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 of related applications

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

[0002]

[0002] Quantum computing is distinguished from "classical" computing by its reliance on structures called "qubits." At its most common level, a qubit is one of two orthogonal states (represented as |0〉 and |1〉 in conventional bra / ket notation) or a superposition of the two states (for example,

number

[0003]

[0003] However, the actual realization of quantum computers remains a challenging task. One challenge is the reliable formation and entanglement of qubits. [Overview of the project]

[0004]

[0004] Certain embodiments described herein relate to circuits for generating entanglement between qubits using a “rasterization” technique. In some embodiments, the circuit may include a resource state generator, a first temporal fusion circuit, a second temporal fusion circuit, and a third temporal fusion circuit. The resource state generator may have circuits for generating a first resource state during a first clock cycle, a second resource state during a second clock cycle, a third resource state during a third clock cycle, and 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, where the first, second, third, and fourth clock cycles are different clock cycles. A first temporal fusion circuit may be configured to generate a first entangled state between a first resource state and a second resource state by performing an entanglement measurement operation between a first qubit in a first resource state and a first qubit in a second resource state. A second temporal fusion circuit may be configured to generate a second entangled state between a first entangled state and a third resource state by performing an entanglement measurement operation between a second qubit in a first resource state and a first qubit in a third resource state. A third temporal fusion circuit may be configured to generate a third entangled state between a second entangled state and a fourth resource state by performing an entanglement measurement operation between a third qubit in a first resource state and a first qubit in a fourth resource state.

[0005]

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

[0006]

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

[0007]

[0007] In some embodiments, the first temporal fusion circuit may include a delay line for delaying the first qubit of the first resource state until the second clock cycle, and the second temporal fusion circuit may include a delay line for delaying the second qubit of the first resource state until the third clock cycle.

[0008]

[0008] In some embodiments, the entanglement measurement operation performed by the first temporal fusion circuit includes a destructive measurement of the first qubit in the first resource state and the first qubit in the second resource state. Similarly, the entanglement measurement operation performed by the second temporal fusion circuit may include a destructive measurement of the second qubit in the first resource state and the first qubit in the third resource state.

[0009]

[0009] Some embodiments relate to a circuit for generating entanglement between qubits, comprising a plurality (N) of unit cells that form a network such that each unit cell is coupled to at least two neighboring 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 photonic circuits 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 which comprises a system of entangled photonic qubits, and the first, second, and third clock cycles are 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 which is configured to perform an entanglement measurement operation between two input qubits. A 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. A second local delay line may be coupled to the first input of a second local fusion circuit and may have a delay of a second number of clock cycles, where the second number is greater than the first number. A third local delay line may be coupled to the first input of a third local fusion circuit and may have a delay of a third number of clock cycles, where the third number is greater than the second number.A first routing switch may be configured to selectively direct the first qubit of each resource state to either the first local delay line of the unit cell or the first input of the first networked fusion circuit of the first adjacent unit cell. A second routing switch may be configured to selectively direct the second qubit of each resource state to either the second input of the first local fusion circuit of the unit cell or the second input of the first networked fusion circuit. A third routing switch may be configured to selectively direct the third qubit of each resource state to either the second local delay line of the unit cell or the first input of the second networked fusion circuit of the second adjacent unit cell. A fourth routing switch may be configured to selectively direct the fourth qubit of each resource state to either the second input of the second local fusion circuit of the unit cell or the second input of the second networked fusion circuit. The first routing path can direct the fifth qubit of each resource state to the third local delay line. The second routing path can direct the sixth qubit of each resource state to the third local fusion circuit.

[0010]

[0010] In some embodiments, resource states define multiple layers in entanglement space, and in some embodiments, the circuit is operable to form a large entangled system of qubits having an entangled structure comprising multiple layers in entanglement space. When layers in entanglement space are defined, a first local resource state, a second local resource state, and a third local resource state can all be associated with the first layer of the multiple layers, while a fourth local resource state can be associated with the second layer of the multiple layers. For example, each layer of a large entangled system of qubits is L 2 If it is a two-dimensional layer with size , then each unit cell is multiple (P) for each layer of the large entanglement system of qubits. 2 ) can generate a resource state, where P 2 =L 2 / N. In these embodiments and other embodiments, the first clock cycle and the second clock cycle 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 clock cycle and the fourth clock cycle are separated by P of the first time interval. 2 They are twice as far apart.

[0011]

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

[0012]

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

[0013]

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

[0014]

[0014] In some embodiments, the circuit may further include boundary circuits configured to receive peripheral qubits corresponding to the boundaries of each layer of entangled qubits, and the boundary circuits include detectors configured to detect peripheral qubits.

[0015]

[0015] In some embodiments, the circuit may also include a boundary circuit configured to receive the peripheral qubits of the resource state at the boundary of each layer of entangled qubits as boundary qubits. The boundary circuit may include a detector configured to detect boundary qubits, a temporal fusion circuit for fusing two boundary qubits from layers generated during two different time periods, and a switch that can be configured to route boundary qubits 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 entanglement measurement operation may include a destructive measurement on qubits on which the entanglement measurement operation is performed between them.

[0017]

[0017] Several embodiments relate to a method for generating entanglement between qubits. The method includes 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 the position in entanglement space of the new resource state, wherein the position is defined within a layer of resource states; routing a first qubit of the new resource state to a first delay line if the position in entanglement space does not correspond to the end of a row of layers; 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 the beginning of a row of layers. The steps may include: routing a third qubit of a new resource state to a second delay line having a longer delay than the first delay line if its position in entanglement space does not correspond to the last row of the layer; performing an entanglement measurement between a fourth qubit of a new resource state and a qubit output from the second delay line if its position in entanglement space does not correspond to the first row of the layer; routing a fifth qubit of a 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 a new resource state and a qubit output from the third delay line.

[0018]

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

[0019]

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

[0020]

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

[0021]

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

[0022]

[0022] In some embodiments, each row of the layer may have a dimension L in the entanglement space, and the second delay line may have a delay corresponding to L times the delay of the first delay line. Furthermore, each layer may have a dimension L in the entanglement space 2 It can have, and the third delay line is L of the delay of the first delay line. 2 It can have a delay that corresponds to twice the number of times.

[0023]

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

[0024]

[0024] Several embodiments relate to a method for generating entanglement between qubits. The method involves: operating a plurality of resource state generators in 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; determining the position of the new resource state in entanglement space for each unit cell, wherein the position is defined in a contiguous patch of layers of resource states; routing a first qubit of the new resource state to a first delay line if the position in entanglement space does not correspond to the end of a row of patches; and routing a second qubit of the new resource state from the first delay line if the position in entanglement space does not correspond to the beginning of a row of patches The steps may include: performing an entanglement measurement with an output qubit; routing a third qubit of a new resource state to a second delay line having a longer delay than the first delay line if its position in entanglement space does not correspond to the last row of the patch; performing an entanglement measurement with a fourth qubit of a new resource state and a qubit output from the second delay line if its position in entanglement space does not correspond to the first row of the patch; routing a fifth qubit of a new resource state to a third delay line having a longer delay than the second delay line; and performing an entanglement measurement with a sixth qubit of a new resource state and a qubit output from the third delay line.

[0025]

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

[0026]

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

[0027]

[0027] In some embodiments, each row of patches can have a dimension P in the 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 2 in the entanglement space, and the third delay line can have a delay corresponding to P 2 times the delay of the first delay line.

[0028]

[0028] In some embodiments, performing each of the entanglement measurements includes performing a fusion operation, and the fusion operation includes a destructive measurement on one or both of the qubits on which the fusion operation is performed with respect to each other.

[0029]

[0029] The following detailed description, along with the accompanying drawings, will provide a better understanding of the nature and merits of the invention described in the claims. [Brief explanation of the drawing]

[0030] [Figure 1] Two representations of a portion of a pair of waveguides corresponding to dual-rail coded photonic qubits are shown. [Figure 2A] A schematic diagram for combining the two modes is shown. [Figure 2B] A schematic physical implementation of mode coupling in a photonic system, which may be used in several embodiments, is shown. [Figure 3A] A schematic example of a physical implementation of a Mach-Zehnder interferometer (MZI) configuration that can be used in several embodiments is shown. [Figure 3B] A schematic example of a physical implementation of a Mach-Zehnder interferometer (MZI) configuration that can be used in several embodiments is shown. [Figure 4A] Another schematic diagram for combining the two modes is shown. [Figure 4B] A schematic physical implementation of the mode coupling in Figure 4A in a photonic system that can be used in several embodiments is shown. [Figure 5] This document describes a four-mode coupling method that performs a "spreader" or "mode information erasure" conversion for four modes according to several embodiments. [Figure 6] An example of an optical device capable of performing the four-mode diffusion conversion schematically shown in Figure 5, according to several embodiments, is shown. [Figure 7] A schematic diagram of a dual-rail coded Bell state generator, which may be used in several embodiments, is shown. [Figure 8A] A schematic diagram of a dual-rail coding type I fusion gate, which may be used in several embodiments, is shown. [Figure 8B] An example of the results of a Type I fusion operation using the gate shown in Figure 8A is presented. [Figure 9A]A schematic diagram of a dual-rail coding type II fusion gate, which may be used in several embodiments, is shown. [Figure 9B] An example of the results of a Type II fusion operation using the gate shown in Figure 9A is presented. [Figure 10A] This shows a resource state entanglement graph that can be used according to several embodiments. [Figure 10B] This shows a resource state entanglement graph that can be used according to several embodiments. [Figure 10C] This shows a resource state entanglement graph that can be used according to several embodiments. [Figure 11A] Examples of resource state layers according to several embodiments are shown. [Figure 11B] Examples of resource state layers according to several embodiments are shown. [Figure 12A] An example of a three-dimensional array containing two layers of resource states according to several embodiments is shown. [Figure 12B] An example of a three-dimensional array containing two layers of resource states according to several embodiments is shown. [Figure 13] An example of a large entanglement system of qubits that can be formed according to several embodiments is shown. [Figure 14A] This section introduces some basic circuit symbols. [Figure 14B] This section introduces some basic circuit symbols. [Figure 14C] This section introduces some basic circuit symbols. [Figure 14D] This section introduces some basic circuit symbols. [Figure 14E] This section introduces some basic circuit symbols. [Figure 14F] This section introduces some basic circuit symbols. [Figure 15] Conceptual diagrams of networked generation of large entanglement systems of qubits according to several embodiments are shown. [Figure 16A] A schematic diagram of a circuit for generating an entanglement structure from resource states using a networked RSG circuit according to several embodiments is shown. [Figure 16B] A schematic diagram of a circuit for generating an entanglement structure from resource states using a networked RSG circuit according to several embodiments is shown. [Figure 17] The following are conceptual diagrams illustrating the rasterization generation of large entangled qubit systems according to several embodiments. [Figure 18] A schematic diagram of a circuit for generating an entangled structure from a resource state using a single RSG circuit according to several embodiments is shown. [Figure 19] A flowchart of the process for generating an entangled structure from resource states according to several embodiments is shown. [Figure 20] The diagrams below illustrate the conceptual generation of raster-based hybrid structures from resource states according to several embodiments. [Figure 21] The schematics of raster-based hybrid unit cells for generating entangled structures from resource states according to several embodiments are shown. [Figure 22] A conceptual diagram of two adjacent patches for a layer according to several embodiments is shown. [Figure 23] This shows an example of a coordinated sequence of resource state generation in different patches relating to layers according to several embodiments. [Figure 24] The flowcharts show other processes for generating entanglement structures from resource states according to several embodiments. [Figure 25] This is a conceptual diagram of layer hybrid generation in an entangled structure using patch-based hybrid circuits according to several embodiments. [Figure 26] The following shows time diagrams that generate large entanglement systems of qubits according to several embodiments. [Figure 27] A simplified conceptual diagram of a linear optical circuit that implements the behavior shown in Figure 26 according to several embodiments is shown. [Figure 28] A conceptual diagram of interleaved generation of two large entangled qubit systems according to several embodiments is shown. [Figure 29]The time diagrams shown generate two interleaved large entanglement systems of qubits according to several embodiments. [Figure 30] A simplified conceptual diagram of a linear optical circuit that implements the behavior shown in Figure 29 according to several embodiments is shown. [Figure 31] This diagram illustrates a conceptual model of two large entangled systems of temporally coexisting qubits. [Figure 32] A conceptual diagram of stitching two large entanglement systems of qubits to form a single larger entanglement system of qubits according to several embodiments is shown. [Figure 33] A conceptual diagram of lattice surgery for two large entangled systems of qubits according to several embodiments is shown. [Figure 34A] A conceptual diagram is shown illustrating the use of interleaving to form a three-dimensional entangled topology with folded layers according to several embodiments. [Figure 34B] A conceptual diagram is shown illustrating the use of interleaving to form a three-dimensional entangled topology with folded layers according to several embodiments. [Figure 34C] A conceptual diagram is shown illustrating the use of interleaving to form a three-dimensional entangled topology with folded layers according to several embodiments. [Figure 34D] A conceptual diagram is shown illustrating the use of interleaving to form a three-dimensional entangled topology with folded layers according to several embodiments. [Figure 35A] This is a conceptual diagram illustrating the use of bending techniques to generate periodic boundary conditions in layers of entangled structures according to several embodiments. [Figure 35B] This is a conceptual diagram illustrating the use of bending techniques to generate periodic boundary conditions in layers of entangled structures according to several embodiments. [Figure 35C] This is a conceptual diagram illustrating the use of bending techniques to generate periodic boundary conditions in layers of entangled structures according to several embodiments. [Figure 36A]This is a conceptual diagram illustrating the use of folding techniques to generate more complex periodic boundary conditions in layers of entangled structures according to several embodiments. [Figure 36B] This is a conceptual diagram illustrating the use of folding techniques to generate more complex periodic boundary conditions in layers of entangled structures according to several embodiments. [Figure 36C] This is a conceptual diagram illustrating the use of folding techniques to generate more complex periodic boundary conditions in layers of entangled structures according to several embodiments. [Figure 36D] This is a conceptual diagram illustrating the use of folding techniques to generate more complex periodic boundary conditions in layers of entangled structures according to several embodiments. [Figure 37A] This is a conceptual diagram showing the use of the techniques described herein to form diagonal folds with respect to layers of an entangled structure according to several embodiments. [Figure 37B] This is a conceptual diagram showing the use of the techniques described herein to form diagonal folds with respect to layers of an entangled structure according to several embodiments. [Figure 37C] This is a conceptual diagram showing the use of the techniques described herein to form diagonal folds with respect to layers of an entangled structure according to several embodiments. [Figure 37D] This is a conceptual diagram showing the use of the techniques described herein to form diagonal folds with respect to layers of an entangled structure according to several embodiments. [Figure 38] An example of a system architecture for a quantum computer system according to several embodiments is shown. [Modes for carrying out the invention]

[0031]

[0072] This specification discloses 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 and in other situations that utilize quantum entanglement (e.g., quantum communication). To facilitate understanding of this disclosure, Section 1 provides an overview of the relevant concepts and terminology. In this regard, Section 2 describes examples of 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. While embodiments are described in specific detail for ease of understanding, the inventions described in the claims can be practiced without these details, as will be apparent to those skilled in the art who have access to this disclosure.

[0032]

[0073] Furthermore, embodiments of systems of qubits that form and function as such systems are described herein, in which the quantum state space of a qubit can be modeled as a two-dimensional vector space. As those skilled in the art will see when accessing this disclosure, the techniques described herein can be applied to systems of “qubits,” in which a qubit can be any quantum system having a quantum state space that can be modeled as a (complex) n-dimensional vector space (with respect to any integer n) that can be used to encode n bits of information. For clarity, the term “qubit” is used herein, but in some embodiments, the system may 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, such as photons, electrons, atoms, ions, molecules, and nanostructures, which follow the rules of quantum theory. In quantum theory, the quantum state of a quantum object is described by a set of physical properties, the entire set of which is called a mode. In some embodiments, a mode is defined by specifying the value (or distribution of values) of one or more properties of the quantum object. For example, if the quantum object is a photon, a mode may be defined by the photon's frequency, the photon's position in space (e.g., which waveguide or superposition of waveguides the photon is propagating through), the associated propagation direction (e.g., the photon's k-vector in free space), the photon's polarization state (e.g., the direction of the photon's electric and / or magnetic field (horizontal or vertical)), the time window through which the photon is propagating, orbital angular momentum, and so on.

[0034]

[0075] In the case of photons propagating within a waveguide, it is convenient to represent the state of a photon as one of a set of discrete spacetime modes. For example, the space mode k of a photon. i This is determined by which of the finite set of discrete waveguides the photon is propagating through, and also by the time mode t j This is determined by which discrete period (referred to herein as a “bin”) the photon lies in. The degree of time discretization can be given by the pulsed laser responsible for generating the photon. In the following examples, spatial modes are used primarily to avoid complicating the explanation. However, as those skilled in the art will see, the systems and methods can be applied to any type of mode, e.g., time modes, polarization modes, and any other modes or sets of modes that help define the quantum state. Furthermore, the following description describes embodiments that use a photonic waveguide to define the spatial mode of a photon. However, as those skilled in the art will see, other types of modes, e.g., time modes, energy states, etc., can be used without departing from the scope of this disclosure. Furthermore, those skilled in the art can carry out the examples using other types of quantum systems, including but not limited to other types of photonic systems.

[0035]

[0076] For quantum systems of multiple indistinguishable particles, it is useful to describe the quantum state of the entire many-body system using the form of Fock states (sometimes called occupation representation), rather than describing the quantum state of each particle in the system. In the description of Fock states, the many-body quantum state is determined by the number of particles in each mode of the system. For example, the multimode two-particle Fock state |1001> 1、2、3、4 This defines a two-particle quantum state in which one particle is in mode 1, zero particles are in mode 2, zero particles are in mode 3, and one particle is in mode 4. Here again, as previously mentioned, the modes can be any properties of the quantum body. In the case of photons, any two modes of the electromagnetic field can be used, and the system can be designed to use modes related to degrees of freedom that can operate passively in a linear optical system, for example. For example, polarization, spatial degrees of freedom, or angular momentum can be used. Two-particle Fock state |1001> 1、2、3、4 The four-mode system represented by |1111| can be physically implemented as four separate waveguides, each containing a single photon within two of the four waveguides. Another example of a state in such a many-body quantum system is the four-particle Fock state |1111|, representing each mode occupied by a single particle. 1、2、3、4 The four-particle Fock states represent modes 1 and 2, each occupied by two particles, and modes 3 and 4, each occupied by zero particles. 1、2、3、4 These are some examples. For modes in which no particles exist, 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”. A Fock state having a single occupancy mode can be represented in an abbreviated form using a subscript to identify the occupancy mode. For example, |0010〉 1、2、3、4 This is equivalent to |13〉.

[0036] 1.1. Quantum Bits

[0077] As used herein, a “qubit” (or “quantum bit”) is a quantum system with associated quantum states that can be used to encode information. If the quantum state space can be modeled as a (complex) two-dimensional vector space, then one bit of information can be encoded using the quantum states, in which case one dimension in the vector space maps to the logical value 0 and the other dimension maps to the logical value 1. In contrast to classical bits, qubits can have states that are superpositions of logical values ​​0 and 1. More generally, a “qubit” 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, the term “qubit” is used herein, although in some embodiments, the system may also use quantum information carriers that encode information in a manner not necessarily associated with binary bits such as qubits. 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 can be "dual-rail coded" such that its logical value is encoded by the occupation of one of the two modes of the quantum system. For example, the logical 0 and theoretical 1 values ​​can be coded as follows: |0〉 L =|10〉 1、2 (1) |1〉 L =|01〉 1、2 (2) Here, the subscript "L" indicates, as before, that the ket represents a logical state (e.g., a qubit value), and the notation on the right side of the above equation |ij〉 1、2 This indicates that there are i particles in the first mode and j particles in the second mode (for example, i and j are integers). In this notation, the logical states are |0〉|1〉 L A 2-qubit system having (representing the states of two qubits, where the first qubit is in the "0" logical state and the second qubit is in the "1" logical state) is |1001> 1、2、3、4 This can be expressed using occupation across four modes (for example, in a photonic system, one photon in the first waveguide, zero photons in the second waveguide, zero photons in the third waveguide, and one photon in the fourth waveguide). Throughout this disclosure, in some examples, 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 generate entangled states in multi-qubit systems. Mathematically speaking, the states |ψ〉 of n quantum objects are:

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 logic basis, the n-GHZ state is a quantum superposition in which all qubits in the first ground state are superimposed with all qubits in the second ground state.

number

number

[0040] 1.3. Physical Implementation

[0081] Qubits (and operations on qubits) can be implemented using a variety of physical systems. In some examples described herein, a qubit is provided by 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 photons are spatiotemporal modes corresponding to the presence of photons in the waveguide. Modes can be coupled using a mode coupler, such as an optical beam splitter, to perform conversion operations, and measurement operations can be performed by coupling a single-photon detector to a particular waveguide. As those skilled in the art will see when accessing this disclosure, modes defined by any suitable set of degrees of freedom, such as polarization modes, time modes, etc., can be used without departing from the scope of this 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, such as a birefringent material such as a waveplate. In the case of other systems such as ion trap systems or neutral atom systems, the mode coupler can be any physical mechanism capable of coupling two modes, such as a pulsed electromagnetic field tuned to couple two internal states of an atom / ion.

[0041]

[0082] In some embodiments of photonic quantum computing systems using dual-rail coding, a pair of waveguides can be used to implement qubits. Figure 1 shows two representations (100, 100') of a pair of waveguides 102, 104 that may be used to give a dual-rail coded photonic bit. At 100, a photon 106 is in waveguide 102 and no photons are in waveguide 104 (also called the vacuum mode), and in some embodiments, this is the |0〉 of the photonic qubit. L Corresponds to the state. In 100', photon 108 is in waveguide 104 and there are no photons in waveguide 102, and in some embodiments this corresponds to the |1〉 of the photonic qubit. LCorresponding to states. To preprocess photonic qubits of known logical states, a photon source (not shown) can be coupled to one end of a waveguide. The photon source can operate to emit single photons into the waveguide to which it is coupled, thereby preprocessing photonic qubits of known states. The photons travel through the waveguide, and by operating the photon source periodically, 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 do not need to have a specific spatial relationship to one another. For example, these waveguides can be arranged in parallel, but they do not necessarily have to be parallel.

[0042]

[0083] Occupied modes can be formed by using a photon source to generate photons that propagate within a desired waveguide. The photon source can be, for example, a resonator-based photon source that emits photon pairs, also known as a leading single-photon source. In one example of such a photon source, the source is driven by a pump, such as an optical pulse, coupled to a system of optical resonators capable of generating 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 is a microring-based spontaneous four-wave mixing (SPFW) messenger photon source (HPS). However, the exact type of photon source used is not critical; 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, such as quantum dot sources or those using atomic and / or artificial atomic systems such as color centers in crystals, can also be used. In some cases, the photon source may or may not be coupled to a photonic cavity, for example, in the case of artificial atomic systems such as quantum dots coupled to a cavity. Other types of photon sources, such as photomechanical systems, also exist in SPWM and SPDC.

[0043]

[0084] In such cases, the operation of the photon source may be non-deterministic (sometimes called “stochastic”), such that a given pump pulse may or may not generate a photon pair. In some embodiments, coherent spatial and / or temporal multiplexing (referred herein to as “active” multiplexing) of several non-deterministic photon sources can be used to allow the probability of one mode being occupied during a given cycle to approach 1. As those skilled in the art will see, many different active multiplexing architectures incorporating spatial and / or temporal multiplexing can be envisioned. For example, active multiplexing schemes can be used that utilize logarithmic trees, generalized Mach-Zehnder interferometers, multimode interferometers, chain sources, dump-to-pump chain sources, asymmetric polycrystalline single-photon sources, or any other type of active multiplexing architecture. In some embodiments, the photon source can use an active multiplexing scheme, such as quantum feedback control. In some embodiments described below, by using multirail coding, the probability of a bandwidth being occupied during a given pulse cycle can approach 1 without involving active multiplexing.

[0044]

[0085] The measurement operation can be carried out by coupling a 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 with sensitivity to single photons can be used. In some embodiments, the detection of a photon (e.g., at the output end of the waveguide) may indicate an occupied mode, while the absence of a detected photon may indicate an unoccupied mode.

[0045]

[0086] The following embodiments relate to physical embodiments of a unified transformation operation that couples the modes of a quantum system, which can be understood as a transformation of the quantum state of the system. For example, if the initial state of a quantum system (before mode coupling) is one in which one mode is occupied with probability 1 and another in which one mode is not occupied with probability 1 (e.g., the state |10〉 in Fock notation as described above), then mode coupling can result in a state with a non-zero probability in which both modes are occupied, e.g., the state a1|10〉+a2|01〉, and |a1| 2 +|a2| 2 = 1. In some embodiments, this type of operation can be achieved by using a beam splitter to couple modes together and a variable phase shifter to apply a phase shift to one or more modes. The amplitudes a1 and a2 depend on the reflectance (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 notation) for coupling two modes. The modes are depicted as horizontal lines 212 and 214, and the mode coupler 216 is indicated by vertical lines terminated with nodes (solid dots) to identify the coupled modes. 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] Figure 2B shows a physical implementation of mode coupling that implements the transfer matrix T of equation (9) with respect to two photonic modes according to several embodiments. In this example, mode coupling is implemented 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 so that the evanescent field of one waveguide can be coupled to the other waveguide. Different couplings between modes can be obtained by adjusting the spacing d between waveguides 202,204 and / or the length l of the coupling region. In this way, the waveguide beam splitter 200 can be configured to have a desired transmittance. For example, the beam splitter can be designed to have a transmittance equal to 0.5 (i.e., a 50 / 50 beam splitter to implement a 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 phase shifts applied to one or more modes. In some photonic implementations, a variable phase shifter can be implemented in an integrated circuit to control the relative phase of the photon states diffused across multiple modes. An example of a transfer matrix defining such a phase shift is given below (for applying +i and -i phase shifts to the second mode, respectively).

number

[0049]

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

[0050]

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

number

number

[0051]

[0092] Similarly, coupling between three or more modes can be implemented using a network of mode couplers and phase shifters. For example, Figure 5 shows a four-mode coupling scheme that performs a “spreader” or “mode information erasure” transform on four modes, i.e., this scheme delocalizes the photon among each of the four output modes such that the probability of the photon being captured in any one of the input modes and detected in any one of the four output modes is equal. (The well-known Hadamard transform is an example of a spreader transform.) As shown in Figure 2A, horizontal lines 512-515 correspond to modes, and mode coupling is indicated by vertical lines 516 with nodes (dots) to identify the coupled modes. In this case, four modes are coupled. Circuit notation 502 is an equivalent representation of circuit diagram 504, which is a network of first-order mode couplings. More generally, if higher-order mode coupling can be implemented as a network of first-order mode couplings, a circuit notation similar to notation 502 (with an appropriate number of modes) may be used.

[0052]

[0093] Figure 6 shows an example of an optical device 600 capable of performing the four-mode spread conversion schematically shown in Figure 5 according to several 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 Figure 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 Figure 6). The second and first material layers are located at different heights on the substrate. As will be apparent to those skilled in the art, if appropriate low-loss waveguide crossings are used, an interferometer like the one shown in Figure 6 can be implemented in a single layer.

[0053]

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

[0054]

[0095] Furthermore, the optical device shown in Figure 6 may include two interlayer optical couplers 614 and 616. Optical coupler 614 enables the transmission of light propagating through a waveguide on a first material layer to a waveguide on a second material layer, and optical coupler 616 enables the transmission of light propagating through a waveguide on the second material layer to a waveguide on the first material layer. Optical couplers 614 and 616 enable the use of optical waveguides located in at least two different layers in a multi-channel optical coupler, thereby enabling a compact multi-channel optical coupler.

[0055]

[0096] Furthermore, the optical device shown in Figure 6 includes an uncoupled waveguide intersection region 626. In some implementations, two waveguides (603 and 605 in this example) intersect each other without a parallel coupling region at the intersection within the uncoupled waveguide intersection region 626 (for example, the waveguides may be two straight waveguides intersecting each other at an angle of approximately 90 degrees).

[0056]

[0097] As those skilled in the art will see, the examples described above are illustrative, and many different transformation matrices can be implemented using photonic circuits employing beam splitters and / or phase shifters, including transformation matrices for real and imaginary Hadamard transforms of any order, discrete Fourier transforms, and so on. One class of photonic circuits, referred to herein as “spreaders” or “mode information erasure (MIE)” circuits, has the property that, given an input is a single photon localized to one input mode, the circuit delocalizes a photon among 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 a spreader or MIE circuit may receive inputs that are not single photons localized in one input mode, and the behavior of the circuit in such cases depends on the specific transporter matrix implemented.) In other examples, a photonic circuit may implement other transporter matrices that, with respect to a single photon in one input mode, have different probability matrices of detecting the photon in different output modes that are not equal.

[0057]

[0098] In some embodiments, entangled states of multiple photonic qubits can be formed by combining modes of two (or more) qubits and performing measurements in other modes. As an example, Figure 7 shows a schematic of a Bell state generator 700 that can be used in several dual-rail coding photonic embodiments. In this example, modes 732(1)-732(4) are initially occupied by photons (indicated by dashed lines), while modes 732(5)-732(8) are initially vacuum modes. (As those skilled in the art will see, other combinations of occupied and unoccupied modes can be used.)

[0058]

[0099] A first-order mode coupling (e.g., implementing the transfer matrix T of equation (9)) is performed on pairs of occupied and unoccupied modes, as shown by the mode couplers 731(1)-731(4). Subsequently, a mode information elimination coupling (e.g., performing a four-mode spread-mode transform as shown in Figure 5) is performed on four of the modes (modes 732(5)-732(8)), as shown by the mode coupler 737. Modes 732(5)-732(8) act as "messenger" modes, which are measured and used to determine whether the Bell state was 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 the 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) indicating whether it has detected a photon (or the number of photons detected). These outputs can be coupled to a classical decision logic circuit 740 that determines whether a Bell state exists in the other four modes 732(1)-732(4). For example, the decision logic circuit 740 can be configured such that a Bell state is confirmed (also called a "success" for the Bell state generator) only if a single photon is detected by exactly two of the detectors 738(1)-738(4). Modes 732(1)-732(4) can be mapped to the logic states of two qubits (qubit 1 and qubit 2), as shown in Figure 7. Specifically, in this example, the logical state of Qubit 1 is based on the occupancy rates of modes 732(1) and 732(2), and the logical state of Qubit 2 is based on the occupancy rates of modes 732(3) and 732(4). It should be noted that the operation of the Bell state generator 700 can be nondeterministic. That is, inputting four photons as shown in the figure does not guarantee that a Bell state will be generated in modes 732(1)-732(4). In one embodiment, 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 a Bell state can be understood as a cluster state of two qubits). One technique for forming larger entangled systems is by using entanglement measurements, which are projection measurements that can be used to generate entanglement between systems of qubits. As used herein, “fuse” (or “fuse operation” or “fuse”) refers to a two-qubit entanglement measurement. A “fuse gate” is a structure that takes two input qubits, each of which is typically part of an entangled system. The fusion gate performs a projection measurement operation on the input qubits that produces an output qubit of either 1 (“Type I fusion”) or 0 (“Type II fusion”) in such a manner that the two initial entangled systems are fused into a single entangled system. Fusion gates are concrete examples of a general class of two-qubit entanglement measurements and are particularly well suited to photonic architectures. Examples of Type I and Type II fusion gates are described below.

[0060]

[0101] Figure 8A shows a schematic diagram illustrating a type I fusion gate 800 according to several embodiments. The diagram shown in Figure 8A is a schematic diagram in which each horizontal line represents a quantum system, such as a mode of photons. In dual-rail coding, each pair of modes corresponds to a qubit. In a photonic implementation of the gate, the modes shown in the diagram as in Figure 8A can be physically realized using a single photon in a photonic waveguide. Most commonly, a type I fusion gate as shown in Figure 8A takes qubit A (physically realized by, for example, photon modes 843 and 845) and qubit B (physically realized by, for example, photon modes 847 and 849) as inputs and outputs a single “fused” qubit that inherits the entanglement of either (or both) input qubits A or B with other qubits already entangled.

[0061]

[0102] For example, Figure 8B shows the result of a Type I fusion of two qubits A and B, each located at the end (i.e., a leaf) of several longer entangled cluster states (only a portion is shown). The qubit 857 remaining after the fusion operation inherits the entanglement from the original qubits A and B, thereby creating a larger linear cluster state. Also, Figure 8B shows the result of a Type I fusion of two qubits A and B, each being an internal qubit belonging to some longer entangled cluster of qubits (only a portion is shown). As mentioned above, the qubit 859 remaining after the fusion inherits the entanglement from the original qubits A and B, thereby creating a fused cluster state. In this case, the qubit remaining after the fusion operation is also entangled with a larger cluster by four other nearest neighbor qubits, as shown in the figure.

[0062]

[0103] Returning to the schematic diagram of the Type I fusion gate 800 shown in Figure 8A, qubit A is dual-rail coded by modes 843 and 845, and qubit B is dual-rail coded by modes 847 and 849. For example, in the case of a path-coded photonic qubit, the logical zero state of qubit A is |0〉 AThe (indicated by ) occurs when mode 843 is a photonic waveguide containing a single photon and mode 845 is a photonic waveguide containing a zero photon (the same applies to qubit B). Thus, the type I fusion gate 800 can take two dual-rail coded photon qubits as input, thereby resulting in a total of four input modes (e.g., modes 843, 845, 847, and 849). To achieve fusion operation, a mode coupler (e.g., a 50 / 50 beam splitter) 853 is applied between each mode of the input qubit, for example, between mode 843 and mode 849, before performing detection operations for both modes using the photon detector 855 (which includes two separate photon detectors coupled to modes 843 and 849, respectively). The detection operations for modes 843 and 849 are destructive measurements. Furthermore, a mode swap operation 851 can be applied to swap 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 located adjacent to each other. In some embodiments, the mode swap can be achieved by the physical waveguide crossing described above, by one or more photonic switches, or by any other type of physical mode swap.

[0063]

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

[0064]

[0105] The Type I fusion gate 800 is a non-deterministic gate, meaning that the fusion operation succeeds with a probability of less than 1, and in other cases, the resulting quantum state is not a larger cluster state containing the original cluster states fused into a larger cluster state. More specifically, gate 800 has a 50% probability of "success" if detector 855 detects only one photon, and "fails" if detector 855 detects zero or two photons. If the gate succeeds, the two cluster states that qubits A and B were part of are fused into a single larger cluster state, and the fused qubits remain as the qubit that links the two previously unlinked cluster states (see, for example, Figure 8B). However, if the fusion gate fails, it has the effect of removing both qubits from the original cluster resource state without generating a larger fused state.

[0065]

[0106] Figure 9A shows a schematic diagram illustrating a Type II fusion gate 900 according to several embodiments. As with other figures in this specification, the diagram in Figure 9A is a schematic diagram, where each horizontal line corresponds to a mode of the quantum system, such as a photon. In dual-rail coding, each pair of modes corresponds to a qubit. In a photonic implementation of the gate, the modes in the diagram as shown in Figure 9A can be physically realized using a single photon in a photonic waveguide. Most commonly, a Type II fusion gate such as gate 900 takes qubit A (physically realized by, for example, photon modes 943 and 945) and qubit B (physically realized by, for example, photon modes 947 and 949) as input and outputs a quantum state that inherits the entanglement of either (or both) input qubit A or input qubit B with other qubits already entangled. (In the case of Type II fusion, if the input quantum state has N qubits, the output quantum state will have N-2 qubits. This differs from Type I fusion, where an N-qubit input quantum state 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 qubit located at the end (i.e., leaf) of several longer entangled cluster states (only a portion of which are shown). The resulting qubit system 971 inherits the entangled coupling from qubits A and B, thereby creating a larger linear cluster state.

[0067]

[0108] Returning to the schematic diagram of the Type II fusion gate 900 shown in Figure 9A, qubit A is dual-rail coded by modes 943 and 945, and qubit B is dual-rail coded by modes 947 and 949. For example, in the case of a path-coded photonic qubit, the logical zero state (|0〉 of qubit A. AThe mode shown is the one that occurs when mode 943 is a photonic waveguide containing a single photon and mode 945 is a photonic waveguide containing a zero photon (the same applies to qubit B). Thus, the type II fusion gate 900 takes two dual-rail coded photon qubits as input, thereby yielding a total of four input modes (e.g., modes 943, 945, 947, and 949). To achieve fusion operation, a first mode coupler (e.g., a 50 / 50 beam splitter) 953 is applied between each mode 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. Detection operation is performed for all four modes using photon detectors 957(1)-957(4). Detection operation is a destructive measurement. In some embodiments, a mode-swap operation (not shown in Figure 9A) can be performed to place modes in adjacent positions before mode coupling. In some embodiments, mode swapping can be achieved by physical waveguide crossing as described above, by one or more photonic switches, or by any other type of physical mode swap. Mode swapping is optional and not necessary if qubits with non-adjacent modes can be handled by tracking which mode belongs to which qubit, for example, by storing this information in classical memory.

[0068]

[0109] Figure 9A shows only an exemplary arrangement of a Type II fusion gate, and as those skilled in the art will see, the position of the mode coupler and the presence or absence of a mode swap area can be changed without departing from the scope of the present disclosure.

[0069]

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

[0070]

[0111] The above description provides examples of how photonic circuits can be used to perform 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 explained in Section 1, qubits can be physically realized using a pair of waveguides into which photons are introduced, and qubits can be operated using mode couplers (e.g., beam splitters), variable phase shifters, photon detectors, etc. For example, by introducing mode couplers between waveguides associated with different qubits, entanglement between two (or more) qubits can be created. In practice, physical qubits can suffer from losses (e.g., when photons may not be detected during measurement due to inefficiencies in photon generation circuits, mode couplers, fusion circuits, or other components) and noise (e.g., when bit-flip errors may occur before measurement). As a result, relying on a single physical qubit (e.g., photons propagating through a pair of waveguides) when performing quantum computations can lead to unacceptably high error rates. 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 entangled structure that enables error correction. (As used in the following sections, the term “qubit” 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 an abbreviation, this disclosure uses the term “entangled space” to refer to a space having dimensions corresponding to the graph 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 construct a larger entanglement structure from a smaller entanglement system of physical qubits, referred to as a “resource state.” As used herein, a “resource state” refers to an entanglement system of multiple (n) qubits that are inseparable entanglement states (entanglement states that cannot be decomposed into smaller, separate entanglement states). In various embodiments, the number n can be a small number (e.g., any number up to about 20) or a larger number (as desired).

[0073]

[0114] Figures 10A–10C show entanglement graph representations of resource states that can be used according to several 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 necessarily correspond to physical dimensions, and in some cases, qubits may be separated temporally rather than spatially. For example, each physical qubit can be implemented using photons propagating through a waveguide, and specific portions of the waveguide may host photons associated with different qubits at different times.

[0074]

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

[0075]

[0116] The geometric shape or topology of resource state entanglement can be modified. As an example, Figure 10B shows an example of a different resource state 1020 having seven physical qubits 1030-1036. Similar to resource state 1000, the 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 further entanglement between peripheral qubits 1030 and 1032.

[0076]

[0117] As another example, Figure 10C shows resource state 1040, known in the art as the Kagome-6 state. Resource state 1040 has six peripheral qubits 1050-1055 (and no central qubit), each peripheral qubit is entangled with two other qubits. Resource state 1040 can be understood to have a three-dimensional entanglement geometry as suggested by the central bidirectional 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 resource states shown in Figures 10A to 10C are illustrative and not limiting. In some embodiments, the entanglement topology / geometry of a resource state can be selected based on the specific computation performed, and different resource states used to generate a single entanglement structure may have different entanglement topologies. Furthermore, 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 may be larger or smaller than those shown and may contain any number of central qubits (including 0 central qubits) and / or peripheral qubits. Additional considerations regarding the size of resource states and the selection of entanglement geometry are described below.

[0078]

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

[0079]

[0120] To form an entanglement structure larger than the resource states, a fusion operation (e.g., the Type II fusion operation described above or other entanglement measurement operations) can be performed to form entanglement between qubits of different resource states within a layer. Figures 11A and 11B use dotted ellipses to show examples of pairs of qubits that can be input to a fusion circuit (e.g., the Type II fusion circuit 900 in Figure 9B). Thus, for example, in layer 1100 of Figure 11A, the +x qubit (1,1) and the -x qubit (1,2) of resource state 1000 can be inputs to a certain fusion operation, as shown by the dotted ellipse 1105, and the -y qubit (1,1) and the +y qubit (2,1) of resource state 1000 can be inputs to another fusion operation, as shown by the dotted ellipse 1107. As shown, this pattern can be repeated across layer 1100. Similarly, in layer 1140 of Figure 11B, the +x qubit (1,1) and the -x qubit (1,2) of resource state 1040 can be inputs to a certain fusion operation, as shown by the dotted ellipse 1145, and the -y qubit (1,1) and the +y qubit (2,1) of resource state 1040 can be inputs to another fusion operation, as shown by the dotted ellipse 1147. As shown in the figure, this pattern can be repeated across layer 1100.

[0080]

[0121] In some embodiments, qubits at the edges or boundaries of layers (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 the boundaries of layers (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 a similar operation on the qubit. Alternatively, boundary qubits may undergo a fusion operation with another boundary qubit, which may be a boundary qubit in the same 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 so that boundary qubits are not generated or are generated selectively.

[0081]

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

[0082]

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

[0083]

[0124] In the following description, fusion operations may be referred to as “spatial” or “temporal.” These terms evoke specific embodiments in which different qubits or resource states are generated at different times; spatial fusion can be performed between qubits generated simultaneously using different hardware instances, and temporal fusion can be performed between qubits generated at different times using instances of the same hardware. In the case of photonic qubits, temporal fusion can be performed by delaying previously generated qubits (e.g., by using longer waveguide material to create a longer propagation path for photons), thereby enabling mode coupling with later generated qubits. By leveraging temporal fusion, it is possible to generate multiple instances of resource states within a layer and / or multiple layers of resource states using the same hardware.

[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 shifts, mode swaps, and / or basis rotations, and can receive (classical) control signals to select specific operations to be performed. For example, different fusion operations can be selectively performed at different locations within a layer, or different fusion operations can be selectively performed on 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 (for example, the examples in Figures 10A, 11A, and 12A), each resource state has a central qubit (i.e., a qubit such as qubit 1016 that does not undergo fusion operation with a qubit in another resource state). Thus, after performing the fusion operation as described above, a large entanglement system of qubits (referred to herein as "LES") can be generated. Figure 13 shows an example of an LES 1300 that can be formed by the fusion operation shown in Figures 11A and 12A, applied to resource state 1000 in Figure 10A according to some embodiments. In this example, resource state 1000 has a single central qubit (qubit 1016 in Figure 10A), and LES 1300 can be understood as having layers, each layer containing an array of R×C qubits 1316. More generally, a resource state can have any number of central qubits, and the number of qubits per layer of the LES can differ from the size of the layers of resource states that contributed to the layers of the LES. An LES is a system of qubits that are physically pre-processed and therefore physically exist in a specific entangled state. An entangled state of qubits (e.g., photonic qubits) can itself be a graph state, a cluster state, some other entangled state that forms a fault-tolerant cluster state corresponding to a quantum error-correcting code (e.g., a topological code, e.g., a leaf code, a volume code, a color code, etc.) using appropriate measurements on the individual qubits, or any part of these entangled states. Thus, one or more logical qubits can be encoded or used as a cluster state (or part of a cluster state) using an LES (or several LES further entangled through processes such as the “stitching” process described later), in which measurements of the individual physical qubits are taken and quantum computations are performed in a measurement-based quantum computing ("MBQC") system, or in any other situation in which quantum computations are performed in which a large system of entangled physical qubits is generated.

[0086]

[0127] In other embodiments (for example, in the examples in Figures 10C, 11B, and 12B), the resource state has no central qubit. In embodiments where the resource state has no central qubit, intralayer and interlayer fusion operations may include destructive measurements on all of all qubits in the resource state, and the final output generating entanglement may 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 state and the fusion operations performed therein. 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 illustrative and are subject to modification and alteration. The size and entanglement geometry of resource states can be changed. In some embodiments, resource states with different sizes and / or entanglement geometries can be used at different locations within a layer or array of layers, and various logical operations can be performed using position-dependent selection of resource state configurations. It should also be understood that fusion operations may be inherently probabilistic and not always successful, and in some embodiments, the entanglement geometry can support fault tolerance for both MBQC and FBQC. Furthermore, while FBQC and MBQC are examples of use cases for the entanglement generation techniques described herein, it should be understood that these techniques can be applied in other contexts and are not limited to quantum computing.

[0088] 2.1. Resource State Generation

[0129] As described above, some embodiments relate to apparatus and methods that can be used to construct a large entanglement structure from a large number of resource states, where each resource state is an entanglement system of n qubits in an inseparable entanglement state.

[0089]

[0130] The specific size and entanglement shape of resource states can be selected as design parameters. In some cases, the optimal size may depend on a particular physical embodiment of the qubit. For example, as mentioned above, qubits can be implemented using photons propagating through a waveguide. The process used to generate photons and create entanglement may be probabilistic (i.e., the probability of successfully generating a photon in any given case is significantly less than 1). If qubit generation or entanglement is probabilistic, multiplexing techniques or other techniques can be used to increase the probability of generating resource states with a defined entanglement structure (for each trial). Given a set of resource states, the process used to form larger entanglement structures (e.g., the fusion process described above) may also be probabilistic, and larger entanglement structures can be defined in a manner that supports fault-tolerant behavior in the presence of probabilistic processes. Thus, the size of resource states can be selected for a particular implementation based on an acceptable error rate in resource state generation and a specific probability of generating resource states with a defined entanglement structure.

[0090]

[0131] In some embodiments, resource states such as resource state 1100 can be generated using photonic circuits and electronic circuits and components (e.g., those of the type described in Section 1.3 above) to generate and operate individual photons. In some implementations, the resource state generator may be a single integrated circuit manufactured, for example, using conventional silicon-based technology. The resource state generator may include a photon source or may receive photons from an external source. The resource state generator may also include a photonic circuit that implements a Bell state generator and fusion operation as described above. To provide robustness, the resource state generator may include multiple parallel instances of various photonic circuits having detectors and electronic control logic for selecting successful instances for propagating photons. Those skilled in the art know various methods for constructing a photonic resource state generator capable of generating resource states having a desired entangled geometry.

[0091]

[0132] In some embodiments, resource states can be generated using techniques other than linear optical systems. For example, various devices are known for generating and bringing about entanglement between systems of "material-based" qubits, such as qubits implemented in ion traps, other qubits encoded at the energy level of atoms or ions, spin-coded qubits, superconducting qubits, or other physical systems. It is also understood in the art that quantum information is interchangeable in the sense that the same information (in this case, quantum states) can be encoded using many different physical systems. Therefore, in principle, it is possible to exchange the quantum state of one system with another by inducing interactions between systems. For example, the state of a qubit (or a collection of quantized qubits) encoded at the energy level of atoms or ions can be exchanged for an electromagnetic field (i.e., photons). It is also possible to use transducer techniques to exchange the state of a superconducting qubit for a photonic state. In some cases, the initial swap may be on photons having microwave frequencies. After the exchange, the frequency of the photons can be increased to the operating frequency of an optical fiber or other optical waveguide. As another example, quantum teleportation can be applied between a material-based qubit and a Bell pair in which 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, a material-based qubit can be used to generate a resource state consisting of photonic qubits, and the specific configuration and configuration of the resource state generator are not relevant to the understanding herein.

[0092] 2.2. Circuits for forming entangled structures from resource states

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

[0093] 2.2.1. Circuit Symbols

[0134] To facilitate understanding of the explanation, Figures 14A–14F introduce a set of schematic circuit symbols used in subsequent figures. These circuit symbols represent circuits operating with physical (photonic) qubits, where each input or output line represents a (physical) qubit. As a convention of the drawings, inputs are shown on the left and outputs on the right, with the separation that the schematic circuit diagram does not need to correspond to a specific physical layout.

[0094]

[0135] Figure 14A shows symbols representing a resource state generator (RSG) circuit 1400. As previously mentioned, an 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 resource states encoded on a non-photonic system of physical qubits and then exchange the quantum states for photonic qubits. Other implementations of resource state generator circuits can form an initial state on a non-photonic system of physical qubits, exchange the initial state for photonic qubits, and then perform a linear optical operation to form a resource state. Regardless of the implementation, the output of the RSG circuit 1400 is a qubit indicated by line 1402, and the number of outputs depends 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 required time may depend on the particular RSG circuit. For example, an 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 can be longer than the time required for the RSG circuit to generate one resource state, and the RSG does not need to operate at its maximum speed. For the purposes of this specification, we assume that the RSG circuit 1400 outputs all qubits of a resource state in the same clock cycle. However, as will be apparent to those skilled in the art accessing this disclosure, the timing can be changed.

[0095]

[0136] Figure 14B shows a symbol representing a Type II fusion circuit 1405. The Type II fusion circuit can be implemented as previously described, for example, in relation to Figures 9A and 9B. The input is two qubits (indicated by line 1404). As previously described, the Type II fusion operation involves a destructive measurement on the two qubits. The Type II fusion circuit 1405 can provide a classical output signal 1406, which can encode measurement data indicating the count of photons detected from each detector and / or other information (e.g., success or failure of the fusion operation).

[0096]

[0137] Figure 14C shows symbols representing the switching circuit 1410. The inputs and outputs to the switching circuit 1410 can contain any number of qubits (line 1408), and the number of inputs does not have to be equal to the number of outputs (line 1409). The switching circuit 1410 can incorporate any combination of one or more active optical switches, mode couplers, mode swap circuits, phase shifters, etc. The switching circuit can be configured to perform active operations that reconfigure the input modes (e.g., to bring about a base change of qubits by coupling the modes of qubits), change the input modes, and / or apply a phase to one or more of the input modes (this may affect subsequent coupling between modes). In some embodiments, the operation of the switching circuit 1410 can be dynamically controlled in response to a classical control signal 1411, and its state can be determined based on the results of previous operations, specific calculations performed, configuration settings, timing counters (e.g., for periodic switching), or any other parameters or information.

[0097]

[0138] Figure 14D shows a symbol representing a delay circuit 1415. The delay circuit delays the propagation of a qubit (input 1412) over a fixed length of time, and then outputs a qubit (output 1414). The length of time (in clock cycles) is indicated by a number, where D=1 indicates a delay of one clock cycle. The delay circuit can be implemented, for example, by providing one or more optical fibers of appropriate length, other waveguide materials, nitride layers, memory, etc., so that the photons of the delayed qubit travel a longer path than the photons of the non-delayed qubit.

[0098]

[0139] Figure 14E shows a symbol representing a reconfigurable fusion circuit 1420. As shown, the reconfigurable fusion circuit includes a switching circuit 1410 followed by a fusion circuit 1405. The reconfigurable fusion circuit can support configurable operations applied by the switching circuit 1410 before the fusion operation by the fusion circuit 1405, such as a base change or a phase shift. 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 classic 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 classic output signal 1406.

[0099]

[0140] Figure 14F shows a symbol representing the offset reconfigurable fusion circuit 1425. As shown in the figure, the offset reconfigurable fusion circuit is similar to the reconfigurable fusion circuit 1420, with the addition of a delay circuit 1415 that delays one input relative to the other by a predetermined number of clock cycles. The offset reconfigurable fusion circuit 1425 may also be called a “temporal” fusion circuit, a term that emphasizes the temporal aspect arising 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, each RSG circuit giving one resource state that is fused with resource states from other RSG circuits to form a layer of entanglement (for example, as shown in Figure 11A or Figure 11B), and the same RSG circuit can successively generate different layers for the entanglement. Figure 15 shows a conceptual diagram of networked generation of layers according to some embodiments. Size L 2 To support the generation of layers, the corresponding number L 2 An RSG circuit 1502 is provided. In the simplified example used herein, L 2 =16, but in reality, L 2 It can be made much larger (for example, about 10 2 , about 10 4 , about 10 6 ). In each clock cycle, enough resource states 1500 can be generated to form a complete two-dimensional (2D) layer of resource states. (In Figure 15, each resource state 1500 is annotated with time "t=1" to show that they are all generated during the same clock cycle.) Spatial fusion operation can be performed on the qubits of adjacent resource states 1500 (for example, as shown in Figures 11A and 11B) using further circuits described below. 2 Using RSG circuit 1502, L 2 By generating layers with different resource states, a three-dimensional entanglement structure can be created, and further circuits, as described later, can be used to perform a temporal fusion operation on the 1500 qubits of different layer resource states (for example, as shown in Figures 12A and 12B).

[0101]

[0142] Figures 16A and 16B are schematic diagrams of a “fully networked” circuit for generating entanglement structures from resource states according to several embodiments. The circuit notation is the same as previously described in relation to Figures 14A to 14F, except that classical inputs and outputs are not shown for clarity in the diagram. Figure 16A shows a typical network cell 1600, and Figure 16B shows the coupling between adjacent instances of network cell 1600 in network 1650. As is most commonly seen in Figure 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 that are not subject to fusion operation (if any). For example, if the RSG circuit 1502 generates resource state 1000 in Figure 10A, the central qubit 1016 can be given as the central qubit 1615, but if the RSG circuit instead generates resource state 1040 in Figure 10C, the central qubit 1615 is not given. RSG 1502 provides two peripheral qubits to adjacent network cells, as indicated by the "x-fusion" output path 1611 and the "y-fusion" output path 1612. Network cell 1600 also receives qubits from two adjacent network cells. Specifically, input path 1611' connects to the x-fusion output path of network cell 1600' (as shown in Figure 16B). Similarly, input path 1612'' connects to the y-fusion output path of network cell 1600'', which is adjacent to network cell 1600 in the +y direction (as shown in Figure 16B).

[0102]

[0143] Furthermore, 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 couples the +y qubit of the "local" resource state generated by the RSG circuit 1502 to the -y qubit of the "networked" resource state generated by the RSG circuit in the adjacent network cell 1600'' in the +y direction. The x+ reconfigurable fusion circuit 1630 couples the +x qubit of the local resource state generated by the RSG circuit 1502 to the -x qubit 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 the RSG circuit 1502. The -z qubit is delayed by one clock cycle and fused with the +z qubit of the resource state generated by the RSG circuit 1502 during the next clock cycle.

[0103]

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

[0104] 2.2.3. Rasterization generation of entanglement

[0145] As mentioned above, generating entanglement using a fully networked RSG circuit provides fast computation, but especially considering the size of each layer (L 2If the size is large, it can be hardware-intensive. Furthermore, the maximum size of the layer may be constrained by the available hardware. Therefore, some embodiments employ a hardware reduction technique called “rasterization” generation of entanglement, in which one instance of the RSG circuit gives multiple resource states within a single layer. In one example of “full rasterization” generation, a single instance of the RSG circuit can be used to generate an entanglement structure with layers of any size by providing appropriate delay and fusion circuits.

[0105]

[0146] Figure 17 is a conceptual diagram of layer rasterization generation for entangled structures according to several embodiments. Size L 2 To support the generation of the layer, a single instance of the RSG circuit 1702 is given. In the simplified example used herein, L 2 =16, but in reality, L 2 It can be made much larger (for example, about 10 2 , about 10 4 , about 10 6 ). In each clock cycle, the RSG circuit 1702 generates a single resource state, and L generates enough resource states to form a complete 2D layer. 2 It can be generated in clock cycles. 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 times "t=1" to "t=16" to indicate the clock cycle in which each resource state 1700 is generated. Temporal fusion operations can be performed on qubits of adjacent resource states 1700 generated in different clock cycles using further circuits described later (e.g., the fusion operations shown in Figures 11A and 11B). The three-dimensional entanglement structure is L layer by layer. 2The resource states can be generated by repeating the process of generating resource states using the same RSG circuit 1702, and the temporal fusion operation can be performed on the qubits of resource states 1700 in different layers using further circuits described later (e.g., the fusion operation shown in Figures 12A and 12B).

[0106]

[0147] Figure 18 shows a schematic diagram of a “fully rasterized” circuit 1800 for generating an entangled structure from resource states according to several embodiments. The circuit notation is the same as described above in relation to Figures 14A to 14F, except that classical inputs and outputs are not shown for clarity in the diagram. 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 fusion operation (if any). For example, if the RSG circuit 1502 generates resource state 1000 in Figure 10A, the central qubit 1016 can be given as the central qubit 1815, but if the RSG circuit instead generates resource state 1040 in Figure 10C, the central qubit 1815 is not given. The offset reconfigurable fusion circuit 1852 delays the -x qubit of each resource state output from the RSG circuit 1702 by one clock cycle, then passes the -x qubit through a switching circuit that can be configured with the (undelayed) +x qubit of the resource state output from the RSG circuit 1702 in the next clock cycle, and then performs a fusion operation on the two qubits output from the switching circuit. The offset reconfigurable fusion circuit 1854 delays the -y qubit of each resource state output from the RSG circuit 1702 by L clock cycle, then passes the -y qubit through a switching circuit that can be configured with the (undelayed) +y qubit of the resource state output from the RSG circuit 1702 after L clock cycle, and then performs a fusion operation on the two qubits output from 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 2 Delay by the clock cycle, then -z qubit, L2 After the clock cycle, the resource state (undelayed) +z qubit output from the RSG circuit 1702 is passed through a configurable switching circuit, and then a fusion operation is performed on the two qubits output from the switching circuit.

[0107]

[0148] In this example, the generation of resource states by the full rasterization circuit 1800 can be understood as proceeding along rows of resource states, as shown in Figure 17. The resource state generation and fusion operation between qubits of adjacent resource states using the offset reconfigurable fusion circuit 1852 proceeds along the +x direction (in the entangled geometry) with respect to the length (L) of one row of the layer. After the completion of the first row, the full rasterization circuit 1800 proceeds again along the +x direction to the next row in the +y direction, generating a second row and using the offset reconfigurable fusion circuit 1854 to perform a fusion operation between the (delayed) +y qubit from the resource state of the first row and the -y qubit from the newly generated resource state of the second row, and so on until the entire layer is generated. The process can then be repeated to generate a second layer and use the offset reconfigurable fusion circuit 1856 to perform a fusion operation between the (delayed) +z qubit from the resource state of the first layer and the -z qubit from the newly generated resource state of the second layer. Therefore, any number of layers can be generated in rasterized form. It should be understood that the term “rasterized” as used herein does not imply a specific physical arrangement of components, and the rasterizing circuit 1800 does not need to move at all to generate resource states corresponding to different locations within the layers. Instead, photons encoding qubits associated with different instances of resource state 1700 can propagate through the same set of waveguides at different times.

[0108]

[0149] Referring again to Figure 18, the switching circuits within the offset reconfigurable fusion circuits 1852, 1854, and 1856 can be controlled to give desired behavior at the array boundaries. 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 (which is 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 to the y and z dimensions. Thus, in some embodiments, the switching circuits within the offset reconfigurable fusion circuits 1852, 1854, and 1856 can be reconfigured to perform a single-qubit Z measurement on the incident qubit during a selected clock cycle (for example, by selectively coupling the input mode to the output mode which couples the input mode to the photon detector). Different behaviors can be implemented for other layer topologies, examples of which are described below. In some embodiments, the RSG circuit 1702 may be reconfigurable such that the resource state at the end of a row does not include any qubits that are not subject to fusion operations with qubits in other resource states.

[0109]

[0150] It should be understood that the circuit 1800 in Figure 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 lengths of various delay lines.) Size L 2 The layer is L (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, there are about three physical delay lines (e.g., 1, L and L). 2It should also be noted that only three optical fibers or other waveguides of a length corresponding to the clock cycle delay are required. More generally, the number of physical delay lines required for a given embodiment may depend on the specific 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, although the fully rasterized circuit requires a longer runtime to generate and operate a given number of resource states.

[0110]

[0151] Figure 19 shows a flowchart of process 1900 that can be carried out using circuit 1800 (or other circuits) of Figure 18 according to several embodiments. Process 1900 can be executed during each clock cycle while the entanglement structure is being generated, or the duration of the clock cycle can be defined according to the time consumed when executing one iteration of process 1900. In this example, we assume that each layer is generated using the RSG circuit 1702, as shown in Figure 17, by generating one row, then the next row, and so on. (It should be understood that terms such as “row,” “column,” and “layer,” as stated elsewhere in this specification, are used in reference to the entanglement shape, which does not need to correspond to the physical arrangement of qubits.)

[0111]

[0152] In block 1902, the RSG circuit 1702 (or other circuit) can 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 the layers of the entanglement structure is determined. For example, a row position counter is incremented each clock cycle to count the position within a row (e.g., 1 to L, where L corresponds to the size of the row) and is resettable at the end of each row, and a column position counter is incremented when each row is completed (e.g., every L clock cycles, or when the row position counter is reset) and may be reset when the layer is completed (e.g., after L rows have been completed). Thus, the current counter value can indicate the position of the new resource state within the layer. Other techniques can be used to define the current position in entanglement space.

[0112]

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

[0113]

[0154] In block 1916, a decision is made as to whether the current position corresponds to the beginning of a row (for example, whether the row position counter has a value of 1). If not, in block 1918, a fusion operation is performed on the second qubit of the new resource state and the qubit output from the O(1) delay line. For example, the offset reconfigurable fusion circuit 1852 can perform a fusion operation on the second qubit of 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. If the current position corresponds to the beginning of a row in block 1916, a layer edge operation can be performed on the second qubit in block 1920. In some embodiments, the layer edge operation may include performing a measurement on the second qubit to remove it from the system without disrupting the entanglement of other qubits. Other options for layer edge operation are described later.

[0114]

[0155] In block 1926, a decision is made as to whether the current position corresponds to the last row of the layer (for example, whether the column position counter has a value L). If not, in block 1928, the third qubit of the new resource state is routed to an "O(L)" delay line that imposes a delay of approximately L clock cycles, such as the delay line of the offset reconfigurable fusion circuit 1854 in Figure 18. In some embodiments, the O(L) delay line can impose a delay of exactly L clock cycles. If the current position in block 1926 corresponds to the last row of the layer, in block 1930, a layer edge operation can be performed on the third qubit. In some embodiments, the layer edge operation may include performing a measurement on the third qubit that removes the third qubit from the system without disrupting the entanglement of other qubits.

[0115]

[0156] In block 1936, a decision is made as to whether the current position corresponds to the first row of the layer (for example, whether the column position counter has a value of 1). If not, in 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 can 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 in the previous row. If the current position in block 1936 corresponds to the first row of the layer, then in block 1940, a layer edge operation can be performed on the fourth qubit. In some embodiments, the layer edge operation may include performing a measurement on the fourth qubit to remove the fourth qubit from the system without disrupting the entanglement of other qubits. Other options for layer edge operation are described later.

[0116]

[0157] In block 1946, the fifth qubit of the new resource state is L, such as the delay line of the offset reconfigurable fusion circuit 1856 in Figure 18. 2 "O(L)" imposes a delay of about a clock cycle. 2 )" can be routed to the delay line. In some embodiments, O(L 2 ) The delay line is precisely L 2 A clock cycle delay can be imposed.

[0117]

[0158] In block 1956, the sixth qubit of the new resource state and O(L 2 ) A fusion operation can be performed on the qubit output from the delay line. For example, the offset reconfigurable fusion circuit 1856 can perform a fusion operation on the second qubit of a new resource state and the O(L) of the offset reconfigurable fusion circuit 1856 during the clock cycle corresponding to the same position in the previous layer. 2A fusion operation can be performed on qubits routed to delay lines. In some embodiments, for a clock cycle corresponding to the generation of the first layer of the entangled structure, the sixth qubit may or may not undergo a different operation, such as a measurement operation that removes the sixth qubit from the system without disrupting the entanglement of the other qubits.

[0118]

[0159] Process 1900 is exemplary and can be modified and altered. For example, various decision and routing operations are shown sequentially, but 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 two systems of qubits. The specific lengths of various delay lines can be varied, and delay lines of different lengths can be used when generating different positions within the layers, depending on the desired entanglement structure. Process 1900 can be repeated over 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) may include measuring qubits at the edges (or boundaries) of a layer. In some embodiments, layer edge processing may also include performing a fusion operation or other entanglement operation on qubits at different edges of the same layer or on qubits at edges of different layers, examples of which are given below.

[0119] 2.2.4. Hybrid generation of entanglement

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

[0120]

[0161] Two different exemplary embodiments of the hybrid approach, namely, a "raster-based hybrid" circuit and a "patch-based hybrid" circuit, are described. In both implementations, a layer of resource states can be viewed as a two-dimensional array of "patches" of consecutive groupings of resource states. For example, if the layer is of size L 2 If so, the layer is size P 2 It can be considered as a two-dimensional array of patches. In raster-based hybrid methods, the number of RSG circuits N is N=L 2 / P 2 This allows each RSG circuit to provide a different patch resource state, enabling the generation of N patches in parallel, and in some embodiments, the layers are P 2 It can be completed in a clock cycle. In a patch-based hybrid method, the number of RSG circuits N is N=P 2 The RSG circuit 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, and layer generation can be completed in N clock cycles.

[0121]

[0162] Turning to raster-based hybrid circuits, Figure 20 shows a conceptual diagram of raster-based hybrid generation of entangled structures from resource states according to several embodiments. Size L 2 To support the generation of the layer, N RSG circuits 2002 are provided. In the simplified example used herein, L 2 =16 and N=4, but in reality, L 2 It can be made much larger (for example, about 10 2 , about 10 4 , about 10 6 ). N can also be made much larger (for example, about 100, about 1000), L 2 / N can be selected as desired, depending on the desired balance between hardware size and operating speed. In each clock cycle, each RSG circuit 2002 generates one instance of resource state 2000 so that a total of N resource states are generated. The number of resource states sufficient to complete the 2D layer is L 2 It can be generated in / N clock cycles. In this example, each instance of resource state 2000 is annotated with time "t=1" to "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. The same temporal fusion operation described in Section 2.2.3 above regarding the rasterization generation of the layer can be performed on qubits of adjacent resource states within the same patch of patches 2011-2014, and the further fusion operation described below can be performed on qubits of adjacent resource states across the entire patch boundary (e.g., the fusion operation shown in Figures 11A and 11B). L2 The complete layer is L 2 It can be generated in / N clock cycles. The 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 resource states 2000 of different layers using further circuits described later (as shown, for example, in Figures 12A and 12B). The three-dimensional entangled structure is L for each layer 2 The resource states can be generated by using the same N RSG circuits 2002 to repeat the process of generating resource states, and the temporal fusion operation can be performed on the qubits of the resource states 1700 in different layers using further circuits described later (e.g., the fusion operation shown in Figures 12A and 12B).

[0122]

[0163] Figure 21 shows a schematic circuit diagram of a “raster-based” hybrid unit cell 2100 for generating entanglement structures from resource states according to several embodiments. The circuit notation is the same as previously described in relation to Figures 14A to 14F, except that classical inputs and outputs are not shown for clarity in the diagram. In this example, the hybrid unit cell 2100 is 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 generates the resource state 1040 of FIG. 10C instead, 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 locally generated resource states within a 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.Routing switches 2116 to 2119 may be reconfigurable switching circuits that selectively route specific +x, -x, +y, and -y qubit resource states to one of circuits 2102, 2104 (to be used in a fusion operation with qubits of different resource states generated by the same RSG circuit 2002), or to one of fusion circuits 2112, 2114 (to be used in a fusion operation with qubits of resource states generated by adjacent instances of the hybrid unit cell 2100).

[0123]

[0164] To further illustrate the operation of routing switch 2116, FIG. 22 shows a conceptual diagram of two adjacent patches 2202, 2204 according to some embodiments. 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 size P 2A patch of =9 is generated. Each instance of resource state 2210 within patch 2202 is labeled with a directional indicator (NW, N, NE, E, SE, S, SW, W, or C) to indicate its position within the patch. The hybrid unit cell 2100 can generate resource states within patch 2202 by moving across the row below in the +x direction, then across the next row in the +y direction, and so on. Routing switches 2116-2119 can be operated so that, with respect to resource state 2210(C), all x and y qubits are routed to “local” offset reconfigurable fusion circuits 2102, 2104 so that they are fused with qubits of other local resource states generated within the hybrid unit cell 2100. In resource state 2210(E) in Figure 22, routing switches 2116-2119 can be operated so 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 unit cell 2100, while all other x and y qubits are routed to local fusion circuits 2102, 2104. In resource state 2210(NE) in Figure 22, routing switches 2116-2119 can be operated so that the +x and +y qubits are routed to the networked fusion circuits 2112, 2114 to be fused with the resource state qubit from an adjacent instance of unit cell 2100, while the -x and -y qubits are routed to local fusion circuits 2102, 2104. Similar logic can be applied to other instances of resource state 2210 and extended to patches of any size. In this example, a routing switch for the z qubit is not required because a given instance of the unit cell 2100 generates the same patch within each layer, and the +z and -z qubits can always be routed to the offset reconfigurable fusion circuit 2106. It should be understood that this form is not necessary, and other embodiments of the hybrid unit cell may include a routing switch for the z qubit.

[0124]

[0165] In the implementation of the hybrid unit cell 2100 shown in Figure 21, qubits given to (or received from) an adjacent unit cell are not affected by the delay circuit. Therefore, it may be desirable to adjust the order in which resource states are generated in different unit cells so that resource states with qubits given to an adjacent unit cell as input to the networked fusion circuits 2112, 2114 are generated in the same clock cycle as adjacent resource states. Figure 23 shows an example of the adjusted order of resource state generation for different patches 2301-2304 according to several embodiments. In this example, each patch 2301-2304 is 4x4 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 from the figure, in all instances where a resource state in one patch is given to a networked fusion circuit associated with an adjacent patch, both resource states (or all four central resource states where all patches 2301-2304 are adjacent) are generated in the same clock cycle. Therefore, position-dependent delays are not required to perform fusion operations on resource state qubits generated in different patches. This principle can be extended to P×P patches of any value 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] Figure 24 shows a flowchart of a process that can be carried out using the hybrid unit cell 2100 of Figure 21 or a similar circuit according to several embodiments. Process 2400 can be performed by each hybrid unit cell 2100 in each clock cycle while the entangled structure is being generated, in which case different hybrid unit cells 2100 operate in parallel. In this example, we assume that the hybrid unit cells 2100 are used to generate layers of the entangled structure, and each hybrid unit cell generates a continuous patch having dimensions P×P within each layer. Each hybrid unit cell generates its patch by generating one row, then the next row, and so on (for example, as shown for each of patches 2301-2304 in Figure 23). (As stated elsewhere in this specification, terms such as “row,” “column,” and “layer” should be understood to be used in reference to the entangled space, which does not need to correspond to the physical arrangement of qubits or hybrid unit cells.)

[0126]

[0167] In block 2402, the RSG circuit 2002 (or other circuit) can 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 position of the new resource state in the patch being generated by the hybrid unit cell is determined. For example, a row position counter is incremented per clock cycle to count the position in a row (e.g., 1 to P, where P corresponds to the size of the row in the patch) and 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 reset when the patch is completed (e.g., after P rows have been completed). Thus, the current counter value can indicate the position of the new resource state in the patch. Other techniques can be used to define the current position in entangled space.

[0127]

[0168] In block 2406, a decision is made as to whether the current position corresponds to the end of a patch row (for example, whether the row position counter has a value P). If not, in block 2408, the first qubit of the new resource state is routed to an O(1) delay line that imposes a delay of approximately one clock cycle, such as the delay line of the offset reconfigurable fusion circuit 2102 in Figure 21. In some embodiments, the O(1) delay line can impose a delay of exactly one clock cycle. If the current position in block 2406 corresponds to the end of a patch row, in block 2410, the first qubit can be routed to the first adjacent unit cell (for example, by the operation of switch 2117 in Figure 21).

[0128]

[0169] In block 2416, a decision is made as to whether the current position corresponds to the beginning of a row in the patch (for example, whether the row position counter has a value of 1). If not, in block 2418, a fusion operation is performed on the second qubit of the new resource state and the qubit output from the O(1) delay line (which may be a qubit routed to the O(1) delay line during the previous clock cycle), for example, using the offset reconfigurable fusion circuit 2102 in Figure 21. If the current position in block 2416 corresponds to the beginning of a row, in block 2420, a fusion operation can be performed on the second qubit of the new resource state and the first networked qubit received from the second neighboring unit cell. Assuming the second neighboring unit cell is also performing process 2400, the first networked qubit may be a qubit routed from the second neighboring unit cell according to block 2410.

[0129]

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

[0130]

[0171] In block 2436, a decision is made as to whether the current position corresponds to the first row of the patch (for example, whether the column position counter has a value of 1). If not, in block 2438, a fusion 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 a qubit routed to the O(P) delay line during the clock cycle corresponding to the position of the previous row). If the current position in block 2436 corresponds to the first row of the patch, in block 2440, a fusion operation can be performed on the fourth qubit of the new resource state and the second networked qubit received from the fourth neighboring unit cell. Assuming that the fourth neighboring unit cell is also performing process 2400, the second networked qubit can be a qubit routed from the fourth neighboring unit cell according to block 2430.

[0131]

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

[0132]

[0173] In block 2456, the sixth qubit of the new resource state and O(P 2 )The qubit output from the delay line (O(P) during the clock cycle corresponding to the position of the previous layer 2 A fusion operation can be performed on the qubit (which may be a qubit routed to a delay line). In some embodiments, for a clock cycle corresponding to the generation of the first layer of the entangled structure, the sixth qubit may or may not undergo a different operation, such as a measurement operation that removes the sixth qubit from the system without breaking the entanglement of the other qubits.

[0133]

[0174] Process 2400 is illustrative and can be modified and altered. For example, various decision and routing operations are shown sequentially, but 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 two systems of qubits. The specific lengths of various delay lines can be varied, and delay lines of different lengths can be used when generating different positions within the layers, depending on the desired entanglement structure. Process 2400 can be repeated over any number of clock cycles to generate an entanglement structure having any number of layers of any desired size. Also, process 2400 is described assuming that the unit cell execution process 2400 has four adjacent unit cells. However, this does not have to be true for all unit cells (or actually any unit cells). Therefore, in any instance where process 2400 shows performing operations involving routing qubits to adjacent unit cells or networked qubits received from adjacent unit cells, if suitable adjacent unit cells are not available, the layer edge processing can be replaced, for example, as described above in relation to Figure 19 or in the following examples.

[0134]

[0175] As described above, in the "patch-based" hybrid circuit, the number N of RSG circuits is N = P 2 and, in a single clock cycle, P 2 The resource states generated by the RSG circuits can form patches of size P 2 within a layer of size L 2 (continuous). FIG. 25 is a conceptual diagram of hybrid generation of a layer for a tangled structure using a patch-based hybrid circuit according to some embodiments. To support the generation of a layer of size L 2 RSG circuits 2002 with a number N = P 2 are provided. In the simplified example used herein, L 2 = 16 and N = 4, but in practice, L 2 can be much larger (e.g., about 10 2 , about 10 4 , about 10 6 ). Also, N can be much larger (e.g., about 100, about 1000), and P 2 can be selected as desired according to the desired balance between hardware size and operating speed. In each clock cycle, each RSG circuit 2502 generates one resource state 2500. (In FIG. 25, each resource state 2500 is annotated at times "t = 1" to "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. The spatial fusion operation can be performed on the qubits of adjacent resource states 2500 within the patch (e.g., as shown in FIGS. 11A and 11B) using additional circuitry that can be the same or identical to the fully networked circuits 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, size L 2By forming layers, further temporal fusion operations can be performed on qubits belonging to resource states within different patches. An example of a circuit that performs a 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 it is not expected that a given instance of the RSG circuit will generate a desired resource state in every clock cycle. Therefore, instead of a dedicated RSG circuit for each hybrid unit cell, some embodiments can provide a number (M) of RSG circuits, where M > N, and M is selected to give a sufficiently high probability that at least N resource states will be generated in a given clock cycle (the "sufficiently high probability" in a given embodiment can be determined based on a particular embodiment of fault tolerance). An example of this is using active multiplexing techniques known in the art, in which N M RSG circuits can be selected in each clock cycle to send resource states to N different instances of the switching and fusion circuits of the hybrid unit cell. Thus, each hybrid unit cell may, but is not required to, have its own dedicated instance of the RSG circuit.

[0136]

[0177] It should be understood that an entanglement structure of any size can be generated using an array of hybrid unit cells as shown in Figure 21. (In some embodiments, the size may be fixed in the hardware design.) Layer size (L 2 A different selection of the number of RSG circuits (N) relative to ) results in different computation times, allowing for selection to achieve the desired balance between hardware size and computation speed.

[0137]

[0178] The aforementioned examples of entanglement generation circuits and processes are illustrative and can be modified as needed. The use of directional labels (e.g., x, y, z, NE, SE, SW, NW, etc.) is for illustrative purposes only and should be understood as referring to the entanglement space, without requiring or implying a specific physical arrangement of components or physical qubits. All numerical embodiments are illustrative and can be modified. Furthermore, while layers and patches are described in relation to the number of squares, it should be understood that non-square layers and / or non-square patches can also be used. For example, a patch or layer may be rectangular. Triangular patches or layers (or patches or layers with other shapes) can also be generated, for example, by varying the number of resource states per row. Furthermore, while the aforementioned examples assume that all instances of resource states have the same entanglement pattern, such uniformity is not required. For example, in some embodiments, an RSG circuit may be reconfigurable to generate resource states with different entanglement patterns at different clock cycles. In addition, an RSG circuit can operate non-deterministically, which can result in stochastic variation between resource states.

[0138] 3. Interleaving 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 the 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 LES as described above in relation to Figure 13. Figure 26 shows a time diagram generating a photonic LES according to some embodiments. The photonic LES in this example is simplified but is similar to an LES that can be used as a logical qubit. Figure 26 should be understood as a diagram in entangled space. For clarity of explanation, only the y ("space") and z ("time") dimensions are shown, as 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, a time step of duration τ is defined, and for example, the time step can correspond to a clock cycle (or the 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 exist at multiple positions along a given waveguide. Thus, Figure 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 a complete layer 2603 of resource states (in time step 2602). In this example, it is assumed that the resource state 2603 includes a central qubit that forms the LES. In some embodiments, a fully networked circuit can be used (e.g., as described in Section 2.2.2), and the time step τ can correspond to a clock cycle. In other embodiments, a rasterized or hybrid network / rasterized circuit can be replaced (e.g., as described in Sections 2.2.3 and 2.2.4), and the time step τ corresponds to a layer (e.g., L 2 Clock cycle or L 2This can correspond to the time required to generate all resource states in terms of / N clock cycles. In time step 2604, a fusion operation is performed, which includes a spatial fusion operation 2606 for adjacent physical qubits in the y dimension (x dimension (not shown)) and a temporal fusion operation 2608 for fusing adjacent qubits in consecutive layers. Optionally, the detector 2610 can be applied to the edge to perform a Z measurement on the peripheral qubits of the resource state at the layer boundary, thereby removing it from the system. In time step 2612 (and any number of subsequent time steps), the LES can be held in abeyance for subsequent operations. In the illustrated example, subsequent operations include a measurement operation on the qubits of the LES using the detector 2614, but subsequent operations performed on the LES may be independent of how the LES is generated, and the LES generated in the manner shown in Figure 26 can be used for various operations.

[0141]

[0182] Figure 27 is a simplified conceptual diagram of a linear optical circuit that implements the behavior of Figure 26 according to several embodiments. For clarity, only the y ("space") and z ("time") axes are shown, but it should be understood that each layer can be two-dimensional (in entangled space). At time t=0, each resource state generator 2702 outputs a resource state 2704, as previously described, for example. 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. The entanglement is shown by curves connecting the qubits, while straight lines represent waveguides (or groups of waveguides to which each qubit is encoded). (Although not illustrated, it should be understood that resource states 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 the 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 boundary 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 qubits of the LES can propagate through further delay circuits 2714 and eventually reach detector 2720 (or another subsequent operation). Depending on the lifetime of the desired LES, any number of delay circuits 2714 can be introduced.

[0142] 3.2 Temporal interleaving for generating multiple entanglement structures

[0183] In the examples of Figures 26 and 27, a single LES is generated using the circuit shown in Figure 27. Although only a single two-dimensional portion of the LES is shown in Figures 26 and 27, those skilled in the art who benefit from this disclosure will understand that a system including further rows of RSG circuits, which may be arranged in the x-direction (in and out of the page), can generate a three-dimensional LES that can be used for fault-tolerant quantum computing. Furthermore, the spatial fusion shown in Figures 26 and 27 can be replaced with temporal fusion, and the aforementioned rasterizing and hybrid circuits can also be used to generate the LES.

[0143]

[0184] In some cases, it may be desirable to use the same circuit to provide multiple temporally coexisting entanglement structures (including, but not limited to, LES) (meaning that photons from both entanglement structures are simultaneously flying, for example, within one or more delay lines). According to some embodiments, the coexistence of multiple entanglement structures can be achieved by "interleaving" the generation of different layers of entanglement structures.

[0144]

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

[0145]

[0186] Interleaved generation of two LES may involve generating layers of both LES, for example, alternately, using the same hardware. In some embodiments, layer generation hardware 2810 (which can be implemented using various circuits as described above) can be used to generate layers 2802a or 2802b in each of a series of time intervals. Entanglement can be generated between layers generated during alternating periods (by performing the fusion operation described above or other entanglement generation operation), as shown by the dotted arc 2815, but not between layers generated during consecutive periods. The result is identical to LES 2804a, 2804b with respect to the entanglement topology, as shown by the mapping arrow 2817.

[0146]

[0187] Figure 29 is a time-lapse diagram showing the generation of two interleaved LES (and optionally, the entanglement of the two interleaved LES at their boundaries) using a single set of resource state generators and downstream circuits according to several embodiments. Figure 29 is similar in many ways to Figure 26. For example, although only the y and z dimensions are shown, it should be understood that each layer of the LES can be two-dimensional (in entanglement space). Similar to Figure 26, Figure 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 a complete set of resource states for the LES layer (at time step 2902). As in Figure 26, the resource states of the layer can be generated using various techniques, and the time step τ can be defined accordingly. At time step 2904, a spatial fusion 2906 occurs to fuse adjacent physical qubits in the y dimension (x dimension (not shown)).

[0148]

[0189] Unlike Figure 26, in this example, resource states generated in alternating time steps are associated with two different LESs. To show the association between qubits and LESs, the qubits are color-coded (gray circles for qubits associated with LES A, and white for qubits associated with LES B). Thus, the temporal fusion 2908 fuses two qubits from resource states generated two time steps apart. At the edge of the layer, boundary qubits can be removed using detector 2910. Alternatively, the fusion circuit 2912 can "stitch" the LESs together at the boundary by fusing the peripheral qubits of the LES B layer with the previously generated peripheral qubits of the LES A layer, as described below. In time step 2914 (and any number of subsequent time steps), the LES persists until subsequent operations, including measurements using detector 2916 in this example.

[0149]

[0190] Figure 30 shows a simplified conceptual diagram of a linear optical circuit that implements the behavior of Figure 29 according to several embodiments, using the same notation as in Figure 27. At time t=0, the resource state generator 3002 outputs a resource state 3004, for example, as previously described. In this example, each resource state 3004 is shown as having 5 qubits, including one propagating central qubit and peripheral qubits associated with the +y, -y, +z, and -z dimensions. (It should be understood that, although not shown, resource states 2704 may also include peripheral qubits associated with the +x and -x dimensions.) Between times t=0 and t=τ, the fusion circuit 3006 performs a fusion operation on the peripheral qubits of adjacent resource states along the y dimension, and the delay circuit 3008 delays the -z peripheral qubit of each resource state by one time step. Between times t=τ and t=2τ, the second delay circuit 3008' delays the -z peripheral qubit of each resource state by another time step.

[0150]

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

[0151]

[0192] After time t=3τ, the physical qubits constituting the two LES can propagate through further delay circuits 3014 and eventually reach the detector 3020 (or some other subsequent operation). Depending on the lifetime of the desired LES, any number of delay circuits 3014 can be introduced.

[0152]

[0193] In some embodiments, a configurable boundary circuit 3030, shown to operate 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) which can direct a qubit to either a detector 3034 or an offset reconfigurable fusion circuit 3036. With respect to 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 the first period, the peripheral qubit associated with the layer of one LES (LES A in this example) is delayed by the delay circuit 3038, and during the next period, the peripheral qubit associated with the layer of the other LES (LES B in this example) is received, and the offset reconfigurable fusion circuit 3036 performs a fusion operation on the received qubit and the delayed qubit. The operation performed by the offset reconfigurable fusion circuit 3036 is also called “boundary stitching.” In some embodiments, boundary stitching can be used to stitch together patches generated over different time periods (for example, patches generated using the patch-based hybrid method in Figure 25) to form a larger layer.

[0153]

[0194] It should be understood that these examples are illustrative and not limiting. Interleaving techniques are 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 having larger layer and / or non-planar layer topology, 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 networking, rasterizing, or hybrid methods described above, and the same RSG circuit can be used to generate the resource states for all interleaved entangled structures. In some embodiments, the RSG circuit may be reconfigurable so that different entangled structures or different layers within a single entangled structure can have different entangled geometries from one another. Furthermore, if interleaving generates multiple entanglement structures, different, simultaneously existing entanglement structures can be selectively entangled with each other using additional circuits.

[0154] 3.3 Grid surgery

[0195] In addition to or instead of interleaving multiple LESs, configurable boundary circuits 3030 and similar circuits may be able to construct 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, switch 3032 can be dynamically configured for each pair of periods to support coupling (or lack thereof) between layers, also called “boundary stitching.” As an example, Figure 31 shows a conceptual diagram of two temporally coexisting LESs 3102, 3104. As shown in Figures 26 and 27, only the y-dimension (vertical axis labeled as “space”) and the z-dimension (horizontal axis labeled as “time”) are shown, but it should be understood that each LES can be three-dimensional. The first LES 3102 and the second LES 3104 overlap in time. The layers of LES 3102 and 3104 (shown as columns, as only the y-dimension is shown) can be temporally offset from each other, as indicated by the time offset of the physical qubits. For example, the layers can be generated using interleaving techniques. In some embodiments, the time offset can be formed by generating the physical qubits of LES 3102 and 3104 during alternating periods τ. Thus, as previously mentioned, both LES can be generated using the same hardware. In the example shown in Figure 31, a first column of LES 3102 can be generated and its photons sent to a delay line. Next, a first column of LES 3104 can be generated and sent to a different (or the same) delay line. Then, a second column of LES 3102 can be generated and subsequently merged with the first column of LES 3102 (but not merged with the first column of LES 3104) that was stored in the delay line, and so on. Figures 31-33 show LES 2152 and LES 3104 offset from each other in the y-direction, and it can be seen that interleaving allows the same set of physical resource state generators to generate the resource states necessary to generate each LES, for example, in alternating clock cycles.

[0155]

[0196] In some embodiments, LES 3102 and 3104 can be joined together to form a single LES with a larger layer, for example. For example, Figure 32 shows a conceptual diagram of "stitching" LES 3102 and 3104 at the boundary to form a single LES with a larger layer size by, for example, 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] Figure 33 shows a conceptual diagram of selective lattice surgery, in which LES 3102 and 3104 are selectively entangled along the boundaries of some layers but not along the boundaries of other layers. Such a form can be generated by controlling the configurable boundary circuit 3030 on a clock cycle basis.

[0157]

[0198] In a scenario where LES 3102 and 3104 are three-dimensional LES representing different logic qubits, a two-qubit logic gate can be implemented between the logic qubits encoded within LES 3102 and 3104 using the lattice surgery disclosed herein. When it is necessary to apply a gate between interleaved logic qubits, an appropriate lattice surgery can be applied by changing the type of resource state generated or by changing the type of measurement performed on the individual physical qubits of the LES. Other applications of lattice surgery can also be envisioned. In some embodiments, the fusion circuit at the boundary may be reconfigurable to change the type of lattice surgery operation.

[0158]

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

[0159] 3.4. Interleaving for constructing hierarchical topology

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

[0160]

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

[0161]

[0202] Therefore, in some embodiments, the folded entanglement structure in Figure 34C (or Figure 32) can be understood as a single layer of entanglement structure generated using a patch-based hybrid raster / networked RSG circuit, similar to the example described in relation to Figure 25. For example, in the embodiment described in relation to Figure 25, P 2 The RSG circuit set is P in one clock cycle 2 It is possible to generate patches of consecutive resource states. In some embodiments, patches generated during different clock cycles can be stitched together at the boundary, and using interleaving techniques, larger layers can be formed, as shown in Figures 34C and 34D. In the examples shown in Figures 34C and 34D, the size of each patch is L × (L / 2). However, smaller patches can be used. Given a fusion circuit to perform stitching between patches along both spatial boundaries, the patch size may be less than L in both (spatial) dimensions. Furthermore, if there are three or more patches per layer, the delay associated with the fusion operation between qubits in different layers can be appropriately adjusted by taking into account the number of patches per layer.

[0162]

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

[0163]

[0204] Figures 36A to 36D are conceptual diagrams of using a folding technique to produce more complex periodic boundary conditions for layers of entangled structures according to several embodiments. Figure 36A shows a layer 3600 of an entangled structure having boundaries 3602, 3603, 3604, and 3605, and this layer 3600 of the structure can be folded to form a layer of entangled structures having a toroidal topology. Specifically, as shown in Figure 36B, boundaries 3604 and 3605 are joined to each other (similar to the cylindrical topology in Figure 35A), and as shown in Figure 36C, boundaries 3602 and 3603 are also joined to each other, thereby forming a torus. Figure 36D shows an interleaving technique that can be used to form a layer having a toroidal topology by selectively joining boundaries along different dimensions of the layer. As in Figure 34C, time progresses along the z-axis (perpendicular to the page), and the layer is shown as a rectangle in the xy-plane. Four layers 3621, 3622, 3623, and 3624 are generated. At the boundaries, layers are stitched together (for example, using temporal fusion). Specific 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 a variable delay 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, as shown in Figure 30.

[0164]

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

[0165]

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

[0166] 4. Implementation of Quantum Computing Operations

[0207] Quantum computing operations using the entanglement structures generated as described above can be implemented using various techniques. One approach is to modify the resource state (and thus the entanglement geometry) based on the computation being performed. For example, resource states at different positions within a 2D layer can be generated with different entanglement geometries. In some embodiments, the RSG circuit can be dynamically reconfigurable to be able to generate resource states with different entanglement geometries.

[0167]

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

[0168] 5. Examples of Quantum Computer Systems

[0209] FIG. 38 shows an example of a system architecture for a quantum computer system 3800 that can implement MBQC or FBQC according to some embodiments. Using photonic physical qubits, some embodiments of the quantum computer system 3800 can generate a fault-tolerant cluster state 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 the entanglement structure 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 can include a single instance or multiple instances of the aforementioned resource state generator circuits. The RSG circuit can operate autonomously without requiring a data input, and each RSG circuit can generate one resource state per clock cycle (which can be, for example, about 1 ns or more). It can generate any of the aforementioned resource states or other resource states. The resource state can 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 circuit. Also, the resource state generator unit 3802 can transmit a classical data output (e.g., indicating the success or failure of various elements of the resource state generation process) to the classical processing unit 3810 via a data path 3822. In some embodiments, the resource state generator unit 3802 can be maintained at cryogenic temperatures (e.g., 4K). The delay circuit 3804 can provide an appropriate delay time, such as the aforementioned 1 clock cycle, L clock cycles, and L, for photons corresponding to an optical fiber, other waveguide, optical memory, or a particular qubit. 2Other components may be included to delay by the clock cycle delay time. As mentioned above, in some embodiments, only one delay line for each duration is required to carry out the rasterization generation of logic qubits. The delay circuit 3804 does not need to operate at cryogenic temperatures. Photons exiting the delay circuit 3804 can be delivered to the switch circuit 3806 via waveguide 3824, which may be an optical fiber, an on-chip waveguide, or any other type of waveguide.

[0170]

[0211] The switch circuit 3806 may include active switches and waveguides for performing mode coupling, mode swapping, and phase shift operations on qubits. In various embodiments, the switch circuit 3806 can perform mode coupling operations related to fusion operations (e.g., the Type II fusion operation described above in relation to Figure 9A), and / or basis selection operations related to the measurement of 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 fusion circuit used in the above examples. The switch circuit 3806 delivers output photons to the detector 3808 via waveguide 3828, which may be optical fiber, on-chip waveguide, or any other type of waveguide.

[0171]

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

[0172]

[0213] The classical processing unit 3810 can be a classical computer system capable of communicating 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 can determine the appropriate settings of the switch circuit 3806 based on a specific quantum computation (or program) being performed. The classical processing unit 3810 can receive feedback signals (e.g., measurement results) from the resource state generator 3802 and the detector 3808, and can determine the results of the computation based on the feedback signals. In some embodiments, the classical processing unit 3810 can use the feedback signals to correct subsequent control signals sent to the switch circuit 3806. The operation of the classical processing unit 3810 can incorporate error correction algorithms and other techniques.

[0173]

[0214] The system 3800 in Figure 38 is illustrative and can be modified and altered. Blocks shown separately can be combined, or a single block can be implemented using multiple distinct components. The resource state generator 3802, delay circuit 3804, switch circuit 3806, and detector 3808 can implement the aforementioned circuits for generating entanglement structures. For example, the delay circuit 3804 can implement all of the delay line portion of the offset reconfigurable fusion circuit described above, the switch circuit 3806 can implement reconfigurable switches and mode couplers related to reconfigurable fusion, and the detector 3808 can implement destructive measurements related to fusion operations. In some embodiments, generating entanglement structures can include generating LES that can perform measurements on individual qubits to implement MBQC. In other embodiments, generating entanglement structures can include performing fusion operations on resource state qubits (e.g., as described above) using measurement results obtained from fusion operations provided to the classical processing unit 3810, thereby implementing FBQC.

[0174]

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

[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 in which a large entanglement structure may be desirable. The size and geometric shape of the entanglement structure can be varied depending on the specific use case. For example, the above description uses an example of an entangled structure from layers that are two-dimensional (in entanglement space), but layers can have more dimensions. Furthermore, the embodiments described herein include references to specific materials and structures (e.g., optical fibers), but other materials and structures capable of generating, propagating, and operating photons can be substituted.

[0176]

[0217] It should be understood that all numerical values ​​used herein are illustrative and subject to change. In some cases, ranges are defined to give a sense of scale, but numbers outside the disclosed range are not excluded.

[0177]

[0218] It should also be understood that all figures in this specification are intended as schematic diagrams. Unless otherwise specifically indicated, the drawings are not intended to imply a specific physical arrangement of the elements shown therein, or that all elements shown are necessary. A person skilled in the art who has access to this disclosure will understand that elements shown in the drawings or otherwise described in this disclosure may be modified or omitted, and other elements not shown or described may be added.

[0178]

[0219] This disclosure provides a description of the invention as claimed in relation to a particular embodiment. A person skilled in the art who accesses this disclosure will see that the embodiments do not encompass the scope of the invention as claimed, and that the scope of the invention as claimed extends to all variations, modifications, and equivalents.

Claims

1. A circuit for routing qubits, wherein the circuit is A first unit cell circuit for receiving a first set of resource states, wherein the first unit cell circuit is A first set of local delays for receiving qubits from different resource states of the first plurality of resource states, A first set of networked delays connected to adjacent unit cell circuits, wherein the adjacent unit cell circuits are adjacent based on resource states received by adjacent unit cell circuits that are adjacent to one or more of the first plurality of resource states in a layer of entangled space, A first unit cell circuit including a first plurality of switches for routing qubits from the first plurality of resource states to a first set of local delays or a first set of networked delays, A second unit cell circuit for receiving a second set of resource states, wherein the second unit cell circuit is A second set of local delays for receiving qubits from different resource states of the second set of resource states, A second set of networked delays connected to an adjacent unit cell circuit, wherein the adjacent unit cell circuit is adjacent to the adjacent unit cell circuit that is adjacent to one or more of the second plurality of resource states in the layer of the entangled space, A second unit cell circuit including a second plurality of switches for routing qubits from the second plurality of resource states to a second set of local delays or a second set of networked delays, A control circuit, wherein the control circuit is Control the first set of switches to route qubits from the first set of resource states to a first set of local delays or a first set of networked delays, according to the location of each of the first set of resource states in the layer of the entangled space. A control circuit controls the second set of switches to route qubits from the second set of resource states to the second set of local delays or the second set of networked delays, according to the location of each of the second set of resource states in the layer of the entangled space. A circuit for routing qubits, equipped with [a specific feature / feature].

2. The circuit according to claim 1, wherein at least some of the first set of networked delays of the first resource state circuit are connected to the second unit cell circuit.

3. The circuit according to claim 2, wherein the control circuit routes qubits from the first plurality of resource states to the second unit cell circuit based on the first plurality of resource states and the second plurality of resource states having adjacent resource states in the layer of the entangled space.

4. The circuit according to claim 2, wherein the control circuit routes qubits from the resource states among the first plurality of resource states to a first set of local delays based on resource states among the first plurality of resource states adjacent to further resource states of the first plurality of resource states in the layer of the entangled space.

5. The circuit according to claim 1, wherein the first set of local delays includes delays of different lengths.

6. The circuit according to claim 5, wherein the first set of local delays includes three subsets of local delays of different lengths, each subset including a first subset of local delays, a second subset of local delays longer than the first subset of local delays, and a third subset of local delays longer than the second subset of local delays.

7. The circuit according to claim 1, wherein the second set of local delays includes delays of different lengths.

8. The circuit according to claim 7, wherein the second set of local delays includes three subsets of local delays of different lengths, comprising a first subset of local delays, a second subset of local delays longer than the first subset of local delays, and a third subset of local delays longer than the second subset of local delays.

9. The circuit according to claim 1, wherein the first set of networked delays includes delays of different lengths.

10. The circuit according to claim 9, wherein the first set of networked delays includes three subsets of networked delays of different lengths, the first subset of networked delays includes a first subset of networked delays, a second subset of networked delays longer than the first subset of networked delays, and a third subset of networked delays longer than the second subset of networked delays.

11. A step of providing a first set of resource states to a first unit cell circuit, The first unit cell circuit includes a first set of local delays for receiving qubits from different resource states of the first plurality of resource states, The first unit cell circuit further includes a first set of networked delays connected to adjacent unit cell circuits, The adjacent unit cell circuits are adjacent based on the resource states output to the adjacent unit cell circuits that are adjacent to one or more of the first plurality of resource states in the layer of entangled space, The first unit cell circuit further includes a first plurality of switches for routing qubits from the first plurality of resource states to a first set of local delays or a first set of networked delays, step, A step of providing a second set of resource states to a second unit cell circuit, The second unit cell circuit includes a second set of local delays for receiving qubits from different resource states of the second plurality of resource states, The second unit cell circuit further includes a second set of networked delays connected to an adjacent unit cell circuit. The adjacent unit cell circuits are adjacent based on the resource states from adjacent unit cell circuits that are adjacent to one or more of the second plurality of resource states in the layer of the entangled space, The second unit cell circuit further includes a second plurality of switches for routing qubits from the second plurality of resource states to a second set of local delays or a second set of networked delays, step, The steps of routing qubits from the first plurality of resource states to a first set of local delays or a first set of networked delays, according to the position of each of the first plurality of resource states in the layer of the entangled space, using the first plurality of switches controlled by one or more classical processing units, The steps include: routing qubits from the second plurality of resource states to a second set of local delays or a second set of networked delays, according to the position of each of the second plurality of resource states in the layer of the entangled space, using the second plurality of switches controlled by one or more classical processing units; Methods that include...

12. The method according to claim 11, wherein at least some of the first set of networked delays of the first resource state circuit are connected to the second unit cell circuit.

13. The method according to claim 12, wherein routing qubits from the first plurality of resource states to the second unit cell circuit is based on the first plurality of resource states and the second plurality of resource states having adjacent resource states in the layer of the entangled space.

14. The method according to claim 12, wherein routing qubits from a resource state among the first plurality of resource states to a first set of local delays is based on a resource state among the first plurality of resource states adjacent to a further resource state of the first plurality of resource states in the layer of the entangled space.

15. The method according to claim 11, wherein the first set of local delays includes delays of different lengths.

16. The method according to claim 15, wherein the first set of local delays includes three subsets of local delays of different lengths, each subset including a first subset of local delays, a second subset of local delays longer than the first subset of local delays, and a third subset of local delays longer than the second subset of local delays.

17. The method according to claim 11, wherein the second set of local delays includes delays of different lengths.

18. The method according to claim 17, wherein the second set of local delays includes three subsets of local delays of different lengths, comprising a first subset of local delays, a second subset of local delays longer than the first subset of local delays, and a third subset of local delays longer than the second subset of local delays.

19. The method according to claim 11, wherein the first set of networked delays includes delays of different lengths.

20. The method according to claim 19, wherein the first set of networked delays includes three subsets of networked delays of different lengths, the first set of networked delays includes a first subset of networked delays, a second subset of networked delays longer than the first subset of networked delays, and a third subset of networked delays longer than the second subset of networked delays.

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