of network 1210 and upstream of detectors 1220-1 through 1220-M. Mode information erasure circuit 1218 can implement an M unitary transform having the property that a photon input on any one of paths 1216-1 through 1216-M has an equal probability of being output on any one of the M output paths of mode information erasure circuit 1218. Detector circuits 1220-1 through 1220-M can be photon-counting detectors that each provide a classical output signal indicating the number of photons counted.

[0108] Classical decision logic circuit 1230 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 1230 can receive the respective photon count signals from detectors 1220-1 through 1220-M, e.g., via a signal path 1222, and can determine whether the pattern of photon counts corresponds to a success pattern. For instance, classical decision logic circuit 1230 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. (As described above, the set of success pattern(s) can be determined given a particular unitary transform and desired seed state; accordingly, success patterns can be stored or programmed into a memory, which can be read-only memory, for use during operation.) When photon-count signals are received via signal path 1222, classical decision logic circuit 1230 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 130 can output a classical decision signal 1232 indicating whether a match has occurred.

[0109] Classical decision logic circuit 1230 can also receive input-photon heralding signals from the photon sources that provide photons to input paths 1202-1 through 1202-N, e.g., via a signal path 1228. In some embodiments, classical decision logic circuit 1230 can also determine, based on the input-photon heralding signals on path 1228, whether the input state was valid. In examples described herein, valid input states have one photon on each input path 1202-1 through 1202-N. Decision logic circuit 1230 can implement the following decision logic: If the signals on signal path 1228 indicate a valid input state and the readout pattern of photon counts on signal path 1222 matches a success pattern, then the decisionsignal on path 1232 is set to indicate success (e.g., Boolean true); otherwise, the decision signal on path 1232 is set to indicate failure (e.g., Boolean false).

[0110] Using circuit 1200 as a model, a number of different seed state generator circuits can be implemented by selecting an appropriate N*N unitary transform to be implemented in network 1210 and assigning each output waveguide to be either a propagation output 1214 or a heralding output 1216. With appropriate selection of the transform and assignment of the output waveguides, robustness against errors in circuit 1200 can be enhanced.[OHl] More specifically, given a particular N*N unitary transform and a specific input state (e.g., all N input waveguides occupied by a photon), a set of possible patterns of photons on the N output waveguides can be computed for the case of an ideal circuit. Based on the set of possible patterns, the N output waveguides can be divided into a first subset of N-M propagation outputs and a second subset of M heralding outputs. The division of the output waveguides into these subsets can be based on various considerations, such as maximizing probability of the seed state being generated on the propagation outputs and on the correlation between seed state (or absence thereof) on the propagation outputs and distinct patterns of photons on the heralding outputs. It should be understood that the division of waveguides into these subsets is a logical division and need not correspond to any particular physical arrangement of waveguides.

[0112] According to some embodiments, the N*N unitary transform can correspond to a unitary matrix that is decomposable into component matrices that correspond to the beam splitters and phase shifters of network 1210. The unitary matrix can possess additional symmetry properties, which can include having the same magnitude (e.g., complex- valued roots of 1) for all elements, or having the same magnitude for elements that couple inputs to the heralding outputs. Such properties allow the output waveguides to be assigned to propagation outputs and heralding outputs in a manner such that destructive interference results in making false indications of success due to errors within circuit 1200 less likely to occur and / or easer to detect.

[0113] While the identification of success patterns and failure pattern can be determined from an ideal circuit, it is expected that in a real circuit occasional errors will occur, including detector inefficiency or photon loss (leading to a photon that is present not being counted) and / or noise or dark counts (leading to a photon being counted where none is present). Such errors can change the pattern of photon counts received by decision logic 1230 (referred toherein as a “readout pattern” or “received pattern”) and can, in some instances, turn a failure pattern into a success pattern, resulting in a false positive signal output from decision logic 1230. False positives can lead to downstream errors if the propagation outputs are treated as carrying the seed state when in fact they are not. (While false negatives can also occur, these are of less concern for purposes of this disclosure.)

[0114] According to some embodiments, false positives in a seed state generator circuit can be reduced by selecting the transform and dividing the output waveguides into heralding and propagation outputs in a manner such that destructive interference results in the circuit having the property that a false positive only occurs if multiple errors occur in the same operating cycle. For instance, if the readout pattern from detectors 1220 is expressed as coordinates (cl, c2, . . ., cM), a coordinate distance between success patterns and failure patterns can be defined. Standard distance metrics such as Manhattan distance, Hamming distance, edit distance, or the like can be used to define coordinate distance.

[0115] According to some embodiments, the downstream effect of a false positive in a seed state generator circuit can be mitigated by selecting the transform and dividing the output waveguides into heralding and propagation outputs in a manner such that when a false positive occurs, two or more (or all) of the “qubits” on the propagation outputs are in invalid states. This type of multi-qubit correlated error can be easier to detect downstream.

[0116] The following subsections describe specific examples of seed state generator circuits that can reduce the occurrence of false positives and / or mitigate the downstream effects of false positives. It should be understood that, while these examples are presented for purposes of illustration, other circuits can also be used.2.1. Bell State Generator Using Se Transform

[0117] FIG. 13 shows a high-level schematic diagram of a Bell state generator circuit 1300 according to some embodiments. Circuit 1300 includes six input paths 1302-1 through 1302- 6 coupled to a network 1310 of linear optical components. Network 1310 implements a unitary transform matrix referred to herein as Se. The Se matrix can be expressed as:where w = e27Tt / 3.

[0118] Network 1310 has six output paths, divided into four propagation outputs 1314-1 through 1314-4 and two heralding outputs 1316-1 and 1316-2. Two 50 / 50 beam splitters 1317-1, 13712-2 can couple pairs of propagation outputs 1314-1, 1314-2 (coupled via beam splitter 1317-1) and 1314-3, 1314-4 (coupled via beam splitter 1317-2). Beam splitters 1317- 1, 1317-2 can convert the output Bell state into a standard state given by (| 1100) + a|0011)) / V2, where a is a phase. Heralding outputs 1316-1 and 1316-2 are coupled to photon-counting detectors 1320-1 and 1320-2, which provide respective photon counts to classical decision logic circuit 1330, which can be similar to classical decision logic circuit 1230 described above.

[0119] Specific implementations of circuit 1300 using beam splitters and phase shifters are illustrated in FIGs. 14A and 14B, which show schematic diagrams of optical circuits 1400 and 1450 implementing Bell state generator circuit 1300 according to various embodiments. In the notation of FIGs. 14A and 14B, phase shifters are indicated by rectangles (e.g., rectangles 1405, 1455) labeled to indicate the amount of phase shift (in radians, in a range from — 7T to TT). Beam splitters are shown as lines connecting pairs of waveguides. A 50 / 50 beam splitter is represented by a line with circles at the end (e.g., lines 1407, 1457). Asymmetric beam splitters are represented by colored lines with diagonal boxes at the ends, with color indicating the transmissivity. In FIG. 14A, beam splitters 1409-1 and 1409-2 have transmissivity 1 / 5; beam splitter 1411 has transmissivity 3 / 8; and beam splitter 1413 has transmissivity 1 / 6. In FIG. 14B, beam splitters 1459-1 and 1459-2 have transmissivity 1 / 5; beam splitter 1461 has transmissivity 3 / 8; and beam splitter 1463 has transmissivity 1 / 6. It should be understood that the circuits in FIGs. 14A and 14B implement the same transform, but with different combinations of components. For instance, circuit 1450 in FIG. 14B uses fewer phase shifters than circuit 1400 of FIG. 14A. It should also be understood that other implementations of the Se transform using linear optical components are also possible.

[0120] In embodiments of Bell state generator circuit 1300, the output waveguides are divided into propagation outputs and heralding outputs such that, when one photon is input on each of input waveguides 1302-1 through 1302-6, a Bell state is probabilistically produced on propagation outputs 1314-1 through 1314-4 (probability 5.5%). Success is heralded by the presence of exactly two photons at each of detectors 1320-1 and 1320-2. The success pattern can be denoted as (2, 2). Patterns (3, 2), (2, 3), (2 ,1), and (1, 2), which haveManhattan distance of 1 from the success pattern, are never produced in an ideal circuit. Put differently, circuit 1300 has no failure patterns with Manhattan distance of 1 from the success pattern. (“Manhattan distance” refers to the sum of differences of coordinates.) In operation of an actual circuit, readout patterns (3,2), (2,3), (2,1), or (1,2) may occasionally occur; however, any such occurrence would be due to an error, such as a dark count or an undetected photon. In practice, the absence of failure patterns with Manhattan distance of 1 from the success pattern means that at least two errors would need to occur in order to convert a failure pattern into the success pattern. This is because the patterns with Manhattan distance of 1 from the only occur if at least one error occurs, and another error would have to occur in order to further modify the pattern to match the success pattern (2, 2). Accordingly, circuit 1300 can be more robust against detector errors than circuits such as Bell state generator circuit 700 described above.2.2. Bell State Generator Using Two DFT.; Circuits

[0121] Circuit 1300 has high robustness against false positives; however, the success probability is 5.5%. In some embodiments, success probability of a Bell state generator circuit can be increased by using a different unitary transform. FIG. 15 shows a high-level schematic diagram of a Bell state generator circuit 1500 according to some embodiments. Circuit 1500 includes six input paths 1502-1 through 1502-6 coupled to a network of linear optical components. In this example, the network of linear optical components implements two three-dimensional Discrete Fourier Transforms (DFT3) 1510-1, 1510-2. Three input paths are coupled to each DFT3 network 1510-1, 1510-2. One output from each DFT3 network 1510-1, 1510-2 is used as a heralding output 1516-1, 1516-2, and the other outputs are used as propagation outputs 1514-1 through 1514-4. Heralding outputs 1516-1 and 1516- 2 are coupled to photon counting detectors 1520-1 and 1520-2 downstream of a mode information erasure circuit 1518 (implemented, e.g., as a 50 / 50 beam splitter). Photon counting detectors 1520-1 and 1520-2 provide respective photon counts to classical decision logic circuit 1530, which can be similar to classical decision logic circuit 1230 described above.

[0122] A specific implementation of Bell state generator circuit 1500 using beam splitters and phase shifters is illustrated in FIG. 16, which shows a schematic diagram of an optical circuit 1600 according to some embodiments. Similarly to FIGs. 14A-14B, phase shifters are indicated by rectangles 1605, 1607, labeled to indicate the amount of phase shift (in this case, TT / 2 radians). Beam splitters are shown as vertical lines connecting pairs of waveguides. A50 / 50 beam splitter is represented by a line with circles at the end (e.g., line 1609, 1610), and a beam splitter with 1 / 3 transmissivity is represented by a line with diagonal boxes at the ends (lines 1611, 1613). It should also be understood that other networks of linear optical components can also be used to implement a DFT3.

[0123] In embodiments of Bell state generator circuit 1500, the output waveguides are divided into propagation outputs and heralding outputs such that, when one photon is input to each of input waveguides 1502-1 through 1502-6, a Bell state is probabilistically produced on propagation outputs 1514-1 through 1514-4 (probability 14.5%). Success is heralded by any of the following detection patterns: (4,0), (0,4) (1,3), and (3,1). Four-photon failure pattern (2,2) is not produced in an ideal circuit. In operation, the four-photon pattern (2, 2) may be received; however, any such occurrence would be due to an error. Further, the ideal circuit has no failure patterns with a total of five photons. In operation, readout patterns with five photons may occur; however, any such occurrence would be due to an error, such as a dark count or an undetected photon. Put differently, for Bell state generator circuit 1500, all success patterns have a number (P) of photons, and failure patterns with P+1 photons do not exist. Consequently, a single undetected photon would not convert a failure pattern into a success pattern. Accordingly, circuit 1500 can be more robust against detector errors than circuits such as Bell state generator circuit 700 described above while providing higher success probability than circuit 1300. In addition, circuit 1500 has the property that if a single dark count results in a false positive, the propagation outputs will have an occupancy pattern of the form 13000), which corresponds to one qubit with one mode containing three photons and one qubit containing zero photons. In other words, if a false positive occurs due to a dark count, the qubit errors are correlated and can be detected by downstream components that operate on the qubits.

[0124] Circuit 1500 produces Bell states with dual -rail-encoded qubits. In some embodiments, a variation of circuit 1500 can be used to produce Bell states with temporally- encoded qubits (e.g., as described above with reference to FIG. 11 A). By way of illustration, FIG. 17 shows a high-level schematic diagram of a Bell state generator circuit 1700 according to some embodiments. Circuit 1700 includes three input paths 1702-1 through 1702-3 coupled to a single DFT3 network 1710, which can be implemented in the manner shown in FIG. 16. Two outputs from DFT3 network 1710 are used as propagation outputs 1714-1, 1714-2, while the third output is used as heralding output 1716. Heralding output 1716 is coupled to a temporal -to-spatial conversion circuit 1717, which can be implementedsimilarly to optical circuit 1150 of FIG. 11C. Outputs of circuit 1717 are coupled to photoncounting detectors 1720-1, 1720-2 downstream of a mode information erasure circuit 1718, which can be implemented using a 50 / 50 beam splitter.

[0125] In operation, one photon is input to each of input waveguides 1702-1 through 1702- 3 during each of two successive time bins, for a total of six photons. Each group of three photons passes through the same DFT3 network 1710 (at different times), resulting in a Bell state of temporally encoded qubits being probabilistically produced on propagation outputs 1714-1, 1714-2. The photons (if present) on heralding output 1716 in the two time bins are temporally aligned using circuit 1717 and pass through a mode information erasure circuit 1718 (implemented, e.g., using a 50 / 50 beam splitter). Photon counting detectors 1720-1 and 1720-2 provide respective photon counts to classical decision logic circuit 1730, which can be similar to classical decision logic circuit 1530 described above. The success patterns and failure patterns (and the success probability) are the same as in circuit 1500. Accordingly, circuit 1700 has the same robustness against error as circuit 1500.

[0126] For some applications, Bell state generator circuit 1700 has the advantage that the Bell state is produced in a temporal encoding. As noted above, this may be convenient if the Bell state is to be stored (e.g., using optical fiber). While Bell states produced using circuit 1500 can be converted to temporal encoding (e.g., using circuit 1120 of FIG. 1 IB), active optical switches can be lossy, and avoiding an extra active optical switch along the propagation path of the qubits may be desirable. In circuit 1700, the only active optical switch is in temporal -to-spatial conversion circuit 1717, which acts on the heralding outputs rather than the propagation outputs. This arrangement can further increase robustness against false positives where temporally encoded qubits are desired.

[0127] In some embodiments, a variation of circuit 1500 operates in a “hybrid” manner that combines temporal and spatial encoding. By way of illustration, FIG. 18 shows a high-level schematic diagram of a Bell state generator circuit 1800 according to some embodiments. Circuit 1800 includes three input paths 1802-1 through 1802-3 coupled to a first DFT3 network 1810-1 and three input paths 1802-4 through 1802-6 coupled to a second DFT3 network 1810-2. DFT3 networks 1810-1, 1810-2 can be implemented in the manner shown in FIG. 16. One output from each DFT3 network 1810-1, 1810-2 is as a heralding output 1816- 1, 1816-2, and the other outputs are intermediate outputs 1813-1 through 1813-4. Heralding outputs 1816-1 and 1816-2 are coupled to photon counting detectors 1820-1 and 1820-2downstream of a mode information erasure circuit 1818 (implemented, e.g., as a 50 / 50 beam splitter). Photon counting detectors 1820-1 and 1820-2 provide respective photon counts to classical decision logic circuit 1830, which can be similar to classical decision logic circuit 1530 described above. Heralding output 1816-1 includes a delay line 1817, which can be, e.g., an extra length of waveguide compared to heralding output 1816-2, upstream of mode information erasure circuit 1818. A first 2-to-l optical switch 1819-1 has inputs coupled to intermediate outputs 1813-1 and 1813-3 (one propagation output of each of DFT3 networks 1810-1 and 1810-2) and a propagation output 1814-1. A second 2-to-l optical switch 1819-2 has inputs coupled to intermediate outputs 1813-2 and 1813-4 and a propagation output 1814- 2. Each of optical switches 1819-1 and 1819-2 can be implemented, e.g., using a MZI.

[0128] In operation, one photon is input to each of input waveguides 1802-1 through 1802- 3 during a first time bin (0), and one photon is input to each of input waveguides 1802-4 through 1802-6 during a second time bin (ti) (later than the first time bin), for a total of six photons. The photons of the first time bin pass through DFT3 network 1810-1. Output 1813- 1 of DFT3 network 1810-1 is delivered via optical switch 1819-1 to propagation output 1814- 1; output 1813-2 of DFT3 network 1810-1 is delivered via optical switch 1819-2 to propagation output 1814-2; and heralding output 1861-1 of DFT3 network 1810-1 is delivered through delay line 1817 to mode information erasure circuit 1818. The photons of the second time bin pass through DFT3 network 1810-2. Output 1813-3 of DFT3 network 1810-2 is delivered via optical switch 1819-1 to propagation output 1814-1; output 1813-4 of DFT3 network 1810-2 is delivered via optical switch 1819-2 to propagation output 1814-2; and output 1816-2 of DFT3 network 1810-2 is delivered to mode information erasure circuit 1818. It should be noted that the temporal offset between 0 and tz can be maintained such that output photons from DFT3 network 1810-1 arrive at optical switches 1819-1 and 1819-2 earlier than output photons from DFT3 network 1810-2, resulting in temporally encoded qubits being output on propagation outputs 1814-1 and 1814-2. Delay line 1817 can impose a delay such that photons for the first and second time bins arrive at mode information erasure circuit 1818 in temporal alignment. Photon counting detectors 1820-1 and 1820-2 provide respective photon counts to classical decision logic circuit 1830, which can be similar to classical decision logic circuit 1530 described above. The success patterns and failure patterns (and the success probability) are the same as in circuit 1500. Accordingly, circuit 1800 has the same robustness against detector error as circuit 1500. Compared to circuit1700, circuit 1800 can allow more flexibility in setting the temporal separation between the time bins.2.3. 3-GHZ State Generator Using Hadamard Transforms

[0129] In the previous examples, the seed states are Bell states. In some embodiments, other seed states can be generated using other circuits. FIG. 19 shows a high-level schematic diagram of a 3-GHZ state generator circuit 1900 according to some embodiments. Circuit 1900 includes eight input paths 1902-1 through 1902-8. Input paths 1902-1 through 1902-4 are coupled to a first beam splitter network 1910-1 implementing a second-order (4^4) Hadamard transform, and input paths 1902-5 through 1902-8 are coupled to a second beam splitter network 1910-2 that also implements a second-order Hadamard transform. One output of each network 1910-1, 1910-2 is used as a heralding output 1916-1, 1916-2, while the other three outputs of each network 1910-1, 1910-2 are used as propagation outputs 1914- 1 through 1914-6. Heralding outputs 1916-1, 1916-2 are coupled to mode information erasure circuit 1918, (implemented, e.g., using a 50 / 50 beam splitter), and the outputs of mode information erasure circuit 1918 are delivered to photon-counting detectors 1920-1, 1920-2. Photon counting detectors 1920-1 and 1920-2 provide respective photon counts to classical decision logic circuit 1930, which can be similar to classical decision logic circuit 1230 described above.

[0130] A specific implementation of circuit 2000 using 50 / 50 beam splitters is illustrated in FIG. 20, which shows a schematic diagram of an optical circuit 2000 according to some embodiments. The Hadamard networks can be implemented using 50 / 50 beam splitters 2005 and mode crossings as shown. Other implementations are also possible.

[0131] In embodiments of 3-GHZ state generator circuit 1900, the output waveguides are divided into propagation outputs and heralding outputs such that, when one photon is input to each of input waveguides 1901-1 through 1901-8, a 3-GHZ state is probabilistically produced on propagation outputs 1914-1 through 1914-6. Success is heralded by a total of five photons counted at detectors 1920-1, 1920-2. In this instance, a failure pattern with a total of six photons can occur, and this may lead to a false positive if one of the six photons is not detected. However, if the six-photon failure pattern occurs, the propagation outputs 1914-1 through 1914-6 would propagate a pattern of photons in which none of the qubits of the supposed 3-GHZ state is in a logically valid state. Specifically, one of the qubits would be a two-photon state while the other two qubits would be vacuum (zero-photon) states (e.g.,|002000)). Accordingly, while a false positive may occur in circuit 1900, qubit errors on the propagation outputs would be correlated, making them easier to detect in downstream components. In addition, if a single dark count results in a false positive, the propagation outputs would propagate a pattern of photons in which none of the qubits of the supposed 3- GHZ state is in a logically valid state. Specifically, the pattern would include one four- photon qubit and two zero-photon qubits (e.g., |400000)) or two two-photon qubits and one zero-photon qubit (e.g., 1202000)). In this manner, the effect of a false positive is mitigated due to correlation of qubit errors. Circuit 1900 thus illustrates another type of enhanced robustness provided in some embodiments: a success pattern has a number P of photons, and a failure pattern with P+1 photons corresponds to a state in which none of the qubits on the propagation outputs are in a logically valid state. In circuit 1900, failure patterns with P+2 photons do not occur.

[0132] Circuit 1900 produces 3 -GHZ states with dual -rail-encoded qubits. In some embodiments, a variation of circuit 1900 can be used to produce 3 -GHZ states with temporally-encoded qubits (e.g., as described above with reference to FIG. 11 A). By way of illustration, FIG. 21 shows a high-level schematic diagram of a photonic circuit 2100 according to some embodiments. Circuit 2100 includes four input paths 2102-1 through 2102-4, coupled to a single beam splitter network 2110 implementing a second-order Hadamard transform, which can be implemented in the manner shown in FIG. 20. Three outputs from network 2110 are used as propagation outputs 2114-1 through 2114-3, while the fourth output is used as a heralding output 2116. Heralding output 2116 is coupled to a temporal -to-spatial conversion circuit 2117, which can be implemented similarly to optical circuit 1150 of FIG. 11C. Outputs of circuit 2117 are coupled to photon-counting detectors 2120-1, 2120-2 downstream of a mode information erasure circuit 2118, which can be implemented using a 50 / 50 beam splitter.

[0133] In operation, one photon is input on each of input paths 2102-1 through 2102-4 during each of two successive time bins, for a total of eight photons. Each group of four photons passes through the same Hadamard network 2110 (at different times), resulting in a 3-GHZ state of temporally encoded qubits being probabilistically produced on propagation outputs 2114-1 through 2114-3. The photons (if present) on heralding output 2116 in the two bins are temporally aligned using circuit 2117 and pass through mode information erasure circuit 2118. Photon counting detectors 2120-1 and 2120-2 provide respective photon counts to classical decision logic circuit 2130, which can be similar to classical decision logic circuit1230 described above. The success paterns and failure paterns (and the success probability) are the same as in circuit 1900.

[0134] For some applications, 3 -GHZ state generator circuit 2100 has the advantage that the 3-GHZ state is produced in a temporal encoding. As noted above, this may be convenient if the state is to be stored (e.g., in optical fiber). While 3-GHZ states produced using circuit 1900 can be converted to temporal encoding (e.g., using circuit 1120 of FIG. 1 IB), avoiding an extra active switch may be desirable.

[0135] FIGS. 22, 23, and 24 show schematic diagrams of additional circuits for generating a 3-GHZ state in a temporal encoding (also referred to as a time bin encoded format), in accordance with some embodiments. FIG. 22 shows a simplified schematic diagram of a spatial Hadamard-4 (“H4”) 3-GHZ state generator circuit 2200 with time-bin encoded outputs according to some embodiments. As illustrated, each of paths 2202-1 to 2202-8 (e.g., rails, waveguides, fibers) can receive a single photon (indicated as | 1)). In some embodiments, the photons are generated using one or more single photon sources (e.g., four wave mixing source, parametric down conversion source) that operates at a pump light rate (e.g., 1-3 gigahertz) and received by circuit 2200 on paths 2202-1 through 2202-8. Paths 2202-1 to 2202-8 couple the photons into the first beam splitter network 2210-1 and the second beam splitter network 2210-2, each of which implements a 4 x 4 Hadamard transform (e.g., four-component Hadamard unitary transform, also referred to as a second-order Hadamard transform or Hadamard-4), as described above with reference to FIG. 5 (see also FIGs. 19 and 21). As described above, each of beam splitter networks 2210-1, 2210-2 can be implemented as a set of beam spliters (e.g., 50 / 50 splitters, directional couplers, half silvered mirrors) that interfere the light (e.g., erasing distinguishable mode information, such as path information, from photons on the respective modes).

[0136] The output paths from beam splitter networks 2210-1, 2210-2 generate a 3-GHZ state in a dual rail spatial encoding, which is then converted to time bin encoding using sets of delays (output delay line 2217-1, output delay line 2217-2, and output delay line 2217-3) and switches (output optical switch 2222-1, output optical switch 2222-2, and output optical switch 2222-3) as described above with reference to FIGs. 11 A-l 1C. For example, a photonic qubit in dual rail path encoding can have one arm delayed by delay 2217-1, and then both portions of the qubit can be switched on to the output 2214-1 in time bin encoding. In some example embodiments, the delays can be implemented as waveguide delays (e.g., spiralwaveguides or meandering waveguides in a photonic integrated circuit) or fiber delays (e.g., spools of fiber). The switches can be implemented, e.g., using 2^ 1 couplers (e.g., Mach- Zehnder interferometers (MZIs), a 2x2 coupler that implements a phase shift to output one of two inputs onto an output. The time-bin encoded quantum state is then output on outputs 2214-1, 2214-2, and 2214-3.

[0137] In some embodiments, the light from beam splitter networks 2210-1, 2210-2 that is output on detection paths 2216-1, 2216-2 passes through mode information erasure circuit 2222 and is detected by detectors 2220-1 and 2220-2, which generate detection data. The detection data can be received by classical decision logic 223 (which, like other classical decision logic described herein can be implemented using a CPU, a microcontroller, or the like) having memory that stores success patterns for circuit 2200. In many respects, circuit 2200 is similar to circuit 1900 described above, with conversion from spatial to temporal encoding applied at the outputs.

[0138] In some embodiments, circuit 2200 has the following success patterns: (4,1), (1,4), (5,0), (0,5). Failure patterns for circuit 2200 can include (4, 2). Further, in some embodiments, some detection patterns that can indicate a successfully generated state are identified by the control circuitry as failure patterns based on the given patterns not being robust (e.g., generating or being associated with too many false positives). Examples of detection patterns that are filtered or otherwise marked as failures (though they may sometimes be associated with successful state generation) include (3, 2) and (2,3).

[0139] FIG. 23 shows a simplified schematic diagram of a hybrid (spatial -temporal) H4 3- GHZ state generator circuit 2300 according to some embodiments. At a high level, the circuit 2300 receives photons in different time intervals, denoted asand t2, generated from one or more photon sources. Photons received in the first time interval (indicated as | ltl)) are coupled to a first Hadamard 2310-2, and the photons received in the second time interval (indicated as |lt2)) are coupled to a second Hadamard 2310-2. Downstream of the Hadamard, the light is output onto detection modes 2302-7 and 2302-8 and output modes 2302-1 to 2302-6.

[0140] Similar to circuit 2200, circuit 2300 includes a converter circuit having switches 2322-1 through 2322-3 (e.g., 2-to-l couplers, MZIs), which function in concert to output a time-bin encoded state from the circuit 2300. In circuit 2300, delay 2317 is implemented upstream of mode information erasure circuit 2322 on the detection paths. The 3-GHZ statethat is generated is both path-encoded and time-bin-encoded. Switches 2314-1, 2314-2, and 2314-3 change the state such that a fully time-bin encoded state is output on the set of spatial modes (outputs 2314-1 to 2314-3). In some embodiments, similar to circuit 2200, the quantum state output from the circuit 2300 can be output with a time bin separation that is shorter or longer than the time period of the pump rate of the single photon sources that provide the input photons. For example, the rate of the pump light may be 1 gigahertz, such that the first photons | ltl) are output at a first time and the second photons | lt2) are output (e.g., by another source) at a second time later than the first time (e.g., approximately 1 nanosecond later). The switching speed of output switches 2322-1 through 2322-3 may be 3 gigahertz such that the generated-3-GHZ state is in a time-bin encoded format that has a shorter period (e.g., approximately .333 nanoseconds) that the period of the pumped sources. In some embodiments, shorter time periods in time bin encoding can be useful for downstream devices (e.g., devices that receive the generated 3-GHZ state) due to the downstream devices having to implement smaller delays (e.g., on-PIC waveguide delays) to process the time-bin encoded state. A further advantage of circuit 2300 is that delay(s) for ensuring a time-bin encoded state is output are omitted from the output modes, and instead are integrated on the detection modes (where the delays are less problematic due to the robustness described above). In this way, having the delays on the detection modes can further reduce overall loss on the output state, in accordance with some example embodiments. FIGs. 26 and 29 (described below) illustrate additional examples of circuits having delays omitted from the output modes.

[0141] In some embodiments, two sets of photon sources are implemented for the inputs of circuit 2300. For example, a first set of pumped photon sources can generate the photons coupled to coupler 2302-1 and a second set of pumped photon sources can generate photons coupled to coupler 2302-2. In some example embodiments, control circuitry (e.g., classical decision logic 2230, which can be similar to classical decision logic 2130 of FIG. 21) receives the detections from detectors 2320-1 and 2320-2 and generates success data indicating that a 3-GHZ state is output when the detected pattern matches a stored pattern. The success patterns for circuit 2300 can be the same as described above with reference to FIG. 22.

[0142] FIG. 24 shows a simplified schematic diagram of a temporal 3-GHZ state generator circuit 2400 according to some embodiments. FIG. 24 is similar to FIG. 21 described above; however, in FIG. 24 the converter circuit components (switch 2422 and delay 2417) areshown, whereas in FIG. 21, these components are illustrated as being in temporal -to-spatial conversion circuit 2117. With reference to FIG. 24, circuit 2400 receives two photons on each input (e.g., single-rail paths 2402-1 through 2402-4) in two different time bins and uses a single switch 2422 and single delay 2417 on the detection paths for path-to-time-bin conversion. No delays or switches need be implemented the output paths 2414-1 to 2414-3. By removing the delays and switches from the output paths, overall loss can be reduced in comparison to the circuits of FIG. 22 and FIG. 23.

[0143] An advantage of the circuits of FIGs. 22 and 23 is that the delays and switches can enable the circuits to output time-bin encoded states, where the time bin separation of the output state can be different from the time period of the pump light that generates the input photons, e.g., in contrast to FIG. 24, where the time bin separation is fixed to the separation between pump light pulses.2.4. 4-GHZ State Generators

[0144] FIGS. 25-27 show circuits for generating a 4-GHZ state (e.g., four qubits in a GHZ entanglement state) in a time bin encoded format, in accordance with some embodiments.

[0145] FIG. 25 illustrates a simplified schematic diagram of a spatial 4*3-DFT 4-GHZ generator circuit 2500 according to some embodiments. In some embodiments, the photons are generated using one or more single photon sources (e.g., four wave mixing, parametric down conversion source) that operate at a pump light rate (e.g., 1-3 gigahertz) and received by circuit 2500 on paths 2502-1 through 2502-12. Paths 2502-1 through 2502-12 couple sets of three photons into separate beam splitter networks 2510-1 through 2510-4, such as four three-dimensional DFT couplers (e.g., four DFT3 networks as described above). A first portion of the interfered light from beam splitter networks 2510-1 through 2510-4 is output on a set of output paths to delays 2517-1 through 2517-4 and switches 2522-1 through 2522- 4, which output a 4-GHZ state (e.g., heralded by a success pattern). A second portion of the interfered light from beam splitter networks 2510-1 through 2510-4 is output on a set of detection paths to beam splitter 2511 (e.g., four way Hadamard), which outputs to a set of detectors 2520 that generate detection data.

[0146] The detection data can be received by control circuitry (e.g., classical decision logic 2530, which can be similar to classical decision logic 2130 of FIG. 21 and can include, e.g., a CPU, a microcontroller, or the like) having memory that stores success patterns. In some embodiments, circuit 2500 has the following success patterns: (3, 3, 1, 1), (7, 1, 0, 0), and (5,2, 1, 0). In some example embodiments, failure patterns for circuit 2500 include: (3, 3, 2, 0), (5, 3, 0, 0), (3, 2, 2, 1), and (6, 1, 1, 0), where one or more of the failure patterns are detection patterns that result from a successful state generation but are still stored and implemented as failure patterns to improve the quality of state output (e.g., they are valid patterns, but are marked as failures, or are otherwise filtered / ignored, so the overall robustness of the quantum state improves).

[0147] In some alternative embodiments, each pattern that can successfully generate the target state is stored as a success pattern, and failure states are detection patterns that indicate the target state was not generated. For example, the success patterns can include (3, 3, 1, 1), (7, 1, 0, 0), (5, 2, 1, 0), (3, 3, 2, 0), (5, 3, 0, 0), (3, 2, 2, 1), and (6, 1, 1, 0); and a failure pattern can include (3, 3, 3, 1) (e.g., a failure pattern that does not have eight photons).

[0148] FIG. 26 shows a simplified schematic diagram of a hybrid 4x3 DFT 4-GHZ generator circuit 2600 according to some embodiments. At a high level, circuit 2600 receives photons in different time intervals, denoted as tq and t2, generated from one or more photon sources. In some embodiments, photons received in the first time interval (indicated as | ltl)) are coupled to DFT3 networks 2610-1 and 2610-3 (which can be implemented as described above), and the photons received in the second time interval (indicated as |lt2)) are coupled to the other DFT3 networks 2610-2 and 2610-4.

[0149] Similar to circuit 2500, circuit 2600 includes a converter circuit having switches 2622-1 through 2622-4 (e.g., 2-to-l couplers, MZIs), which function in concert to output a time-bin encoded state from circuit 2600. In circuit 2600, delays 2617-1 and 2617-2 are implemented upstream of mode information erasure circuit 2616 on the detection paths. The 4-GHZ state that is generated is both path-encoded and time-bin-encoded. Switches 2622-1 to 2622-4 change the state such that a fully time-bin encoded state is output on the set of spatial modes (outputs 2614-1 to 2614-4). In some example embodiments, similar to circuit 2500, the quantum state output from the circuit 2600 can be output with a time bin separation that is shorter or longer than the time period of the pump rate of the single photon sources that provide the input photons. For example, the rate of the pump light may be 1 gigahertz, such that the first photons | ltl) are output at a first time and the second photons | lt2) are output (e.g., by another source) at a second time later than the first time (e.g., approximately 1 nanosecond later). The switching speed of output switches 2622-1 through 2622-4 may be 3 gigahertz such that the generated 4-GHZ state is in a time bin encoded format that has ashorter period (e.g., approximately .333 nanoseconds) that the period of the pumped sources. In some embodiments, control circuitry (e.g., classical decision logic 2630, which can be similar to classical decision logic 2130 of FIG. 21) receives the detections from detectors 2620 and generates success data indicating that a 4-GHZ state is output when the detected pattern matches a stored success pattern. The success (and failure) patterns can be the same as described above with reference to FIG. 25.

[0150] FIG. 27 shows a simplified schematic diagram of a temporal 4*3-DFT 4-GHZ generator circuit 2700 according to some embodiments. At a high level, circuit 2700 receives two photons on each input (e.g., single rail paths 2702-1 through 2702-6) in two different time bins and uses a converter circuit (e.g., switches 2722-1, 2722-2 and delays 2713-1, 2713-2) on the detection paths for path-to-time-bin conversion, thereby removing lossy components (e.g., delays and switches) from output paths 2714-1 through 2714-4. Similarly to circuits 2400 and 2500, light on the detection paths is detected by detectors 2720 to generate detection data, where success data indicating that a 4-GHZ state is output is generated when the detection data matches one of the success patterns. The success patterns for circuit 2700 can be the same as described above with reference to FIG. 25.2.5. 4-Line State Generators

[0151] FIGs. 28A-30 show circuits for generating a 4-line entangled state according to some embodiments. The circuits of FIGs. 28A-30 can generate an entangled state equivalent to a 4-line state, up to local operations.

[0152] FIG. 28 A shows a simplified schematic diagram of a spatial 4x3 DFT 4-Line generator circuit 2800 according to some embodiments. As illustrated, each of the twelve paths 2802-1 to 2802-12 can receive a single photon (denoted as | 1)). In some embodiments, the photons are generated using one or more single photon sources (e.g., four wave mixing source, parametric down conversion source) that operate at a pump light rate (e.g., 1-3 gigahertz) and received by circuit 2800 on paths 2802-1 through 2802-12. Paths 2802-1 through 2802-12 couple sets of three photons into separate beam splitter networks 2810-1 to 2810-4, such as four three-dimensional DFT couplers (e.g., four DFT3 networks as described above), which interfere the photons and output interfered light on paths (e.g., intermediate paths) coupled to further optical components. In some embodiments, a first portion of the interfered light from beam splitter networks 2810-1 to 2810-4 is output on a set of output paths to delays 2817-1 to 2817-4 and switches 2822-1 to 2822-4, which output a 4-Line state(e.g., heralded by a success pattern). A second portion of the interfered light from beam splitter networks 2810-1 to 2810-4 is output on a set of detection paths to a beam splitter network 2811 (e.g., a four-dimensional DFT network), which outputs to a set of detectors 2820 that generate detection data. In the embodiment shown in FIG. 28A, beam splitter network 2811 implements a unitary transform that corresponds to a four-dimensional Discrete Fourier Transform matrix, such as:The four-dimensional DFT coupler can be decomposed into different optical components. One example is shown in circuit diagram 2850 in FIG. 28B, which is similar to circuit diagram 504 (FIG. 5) with an additional phase shift of TT / 2 on the third rail.

[0153] The detection data can be received by control circuitry (e.g., classical decision logic 2830, which can be similar to classical decision logic 2130 of FIG. 21 and can include, e.g., a CPU, a microcontroller, or the like) having memory that stores success patterns. In some embodiments, circuit 2800 has the following success patterns: (4, 2, 1, 1), (7, 1, 0, 0), (5, 2, 1, 0), and (1, 4, 3, 0). In some embodiments, failure patterns for circuit 2800 include: (3, 3, 2, 0), (5, 3, 0, 0), (3, 2, 2, 1), (6, 1, 1, 0), where as in examples above, one or more of the failure patterns may be valid detection patterns of a successfully generated state but are nevertheless marked or otherwise treated failure patterns to improve the overall robustness of state generation.

[0154] In some alternative example embodiments, each pattern that can successfully generate the target state is stored as a success pattern, and failure states are detection patterns that indicate the target state was not generated. For example, the success patterns can include (4, 2, 1, 1), (7, 1, 0, 0), (5, 2, 1, 0), (1, 4, 3, 0), (3, 3, 2, 0), (5, 3, 0, 0), (3, 2, 2, 1), and (6, 1, 1, 0); and a failure pattern can include (3, 3, 3, 1) (e.g., a failure pattern that does not have eight photons).

[0155] FIG. 29 shows a simplified schematic diagram of a hybrid 4x3 DFT 4-Line generator circuit 2900 according to some embodiments. At a high level, circuit 2900 receives photons in different time intervals, denoted as G and t2, generated from one or more photon sources. In some embodiments, photons received in the first time interval (indicated as | ltl))are coupled to DFT3 networks 2910-1 and 2910-3, and the photons received in the second time interval (indicated as | lt2)) are coupled to the other DFT3 networks 2910-2, and 2910-4.

[0156] Similar to circuit 2800, circuit 2900 comprises a converter circuit having switches 2922-1 through 2922-4 (e.g., 2-to-l couplers, MZIs), which function in concert to output a time-bin encoded state from circuit 2900. In circuit, delays 2913-1 and 2913-2 are implemented upstream of DFT4 network 2911 on the detection paths. The 4-Line state that is generated is both path-encoded and time-bin-encoded. Switches 2922-1 to 2922-4 change the state such that a fully time-bin-encoded state is output on the set of spatial modes (outputs 2922-1 to 2922-4). In some example embodiments, similar to circuit 2800, the quantum state output from the circuit 2900 can be output with a time bin separation that is shorter or longer than the time period of the pump rate of the single photon sources that provide the input photons. For example, the rate of the pump light may be 1 gigahertz, such that the first photons | ltl) are output at a first time and the second photons | lt2) are output (e.g., by another source) at a second time later than the first time (e.g., approximately 1 nanosecond later). The switching speed of output switches 2922-1 through 292204 may be 3 gigahertz such that the generated-4-line state is in a time bin encoded format that has a shorter period (e.g., approximately .333 nanoseconds) that the period of the pumped sources. In some embodiments, control circuitry (e.g., classical decision logic 2930, which can be similar to classical decision logic 2130 of FIG. 21) receives the detections from detectors 2620 and generates success data indicating that a 4-line state is output when the detected pattern matches a stored success pattern. The success (and failure) patterns can be the same as described above with reference to FIG. 28.

[0157] FIG. 30 shows a simplified schematic diagram of a temporal 4*3-DFT 4-line generator circuit 3000 according to some embodiments. At a high level, circuit 3000 receives two photons on each input (e.g., single rail paths 3002-1 through 3002-6) in two different time bins and uses a converter circuit (e.g., switches 3022-1 and 3022-2, delays 3013-1 and 3013-2) on the detection paths for path-to-time-bin conversion, thereby removing lossy components (e.g., delays and switches) from output paths 3014-1 through 3014-4. Similarly to circuits 2800 and 2900, light on the detection paths is detected by detectors 3020 to generate detection data, where success data indicating that a 4-line state is output is generated when the detection data matches one of the success patterns. The success patterns for circuit 3000 can be the same as described above with reference to FIG. 28 A.2.6. Expanded Seed State Generators

[0158] As noted above, photon sources may operate non-deterministically, and it is not guaranteed that a photon will be received on each input path of a seed state generation circuit such as the spatial circuits of FIGs. 22, 25, and 28 A. Various techniques can be used to increase the probability, including multiplexing outputs of a number of photon sources that exceeds the number of input paths of the seed state generation circuit. Another approach is to use an “expanded” seed state generation circuit, which refers to a seed state generation circuit that includes additional rails to provide multiple copies of the input rails (e.g., waveguides or other optical paths) and circuit components and that produces the target seed state on a subset of its output rails (e.g., waveguides).

[0159] By way of example, FIG. 31 shows a simplified schematic diagram of a circuit 3130 that implements an expanded version of spatial 4*3-DFT 4-GHZ generator circuit 2500 according to some embodiments. Circuit 3100 (at the left of FIG. 31) corresponds to circuit 2500 (FIG. 25), with delays and switches omitted for clarity in mapping to circuit 3130. As illustrated in FIG. 31, expanded circuit 3130 has 24 input rails 3102, eight DFT3 networks 3110, and sixteen propagation output rails 3114. Each input path 3102 includes a blocking switch 3132 (shown as black or white squares, with white indicating that the blocking switch is open and passes through input photons and black indicating that the blocking switch is closed and blocks input light). Via blocking switches 3132, the inputs to a first subset (four in this case) of DFT3 networks 3110 receive three single photons (indicated as 11) and white squares) while inputs to other DFT3 networks 3110 are blocked by blocking switches 3132 on the respective paths 3102 (as indicated by black squares). Downstream of DFT3 networks 3110, the heralding outputs pass through a Hadamard-8 network, 3116, and photons are detected by detectors 3120. Hadamard-8 network 3116 can be replaced by other optical switching networks having the property that a photon entering on any one of the inputs to the network has an equal probability of exiting on any one of the outputs of the optical network. (In other words, detection of a photon by a particular detector 3120 provides no information as to which one of DFT3 networks 3110 the photon passed through.) The creation of a 4- GHZ state on a subset of propagation outputs 3114 can be heralded by detection of an appropriate pattern of photons in detectors 3120, by analogy to the detection patterns described above. While detectors 3120 do not provide information about which subset of propagation outputs 3114 carry the 4-GHZ state, that information can be determined based on which input waveguides 3102 passed photons through blocking switches 3132. In someembodiments, increasing the number of paths of the circuit can improve the probabilistic performance of the sources without increasing the switching complexity, since only blocking switches are implemented.

[0160] According to various embodiments, any of the seed state generator circuits described herein can be expanded to a larger number of rails in a similar manner. In addition, seed state generator circuits can be expanded to larger numbers of rails (e.g., by an expansion factor that can be but need not be a power of 2).3. Additional Embodiments

[0161] The foregoing examples of seed state generator circuits and their applications 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 seed state generation 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 photons can be varied as desired.

[0162] The particular transform matrix can be chosen as desired, based in part on the seed state to be generated and in part on whether the success patterns and failure patterns (for a particular division of the outputs into heralding and propagation outputs) provide a desired degree of robustness against error. For instance, symmetric matrices that provide destructive interference to preclude failure patterns that differ by just one photon count in one detector can be used. A given transform can have multiple different implementations using networks of linear optical components, and such components can include any combination of beam splitters and / or phase shifters. In some embodiments, the linear optical components can be passive components, which can further reduce photon loss.

[0163] Various robustness criteria can be applied. For instance, in some embodiments described above, a coordinate distance (e.g., Manhattan distance or edit distance) between any success pattern and any failure pattern is at least 2. In some embodiments, each of the success patterns has a total number of photons equal to a first number (P), and none of the failure patterns has a total number of photons equal to P-1. In some embodiments, each of the success patterns has a total number of photons equal to a first number (P), and any failure pattern having a total number of photons equal to P+1 corresponds to an output state on the propagation outputs with correlated qubit errors (e.g., none of the qubits is in a logically validstate). In some embodiments, each of the success patterns has a total number of photons equal to a first number (P), and at least some of the failure patterns having a total number of photons equal to P+1 correspond to an output states on the propagation outputs with correlated qubit errors (e.g., none of the qubits is in a logically valid state).

[0164] 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.

[0165] 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.

[0166] The following are example embodiments:

[0167] Example 1 : A circuit comprising: a plurality of input waveguides to receive photons; a plurality of output waveguides including a first subset of output waveguides and a second subset of output waveguides; a network of linear optical components coupling the plurality of input waveguides to the plurality of output waveguides, the network of linear optical components implementing a unitary transform operation that probabilistically produces, in response to an input event that includes a photon being input to each of the plurality of input waveguides, an output state that is either: a success output state in which the first subset of output waveguides propagates one of a set of target entangled qubit states while the second subset of output waveguides propagates a corresponding one of a set of success patterns of photons; or a failure output state in which the first subset of output waveguides propagates one of a set of invalid qubit states while the second subset of output waveguides propagates a corresponding one of a set of failure patterns of photons; a plurality of photon detectors, each photon detector coupled to one of the output waveguides in thesecond subset of output waveguides and configured to produce a classical output signal indicating a number of photons detected; and a classical decision logic circuit coupled to the photon detectors and configured to receive a readout pattern comprising the respective classical output signals from the photon detectors, to determine whether each of the input waveguides received a photon, to compare the readout pattern to the set of success patterns, and to output a classical decision signal that indicates success in the event that each of the input waveguides received a photon and the readout pattern matches one of the success patterns and indicates failure otherwise, wherein the unitary transform operation satisfies one or both of the following criteria: (a) a coordinate distance between any one of the success patterns and any one of the failure patterns is at least 2; or (b) for any failure pattern with a coordinate distance less than 2 from one of the success patterns, the first subset of waveguides propagates an invalid qubit state in which none of the qubits is in a valid logical state.

[0168] Example 2: The circuit of Example 1 wherein the success patterns and the failure patterns are defined such that each of the success patterns has a total number of photons equal to a first number (N) and none of the failure patterns has a total number of photons equal to N or N+l.

[0169] Example 3: The circuit of Example 2 wherein the success patterns and the failure patterns are further defined such that none of the failure patterns has a total number of photons equal to N-l.

[0170] Example 4: The circuit of any one of Examples 1-3 wherein the success patterns and the failure patterns are defined such that a coordinate distance between any one of the success patterns and any one of the failure patterns is at least 2.

[0171] Example 5: The circuit of Example 4 wherein the coordinate distance is a Manhattan distance.

[0172] Example 6: The circuit of Example 1 wherein the success patterns and the failure patterns are defined such that each of the success patterns has a total number of photons equal to a first number (N) and any failure pattern having a total number of photons equal to N+l corresponds to an invalid qubit state in which none of the qubits is in a logically valid state.

[0173] Example 7: The circuit of any one of Examples 1-6 wherein the network of linear optical components comprises beam splitters and phase shifters.

[0174] Example 8: The circuit of Example 7 wherein the beam splitters and phase shifters implement a plurality of integrated interferometers corresponding to the unitary transform.

[0175] Example 9: The circuit of Example 8 wherein the unitary transform corresponds to a matrix that is decomposable into component matrices that correspond to the beam splitters and phase shifters.

[0176] Example 10: The circuit of Example 9 wherein all elements of the matrix have a same magnitude.

[0177] Example 11 : The circuit of Example 9 or 10 wherein the output waveguides are assigned to the first subset and the second subset in a manner such that destructive interference results in all of the success patterns being sufficiently different from all of the failure patterns that a false indication of success occurs only if two or more errors occur in the circuit.

[0178] Example 12: The circuit of any one of Examples 1-11 wherein the set of target entangled qubits comprises a plurality of dual-rail encoded qubits, each dual-rail encoded qubit propagating in a different pair of the output waveguides in the first subset of output waveguides.

[0179] Example 13: The circuit of any one of Examples 1-11 wherein the input event includes a photon being input to each of the plurality of input waveguides during each of a pair of consecutive time bins and wherein the set of target entangled qubits comprises a plurality of temporally encoded qubits, each temporally encoded qubit propagating in a pair of consecutive time bins in a different one of the output waveguides in the first subset of output waveguides.

[0180] Example 14: The circuit of any one of Examples 1-13 wherein the readout pattern has a nonzero probability of being affected by a dark count error or a photonic loss error in one or more of the photon detectors.

[0181] Example 15: The circuit of any one of Examples 1-14 further comprising: a plurality of heralded single photon source circuits coupled to the input waveguides, wherein each of the heralded single photon source circuits is configured to provide a heralding signal to the classical decision logic circuit, each heralding signal indicating whether a photon was generated in the heralded single photon source circuit, wherein the classical decision logiccircuit is configured to determine whether each of the input waveguides received a photon based at least in part on the heralding signals.

[0182] Example 16: The circuit of any one of Examples 1-15 wherein the unitary transform operation comprises an Se transform and the target entangled state is a Bell state.

[0183] Example 17: The circuit of any one of Examples 1-15 wherein the unitary transform operation comprises two three-component Discrete Fourier Transforms and the target entangled state is a Bell state.

[0184] Example 18: The circuit of any one of Examples 1-15 wherein the unitary transform operation comprises two second-order Hadamard transforms and the target entangled state is a 3-GHZ state.

[0185] Example 19: A method comprising: receiving, on a photonic circuit, a plurality of single photons; interfering, on a beam splitter network in the photonic circuit, the plurality of single photons to form interfered light, a first portion of the interfered light being coupled to output paths that are connected to outputs of the photonic circuits, a second portion of the interfered light being coupled to detection paths that are connected to detectors; determining, on control circuitry, the interfered light comprises a photonic quantum state, the photonic quantum state being in path encoded format; converting the photonic quantum state from the path encoded format to a time bin encoded format; and outputting, on the outputs of the photonic circuit, the photonic quantum state in the time bin encoded format.

[0186] Example 20: The method of Example 19, wherein converting the photonic quantum state from the path encoded format to the time bin encoded format comprises delaying the interfered light that is from the beam splitter network.

[0187] Example 21 : The method of Example 20, wherein converting the photonic quantum state from the path encoded format to the time bin encoded format comprises switching the interfered light that is from the beam splitter network, wherein the interfered light propagates on a plurality of intermediate paths in the photonic circuit and light from the interfered light is switched between at least two of the plurality of intermediate paths.

[0188] Example 22: The method of any of Examples 19-21, wherein the interfered light comprises the photonic quantum state that is both in the path encoded format and further in the time bin encoded format, and wherein the photonic quantum state is converted such that the photonic quantum state is only in the time bin encoded format.

[0189] Example 23 : The method of Example 22, wherein the photonic circuit comprises a delay on the detection paths to delay the second portion of the interfered light, wherein the photonic circuit comprises an optical switch on the detection paths to apply switching to the second portion of the interfered light.

[0190] Example 24: The method of any of Examples 22 or 23, wherein the photonic circuit further comprises an additional beam splitter network that further interferes the second portion of the interfered light before detection.

[0191] Example 25: The method of any of Examples 21-24, wherein the photonic circuit comprises a plurality of optical switches on the output paths to apply switching to the first portion of the interfered light.

[0192] Example 26: The method of Example 25, wherein the photonic circuit comprises a plurality of delays on the output paths to delay the first portion of the interfered light.

[0193] Example 27: The method any of Examples 19-26, wherein the plurality of single photons are generated by a single photon source.

[0194] Example 28: The method of Example 27, wherein the single photon source is pumped at a rate comprises a first time period, and wherein the time bin encoded format of the photonic quantum state comprises a second time period that is different than the first time period.

[0195] Example 29: The method of Example 28, wherein the second time period is shorter than the first time period.

[0196] Example 30: The method of any of Examples 19-29, wherein the beam splitter network comprises a plurality of linear optical components that implement a unitary transform that interferes light to probabilistically produce the photonic quantum state.

[0197] Example 31 : The method of Example 30, wherein the unitary transform comprises one or more of: a four dimensional Hadamard transform; a three dimensional Discrete Fourier Transform; or a four dimensional Discrete Fourier Transform.

[0198] Example 32: The method of any of Examples 19-31, wherein the photonic output state is one of: a 3-GHZ state, a 4-GHZ state, or a line state.

[0199] Example 33: The method of any of Examples 19-32, wherein the plurality of single photons comprises a first set of photons in a earlier period and a second set of photons in alater period, wherein the first set of photons in the earlier period are received by the photonic circuit before the photonic circuit receives the second set of photons.

[0200] Example 34: The method of any of Examples 19-33, wherein the control circuitry receives detection data from the detectors detecting the second portion of the interfered light, and wherein determining that the interfered light comprises the photonic quantum state is based on the detection data matching a success pattern stored in memory.

[0201] Example 35: The method of Example 34, wherein the success patterns is from a plurality of success patterns in the memory, and wherein the plurality of success patterns exclude one or more valid patterns that correspond to the photonic circuit successfully generating a target photonic state, wherein the one or more valid patterns are excluded based on the one or more valid patterns being associated with detection errors from the detectors.

[0202] Example 36: A photonic circuit comprising: a plurality of input paths to receive a plurality of single photons; a beam splitter network that interferes the plurality of single photons to form interfered light; a plurality of output paths coupled to the beam splitter network that receives a first portion of the interfered light; a plurality of detection paths coupled to the beam splitter network that receives a second portion of the interfered light; a plurality of detectors connected to the plurality of detection paths; control circuitry to determine that the interfered light comprises a photonic quantum state, the photonic quantum state being in a path encoded format; a converter to convert the photonic state from the path encoded format to a time bin encoded format; and a plurality of input paths to output the photonic quantum state in the time bin encoded format.

[0203] Example 37: The photonic circuit of Example 36, wherein the beam splitter network comprises a plurality of linear optical components that implement a unitary transform that interferes light to probabilistically produce the photonic quantum state.

[0204] Example 38: The photonic circuit of claim 37, wherein the unitary transform comprises one or more of: a four dimensional Hadamard transform, a three dimensional Discrete Fourier Transform, a four dimensional Discrete Fourier Transform.

[0205] 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 theengineering 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), and successive 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).

[0206] 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. 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.

[0207] 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 input waveguides to receive photons; a plurality of output waveguides including a first subset of output waveguides and a second subset of output waveguides; a network of linear optical components coupling the plurality of input waveguides to the plurality of output waveguides, the network of linear optical components implementing a unitary transform operation that probabilistically produces, in response to an input event that includes a photon being input to each of the plurality of input waveguides, an output state that is either: a success output state in which the first subset of output waveguides propagates one of a set of target entangled qubit states while the second subset of output waveguides propagates a corresponding one of a set of success patterns of photons; or a failure output state in which the first subset of output waveguides propagates one of a set of invalid qubit states while the second subset of output waveguides propagates a corresponding one of a set of failure patterns of photons; a plurality of photon detectors, each photon detector coupled to one of the output waveguides in the second subset of output waveguides and configured to produce a classical output signal indicating a number of photons detected; and a classical decision logic circuit coupled to the photon detectors and configured to receive a readout pattern comprising the respective classical output signals from the photon detectors, to determine whether each of the input waveguides received a photon, to compare the readout pattern to the set of success patterns, and to output a classical decision signal that indicates success in the event that each of the input waveguides received a photon and the readout pattern matches one of the success patterns and indicates failure otherwise, wherein the unitary transform operation satisfies one or both of the following criteria:(a) a coordinate distance between any one of the success patterns and any one of the failure patterns is at least 2; or(b) for any failure pattern with a coordinate distance less than 2 from one of the success patterns, the first subset of waveguides propagates an invalid qubit state in which none of the qubits is in a valid logical state.

2. The circuit of claim 1 wherein the success patterns and the failure patterns are defined such that each of the success patterns has a total number of photons equal to a first number (N) and none of the failure patterns has a total number of photons equal to N or N+l.

3. The circuit of claim 2 wherein the success patterns and the failure patterns are further defined such that none of the failure patterns has a total number of photons equal to N-l.

4. The circuit of claim 1 wherein the success patterns and the failure patterns are defined such that a coordinate distance between any one of the success patterns and any one of the failure patterns is at least 2.

5. The circuit of claim 4 wherein the coordinate distance is a Manhattan distance.

6. The circuit of claim 1 wherein the success patterns and the failure patterns are defined such that each of the success patterns has a total number of photons equal to a first number (N) and any failure pattern having a total number of photons equal to N+l corresponds to an invalid qubit state in which none of the qubits is in a logically valid state.

7. The circuit of claim 1 wherein the network of linear optical components comprises beam splitters and phase shifters.

8. The circuit of claim 7 wherein the beam splitters and phase shifters implement a plurality of integrated interferometers corresponding to the unitary transform.

9. The circuit of claim 8 wherein the unitary transform corresponds to a matrix that is decomposable into component matrices that correspond to the beam splitters and phase shifters.

10. The circuit of claim 9 wherein all elements of the matrix have a same magnitude.

11. The circuit of claim 9 wherein the output waveguides are assigned to the first subset and the second subset in a manner such that destructive interference results in all of the success patterns being sufficiently different from all of the failure patterns that a false indication of success occurs only if two or more errors occur in the circuit.

12. The circuit of claim 1 wherein the set of target entangled qubits comprises a plurality of dual-rail encoded qubits, each dual-rail encoded qubit propagating in a different pair of the output waveguides in the first subset of output waveguides.

13. The circuit of claim 1 wherein the input event includes a photon being input to each of the plurality of input waveguides during each of a pair of consecutive time bins and wherein the set of target entangled qubits comprises a plurality of temporally encoded qubits, each temporally encoded qubit propagating in a pair of consecutive time bins in a different one of the output waveguides in the first subset of output waveguides.

14. The circuit of claim 1 wherein the readout pattern has a nonzero probability of being affected by a dark count error or a photonic loss error in one or more of the photon detectors.

15. The circuit of claim 1 further comprising: a plurality of heralded single photon source circuits coupled to the input waveguides, wherein each of the heralded single photon source circuits is configured to provide a heralding signal to the classical decision logic circuit, each heralding signal indicating whether a photon was generated in the heralded single photon source circuit, wherein the classical decision logic circuit is configured to determine whether each of the input waveguides received a photon based at least in part on the heralding signals.

16. The circuit of claim 1 wherein the unitary transform operation comprises an Se transform and the target entangled state is a Bell state.

17. The circuit of claim 1 wherein the unitary transform operation comprises two three-component Discrete Fourier Transforms and the target entangled state is a Bell state.

18. The circuit of claim 1 wherein the unitary transform operation comprises two second-order Hadamard transforms and the target entangled state is a 3-GHZ state.

19. A method comprising: receiving, on a photonic circuit, a plurality of single photons; interfering, on a beam splitter network in the photonic circuit, the plurality of single photons to form interfered light, a first portion of the interfered light being coupled to output paths that are connected to outputs of the photonic circuits, a second portion of the interfered light being coupled to detection paths that are connected to detectors; determining, on control circuitry, the interfered light comprises a photonic quantum state, the photonic quantum state being in path encoded format; converting the photonic quantum state from the path encoded format to a time bin encoded format; and outputting, on the outputs of the photonic circuit, the photonic quantum state in the time bin encoded format.

20. The method of claim 19, wherein converting the photonic quantum state from the path encoded format to the time bin encoded format comprises delaying the interfered light that is from the beam splitter network.

21. The method of claim 20, wherein converting the photonic quantum state from the path encoded format to the time bin encoded format comprises switching the interfered light that is from the beam splitter network, wherein the interfered light propagates on a plurality of intermediate paths in the photonic circuit and light from the interfered light is switched between at least two of the plurality of intermediate paths.

22. The method of claim 19, wherein the interfered light comprises the photonic quantum state that is both in the path encoded format and further in the time bin encoded format, and wherein the photonic quantum state is converted such that the photonic quantum state is only in the time bin encoded format.

23. The method of claim 22, wherein the photonic circuit comprises a delay on the detection paths to delay the second portion of the interfered light, wherein the photonic circuit comprises an optical switch on the detection paths to apply switching to the second portion of the interfered light.

24. The method of claim 22, wherein the photonic circuit further comprises an additional beam splitter network that further interferes the second portion of the interfered light before detection.

25. The method of claim 21, wherein the photonic circuit comprises a plurality of optical switches on the output paths to apply switching to the first portion of the interfered light.

26. The method of claim 25, wherein the photonic circuit comprises a plurality of delays on the output paths to delay the first portion of the interfered light.

27. The method of claim 19, wherein the plurality of single photons are generated by a single photon source.

28. The method of claim 27, wherein the single photon source is pumped at a rate comprises a first time period, and wherein the time bin encoded format of the photonic quantum state comprises a second time period that is different than the first time period.

29. The method of claim 28, wherein the second time period is shorter than the first time period.

30. The method of claim 19, wherein the beam splitter network comprises a plurality of linear optical components that implement a unitary transform that interferes light to probabilistically produce the photonic quantum state.

31. The method of claim 30, wherein the unitary transform comprises one or more of: a four dimensional Hadamard transform; a three dimensional Discrete Fourier Transform; or a four dimensional Discrete Fourier Transform.

32. The method of claim 19, wherein the photonic output state is one of: a 3 -GHZ state, a 4-GHZ state, or a line state.

33. The method of claim 19, wherein the plurality of single photons comprises a first set of photons in a earlier period and a second set of photons in a laterperiod, wherein the first set of photons in the earlier period are received by the photonic circuit before the photonic circuit receives the second set of photons.

34. The method of claim 19, wherein the control circuitry receives detection data from the detectors detecting the second portion of the interfered light, and wherein determining that the interfered light comprises the photonic quantum state is based on the detection data matching a success pattern stored in memory.

35. The method of claim 34, wherein the success patterns is from a plurality of success patterns in the memory, and wherein the plurality of success patterns exclude one or more valid patterns that correspond to the photonic circuit successfully generating a target photonic state, wherein the one or more valid patterns are excluded based on the one or more valid patterns being associated with detection errors from the detectors.

36. A photonic circuit comprising: a plurality of input paths to receive a plurality of single photons; a beam splitter network that interferes the plurality of single photons to form interfered light; a plurality of output paths coupled to the beam splitter network that receives a first portion of the interfered light; a plurality of detection paths coupled to the beam splitter network that receives a second portion of the interfered light; a plurality of detectors connected to the plurality of detection paths; control circuitry to determine that the interfered light comprises a photonic quantum state, the photonic quantum state being in a path encoded format; a converter to convert the photonic state from the path encoded format to a time bin encoded format; and a plurality of input paths to output the photonic quantum state in the time bin encoded format.

37. The photonic circuit of claim 36, wherein the beam splitter network comprises a plurality of linear optical components that implement a unitary transform that interferes light to probabilistically produce the photonic quantum state.

38. The photonic circuit of claim 37, wherein the unitary transform comprises one or more of: a four dimensional Hadamard transform, a three dimensional Discrete Fourier Transform, a four dimensional Discrete Fourier Transform.

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