Photonic quantum computer architecture

Abstract

To provide a circuit and a method which generate entanglement among quantum bits using "rasterized" and interleaving techniques.SOLUTION: A circuit includes a resource state generator 2702 that generates one resource state per clock cycle. Each resource state 2704 has multiple entangled quantum bits. The circuit also includes delay circuits 2708 and delay lines to perform entangling measurement operations on quantum bits in the resource states generated by the same resource state generator at different clock cycles. By appropriate selection of the delay lines, the single resource state generator generates all of the necessary resource states for generating a large entanglement structure. Hybrid techniques can also be used. According to these techniques, the number of resource state circuits is greater than one, but is smaller than the number of necessary resource states for generating the entanglement structure.SELECTED DRAWING: Figure 27

Description

[Technical field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Provisional Application No. 62 / 865,058, filed June 21, 2019, U.S. Provisional Application No. 62 / 926,383, filed October 25, 2019, and U.S. Provisional Application No. 63 / 006,590, filed April 7, 2020. The disclosures of all three applications are incorporated herein by reference. [Background technology]

[0002] Quantum computing is distinguished from "classical" computing by its reliance on structures called "qubits". At the most general level, a qubit can exist in one of two orthogonal states (denoted in traditional bra / ket notation as |0〉 and |1〉) or in a superposition of two states (e.g.,

number

[0003]

[0003] However, the practical realization of quantum computers remains a formidable challenge. One challenge is the reliable formation and entanglement of qubits. Summary of the Invention

[0004]

[0004] Certain embodiments described herein relate to a circuit for generating entanglement between qubits using a "rasterization" technique. In some embodiments, the circuit may include a resource state generator, a first time-fusion circuit, a second time-fusion circuit, and a third time-fusion circuit. The resource state generator may have circuitry for generating a first resource state during a first clock cycle, generating a second resource state during a second clock cycle, generating a third resource state during a third clock cycle, and generating a fourth resource state during a fourth clock cycle, each of the first, second, third, and fourth resource states comprising a system of entangled photonic qubits, and the first, second, third, and fourth clock cycles being different clock cycles. The first time-fusion circuit may be configured to generate a first entangled state between the first resource state and the second resource state by performing an entangled measurement operation between a first qubit of the first resource state and a first qubit of the second resource state. The second time-fusion circuit may be configured to generate a second entangled state between the first entangled state and the third resource state by performing an entangled measurement operation between a second qubit of the first resource state and a first qubit of the third resource state. The third time-fusion circuit may be configured to generate a third entangled state between the second entangled state and the fourth resource state by performing an entangled measurement operation between the third qubit of the first resource state and a first qubit of the fourth resource state.

[0005] In some embodiments, the first and second clock cycles are consecutive clock cycles.

[0006]

[0006] In some embodiments, the resource states define a plurality of layers in the entanglement space, and in some embodiments, the circuit is operable to form a large entangled system of qubits having an entanglement structure comprising a plurality of layers in the entanglement space. When layers in the entanglement space are defined, the first resource state, the second resource state, and the third resource state may all be associated with a first layer of the plurality of layers, while the fourth resource state is associated with a second layer of the plurality of layers. For example, each layer in the entanglement space may be a two-dimensional layer with a first linear dimension of size L, the first clock cycle and the second clock cycle may be separated by a first time interval, and the first clock cycle and the third clock cycle may be separated by L times the first time interval. Furthermore, each layer in the entanglement space may be a two-dimensional layer with a second linear dimension of size L, and the first clock cycle and the fourth clock cycle may be separated by L times the first time interval. 2 You can separate them by twice as much.

[0007]

[0007] In some embodiments, the first time fusion circuit may include a delay line for delaying the first quantum bit in the first resource state until the second clock cycle, and the second time fusion circuit may include a delay line for delaying the second quantum bit in the first resource state until a third clock cycle.

[0008] In some embodiments, the entangled measurement operation performed by the first time-fusing circuitry includes a destructive measurement on the first qubit in a first resource state and the first qubit in a second resource state. Similarly, the entangled measurement operation performed by the second time-fusing circuitry can include a destructive measurement on the second qubit in the first resource state and the first qubit in a third resource state.

[0009] Some embodiments relate to a circuit for generating entanglement between qubits, the circuit including a plurality (N) of unit cells forming a network such that each unit cell is coupled to at least two adjacent unit cells. Each unit cell may comprise a resource state generator, a plurality of fusion circuits, a first local delay line, a second local delay line, a third local delay line, a first routing switch, a second routing switch, a third routing switch, a fourth routing switch, a first routing path, and a second routing path. The resource state generator may have a photonic circuit for generating a first local resource state during a first clock cycle, a second local resource state during a second clock cycle, a third local resource state during a third clock cycle, and a fourth local resource state during a fourth clock cycle, each of the first, second, third, and fourth local resource states comprising a system of entangled photonic qubits, the first, second, and third clock cycles being different clock cycles. The plurality of fusion circuits may include a first local fusion circuit, a second local fusion circuit, a third local fusion circuit, a first networked fusion circuit, and a second networked fusion circuit, each of the plurality of fusion circuits configured to perform an entanglement measurement operation between two input qubits. The first local delay line may be coupled to a first input of the first local fusion circuit and may have a delay of a first number of clock cycles. The second local delay line may be coupled to a first input of the second local fused circuit and may have a delay of a second number of clock cycles, the second number being greater than the first number. The third local delay line may be coupled to a first input of the third local fused circuit and may have a delay of a third number of clock cycles, the third number being greater than the second number.The first routing switch may be configured to selectively direct the first qubit of each resource state to one of the first local delay line of the unit cell or the first input of the first networked fused circuit of the first adjacent unit cell. The second routing switch may be configured to selectively direct the second qubit of each resource state to one of the second input of the first local fused circuit of the unit cell or the second input of the first networked fused circuit. The third routing switch may be configured to selectively direct the third qubit of each resource state to one of the second local delay line of the unit cell or the first input of the second networked fused circuit of the second adjacent unit cell. The fourth routing switch may be configured to selectively direct the fourth qubit of each resource state to one of the second input of the second local fused circuit of the unit cell or the second input of the second networked fused circuit. The first routing path may direct the fifth qubit of each resource state to the third local delay line. A second routing path can direct the sixth qubit of each resource state to a third local fused circuit.

[0010] In some embodiments, the resource states define multiple layers in the entanglement space, and in some embodiments, the circuit is operable to form a large entangled system of qubits having an entanglement structure comprising multiple layers in the entanglement space. When layers in the entanglement space are defined, a first local resource state, a second local resource state, and a third local resource state may all be associated with a first layer of the multiple layers, while a fourth local resource state is associated with a second layer of the multiple layers. For example, each layer of the large entangled system of qubits may be L 2 For a two-dimensional layer with a size of , each unit cell can have multiple (P 2 ) resource state, where P 2 =L 2In these and other embodiments, the first and second clock cycles are separated by a first time interval, while the first and third clock cycles are separated by P times the first time interval. Furthermore, the first and fourth clock cycles are separated by P times the first time interval. 2 It's twice as far away.

[0011]

[0011] In some embodiments, each of the multiple fused circuits can be configured such that the entanglement measurement operation includes a destructive measurement on both of the input qubits.

[0012]

[0012] Some embodiments relate to a circuit for generating a plurality of entangled structures, where each entangled structure can be represented as a plurality of layers in an entanglement space. The circuit can include a layer generating circuit and a plurality of temporal blending circuits. The layer generating circuit can be configured to generate a first layer during a first period, generate a second layer during a second period, and generate a third layer during a third period, where each of the first, second, and third layers comprises a system of photonic qubits entangled in at least two dimensions in the entanglement space, and the second period is between the first period and the third period. Each temporal blending circuit can be configured to perform an entanglement measurement operation between the qubits of the first layer and the qubits of the third layer during a fourth period following the third period.

[0013]

[0013] In some embodiments, the layer generation circuitry is further configured to generate a fourth layer during a fourth period, and the multiple temporal fusion circuits are configured to perform an entanglement measurement operation between one or more quantum bits of the second layer and one or more quantum bits of the fourth layer during a fifth period following the fourth period.

[0014]

[0014] In some embodiments, the circuit may further include a boundary circuit configured to receive surrounding quantum bits corresponding to the boundary of each layer of entangled quantum bits, the boundary circuit including a detector configured to detect the surrounding quantum bits.

[0015]

[0015] In some embodiments, the circuit may also include a boundary circuit configured to receive a peripheral qubit of a resource state at the boundary of each layer of entangled qubits as a boundary qubit. The boundary circuit may include a detector configured to detect the boundary qubit, a temporal fusion circuit for fusing two boundary qubits from layers generated during two different time periods, and a switch configurable to route the boundary qubit to either the detector or the temporal fusion circuit. The switch may be dynamically reconfigurable for each time period.

[0016]

[0016] In some embodiments, the entangled measurement operation can include a destructive measurement on the qubits between which the entangled measurement operation is performed.

[0017]

[0017] Some embodiments relate to a method for generating entanglement between qubits, the method comprising the steps of: operating a resource state generator to generate a new resource state comprising a system of entangled photonic qubits during each clock cycle of a plurality of clock cycles; determining a position in entanglement space for the new resource state, the position being defined within a layer of the resource state; routing a first qubit of the new resource state to a first delay line if the position in entanglement space does not correspond to an end of a row of the layer; and performing an entanglement measurement between a second qubit of the new resource state and a qubit output from the first delay line if the position in entanglement space does not correspond to a beginning of a row of the layer. routing a third qubit of the new resource state to a second delay line having a longer delay than the first delay line if the location in entanglement space does not correspond to the last row of the layer; performing an entanglement measurement between a fourth qubit of the new resource state and a qubit output from the second delay line if the location in entanglement space does not correspond to the first row of the layer; routing a fifth qubit of the new resource state to a third delay line having a longer delay than the second delay line; and performing an entanglement measurement between a sixth qubit of the new resource state and a qubit output from the third delay line if the location in entanglement space does not correspond to the first row of the layer.

[0018] In some embodiments, the method may further include performing a layer edge processing operation on the first qubit of the new resource state when the location in the entanglement space corresponds to the end of a row of the layer. The layer edge processing operation may include, for example, performing a measurement operation on the first qubit of the new resource state or performing an entanglement measurement between the first qubit of the new resource state and a qubit associated with an edge of a different layer of the large entangled system.

[0019]

[0019] In some embodiments, the method may further include performing a layer edge processing operation on a second quantum bit of the new resource state when the position in the entanglement space corresponds to the beginning of a row of the layer.

[0020]

[0020] In some embodiments, the method may further include performing a layer edge processing operation on a third quantum bit of the new resource state when the position in the entanglement space corresponds to the end of a row of the layer.

[0021]

[0021] In some embodiments, the method may further include performing a layer edge processing operation on a fourth quantum bit of the new resource state if the position in the entanglement space corresponds to the beginning of a row of the layer.

[0022] In some embodiments, each row of layers can have a dimension L in the entanglement space, and the second delay line can have a delay corresponding to L times the delay of the first delay line. 2 and the third delay line has a delay L of the first delay line. 2 The delay may be a corresponding factor of 1.

[0023]

[0023] In some embodiments, the step of performing each of the entanglement measurements includes performing a fusion operation, the fusion operation including a destructive measurement on one or both of the quantum bits between which the fusion operation is performed.

[0024]

[0024] Some embodiments relate to a method for generating entanglement between qubits, the method comprising the steps of: operating a plurality of resource state generators at a plurality of unit cells such that during each clock cycle of a plurality of clock cycles, each unit cell generates a new resource state comprising a system of entangled photonic qubits; for each unit cell, determining a location in entanglement space of the new resource state, the locations being defined within successive patches of a layer of resource states; routing a first qubit of the new resource state to a first delay line if the location in entanglement space does not correspond to an end of a row of patches; and routing a second qubit of the new resource state from the first delay line if the location in entanglement space does not correspond to a beginning of a row of patches. performing an entanglement measurement between the fourth qubit of the new resource state and the qubit output from the second delay line, routing the third qubit of the new resource state to a second delay line having a longer delay than the first delay line if the position in the entanglement space does not correspond to the last row of the patch, performing an entanglement measurement between the fourth qubit of the new resource state and the qubit output from the second delay line if the position in the entanglement space does not correspond to the first row of the patch, routing the fifth qubit of the new resource state to a third delay line having a longer delay than the second delay line, and performing an entanglement measurement between the sixth qubit of the new resource state and the qubit output from the third delay line.

[0025] In some embodiments, the method may further include, for at least one of the unit cells, routing a first qubit of the new resource state to a first adjacent unit cell if the location in the entanglement space corresponds to an end of a row of patches. The method may further include, for at least one other of the unit cells, performing an entanglement measurement operation between a second qubit of the new resource state and a networked qubit received from a second adjacent unit cell if the location in the entanglement space corresponds to a beginning of a row of patches.

[0026] In some embodiments, the method may further include, for at least one of the unit cells, routing the third qubit of the new resource state to a first adjacent unit cell if the location in the entanglement space corresponds to an end of a row of the patch. The method may further include, for at least one of the unit cells, performing an entanglement measurement operation between a fourth qubit of the new resource state and a networked qubit received from a second adjacent unit cell if the location in the entanglement space corresponds to a first row of the patch.

[0027] In some embodiments, each row of patches can have a dimension P in entanglement space, and the second delay line can have a delay corresponding to P times the delay of the first delay line. In these and other embodiments, each patch can have a size P in entanglement space. 2 and the third delay line has a delay P of the first delay line. 2 The delay may be a corresponding factor of 1.

[0028]

[0028] In some embodiments, the step of performing each of the entanglement measurements includes performing a fusion operation, the fusion operation including a destructive measurement on one or both of the quantum bits between which the fusion operation is performed.

[0029]

[0029] The following detailed description, taken together with the accompanying drawings, provides a better understanding of the nature and advantages of the invention as set forth in the claims. [Brief description of the drawings]

[0030] [Figure 1] 1 shows two representations of a portion of a pair of waveguides corresponding to a dual-rail encoded photonic qubit. [Figure 2A] A schematic diagram for coupling two modes is shown. [Figure 2B] 1 illustrates a schematic of a physical implementation of mode coupling in a photonic system that may be used in some embodiments. [Figure 3A] 1 illustrates a schematic example of a physical implementation of a Mach-Zehnder Interferometer (MZI) configuration that may be used in some embodiments. [Figure 3B] 1 illustrates a schematic example of a physical implementation of a Mach-Zehnder Interferometer (MZI) configuration that may be used in some embodiments. [Figure 4A] FIG. 1 shows another schematic diagram for coupling two modes. [Figure 4B] 4B illustrates a schematic diagram of a physical implementation of the mode coupling of FIG. 4A in a photonic system that may be used in some embodiments. [Diagram 5] 1 illustrates a four-mode coupling scheme implementing a "spreader" or "modal information erasure" transformation for the four modes according to some embodiments. [Figure 6] 6 illustrates an example of an optical device capable of implementing the four-mode diffusive conversion shown generally in FIG. 5 according to some embodiments. [Figure 7] 1 shows a circuit diagram for a dual-rail encoded Bell state generator that may be used in some embodiments. [Figure 8A] 1 shows a circuit diagram for a dual-rail encoded type I fused gate that may be used in some embodiments. [Figure 8B] 8B shows an example of the result of an I-type fusion operation using the gate of FIG. 8A. [Figure 9A]1 shows a circuit diagram for a dual-rail encoded type-II fused gate that may be used in some embodiments. [Figure 9B] 9B shows an example of the results of a Type II fusion operation using the gates of FIG. 9A. [Figure 10A] 1 illustrates an entanglement graph representation of resource states that may be used in accordance with some embodiments. [Figure 10B] 1 illustrates an entanglement graph representation of resource states that may be used in accordance with some embodiments. [Figure 10C] 1 illustrates an entanglement graph representation of resource states that may be used in accordance with some embodiments. [Figure 11A] 1 illustrates an example of a tier of resource states according to some embodiments. [Figure 11B] 1 illustrates an example of a tier of resource states according to some embodiments. [Figure 12A] 1 illustrates an example of a three-dimensional array including two layers of resource states according to some embodiments. [Figure 12B] 1 illustrates an example of a three-dimensional array including two layers of resource states according to some embodiments. [Figure 13] 1 illustrates an example of a large entangled system of qubits that may be formed in accordance with some embodiments. [Figure 14A] Introducing schematic circuit symbols. [Figure 14B] Introducing schematic circuit symbols. [Figure 14C] Introducing schematic circuit symbols. [Figure 14D] Introducing schematic circuit symbols. [Figure 14E] Introducing schematic circuit symbols. [Figure 14F] Introducing schematic circuit symbols. [Figure 15] FIG. 1 illustrates a conceptual diagram of networked creation of a large entangled system of qubits according to some embodiments. [Figure 16A] FIG. 1 illustrates a schematic diagram of a circuit for generating an entanglement structure from resource states using networked RSG circuits in accordance with some embodiments. [Figure 16B] FIG. 1 illustrates a schematic diagram of a circuit for generating an entanglement structure from resource states using networked RSG circuits in accordance with some embodiments. [Figure 17] FIG. 1 illustrates a conceptual diagram of a rasterized generation of a large entangled system of qubits according to some embodiments. [Figure 18] 1 illustrates a schematic diagram of a circuit for generating an entanglement structure from resource states using a single RSG circuit in accordance with some embodiments. [Figure 19] FIG. 1 illustrates a flow diagram of a process for generating an entanglement structure from resource states in accordance with some embodiments. [Figure 20] FIG. 1 illustrates a conceptual diagram of raster-based hybrid generation of entanglement structures from resource states according to some embodiments. [Figure 21] FIG. 1 illustrates a circuit diagram of a raster-based hybrid unit cell for generating entanglement structures from resource states in accordance with some embodiments. [Figure 22] 1 shows a conceptual diagram of two adjacent patches for a layer according to some embodiments. [Figure 23] 1 illustrates an example of a coordinated order of resource state generation in different patches for a layer according to some embodiments. [Figure 24] 1 illustrates a flow diagram of another process for generating an entanglement structure from resource states in accordance with some embodiments. [Diagram 25] FIG. 1 is a conceptual diagram of hybridization of layers in an entangled structure using a patch-based hybrid circuit according to some embodiments. [Figure 26] 1 illustrates a time diagram for generating a largely entangled system of qubits according to some embodiments. [Figure 27] FIG. 27 shows a simplified conceptual diagram of a linear optical circuit implementing the behavior of FIG. 26 in accordance with some embodiments. [Figure 28] FIG. 1 illustrates a conceptual diagram of interleaved generation of two large entangled systems of qubits according to some embodiments. [Figure 29]1 illustrates a time diagram for generating two interleaved large entangled systems of qubits according to some embodiments. [Diagram 30] FIG. 30 shows a simplified conceptual diagram of a linear optical circuit implementing the behavior of FIG. 29 in accordance with some embodiments. [Diagram 31] A conceptual diagram of two large entangled systems of qubits coexisting in time is shown. [Diagram 32] FIG. 1 illustrates a conceptual diagram of stitching two larger entangled systems of qubits to form a single larger entangled system of qubits in accordance with some embodiments. [Diagram 33] FIG. 1 shows a conceptual diagram of lattice surgery on two large entangled systems of qubits according to some embodiments. [Figure 34A] FIG. 1 illustrates a conceptual diagram of using interleaving to form a 3D entangled topology with folded layers according to some embodiments. [Figure 34B] FIG. 1 illustrates a conceptual diagram of using interleaving to form a 3D entangled topology with folded layers according to some embodiments. [Figure 34C] FIG. 1 illustrates a conceptual diagram of using interleaving to form a 3D entangled topology with folded layers according to some embodiments. [Fig. 34D] FIG. 1 illustrates a conceptual diagram of using interleaving to form a 3D entangled topology with folded layers according to some embodiments. [Figure 35A] FIG. 13 is a conceptual diagram of the use of folding techniques to create periodic boundary conditions in layers of an entangled structure according to some embodiments. [Figure 35B] FIG. 13 is a conceptual diagram of the use of folding techniques to create periodic boundary conditions in layers of an entangled structure according to some embodiments. [Figure 35C] FIG. 13 is a conceptual diagram of the use of folding techniques to create periodic boundary conditions in layers of an entangled structure according to some embodiments. [Figure 36A]FIG. 13 is a conceptual diagram of the use of folding techniques to create more complex periodic boundary conditions in layers of an entangled structure according to some embodiments. [Figure 36B] FIG. 13 is a conceptual diagram of the use of folding techniques to create more complex periodic boundary conditions in layers of an entangled structure according to some embodiments. [Figure 36C] FIG. 13 is a conceptual diagram of the use of folding techniques to create more complex periodic boundary conditions in layers of an entangled structure according to some embodiments. [Figure 36D] FIG. 13 is a conceptual diagram of the use of folding techniques to create more complex periodic boundary conditions in layers of an entangled structure according to some embodiments. [Figure 37A] FIG. 13 is a conceptual diagram of the use of techniques described herein to form diagonal folds for layers of an entangled structure according to some embodiments. [Figure 37B] FIG. 13 is a conceptual diagram of the use of techniques described herein to form diagonal folds for layers of an entangled structure according to some embodiments. [Figure 37C] FIG. 13 is a conceptual diagram of the use of techniques described herein to form diagonal folds for layers of an entangled structure according to some embodiments. [Figure 37D] FIG. 13 is a conceptual diagram of the use of techniques described herein to form diagonal folds for layers of an entangled structure according to some embodiments. [Figure 38] 1 illustrates an example of a system architecture for a quantum computer system according to some embodiments. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0031]

[0072] Disclosed herein are examples of systems and methods (also referred to as "embodiments") for forming qubits and superposition states of qubits (including entangled states) based on various physical quantum systems, including photonic systems. Such embodiments may be used, for example, in quantum computing, as well as in other contexts that utilize quantum entanglement (e.g., quantum communication). To facilitate understanding of the present disclosure, an overview of relevant concepts and terminology is provided in Section 1. In this regard, Section 2 describes example circuits and methods for generating entangled structures, and Section 3 describes further examples of interleaving techniques that may be used to generate entangled structures. In some embodiments, entanglement generated using the techniques described herein may be used to support fault-tolerant quantum computing. Although embodiments are described in specific detail for ease of understanding, those skilled in the art having access to this disclosure will recognize that the claimed invention may be practiced without these details.

[0032]

[0073] Further described herein are embodiments forming and functioning in systems of qubits, where the quantum state space of the qubits can be modeled as a two-dimensional vector space. As will be appreciated by those of ordinary skill in the art with access to this disclosure, the techniques described herein can be applied to systems of "qubits," where the qubits can be any quantum system having a quantum state space that can be modeled as a (complex) n-dimensional vector space (for any integer n) that can be used to encode n bits of information. For clarity of explanation, the term "qubit" is used herein, although in some embodiments the system can also use quantum information carriers that encode information in a manner not necessarily associated with binary bits, such as qubits.

[0033] 1. Overview of Quantum Computing

[0074] Quantum computing relies on the dynamics of quantum objects, e.g., photons, electrons, atoms, ions, molecules, nanostructures, etc., that 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 entire set of which is referred to as a mode. In some embodiments, a mode is defined by defining the value (or distribution of values) of one or more properties of the quantum object. For example, if the quantum object is a photon, the mode may 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 in), the associated propagation direction (e.g., the k-vector of the photon in free space), the polarization state of the photon (e.g., the direction of the electric and / or magnetic fields of the photon (horizontal or vertical)), the time window in which the photon is propagating, the orbital angular momentum, etc.

[0034]

[0075] For a photon propagating in a waveguide, it is convenient to represent the state of the photon as one of a set of discrete spatiotemporal modes. For example, the spatial mode k of the photon i is determined according to which of a finite set of discrete waveguides the photon is propagating through, and the time mode t j is determined by which of the discrete time periods (referred to herein as "bins") the photon resides in. The degree of time discretization may be provided by a pulsed laser responsible for generating the photons. In the following examples, spatial modes are used primarily to avoid complexity of description. However, as will be appreciated by those skilled in the art, the systems and methods may be applied to any type of mode, e.g., time modes, polarization modes, and any other mode or set of modes that serve to define a quantum state. Furthermore, the following description describes an embodiment that uses a photonic waveguide to define the spatial mode of the photons. However, as will be appreciated by those skilled in the art with access to this disclosure, other types of modes, e.g., time modes, energy states, etc., may be used without departing from the scope of the present disclosure. Furthermore, those skilled in the art may implement the examples using other types of quantum systems, including but not limited to other types of photonic systems.

[0035]

[0076] For quantum systems of indistinguishable particles, it is useful to describe the quantum state of the many-body system as a whole using the Fock state formalism (sometimes called the occupation number representation) rather than describing the quantum state of each particle in the system. In the Fock state description, the many-body quantum state is defined by how many particles are in each mode of the system. For example, the multi-mode two-particle Fock state |1001〉 1、2、3、4 defines 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 before, the modes can be any property of a quantum object. In the case of photons, any two modes of the electromagnetic field can be used, and for example, systems can be designed to use modes associated with degrees of freedom that can be operated passively with linear optics. For example, polarization, spatial degrees of freedom, or angular momentum can be used. Two-particle Fock state |1001〉 1、2、3、4 A four-mode system represented by can be physically implemented as four separate waveguides with one photon traveling inside two of the four waveguides. Another example of such a state of a many-body quantum system is the four-particle Fock state |1111〉, which represents each mode occupied by one particle. 1、2、3、4 and a four-particle Fock state |2200〉, which represents modes 1 and 2 occupied by two particles and modes 3 and 4 occupied by zero particles. 1、2、3、4 For modes in which no particles are present, the term "vacuum mode" is used. For example, the four-particle Fock state |2200〉 1、2、3、4 In this case, modes 3 and 4 are referred to herein as "vacuum modes". Fock states with a single occupied mode can be represented in shorthand notation using a subscript to identify the occupied mode. For example, |0010〉 1、2、3、4 is equivalent to |13〉.

[0036] 1.1. Quantum Bits

[0077] 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, where one dimension in the vector space is mapped to a logical value of 0 and the other dimension is mapped to a logical value of 1. In contrast to classical bits, quantum bits can have states that are superpositions of logical values ​​0 and 1. More generally, a "qubit" can be any quantum system with a quantum state space that can be modeled as a (complex) n-dimensional vector space (for any integer n) that can be used to encode n bits of information. For clarity of explanation, the term "qubit" is used herein, although in some embodiments, the system can also use quantum information carriers that encode information in a manner not necessarily associated with binary bits, such as quantum bits. Qubits (or qudits) can be implemented in a variety of quantum systems. Examples of qubits include the polarization state of a photon, the presence of a photon in a waveguide, or the energy state of an atom, ion, nucleus, or photon. Other examples include flux qubits, phase qubits, or other artificial quantum systems such as charge qubits (e.g., formed from superconducting Josephson junctions), topological qubits (e.g., Majorana fermions), or spin qubits formed from vacancy centers (e.g., nitrogen vacancies in diamond).

[0037]

[0078] A qubit may be "dual-rail encoded" such that the logical value of the qubit is encoded by the occupancy of one of two modes of the quantum system. For example, a logical 0 value and a logical 1 value may be encoded as follows: |0〉 L =|10〉 1、2 (1) |1〉 L =|01〉 1、2 (2) Here, the subscript “L” again indicates that the ket represents a logical state (e.g., a qubit value), and the notation |ij〉 on the right-hand side of the above equation 1、2 indicates that there are i particles in the first mode and j particles in the second mode (e.g., i and j are integers). In this notation, the logical states |0〉|1〉 L (representing the states of two qubits, the first qubit being in a "0" logical state and the second qubit being in a "1" logical state) is |1001〉 1、2、3、4 (e.g., in a photonic system, 1 photon in the first waveguide, 0 photons in the second waveguide, 0 photons in the third waveguide, and 1 photon in the fourth waveguide). In some examples throughout this disclosure, various subscripts have been omitted to avoid unnecessary mathematical clutter.

[0038] 1.2. Entangled state

[0079] Many of the advantages of quantum computing over "classical" computing (e.g., conventional digital computers using binary logic) stem from its ability to create entangled states of multi-qubit systems. Mathematically speaking, the state |ψ〉 of n quantum objects can be expressed as:

number

number

[0039]

[0080] More generally, an n-qubit Greenberger-Horne-Zeilinger (GHZ) state (or "n-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 in a first basis state with all qubits in a second basis state,

number

number

[0040] 1.3. Physical Implementation

[0081] The qubits (and operations on the qubits) can be implemented using various physical systems. In some examples described herein, the qubits are provided in an integrated photonic system using a waveguide, a beam splitter, a photonic switch, and a single photon detector, and the modes that can be occupied by the photons are spatiotemporal modes corresponding to the presence of the photons in the waveguide. To perform the conversion operation, the modes can be coupled using a mode coupler, e.g., an optical beam splitter, and a measurement operation can be performed by coupling a single photon detector to a particular waveguide. As will be appreciated by those skilled in the art with access to this disclosure, modes defined by any suitable set of degrees of freedom, e.g., polarization modes, time modes, etc., can be used without departing from the scope of the present disclosure. For example, in the case of modes that differ only in polarization (e.g., horizontal (H) and vertical (V)), the mode coupler can be any optical element that coherently rotates the polarization, e.g., a birefringent material such as a wave plate. For other systems such as ion trap systems or neutral atom systems, the mode coupler can be any physical mechanism capable of coupling two modes, for example a pulsed electromagnetic field tuned to couple the two internal states of the atoms / ions.

[0041]

[0082] In some embodiments of a photonic quantum computing system using dual-rail encoding, a pair of waveguides may be used to implement a qubit. Figure 1 shows two views (100, 100') of a portion of a pair of waveguides 102, 104 that may be used to provide a dual-rail encoded photonic bit. In 100, a photon 106 is in waveguide 102 and there are no photons in waveguide 104 (also referred to as the vacuum mode), which in some embodiments represents the |0> mode of the photonic qubit. L In 100′, a photon 108 is in waveguide 104 and there is no photon in waveguide 102, which in some embodiments corresponds to the |1> state of the photonic qubit. LA photon source (not shown) can be coupled to one end of one of the waveguides to precondition a photonic qubit of a known logical state. The photon source can be operated to emit a single photon into the waveguide to which it is coupled, thereby preconditioning a photonic qubit of a known state. The photon travels through the waveguide, and by periodically operating the photon source, a quantum system can be formed in the same waveguide pair having qubits whose logical states are mapped to different time modes of the photonic system. Furthermore, by providing multiple pairs of waveguides, a quantum system can be formed having qubits whose logical states correspond to different spatiotemporal modes. It should be understood that the waveguides in such a system need not have a particular spatial relationship to one another. For example, the waveguides can be arranged in parallel, but are not necessarily arranged in parallel.

[0042]

[0083] The occupied modes can be formed by using a photon source to generate photons that propagate in the desired waveguide. The photon source can be, for example, a cavity-based photon source that emits photon pairs, also referred to as leading single photon sources. In one example of such a photon source, the photon source is driven by a pump, e.g., an optical pulse, that is coupled to a system of optical resonators that can generate photon pairs by a nonlinear optical process (e.g., spontaneous four-wave mixing (SFWM), spontaneous parametric down-conversion (SPDC), second harmonic generation, etc.). Many different types of photon sources can be used. An example of a photon pair source can be a microring-based spontaneous four-wave mixing (SPFW) heralded photon source (HPS). However, the exact type of photon source used is not critical and any type of nonlinear source using any process, such as SPFW, SPDC, or any other process, can be used. Other classes of photon sources that do not necessarily require nonlinear materials can also be used, such as quantum dot sources, those that use atoms and / or artificial atomic systems such as color centers in crystals. In some cases, the photon source may or may not be coupled to a photonic cavity, as is the case for example with artificial atomic systems such as quantum dots coupled to a cavity. Other types of photon sources, such as optomechanical systems, also exist in SPWM and SPDC.

[0043]

[0084] In such cases, the operation of the photon source may be non-deterministic (sometimes referred to as "stochastic") such that a given pump pulse may or may not generate a photon pair. In some embodiments, coherent spatial and / or temporal multiplexing (herein referred to as "active" multiplexing) of several non-deterministic photon sources may be used to allow the probability of one mode being occupied during a given cycle to approach unity. As will be appreciated by those skilled in the art, many different active multiplexing architectures incorporating spatial and / or temporal multiplexing are envisioned. For example, active multiplexing schemes using logarithmic trees, generalized Mach-Zehnder interferometers, multimode interferometers, chained sources, dump-to-pump chained sources, asymmetric polycrystalline single photon sources, or any other type of active multiplexing architecture may be used. In some embodiments, the photon source may use active multiplexing schemes using quantum feedback control or the like. In some embodiments described below, the use of multi-rail encoding allows the probability of one band being occupied by one mode during a given pulse cycle to approach unity without active multiplexing.

[0044]

[0085] The measurement operation can be performed by coupling the waveguide to a single-photon 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, the detection of a photon (e.g., at the output end of the waveguide) can indicate an occupied mode, while the absence of a detected photon can indicate an unoccupied mode.

[0045]

[0086] Some embodiments described below relate to the physical implementation of a unitary transformation operation that couples modes of a quantum system, which can be understood as transforming the quantum state of the system. For example, if the initial state of the quantum system (before mode coupling) is one in which one mode is occupied with probability 1 and another mode is unoccupied with probability 1 (e.g., state |10〉 in the Fock notation introduced above), then mode coupling can result in a state in which both modes have a nonzero probability of being occupied, e.g., state a1|10〉+a2|01〉, where |a1| 2 +|a2| 2 = 1. In some embodiments, this type of operation can be implemented by using a beam splitter to couple the modes together and a variable phase shifter to apply a phase shift to one or more of the modes. The amplitudes a1 and a2 depend on the reflectivity (or transmittance) of the beam splitter and any phase shift introduced.

[0046]

[0087] Figure 2A shows a schematic diagram 210 (also called a circuit diagram or circuit representation) for coupling two modes. The modes are depicted as horizontal lines 212, 214, and the mode coupler 216 is indicated by a vertical line terminating in a node (solid dot) to identify the modes being coupled. In the more specific language of linear quantum optics, the mode coupler 216 shown in Figure 2A represents a 50 / 50 beam splitter that implements a transfer matrix.

number

number

number

number

number

[0047]

[0088] FIG. 2B shows a physical implementation of mode coupling that implements the transfer matrix T of equation (9) for two photonic modes according to some embodiments. In this example, the mode coupling is performed using a waveguide beam splitter 200, sometimes called a directional coupler or mode coupler. The waveguide beam splitter 200 can be realized by placing two waveguides 202, 204 close enough together so that the evanescent field of one waveguide can couple into the other. By adjusting the spacing d between the waveguides 202, 204 and / or the length l of the coupling region, different couplings between the modes can be obtained. In this way, the waveguide beam splitter 200 can be configured to have a desired transmission. For example, the beam splitter can be designed to have a transmission equal to 0.5 (i.e., a 50 / 50 beam splitter to implement the particular form of the transfer matrix T introduced above). If other transfer matrices are desired, the reflectance (or transmittance) can be designed 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 this disclosure.

[0048]

[0089] In addition to mode coupling, some unitary transformations may include a phase shift applied to one or more modes. In some photonic implementations, variable phase shifters can be implemented in an integrated circuit to control the relative phase of photon states spread across multiple modes. An example of a transfer matrix defining such a phase shift is given by (to apply +i and -i phase shifts to the second mode, respectively):

number

[0049]

[0090] A beam splitter with variable transmission and arbitrary phase relationship between output modes can also be achieved by combining directional couplers and variable phase shifters in a Mach-Zehnder interferometer (MZI) configuration 300, for example as shown in FIG. 3A. Full control over the relative phase and amplitude of the two modes 302a, 302b in dual rail encoding can be achieved by varying the phase provided by the phase shifters 306a, 306b and 306c, as well as the length and proximity of the coupling regions 304a and 304b. FIG. 3B shows a slightly simpler example of an MZI 310 that allows variable transmission between modes 302a, 302b by varying the phase provided by the phase shifter 306. Although FIGS. 3A and 3B are examples of how a mode coupler can be implemented in a physical device, any type of mode coupler / beam splitter can be used without departing from the scope of this disclosure.

[0050]

[0091] In some embodiments, beam splitters and phase shifters can be used in combination to implement various transfer matrices. For example, Figure 4A shows a mode coupler 400, in a schematic form similar to that of Figure 2A, that implements the following transfer matrix:

number

number

[0051]

[0092] Similarly, a network of mode couplers and phase shifters can be used to implement coupling between 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., the scheme takes a photon in any one of the input modes and delocalizes it among each of the four output modes such that there is an equal probability that the photon will be detected in any one of the four output modes. (The well-known Hadamard transform is an example of a spreader transform.) As in FIG. 2A, horizontal lines 512-515 correspond to modes, and mode coupling is indicated by vertical line 516 with nodes (dots) to identify the modes that are coupled. In this case, four modes are coupled. Circuit representation 502 is an equivalent representation to circuit diagram 504, which is a network of first-order mode coupling. More generally, where higher-order mode coupling can be implemented as a network of first-order mode coupling, a circuit representation similar to representation 502 (with the appropriate number of modes) can be used.

[0052]

[0093] FIG. 6 illustrates an example of an optical device 600 capable of implementing the four-mode diffusive conversion shown in FIG. 5 according to some embodiments. The optical device 600 includes a first set of optical waveguides 601, 603 formed in a first material layer (represented by solid lines in FIG. 6) and a second set of optical waveguides 605, 607 formed in a separate second material layer different from the first material layer (represented by dashed lines in FIG. 6). The second material layer and the first material layer are located at different heights above the substrate. As will be appreciated by those skilled in the art, an interferometer such as that shown in FIG. 6 can be implemented in a single layer if appropriate low-loss waveguide crossings are used.

[0053]

[0094] At least one optical waveguide 601, 603 of the first set of optical waveguides is coupled to an optical waveguide 605, 607 of the second set of optical waveguides with any type of suitable optical coupler, such as a directional coupler as described herein (e.g., the optical couplers shown in Figures 2B, 3A, 3B). For example, the optical device shown in Figure 6 includes four optical couplers 618, 620, 622, 624. Each optical coupler can have a coupling region in which the two waveguides propagate in parallel. Although the two waveguides are shown in Figure 6 as being offset from each other in the coupling region, the two waveguides may be located directly above and below each other in the coupling region without any offset. In some embodiments, one or more of the optical couplers 618, 620, 622, 624 are configured to have approximately a 50% coupling efficiency between the two waveguides (e.g., a coupling efficiency of 49%-51%, a coupling efficiency of 49.9%-50.1%, a coupling efficiency of 49.99%-50.01%, a coupling efficiency of 50%, etc.). For example, the lengths 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 the two waveguides, and the distance between the two waveguides are selected to provide a 50% coupling efficiency between the two waveguides. This allows the optical coupler to operate like a 50 / 50 beam splitter.

[0054]

[0095] 6 may further include two interlayer optical couplers 614 and 616. Optical coupler 614 allows for the transmission of light propagating in a waveguide on a first material layer to a waveguide on a second material layer, and optical coupler 616 allows for the transmission of light propagating in a waveguide on a second material layer to a waveguide on a first material layer. Optical couplers 614 and 616 allow for the use of optical waveguides located in at least two different layers in a multi-channel optical coupler, which allows for a compact multi-channel optical coupler.

[0055]

[0096] 6 further includes an uncoupled waveguide intersection region 626. In some implementations, two waveguides (603 and 605 in this example) cross each other without the presence of a parallel coupling region at the intersection in the uncoupled waveguide intersection region 626 (e.g., the waveguides can be two straight waveguides that cross each other at approximately a 90 degree angle).

[0056]

[0097] As will be appreciated by those skilled in the art, the foregoing examples are illustrative and photonic circuits using beam splitters and / or phase shifters can be used to implement many different transform matrices, including transform matrices for real and imaginary Hadamard transforms of any order, discrete Fourier transforms, etc. One class of photonic circuits, referred to herein as "spreader" or "mode information elimination (MIE)" circuits, has the property that if the input is a single photon localized in one input mode, the circuit delocalizes the photon among each of several output modes such that the photon has an equal probability of being detected in any one of the output modes. Examples of spreader or MIE circuits include circuits that implement Hadamard transfer matrices. (It should be understood that the spreader or MIE circuit can receive inputs that are not single photons localized in one input mode, and the behavior of the circuit in such cases depends on the particular transfer matrix implemented.) In other examples, the photonic circuit can implement other transfer matrices, including transfer matrices that have unequal probabilities of detecting a photon in different output modes for a single photon in one input mode.

[0057]

[0098] In some embodiments, an entangled state of multiple photonic qubits can be formed by coupling the modes of two (or more) qubits and performing a measurement in the other mode. By way of example, FIG. 7 shows a circuit diagram of a Bell state generator 700 that can be used in some dual-rail encoded photonic embodiments. In this example, modes 732(1)-732(4) are initially occupied by photons (shown as dashed lines), respectively, and modes 732(5)-732(8) are initially vacuum modes. (Those skilled in the art will recognize that other combinations of occupied and unoccupied modes can be used.)

[0058]

[0099] First-order mode coupling (e.g., implementing the transfer matrix T of equation (9)) is performed for pairs of occupied and unoccupied modes as shown by mode couplers 731(1)-731(4). Then, mode information cancellation coupling (e.g., implementing a four-mode mode spreading transformation as shown in FIG. 5) is performed for four of the modes (modes 732(5)-732(8)) as shown by mode coupler 737. Modes 732(5)-732(8) act as “messenger” modes that are measured and used to determine whether Bell states have been successfully generated in the other four modes 732(1)-732(4). For example, detectors 738(1)-738(4) can be coupled to modes 732(5)-732(8) after second-order mode coupler 737. Each detector 738(1)-738(4) can output a classical data signal (e.g., a voltage level on a conductor) that indicates whether it detected a photon (or the number of photons detected). These outputs can be coupled to classical decision logic 740 that determines whether a Bell state exists in the other four modes 732(1)-732(4). For example, decision logic 740 can be configured such that a Bell state is confirmed (also referred to as a "success" of the Bell state generator) if and only if a single photon is detected by each of exactly two detectors 738(1)-738(4). The modes 732(1)-732(4) can be mapped to the logical states of two qubits (qubit 1 and qubit 2) as shown in FIG. 7. Specifically, in this example, the logical state of Qubit 1 is based on the occupancy of modes 732(1) and 732(2), and the logical state of Qubit 2 is based on the occupancy of modes 732(3) and 732(4). It should be noted that the operation of Bell state generator 700 may be non-deterministic; that is, inputting four photons as shown does not guarantee that Bell states in modes 732(1)-732(4) will be generated. In one implementation, the probability of success is 4 / 32.

[0059]

[0100] In some embodiments, it is desirable to form cluster states of multiple entangled qubits (typically three or more qubits, although Bell states can be understood as two-qubit cluster states). One technique for forming larger entangled systems is through the use of entanglement measurements, which are projective measurements that can be used to generate entanglement between systems of qubits. As used herein, "fusion" (or "fusion operation" or "fusing") refers to a two-qubit entangled measurement. A "fusion gate" is a structure that receives two input qubits, each of which is typically part of an entangled system. The fusion gate performs a projective measurement operation on the input qubits that produces an output qubit of either a 1 ("type I fusion") or a 0 ("type II fusion"), in such a manner that the initial two entangled systems are fused into a single entangled system. Fusion gates are an example of a general class of two-qubit entangled measurements, and are particularly suited to photonic architectures. Examples of type I and type II fusion gates are now described.

[0060]

[0101] FIG. 8A shows a circuit diagram illustrating a type I fused gate 800 according to some embodiments. The diagram shown in FIG. 8A is a schematic diagram in which each horizontal line represents a mode of a quantum system, e.g., photons. In dual-rail encoding, each pair of modes corresponds to a qubit. In a photonic implementation of the gate, the modes of the diagram as shown in FIG. 8A may be physically realized using a single photon in a photonic waveguide. Most generally, a type I fused gate as shown in FIG. 8A takes as inputs qubit A (e.g., physically realized by photon modes 843 and 845) and qubit B (e.g., physically realized by photon modes 847 and 849), and outputs a single "fused" qubit that inherits the entanglement of either input qubit A or input qubit B (or both) with the other qubit that was already entangled.

[0061]

[0102] For example, FIG. 8B shows the result of a type I fusion of two qubits A and B, each of which is 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 remaining after the fusion operation inherits entanglement coupling from the original qubits A and B, thereby producing a larger linear cluster state. FIG. 8B also shows the result of a type I fusion of two qubits A and B, each of which is an interior qubit belonging to some longer entangled cluster of qubits (only a portion of which is shown). As before, the qubit 859 remaining after the fusion inherits entanglement coupling from the original qubits A and B, thereby producing a fused cluster state. In this case, the qubit remaining after the fusion operation is entangled with the larger cluster by four other nearest neighbor qubits, as shown.

[0062]

[0103] Returning to the schematic diagram of type I fused 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, for a path-encoded photonic qubit, the logical zero state of qubit A, |0〉, A) occurs when mode 843 is a photonic waveguide containing a single photon and mode 845 is a photonic waveguide containing zero photons (similarly for qubit B). Thus, type I fusion gate 800 can take two dual-rail encoded photon qubits as input, thereby resulting in a total of four input modes (e.g., modes 843, 845, 847, and 849). To achieve the fusion operation, a mode coupler (e.g., a 50 / 50 beam splitter) 853 is applied between each mode of the input qubit, e.g., between mode 843 and mode 849, before performing a detection operation on both modes using photon detector 855 (which includes two separate photon detectors coupled to modes 843 and 849, respectively). The detection operation on modes 843 and 849 is a destructive measurement. Furthermore, a mode swap operation 851 can be applied that swaps the position of the second mode of qubit A (mode 845) with the position of the second mode of qubit B (mode 849) so that the output modes are adjacently located. In some embodiments, the mode swap can be achieved by a physical waveguide crossing as described above, or by one or more photonic switches, or by any other type of physical mode swap.

[0063]

[0104] 8A shows only an example arrangement of a Type I fused gate, and those skilled in the art will appreciate that the location of the mode coupler and the presence of mode swap region 851 can be modified without departing from the scope of this disclosure. For example, beam splitter 853 can be applied between mode 845 and mode 847. Mode swapping is optional and is not necessary if qubits with non-adjacent modes can be handled by keeping track of which mode belongs to which qubit, for example by storing this information in a classical memory.

[0064]

[0105] Type I fusion gate 800 is a non-deterministic gate, i.e., the fusion operation succeeds with some probability less than 1, and otherwise the resulting quantum state is not a larger cluster state that includes the original cluster states fused to the larger cluster state. More specifically, gate 800 "succeeds" with a 50% probability if only one photon is detected by detector 855, and "fails" if zero or two photons are detected by detector 855. If the gate is successful, the two cluster states of which qubit A and qubit B were part are fused into a single larger cluster state, and the fused qubit remains as the qubit that links the two previously unlinked cluster states (see, e.g., FIG. 8B). However, if the fusion gate fails, it has the effect of removing both qubits from the original cluster resource state without producing a larger fused state.

[0065]

[0106] FIG. 9A shows a circuit diagram illustrating a type II fused gate 900 according to some embodiments. As with other figures herein, the diagram shown in FIG. 9A is a schematic diagram, where each horizontal line corresponds to a mode of a quantum system, e.g., a photon. In dual-rail encoding, each pair of modes corresponds to a qubit. In a photonic implementation of the gate, the modes of the diagram as shown in FIG. 9A can be physically realized using a single photon in a photonic waveguide. Most generally, a type II fused gate such as gate 900 takes as inputs qubit A (e.g., physically realized by photon modes 943 and 945) and qubit B (e.g., physically realized by photon modes 947 and 949), and outputs a quantum state that inherits the entanglement of either (or both) input qubit A or input qubit B with the other qubit that is already entangled. (For Type II fusion, if an input quantum state has N qubits, then the output quantum state has N-2 qubits. This differs from Type I fusion, in which an input quantum state of N qubits results in an output quantum state with N-1 qubits.)

[0066]

[0107] For example, Figure 9B shows the result of a Type II fusion of two qubits A and B, each of which is a qubit located at the end (i.e., leaf) of several longer entangled cluster states (only some of which are shown). The resulting qubit system 971 inherits entanglement couplings from qubit A and qubit B, thereby producing a larger linear cluster state.

[0067]

[0108] Returning to the schematic diagram of Type II fused gate 900 shown in FIG. 9A, qubit A is dual-rail encoded by modes 943 and 945, and qubit B is dual-rail encoded by modes 947 and 949. For example, for a path-encoded photonic qubit, the logical zero state (|0〉 qubit A) of qubit A is dual-rail encoded by modes 943 and 945, and qubit B is dual-rail encoded by modes 947 and 949. AA type II fusion gate 900, denoted by 943(1)-945(4), occurs when mode 943 is a photonic waveguide containing a single photon and mode 945 is a photonic waveguide containing zero photons (similarly for qubit B). Thus, the type II fusion gate 900 takes two dual-rail encoded photon qubits as input, thereby resulting in a total of four input modes (e.g., modes 943, 945, 947, and 949). To achieve the fusion operation, a first mode coupler (e.g., a 50 / 50 beam splitter) 953 is applied between each of the modes of the input qubits, e.g., between mode 943 and mode 949, and a second mode coupler (e.g., a 50 / 50 beam splitter) 955 is applied between each of the other modes of the input qubits, e.g., between mode 945 and mode 947. A detection operation is performed on all four modes using photon detectors 957(1)-957(4). The detection operation is a destructive measurement. In some embodiments, a mode-swapping operation (not shown in FIG. 9A ) can be performed to place the modes in adjacent positions prior to mode coupling. In some embodiments, mode-swapping can be achieved by a physical waveguide crossing as described above, or by one or more photonic switches, or by any other type of physical mode-swapping. Mode-swapping is optional and is not necessary in cases where qubits with non-adjacent modes can be handled by, for example, keeping track of which modes belong to which qubits by storing this information in classical memory.

[0068]

[0109] FIG. 9A shows only an example arrangement of a Type II fused gate; one skilled in the art will appreciate that the location of the mode couplers and the presence or absence of mode swap regions may be varied without departing from the scope of this disclosure.

[0069]

[0110] The Type II fusion gate shown in FIG. 9A is a non-deterministic gate, i.e., the fusion operation succeeds with a certain probability less than one, and in other cases the resulting quantum state is not a larger cluster state that includes the original cluster states fused into the larger cluster state. More specifically, the gate "succeeds" if one photon is detected at either detector 957(1), 957(4) and one photon is detected at either detector 957(2), 957(3), and in all other cases the gate "fails." If the gate is successful, the two cluster states of which qubits A and B were part are fused into a single larger cluster state. Unlike Type I fusion, no fused qubits remain (compare FIG. 8B with FIG. 9B). If the fusion gate fails, it has the effect of removing both qubits from the original cluster resource state without producing a larger fused state.

[0070]

[0111] The above description provides examples 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. The examples described below can be realized using similar photonic circuit elements.

[0071] 2. Generation of entangled structures

[0112] As described in Section 1, a qubit can be physically realized using a pair of waveguides into which photons are introduced, and the qubit can be operated using mode couplers (e.g., beam splitters), variable phase shifters, photon detectors, etc. For example, entanglement between two (or more) qubits can be created by providing a mode coupler between the waveguides associated with different qubits. In practice, physical qubits can suffer from losses (e.g., when a photon may not be detected during measurement due to inefficiencies in the photon generation circuit, mode coupler, fusion circuit, or other components) and noise (e.g., when a bit-flip error may occur before the measurement). As a result, relying on a single physical qubit (e.g., a photon propagating in a pair of waveguides) when performing a quantum computation can result in an unacceptably high error rate. To provide fault tolerance, photonic quantum computers can be designed to operate with one or more logical qubits, where a "logical qubit" is a topological cluster state with an entanglement structure that allows for error correction. (The term "qubit" as used in the following sections refers to a physical qubit, and all references to a logical qubit include the modifier "logical.") For example, in some embodiments, the entangled structure of logical qubits can be represented as a three-dimensional graph. As shorthand, this disclosure uses the term "entanglement space" to refer to a space having dimensions corresponding to the graphical representation of the entangled structure. In the context of quantum computing, logical qubits can improve robustness by supporting error detection and error correction. Logical qubits may also be used in other contexts, such as quantum communication.

[0072]

[0113] Some embodiments described herein relate to apparatus and methods that can be used to build large entangled structures from smaller entangled systems of physical qubits, referred to as “resource states.” As used herein, “resource state” refers to an entangled system of multiple (n) qubits in a non-separable entangled state (an entangled state that cannot be decomposed into smaller separate entangled states). In various embodiments, the number n can be a small number (e.g., 2 or more, or any number up to about 20), or a larger number (as large as desired).

[0073]

[0114] 10A-10C show entanglement graph representations of resource states that can be used in accordance with some embodiments. In the graph representations used herein, physical qubits are represented as dots, and entanglement between physical qubits is represented by lines connecting pairs of dots. In these examples, the entanglement geometry defines a three-dimensional space, and the labels x, y, and z are used to define different dimensions within this entanglement space. It should be understood that these dimensions do not have to correspond to physical dimensions, and in some cases, qubits may be separated in time rather than spatial dimensions. For example, each physical qubit may be implemented using photons propagating in a waveguide, and particular portions of the waveguide may host photons associated with different qubits at different times.

[0074]

[0115] FIG. 10A illustrates an example of a resource state 1000 having seven physical qubits 1010-1016. In resource state 1000, a "central" qubit 1016 is entangled with six "peripheral" qubits 1010-1015. For ease of explanation, the six peripheral qubits are distinguished from one another using directional identifiers +x, -x, +y, -y, +z, -z (as shown by coordinate axes 1001). Thus, for example, qubit 1012 may be referred to as a +x qubit, qubit 1013 may be referred to as a -x qubit, and so on. It should be understood that these identifiers refer to the entanglement geometry and do not necessarily correspond to actual physical directions. As will be clear, the terms "central" qubit and "peripheral" qubit are used herein to distinguish qubits that are subject to fusion operations with qubits from other resource states ("peripheral qubits") from qubits that are not subject to fusion operations with qubits from other resource states ("central qubits").

[0075]

[0116] The entanglement geometry or topology of the resource states can be varied. As an example, FIG. 10B shows an example of a different resource state 1020 having seven physical qubits 1030-1036. Similar to resource state 1000, central qubit 1036 is entangled with six peripheral qubits 1030-1035. Resource state 1020 differs from resource state 1010 in that resource state 1010 has additional entanglement between peripheral qubits 1030 and 1032.

[0076]

[0117] As another example, Figure 10C shows resource state 1040, known in the art as a Kagome-6 state. Resource state 1040 has six peripheral qubits 1050-1055 (and no central qubit), where each peripheral qubit is entangled with two other qubits. Resource state 1040 can be understood to have a three-dimensional entangled geometry as suggested by the central double-headed arrow, where qubit 1050 is a +y qubit, qubit 1051 is a -y qubit, qubit 1052 is a +x qubit, qubit 1053 is a -x qubit, qubit 1054 is a +z qubit, and qubit 1055 is a -z qubit.

[0077]

[0118] The example resource states in Figures 10A-10C are illustrative and not limiting. In some embodiments, the entanglement topology / geometry of the resource states can be selected based on the particular computation being performed, and different resource states used in generating a single entangled structure can have different entanglement topologies. Additionally, while the examples shown include resource states with 6 or 7 qubits, the number of qubits in each resource state can also be varied. Thus, resource states can be larger or smaller than the examples shown and can include any number of central qubits (including zero central qubits) and / or peripheral qubits. Additional considerations regarding the selection of resource state size and entanglement geometry are discussed below.

[0078]

[0119] According to various embodiments, a "layer" of some number of resource states can be generated using one or more resource state generators. (It should be understood that, as with other geometric or spatial terms used herein, "layer" refers to a graphical representation of quantum entanglement of physical qubits and does not imply a particular physical arrangement of waveguides or other components.) Figures 11A and 11B show examples of layers of resource states according to some embodiments. In Figure 11A, layer 1100 is formed from multiple instances of resource states 1000 of Figure 10A, and in Figure 11B, layer 1140 is formed from multiple instances of resource states 1040 of Figure 10C. Layers 1100 and 1140 have a size defined as the number of resource states contained in the layer. In the example used herein, each layer has a regular array structure with rows and columns. (The terms "row" and "column" are used herein to distinguish dimensions of entanglement space, and do not necessarily correspond to physical dimensions.) Thus, as shown in FIG. 11A, layer 1100 includes R×C resource states, where R is the number of rows and C is the number of columns. In some cases (e.g., as shown in FIG. 11B), R=C=L, and layer 1100 is of size L. 2 In some embodiments, L 2 (or R × C) is a large number, for example, about 100 to about 10 6 It could be.

[0079]

[0120] To form an entanglement structure larger than the resource states, a fusion operation (e.g., a type II fusion operation as described above or other entanglement measurement operations) can be performed to form entanglement between qubits in different resource states in a layer. Figures 11A and 11B use dotted ovals to show example pairs of qubits that may be input to a fusion circuit (e.g., type II fusion circuit 900 of Figure 9B). Thus, for example, in layer 1100 of Figure 11A, the +x qubit (1,1) of resource state 1000 and the -x qubit (1,2) of resource state 1000 may be inputs to one fusion operation, as shown by dotted oval 1105, and the -y qubit (1,1) of resource state 1000 and the +y qubit (2,1) of resource state 1000 may be inputs to another fusion operation, as shown by dotted oval 1107. As shown, this pattern can be repeated across layer 1100. 11B , the +x qubit (1,1) of resource state 1040 and the -x qubit (1,2) of resource state 1040 may be inputs to one fusion operation, as indicated by dotted oval 1145, and the -y qubit (1,1) of resource state 1040 and the +y qubit (2,1) of resource state 1040 may be inputs to another fusion operation, as indicated by dotted oval 1147. As shown, this pattern may be repeated across layer 1100.

[0080]

[0121] In some embodiments, qubits at the edges or boundaries of a layer (e.g., qubits 1106 and 1108 in layer 1100, or qubits 1146 and 1148 in layer 1140) can be treated as a special case. For example, qubits at layer boundaries (also called "boundary qubits") can be removed from the system by performing a Z measurement (i.e., a measurement in the Pauli Z basis) or similar operation on the qubit. Alternatively, the boundary qubit may undergo a merging operation with another boundary qubit, which may be a boundary qubit of the same layer or a different layer, as desired. Examples of operations on boundary qubits are described below. In some embodiments, the resource state generator can be configured such that boundary qubits are not generated or are selectively generated.

[0081]

[0122] In some embodiments, multiple layers of resource states can be formed, and further fusion operations (e.g., the Type II fusion operations described above) can be performed to form entanglement between qubits associated with resource states of different layers. For example, FIGS. 12A and 12B show examples of three-dimensional arrays including two layers of resource states according to some embodiments. In FIG. 12A, array 1200 includes two instances of layer 1100 of FIG. 11B, and in FIG. 12B, array 1240 includes two instances of layer 1140 of FIG. 11B. For clarity of illustration, in FIGS. 12A and 12B, layers 1100(1) and 1140(1) are shown using black dots to represent qubits, and layers 1100(2) and 1140(2) are shown using white dots to represent qubits. FIGS. 12A and 12B use dashed ovals to show examples of pairs of qubits from different layers that can be input to a fused circuit (e.g., Type II fused circuit 900 of FIG. 9B). Thus, for example, as shown in FIG. 12A, the -z qubit (1,1,1) of resource state 1000 and the +z qubit (1,1,2) of resource state 1000 may be inputs to a fusion operation, as indicated by dashed oval 1205. Similarly, the -z qubit of each other resource state in layer 1100(1) may be fused with the +z qubit of a resource state in a corresponding position in layer 1100(2). Similarly, as shown in FIG. 12B, the -z qubit of each resource state 1040(i,j,1) of layer 1140(1) and the +z qubit of a corresponding resource state 1040(i,j,2) of layer 1140(2) may be inputs to a fusion operation, as indicated by dashed oval 1245. For clarity of illustration, fusion operations between adjacent qubits within a layer are not shown in FIGS. 12A and 12B. However, it should be understood that blending operations within each layer (e.g., as shown in Figures 11A and 11B) can also be performed. The same pattern of blending operations can be extended to any number of layers. The number of layers created can be independent of the size of the layers and can be determined, for example, based on the particular quantum computation being performed.

[0082]

[0123] In some embodiments, fusion operations between resource state qubits within a layer (e.g., as shown in FIGS. 11A and 11B) and between resource state qubits in different layers (e.g., as shown in FIGS. 12A and 12B) are Type II fusion operations (as described above in connection with FIGS. 9A and 9B) performed on pairs of input qubits. A successful Type II fusion removes the input qubit from the system and generates entanglement between the remaining qubit (in this case, the central qubit). Furthermore, Type II fusion (whether successful or not) involves making destructive measurements, and the results of those measurements (e.g., the number of photons detected by each of the detectors 957 in the fusion circuit 900 of FIG. 9A) can be provided as (classical) data to a classical computer, which can interpret the results to extract information reflective of the entanglement structure. For example, the classical computer can use the measurement data to determine the outcome of a quantum computation.

[0083]

[0124] In the following description, fusion operations may be referred to as "spatial" or "temporal." This term recalls the particular implementation in which different qubits or resource states are generated at different times; spatial fusion may be performed between qubits generated simultaneously using different hardware instances, and temporal fusion may be performed between qubits generated at different times using the same hardware instance. In the case of photonic qubits, temporal fusion may be performed by delaying the previously generated qubit (e.g., using additional lengths of waveguide material to create a longer propagation path for the photons), thereby allowing mode coupling with the later generated qubit. By utilizing temporal fusion, the same hardware may be used to generate multiple instances of a resource state within a layer and / or to generate multiple layers of resource states.

[0084]

[0125] In some embodiments, some or all of the fusion operations can be performed using a reconfigurable fusion circuit. The reconfigurable fusion circuit can incorporate various pre-fusion operations such as phase shifting, mode swapping, and / or basis rotation, and can receive (classical) control signals to select the particular operations to be performed. For example, different fusion operations can be selectively performed at different positions within a layer, or different fusion operations can be selectively performed for different layers. For example, a reconfigurable fusion circuit can be used to implement a particular quantum computing algorithm using an array of resource states.

[0085]

[0126] In some embodiments (e.g., the examples of FIGS. 10A, 11A, and 12A), each resource state has a central qubit (i.e., a qubit such as qubit 1016 that does not undergo a merging operation with qubits of another resource state). Thus, after performing a merging operation as described above, a large entangled system of qubits (referred to herein as a "LES") can be generated. FIG. 13 illustrates an example of an LES 1300 that may be formed by a merging operation as shown in FIGS. 11A and 12A applied to the resource state 1000 of FIG. 10A according to some embodiments. In this example, the resource state 1000 has a single central qubit (qubit 1016 of FIG. 10A), and the LES 1300 may be understood to have layers, with each layer comprising an array of R×C qubits 1316. More generally, a resource state may have any number of central qubits, and the number of qubits per layer of the LES may differ from the size of the layers of resource states that contributed to the layer of the LES. A LES is a system of qubits that are physically preprocessed and thus physically exist in a particular entangled state. The entangled state of qubits (e.g., photonic qubits) may itself be a graph state, a cluster state, some other entangled state that forms a fault-tolerant cluster state that corresponds to a quantum error-correcting code (e.g., a topological code, e.g., a leaf surface code, a volume code, a color code, etc.) with appropriate measurements on individual qubits, or any part of these entangled states. Thus, a LES (or several LESs that are further entangled with each other via a process such as the "stitching" process described below) can be used to encode one or more logical qubits or be used as a cluster state (or part of a cluster state) where measurements of individual physical qubits are made to perform quantum computations in a measurement-based quantum computing ("MBQC") system, or in any other context where a large entangled system of physical qubits is generated.

[0086]

[0127] In other embodiments (e.g., the examples of Figures 10C, 11B, 12B), the resource state does not have any central qubit. In embodiments where the resource state does not have a central qubit, the intra-layer and inter-layer fusion operations can include destructive measurements on all of the qubits of all of the resource states, and the final output that generates the entanglement can be a set of (classical) measurement result data from the fusion operations. In some embodiments, this measurement result data can be interpreted as the result of a computation involving one or more error-corrected logical qubits having an entanglement structure defined by the resource states and the fusion operations performed thereon. This technique is referred to herein as "fusion-based quantum computing" or "FBQC."

[0087]

[0128] It should be understood that the resource states and arrays shown herein are exemplary and variations and modifications are possible. The size and entanglement shapes of the resource states can be changed. In some embodiments, resource states having different sizes and / or entanglement geometries can be used at different locations within a layer or array of layers, and location-dependent selection of resource state configurations can be used to perform various logical operations. It should also be understood that the fusion operation may be probabilistic in nature and may not always be successful, and in some embodiments, the entanglement shapes can support fault tolerance of both MBQC or FBQC. Furthermore, it should be understood that while FBQC and MBQC are example use cases for the entanglement generation techniques described herein, these techniques can be applied in other contexts and are not limited to quantum computing.

[0088] 2.1. Resource State Creation

[0129] As mentioned above, some embodiments relate to apparatus and methods that can be used to construct large entangled structures from multiple resource states, where each resource state is an entangled system of quantum bits, number n of non-separable entangled states.

[0089]

[0130] The particular size and entanglement shape of the resource states can be selected as design parameters. In some cases, the optimal size may depend on the particular physical implementation of the qubits. For example, as previously described, the qubits may be implemented using photons propagating in a waveguide. The process used to generate the photons and create entanglement may be stochastic (i.e., the probability of successfully generating a photon in any given instance is significantly less than one). If the generation or entanglement of the qubits is stochastic, multiplexing or other techniques may be used to increase the probability of generating (for each attempt) a resource state with a defined entanglement structure. Given a set of resource states, the process used to form a larger entanglement structure (e.g., the fusion process described above) may be stochastic, and the larger entanglement structure may be defined in a manner that supports fault-tolerant behavior in the presence of stochastic processes. Thus, the size of the resource states can be selected for a particular implementation based on an acceptable error rate for resource state generation and the particular probability of generating a resource state with a defined entanglement structure.

[0090]

[0131] In some embodiments, resource states such as resource state 1100 can be generated using photonic and electronic circuits and components (e.g., of the type described in section 1.3 above) to generate and manipulate individual photons. In some implementations, the resource state generator can be a single integrated circuit, fabricated, for example, using conventional silicon-based technology. The resource state generator can include a photon source or can receive photons from an external source. The resource state generator can also include photonic circuits that implement Bell state generators and fusion operations as described above. To provide robustness, the resource state generator can include multiple parallel instances of different photonic circuits with detectors and electronic control logic to select successful instances to propagate photons. Those skilled in the art are aware of various methods for constructing photonic resource state generators capable of generating resource states with desired entanglement geometries.

[0091]

[0132] In some embodiments, the resource states can be generated using techniques other than linear optical systems. For example, various devices are known for generating and creating entanglement between systems of "material-based" qubits, such as qubits implemented in ion traps, other qubits encoded at the energy levels of atoms or ions, spin-encoded qubits, superconducting qubits, or other physical systems. It is also understood in the art that quantum information is fungible, in the sense that many different physical systems can be used to encode the same information (in this case, quantum states). Thus, in principle, it is possible to swap the quantum state of one system for another by inducing an interaction between the systems. For example, the state of a qubit (or a collection of quantized qubits) encoded at the energy levels of atoms or ions can be swapped into an electromagnetic field (i.e., photons). It is also possible to use transducer techniques to swap the state of a superconducting qubit into a photonic state. In some cases, the initial swap may be on photons with microwave frequencies. After the swap, the frequency of the photons can be increased to the operating frequency of the optical fiber or other optical waveguide. As another example, quantum teleportation may be applied between a material-based qubit and a Bell pair, where one qubit of the Bell pair is a photon having a frequency suitable for an optical fiber (or other optical waveguide), thereby transferring the quantum state of the material-based qubit to a system of photonic qubits. Thus, in some embodiments, material-based qubits may be used to generate resource states comprised of photonic qubits, and the particular configuration and arrangement of the resource state generator is not relevant to an understanding of this specification.

[0092] 2.2. Circuit for forming entanglement structure from resource states

[0133] We now describe examples of circuits and techniques that can be used to form an entangled structure by performing a fusion operation as described above between qubits in resource states generated by one or more resource state generators. For ease of explanation, two cases are considered. One case includes the examples of Figures 10A, 11A, and 12A, where each resource state includes a central qubit, and an LES as shown in Figure 13 is generated. The other case includes the examples of Figures 10C, 11B, and 12B, where each resource state does not include a central qubit, and it should be understood that other resource state configurations can be used, including configurations with any number (0 or more) of central qubits, where the result of the aforementioned fusion operation is (classical) measurement result data that reflects the entangled structure.

[0093] 2.2.1. Circuit symbols

[0134] To facilitate understanding of the description, Figures 14A-14F introduce a set of schematic circuit symbols that are used in subsequent figures. These circuit symbols represent circuits that operate on physical (photonic) quantum bits, with each input or output line representing a (physical) quantum bit. As a drawing convention, inputs are shown on the left and outputs on the right, with the distinction being that the schematic circuit diagram does not necessarily correspond to a particular physical layout.

[0094]

[0135] FIG. 14A depicts a symbol for a resource state generator (RSG) circuit 1400. As previously mentioned, the RSG circuit can be implemented using any circuit or device that generates resource states encoded on photonic qubits. Examples include photonic / electronic circuits as well as devices that form the resource states encoded on a non-photonic system of physical qubits and then exchange the quantum states to photonic qubits. Other implementations of resource state generator circuits can form initial states in non-photonic systems of physical qubits, exchange the initial states to photonic qubits, and then perform linear optical operations to form the resource states. Regardless of the implementation, the output of the RSG circuit 1400 is a qubit, as indicated by line 1402, with the number of outputs depending on the particular resource state. In the embodiments described herein, it is assumed that the RSG circuit generates one resource state per clock cycle, and the length of the clock cycle can be defined based on the time required for one RSG circuit to generate one resource state. The time required can depend on the particular RSG circuit. For example, the RSG circuit may generate a resource state in 1 ns (or 100 ns) and the clock cycle may be 1 ns (or 100 ns). In some embodiments, the clock cycle may be longer than the time required for the RSG circuit to generate one resource state, and the RSG need not run at full speed. For purposes of this specification, it is assumed that the RSG circuit 1400 outputs all qubits of a resource state in the same clock cycle. However, timing can vary, as will be appreciated by those of ordinary skill in the art with access to this disclosure.

[0095]

[0136] FIG. 14B shows a symbol for a Type II fusion circuit 1405. The Type II fusion circuit may be implemented, for example, as described above in connection with FIGS. 9A and 9B. The input is two qubits (indicated by lines 1404). As described above, a Type II fusion operation involves a destructive measurement on two qubits. The Type II fusion circuit 1405 may provide a classical output signal 1406, which may encode measurement data indicative of detected photon counts from each detector and / or other information (e.g., success or failure of the fusion operation).

[0096]

[0137] FIG. 14C depicts a symbol for a switching circuit 1410. The inputs and outputs to the switching circuit 1410 can include any number of qubits (lines 1408), and the number of inputs need not equal the number of outputs (lines 1409). The switching circuit 1410 can incorporate any combination of one or more active optical switches, mode couplers, mode swap circuits, phase shifters, and the like. The switching circuit can be configured to perform active operations to reconfigure the input modes (e.g., to effect a basis change of the qubit by coupling the modes of the qubit), change the input modes, and / or apply a phase to one or more of the input modes (which may affect subsequent coupling between the modes). In some embodiments, the operation of the switching circuit 1410 can be dynamically controlled in response to classical control signals 1411, and its state can be determined based on the results of previous operations, a particular calculation to be performed, configuration settings, timing counters (e.g., for periodic switching), or any other parameters or information.

[0097]

[0138] 14D depicts a symbol for a delay circuit 1415. The delay circuit delays the propagation of a quantum bit (input 1412) for a fixed amount of time and then outputs a quantum bit (output 1414). The amount of time (in clock cycles) is indicated by a number, with D=1 indicating a delay of one clock cycle. The delay circuit may be implemented, for example, by providing one or more appropriate lengths of optical fiber, other waveguiding material, nitride layers, memory, etc., such that photons of a delayed quantum bit travel a longer path than photons of a non-delayed quantum bit.

[0098]

[0139] FIG. 14E illustrates a symbol for a reconfigurable fusion circuit 1420. As illustrated, the reconfigurable fusion circuit includes a switching circuit 1410 followed by a fusion circuit 1405. The reconfigurable fusion circuit can support configurable operations, such as basis changes or phase shifts, applied by the switching circuit 1410 prior to the fusion operation by the fusion circuit 1405. As with other examples of the switching circuit 1410, the operation of the switching circuit 1410 in the reconfigurable fusion circuit 1420 can be dynamically controlled in response to a classical control signal 1411. As with other examples of the fusion circuit 1405, the fusion circuit 1405 in the reconfigurable fusion circuit 1420 can provide a classical output signal 1406.

[0099]

[0140] 14F depicts a symbol for an offset reconfigurable fusion circuit 1425. As shown, the offset reconfigurable confusion circuit is similar to the reconfigurable fusion circuit 1420 with the addition of a delay circuit 1415 to delay one input relative to the other by a defined number of clock cycles. The offset reconfigurable fusion circuit 1425 may also be referred to as a "temporal" fusion circuit, a term that emphasizes the temporal aspect resulting from the delay circuit. 2.2.2. Network generation of entanglements

[0100]

[0141] In some embodiments, a set of networked RSG circuits can be provided, with each RSG circuit contributing one resource state that is merged with resource states from other RSG circuits to form a layer of an entanglement structure (e.g., as shown in FIG. 11A or FIG. 11B ), and the same RSG circuit can successively generate different layers for the entanglement structure. FIG. 15 shows a conceptual diagram of networked generation of layers according to some embodiments. Size L 2 To support the generation of layers, a corresponding number L 2 In the simplified example used herein, the RSG circuit 1502 is 2 = 16, but in reality, L 2 can be much larger (e.g., about 10 2 , about 10 4 , about 10 6 ). At each clock cycle, enough resource states 1500 can be generated to form a complete two-dimensional (2D) layer of resource states. (In FIG. 15, each resource state 1500 is annotated with time “t=1” to indicate that they are all generated during the same clock cycle.) Spatial blending operations can be performed on qubits of adjacent resource states 1500 (e.g., as shown in FIGS. 11A and 11B) using additional circuitry described below. 2 Using RSG circuit 1502, 2 By generating different layers of resource states, a three-dimensional entangled structure can be generated, and additional circuitry described below can be used to perform temporal fusion operations on quantum bits in the different layers of resource states 1500 (e.g., as shown in Figures 12A and 12B).

[0101]

[0142] 16A and 16B are schematic diagrams of a "fully networked" circuit for generating entanglement structures from resource states according to some embodiments. The circuit notation is as previously described in connection with FIGS. 14A-14F, except that classical inputs and outputs are not shown for clarity of illustration. FIG. 16A shows a representative network cell 1600, and FIG. 16B shows the coupling between adjacent instances of the network cell 1600 in a network 1650. As best seen in FIG. 16A, each network cell 1600 includes an RSG circuit 1502 that generates a resource state having six peripheral qubits (solid lines) and, optionally, one or more central qubits 1615 (if present) that are not subject to a fusion operation. For example, if the RSG circuit 1502 generates the resource state 1000 of FIG. 10A, the central qubit 1016 may be provided as the central qubit 1615, but if the RSG circuit instead generates the resource state 1040 of FIG. 10C, the central qubit 1615 is not provided. RSG 1502 provides two peripheral qubits to adjacent network cells, as indicated by "x-fused" output path 1611 and "y-fused" output path 1612. Network cell 1600 also receives qubits from two adjacent network cells. Specifically, input path 1611' couples to the x-fused output path of network cell 1600' (as shown in FIG. 16B). Similarly, input path 1612'' couples to the y-fused output path of network cell 1600'' adjacent to network cell 1600 in the +y direction (as shown in FIG. 16B).

[0102]

[0143] Each instance of network cell 1600 also includes a y+ reconfigurable fusion circuit 1620, an x+ reconfigurable fusion circuit 1630, and a z+ / - offset reconfigurable fusion circuit 1640. The y+ reconfigurable fusion circuit 1620 combines +y qubits of the “local” resource state generated by RSG circuit 1502 with −y qubits of the “networked” resource state generated by an RSG circuit in the adjacent network cell 1600″ in the +y direction. The x+ reconfigurable fusion circuit 1630 combines +x qubits of the local resource state generated by RSG circuit 1502 with −x qubits of the network resource state generated by the adjacent network cell 1600′ in the +x direction. The z+ / - offset reconfigurable fusion circuit receives the +z and −z qubits of the local resource state generated by RSG circuit 1502. The −z qubits are delayed by one clock cycle and fused with the +z qubits of the resource state generated by RSG circuit 1502 during the next clock cycle.

[0103]

[0144] The connectivity shown in Figures 16A and 16B can be expanded to any number of network cells, thereby creating layers of any size (which may be fixed in the hardware design).

[0104] 2.2.3. Rasterization of entanglements

[0145] Generating entanglements using a fully networked RSG circuit as described above provides fast computation, but it is difficult to achieve high speed computation, especially with the size of each layer (L 2) can be hardware intensive. Furthermore, the maximum size of a layer may be constrained by available hardware. Therefore, some embodiments employ a hardware reduction approach, referred to herein as "rasterized" generation of entanglements, in which one instance of an RSG circuit provides multiple resource states in a single layer. In one example of "fully rasterized" generation, a single instance of an RSG circuit can be used to generate entanglement structures with layers of any size by providing appropriate delay and fusion circuits.

[0105]

[0146] 17 is a conceptual diagram of rasterized generation of layers for an entangled structure according to some embodiments. 2 A single instance of the RSG circuit 1702 is provided to support the generation of layers of L 2 = 16, but in reality, L 2 can be much larger (e.g., about 10 2 , about 10 4 , about 10 6 Each clock cycle, the RSG circuit 1702 generates a single resource state and allocates enough resource states to form a complete 2D layer. 2 In this example, each instance of resource state 1700 is generated in a different clock cycle, and each instance of resource state 1700 is annotated with a time "t=1" through "t=16" to indicate the clock cycle in which each resource state 1700 is generated. A temporal fusion operation can be performed on qubits of adjacent resource states 1700 generated during different clock cycles (e.g., the fusion operation shown in Figures 11A and 11B) using additional circuitry described below. The three-dimensional entanglement structure is structured as L 2The resource states can be generated by using the same RSG circuitry 1702 to repeat the process of generating the resource states, and a temporal fusion operation can be performed on the quantum bits of the resource states 1700 of different layers using additional circuitry described below (e.g., the fusion operation shown in Figures 12A and 12B).

[0106]

[0147] FIG. 18 shows a schematic diagram of a "fully rasterized" circuit 1800 for generating entanglement structures from resource states according to some embodiments. The circuit notation is as described above in connection with FIGS. 14A-14F, except that classical inputs and outputs are not shown for clarity of illustration. The RSG circuit 1702 generates a resource state having six peripheral qubits and optionally one or more central qubits 1815, which are not subject to a fusion operation (if present). For example, if the RSG circuit 1502 generates the resource state 1000 of FIG. 10A, the central qubit 1016 may be provided as the central qubit 1815, but if the RSG circuit instead generates the resource state 1040 of FIG. 10C, the central qubit 1815 is not provided. The offset reconfigurable fusion circuit 1852 delays the -x qubit of each resource state output by the RSG circuit 1702 by one clock cycle, then passes the -x qubit through a configurable switching circuit along with the (non-delayed) +x qubit of the resource state output by the RSG circuit 1702 in the next clock cycle, after which a fusion operation is performed on the two qubits output by the switching circuit. The offset reconfigurable fusion circuit 1854 delays the -y qubit of each resource state output by the RSG circuit 1702 by L clock cycles, then passes the -y qubit through a configurable switching circuit along with the (non-delayed) +y qubit of the resource state output by the RSG circuit 1702 L clock cycles later, after which a fusion operation is performed on the two qubits output by the switching circuit. The offset reconfigurable fusion circuit 1856 delays the -z qubit of each resource state output by the RSG circuit 1702 by L clock cycles, then passes the -y qubit through a configurable switching circuit along with the (non-delayed) +y qubit of the resource state output by the RSG circuit 1702 L clock cycles later, after which a fusion operation is performed on the two qubits output by the switching circuit. 2 Delay by a clock cycle and then add the -z qubit, L2 After a clock cycle, the resource state output from the RSG circuit 1702 is passed through a configurable switching circuit along with the (non-delayed) +z quantum bit, and then a fusion operation is performed on the two quantum bits output from the switching circuit.

[0107]

[0148] In this example, the generation of resource states by the full rasterization circuit 1800 can be understood to proceed along the rows of a layer of resource states, as shown in FIG. 17. Resource state generation and fusion operations between qubits of adjacent resource states using the offset reconfigurable fusion circuit 1852 proceed along the +x direction (in the entangled geometry) for the length (L) of one row of the layer. After completing the first row, the full rasterization circuit 1800 continues to the next row in the +y direction, again proceeding along the +x direction to generate a second row, and using the offset reconfigurable fusion circuit 1854 to perform fusion operations between (delayed) +y qubits from the resource states of the first row and -y qubits from the newly generated resource states of the second row, and so on until an entire layer has been generated. The process can then be repeated to generate a second layer, and using the offset reconfigurable fusion circuit 1856 to perform fusion operations between (delayed) +z qubits from the resource states of the first layer and -z qubits from the newly generated resource states of the second layer. Thus, any number of layers may be generated in a rasterized fashion. It should be understood that the term "rasterized" as used herein does not imply a particular physical arrangement of components, and the rasterization circuitry 1800 need not move at all to generate resource states that correspond to different locations in a layer. Instead, photons encoding quantum bits associated with different instances of resource state 1700 may propagate through the same set of waveguides at different times.

[0108]

[0149] Referring again to FIG. 18, the switching circuits in the offset reconfigurable fusion circuits 1852, 1854, 1856 can be controlled to provide desired behavior at the boundaries of the array. For example, to form a layer with a planar topology, the +x qubit of the resource state at the end of a given row should not be fused with the -x qubit of the next resource state (in a different row); instead, the +x qubit of the resource state and the -x qubits of the resource states at the end and beginning of each row can be removed from the system, which can be done, for example, by measuring each qubit in the Z basis. Similar considerations apply in the y and z dimensions. Thus, in some embodiments, the switching circuits in the offset reconfigurable fusion circuits 1852, 1854, 1856 can be reconfigured to perform single qubit Z measurements on an incoming qubit during a selected clock cycle (e.g., by selectively coupling an input mode to an output mode that couples to a photon detector). For other layer topologies, different behaviors can be implemented, examples of which are described below. In some embodiments, the RSG circuit 1702 may be reconfigurable such that the resource states at the end of a row do not include qubits that are not subject to a fusion operation with qubits in other resource states.

[0109]

[0150] It should be understood that the circuit 1800 of FIG. 18 can be used to generate layers of any size. (In some embodiments, the maximum size may be fixed in the hardware design, for example, by the length of the various delay lines.) 2 The layer of L is (assuming one resource state is generated during each clock cycle) 2 It can be generated in a clock cycle. Since many photons can coexist in the delay line, it is possible to use as few as three physical delay lines (e.g., 1, L and L 2It should also be noted that only three optical fibers or other waveguides (of lengths corresponding to the delay in clock cycles) are required. More generally, the number of physical delay lines required for a given implementation may depend on the particular structure and layer dimensions of the resource states. Thus, a hardware implementation using a fully rasterized circuit can be significantly smaller than the fully networked circuit described above, but the fully rasterized circuit will require a longer execution time to generate and operate a given number of resource states.

[0110]

[0151] FIG. 19 illustrates a flow diagram of a process 1900 that can be implemented using the circuit 1800 of FIG. 18 (or other circuitry) according to some embodiments. The process 1900 can be performed during each clock cycle while the entangled structure is being generated, or the duration of the clock cycle can be defined according to the time consumed in performing one iteration of the process 1900. In this example, assume that the RSG circuit 1702 is used to generate one row, then the next row, and so on, to generate each layer, as shown in FIG. 17. (As discussed elsewhere herein, it should be understood that terms such as "row," "column," and "layer" are used in reference to entangled shapes that do not necessarily correspond to a physical arrangement of qubits.)

[0111]

[0152] In block 1902, the RSG circuit 1702 (or other circuitry) may operate to generate new resource states. In some embodiments, the RSG circuit 1702 generates one new resource state per clock cycle. In block 1904, the position (in entanglement space) of the new resource state within a layer of the entanglement structure is determined. For example, a row position counter may be incremented at each clock cycle to count positions within a row (e.g., 1 to L, where L corresponds to the size of the row) and reset at the end of each row, and a column position counter may be incremented when each row is completed (e.g., every L clock cycles or when the row position counter is reset) and reset when a layer is completed (e.g., after L rows are completed). Thus, the current counter value may indicate the position of the new resource state within the layer. Other techniques for defining the current position in the entanglement space may be used.

[0112]

[0153] At block 1906, a determination is made as to whether the current position corresponds to the end of a row (e.g., whether the row position counter has a value L). If not, at block 1908, the first qubit of the new resource state is routed to an "O(1)" delay line that imposes a delay on the order of one clock cycle, such as the delay line of the offset reconfigurable fused circuit 1852 of FIG. 18. In some embodiments, the delay line may impose a delay of exactly one clock cycle. If, at block 1906, the current position corresponds to the end of a row, at block 1910, layer edge processing may be performed on the first qubit. In some embodiments, the layer edge processing may include performing a measurement on the first qubit that removes the first qubit from the system without destroying the entanglement of other qubits. Other options for layer edge processing are described below.

[0113]

[0154] At block 1916, a determination is made as to whether the current position corresponds to the beginning of a row (e.g., whether the row position counter has a value of 1). If not, at block 1918, a fusion operation is performed on the second qubit in the new resource state and the qubit output from the O(1) delay line, e.g., the offset reconfigurable fusion circuit 1852 may perform a fusion operation on the second qubit in the new resource state and the qubit routed to the O(1) delay line of the offset reconfigurable fusion circuit 1852 during the previous clock cycle. At block 1916, if the current position corresponds to the beginning of a row, at block 1920, a layer edge operation may be performed on the second qubit. In some embodiments, the layer edge operation may include performing a measurement on the second qubit that removes the second qubit from the system without destroying the entanglement of other qubits. Other options for layer edge operations are described below.

[0114]

[0155] At block 1926, a determination is made as to whether the current position corresponds to the last row of the layer (e.g., whether the column position counter has a value L). If not, at block 1928, the third qubit in the new resource state is routed to an “O(L)” delay line that imposes a delay on the order of L clock cycles, such as the delay line of the offset reconfigurable fused circuit 1854 of FIG. 18. In some embodiments, the O(L) delay line can impose a delay of exactly L clock cycles. At block 1926, if the current position corresponds to the last row of the layer, then at block 1930, layer edge processing can be performed on the third qubit. In some embodiments, the layer edge processing can include performing a measurement on the third qubit that removes the third qubit from the system without destroying the entanglement of other qubits.

[0115]

[0156] At block 1936, a determination is made as to whether the current position corresponds to the first row of the layer (e.g., whether the column position counter has a value of 1). If not, at block 1938, a fusion operation is performed on the fourth qubit of the new resource state and the qubit output from the O(L) delay line. For example, the offset reconfigurable fusion circuit 1854 may perform a fusion operation on the second qubit of the new resource state and the qubit routed to the O(L) delay line of the offset reconfigurable fusion circuit 1854 during the clock cycle corresponding to the same position of the previous row. At block 1936, if the current position corresponds to the first row of the layer, then at block 1940, a layer edge operation may be performed on the fourth qubit. In some embodiments, the layer edge operation may include performing a measurement on the fourth qubit that removes the fourth qubit from the system without destroying the entanglement of the other qubits. Other options for layer edge operations are described below.

[0116]

[0157] In block 1946, the fifth qubit of the new resource state is a delay line of the offset reconfigurable fused circuit 1856 of FIG. 2 O(L) 2 )" delay line. In some embodiments, 2 ) The delay line is exactly L 2 A delay of a clock cycle can be imposed.

[0117]

[0158] In block 1956, the sixth qubit of the new resource state and O(L 2 ) delay line. For example, the offset reconfigurable fusion circuit 1856 may perform a fusion operation on the qubits output from the delay line during the clock cycle corresponding to the same position in the previous layer, by merging the second qubit in the new resource state and the O(L 2) a fusion operation may be performed on the qubits routed to the delay line. In some embodiments, for the clock cycle corresponding to the creation of the first layer of the entanglement structure, the sixth qubit may instead undergo a different operation, such as a measurement operation that removes the sixth qubit from the system without destroying the entanglement of the other qubits, or no operation.

[0118]

[0159] Process 1900 is illustrative and variations and modifications are possible. For example, while the various determination and routing operations are shown sequentially, some or all of these operations can be performed in parallel or in a different order than described. The fusion operation can be replaced with other entanglement measurement operations that create entanglement between the two systems of qubits. The specific lengths of the various delay lines can be varied, and delay lines of different lengths can be used when generating different positions within a layer, depending on the entanglement structure desired. Process 1900 can be repeated for any number of clock cycles to generate an entanglement structure having any number of layers of any desired size. Layer edge processing (also referred to herein as boundary processing) can include measuring qubits at the edges (or boundaries) of a layer. In some embodiments, layer edge processing can also include performing fusion or other entanglement operations on qubits at different edges of the same layer or on qubits at edges of different layers, examples of which are described below.

[0119] 2.2.4. Hybridization of Entanglement

[0160] The embodiments described in Sections 2.2.2 and 2.2.3 represent extreme examples of design tradeoffs between hardware size and computational speed. Other embodiments provide a "hybrid" approach to generating entangled structures, thereby balancing between hardware size and computational speed. In the hybrid approach, 2 The layer of resource states of is generated using multiple (N) RSG circuits, where N is greater than 1 but L 2 Less than.

[0120]

[0161] Two different illustrative implementations of the hybrid approach are described: a "raster-based hybrid" circuit and a "patch-based hybrid" circuit. In both implementations, a layer of resource states can be viewed as a two-dimensional array of "patches" of contiguous groupings of resource states. For example, if a layer is of size L 2 If , the layer is of size P 2 In the raster-based hybrid approach, the number of RSG circuits, N, is N = L 2 / P 2 Each RSG circuit may provide a different patch resource state, allowing N patches to be generated in parallel, and in some embodiments, a layer may be P 2 In the patch-based hybrid approach, the number of RSG circuits, N, is N = P 2 and the RSG circuits can be used together (similar to the fully networked unit cells described in Section 2.2.2) to generate a patch in just one clock cycle. Generation of a layer can be completed in N clock cycles.

[0121]

[0162] Turning first to the raster-based hybrid circuit, FIG. 20 shows a conceptual diagram of raster-based hybrid generation of entanglement structures from resource states according to some embodiments. 2 In order to support the generation of layers of N, N RSG circuits 2002 are provided. In the simplified example used here, 2 = 16 and N = 4, but in reality, L 2 can be much larger (e.g., about 10 2 , about 10 4 , about 10 6 ). N can be much larger (e.g., about 100, about 1000), and L 2 / N can be selected as desired depending on the desired balance between hardware size and speed of operation. At each clock cycle, each RSG circuit 2002 generates one instance of resource state 2000 for a total of N resource states. The resource states sufficient to complete a 2D layer are L 2 11A and 11B. In this example, each instance of resource state 2000 is annotated with a time "t=1" through "t=4" to indicate the clock cycle in which that instance of resource state 2000 is generated. In this example, one resource state 2000 is generated for each of the four patches 2011-2014 during each clock cycle. Temporal blending operations similar to those described in section 2.2.3 above with respect to the rasterized generation of layers can be performed on qubits of adjacent resource states within the same patch of patches 2011-2014, and further blending operations described below can be performed on qubits of adjacent resource states across patch boundaries (e.g., the blending operations shown in Figures 11A and 11B). Size L2 The complete layer of L 2 A three-dimensional entangled structure can be generated in L / N clock cycles. A three-dimensional entangled structure can be generated by using the same RSG circuit 2002 to repeat the process of generating patches for each layer, and the temporal fusion operation can be performed on the qubits of the resource states 2000 of the different layers using additional circuitry described below (e.g., as shown in Figures 12A and 12B). A three-dimensional entangled structure can be generated in L for each 2 The resource states can be generated by using the same N RSG circuits 2002 to repeat the process of generating the resource states, and a temporal fusion operation can be performed on the quantum bits of the resource states 1700 of different layers using additional circuitry described below (e.g., the fusion operation shown in Figures 12A and 12B).

[0122]

[0163] 21 shows a schematic circuit diagram of a "raster-based" hybrid unit cell 2100 for generating entanglement structures from resource states according to some embodiments. The circuit notation is as described above in connection with FIGS. 14A-14F, except that classical inputs and outputs are not shown for clarity of illustration. In this example, the hybrid unit cell 2100 generates a series of P 2Over a clock cycle, continuous patches of size N = P×P (P < L) can be generated, and N instances of the hybrid unit cell 2100 can be network-connected to generate all layers of the LES. Therefore, some aspects of the hybrid unit cell 2100 can be similar to the aforementioned fully rasterized circuit 1800, and other aspects can be similar to the aforementioned fully networked cell 1600. Each hybrid unit cell 2100 includes an RSG circuit 2002 that generates a resource state having six peripheral qubits and optionally one or more central qubits 2115 that do not undergo a fusion operation (if present). For example, when the RSG circuit 2002 generates the resource state 1000 of FIG. 10A, the central qubit 1016 can be provided as the central qubit 2115, but when the RSG circuit instead generates the resource state 1040 of FIG. 10C, the central qubit 2115 is not provided. The offset reconfigurable fusion circuits 2102, 2104, 2106 can operate in the same manner as the offset reconfigurable fusion circuits 1852, 1854, 1856 of FIG. 18 to generate entanglement between the resource states locally generated within the patch. Further, additional "networked" reconfigurable fusion circuits 2112, 2114 can be provided to create entanglement between the patch generated by the hybrid unit cell 2100 and the patch generated by an adjacent instance of the hybrid unit cell 2100. The reconfigurable fusion circuits 2112, 2114 can operate in the same manner as the reconfigurable fusion circuits 1620 and 1630 within the network cell 1600 of FIG. 16A to perform a fusion operation on the qubits of the locally generated resource state and the qubits of the network resource state received from an adjacent instance of the hybrid unit cell 2100.The routing switches 2116-2119 may be reconfigurable switching circuits that operate to selectively route +x, -x, +y, and -y quantum bits of a particular resource state to one of the circuits 2102, 2104 (to be used in a fusion operation with quantum bits of a different resource state generated by the same RSG circuit 2002) or to one of the fusion circuits 2112, 2114 (to be used in a fusion operation with quantum bits of a resource state generated by an adjacent instance of the hybrid unit cell 2100).

[0123]

[0164] To further illustrate the operation of the routing switch 2116, Figure 22 shows a conceptual diagram of two adjacent patches 2202, 2204 according to some embodiments. The patches 2202 and 2204 are generated by two different instances of the hybrid unit cell 2100. In this example, each instance of the hybrid unit cell 2100 has a size P 2= 9 patches. Each instance of resource state 2210 in patch 2202 is labeled with a directional indicator (NW, N, NE, E, SE, S, SW, W, or C) to indicate its location within the patch. Hybrid unit cell 2100 may generate resource states in patch 2202 by stepping across the bottom row in the +x direction, then across the next row in the +y direction, and so on. Routing switches 2116-2119 may operate such that, for resource state 2210(C), all x and y qubits are routed to a "local" offset reconfigurable fusion circuit 2102, 2104 to be fused with qubits of other local resource states generated within hybrid unit cell 2100. For resource state 2210(E) of Figure 22, the routing switches 2116-2119 may operate such that the +x qubit is routed to the networked reconfigurable fusion circuit 2112 to be fused with the -x qubit of the resource state generated in an adjacent instance of the unit cell 2100, while all other x and y qubits are routed to the local fusion circuit 2102, 2104. For resource state 2210(NE) of Figure 22, the routing switches 2116-2119 may operate such that the +x and +y qubits are routed to the networked fusion circuit 2112, 2114 to be fused with the qubit of the resource state from an adjacent instance of the unit cell 2100, and the -x and -y qubits are routed to the local fusion circuit 2102, 2104. Similar logic may be applied to other instances of resource state 2210 and may be extended to patches of any size. In this example, no z qubit routing switch is required because a given instance of unit cell 2100 generates the same patch in each layer, and the +z and -z qubits can always be routed to the offset reconfigurable fused circuit 2106. It should be understood that this configuration is not required and other embodiments of the hybrid unit cell can include a z qubit routing switch.

[0124]

[0165] In the implementation of the hybrid unit cell 2100 shown in FIG. 21, the quantum bits provided to (or received from) adjacent unit cells are not affected by the delay circuit. Therefore, it may be desirable to coordinate the order in which resource states are generated in different unit cells, such that a resource state having a quantum bit provided to an adjacent unit cell as an input to the networked fusion circuit 2112, 2114 is generated during the same clock cycle as the adjacent resource state. FIG. 23 illustrates an example of an coordinated order of resource state generation for different patches 2301-2304 according to some embodiments. In this example, each patch 2301-2304 is 4×4 in size. Within each patch 2301-2304, the numbers (1-16) indicate the order of resource state generation, and all resource states with the same number are generated in the same clock cycle. As can be seen, in all instances in which a resource state in one patch is provided to a networked fusion circuit associated with an adjacent patch, both resource states (or all four resource states in the central position where all patches 2301-2304 are adjacent) are generated in the same clock cycle. Thus, no position-dependent delays are required to perform fusion operations on resource state qubits generated in different patches. This principle can be extended to P×P patches for any value of P and any number of patches. In other embodiments, position-dependent delay circuits and switches can be provided to synchronize qubits between different patches.

[0125]

[0166] FIG. 24 illustrates a flow diagram of a process that can be implemented using the hybrid unit cell 2100 of FIG. 21 or a similar circuit according to some embodiments. The process 2400 can be executed by each hybrid unit cell 2100 at each clock cycle while the entangled structure is being generated, where different hybrid unit cells 2100 operate in parallel. In this example, assume that the hybrid unit cells 2100 are used to generate layers of the entangled structure, with each hybrid unit cell generating successive patches having dimensions P×P within each layer. Each hybrid unit cell generates one row, followed by the next row, and so on to generate its patches (e.g., as shown for each of the patches 2301-2304 of FIG. 23). (As stated elsewhere herein, it should be understood that terms such as "row," "column," and "layer" are used in reference to the entanglement space that do not necessarily correspond to a physical arrangement of qubits or hybrid unit cells.)

[0126]

[0167] In block 2402, the RSG circuit 2002 (or other circuitry) may operate to generate new resource states. In some embodiments, the RSG circuit 2002 generates one new resource state per clock cycle. In block 2404, the location of the new resource state within the patch being generated by the hybrid unit cell is determined. For example, a row position counter is incremented every clock cycle to count positions within a row (e.g., from 1 to P, where P corresponds to the size of the rows in the patch) and is reset at the end of each row, and a column position counter is incremented as each row is completed (e.g., every P clock cycles) and is reset when the patch is completed (e.g., after completing P rows). Thus, the current counter value may indicate the location of the new resource state within the patch. Other techniques for defining the current position in the entanglement space may be used.

[0127]

[0168] At block 2406, a determination is made as to whether the current position corresponds to the end of a row of the patch (e.g., whether the row position counter has value P). If not, at block 2408, the first qubit of the new resource state is routed to an O(1) delay line that imposes a delay on the order of one clock cycle, such as the delay line of the offset reconfigurable fused circuit 2102 of FIG. 21. In some embodiments, the O(1) delay line can impose a delay of exactly one clock cycle. If at block 2406, the current position corresponds to the end of a row of the patch, then at block 2410, the first qubit can be routed to a first adjacent unit cell (e.g., by operation of switch 2117 of FIG. 21).

[0128]

[0169] At block 2416, a determination is made as to whether the current position corresponds to the beginning of a row of the patch (e.g., whether the row position counter has a value of 1). If not, at block 2418, a fusion operation is performed on the second qubit in the new resource state and the qubit output from the O(1) delay line (which may be the qubit routed to the O(1) delay line during the previous clock cycle), e.g., using the offset reconfigurable fusion circuit 2102 of FIG. 21. If at block 2416, the current position corresponds to the beginning of a row, then at block 2420, a fusion operation may be performed on the second qubit in the new resource state and the first networked qubit received from the second adjacent unit cell. The first networked qubit may be the qubit routed from the second adjacent unit cell according to block 2410, assuming that the second adjacent unit cell is also performing process 2400.

[0129]

[0170] At block 2426, a determination is made as to whether the current position corresponds to the last row of the patch (e.g., whether the column position counter has a value P). If not, at block 2428, the third qubit in the new resource state is routed to an O(P) delay line that imposes a delay on the order of P clock cycles. In some embodiments, the O(P) delay line can impose a delay of exactly P clock cycles. At block 2426, if the current position corresponds to the last row of the patch, then at block 2430, the third qubit can be routed to a third adjacent unit cell (e.g., by operation of switch 2118 of FIG. 21).

[0130]

[0171] At block 2436, a determination is made as to whether the current position corresponds to the first row of the patch (e.g., whether the column position counter has a value of 1). If not, at block 2438, a merge operation is performed on the fourth qubit of the new resource state and the qubit output from the O(P) delay line (which may be the qubit routed to the O(P) delay line during the clock cycle corresponding to the previous row position). At block 2436, if the current position corresponds to the first row of the patch, at block 2440, a merge operation may be performed on the fourth qubit of the new resource state and the second networked qubit received from the fourth adjacent unit cell. Assuming that the fourth adjacent unit cell is also performing process 2400, the second networked qubit may be the qubit routed from the fourth adjacent unit cell according to block 2430.

[0131]

[0172] In block 2446, the fifth qubit of the new resource state is P 2 O(P 2 ) delay line. In some embodiments, 2 ) delay line is exactly P 2 A delay of a clock cycle can be imposed.

[0132]

[0173] At block 2456, the sixth qubit of the new resource state and O(P 2 ) the qubit output from the delay line (during the clock cycle corresponding to the position of the previous layer, O(P 2 ) may be a fusion operation performed on the sixth qubit (which may be a qubit routed to a delay line). In some embodiments, for a clock cycle corresponding to the creation of a first layer of an entanglement structure, the sixth qubit may instead undergo a different operation, such as a measurement operation that removes the sixth qubit from the system without destroying the entanglement of the other qubits, or no operation.

[0133]

[0174] Process 2400 is illustrative and variations and modifications are possible. For example, while various determination and routing operations are shown sequentially, some or all of these operations can be performed in parallel or in a different order than described. The fusion operation can be replaced with other entanglement measurement operations that create entanglement between the two systems of qubits. The specific lengths of the various delay lines can be varied, and delay lines of different lengths can be used when generating different positions within a layer, depending on the entanglement structure desired. Process 2400 can be repeated for any number of clock cycles to generate an entanglement structure having any number of layers of any desired size. Also, process 2400 is described as having four adjacent unit cells. However, this need not be the case for all unit cells (or indeed any unit cells). Thus, in any instance where process 2400 illustrates routing a qubit to an adjacent unit cell or performing an operation involving a networked qubit being received from an adjacent unit cell, if no suitable adjacent unit cell exists, then layer edge processing can be substituted, for example, as described above in connection with FIG. 19 or in the examples below.

[0134]

[0175] As mentioned before, in a "patch-based" hybrid circuit, the number of RSG circuits, N, is N=P 2 and P in a single clock cycle. 2 The resource state generated by the RSG circuit is of size L 2 In the layer of size P 2 25 is a schematic diagram of layer hybridization for an entangled structure using a patch-based hybrid circuit according to some embodiments. 2 To support the generation of layers, we use a number N=P 2 In the simplified example used herein, an RSG circuit 2002 is provided. 2 = 16 and N = 4, but in reality, L 2 can be much larger (e.g., about 10 2 , about 10 4 , about 10 6 ) We can also make N much larger (e.g., about 100, about 1000) and P 2 can be selected as desired depending on the desired balance between hardware size and speed of operation. In each clock cycle, each RSG circuit 2502 generates one resource state 2500. (In FIG. 25, each resource state 2500 is annotated with time “t=1” through “t=4” to indicate which resource state 2500 is generated during each clock cycle.) As shown, patch 2511 is formed during the first clock cycle, patch 2512 is formed during the second clock cycle, patch 2513 is formed during the third clock cycle, and patch 2514 is formed during the fourth clock cycle. Spatial fusion operations can be performed on qubits of adjacent resource states 2500 in a patch (e.g., as shown in FIGS. 11A and 11B) using additional circuitry, which may be similar or identical to the fully networked circuitry of FIGS. 16A and 16B. For example, a delay-offset reconfigurable fusion circuit or other circuitry can be used to “stitch” the patches together, thereby forming a cluster of size L. 2By forming a layer of , further temporal fusion operations can be performed on qubits belonging to resource states in different patches. An example of a circuit that implements the fusion operation to stitch patches together into a layer is described in Section 3.3 below.

[0135]

[0176] In the hybrid embodiments described above, each hybrid unit cell has its own dedicated RSG circuit. In some embodiments, the operation of the RSG circuit is non-deterministic, meaning that a given instance of the RSG circuit is not expected to generate the desired resource state at every clock cycle. Thus, rather than a dedicated RSG circuit for each hybrid unit cell, some embodiments may provide a number (M) of RSG circuits, where M>N and M is selected to provide a sufficiently high probability that at least N resource states are generated during a given clock cycle (the "sufficiently high probability" in a given implementation may be determined based on the particular implementation of the fault tolerance). Using active multiplexing techniques, examples of which are known in the art, the N M RSG circuits may be selected at each clock cycle to deliver resource states to the N different instances of the switching and fusion circuits of the hybrid unit cell. Thus, each hybrid unit cell may, but need not, have its own dedicated instance of the RSG circuit.

[0136]

[0177] It should be understood that an array of hybrid unit cells such as that shown in FIG. 21 can be used to generate entangled structures of any size. (In some embodiments, the size may be fixed in the hardware design.) The layer size (L 2 Different choices of the number of RSG circuits (N) for N = 1 result in different computation times, and a choice can be made to achieve a desired balance between hardware size and computation speed.

[0137]

[0178] The foregoing examples of entanglement generation circuits and processes are illustrative and may be modified as necessary. The use of directional labels (e.g., x, y, z, NE, SE, SW, NW, etc.) is for convenience of explanation and should be understood to refer to entanglement space without requiring or implying any particular physical arrangement of components or physical qubits. All numerical examples are illustrative and may be modified. Additionally, while the layers and patches are described with reference to square numbers, it should be understood that non-square layers and / or non-square patches may also be used. For example, the patches or layers may be rectangular. Triangular patches or layers (or patches or layers having other shapes) may also be generated by, for example, varying the number of resource states per row. Additionally, while the foregoing examples assume that all instances of resource states have the same entanglement pattern, such uniformity is not required. For example, in some embodiments, the RSG circuit may be reconfigurable to generate resource states with different entanglement patterns at different clock cycles. Additionally, the RSG circuit may operate non-deterministically, which may result in stochastic fluctuations between resource states.

[0138] 3. Interleaved generation of entangled structures

[0179] The embodiments described in Section 2 support the generation of entangled structures over time. As previously mentioned, entangled structures can be used as logical qubits (e.g., for fault-tolerant quantum computing). In some cases, it is desirable to generate multiple entangled structures simultaneously (e.g., so that two or more logical qubits can be coupled together). One option is to provide a separate hardware instance for each entangled structure. Alternatively, some embodiments support interleaved generation of multiple entangled structures using the same hardware.

[0139] 3.1. Overview of LES generation

[0180] In some embodiments, the entangled structure may include a LES as described above in connection with FIG. 13. FIG. 26 illustrates a time diagram for generating a photonic LES according to some embodiments. The photonic LES in this example is simplified, but is similar to the LES that can be used as a logical qubit. FIG. 26 should be understood as a diagram in entangled space. For clarity of explanation, only the y ("space") and z ("time") dimensions are shown, such that each layer is one-dimensional, but it should be understood that each layer can be two-dimensional or higher dimensional (in entangled space). For convenience of explanation, time steps of duration τ are defined, where, for example, a time step can correspond to a clock cycle (or time to generate a layer of resource states). The qubits are implemented as photons propagating through a waveguide, and at any given time, the photons can be in multiple positions along a given waveguide. Thus, FIG. 26 can be understood as a snapshot view showing the positions of many different (physical) qubits at a single time, or as a time-lapse view showing the positions of the same (physical) qubit at different points in time.

[0140]

[0181] Block 2600 represents a resource state generator 2601 that generates (at time step 2602) a complete layer 2603 of resource state. In this example, the resource state 2603 is assumed to include a central qubit forming a LES. In some embodiments, a fully networked circuit may be used (e.g., as described in Section 2.2.2), and the time step τ may correspond to a clock cycle. In other embodiments, a rasterized or hybrid network / rasterized circuit may be replaced (e.g., as described in Sections 2.2.3 and 2.2.4), and the time step τ may correspond to a number of clock cycles. 2 Clock cycle or L 226 may correspond to the time required to generate all resource states over a y dimension (x dimension (not shown)). At time step 2604, fusion operations are performed, including spatial fusion operation 2606 on adjacent physical qubits in the y dimension (x dimension (not shown)) and temporal fusion operation 2608 to fuse adjacent qubits in successive layers. Optionally, a detector 2610 may be applied to the edge to perform Z measurements on the surrounding qubits of the resource state at the boundary of the layer, thereby removing it from the system. At time step 2612 (and any number of time steps thereafter), the LES may continue pending subsequent operations. In the illustrated example, subsequent operations include measurement operations on the qubits of the LES using detector 2614, although subsequent operations performed on the LES may be independent of how the LES is generated, and the LES generated in the manner shown in FIG. 26 may be used in a variety of operations.

[0141]

[0182] FIG. 27 is a simplified conceptual diagram of a linear optical circuit implementing the behavior of FIG. 26 according to some embodiments. For clarity of illustration, only the y ("space") and z ("time") axes are shown, but it should be understood that each layer can be two-dimensional (in entanglement space). At time t=0, each resource state generator 2702 outputs a resource state 2704, e.g., as described above. In this example, each resource state 2704 is shown as having five qubits (dots), including a propagating central qubit 2706 and peripheral qubits associated with the +y, -y, +z, and -z dimensions. Entanglement is indicated by the curved lines connecting the qubits, while straight lines indicate the waveguides (or groups of waveguides on which each qubit is encoded). (Although not shown, it should be understood that resource state 2704 may also include peripheral qubits associated with the +x and -x dimensions.) Between times t=0 and t=τ, fusion circuit 2706 (which may be, for example, the reconfigurable Type II fusion circuit described above) performs a fusion operation on peripheral qubits of adjacent resource states along the y dimension, and delay circuit 2708 delays the -z qubit of each resource state by one time step. Detector 2710 operates at the layer boundaries to remove peripheral qubits at the edges of each layer. Between times t=τ and t=2τ, fusion circuit 2712 (for example, the offset fusion circuit described above) fuses the delayed -z qubit with the +z qubit generated one time step later by the same RSG 2702. After time t=2τ, the qubit of the LES may propagate through further delay circuit 2714 and eventually reach detector 2720 (or another subsequent operation). Any number of delay circuits 2714 may be introduced depending on the desired lifetime of the LES.

[0142] 3.2. Temporal interleaving to generate multiple entangled structures

[0183] In the example of Figures 26 and 27, a single LES is generated using the circuitry shown in Figure 27. Although only a single two-dimensional portion of the LES is shown in Figures 26-27, one skilled in the art with the benefit of this disclosure will appreciate that a system including additional rows of RSG circuits that may be placed in the x-direction (into or out of the page) can generate a three-dimensional LES that can be used for fault-tolerant quantum computing. Additionally, the spatial fusion shown in Figures 26 and 27 can be replaced with temporal fusion, and the rasterization and hybrid circuits described above can also be used to generate the LES.

[0143]

[0184] In some cases, it may be desirable to use the same circuitry to provide multiple entangled structures (including but not limited to LES) that coexist in time (in the sense that photons of both entangled structures are flying at the same time, e.g., within one or more delay lines). According to some embodiments, the coexistence of multiple entangled structures may be brought about by "interleaving" the generation of layers of different entangled structures.

[0144]

[0185] 28 is a conceptual diagram of the interleaved generation of two entangled structures (in this case LESs) according to some embodiments. Using the techniques described above (or other techniques), a layer 2802a of entangled qubits can be generated, and then qubits from different layers 2802a can be entangled (using an operation such as the merging operation described above) to generate a first LES 2804a. Similarly, a layer 2802b can be generated, and qubits from different layers 2802b can be entangled to generate a second LES 2804b. (Different line types are used for LES 2804a and LES 2804b to aid in visualization.) Each LES 2804a, 2804b is shown as having five layers, although it should be understood that a LES may have any number of layers.

[0145]

[0186] Interleaved generation of two LESs may include using the same hardware to generate layers of both LESs, e.g., alternately. In some embodiments, layer generation hardware 2810 (which may be implemented using various circuits as described above) may be used to generate layers 2802a or 2802b in each of a series of time intervals. As indicated by dotted arc 2815, entanglement may be generated between layers generated during alternating periods (by performing a merging or other entanglement operation as described above), but no entanglement is generated between layers generated during successive periods. The result is identical in terms of entanglement topology to LESs 2804a, 2804b, as indicated by mapping arrow 2817.

[0146]

[0187] FIG. 29 is a time diagram of generating two interleaved LESs (and optionally entangling the two interleaved LESs with each other at the boundary) using a single set of resource state generators and downstream circuitry according to some embodiments. FIG. 29 is similar in many ways to FIG. 26. For example, although only y and z dimensions are shown, it should be understood that each layer of LESs can be two-dimensional (in entangled space). Like FIG. 26, FIG. 29 can be understood as a snapshot view or a time lapse view.

[0147]

[0188] Block 2900 represents a resource state generator 2901 that generates (at time step 2902) a complete set of resource states for a layer of the LES. Similar to Figure 26, various techniques can be used to generate the resource states for the layer, and the time step τ can be defined accordingly. At time step 2904, spatial fusion 2906 occurs to fuse adjacent physical qubits in the y dimension (x dimension (not shown)).

[0148]

[0189] Unlike FIG. 26, in this example the resource states generated at alternating time steps are associated with two different LESs. The qubits are color coded (grey circles for qubits associated with LES A, white for qubits associated with LES B) to indicate the association of the qubits with the LESs. Thus, temporal fusion 2908 fuses two qubits from resource states generated two time steps apart. At the edge of the layer, a detector 2910 can be used to remove the border qubits. Alternatively, fusion circuitry 2912 can fuse the peripheral qubits of the layer of LES B with previously generated peripheral qubits of the layer of LES A to "stitch" the LESs together at the boundary, as described below. At time step 2914 (and any number of time steps thereafter), the LES persists until a subsequent operation, which in this example includes a measurement using detector 2916.

[0149]

[0190] FIG. 30 illustrates a simplified conceptual diagram of a linear optical circuit implementing the behavior of FIG. 29 according to some embodiments, using notation similar to FIG. 27. At time t=0, a resource state generator 3002 outputs a resource state 3004, e.g., as described above. In this example, each resource state 3004 is shown as having five qubits, including one propagating central qubit and peripheral qubits associated with the +y, -y, +z, and -z dimensions. (Although not shown, it should be understood that resource state 2704 may also include peripheral qubits associated with the +x and -x dimensions.) Between times t=0 and t=τ, a merging circuit 3006 performs a merging operation on the peripheral qubits of adjacent resource states along the y dimension, and a delay circuit 3008 delays the -z peripheral qubits of each resource state by one time step. Between times t=τ and t=2τ, a second delay circuit 3008' delays the -z peripheral qubits of each resource state by another time step.

[0150]

[0191] Between times t=2τ and t=3τ, a fusion circuit 3012 (e.g., the offset fusion circuit described above) performs a fusion operation on the delayed (by 2τ) −z and +z qubits generated by the same RSG 3002 two time steps later. In this manner, entanglement can be created between layers of LESs formed during alternating time steps, allowing the same hardware to generate two LESs by time interleaving.

[0151]

[0192] After a time t=3τ, the physical qubits that make up the two LESs can propagate through further delay circuits 3014 and finally reach a detector 3020 (or some other subsequent operation). Any number of delay circuits 3014 can be introduced depending on the desired lifetime of the LES.

[0152]

[0193] In some embodiments, a configurable boundary circuit 3030, shown as operating between times t=0 and t=2τ, can be used to perform various boundary operations on the boundary qubits of a layer. The configurable boundary circuit 3030 includes a switch 3032 (similar to the active switch described above) that can direct qubits to either a detector 3034 or an offset reconfigurable fusion circuit 3036. For a given time step, if the switch 3032 selects the detector 3034, the boundary qubit is removed from the layer currently propagating between t=0 and t=2τ. If the switch 3032 instead selects the offset reconfigurable fusion circuit 3036, during a first period, the peripheral qubits associated with one LES layer (LES A in this example) are delayed by the delay circuit 3038, and during a next period, the peripheral qubits associated with the other LES layer (LES B in this example) are received, and the offset reconfigurable fusion circuit 3036 performs a fusion operation on the received qubits and the delayed qubits. The operation performed by the offset reconfigurable fusion circuit 3036 is also referred to as “boundary stitching.” In some embodiments, boundary stitching can be used to stitch together patches generated during different time periods (e.g., patches generated using the patch-based hybrid approach of FIG. 25) to form larger layers.

[0153]

[0194] It should be understood that these examples are illustrative and not limiting. The interleaving technique is not limited to the formation of LES. Similar techniques can be used when entangled structures are generated from resource states that do not have a central qubit, to allow multiple entangled structures to coexist in time or to support the generation of entangled structures with larger layers and / or non-planar layer topologies, examples of which are described below. The interleaving techniques described herein can be modified to provide any number (two or more) of simultaneous entangled structures, and the size of the entangled structures can be selected as needed. The layers of resource states used for interleaving can be generated using any of the networked, rasterized, or hybrid approaches described above, and the same RSG circuit can be used to generate the resource states of all entangled structures that are interleaved. In some embodiments, the RSG circuit can be reconfigurable to allow different entangled structures or different layers within a single entangled structure to have different entanglement geometries from each other. Furthermore, if the interleaving produces multiple entangled structures, the different simultaneously existing entangled structures can be selectively entangled with one another using additional circuitry.

[0154] 3.3 Lattice Surgery

[0195] In addition to or instead of interleaved generation of multiple LESs, the configurable boundary circuit 3030 and similar circuits may enable construction of entangled structures with various layer topologies by selectively performing (or not performing) fusion operations on qubits at layer boundaries. Such selective boundary fusion is also referred to herein as "lattice surgery." For example, in some embodiments, the switch 3032 can be dynamically configured for each pair of periods to support coupling (or lack of coupling between layers), also referred to as "boundary stitching." As an example, FIG. 31 shows a conceptual diagram of two LESs 3102, 3104 coexisting in time. As in FIGS. 26 and 27, only the y dimension (vertical axis labeled as "space") and z dimension (horizontal axis labeled as "time") are shown, but it should be understood that each LES can be three-dimensional. A first LES 3102 and a second LES 3104 overlap in time. The layers of LES 3102 and 3104 (shown as columns since only the y dimension is shown) may be offset in time from one another, as indicated by the time offset of the physical qubits. For example, the layers may be generated using interleaving techniques. In some embodiments, the time offset may be formed by generating the physical qubits of LES 3102 and 3104 during alternating periods τ. Thus, as previously described, the same hardware may be used to generate both LESs. In the example shown in FIG. 31, a first column of LES 3102 may be generated and its photons sent to a delay line. Then, a first column of LES 3104 may be generated and sent to a different (or the same) delay line. Then, a second column of LES 3102 may be generated and subsequently fused with the first column of LES 3102 (but not fused with the first column of LES 3104) that was stored in the delay line, and so on. Although Figures 31-33 show LES 2152 and LES 3104 offset in the y-direction relative to each other, it will be appreciated that interleaving allows the same set of physical resource state generators to generate the resource states required to generate each LES, for example, on alternating clock cycles.

[0155]

[0196] In some embodiments, the LESs 3102 and 3104 can be coupled together, e.g., to form a single LES with a larger layer. For example, FIG. 32 shows a conceptual diagram of "stitching" the LESs 3102 and 3104 at their boundaries to form a single LES with a larger layer size, e.g., by performing a fusion operation between boundary qubits on one side of the boundary of each layer. This technique can be used, for example, to stitch together patches generated in a hybrid circuit at different times or to increase the size of a layer by stitching layers together.

[0156]

[0197] 33 shows a conceptual diagram of selective lattice surgery, where LES 3102 and 3104 are selectively entangled along some layer boundaries but not along other layer boundaries. Such a configuration can be created by controlling configurable boundary circuit 3030 on a clock cycle-by-clock basis.

[0157]

[0198] In a scenario where the LESs 3102 and 3104 are three-dimensional LESs representing different logical qubits, the lattice surgery disclosed herein can be used to implement a two-qubit logic gate between the logical qubits encoded in the LESs 3102 and 3104. If a gate needs to be applied between interleaved logical qubits, the appropriate lattice surgery can be applied by changing the type of resource state that is generated or by changing the type of measurement made on the individual physical qubits of the LES. Other applications of lattice surgery may also be envisioned. In some embodiments, the fused circuitry at the boundary may be reconfigurable to change the type of lattice surgery operation.

[0158]

[0199] It should also be understood that while a simple LES is used for purposes of illustration, interleaving, boundary stitching, and lattice surgery are not limited to the context of forming an LES. Any entangled structure that can be generated from a layer of resource states (including entangled structures without a central qubit) can have that layer interleaved with one or more other entangled structures generated in the same manner, and boundary stitching and / or lattice surgery can be performed between layers of such structures.

[0159] 3.4. Interleaving to Create Hierarchical Topologies

[0200] In some embodiments, time interleaving techniques can be used to generate entangled structures with layers having various topologies depending on how the boundary qubits are coupled. For example, as shown in FIG. 29, a single "folded" layer can be generated by generating two layers on successive clock cycles and stitching the layers together at the boundary using a merging circuit. FIGS. 34A-34D are conceptual diagrams of using interleaving to form a three-dimensional entangled topology with a folded layer, according to some embodiments. FIG. 34A shows a layer 3400 in the xy plane in entanglement space. Layer 3400 can be a layer of resource states that are entangled with each other using the merging operation described above. Any of the techniques described in Section 2 or other techniques can be used to form layer 3400. FIG. 34B shows a "folded" topology 3410 that can be formed for layer 3400. FIG. 34C shows an interleaving technique that can be used to form a three-dimensional entangled structure with a layer having a folded topology 3410. In FIG. 34C, time progresses along the z-axis (vertical on the page). Four layers (or patches) 3411, 3412, 3413, 3414, each of which may be part of layer 3400, are shown in the xy plane. Each layer (or patch 3411, 3412, 3413, 3414) may be generated by the same hardware for a different period τ. Entanglement is generated between the qubits of alternating layers. For example, some or all of the qubits of layer 3411 may be entangled with the corresponding qubits of layer 3413, as shown by vertical line 3420, and some or all of the qubits of layer 3412 may be entangled with the corresponding qubits of layer 3414, as shown by vertical line 3422. The merging operation for the qubits of alternating layers (time interval 2τ) may be performed, for example, using the delay circuit of FIG. 30.

[0160]

[0201] Furthermore, pairs of consecutively generated layers are "stitched" together at their boundaries, as shown by curves 3416, 3418. Stitching can be performed, for example, by performing a merging operation on the boundary qubits of the two layers using offset merging circuit 3036 of FIG. 30 or a similar circuit to form entanglement at the edges of the layers. As shown by line 3416, consecutively generated layers 3411 and 3412 are stitched together, and as shown by line 3418, consecutively generated layers 3413, 3414 are stitched together. FIG. 34D shows an "unfolded" view of the entangled structure of FIG. 34C.

[0161]

[0202] Thus, in some embodiments, the folded entangled structure of FIG. 34C (or FIG. 32) can be understood as a single layer of entangled structures generated using a patch-based hybrid raster / networked RSG circuit, similar to the example described above in connection with FIG. 25. For example, in the embodiment described in connection with FIG. 2 The RSG circuit is set in one clock cycle. 2 A patch of consecutive resource states can be generated. In some embodiments, patches generated during different clock cycles can be stitched together at the boundaries, and an interleaving technique can be used to form larger layers, as shown in Figures 34C and 34D. In the example shown in Figures 34C and 34D, the size of each patch is L x (L / 2). However, smaller patches can be used. If a fusion circuit is provided to perform stitching between patches along both spatial boundaries, the size of the patch can be less than L in both (spatial) dimensions. Furthermore, if there are more than two patches per layer, the delay associated with the fusion operation between qubits of different layers can be appropriately adjusted to account for the number of patches per layer.

[0162]

[0203] Although Figs. 34A-34D show an entangled structure with a planar layer topology, folding techniques can also be used to form other layer topologies. Figs. 35A-35C are conceptual diagrams of using folding techniques to create periodic boundary conditions for layers of an entangled structure according to some embodiments. Fig. 35A shows a layer 3500 as a rectangle in the xy plane. Fig. 35B shows a cylindrical layer topology that can be formed by performing a merge operation on boundary qubits at the +x boundary 3502 and corresponding qubits at the -x boundary 3504 of layer 3500, as shown by curve 3510. As another example, Fig. 35C shows an interleaving technique that can be used to form a cylindrical layer topology by forming two layers 3522, 3524 and performing a merge operation on corresponding boundary qubits at the +x boundary (as shown by curve 3526) and corresponding boundary qubits at the -x boundary (as shown by curve 3528).

[0163]

[0204] 36A-36D are conceptual diagrams of the use of folding techniques to create more complex periodic boundary conditions of entangled layers according to some embodiments. FIG. 36A shows an entangled layer 3600 with boundaries 3602, 3603, 3604, 3605 that can be folded to form an entangled layer with a toroidal topology. Specifically, as shown in FIG. 36B, boundaries 3604, 3605 are bonded together (similar to the cylindrical topology of FIG. 35A), and as shown in FIG. 36C, boundaries 3602, 3603 are also bonded together, thereby forming a torus. FIG. 36D shows an interleaving technique that can be used to form a layer with a toroidal topology by selectively bonding boundaries along different dimensions of the layer. As in FIG. 34C, time progresses along the z-axis (vertical on the page) and the layers are shown as rectangles in the xy plane. Four layers 3621, 3622, 3623, 3624 are generated. At the boundaries, the layers are stitched together (e.g., using temporal fusion). Particular patterns of temporal fusion are shown by curves 3631 (between layers 3621 and 3624), 3632 (between layers 3622 and 3623), 3633 (between layers 3621 and 3622), and 3634 (between layers 3623 and 3624), and include variable delays of up to 4τ (depending on which layers are being fused). The variable delay length can be implemented using active switches and multiple delay circuits, similar to FIG. 30.

[0164]

[0205] 37A-37D are conceptual diagrams of using the techniques described herein to form diagonal folds of an entangled layer according to some embodiments. FIG. 37A shows an entangled layer 3700 with a +x boundary 3702 and a -y boundary 3704. In this example, the layer 3700 is a square layer. In some embodiments, the layer 3700 can be formed with diagonal folds as shown in FIG. 37B. For example, as shown in FIG. 37C, four triangular patches 3711, 3712, 3713, and 3714 can be generated during four different time steps (each time step can be a clock cycle or a longer time step). Successive patches 3711, 3712 are stitched together at their diagonal boundaries (as shown by curve 3721) to form a first square layer, and successive patches 3713, 3714 are stitched together at their diagonal boundaries as shown by curve 3722 to form a second square layer. An entanglement between corresponding locations in the first and second layers may be formed as shown by line 3724 (representing an entanglement between patch 3711 of the first square layer and patch 3713 of the second square layer) and line 3726 (representing an entanglement between patch 3712 of the first square layer and patch 3714 of the second square layer). In some embodiments, triangular patches may be generated using a network of unit cells having different numbers of unit cells corresponding to different rows or using rasterized unit cells that generate different numbers of resource states per row. Furthermore, a square network of unit cells or rasterized unit cells that generate a fixed number of resource states per row may be used to simultaneously generate triangular patches for two different structures that may later be entangled with each other (e.g., by appropriately configuring x- and y-dimension merging circuits). In some embodiments, the diagonal folding of layers can support logical operations that can be implemented using merge operations on pairs of qubits that are close in space and time, or logical operations can be performed between multiple logical qubits by performing merge operations on pairs of qubits that are close in space and time as a result of the diagonal folded layer topology.For example, Figure 37D shows an example of merging between qubits (indicated by lines 3734) in different parts of a diagonally folded layer made from triangular patches 3731, 3732. In some embodiments, this type of merging operation can be used to implement lateral gates.

[0165]

[0206] These examples of layer topologies are illustrative. It should be understood that a variety of layer topologies can be generated without being limited to the examples shown. Furthermore, the generation of multiple entangled structures can be performed using interleaving techniques regardless of the layer topology of any particular entangled structure.

[0166] 4. Performing quantum computing operations

[0207] Quantum computation operations using the entangled structures generated as described above can be performed using various techniques. One approach is to modify the resource state (and therefore the entangled geometry) based on the computation being performed. For example, resource states at different locations in a 2D layer can be generated with different entangled geometries. In some embodiments, the RSG circuitry can be dynamically reconfigurable to enable the generation of resource states with different entangled geometries.

[0167]

[0208] Another approach involves modifying the fusion operation when resource states are fused. For example, using a reconfigurable fusion circuit such as that described above in connection with FIG. 14E, an MZI circuit with a variable phase shift (e.g., as described in section 1.3 above) can be selectively applied to different qubits (or individual modes) before fusion, thereby performing different quantum logic operations. In various embodiments, these approaches can be combined.

[0168] 5. Examples of quantum computer systems

[0209] 38 illustrates an example of a system architecture for a quantum computer system 3800 that can implement MBQC or FBQC according to some embodiments. Using photonic physics qubits, some embodiments of the quantum computer system 3800 can generate fault-tolerant cluster states that can be used to represent logical qubits for MBQC, and other embodiments of the quantum computer system 3800 can generate measurement data that reflects entanglement structures for fault-tolerant FBQC. The system 3800 includes a resource state generator 3802, a delay circuit 3804, a switch circuit 3806, a detector 3808, and a classical processing unit 3810.

[0169]

[0210] The resource state generator 3802 may include a single instance or multiple instances of the resource state generator circuits described above. The RSG circuits may operate autonomously without requiring data input, with each RSG circuit generating one resource state per clock cycle (which may be, for example, on the order of 1 ns or more). Any of the resource states described above or other resource states may be generated. The resource states may be output onto an optical fiber (or other waveguide) 3820, for example, at a rate of n*N photons per clock cycle, where n is the number of qubits in each resource state and N is the number of instances of the RSG circuits. The resource state generator unit 3802 may also send classical data outputs (e.g., indicating success or failure of various elements of the resource state generation process) to the classical processing unit 3810 via data path 3822. In some embodiments, the resource state generator unit 3802 may be maintained at cryogenic temperatures (e.g., 4K). The delay circuit 3804 may delay the photons corresponding to a particular quantum bit through an optical fiber, other waveguide, optical memory, or by an appropriate delay time, such as the aforementioned 1 clock cycle, L clock cycles, and L 2Other components may be included to delay by a delay time of a clock cycle. As previously mentioned, in some embodiments, only one delay line of each duration is needed to perform the rasterized generation of logical qubits. The delay circuit 3804 does not need to operate at cryogenic temperatures. The photons exiting the delay circuit 3804 may be delivered to the switch circuit 3806 via a waveguide 3824, which may be an optical fiber, an on-chip waveguide, or any other type of waveguide.

[0170]

[0211] The switch circuit 3806 can include active switches and waveguides for performing mode-coupling, mode-swapping, and phase-shifting operations on the qubits. In various embodiments, the switch circuit 3806 can perform mode-coupling operations associated with fusion operations (e.g., the type II fusion operation described above in connection with FIG. 9A ) and / or basis selection operations associated with measuring individual qubits. In some embodiments, the switch circuit 3806 is dynamically reconfigurable in response to control signals from the classical processing unit 3810, and the quantum computer 3800 can perform different calculations by reconfiguring the switches in the switch circuit 3806. In some embodiments, the switch circuit 3806 can implement all of the reconfigurable switches and mode couplers for the reconfigurable fused circuit used in the examples above. The switch circuit 3806 delivers output photons to the detector 3808 via a waveguide 3828, which can be an optical fiber, an on-chip waveguide, or any other type of waveguide.

[0171]

[0212] Detectors 3808 may include photonic detectors capable of detecting photons in the waveguides. Each photonic detector is coupled to one waveguide and generates an output (classical) signal indicative of whether a photon is detected. In some embodiments, some or all of the photonic detectors may count photons, and the output signal from each photonic detector may include the number of photons detected by that photonic detector. In some embodiments, detectors 3808 may operate at cryogenic temperatures. Detectors 3808 may provide a classical output signal indicative of the number of photons (or a binary signal indicative of whether a photon is detected) to classical processing unit 3810 via signal path 3830.

[0172]

[0213] The classical processing unit 3810 may be a classical computer system that may communicate with the resource state generator 3802, the switch circuit 3806, and the detector 3808 using classical digital logic signals. In some embodiments, the classical processing unit 3810 may determine the appropriate settings of the switch circuit 3806 based on the particular quantum computation (or program) being executed. The classical processing unit 3810 may receive feedback signals (e.g., measurement results) from the resource state generator 3802 and the detector 3808 and may determine the results of the computation based on the feedback signals. In some embodiments, the classical processing unit 3810 may use the feedback signals to modify subsequent control signals sent to the switch circuit 3806. The operation of the classical processing unit 3810 may incorporate error correction algorithms and other techniques.

[0173]

[0214] The system 3800 of FIG. 38 is illustrative and variations and modifications are possible. Blocks shown separately can be combined or single blocks can be implemented using multiple separate components. The resource state generator 3802, the delay circuit 3804, the switch circuit 3806, and the detector 3808 can implement the circuits described above for generating the entanglement structure. For example, the delay circuit 3804 can implement all of the delay line portions of the offset reconfigurable fusion circuit described above, the switch circuit 3806 can implement the reconfigurable switches and mode couplers associated with the reconfigurable fusion, and the detector 3808 can implement the destruction measurements associated with the fusion operation. In some embodiments, generating the entanglement structure can include generating an LES that can perform measurements of individual qubits to implement MBQC. In other embodiments, generating the entanglement structure can include performing a fusion operation on the qubits of the resource state (e.g., as described above) with the measurement results obtained in the fusion operation provided to the classical processing unit 3810, thereby implementing FBQC.

[0174]

[0215] System 3800 is but one example of a quantum computer system that can incorporate the rasterization and / or interleaving techniques described herein to generate one or more logical quantum bits or other cluster states or other entangled structures, and many different systems can be implemented, as will be appreciated by those of ordinary skill in the art with access to this disclosure.

[0175] 6. Further embodiments

[0216] The embodiments described herein provide examples of systems and methods for generating entangled structures that can be used, for example, as fault-tolerant cluster states (which can be used to form and operate logical qubits) or in any other operation where a large entangled structure may be desirable. The size of the entangled structure and the entanglement geometry can be varied depending on the particular use case. For example, the above description uses an example of an entangled structure from a layer that is two-dimensional (in entanglement space), but the layer can have more dimensions. Additionally, the embodiments described above include references to specific materials and structures (e.g., optical fibers), but other materials and structures that can generate, propagate, and operate on photons can be substituted.

[0176]

[0217] It should be understood that all numerical values ​​used herein are for illustrative purposes and may be varied. In some cases, ranges are provided to give a sense of scale, but values ​​outside the disclosed ranges are not excluded.

[0177]

[0218] It should also be understood that all figures herein are intended as schematic diagrams. Unless specifically indicated otherwise, the drawings are not intended to imply a particular physical arrangement of elements shown therein, or that all elements shown are required. Those skilled in the art with access to this disclosure will understand that elements shown in the drawings or otherwise described in this disclosure can be modified or omitted, and other elements not shown or described can be added.

[0178]

[0219] This disclosure provides a description of the claimed invention with reference to certain embodiments, and those skilled in the art with access to this disclosure will appreciate that the embodiments do not exhaust the scope of the claimed invention, which scope extends to all variations, modifications, and equivalents.

Claims

1. 1. A circuit for generating entanglement between quantum bits, comprising: a resource state generator having circuitry for generating a first resource state during a first clock cycle, generating a second resource state during a second clock cycle, generating a third resource state during a third clock cycle, and generating a fourth resource state during a fourth clock cycle, each of the first, second, third, and fourth resource states comprising a system of entangled photonic qubits, and the first, second, third, and fourth clock cycles being different clock cycles; a first time-fusion circuit configured to generate a first entangled state between the first resource state and the second resource state by performing an entanglement measurement operation between a first qubit of the first resource state and a first qubit of the second resource state; a second time-fusion circuit configured to generate a second entangled state between the first entangled state and the third resource state by performing an entanglement measurement operation between a second qubit of the first resource state and a first qubit of the third resource state; a third time-fusion circuit configured to generate a third entangled state between the second entangled state and the fourth resource state by performing an entanglement measurement operation between a third qubit of the first resource state and a first qubit of the fourth resource state; A circuit comprising:

2. 2. The circuit of claim 1, wherein the first and second clock cycles are consecutive clock cycles.

3. the circuit is operable to form a large entangled system of qubits having an entanglement structure with multiple layers in an entanglement space; the first resource state, the second resource state, and the third resource state are all associated with a first tier of the plurality of tiers; the fourth resource state is associated with a second tier of the plurality of tiers; The circuit of claim 1 .

4. the resource states define multiple layers in the entanglement space; the first resource state, the second resource state, and the third resource state are all associated with a first tier of the plurality of tiers; the fourth resource state is associated with a second tier of the plurality of tiers; The circuit of claim 1 .

5. Each layer in the entanglement space is a two-dimensional layer with a first linear dimension of size L; the first clock cycle and the second clock cycle are separated by a first time interval; the first clock cycle and the third clock cycle are spaced apart by L times the first time interval; 5. The circuit of claim 4.

6. Each layer in the entanglement space is a two-dimensional layer with a second linear dimension of size L, and the first clock cycle and the fourth clock cycle are L times the first time interval. ２ 6. The circuit of claim 5, wherein the first and second inputs are spaced apart by a factor of 1.

7. 2. The circuit of claim 1 , wherein the first temporal fusion circuit includes a delay line for delaying the first quantum bit in the first resource state until the second clock cycle, and the second temporal fusion circuit includes a delay line for delaying the second quantum bit in the first resource state until the third clock cycle.

8. 2. The circuit of claim 1 , wherein the entanglement measurement operation performed by the first time-fusing circuit comprises a destructive measurement on the first qubit in the first resource state and the first qubit in the second resource state.

9. 9. The circuit of claim 8, wherein the entanglement measurement operation performed by the second time-fusing circuit comprises a destructive measurement on the second qubit in the first resource state and the first qubit in the third resource state.

10. 1. A circuit for generating entanglement between quantum bits, the circuit comprising a plurality (N) of unit cells forming a network such that each unit cell is coupled to at least two adjacent unit cells, each unit cell comprising: a resource state generator having a photonic circuit for generating a first local resource state during a first clock cycle, a second local resource state during a second clock cycle, a third local resource state during a third clock cycle, and a fourth local resource state during a fourth clock cycle, each of the first, second, third, and fourth local resource states comprising a system of entangled photonic qubits, and the first, second, and third clock cycles being different clock cycles; a plurality of fusion circuits, including a first local fusion circuit, a second local fusion circuit, a third local fusion circuit, a first networked fusion circuit, and a second networked fusion circuit, each of the plurality of fusion circuits configured to perform an entanglement measurement operation between two input qubits; a first local delay line coupled to a first input of the first local fused circuit, the first local delay line having a delay of a first number of clock cycles; a second local delay line coupled to a first input of the second local fused circuit, the second local delay line having a delay of a second number of clock cycles, the second number being greater than the first number; a third local delay line coupled to a first input of the third local fused circuit, the third local delay line having a delay of a third number of clock cycles, the third number being greater than the second number; a first routing switch configured to selectively direct a first qubit of each resource state to one of the first local delay line of the unit cell or a first input of the first networked fused circuit of a first adjacent unit cell; a second routing switch configured to selectively direct a second qubit of each resource state to one of a second input of the first local fused circuit of the unit cell or a second input of the first networked fused circuit; a third routing switch configured to selectively direct a third qubit of each resource state to one of the second local delay line of the unit cell or a first input of the second networked fused circuit of a second adjacent unit cell; and a fourth routing switch configured to selectively direct a fourth qubit of each resource state to one of a second input of a second local fused circuit of the unit cell or a second input of the second networked fused circuit; a first routing path for directing a fifth qubit of each resource state to the third local delay line; a second routing path for directing a sixth qubit of each resource state to the third local fused circuit; A circuit comprising:

11. the circuit is operable to form a large entangled system of qubits having an entanglement structure with multiple layers in an entanglement space; the first local resource state, the second local resource state, and the third local resource state are all associated with a first tier of the plurality of tiers; the fourth local resource state is associated with a second tier of the plurality of tiers; The circuit of claim 10.

12. the resource states define multiple layers in the entanglement space; the first local resource state, the second local resource state, and the third local resource state are all associated with a first tier of the plurality of tiers; the fourth local resource state is associated with a second tier of the plurality of tiers; The circuit of claim 10.

13. Each layer of the large entangled system of qubits is ２ , and each unit cell is made up of multiple (P ２ ) resource state, and P ２ = L ２ 13. The circuit of claim 12, wherein:

14. the first clock cycle and the second clock cycle are separated by a first time interval; the first and third clock cycles being separated by P times the first time interval; 14. The circuit of claim 13.

15. The first clock cycle and the fourth clock cycle are P of the first time interval. ２ 15. The circuit of claim 14, wherein the first and second inputs are spaced apart by a factor of 1.

16. 11. The circuit of claim 10, wherein each of the plurality of fused circuits is configured such that the entangled measurement operation includes a destructive measurement on both of the input qubits.

17. 1. A circuit for generating a plurality of entangled structures, each of which can be represented as a plurality of layers in an entanglement space, the circuit comprising: a layer generation circuit configured to generate a first layer during a first time period, generate a second layer during a second time period, and generate a third layer during a third time period, each of the first, second, and third layers comprising a system of photonic qubits that are at least two-dimensionally entangled in an entanglement space, and the second time period is between the first time period and the third time period; a plurality of time-fused circuits, each time-fused circuit configured to perform an entanglement measurement operation between the first layer qubits and the third layer qubits during a fourth time period following the third time period; A circuit comprising:

18. the layer generating circuitry is further configured to generate a fourth layer during the fourth time period; the plurality of temporal fusion circuits are configured to perform an entanglement measurement operation between one or more qubits in the second layer and one or more qubits in the fourth layer during a fifth time period subsequent to the fourth time period.

20. The circuit of claim 17.

19. and a boundary circuit configured to receive a peripheral qubit corresponding to a boundary of each layer of entangled qubits, the boundary circuit including a detector configured to detect the peripheral qubit.

20. The circuit of claim 18.

20. and a boundary circuit configured to receive a peripheral qubit of a resource state at a boundary of each layer of entangled qubits as a boundary qubit, the boundary circuit comprising: a detector configured to detect the boundary qubit; a temporal fusion circuit for fusing two boundary qubits from the layers generated during two different time periods; a switch configurable to route the boundary qubit to either the detector or the temporal fusion circuit; 20. The circuit of claim 18, comprising:

21. 21. The circuit of claim 20, wherein the switches are dynamically reconfigurable for each period.

22. 20. The circuit of claim 17, wherein the entanglement measurement operation comprises a destructive measurement on the qubits between which the entanglement measurement operation is performed.

23. 1. A method for generating entanglement between quantum bits, the method comprising, during each clock cycle of a plurality of clock cycles: operating the resource state generator to generate a new resource state comprising a system of entangled photonic qubits; determining a location in entanglement space for the new resource state, the location being defined within a layer of resource states; if the position in the entanglement space does not correspond to an end of a row of the layer, routing a first qubit of the new resource state to a first delay line; performing an entanglement measurement between a second qubit in the new resource state and a qubit output from the first delay line if the position in the entanglement space does not correspond to the beginning of a row of the layer; if the location in the entanglement space does not correspond to a last row of the layer, routing a third qubit of the new resource state to a second delay line having a longer delay than the first delay line; performing an entanglement measurement between a fourth qubit in the new resource state and a qubit output from the second delay line if the position in the entanglement space does not correspond to a first row of the layer; routing a fifth qubit of the new resource state to a third delay line having a longer delay than the second delay line; performing an entanglement measurement between a sixth qubit in the new resource state and a qubit output from the third delay line; The method includes:

24. performing a layer edge processing operation on the first qubit in the new resource state if the position in the entanglement space corresponds to the end of a row of the layer.

24. The method of claim 23, further comprising:

25. 25. The method of claim 24, wherein the layer edge processing operation comprises performing a measurement operation on the first qubit in the new resource state.

26. 25. The method of claim 24, wherein the layer edge processing operation comprises performing an entanglement measurement between the first qubit in the new resource state and a qubit associated with an edge of a different layer of the large entanglement system.

27. performing a layer edge processing operation on the second qubit in the new resource state if the position in the entanglement space corresponds to the beginning of a row of the layer.

24. The method of claim 23, further comprising:

28. performing a layer edge processing operation on the third qubit in the new resource state if the position in the entanglement space corresponds to the end of a row of the layer.

24. The method of claim 23, further comprising:

29. performing a layer edge processing operation on the fourth qubit in the new resource state if the location in the entanglement space corresponds to a first one of the rows of the layer.

24. The method of claim 23, further comprising:

30. 24. The method of claim 23, wherein each row of the layer has a size L in the entanglement space, and the second delay line has a delay corresponding to L times the delay of the first delay line.

31. Each layer has size L ２ and the third delay line has a delay L of the first delay line. ２ 31. The method of claim 30, having a delay corresponding to a factor of 1.

32. 24. The method of claim 23, wherein the step of performing each of the entanglement measurements comprises performing a fusion operation, the fusion operation comprising a destructive measurement on one or both of the qubits between which the fusion operation is performed.

33. 1. A method for generating entanglement between quantum bits, the method comprising, during each clock cycle of a plurality of clock cycles: operating a plurality of resource state generators at a plurality of unit cells such that each unit cell generates a new resource state comprising a system of entangled photonic qubits; For each unit cell, determining a location in entanglement space for the new resource state, the location being defined within a contiguous patch of a layer of resource states; if the position in the entanglement space does not correspond to an end of a row of the patch, routing a first qubit of the new resource state to a first delay line; performing an entanglement measurement between a second qubit in the new resource state and a qubit output from the first delay line if the position in the entanglement space does not correspond to the beginning of a row of the patch; if the position in the entanglement space does not correspond to a last row of the patch, routing a third qubit of the new resource state to a second delay line having a longer delay than the first delay line; performing an entanglement measurement between a fourth qubit in the new resource state and a qubit output from the second delay line if the position in the entanglement space does not correspond to a first row of the patch; routing a fifth qubit of the new resource state to a third delay line having a longer delay than the second delay line; performing an entanglement measurement between a sixth qubit in the new resource state and a qubit output from the third delay line; The method includes:

34. For at least one of the unit cells, 34. The method of claim 33, further comprising: if the position in the entanglement space corresponds to an end of a row of the patch, routing the first qubit in the new resource state to a first adjacent unit cell.

35. With respect to at least one other one of the unit cells, 35. The method of claim 34, further comprising: when the position in the entanglement space corresponds to the beginning of a row of the patch, performing an entanglement measurement operation between the second qubit in the new resource state and a networked qubit received from a second adjacent unit cell.

36. For at least one of the unit cells, 34. The method of claim 33, further comprising: if the location in the entanglement space corresponds to a last row of the patch, routing the third qubit in the new resource state to a first neighboring unit cell.

37. For at least one of the unit cells, 37. The method of claim 36, further comprising: if the location in the entanglement space corresponds to a first row of the patch, performing an entanglement measurement operation between the fourth qubit in the new resource state and a networked qubit received from a second adjacent unit cell.

38. 34. The method of claim 33, wherein each row of the patches has a size P in the entanglement space, and the second delay line has a delay corresponding to P times the delay of the first delay line.

39. Each patch has size P ２ and the third delay line has a delay P of the first delay line. ２ 39. The method of claim 38, having a delay corresponding to a factor of 1.

40. 34. The method of claim 33, wherein the step of performing each of the entanglement measurements comprises performing a fusion operation, the fusion operation comprising a destructive measurement on one or both of the qubits between which the fusion operation is performed.

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