N-way fusion networks based on fusion complexes

Abstract

A system having reconfigurable fusion circuits can receive, obtain, or generate resource states, where each resource state includes multiple qubits that are entangled with each other and qubits of different resource states are not entangled with each other. Qubits of the resource states can be routed to the reconfigurable fusion circuits such that each fusion circuit receives a qubit from each of three or more of the resource states. The reconfigurable fusion circuits can selectably perform one of a number of measurement operations on the received qubits to produce measurement data. At least some of the selected measurement operations can be n-way fusion operations on all of the received qubits.

Description

N-WAY FUSION NETWORKS BASED ON FUSION COMPLEXESCROSS-REFERENCES TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Provisional Application No. 63 / 531,805, filed August 10, 2023, the disclosure of which is incorporated by reference.BACKGROUND

[0002] Quantum computing is distinguished from “classical” computing by its reliance on structures referred to as “qubits.” At the most general level, a qubit is a quantum system that can exist in one of two orthogonal states (denoted as |0) and |1) in the conventional bra / ket notation) or in a superposition of the two states (e.g., -= (10) + 11)). By operating on aV2 system (or ensemble) of qubits, a quantum computer can quickly perform certain categories of computations that would require impractical amounts of time in a classical computer.

[0003] Practical realization of a quantum computer, however, remains a daunting task, in part because the physical systems implementing qubits and operations on qubits are often non-determini stic and / or noisy. For example, some techniques for creating entanglement between qubits succeed non-deterministically, with a probability of success that may be considerably less than 1. In addition or instead, the physical systems may be noisy (e.g., affected by interactions with the environment or with physical media in which the qubits are contained). For reasons such as these, fault tolerant quantum computing is a desirable goal.SUMMARY

[0004] According to some embodiments, fusion-based quantum computations can be implemented using a network (also referred to as a network array) of interleaving modules. Each interleaving module can receive or produce resource states consisting of entangled physical qubits and can include a set of reconfigurable fusion circuits that can be controlled to perform either fusion operations or single qubit measurements on sets of three or more qubits from different resource states, routing paths connected to the reconfigurable fusion circuits, and delay lines and routing switches that operate to select routing paths for qubits ofthe resource states, thereby implementing a desired combination of fusion operations and single qubit measurements.

[0005] Certain embodiments relate to methods that can include: providing, to a system having a plurality of reconfigurable fusion circuits, a plurality of resource states, each resource state including a plurality of qubits that are entangled with each other, wherein qubits of different resource states are not entangled with each other; routing qubits of the resource states to the reconfigurable fusion circuits such that each fusion circuit receives a qubit from each of three or more of the resource states; and performing, by each reconfigurable fusion circuit, a selected one of a plurality of measurement operations on the received qubits to produce measurement data, wherein at least some of the selected measurement operations are three-way fusion operations.

[0006] Certain embodiments relate to circuits that can include: a plurality of resource state interconnects, each resource state interconnect having a plurality of output paths to output a resource state during each of a plurality of operating cycles, wherein each resource state is a quantum system of multiple entangled qubits, wherein different qubits of the resource state are output on a different ones of the output paths, and wherein different resource state interconnects are associated with different directions in a multi-dimensional entanglement space; a plurality of first routing switches, each of the first routing switches having an input path coupled to a different one of the output paths of the resource state interconnects and a plurality of output paths, wherein each of the first routing switches is configured to receive a different qubit of one of the resource states on the input path and to selectably route the received qubit to one of the plurality of output paths; a plurality of routing paths including a plurality of local routing paths, a plurality of outbound network routing paths, and a plurality of inbound network routing paths, wherein each of the outbound network routing paths exits the circuit, and wherein each of the inbound network routing paths enters the circuit from an external source, wherein at least some of the routing paths include delay lines having different delay lengths; a plurality of reconfigurable fusion circuits, each of the plurality of reconfigurable fusion circuits being configured to receive three input qubits on three input paths and to selectably perform one of a plurality of measurement operations on the input qubits, thereby producing measurement outcome data, wherein the plurality of measurement operations including a projective entangling measurement between three input qubits and a plurality of single-qubit measurements on each of the three input qubits, wherein the local routing paths are coupled between the routing switches and the reconfigurable fusion circuitssuch that each of the routing switches is coupled to at least one of the reconfigurable fusion circuits; and a plurality of second routing switches, each of the second routing switches having a first input path coupled to one of the local routing paths, a second input path coupled to one of the inbound network routing paths, and an output path coupled to one of the input paths of one of the reconfigurable fusion circuits.

[0007] Certain embodiments relate to systems that can include a network of interleaving modules, wherein each interleaving module includes: a plurality of resource state interconnects, each resource state interconnect having a plurality of output paths to output a resource state during each of a plurality of operating cycles, wherein each resource state is a quantum system of multiple entangled qubits, wherein different qubits of the resource state are output on a different ones of the output paths, and wherein different resource state interconnects are associated with different directions in a multi-dimensional entanglement space; a plurality of first routing switches, each of the first routing switches having an input path coupled to a different one of the output paths of the resource state interconnects and a plurality of output paths, wherein each of the first routing switches is configured to receive a different qubit of one of the resource states on the input path and to selectably route the received qubit to one of the plurality of output paths; a plurality of routing paths including a plurality of local routing paths, a plurality of outbound network routing paths, and a plurality of inbound network routing paths, wherein each of the outbound network routing paths exits the interleaving module, and wherein each of the inbound network routing paths enters the interleaving module from another interleaving module, wherein at least some of the routing paths include delay lines having different delay lengths; a plurality of reconfigurable fusion circuits, each of the plurality of reconfigurable fusion circuits being configured to receive three input qubits on three input paths and to selectably perform one of a plurality of measurement operations on the input qubits, thereby producing measurement outcome data, wherein the plurality of measurement operations including a projective entangling measurement between three input qubits and a plurality of single-qubit measurements on each of the three input qubits, wherein the local routing paths are coupled between the routing switches and the reconfigurable fusion circuits such that each of the routing switches is coupled to at least one of the reconfigurable fusion circuits; and a plurality of second routing switches, each of the second routing switches having a first input path coupled to one of the local routing paths, a second input path coupled to one of the inbound network routing paths, and an output path coupled to one of the input paths of one of the reconfigurable fusioncircuits. The system can also include classical control logic coupled to the network of interleaving modules and configured to control the first and second routing switches and the reconfigurable fusion circuits and to receive classical data signals representing the measurement outcome data from the reconfigurable fusion circuits.

[0008] The following detailed description, together with the accompanying drawings, will provide a better understanding of the nature and advantages of the claimed invention.BRIEF DESCRIPTION OF THE DRAWINGS

[0009] FIG. 1 illustrates an example of a qubit entangling system in accordance with some embodiments.

[0010] FIG. 2 shows two representations of a portion of a pair of waveguides corresponding to a dual-rail-encoded photonic qubit.

[0011] FIG. 3A shows a schematic diagram for coupling of two modes.

[0012] FIG. 3B shows, in schematic form, a physical implementation of mode coupling in a photonic system that can be used in some embodiments.

[0013] FIG. 4A shows a simplified schematic diagram for a dual-rail encoded Bell state generator that can be used in some embodiments.

[0014] FIG. 4B shows a four-mode coupling scheme that implements a “spreader,” or “mode-information erasure,” transformation on four modes in accordance with some embodiments.

[0015] FIG. 5 shows a simplified schematic diagram of a 3 -GHZ state generator circuit according to some embodiments.

[0016] FIG. 6A shows a simplified schematic diagram for a dual-rail-encoded type II fusion circuit that can be used in some embodiments.

[0017] FIG. 6B shows an example result of a type II fusion operation using the circuit of FIG. 6A.

[0018] FIG. 7A shows a simplified schematic diagram of a three-way fusion circuit that can be used in some embodiments.

[0019] FIG. 7B shows a table listing success patterns for the circuit of FIG. 7A.

[0020] FIGs. 8A-8F show examples of quantum node diagrams, in accordance with some example embodiments.

[0021] FIG. 9 shows another notational scheme that can be mapped to node diagrams in accordance with some example embodiments.

[0022] FIGs. 10A-10C show diagrams mapping relations between node diagrams and hardware schematics, in accordance with some example embodiments.

[0023] FIG. 11 A shows a graphical representation of a “4-star” resource state for a fusion complex according to some embodiments.

[0024] FIG. 1 IB shows a ZX diagram corresponding to the resource state of FIG. 11 A.

[0025] FIG. 12A shows a graphical representation of a “cuboctahedral” resource state for a fusion complex according to some embodiments.

[0026] FIG. 12B shows a ZX diagram corresponding to the resource state of FIG. 12A.

[0027] FIG. 13 shows a schematic diagram of a unit cell for a fusion network according to some embodiments.

[0028] FIG. 14 shows a schematic diagram of a layer of a fusion network according to some embodiments.

[0029] FIG. 15 shows resource states for the fusion network of FIG. 14 according to some embodiments.

[0030] FIGs. 16A and 16B further illustrate an implementation of a resource state generator circuit that can generate resource states for the fusion network of FIG. 14 according to some embodiments.

[0031] FIGs. 17A and 17B show schematics for additional photonic resource state generator circuits according to various embodiments.

[0032] FIG. 18 shows a view of the fusion network of FIG. 14 with an “interleaving order” assigned in accordance with some embodiments.

[0033] FIG. 19 shows a high-level schematic diagram of an interleaving module according to some embodiments.

[0034] FIG. 20 shows an example of a reconfigurable three-way fusion circuit according to some embodiments.

[0035] FIG. 21 shows a high-level schematic diagram of an interleaving module according to some embodiments.

[0036] FIG. 22 shows a simplified representation of a network of interleaving modules according to some embodiments.

[0037] FIG. 23 is a flow diagram of a process for performing an RSI cycle in one or more interleaving modules according to some embodiments.

[0038] FIG. 24 shows an example system architecture for a quantum computer system according to some embodiments.

[0039] FIG. 25 is a flow diagram of a process for operating an array of interleaving modules according to some embodiments.DETAILED DESCRIPTION

[0040] The following description of exemplary embodiments is presented for the purpose of illustration and description. It is not intended to be exhaustive or to limit the claimed embodiments to the precise form described, and persons skilled in the art will appreciate that many modifications and variations are possible. The embodiments have been chosen and described in order to best explain their principles and practical applications to thereby enable others skilled in the art to best make and use various embodiments and with various modifications as are suited to the particular use contemplated.

[0041] Disclosed herein are examples (also referred to as “embodiments”) of systems and methods for performing operations on ensembles of qubits based on various physical quantum systems, including photonic systems. Such embodiments can be used, for example, in quantum computing as well as in other contexts (e.g., quantum communication) that exploit quantum entanglement. To facilitate understanding of the disclosure, an overview of relevant concepts and terminology is provided in Section 1. With this context established, Section 2 describes examples of fusion networks and resource states according to various embodiments, and Section 3 describes examples of interleaving modules according to various embodiments that can implement fusion networks of the kind described in Section 2. Section 4 describes an example embodiment of a computing system that can implement fusion-basedquantum computing (FBQC) using a network of interleaving modules. Although embodiments are described with specific detail to facilitate understanding, those skilled in the art with access to this disclosure will appreciate that the claimed invention can be practiced without these details.

[0042] Further, embodiments are described herein as creating and operating on systems of qubits, where the quantum state space of a qubit can be modeled as a 2-dimensional vector space. Those skilled in the art with access to this disclosure will understand that techniques described herein can be applied to systems of “qudits,” where a qudit can be any quantum system having a quantum state space that can be modeled as a (complex) / / -dimensional vector space (for any integer ri), which can be used to encode n bits of information. For the sake of clarity of description, the term “qubit” is used herein, although in some embodiments the system can also employ quantum information carriers that encode information in a manner that is not necessarily associated with a binary bit, such as a qudit.1. Overview of Quantum Computing

[0043] FIG. 1 illustrates an example of a qubit entangling system 101 in accordance with some embodiments. Such a system can be used to generate qubits (e.g., photons) in an entangled state (e.g., a GHZ state, Bell pair, and the like), in accordance with some embodiments. In some embodiments, qubit entangling system 101 can operate as a resource state generator as described below.

[0044] In an illustrative photonic architecture, qubit entangling system 101 can include a photon source module 105 that is optically connected to entangled state generator 100. Both the photon source module 105 and the entangled state generator 100 may be coupled to a classical computer system 103 such that the classical computer system 103 can communicate and / or control (e.g., via the classical information channels 130a-b) the photon source module 105 and / or the entangled state generator 100. Photon source module 105 may include a collection of single-photon sources that can provide output photons to entangled state generator 100 by way of interconnecting waveguides 132. Entangled state generator 100 may receive the output photons and convert them to one or more entangled photonic states and then output these entangled photonic states into output waveguides 140. In some embodiments, output waveguide 140 can be coupled to some downstream quantum photonic circuit that may use the entangled states, e g., for performing a quantum computation. Forexample, the entangled states generated by the entangled state generator 100 may be used as resource states for one or more interleaving modules as described below.

[0045] In some embodiments, system 101 may include classical channels 130 (e.g., classical channels 130-a through 130-d) for interconnecting and providing classical information between components. It should be noted that classical channels 130-a through 130-d need not all be the same. For example, classical channel 130-a through 130-c may comprise a bi-directional communication bus carrying one or more reference signals, e.g., one or more clock signals, one or more control signals, or any other signal that carries classical information, e.g., heralding signals, photon detector readout signals, and the like.

[0046] In some embodiments, qubit entangling system 101 includes the classical computer system 103 that communicates with and / or controls the photon source module 105 and / or the entangled state generator 100. For example, in some embodiments, classical computer system103 can be used to configure one or more circuits, e.g., using a system clock that may be provided to photon source module 105 and entangled state generator 100 as well as any downstream quantum photonic circuits used for performing quantum computation. In some embodiments, the quantum photonic circuits can include optical circuits, electrical circuits, or any other types of circuits. In some embodiments, classical computer system 103 includes memory 104, one or more processor(s) 102, a power supply, an input / output (I / O) subsystem, and a communication bus or interconnecting these components. The processor(s) 102 may execute modules, programs, and / or instructions stored in memory 104 and thereby perform processing operations.

[0047] In some embodiments, memory 104 stores one or more programs (e.g., sets of instructions) and / or data structures. For example, in some embodiments, entangled state generator 100 can attempt to produce an entangled state over successive stages, any one of which may be successful in producing an entangled state. In some embodiments, memory104 stores one or more programs for determining whether a respective stage was successful and configuring the entangled state generator 100 accordingly (e.g., by configuring entangled state generator 100 to switch the photons to an output if the stage was successful, or pass the photons to the next stage of the entangled state generator 100 if the stage was not yet successful). To that end, in some embodiments, memory 104 stores detection patterns (described below) from which the classical computer system 103 may determine whether a stage was successful. In addition, memory 104 can store settings that are provided to thevarious configurable components (e.g., switches) described herein that are configured by, e.g., setting one or more phase shifts for the component.

[0048] In some embodiments, some or all of the above-described functions may be implemented with hardware circuits on photon source module 105 and / or entangled state generator 100. For example, in some embodiments, photon source module 105 includes one or more controllers 107-a (e.g., logic controllers) (e.g., which may comprise field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a “system on a chip” that includes classical processors and memory, or the like). In some embodiments, controller 107-a determines whether photon source module 105 was successful (e.g., for a given attempt on a given clock cycle, described below) and outputs a reference signal indicating whether photon source module 105 was successful. For example, in some embodiments, controller 107-a outputs a logical high value to classical channel 130-a and / or classical channel 130-c when photon source module 105 is successful and outputs a logical low value to classical channel 130-a and / or classical channel 130-c when photon source module 105 is not successful. In some embodiments, the output of control 107-a may be used to configure hardware in controller 107-b.

[0049] Similarly, in some embodiments, entangled state generator 100 includes one or more controllers 107-b (e.g., logical controllers) (e.g., which may comprise field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), or the like) that determine whether a respective stage of entangled state generator 100 has succeeded, perform the switching logic described above, and output a reference signal to classical channels 130-b and / or 130-d to inform other components as to whether the entangled state generator 100 has succeeded.

[0050] In some embodiments, a system clock signal can be provided to photon source module 105 and entangled state generator 100 via an external source (not shown) or by classical computer system 103 generates via classical channels 130-a and / or 130-b. In some embodiments, the system clock signal provided to photon source module 105 triggers photon source module 105 to attempt to output one photon per waveguide. In some embodiments, the system clock signal provided to entangled state generator 100 triggers, or gates, sets of detectors in entangled state generator 100 to attempt to detect photons. For example, in some embodiments, triggering a set of detectors in entangled state generator 100 to attempt to detect photons includes gating the set of detectors.

[0051] It should be noted that, in some embodiments, photon source module 105 and entangled state generator 100 may have internal clocks. For example, photon source module 105 may have an internal clock generated and / or used by controller 107-a and entangled state generator 100 has an internal clock generated and / or used by controller 107-b. In some embodiments, the internal clock of photon source module 105 and / or entangled state generator 100 is synchronized to an external clock (e.g., the system clock provided by classical computer system 103) (e.g., through a phase-locked loop). In some embodiments, any of the internal clocks may themselves be used as the system clock, e.g., an internal clock of the photon source may be distributed to other components in the system and used as the master / system clock.

[0052] In some embodiments, photon source module 105 includes a plurality of probabilistic photon sources that may be spatially and / or temporally multiplexed, i.e., a so- called multiplexed single photon source. In one example of such a source, the source is driven by a pump, e.g., a light pulse, that is coupled into an optical resonator that, through some nonlinear process (e.g., spontaneous four wave mixing, second harmonic generation, and the like) may generate zero, one, or more photons. As used herein, the term “attempt” is used to refer to the act of driving a photon source with some sort of driving signal, e g., a pump pulse, that may produce output photons non-deterministically (i.e., in response to the driving signal, the probability that the photon source will generate one or more photons may be less than 1). In some embodiments, a respective photon source may be most likely to, on a respective attempt, produce zero photons (e.g., there may be a 90% probability of producing zero photons per attempt to produce a single-photon). The second most likely result for an attempt may be production of a single-photon (e.g., there may be a 9% probability of producing a single-photon per attempt to produce a single-photon). The third most likely result for an attempt may be production of two photons (e.g., there may be an approximately 1% probability of producing two photons per attempt to produce a single photon). In some circumstances, there may be less than a 1% probability of producing more than two photons.

[0053] In some embodiments, the apparent efficiency of the photon sources may be increased by using a plurality of single-photon sources and multiplexing the outputs of the plurality of photon sources.

[0054] The precise type of photon source used is not critical and any type of source can be used, employing any photon generating process, such as spontaneous four wave mixing(SPFW), spontaneous parametric down-conversion (SPDC), or any other process. Other classes of sources that do not necessarily require a nonlinear material can also be employed, such as those that employ atomic and / or artificial atomic systems, e.g., quantum dot sources, color centers in crystals, and the like. In some cases, sources may or may be coupled to photonic cavities, e.g., as can be the case for artificial atomic systems such as quantum dots coupled to cavities. Other types of photon sources also exist for SPWM and SPDC, such as optomechanical systems and the like. In some examples the photon sources can emit multiple photons already in an entangled state in which case the entangled state generator 100 may not be necessary, or alternatively may take the entangled states as input and generate even larger entangled states.

[0055] For the sake of illustration, an example which employs spatial multiplexing of several non-deterministic photon sources is described as an example of a MUX photon source. However, many different spatial MUX architectures are possible without departing from the scope of the present disclosure. Temporal MUXing can also be implemented instead of or in combination with spatial multiplexing. MUX schemes that employ log-tree, generalized Mach-Zehnder interferometers, multimode interferometers, chained sources, chained sources with dump-the-pump schemes, asymmetric multi-crystal single photon sources, or any other type of MUX architecture can be used. In some embodiments, the photon source can employ a MUX scheme with quantum feedback control and the like.

[0056] The foregoing description provides an example of how photonic circuits can be used to implement physical qubits and operations on physical qubits using mode coupling between waveguides. In these examples, a pair of modes can be used to represent each physical qubit. Examples described below can be implemented using similar photonic circuit elements.

[0057] In some embodiments, an entangled system of multiple physical qubits can be mapped to one or more “logical qubits,” and operations associated with a quantum computation can be defined as logical operations on logical qubits, which in turn can be mapped to physical operations on physical qubits.

[0058] Quantum computing relies on the dynamics of quantum objects, e.g., photons, electrons, atoms, ions, molecules, nanostructures, and the like, which follow the rules of quantum theory. In quantum theory, the quantum state of a quantum object is described by a set of physical properties, the complete set of which is referred to as a mode. In some embodiments, a mode is defined by specifying the value (or distribution of values) of one ormore properties of the quantum object. For example, in the case where the quantum object is a photon, modes can be defined by the frequency of the photon, the position in space of the photon (e.g., which waveguide or superposition of waveguides the photon is propagating within), the associated direction of propagation (e.g., the k- vector for a photon in free space), the polarization state of the photon (e.g., the direction (horizontal or vertical) of the photon’s electric and / or magnetic fields), a time window in which the photon is propagating, the orbital angular momentum state of the photon, and the like.

[0059] For the case of photons propagating in a waveguide, it is convenient to express the state of the photon as one of a set of discrete spatio-temporal modes. For example, the spatial mode ki of the photon is determined according to which one of a finite set of discrete waveguides the photon is propagating in, and the temporal mode t}is determined by which one of a set of discrete time periods (referred to herein as “bins”) the photon is present in. In some photonic implementations, the degree of temporal discretization can be provided by a pulsed laser which is responsible for generating the photons. In examples below, spatial modes will be used primarily to avoid complication of the description. However, one of ordinary skill will appreciate that the systems and methods can apply to any type of mode, e.g., temporal modes, polarization modes, and any other mode or set of modes that serves to specify the quantum state. Further, in the description that follows, embodiments will be described that employ photonic waveguides to define the spatial modes of the photon.However, persons of ordinary skill in the art with access to this disclosure will appreciate that other types of mode, e.g., temporal modes, energy states, and the like, can be used without departing from the scope of the present disclosure. In addition, persons of ordinary skill in the art will be able to implement examples using other types of quantum systems, including but not limited to other types of photonic systems.

[0060] For quantum systems of multiple indistinguishable particles, rather than describing the quantum state of each particle in the system, it is useful to describe the quantum state of the entire many-body system using the formalism of Fock states (sometimes referred to as the occupation number representation). In the Fock state description, the many-body quantum state is specified by how many particles there are in each mode of the system. For example, a multi-mode, two particle Fock state | 1001)1 234specifies a two-particle quantum state with one particle in mode 1, zero particles in mode 2, zero particles in mode 3, and one particle in mode 4. Again, as introduced above, a mode can be any property of the quantum object. For the case of a photon, any two modes of the electromagnetic field can be used, e g., one maydesign the system to use modes that are related to a degree of freedom that can be manipulated passively with linear optics. For example, polarization, spatial degree of freedom, or angular momentum could be used The four-mode system represented by the two-particle Fock state |1001)1;2;3;4can be physically implemented as four distinct waveguides with two of the four waveguides having one photon travelling within them. Other examples of a state of such a many -body quantum system include the four-particle Fock state 11111)! ,2,3, that represents each mode occupied by one particle and the four- particle Fock state |2200)12 3 4that represents modes 1 and 2 respectively occupied by two particles and modes 3 and 4 occupied by zero particles. For modes having zero particles present, the term “vacuum mode” is used. For example, for the four-particle Fock state |2200)12 3 4modes 3 and 4 are referred to herein as “vacuum modes.” Fock states having a single occupied mode can be represented in shorthand using a subscript to identify the occupied mode. For example, |0010)12 3 4is equivalent to |13).1.1. Qubits

[0061] As used herein, a “qubit” (or quantum bit) is a quantum system with an associated quantum state that can be used to encode information. A quantum state can be used to encode one bit of information if the quantum state space can be modeled as a (complex) two- dimensional vector space, with one dimension in the vector space being mapped to logical value 0 and the other to logical value 1. In contrast to classical bits, a qubit can have a state that is a superposition of logical values 0 and 1. More generally, a “qudit” can be any quantum system having a quantum state space that can be modeled as a (complex) n- dimensional vector space (for any integer ri), which can be used to encode n bits of information. For the sake of clarity of description, the term “qubit” is used herein, although in some embodiments the system can also employ quantum information carriers that encode information in a manner that is not necessarily associated with a binary bit, such as a qudit. Qubits (or qudits) can be implemented in a variety of quantum systems. Examples of qubits include: polarization states of photons; presence of photons in waveguides; or energy states of molecules, atoms, ions, nuclei, or photons. Other examples include other engineered quantum systems such as flux qubits, phase qubits, or charge qubits (e g., formed from a superconducting Josephson junction); topological qubits (e.g., Majorana fermions); or spin qubits formed from vacancy centers (e.g., nitrogen vacancies in diamond).

[0062] A qubit can be “dual-rail encoded” such that the logical value of the qubit is encoded by occupation of one of two spatial modes (e.g., waveguides) of the quantum system. For example, the logical 0 and 1 values can be encoded as follows:|0)L= |10>x 2(1)|1>L = |01>lj2(2) where the subscript “L” indicates that the ket represents a logical state (e.g., a qubit value) and, as before, the notation 117)1,2onthe right-hand side of the equations above indicates that there are i particles in a first mode and j particles in a second mode, respectively (e.g., where i and j are integers). In this notation, a two-qubit system having a logical state |0)|l)L(representing a state of two qubits, the first qubit being in a ‘0’ logical state and the second qubit being in a ‘ 1’ logical state) may be represented using occupancy across four spatial modes by | 1001)1 2J3J4 (e.g., in a photonic system, one photon in a first waveguide, zero photons in a second waveguide, zero photons in a third waveguide, and one photon in a fourth waveguide). In some instances throughout this disclosure, the various subscripts are omitted to avoid unnecessary mathematical clutter.

[0063] A qubit can also be “temporally encoded” such that the logical value of a qubit is encoded based on presence or absence of a photon at a particular location along a waveguide at a particular time. For temporal encoding of qubits, it can be useful to define time bins of fixed duration (e.g., measured using a clock circuit), and two consecutive time bins can be mapped to the logical states of a qubit.

[0064] As used herein, a “physical qubit” refers to a physical system that can be used to encode one bit of physical information. For purposes of constructing fault-tolerant quantum computing and communication systems, it can be useful to consider an “encoded qubit,” which refers to a set of entangled physical qubits that can be treated as a single qubit, where operations on the single (encoded) qubit are implemented using collective operations on the physical qubits. A “resource state” refers to a system of entangled qubits that may be suitable for use in quantum computations; in some embodiments, a resource state can include multiple encoded qubits. Examples of resource states and encoded qubits (and applications thereof, including but not limited to quantum computing and quantum communication) have been described elsewhere. Some additional examples are described below.1.2. Entangled States

[0065] Many of the advantages of quantum computing relative to “classical” computing (e g., conventional digital computers using binary logic) stem from the ability to create entangled states of multi-qubit systems. In mathematical terms, a state \i ) of n quantum objects is a separable state if | =and an entangled state is a state that is not separable. One example is a Bell state, which, loosely speaking, is a type of maximally entangled state for a two-qubit system, and qubits in a Bell state may be referred to as a Bell pair. For example, for qubits encoded by single photons in pairs of modes (a dual -rail encoding), examples of Bell states include:

[0066] More generally, an / / -qubit Greenberger-Horne-Zeilinger (GHZ) state (or “ / -GHZ state”) is an entangled quantum state of n qubits. For a given orthonormal logical basis, an n- GHZ state is a quantum superposition of all qubits being in a first basis state superposed with all qubits being in a second basis state:where the kets above refer to the logical basis. For example, for qubits encoded by single photons in pairs of modes (a dual-rail encoding), a 3-GHZ state can be written:where the kets above refer to photon occupation number in six respective modes (with mode subscripts omitted).1.3.Photonic Implementations

[0067] Qubits (and operations on qubits) can be implemented using a variety of physical systems. In some examples described herein, qubits can be provided in an integrated photonic system employing waveguides, beam splitters, photonic switches, and single photon detectors, and the modes that can be occupied by photons are spatiotemporal modes that correspond to presence of a photon in a waveguide. Modes can be coupled using mode couplers, e.g., optical beam splitters, to implement transformation operations, and measurement operations can be implemented by coupling single-photon detectors to specific waveguides. One of ordinary skill in the art with access to this disclosure will appreciate that modes defined by any appropriate set of degrees of freedom, e.g., polarization modes, temporal modes, and the like, can be used without departing from the scope of the present disclosure. For instance, for modes that only differ in polarization (e.g., horizontal (H) and vertical (V)), a mode coupler can be any optical element that coherently rotates polarization, e.g., a birefringent material such as a waveplate. For other systems such as ion trap systems or neutral atom systems, a mode coupler can be any physical mechanism that can couple two modes, e.g., a pulsed electromagnetic field that is tuned to couple two internal states of the atom / ion.

[0068] In some embodiments of a photonic quantum computing system using dual-rail encoding, a qubit can be implemented using a pair of waveguides. FIG. 2 shows two representations (200, 200') of a portion of a pair of waveguides 202, 204 that can be used to provide a dual-rail-encoded photonic qubit. At 200, a photon 206 is in waveguide 202 and no photon is in waveguide 204 (also referred to as a vacuum mode); in some embodiments, this corresponds to the |0) state (or “logical 0 state”) of a photonic qubit. At 200', a photon 208 is in waveguide 204, and no photon is in waveguide 202; in some embodiments this corresponds to the |1)Lstate (or “logical 1 state”) of the photonic qubit. To prepare a photonic qubit in a known logical state, a photon source (not shown) can be coupled to one end of one of the waveguides. The photon source can be operated to emit a single photon into the waveguide to which it is coupled, thereby preparing a photonic qubit in a known state. Photons travel through the waveguides, and by periodically operating the photon source, a quantum system having qubits whose logical states map to different temporal modes of the photonic system can be created in the same pair of waveguides. In addition, by providing multiple pairs of waveguides, a quantum system having qubits whose logical states correspond to different spatiotemporal modes can be created. It should be understood that thewaveguides in such a system need not have any particular spatial relationship to each other. For instance, they can be but need not be arranged in parallel.

[0069] Occupied modes can be created by using a photon source to generate a photon that then propagates in the desired waveguide. A photon source can be, for instance, a resonatorbased source that emits photon pairs, also referred to as a heralded single photon source. In one example of such a source, the source is driven by a pump, e.g., a light pulse, that is coupled into a system of optical resonators that, through a nonlinear optical process (e.g., spontaneous four wave mixing (SFWM), spontaneous parametric down-conversion (SPDC), second harmonic generation, or the like), can generate a pair of photons. Many different types of photon sources can be employed. Examples of photon pair sources can include a microring-based spontaneous four wave mixing (SPFW) heralded photon source (HPS). However, the precise type of photon source used is not critical and any type of source, employing any process, such as SPFW, SPDC, or any other process can be used. Other classes of sources can also be employed, such as those that employ atomic and / or artificial atomic systems, e.g., quantum dot sources, color centers in crystals, and the like, and sources can incorporate nonlinear optical materials and / or other materials as desired. In some cases, sources may or may not be coupled to photonic cavities, e.g., as can be the case for artificial atomic systems such as quantum dots coupled to cavities. Other types of photon sources also exist for SPWM and SPDC, such as optomechanical systems and the like.

[0070] In such cases, operation of the photon source may be non-deterministic (also sometimes referred to as “stochastic”) such that a given pump pulse may or may not produce a photon pair. In some embodiments, coherent spatial and / or temporal multiplexing of several non-deterministic sources (referred to herein as “active” multiplexing) can be used to allow the probability of having one mode become occupied during a given cycle to approach 1. One of ordinary skill will appreciate that many different active multiplexing architectures that incorporate spatial and / or temporal multiplexing are possible. For instance, active multiplexing schemes that employ log-tree, generalized Mach-Zehnder interferometers, multimode interferometers, chained sources, chained sources with dump-the-pump schemes, asymmetric multi-crystal single photon sources, or any other type of active multiplexing architecture can be used. In some embodiments, the photon source can employ an active multiplexing scheme with quantum feedback control and the like.

[0071] Measurement operations can be implemented by coupling a waveguide to a singlephoton detector that generates a classical signal (e.g., a digital logic signal) indicating that a photon has been detected by the detector. Any type of photodetector that has sensitivity to single photons can be used. In some embodiments, detection of a photon (e.g., at the output end of a waveguide) indicates an occupied mode while absence of a detected photon can indicate an unoccupied mode

[0072] Some embodiments described below relate to physical implementations of unitary transform operations that couple modes of a quantum system, which can be understood as transforming the quantum state of the system. For instance, if the initial state of the quantum system (prior to mode coupling) is one in which one mode is occupied with probability 1 and another mode is unoccupied with probability 1 (e.g., a state 110) in the Fock notation introduced above), mode coupling can result in a state in which both modes have a nonzero probability ofbeing occupied, e.g., a state a 110) += 1. In some embodiments, operations of this kind can be implemented by using beam splitters to couple modes together and variable phase shifters to apply phase shifts to one or more modes. The amplitudes a and az depend on the reflectivity (or transmissivity) of the beam splitters and on any phase shifts that are introduced.

[0073] FIG. 3 A shows a schematic diagram 310 (also referred to as a circuit diagram or circuit notation) for coupling of two modes. The modes are drawn as horizontal lines 312, 314, and the mode coupler 316 is indicated by a vertical line that is terminated with nodes (solid dots) to identify the modes being coupled. In the more specific language of linear quantum optics, the mode coupler 316 shown in FIG. 3A represents a 50 / 50 beam splitter that implements a transfer matrix:where T defines the linear map for the photon creation operators on two modes. (In certain contexts, transfer matrix T can be understood as implementing a first-order imaginary Hadamard transform.) By convention the first column of the transfer matrix corresponds to creation operators on the top mode (referred to herein as mode 1, labeled as horizontal line 312), and the second column corresponds to creation operators on the second mode (referred to herein as mode 2, labeled as horizontal line 314), and so on if the system includes more than two modes. More explicitly, the mapping can be written as:where subscripts on the creation operators indicate the mode that is operated on, the subscripts input and output identify the form of the creation operators before and after the beam splitter, respectively and where:For example, the application of the mode coupler shown in FIG. 3 A leads to the following mappings:Thus, the action of the mode coupler described by Eq. (9) is to take the input states 110), |01), and 111) to

[0074] FIG. 3B shows a physical implementation of a mode coupling that implements the transfer matrix T of Eq. (9) for two photonic modes in accordance with some embodiments. In this example, the mode coupling is implemented using a waveguide beam splitter 300, also sometimes referred to as a directional coupler or mode coupler. Waveguide beam splitter 300 can be realized by bringing two waveguides 302, 304 into close enough proximity that theevanescent field of one waveguide can couple into the other. By adjusting the separation d between waveguides 302, 304 and / or the length I of the coupling region, different couplings between modes can be obtained. In this manner, a waveguide beam splitter 300 can be configured to have a desired transmissivity. For example, the beam splitter can be engineered to have a transmissivity equal to 0.5 (i.e., a 50 / 50 beam splitter for implementing the specific form of the transfer matrix T introduced above). If other transfer matrices are desired, the reflectivity (or the transmissivity) can be engineered to be greater than 0.6, greater than 0.7, greater than 0.8, or greater than 0.9 without departing from the scope of the present disclosure.

[0075] In addition to mode coupling, some unitary transforms may involve phase shifts applied to one or more modes. In some photonic implementations, variable phase- shifters can be implemented in integrated circuits, providing control over the relative phases of the state of a photon spread over multiple modes. Examples of transfer matrices that define such a phase shifts are given by (for applying a +i and -i phase shift to the second mode, respectively):For silica-on-silicon materials some embodiments implement variable phase-shifters using thermo-optical switches. The thermo-optical switches use resistive elements fabricated on the surface of the chip, that via the thermo-optical effect can provide a change of the refractive index n by raising the temperature of the waveguide by an amount of the order of 10'5K. One of skill in the art with access to the present disclosure will understand that any effect that changes the refractive index of a portion of the waveguide can be used to generate a variable, electrically tunable, phase shift. For example, some embodiments use beam splitters based on any material that supports an electro-optic effect, so-called x2and x3materials such as lithium niobite, BBO, KTP, and the like and even doped semiconductors such as silicon, germanium, and the like.

[0076] In some embodiments, entangled states of multiple photonic qubits can be created by coupling modes of two (or more) qubits and performing measurements on other modes. By way of example, FIG. 4A shows a circuit diagram for a Bell state generator circuit 400that can be used in some dual-rail-encoded photonic embodiments. In this example, waveguides (or modes) 432-1 through 432-4 are initially each occupied by a photon (indicated by a wavy line); waveguides (or modes) 432-5 through 432-8 are initially vacuum (unoccupied) modes. (Those skilled in the art will appreciate that other combinations of occupied and unoccupied modes can be used.)

[0077] A first-order mode coupling (e.g., implementing transfer matrix T of Eq. (9)) is performed on pairs of occupied and unoccupied modes as shown by mode couplers 431-1 through 431-4, with each mode coupler 431 having one input waveguide receiving a photon and one input waveguide receiving vacuum. Mode couplers 431 can be, e g., 50 / 50 beam splitters so that, for example, a photon entering on waveguide 432-1 (or a photon entering on waveguide 432-5) has a 50% probability of emerging on either output of mode coupler 431-1. In some embodiments, mode couplers 431 are implemented as directional couplers. Thereafter, a mode-information erasure coupling (e.g., implementing a second-order Hadamard transfer matrix) is performed on one output mode of each mode coupler 431 (in this example, waveguides 433-5 through 433-8 provide inputs to the mode-information erasure coupling), as shown by mode coupler 437. FIG. 4B shows how mode coupler 437 can be implemented using a network of couplings 451-454 between pairs of modes. Each of couplings 451-454 can be implemented as described above with reference to FIGs. 3 A and 3B. In the following description, mode coupler 437 may also be referred to as a “mode coupler network” or “Hadamard network.”

[0078] Referring again to FIG. 4A, waveguides 433-5 through 433-8 act as “heralding” modes that are measured and used to determine whether a Bell state was successfully generated on the four output waveguides 433-1 through 433-4. For instance, detectors 438-1 through 438-4 can be coupled to the waveguides 433-5 through 433-8 after second-order mode coupler 437. Each detector 438-1 through 438-4 can output a classical data signal (e.g., a voltage level on a conductor) indicating whether it detected a photon (or the number of photons detected). These outputs can be coupled to classical decision logic circuit 440, which determines whether a Bell state is present on the other four waveguides 433-1 through 433-4. For example, classical decision logic circuit 440 can be configured such that a Bell state is confirmed (also referred to as “success” of the Bell state generator) if and only if a single photon was detected by each of exactly two of detectors 438-1 through 438-4. In some embodiments, output modes (or waveguides) 433-1 through 433-4 can be mapped to the logical states of two qubits (Qubit 1 and Qubit 2), as indicated in FIG. 4A. Specifically, inthis example, the logical state of Qubit 1 is based on occupancy of modes 433-1 and 433-2, and the logical state of Qubit 2 is based on occupancy of modes 433-3 and 433-4. It should be noted that generation of a Bell state by Bell state generator circuit 400 is a non- deterministic (or stochastic) process; that is, inputting four photons as shown does not guarantee that a Bell state will be created on modes 433-1 through 433-4. In one implementation, the probability of success is 4 / 32; in another implementation, the success probability is 3 / 16. It should also be noted that there are six detection patterns with one photon in each of two of detectors 438, and that Bell state generator circuit 400 can be expected to produce a Bell state in all six possible arrangements of the four output modes. For a given choice of assignment of modes to dual-rail qubits (e.g., as shown in FIG. 4A), Bell state generator circuit 400 can produce any of the four two-qubit Bell states defined in Eqs. (3)-(6) above, as well as a “non-qubif ’ maximally entangled state. Different detection patterns at detectors 438 can correspond to different types of Bell states being produced. In some embodiments, based on the particular detection pattern at detectors 438, mode swaps can be selectably applied to modes 433 in order to cast the Bell state into a particular type (e.g., a particular one of the four two-qubit Bell states defined above). In some embodiments, the mode swap can be subsumed into subsequent operations without the need for active optical switches to implement selectable mode swapping at the output of Bell state generator circuit 400.

[0079] FIG. 5 shows a high-level schematic diagram of a 3-GHZ state generator circuit 500 according to some embodiments. Circuit 500 includes eight input paths 502-1 through 502-8. Input paths 502-1 through 502-4 are coupled to a first beam splitter network 510-1 implementing a second-order (4x4) Hadamard transform, and input paths 502-5 through 502- 8 are coupled to a second beam splitter network 510-2 that also implements a second-order Hadamard transform. Each of beam splitter networks 510-1 and 510-2 can be implemented using 50 / 50 beam splitters, e.g., as shown in FIG. 4B. One output of each beam splitter network 510-1, 510-2 is used as a heralding output 516-1, 516-2, while the other three outputs of each beam splitter network 510-1, 510-2 are used as propagation outputs 514-1 through 514-6. Heralding outputs 516-1, 516-2 are coupled to mode information erasure circuit 518, (implemented, e.g., using a 50 / 50 beam splitter), and the outputs of mode information erasure circuit 518 are delivered to photon counting detectors 520-1, 520-2. Photon counting detectors 520-1 and 520-2 provide respective photon counts to classicaldecision logic circuit 530, which can be similar to classical decision logic circuit 1230 described above.

[0080] In embodiments of 3-GHZ state generator circuit 500, the output waveguides are divided into propagation outputs and heralding outputs such that, when one photon is input to each of input waveguides 501-1 through 501-8, a 3-GHZ state is probabilistically produced on propagation outputs 514-1 through 514-6. Success is heralded by a total of five photons counted at photon counting detectors 520-1, 520-2. In this instance, a failure pattern with a total of six photons can occur, and this may lead to a false positive if one of the six photons is not detected. However, if the six-photon failure pattern occurs, the propagation outputs 514- 1 through 514-6 would propagate a pattern of photons in which none of the qubits of the supposed 3-GHZ state is in a logically valid state. Specifically, one of the qubits would be a two-photon state while the other two qubits would be vacuum (zero-photon) states (e.g., 1002000)). Accordingly, while a false positive may occur in 3-GHZ state generator circuit 500, qubit errors on the propagation outputs would be correlated, making them easier to detect in downstream components. In addition, if a single dark count (where the detector counts a photon that is not actually present) results in a false positive, the propagation outputs would propagate a pattern of photons in which none of the qubits of the supposed 3-GHZ state is in a logically valid state. Specifically, the pattern would include one four-photon qubit and two zero-photon qubits (e.g., 1400000)) or two two-photon qubits and one zerophoton qubit (e.g., |202000)). In this manner, the effect of a false positive can be mitigated due to correlation of qubit errors.

[0081] Bell state generator circuit 400 and 3-GHZ state generator circuit 500 produce entangled states of two qubits and three qubits, respectively. In some instances, it is desirable to form quantum systems of more than two or three entangled qubits. One technique for forming multi-qubit quantum systems is through the use of an entangling measurement, which is a projective measurement that can be employed to create entanglement between systems of qubits. As used herein, “fusion” (or “a fusion operation” or “fusing”) refers to a projective entangling measurement. A “fusion circuit” (or “fusion gate”) is a structure that receives two (or more) input qubits, each of which is typically part of a different quantum system. Prior to applying the fusion gate, the different quantum systems need not be entangled with each other. In the case of two input qubits, the fusion gate performs a proj ective measurement operation on the input qubits that produces either one (“type I fusion”) or zero (“type II fusion”) output qubits in a manner such that the initial two quantumsystems are fused into a single quantum system of entangled qubits. In the case of more than two qubits (e.g., any number n of qubits for n > 2), an “ / / -way” fusion operation can consume all of the input qubits in a manner such that the initial n quantum systems are fused into a single quantum system of entangled qubits. Fusion gates are specific examples of a general class of projective entangling measurements and are particularly suited for photonic architectures. Examples of fusion circuits will now be described.

[0082] FIG. 6A shows a circuit diagram illustrating a type II fusion circuit 600 in accordance with some embodiments. The diagram shown in FIG. 6A is schematic with each horizontal line representing a mode of a quantum system, e g., a photon. In a dual-rail encoding, each pair of modes represents a qubit. In a photonic implementation of the circuit, the modes in diagrams such as that shown in FIG. 6A can be physically realized using single photons in photonic waveguides. Most generally, a type II fusion circuit such as circuit 600 takes qubit A (physically realized, e.g., by a photon in one of modes 643 and 645 or in a superposition state across modes 643 and 645) and qubit B (physically realized, e.g., by a photon in one of modes 647 and 649 or in a superposition state across modes 647 and 649) as input and outputs a quantum state that inherits the entanglement with other qubits that were previously entangled with either (or both) of input qubit A or input qubit B. (For type II fusion, if the input quantum states had a total of A qubits between them, the output quantum state has N~ 2 qubits.)

[0083] For example, FIG. 6B shows the result of type II fusing of two qubits A and B that are each, respectively, a qubit located at the end (i.e., a leaf) of some longer entangled state (only a portion of which is shown). The resulting quantum system 671 inherits the entangling bonds from qubits A and B thereby creating a larger linear quantum system.

[0084] Returning to the schematic illustration of type II fusion circuit 600 shown in FIG. 6A, qubit A is dual-rail encoded by modes 643 and 645, and qubit B is dual-rail encoded by modes 647 and 649. For example, in the case of path encoded photonic qubits, the logical zero state of qubit A (denoted 10)^) occurs when mode 643 is a photonic waveguide that includes a single photon and mode 645 is a photonic waveguide that includes zero photons (and likewise for qubit B). Thus, type II fusion circuit 600 takes as input two dual-rail- encoded photon qubits thereby resulting in a total of four input modes (e.g., modes 643, 645, 647, and 649). To accomplish the fusion operation, a first mode coupler (e.g., 50 / 50 beam splitter) 653 is applied between a mode of each of the input qubits, e.g., between mode 643and mode 649, and a second mode coupler (e.g., 50 / 50 beam splitter) 655 is applied between the other modes of each of the input qubits, e.g., between modes 645 and 647. A detection operation is performed on all four modes using photon counting detectors 657(l)-657(4). In some embodiments, mode swap operations (not shown in FIG. 6A) can be performed to place modes in adjacent positions prior to mode coupling. In some embodiments, mode swapping can be accomplished through a physical waveguide crossing or by one or more photonic switches or by any other type of physical mode swap. Mode swaps are optional and are not necessary if qubits having non-adjacent modes can be dealt with, e.g., by tracking which modes belong to which qubits by storing this information in a classical memory.

[0085] FIG. 6A shows only an example arrangement for a type II fusion circuit, and one of ordinary skill will appreciate that the positions of the mode couplers and the presence or absence of mode swap regions can be altered without departing from the scope of the present disclosure.

[0086] The type II fusion circuit shown in FIG. 6A is a nondeterministic gate, i.e., the fusion operation succeeds with a certain probability less than 1, and in other cases the quantum state that results is not a larger quantum system that comprises the original quantum systems fused together to a larger quantum system. More specifically, the gate “succeeds” in the case where one photon is detected by one of photon counting detectors 657(1) and 657(4) and one photon is detected by one of photon counting detectors 657(2) and 657(3); in all other cases, the gate “fails.” When the gate succeeds, the two quantum systems that qubits A and B were a part of become fused into a single larger quantum system; unlike type I fusion, no fused qubit remains. When the fusion gate fails, it has the effect of removing both qubits from the original quantum systems without generating a larger quantum system.

[0087] FIG. 7A shows a simplified schematic diagram of a three-way fusion circuit 700 according to some embodiments. Three-way fusion circuit 700 receives three dual-rail- encoded input qubits QI, Q2, Q3, each of which can initially be part of a different entangled system of qubits. (For simplicity of illustration, the remaining qubits of the initial entangled systems are not shown.) Success of three-way fusion circuit 700 results in the three entangled systems being entangled with each other to form one output entangled system.

[0088] Input qubit QI is encoded on a first pair of waveguides 702, 704 such that the logical-0 rail maps to waveguide 702 and the logical-1 rail maps to waveguide 704. Similarly, input qubit Q2 is encoded on a second pair of waveguides 706, 708 such that thelogical-0 rail maps to waveguide 706 and the logical-1 rail maps to waveguide 708, and input qubit Q3 is encoded on a third pair of waveguides 710, 712 such that the logical-0 rail maps to waveguide 710 and the logical- 1 rail maps to waveguide 712. A first mode coupler 722 couples waveguides 704 and 706; a second mode coupler 724 couples waveguides 708 and 710; and a third mode coupler 726 couples waveguides 702 and 712. Each of mode couplers 722, 724, and 726 can be implemented, e g., using a 50 / 50 beam splitter as described above. Thus, qubit QI has one logical rail coupled to a logical rail of qubit Q2 and the other logical rail coupled to a logical rail of qubit Q3, and so on for the other qubits. Downstream of mode couplers 722, 724, 726, each of waveguides 702, 704, 706, 708, 710, 712 is coupled to one of a set of photon-counting detectors 731-1 through 731-6 (also labeled DI through D6). Components of three-way fusion circuit 700 can be arranged such that all detectors 731-1 through 731-6 operate concurrently (or within the same time bin). Detectors 731-1 through 731-can output classical logic signals indicating the number of photons detected at each detector to classical decision logic circuit 750. In some embodiments, output data from all detectors is concurrently sent to classical decision logic circuit 750.

[0089] Classical decision logic circuit 750 can be implemented using, e.g., field programmable gate arrays (FPGAs), application specific integrated circuits (ASICS), a system on a chip that includes classical processors and memory, or the like. In operation, classical decision logic circuit 750 can receive the respective photon count signals from detectors 731-1 through 731-6 and can determine whether the pattern of photon counts corresponds to a success pattern.

[0090] As noted above, all signals from detectors 731-1 through 731-6 are received concurrently, providing a detection pattern having six elements (one photon count from each detector). Success patterns can correspond to one detector of the first pair of detectors 731-1, 731-2 detecting one photon in combination with one detector of the second pair of detectors 731-3, 731-4 detecting one photon and one detector of the third pair of detectors 731-5, 731-6 detecting one photon. For instance, classical decision logic circuit 750 can include or have access to a lookup table or other memory structure that stores a representation of each success pattern. When photon-count signals are received from detectors 731-1 through 731-6, classical decision logic circuit 750 can compare the “readout” pattern of signals received from detectors 731-1 through 731-6 to the stored success patterns to determine whether or not a match has occurred. In some embodiments, classical decision logic circuit 750 can output a classical decision signal (not shown) indicating whether a match has occurred.

[0091] In some embodiments, classical decision logic circuit 750 can also receive heralding signals from upstream classical circuitry, such as circuits that generated the entangled systems that include QI, Q2, and Q3. Where such heralding signals are provided, classical decision logic circuit 750 can also determine, based on the received heralding signals, whether the input qubits were in valid logical states. (In examples described herein, a dualrail-encoded qubit is in a valid logical state if there is exactly one photon propagating in one or the other of the pair of waveguides mapped to the rails; for qubits in entangled states, it is generally not known which of the two waveguides has the photon.) Classical decision logic circuit 750 can implement the following decision logic: If the received heralding signals indicate a valid input state and the readout pattern of photon counts from detectors 731-1 through 731-6 matches a success pattern, the result is success (e.g., Boolean true); otherwise, the result is failure (e.g., Boolean false). It should be understood that heralding signals from upstream classical circuitry can be incorporated in a similar manner into the logic in any of the classical decision logic circuits described herein.

[0092] Success patterns for three-way fusion circuit 700 can correspond to combinations where exactly one photon is detected in each pair of detectors coupled to the outputs of the same one of mode couplers 722, 724, 726. If all input qubits are in logically valid states, each input qubit is maximally entangled with other qubits in its initial quantum system, and the initial quantum systems are independent of each other, then in the ideal case (with no photon loss or detector noise), the probability of success for three-way fusion circuit 700 is 25%.

[0093] Further illustrating operation of classical decision logic circuit 750, FIG. 7B shows a table 760 listing the success patterns for three-way fusion circuit 700 according to some embodiments. Success patterns are listed in column 762. (In the notation of FIG. 7B, detectors D1-D6 correspond to detectors 731-1 through 731-6 respectively.) These success patterns correspond to combinations where exactly one photon is detected in each pair of detectors coupled to the outputs of the same one of mode couplers 722, 724, 726. Each success pattern corresponds to a projective entangling measurement on qubits QI, Q2, and Q3, and column 764 lists the corresponding projection for each success pattern. In some embodiments, classical decision logic circuit 750 can determine, based on the detection pattern, which projection was detected. In the example of table 760, there are two possible projections (differing by a sign), and classical decision logic circuit 750 can represent theoutcome, e.g., using a three-state signal that indicates failure or, in the case of success, which of the two projections was detected.

[0094] It should be understood that the example photonic circuits described herein are illustrative and that variations and modifications are possible. Three-way fusion circuits of the kind shown in Fig. 7A can be generalized to / / -way fusion circuits acting on n input qubits (for n > 2). Each of these photonic circuits can be implemented as a photonic integrated circuit (PIC) having appropriate optical input and output ports (e.g., couplers for fiber optics) and classical (e.g., digital) logic signal ports (e.g., electrical contact pads). Different circuits can be implemented in different PICs connected by optical fibers, or multiple circuits can be implemented in the same PIC, coupled by integrated waveguides, as desired. Thus, circuits described above or similar circuits implementing the same entangling operations can be used as building blocks in additional circuits described below. It should be understood that other circuits can also be substituted without departing from the scope of this disclosure.1.4. Notational Schemes for Quantum Systems

[0095] In the following, different physical items of quantum hardware (or hardware that generates, stores, and / or operates on qubits) and tasks are described using quantum node diagrams (e g., spacetime diagrams in C* space; e.g., C2for a single qubit; ZX diagrams of ZX calculus, quantum instruments, logical blocks, tensor network diagrams, and the like), in accordance with some example embodiments. Examples of quantum hardware that can be described with quantum node diagrams include photonic circuits (e g., photonic integrated circuits, or PICs), superconducting circuits, electrical controller circuits (e.g., ASICs), which can be interconnected to implement and manage one or more qubits. Example of tasks that can be described with quantum node diagrams includes gates, such as X-gate, Z-gate, T-gate, CNOTs, Toffoli gates, quantum oracles, quantum Fourier transforms (and inverse thereof), and other types of tasks known in the art. Such tasks can be combined into a quantum algorithm or the like.

[0096] FIG. 8A shows an example of nodes of a quantum node diagram, in accordance with some example embodiments. In the illustrated example, a scalar node 805 (e g., a number; zeroth order tensor), a vector node 810, a matrix node 815, and a tensor node 820 are shown. The nodes can have lines (e.g., legs, bonds, qubits) to connect nodes to other nodes. Diagrams of this kind are sometimes referred to as “spider diagrams” or “ZX diagrams.” At a high level, the scalar node 805 (shown as a circle with no legs) corresponds a zeroth ordertensor, the vector node 810 (shown as a circle with one leg (one bond)) corresponds to a list of numbers having one index (e.g., first order tensor), the matrix node 815 (shown as a circle with 2 legs (2 bonds)) corresponds to an array of numbers having two indices (e.g., a second order tensor), and the tensor node 820 (shown as a circle or dot with more than two legs (more than two bonds)) corresponds to a set of numbers having / V number of indices (e.g., an JV-th order tensor, a linear map). In the following, assorted types of nodes will be referred to as “nodes” where the leg count specifies what kind of tensor the node is (e.g., array of numbers with A-indexes where 0 < N).

[0097] Generally, in quantum information processing (e g., quantum computing), and as used in the example diagrams, the nodes can be colored two different ways — such as green nodes and red nodes, or white nodes and black nodes, or non-hatched nodes and hatched nodes — which can be connected and arranged to create networks of quantum nodes. At a high level, the white nodes correspond to states and measurements (e g., projections) with repetition in the 0 / 1 basis, while black nodes correspond to states and measurements with repetition in the orthogonal + / - basis. For instance, the stabilizers of a black (white) node are products of X (Z) on all qubits and all pairwise Z (X) operators

[0098] FIG. 8B shows an example of a white node 830 (e.g., |0) ) and a black node 835 (e.g., |+)) shown as states with legs extending up (e.g., quantum states, sources, vectors). FIG. 8B further shows their counterparts, white node 840 and black node 845, arranged as measurements (e.g., projections, tests) having downward facing legs.

[0099] FIG. 8C shows example quantum node diagrams with nodes having legs facing up and down. In FIG. 8C, white node 850 (e g., a Z spider) has three downward legs and two upward legs and corresponds to the following operator: |00)(000| + 111)( 111| (in bra-ket notation); whereas black node 855 (e.g., an X spider) also has three downward legs and two upward legs but corresponds to the following operator: | + +)(+ + + 1 + | - )( - 1.

[0100] FIG. 8D shows an example Hadamard gate 860, which corresponds to a basis rotation operator |0)(+| + |1)(— |, or, equivalently, |+)(0| + |— )(1|. Note, in FIGs. 8A-8C the general orientation is from down to up, whereas the orientation of the examples of FIG. 8D and FIG. 8E is from left to right (consistent with quantum circuits that are commonly read left to right; see FIG. 8E discussed below) However, it is appreciated that the direction of quantum node diagrams can be direction agnostic or direction dependent according to therules of the convention being implemented (e.g., according to tensor networks, ZX calculus, multi-linear algebras rules).

[0101] Continuing, with reference to FIG. 8D, a Hadamard box (or gate) 860 can be connected and arranged with black or white nodes, which can correspond to different entangled states depending on which states are connected in a given arrangement (e.g., and / or additionally dependent on phases of the tensors / spiders). For example, in the quantum node diagram 865, a white node has a single Hadamard gate on one leg and corresponds to |0)(0 + | + 11)(1 — |. Similarly, in the quantum node diagram 870, a black node has a single Hadamard gate on one leg and corresponds to | +)(+01 + |— )(— 1|.

[0102] FIG. 8E shows an example of correspondence between quantum circuit notation and quantum node diagrams (e.g., ZX diagrams, tensor network diagrams). In the illustrated example, the quantum circuit diagram 875 represents a CNOT gate in quantum circuit notation, which can map to quantum node diagram 880. FIG. 8F shows example node relations, in accordance with some example embodiments. In some example embodiments, two nodes of the same shape and color can be merged with one another (e.g., spider fusing, contraction) as shown in diagram 890, read from left to right. Alternatively, a single node can be expanded to two nodes (read from right to left). Further, a white node with no Hadamard boxes on its legs is equivalent to a black node with Hadamard boxes on all of its legs.Diagram 892 shows an additional illustrative example; in diagram 892 a white node with two bare legs and three legs with Hadamard boxes is equivalent to a black node with two bare legs and three legs having Hadamard boxes. For instance, as a illustrative visualization, one of the Hadamard boxes of the white node can be “pushed through” the white node to color change it to black, and replicate the Hadamard box on all other legs, where two Hadamard boxes on the same leg cancel to leave a bare leg, thereby leaving a black node with three bare legs and two legs with Hadamard boxes. Further, with reference to diagram 894, a white node or a black node can be equivalent to a bare wire (e g , identity). Additional relations for notations that can be mapped to node diagrams (e.g., ZX diagrams) and relations thereof (e.g., rewrite rules, etc.) are known or available to one of ordinary skill in the art (e.g., in Backens et al., arXiv: 1602.04744, 2016).

[0103] FIG. 9 shows another notational scheme that can be mapped to node diagrams in accordance with some example embodiments. In this notational scheme, nodes can have different shapes (e.g., circle nodes, square nodes). Square nodes 900 correspond to states(e.g., quantum states, entangled states), such as a Bell state (having XX, ZZ stabilizer generators) with two legs, a white or black 3GHZ state having three legs (having XXX, ZZI, ZIZ, or ' LL' , XXI, XIX stabilizer generators, respectively), and so on. Circle nodes 905 correspond to fusion measurements (e.g., projective measurements), such as a 2-way fusion in the white basis (projecting onto states with XX, ZZ stabilizer generators) with two legs; a white or black three-way fusion having three legs (projecting onto a state with XXX, ZZI, ZIZ, or ZZZ, XXI, XIX stabilizer generators, respectively), and so on. Examples of circuits that can be used to implement square nodes 900 and circle nodes 905 are described above. As described above, a fusion measurement is a joint measurement that jointly measures multiple qubits (e.g., projections onto an entangled state whereby multiple qubits, two or more, are measured).

[0104] It is noted that in contrast to the merging of similarly shaped nodes described above (e.g., merging two circles of the same color in a ZX diagram), node diagrams using the notational scheme of FIG. 9 have two different node shapes (e.g., square and circle nodes), where merging or expansion is further based on the shapes. In particular, two nodes of a same shape do not merge.

[0105] FIG. 9 further shows a Hadamard box 910, signifying a basis rotation. Hadamard box 910 can be attached to one or more legs of any of square nodes 900 or circle nodes 905. A node of a given color with a Hadamard box on one (or more) legs can be equivalent to another node of a different color with Hadamard boxes on different legs (or no legs), as described above with reference to FIG. 8F. In some example embodiments, a Hadamard box can be implemented in hardware as a optical directional coupler (e.g., 50:50 coupler), a waveguide crossing, or networks thereof.

[0106] FIG. 9 further shows a multiplexer (mux) box 915 (e g., Nx1 switch or NXM switch) having dashed lines and solid connecting lines. In some example embodiments, the mux box corresponds to a switch (e.g., optical switch, Mach Zehnder Interferometer) to multiplex a light at one or more of the inputs (e.g., N input ports) to a given set of outputs (e.g., 1 output port, M output ports). In the illustrated example, the dashed line that inputs into the mux box indicates a multi-channel line (e.g., 12 optical fibers input into the switch, 64 integrated waveguides, etc.) which are output to a single output channel (e.g., single fiber output, single waveguide) indicated by a solid line. (A mux box with multiple outputs can be indicated by a dashed output line.)

[0107] FIGs. 10A-10C show diagrams mapping relations between node diagrams and hardware schematics, in accordance with some example embodiments. FIG. 10A shows a node diagram 1000 that can represent an encoded qubit in which one qubit is encoded using eight entangled qubits (Q1-Q8). Node diagram 1000 comprises six square nodes 1001 that generate Bell pairs in the white basis, three square nodes 1002 that generate 3-GHZ states in the black basis, two Hadamard boxes labeled with a “H”, and four circle nodes 1003 that correspond to three-way fusions . Square nodes 1001 can be implemented as Bell pair sources, such as Bell state generator circuit 400 of FIG. 4A or other Bell pair generator PICs. Square nodes 1002 can be implemented as 3-GHZ sources, such as 3-GHZ state generator circuit 500 of FIG. 5 or other 3-GHZ generator PICs). Circle nodes 1003 can be implemented using three-way fusion circuits such as circuit 700 of FIG. 7A or other three- way fusion PICs. The solid lines are qubit lines. Diagram 1000 has nine qubit ports: qubits QI to Q8 that correspond to an encoded 8-qubit state, and an attachment qubit QA that can be utilized to fuse the entangled qubit of the node diagram 1000 to additional nodes or collections of nodes (e.g., additional instances of node diagram 1000, where the QA qubit can be fused with one or more other QA qubits of one or more other encoded qubits). It should be understood that FIG. 10A represents one of many possible schemes for preparing an encoded qubit.

[0108] In some example embodiments, circle nodes 1003 provide probabilistic measurements (as in examples described above). Circle nodes 1003 are executed in the order indicated by the numbers in the circles: upon the fusion measurements corresponding to the three circle nodes numbered “1” succeeding in their respective projective measurements, the fusion measurement corresponding to the circle node numbered “2” is performed. In some embodiments, the Hadamard boxes on qubit ports Q5 and Q8 are optional and can be omitted based upon a given node diagram design (e.g., the qubits from those respective ports may or may not need a rotation provided by the Hadamard box, or a rotation may be performed downstream of the ports).

[0109] FIG. 10B shows a hardware schematic diagram 1005 that corresponds to node diagram 1000, in accordance with some example embodiments. In hardware schematic diagram 1005, the square nodes (sources) are arranged on a left side of the diagram and qubit output ports are directed to the left (QA, and Q1-Q8), and the circle nodes (measurement sites) are arranged on the right. A plurality of multiplexers 1007 (e.g., optical switches, generalized MZIs, N-z1 switches, N*M switches) connect the square nodes and the circlenodes. As such, in FIG. 10B, some qubits from each source (square node) are directed to the left, and correspond to an output entangled state, whereby other qubits from each source are directed to the right in FIG. 10B, e.g., for multiplexing and projective measurements.

[0110] In some example embodiments, each square node in FIG. 1 OB corresponds to multiple instances of the physical components of the square node, which have their physical outputs connected to the multiplexer switches, shown as dashed lines in hardware schematic diagram 1005. FIG. 10C shows a more detailed example of a square node 1010 corresponding to multiple instances of physical components, from which one instance can be selected to output its qubits of the given state, in accordance with some example embodiments. In FIG. 10C, top square node 1010 and top two multiplexers 1015 and 1020 are expanded and shown in the expansion diagram 1025.

[0111] In the illustrated example, square node 1010 in hardware schematic diagram 1005 corresponds to a set of three square nodes 1010-1, 1010-2, 1010-3, each of which can be implemented, e.g., as a separate 3-GHZ photonic integrated circuit that has three channel outputs: one QA line, and two qubit lines that are physically connected to multiplexers 1015 and 1020. It should be understood that a “channel” in this context corresponds to a path for a physical qubit and can be implemented as a single-rail qubit line (e.g., polarization based encoding, time bin encoding) or dual-rail path encoding, as discussed above. Each of multiplexers 1015, 1020 can be, e.g., a switch PIC that selects one of its inputs to couple to the output in response to a control signal. In some example embodiments, each of square nodes 1010-1, 1010-2, 1010-3 generates success or failure data (e.g., as described above with reference to FIG. 5), and multiplexers 1015, 1020 can be switched according to the success data to select a given one of square nodes 1010-1, 1010-2, 1010-3 that was successful in a given cycle. For example, square node 1010-1 can generate a success signal, and, in response to the success signal, switches 1015 and 1020 can be switched to a state in which the input port from square node 1010-1 becomes coupled to circle nodes “1” and “2” for measurement (where the QA-1 line then becomes the attachment qubit line “QA” in node diagram 1000 and 1005). Further, in the same switch settings that selected square node 1010-1, multiplexers 1015 and 1020 can discard the couplings from square nodes 1010-2 and 1010-3 such that any light in those input ports is blocked during that cycle. Further, although a specific type of multiplexers is shown, in some embodiments, the switches can be implemented as N*M switches. For instance, with reference to expansion diagram 1025, the multiplexers 1015 and 1020 are 3x 1 switches, but each can be replaced by aN*M switch(e.g., multiplexer 1015 is replaced with a N*M multiplexer having multiple outputs) to copy a given input site to a plurality of fusion sites. For example, a number N of instances of square node 1010 can be used collectively to supply qubits to all of the muxes in FIG 10C that receive qubits from black square nodes.1.5. Fusion-Based Quantum Computing (FBQC)

[0112] “Quantum computation,” as used herein, refers generally to performing a sequence of operations (a “computation”) on an ensemble of qubits. Quantum computation is often considered in the framework of “circuit-based quantum computation” (CBQC), in which the operations are specified as a sequence of logical “gates” performed on qubits. Gates can be either single-qubit unitary operations (rotations), two-qubit entangling operations such as the CNOT gate, or other multi-qubit gates such as the Toffoli gate.

[0113] One challenge for CBQC, and for quantum computation generally, is that physical systems implementing qubits and operations on qubits are often non-determini Stic and noisy. For example, the photonic circuits described above can create entanglement between photonic qubits, but they do so non-deterministically, with a probability of success that may be considerably less than 1. In addition, the physical systems may be “noisy”; for instance, a waveguide propagating a photon may be somewhat less than perfectly efficient, resulting in occasional loss of photons. For reasons such as these, fault tolerant quantum computing is a desirable goal.

[0114] “Measurement-based quantum computation” (MBQC) is an approach to implementing quantum computing that allows for fault-tolerance. In MBQC, computation proceeds by first preparing a particular entangled state of many physical qubits, commonly referred to as a “cluster state,” then carrying out a series of single-qubit measurements to enact (or execute) the quantum computation. For instance, rather than implementing a sequence of gates operating on one or two physical qubits, a subset of the physical qubits in the cluster state can be mapped to a “logical” qubit, and a gate operation on logical qubits can be mapped to a particular set of measurements on physical qubits associated with one or more logical qubits. Entanglement between the physical qubits results in expected correlations among measurements on different physical qubits, which enables error correction. The cluster state can be prepared in a manner that is not specific to a particular computation (other than, perhaps, the size of the cluster state), and the choice of single-qubit measurements is determined by the particular computation. In the MBQC approach, fault tolerance can beachieved by careful design of the cluster state and by using the topology of the cluster state to encode logical qubits in a manner that protects against any logical errors that may be caused by errors on any of the physical qubits that make up the cluster state. The value (or state) of the logical qubit(s) can be determined, i.e., read out, based on the results (also referred to herein as measurement outcomes) of the single-particle measurements that are made on the cluster state’s physical qubits as the computation proceeds.

[0115] For example, a cluster state suitable for MBQC can be defined by preparing a collection of physical qubits in a particular state (e.g., the |+) state) and applying a controlled-phase gate (sometimes referred to as a “CZ gate”) between pairs of physical qubits to generate the cluster state. Graphically, a cluster state formed in this manner can be represented by a graph with vertices representing the physical qubits and edges that represent entanglement (e.g., the application of CZ gates) between pairs of qubits. The graph can be a three-dimensional graph having a regular structure formed from repeating unit cells and is sometimes referred to as a “lattice.” One example of a lattice is the Raussendorf lattice, which is described in detail in R. Raussendorf et al., “Fault-Tolerant One-Way Quantum Computer,” Annals of Physics 321(9):2242-2270 (2006). In such representations, two- dimensional boundaries of the lattice can be identified. Qubits belonging to those boundaries are referred to as “boundary qubits” while all other qubits are referred to as “bulk qubits.” Other cluster state structures can also be used. Logical operations are performed by making single-qubit measurements on qubits of the cluster state, with each measurement being made in a particular logical basis that is selected according to the particular quantum computation to be performed. The collection of measurement results across the cluster state can be interpreted as the result of a quantum computation on a set of logical qubits through the use of a decoder. Numerous examples of decoder algorithms are available, including the Union- Find decoder as described in International Patent Application Publication No. WO 2019 / 002934 Al.

[0116] However, the generation and maintenance of long-range entanglement across the cluster state and subsequent storage of large cluster states can be a challenge. For example, for any physical implementation of the MBQC approach, a cluster state containing many thousands, or more, of mutually entangled qubits must be prepared and then stored for some period of time before the single-qubit measurements are performed.

[0117] “Fusion-based quantum computing” (FBQC) is a technique related to MBQC in that a computation on a set of logical qubits can be defined as a set of measurements on a (generally much larger) number of physical qubits, with correlations among measurement results on the physical qubits enabling error correction. FBQC, however, avoids the need to first create, then subsequently manipulate, a large cluster state. In a photonic implementation of FBQC, entangled states consisting of a few physical qubits or a few encoded qubits (referred to as “resource states”) are periodically generated and transported (via waveguides) to circuits that can perform measurement operations (e.g., fusion operations as described above and / or single-qubit measurements). The measurements destroy the measured qubits; however, the quantum information is preserved as it is transferred (teleported) to other qubits of other resource states. Thus, quantum information is not stored in a static array of physical qubits but is instead periodically teleported to freshly generated physical qubits.

[0118] In FBQC, somewhat similarly to MBQC, a computation can be mapped to an undirected graph, referred to as a fusion graph or fusion network, that can have a lattice-like structure. The fusion graph can define operations to be performed on the physical qubits of the resource states, including fusion operations on selected qubits of different resource states (e.g., in the “bulk” region of a lattice) and individual qubit measurements (e.g., at boundaries of the lattice). Examples of FBQC techniques are described in WO 2021 / 155289, “Fusion Based Quantum Computing,” published August 5, 2021, and WO 2022 / 241336, “Interleaving Module for Fault-Tolerant Quantum Computer,” published November 17, 2022.2. Example Fusion Networks for FBQC

[0119] In FBQC, fault tolerance is achieved by constructing a fusion network (or fusion graph). The fusion network defines a set of measurements (generally fusions, or joint measurements on two or more qubits) made on a collection of constant-sized entangled states, referred to as “resource states.” A fusion network can be represented as a cell complex (or 3D cell complex or fusion complex) having vertices, edges, and faces, where the vertices correspond to resource states, the edges correspond to fusion measurements, and the faces correspond to check operators. Such graphs can also be expressed using the notational schemes defined in Section 1.4 above, which can be translated into hardware (e.g., photonic) implementations. In some fusion networks, a basic structure (referred to as a “unit cell”) having one or more faces, edges and / or vertices is repeated to form a cell complex. For example, three-dimensional fusion networks based on a Kagome-6, or 6-ring, unit cell (e.g., a “cubic fusion network”) exhibit high error tolerance. It can be shown that any three-dimensional fusion network where the cell complex has four faces incident at every edge in the bulk defines a surface-code fusion network.

[0120] For scaling to large numbers of logical qubits, loss and Pauli thresholds need to be high. According to some embodiments, a class of fusion complexes whose unit cells are polyhedrons with only two faces can achieve error tolerance comparable to or better than existing fusion networks (e.g., at least the same as cubic fusion networks) but with the same or lower physical footprint, thereby enabling further scalability and performance of fusion networks for use in quantum information based processing.2.1. Example Resource States

[0121] FIG. 11 A shows a graphical representation of a “4-star” resource state 1100 for a fusion complex according to some embodiments. Resource state 1100 is depicted as a surface code on a closed surface in a cell complex (or plaquette) representation. That is, each vertex 1101-1104 represents a qubit, and each face 1111-1116 represents a stabilizer of the resource state. In some example embodiments, the faces can be one of two colors (bi- colorable), such as a darker face (e.g., face 1111 or 1112), which corresponds to one type of stabilizer (e.g., X-type check) acting on all four qubits (vertices 1101-1104), and a lighter face (e.g., any of faces 1113-1116), which corresponds to another type of stabilizer (e.g., Z- type check) acting on pairs of qubits (e.g., face 1113 represents a stabilizer acting on the qubits represented by vertices 1101 and 1103. For instance, resource state 1100 can correspond to a 4-GHZ state. FIG. 1 IB shows a ZX diagram 1150 corresponding to resource state 1100, using the notation introduced in FIGs. 8A-8F. It should be understood that although FIG. 1 IB shows just one example, there are multiple possible ZX diagrams corresponding to resource state 1100.

[0122] FIG. 12A shows a graphical representation of a “cuboctahedral” resource state 1200 for a fusion complex according to some embodiments. As in FIG. 11 A, the resource state is depicted as a surface code on a closed surface in a cell complex (or plaquette) representation, where each vertex 1201-1208 represents a qubit (resource state 1200 has eight qubits), while the faces 1211-1216 represent stabilizers of the resource state. In some example embodiments, the faces can be bi-colorable, such as a lighter face (e.g., face 1211, face 1212) which corresponds to a first type of stabilizer (e.g., Z-type check); and a darker face (e.g., face 1213) which corresponds to a second type of stabilizer (e.g., X-type check). FIG. 12B shows a ZX diagram 1250 corresponding to resource state 1200, using the notationintroduced in FIGs. 8A-8F. The XX and ZZ syndrome graphs are isomorphic (under periodic boundary conditions), with 2 / 3 of the checks having degree 3, 1 / 6 having degree 12, and 1 / 6 having degree 24; the average check degree is 8. It should be understood that although FIG. 12B shows just one example, there are multiple possible ZX diagrams corresponding to resource state 1200. While these examples can be implemented using two way fusions, other n-way stabilizers (e.g., XXX, ZZI, ZIZ) can be implemented as discussed above with reference to FIG. 9.

[0123] Some embodiments can include resource states whose cell complex (or plaquette) representation includes polyhedrons with exactly two faces (e.g., a bulged cell, a ravioli cell, a cell formed by a pair of faces joined at their boundaries to create a volume).

[0124] In some embodiments, resource states can be prepared using encoded qubits. For example, referring to the encoded qubit represented in FIG. 10A, a resource state of a number (q) of encoded qubits can be prepared by preparing a “core” / / -qubit resource state using single physical qubits, preparing q encoded qubits (e.g., q instances of the encoded qubit represented in FIG. 10A or any other encoded qubit), and performing a type II fusion between each of the qubits of the core resource state and the attachment qubit QA of a different one of the encoded qubits. Other techniques for preparing encoded resource states can also be used, with the qubit encoding represented in FIG. 10A or a different qubit encoding as desired. Subsequent two-way or / / -way fusion operations between qubits of different resource states can be performed on corresponding pairs or groups of physical qubits of the encoded qubits. It should be understood that fusion networks of the kind described herein can be implemented using encoded qubits.2.2. Example Fusion Networks

[0125] FIG. 13 shows a schematic diagram of a unit cell 1300 for a fusion network according to some embodiments. Unit cell 1300 can correspond to a cuboctahedral resource state. The notation corresponds to the notation shown in FIG. 9. Thus, unit cell 1300 includes six 3-GHZ states 1311-1316, each of which has three qubits. (Qubits 1311-1 through 1311-3 belong to 3-GHZ state 1311, and so on). For convenience of description, each of 3-GHZ states 1311-1316 is associated with a different coordinate direction in a three- dimensional entanglement space defined by orthogonal E-W, N-S, and U-D axes as shown in FIG. 13. (It should be understood that the entanglement space need not correspond to the physical layout of any implementation of unit cells or fusion networks described herein.)Selected qubits of 3-GHZ states 1311-1316 participate in three-way fusion operations 1321- 1324, each of which can be implemented, e.g., using three-way fusion circuit 700 described above. Specifically, qubits 1311-1, 1312-1, and 1313-1 are input to fusion operation 1321; qubits 1312-2, 1341-1, and 1316-1 are input to fusion operation 1322; qubits 1311-2, 1315,-1, and 1316-2 are input to fusion operation 1323; and qubits 1313-2, 1314-2, and 1315-2 are input to fusion operation 1324.

[0126] “Outer” qubits 1311-3, 1312-3, 1313-3, 1314-3, 1315-3, and 1316-3 can connect to other instances of unit cell 1300 to form a fusion network. Connections can be made along the coordinate directions. For instance, “N” qubit 1311-3 in a first instance of unit cell 1300 can connect to “S” qubit 1314-3 in a second instance of unit cell 1300; “U” qubit 1312-3 in the first instance of unit cell 1300 can connect to “D” qubit 1315-3 in a third instance of unit cell 1300; and “E” qubit 1313-3 in the first instance of unit cell 1300 can connect to “W” qubit 1316-3 in a fourth instance of unit cell 1300. A Hadamard gate 1321-1323 can be included in each cell-to-cell coupling. Unit cell 1300 can be understood as similar to a 6-ring unit cell, with an additional (redundant) measurement of the XIXIXI stabilizer of the 6-ring. Unlike a 6-ring unit cell, unit cell 1300 includes three-way fusion operations. Thus, the additional measurement can allow for a reduced hardware footprint as in examples below.

[0127] FIG. 14 shows a schematic diagram of a layer of a fusion network 1400 according to some embodiments. Fusion network 1400 is formed as a lattice of identical instances of unit cell 1300 connected along the E-W and N-S directions. (A dashed box has been drawn around one instance of unit cell 1300 for ease of visualization.) Although only one layer is shown in FIG. 14, fusion network 1400 can have additional layers, with the U direction of one unit cell 1300 connecting to the D direction of another unit cell 1300 at the corresponding position of the lattice in the next layer.

[0128] As stated above, a fusion network defines a set of measurements (generally fusions, or joint measurements on two or more qubits) made on a collection of constant-sized entangled states, referred to as “resource states.” In fusion network 1400, resource states 1411-1413 each provide four entangled qubits, with each resource state 1411-1413 providing two qubits to two different unit cells 1300. Three-way fusions 1410 (corresponding to three- way fusion operations 1321-1324) are performed on qubits from three different resource states. It should be understood that fusion network 1400 is illustrative and can include any number of unit cells and any number of layers. It should also be understood that, as withother fusion networks, quantum logic can be implemented in fusion network 1400 by selectively changing some of the three-way fusion operations to other types of operations (e.g., single-qubit measurement on each of the three qubits or two-way fusion on two of the three qubits combined with a single-qubit measurement on the third qubit).

[0129] FIG. 15 shows resource states 1411-1413 in isolation. Each of resource states 1411- 1413 can be viewed as two 3-GHZ states (square nodes) entangled with a basis rotation (the “H” box). Resource state 1411 includes 3-GHZ states 1511, 1512, and H box 1513; resource state 1412 includes 3-GHZ states 1521, 1522, and H box 1523; and resource state 1413 includes 3-GHZ states 1531, 1532, and H box 1533. (These H boxes are not shown in FIG.14 for simplicity of illustration but should be understood to be present.) Thus, fusion network 1400 can be implemented with a single type of resource state generator that generates all of the resource states, which can simplify the design.

[0130] FIGs. 16A and 16B further illustrate an implementation of a resource state generator circuit that can generate resource states 1411-1413 according to some embodiments. FIG. 16A shows resource state 1412, which includes 3-GHZ states 1521, 1522, entangled with a basis rotation indicated by H box 1523. Creating entanglement between the two 3-GHZ states consumes one qubit of each 3-GHZ state, producing a four-qubit resource state. The four qubits of the resource state are labeled El, E2, Wl, and W2. A variety of different hardware implementations are possible.

[0131] FIG. 16B shows a high-level schematic for a photonic resource state generator circuit (RSG) 1600 according to some embodiments. RSG 1600 can generate resource state 1412. (Since resource states 1411 and 1413 are identical to resource state 1412 apart from directional labels, RSG 1600 can generate all resource states for fusion network 1400.) RSG 1600 can include two 3-GHZ circuits 1611, 1612, a Hadamard gate 1623, and a two-way (type II) fusion circuit 1625. Each of 3-GHZ circuits 1611, 1612 can be, e.g., an instance of 3-GHZ state generator circuit 500 of FIG. 5 (e.g., implemented in separate PICs) and can produce a three-qubit entangled state from input photons (as well as classical measurement outcome data as described above). Hadamard gate 1623 can be implemented, e.g., using a 50 / 50 beamsplitter as described above. Hadamard gate 1623 can be disposed in the optical path of one of the output qubits of 3-GHZ circuit 1612. Two-way fusion circuit 1625 has one input coupled to receive a qubit from 3-GHZ circuit 1611 and the other input coupled to receive the output qubit of Hadamard gate 1623. Two-way fusion circuit 1625 can be, e g.,an instance of type II fusion circuit 600 of FIG. 6 A and can perform a joint projective measurement on two input qubits to produce classical measurement outcome data as described above In some embodiments, RSG 1600 can produce an encoded 4-GHZ state on qubit output ports El, E2, Wl, and W2.

[0132] Other implementations are also possible. By way of example, FIG. 17A shows a high-level schematic for a photonic resource state generator circuit (RSG) 1700 according to some embodiments. RSG 1700 includes six Bell state generator circuits 1711-1713, 1721- 1723, each of which can be, e.g., an instance of Bell state generator circuit 400 described above. Each Bell state generator circuit 1711-1713, 1721-1723 can generate a pair of qubits in a Bell state from single-photon inputs. Three-way fusion circuits 1715, 1725 can each be, e.g., an instance of three-way fusion circuit 700 described above. Three-way fusion circuits 1715, 1725 can perform a joint projective measurement operation on three input qubits and generate classical measurement outcome data. (The input qubits are consumed in the measurement operation.) For instance, operation of Bell state generator circuits 1711-1713 and three-way fusion circuit 1715 can produce a first 3 -GHZ state, two qubits of which correspond to the El and E2 qubits of the resource state. Similarly, operation of Bell state generator circuits 1721-1723 and three-way fusion circuit 1725 can produce a second 3 -GHZ state, two qubits of which correspond to the Wl and W2 qubits of the resource state. A two- way fusion circuit 1735 can be used to entangle the two 3-GHZ states. For instance, one output of Bell state generator circuit 1713 can be provided as one input to two-way fusion circuit 1735. A Hadamard gate 1733 can apply a phase rotation to one output qubit of Bell state generator 1723, and the output of Hadamard gate 1733 can be provided as the other input to two-way fusion circuit 1735. Two-way fusion circuit 1735 can be, e.g., an instance of type II fusion circuit 600 of FIG. 6A and can perform a joint projective measurement on two input qubits to produce classical measurement outcome data as described above. The resulting entanglement structure is the same as that produced by RSG 1600 of FIG. 16B, using Bell state generators rather than 3-GHZ state generators.

[0133] FIG. 17B shows a high-level schematic for another photonic resource state generator circuit (RSG) 1750 according to some embodiments. RSG 1750 is a simplified variation of RSG 1700 that eliminates one Bell state generator and the two-way fusion circuit. RSG 1750 includes five Bell state generator circuits 1761, 1762, 1771-1773, each of which can be, e.g., an instance of Bell state generator circuit 400 described above. Each Bell state generator circuit 1761, 1762, 1771-1773 can generate a pair of qubits in a Bell statefrom single-photon inputs. Three-way fusion circuits 1765, 1775 can each be, e.g., an instance of three-way fusion circuit 700 described above. Three-way fusion circuits 1765, 1775 can perform a joint projective measurement operation (also referred to as a “fusion” operation) on three input qubits and generate classical measurement outcome data. (The input qubits are consumed in the measurement operation.) Hadamard gate 1783 can be, e.g., a 50 / 50 beam splitter as described above, or another circuit that performs a basis rotation on a qubit. In operation, each of Bell state generator circuits 1761, 1762, 1771-1773 produces a pair of qubits in a Bell state. Three-way fusion circuit 1775 performs a three-way fusion operation on one qubit of each of the Bell pairs from Bell state generator circuits 1771-1773. The other output qubits from Bell state generators 1771 and 1772 correspond to qubits W1 and W2 of the resource state. The other output qubit from Bell state generator 1773 propagates through Hadamard gate 1783. Three-way fusion circuit 1765 performs a three- way fusion operation on one qubit of each of the Bell pairs from Bell state generator circuits 1761, 1762 and the qubit output from Hadamard gate 1783. The other output qubits from Bell state generator circuits 1761, 1762 correspond to qubits El and E2 of the resource state.

[0134] It should be understood that the RSG circuits described herein are illustrative and that variations and modifications are possible. For instance, as described above with reference to FIG. 10C, each of 3-GHZ circuits 1611, 1612 can be implemented using multiple instances of a 3-GHZ circuit, with the outputs being multiplexed onto the input paths of type II fusion circuit 1625. Classical control logic can be used to select instances of the 3-GHZ circuit that succeeded, increasing the probability of successful generation of resource state 1412. Similarly, in FIGs. 17A and 17B, each of the Bell state generator circuits can be implemented using multiple instances of a Bell state generator circuit, with the outputs being multiplexed onto the input paths of the fusion circuits. In some embodiments, resource states 1411-1413 can be constructed using encoded qubits, and the qubit lines in fusion network 1400 can be understood as representing encoded qubits (e.g., using the encoding shown in FIG. 10A) rather than single physical qubits.3. Interleaving Modules for Fusion Networks

[0135] Fusion networks such as fusion network 1400 can be implemented using a wide variety of technologies and circuits, including but not limited to photonic circuits. Those skilled in the art with the benefit of this disclosure will appreciate that implementing fusion network 1400 (or similar fusion networks) can be hardware-intensive. According to some embodiments, hardware requirements can be greatly reduced (with the primary designtradeoff being increased computation time) by employing interleaving techniques, in which unit cells are formed and measurement data extracted sequentially rather than in parallel. Examples of interleaving modules that can be implemented as photonic circuits and used to execute fusion networks that incorporate three-way fusions, such as fusion network 1400, will now be described. For simplicity of description, the layers of fusion network 1400 are assumed to be square (L rows of L unit cells each) Other geometries can be used.

[0136] “Interleaving,” as used herein, refers generally to hardware and processes in which resource states are produced at regular intervals (referred to herein as “cycles” or “RSI cycles”), and unit cells within a layer of a fusion network are executed sequentially rather than in parallel as the qubits from different resource states become available. To allow different qubits of a given resource state to be operated on at different times, interleaving hardware can include storage for qubits that are produced during one cycle and operated on during a later cycle. Such storage is sometimes referred to as “qubit memory” and can be of a first-in, first-out type rather than addressable memory. For example, in photonic implementations, optical fiber or other waveguides that delay propagation of a qubit can provide the memory.

[0137] FIG. 18 shows a view of fusion network 1400 with an “interleaving order” assigned in accordance with some embodiments. In this example, the interleaving order proceeds from west to east along a “row” 1810 of unit cells. (The rows are demarcated by dashed lines to aid visualization.) After the end of the row is reached, the interleaving order proceeds to the next row 1820, again from west to east. Rows 1830 and 1840 are executed in turn. After the last row in a layer (e.g., row 1840) is completed, interleaving can proceed to the next layer, e.g., beginning with row 1810 in the next layer. For convenience, it can be assumed that network 1400 has a size I. '!. (where L is an integer greater than 1), that is, each row includes L unit cells, and there are L rows.

[0138] For purposes of interleaving, a “cycle” (also referred to as an “RSI cycle”) can be defined. During each cycle, a set of resource states is generated, and fusion operations for which all input qubits are available are performed. Other resource-state qubits can be stored in qubit memory (e.g., optical fibers) to be consumed in a fusion operation in a subsequent cycle. In FIG. 18, each resource state is annotated with an interleaving coordinate (1-16) indicating the cycle in which that resource state is generated. For instance, resource states 1411, 1412, 1413 are all generated during cycle 7. Interleaving for fusion network 1400 caninclude generating three resource states per cycle and performing fusion operations that can be performed during that cycle, while storing the remaining qubits of the resource states (e.g., in a delay line such as an optical fiber) for use in fusion operations in subsequent cycles.

[0139] FIG. 19 shows a high-level schematic diagram of an interleaving module 1900 according to some embodiments. (Unit cell 1300 of FIG. 13 is shown next to interleaving module 1900 for convenience.) Interleaving module 1900 includes three resource state interconnects (RSI) 1902, 1904, 1906 and four three-way fusion circuits 1911-1914. Each RSI 1902, 1904, 1906 outputs a four-qubit resource state (e.g., resource states 1411, 1412, 1413 described above) per cycle. In some embodiments, RSIs 1902, 1904, 1906 can incorporate RSGs implemented in accordance with examples described above (e.g., RSG 1600, RSG 1700, or RSG 1750), and the four-qubit resource states can be produced within RSIs 1902, 1904, 1906. In other embodiments, the RSGs can be external to interleaving module 1900. For example, resource state generation may be a non-determini stic (or stochastic) process that succeeds with probability less than 1. In some embodiments, a number (more than three) of RSG circuits can be operated in parallel, with multiplexers or similar circuits being used to selectably couple photons from RSG circuits that succeed into RSIs 1902, 1904, 1906. Where RSGs are external to interleaving module 1900, RSIs 1902, 1904, 1906 can be implemented as optical fibers, optical port couplings, or other passive waveguides that route the qubits from the RSGs to the appropriate routing paths to implement unit cell 1300. In FIG. 19, labels DI, D2, Ul, U2, Wl, W2, El, E2, Nl, N2, SI, and S2 are used to identify routing paths for the qubits at the outputs of RSIs 1902, 1904, 1906, and corresponding qubit edges are labeled in unit cell 1300. Routing paths Nl, Ul, and El are coupled to three-way fusion circuit 1911; routing paths U2, Wl, and SI are coupled to three- way fusion circuit 1912; routing paths DI, W2, and N2 are coupled to three-way fusion circuit 1913; and routing paths D2, E2, and S2 are routed to three-way fusion circuit 1914. It should be understood that, like other paths described herein, each routing path corresponds to a qubit and may include one or more waveguides (e.g., one waveguide for polarization encoding or temporal encoding, or two waveguides for a dual-rail spatial encoding).

[0140] Three-way fusion circuits 1911-1914 can operate on sets of input qubits generated during different cycles. For example, routing paths Wl and W2 include respective delay lines 1921, 1922 that each introduce a delay of one cycle. Routing paths Nl and N2 include respective delay lines 1923, 1924 that each introduce a delay of L cycles (where L is thenumber of unit cells in a row of fusion network 1400). Routing paths DI and D2 include respective delay lines 1925, 1926 that each introduce a delay of L2cycles.

[0141] Three-way fusion circuits 1911-1914 can be reconfigurable fusion circuits that can selectably perform different operations on their input qubits. FIG. 20 shows an example of a reconfigurable three-way fusion circuit 2040 according to some embodiments.Reconfigurable three-way fusion circuit 2040 or similar circuits can be used to implement three-way fusion circuits 1911-1914 and any other reconfigurable fusion circuits described herein. Reconfigurable three-way fusion circuit 2040 receives input qubits on three routing paths 2041, 2042, 2043. Each input routing path 2041-2043 is coupled to an active switch 2050, 2060, 2070. Each of switches 2050, 2060, 2070 can be, e.g., a 1><5 routing switch that selectably couples the input to one of five possible output paths, depending on the particular measurement desired. Switch 2050 has output paths coupled to each of five “destinations”: three-way fusion circuit 2080, Pauli X measurement circuit 2051, Pauli Y measurement circuit 2052, Pauli Z measurement circuit 2053, and phase rotation circuit 2054, which provides its output to Pauli Z measurement circuit 2055. Switch 2060 also has output paths coupled to each of five “destinations”: three-way fusion circuit 2080, Pauli X measurement circuit 2061, Pauli Y measurement circuit 2062, Pauli Z measurement circuit 2063, and phase rotation circuit 2064, which provides its output to Pauli Z measurement circuit 2065. Switch 2070 also has output paths coupled to each of five “destinations”: three-way fusion circuit 2080, Pauli X measurement circuit 2071, Pauli Y measurement circuit 2072, Pauli Z measurement circuit 2073, and phase rotation circuit 2074, which provides its output to Pauli Z measurement circuit 2075. The Pauli X, Y, and Z measurements are defined for qubits, and each Pauli measurement circuit 2051-2053, 2055, 2061-2063, 2065, 2071-2073, 2075 can include a basis rotation (for the X, Y, or Z basis as appropriate), which can be implemented using mode couplers and phase shifters, followed by a detector coupled to each mode. For instance, where qubits are represented in a dual-rail encoding, a detector can be coupled to the end of each of the two waveguides representing a qubit. The measurement result can include a number of photons detected by each detector, or a binary-valued signal from each detector indicating whether a photon was detected or not.

[0142] Three-way fusion circuit 2080 can be, e.g., a three-way fusion circuit as described above with reference to FIGs. 7A and 7B. Three-way fusion circuit 2080 can provide stabilizer measurements (e.g., in the X basis or the Z basis) on a set of three input qubits. For instance, three-way fusion circuit 2080 can include photon-counting detectors coupled to theends of the waveguides as described above, and can provide a measurement result indicating the number of photons in each detector.

[0143] Phase shift circuits 2054, 2064, 2074 each apply a phase shift of et7T / 8prior to a Pauli Z measurement circuits 2055, 2065, 2076. In some embodiments, this phase shift path can be used in generating magic states.

[0144] The example destinations shown for reconfigurable three-way fusion circuit 2040 are illustrative, and different destinations and / or combinations of destinations can be provided. For instance, in some embodiments, single-qubit Pauli Y measurements can be omitted. As another example, multiple instances of three-way fusion circuit 2080 configured to measure in different bases can be provided. As yet another example, it may be desirable to provide destinations that perform two-way fusion operations on a pair of the qubits (with the remaining qubit in any given instance being routed to a single-qubit measurement).

[0145] Switches 2050, 2060, 2070 are controlled by classical control logic 2090. Classical control logic 2090 can be implemented as a digital logic circuit with an arrangement of classical logic gates (AND, OR, NOR, XOR, NAND, NOT, etc.), such as a field programmable gate array (FPGA) or system-on-a-chip (SOC) having a programmable processor and memory, or an on-chip hard-wired circuit, such as an application specific integrated circuit (ASIC). In some embodiments, switches 2050, 2060, 2070 are coupled to an off-chip classical computer having a processor and a memory, and the off-chip classical computer is programmed to perform some or all of the operations of classical control logic 2090. In some embodiments, classical control logic 2090 (which can include an off-chip classical computer) can be provided with program code indicating the type of measurement desired for each triplet of qubits input to reconfigurable three-way fusion circuit 2040. The particular type of measurement can be determined from a fusion graph defining a particular quantum computation to be executed; those skilled in the art will be familiar with techniques for generating fusion graphs. Classical control logic 2090 can send control signals to switches 2050, 2060, 2070 to configure reconfigurable three-way fusion circuit 2040 to perform the desired measurements at the desired time.

[0146] Classical control logic 2090 can also receive the classical output signals (measurement outcome data) from measurement circuits 2051-2053, 2055, 2061-2063, 2065, 2071-2073, 2075, and three-way fusion circuit 2080. In some embodiments, classical control logic 2090 can execute decoding logic to interpret the results of quantum computations basedon the measurement outcome data, and in some instances, results of the decoding logic can be used as inputs to determine subsequent settings for switches 2050, 2060, 2070. In addition or instead, classical control logic 2090 can provide measurement outcome data to other systems or devices, which can decode the measurement outcome data and / or perform other operations using the measurement outcome data.

[0147] Referring again to FIG. 19, operation of interleaving module 1900 will now be described. During each cycle, each of RSIs 1902, 1904, 1906 provides a resource state having four qubits. RSI 1902 provides its four qubits to routing paths DI, D2, Ul, U2; RSI 1904 provides its four qubits to routing paths Wl, W2, El, E2; and RSI 1906 provides its four qubits to routing paths Nl, N2, SI, and S2. During the cycle, three-way fusion circuit 1911 performs a selected operation (e.g., three-way fusion or other operation as described with reference to FIG. 20) on qubits from routing paths Ul and El that are routed from RSIs 1902, 1904, 1906 during the current cycle and a qubit on routing path Nl that was routed from RSI 1906 L cycles previously and delayed by delay line 1923. (Referring to FIG. 18, the Nl qubit “comes from” the neighboring unit cell in the N direction in fusion network 1400.) Three-way fusion circuit 1912 performs a selected operation (e.g., three-way fusion or other operation as described with reference to FIG. 20) on qubits from routing paths U2 and SI that are routed from RSIs 1902, 1906 during the current cycle and a qubit from routing path Wl that was routed from RSI 1904 during the immediately preceding cycle and delayed by delay line 1921. (Referring to FIG. 18, the Wl qubit “comes from” the neighboring unit cell in the W direction in fusion network 1400.) Three-way fusion circuit 1913 performs the selected operation (e.g., three-way fusion or other operation as described with reference to FIG. 20) on a qubit from routing path W2 that was routed from RSI 1904 during the immediately preceding cycle and delayed by delay line 1922, a qubit on routing path N2 that was routed from RSI 1906 L cycles previously and delayed by delay line 1924, and a qubit on routing path DI that was routed from RSI 19021 cycles previously and delayed by delay line 1925. (Referring to FIG. 18, the W2 qubit “comes from” the neighboring unit cell in the W direction in fusion network 1400, the N2 qubit “comes from” the neighboring unit cell in the N direction in fusion network 1400, and the DI qubit “comes from” the corresponding unit cell in the previous layer in fusion network 1400.) Three-way fusion circuit 1914 performs the selected operation (e.g., three-way fusion or other operation as described with reference to FIG. 20) on qubits from routing paths E2 and S2 that are routed from RSIs 1904, 1906 during the current cycle and a qubit from routing path D2 thatwas routed from RSI 1902 / ? cycles previously and delayed by delay line 1926. (Referring to FIG. 18, the D2 qubit “comes from” the corresponding unit cell in the previous layer in fusion network 1400.) Over a sequence of L2cycles, interleaving module 1900 can generate a layer of fusion network 1400, then proceed to the next layer.

[0148] As described above, the particular measurement operation performed by each of three-way fusion circuits 1911-1914 can be selected for each cycle using classical control logic 2090 or the like. In a given cycle, operations can be selected independently for each of three-way fusion circuits 1911-1914. In some embodiments, three-way fusions are selected to generate the bulk of the fusion network, with single-qubit measurements, phase rotations, and / or other operations being selected at boundaries of the fusion network.

[0149] In some embodiments, interleaving module 1900 can generate a fusion network of arbitrary size, at a rate of one unit cell per cycle. Accordingly, if the fusion network graph for a given quantum computation has size I M (where A is the number of layers), interleaving module 1900 can perform the computation in L2M cycles.

[0150] To speed up computations, it may be desirable to provide a network of connected interleaving modules. FIG. 21 shows a high-level schematic diagram of an interleaving module 2100 according to some embodiments. Interleaving module 2100 is similar in many respects to interleaving module 1900 described above. For example, interleaving module 2100 can include three resource state interconnects (RSI) 2102, 2104, 2106 and four reconfigurable three-way fusion circuits 2111-2114, which can be similar or identical to corresponding components of interleaving module 1900. Interleaving module 2100 also includes a first set of routing switches 2121-2124 coupled to the outputs of RSI 2104 and a second set of routing switches 2131-2134 coupled to the outputs of RSI 2106. Each of routing switches 2121-2124 and 2134-2134 can be a separately controllable active optical switch (e.g., a 1 x2 switch) that can route the qubit onto either a “local” routing path (routing paths labeled WILoc, L2Loc, and so on) or a “network” routing path (labeled WINet, W2Net, and so on). The local routing paths correspond to the routing paths of interleaving module 1900 and include corresponding delays. That is, each of local routing paths W ILoc and W2Loc includes a 1-cycle delay, and each of local routing paths NILoc and N2Loc includes a delay of P cycles. (As described below, P is a “patch size” corresponding to the portion of each layer of fusion network 1400 that is executed by interleaving module 2100. In general, P is a smaller number than the layer size Z.) Each local routing path has acorresponding network path. The configuration of delays in this example is such that if the local routing path includes a delay (e.g., local routing paths WILoc, W2Loc, NILoc, N2Loc), then the corresponding network routing path includes no delay (e g., network routing paths WINet, W2Net, NINet, N2Net), and if the local routing path does not include a delay (e.g., local routing paths ElLoc, E2Loc, SILoc, S2Loc), then the corresponding network routing path includes a delay (e g., network routing paths ElNet, E2Net include 1 -cycle delays; network routing paths SINet, S2Net include T’-cycle delays). The network routing paths can connect to other instances of interleaving module 2100. For instance, network routing paths WINet, W2Net, ElNet, E2Net can exit one instance of interleaving module 2100 at exit port 2140 and enter another instance of interleaving module 2100 at entrance port 2142.Similarly, network routing paths NINet, N2Net, SINet, S2Net can exit one instance of interleaving module 2100 at exit port 2141 and enter another instance of interleaving module 2100 at entrance port 2143. Routing paths DI, D2, Ul, and U2 are similar to the corresponding routing paths of interleaving module 1900, except that the delay is P2cycles rather than L2cycles.

[0151] Additional routing switches 2151-2158 can be provided to deliver local or networked qubits to reconfigurable three-way fusion circuits 2111-2114. Each of routing switches 2151-2158 can be a separately controllable active optical switch (e.g., a 2>< 1 switch) that can route a qubit from either a local routing path a corresponding network routing path from entrance port 2142 or entrance port 2143 to inputs of reconfigurable three-way fusion circuits 2111-2114. For example, routing switch 2151 selects a qubit from either local routing path ElLoc or network routing path ElNet (from entrance port 2142) as the El input to reconfigurable three-way fusion circuit 2111. Routing switch 2152 selects a qubit from either local routing path NILoc or network routing path NINet (from entrance port 2143) as the N1 input to reconfigurable three-way fusion circuit 2111. Routing switch 2153 selects a qubit from either local routing path WILoc or network routing path WINet (from entrance port 2142) as the W1 input to reconfigurable three-way fusion circuit 2112. Routing switch 2154 selects a qubit from either local routing path SILoc or network routing path SINet (from entrance port 2143) as the SI input to reconfigurable three-way fusion circuit 2112. Routing switch 2155 selects a qubit from either local routing path W2Loc or network routing path W2Net (from entrance port 2142) as the W2 input to reconfigurable three-way fusion circuit 2113. Routing switch 2156 selects a qubit from either local routing path N2Loc or network routing path N2Net (from entrance port 2143) as the N2 input to reconfigurablethree-way fusion circuit 2113. Routing switch 2157 selects a qubit from either local routing path E2Loc or network routing path E2Net (from entrance port 2142) as the E2 input to reconfigurable three-way fusion circuit 2114. Routing switch 2158 selects a qubit from either local routing path S2Loc or network routing path S2Net (from entrance port 2143) as the S2 input to reconfigurable three-way fusion circuit 2114.

[0152] By operating routing switches 2121-2124, 2134-2134, and 2151-2158, a network of interleaving modules 2100 connected via exit ports 2140, 2141 and entrance ports 2142, 2143 can cooperate to generate layers of a fusion network such as fusion network 1400.

[0153] FIG. 22 shows a simplified representation of a network 2200 of interleaving modules according to some embodiments. In this example, network 2200 includes nine instances of interleaving module 2100 (numbered as 2100-1 through 2100-9).

[0154] Interleaving modules 2100 can be connected to neighboring interleaving modules 2100 via exit ports 2140, 2141 and entrance ports 2142, 2143. For example, interleaving module 2100-5 has exit port 2140 coupled to entrance port 2142 of interleaving module 2100-6, exit port 2141 coupled to entrance port 2143 of interleaving module 2100-8, entrance port 2142 coupled to exit port 2140 of interleaving module 2100-4, and entrance port 2143 coupled to exit port 2141 of interleaving module 2100-2.

[0155] Interleaving modules 2100-1 through 2100-9 can operate in parallel, with each interleaving module 2100-1 through 2100-9 computing a different contiguous patch of unit cells for a layer of fusion network 1400. For example, if a layer includes I. '!. unit cells, each of interleaving modules 2100-1 through 2100-9 can compute a patch ofPxP unit cells, where P is an integer less than L. (Thus the delays of 1, P and P2In some instances L / P can be an integer, although this is not required.

[0156] Operation of each of interleaving modules 2100-1 through 2100-9 can proceed such that local routing paths are selected in each cycle (which provides the same behavior as described with reference to interleaving module 1900), except in instances where the resource states cross patch boundaries. In that case, network routing paths can be selected to transfer qubits between different ones of interleaving modules 2100-1 through 2100-9. For instance, at the “E” edge of a patch, interleaving module 2100-5 can operate its routing switches 2123, 2124 (shown in FIG. 21) to select network routing paths ElNet, E2Net and interleaving module 2100-6 can operate its routing switches 2151, 2157 to select network routing paths ElNet, E2Net, thereby transferring qubits provided by RSI 2104 in interleaving module2100-5 to interleaving module 2100-6, where fusion operations can be performed. The configuration of delay lines shown in FIG. 21 can provide appropriate timing for the fusion operations.

[0157] Using a network of interleaving modules as shown in FIG. 22, rather than a single interleaving module as shown in FIG. 20, can speed up computation time, e.g., by a factor of M, where / V / is the number of interleaving modules. The design tradeoff is that more interleaving modules require more optical circuitry (by approximately a factor of M). The optimum tradeoff between computational speed and hardware is a matter of design choice.

[0158] FIG. 23 is a flow diagram of a process 2300 for performing an RSI cycle in one or more interleaving modules according to some embodiments. Process 2300 can be implemented, e.g., using interleaving module 1900 or interleaving module 2100 described above. At block 2302, resource states are provided to the interleaving module(s). As described above, each resource state can include a system of qubits that are entangled with each other. The resource states are separate quantum systems; in other words, qubits of different resource states are not entangled with each other. The number of resource states provided per cycle depends on the number of interleaving modules and the number of RSIs per interleaving module. At block 2304, resource states are routed to the reconfigurable fusion circuits, such that each reconfigurable fusion circuit receives a qubit from each of three (or more) of the resource states. Depending on implementation the routing for a particular qubit can be local (e.g., within one interleaving module) or networked (e.g., transferring the qubit from one interleaving module to another as described above with reference to FIGs. 21 and 22). As described above, routing of qubits can include introducing delays, such that a qubit received at a particular fusion circuit was provided in a previous cycle. At block 2306, each reconfigurable fusion circuit performs a selected one of a plurality of measurement operations on the received qubits to produce measurement data. At least some of the selected measurement operations can be three-way fusion operations. For instance, three-way fusion operations can be used for the bulk of a fusion network graph, with other operations (e.g., single-qubit measurements and / or two-way fusion operations) performed at the boundaries. The measurement operations selected for different reconfigurable fusion circuits in a given cycle can be the same or different. As described above, the measurement operations produce (classical) measurement outcome data. This data can be provided to classical control logic, which can use the data to determine a result of the quantum computation and / or to determine measurement operations to be performed in futurecycles. For instance, the classical control logic can implement appropriate decoder algorithms. (Examples of techniques for decoding measurement data produced by executing fusion network graphs are known in the art.)

[0159] It should be understood that the interleaving modules and processes described herein are illustrative and that variations and modifications are possible. For example, the qubits can be individual physical qubits or encoded qubits as described above. (The latter may involve additional hardware, with the tradeoff being improvements in fault-tolerance.) A network (or array) of interleaving modules can include any number of interleaving modules, and connectivity of interleaving modules can be increased beyond the examples described herein.4. Computing Systems Implementing FBQC

[0160] FIG. 24 shows an example system architecture for a quantum computer system 2400 that can implement FBQC according to some embodiments. Using photonic physical qubits, some embodiments of quantum computer system 2400 can generate measurement data reflecting entanglement structures (e.g. fusion graphs) for fault-tolerant FBQC. System 2400 includes classical control logic 2410, a resource state generator subsystem 2402, and a network 2412 of interleaving modules 2420. For clarity of illustration, classical signal paths 2432-2437 are shown connected to only one instance of interleaving module 2420. It should be understood that classical control logic 2410 can communicate with components in each instance of interleaving module 2420 in the manner described herein.

[0161] Classical control logic 2410 can be implemented as a digital logic circuit with an arrangement of classical logic gates (AND, OR, NOR, XOR, NAND, NOT, etc.), such as a field programmable gate array (FPGA) or system-on-a-chip (SOC) having a programmable processor and memory, or an on-chip hard-wired circuit, such as an application specific integrated circuit (ASIC). In some embodiments, classical control logic 2410 (or portions thereof) can be implemented in an off-chip classical computer having a processor and a memory, and the off-chip classical computer can be programmed to perform some or all of the operations of classical control logic 2410.

[0162] In operation, classical control logic 2410 (which can include a classical computer) can receive “program code” 2401 specifying a quantum computation (or other fault-tolerant quantum operation) to be executed. For example, the program code can include a machine- readable data file defining a fusion graph that represents the computation. Classical controllogic 2410 can read the program code and generate control signals for resource state generator subsystem 2402 and interleaving modules 2420 to perform the computation.

[0163] Resource state generator subsystem 2402 can include any circuit(s) or other components capable of generating resource states, which can be systems of photonic qubits (e.g., using dual-rail encoding as described above). For instance, resource state generator subsystem 2402 can be an implementation of qubit entangling system 101 of FIG. 1 and can include multiple instances of RSG circuits such as RSG 1600 (FIG.16B), RSG 1700 (FIG. 17A), or RSG 1750 (FIG. 17B) described above. In various embodiments, resource state generator subsystem 2402 can generate resource states for a particular fusion network such as resource states 1411, 1412, 1413 for fusion network 1400 (or other resource states having an appropriate number of qubits and entanglement pattern for a given fusion network). In some embodiments, resource state generator subsystem 2402 can be reconfigurable to generate resource states having different entanglement patterns during different operating cycles, and classical control logic 2410 can send classical control signals via signal path 2430 to resource state generator subsystem 2402, e.g., to start and stop resource state generation and / or to select the number or type of resource states to generate during each operating cycle. (The operating cycle of resource state generator subsystem 2402 can correspond to the RSI cycle of interleaving modules 2420.) In some embodiments, resource state generator subsystem 2402 may succeed in generating the desired number of resource states for a given operating cycle with probability less than 1, and resource state generator subsystem 2402 can provide classical heralding signals to classical control logic 2410 via signal path 2431. The classical heralding signals can include, e.g., signals from detectors associated with heralded photon sources and / or entanglement-generating circuits such as the Bell state generator and / or fusion circuits described above. Classical control logic 2410 can use heralding signals received via signal path 2431 to determine whether each instance of resource state generation succeeded or failed. For instance, particular patterns of presence or absence of photons in detectors can be indicative of success or failure. In some embodiments, the number of RSG circuits in resource state generator subsystem 2402 can be larger than the number of resource states consumed per RSI cycle of interleaving modules 2420, and resource state generator subsystem 2402 can include multiplexers, switch networks, or the like to route resource states to interleaving modules 2420. In some embodiments, resource state generator subsystem 2402 can be maintained at cryogenic temperature (e.g., 4 K) while interleaving modules 2420 (or portions thereof) can operate at higher temperatures (e.g., 300 K). Resource stategenerator subsystem 2402 can be coupled to interleaving module network 2412 using optical fiber or other waveguides and can provide one or more resource state per operating cycle to each interleaving module 2420, depending on the particular fusion network.

[0164] Each interleaving module 2420 can be an instance of interleaving module 2100 of FIG. 21 or any other interleaving module, including examples described above. As shown in FIG. 24, each interleaving module 2420 can include one or more RSIs 2422, a set of routing switches 2424, and a set of reconfigurable fusion circuits 2426. Details of couplings between components within each interleaving module 2420 and between interleaving modules 2420 are not shown in FIG. 24. It should be understood that any of the coupling schemes described above or other schemes that support execution of fusion networks having a particular topological form can be used.

[0165] Each RSI 2422 can receive resource states as described above. In some embodiments, the RSIs 2422 can operate autonomously, with no data input required, and each RSI 2422 circuit can receive one resource state per operating cycle (also referred to as an RSI cycle). Any of the RSI circuit configurations described above or other configurations can be used. If desired, resource state generation can be implemented internally to each RSI 2422 rather than in a separate resource state generator subsystem 2402.

[0166] Optical fibers (or other waveguides) 2442 can be used to couple each RSI 2422 to its associated routing switches 2424. In some embodiments, the optical fibers (or other waveguides) 2442 can introduce appropriate relative delay into the propagation paths of different qubits of the same resource state. For example, optical fibers 2442 can implement the delay lines of lengths 1, P, and P2shown in FIG. 21.

[0167] Classical control logic 2410 can generate control signals for routing switches 2424 in each instance of interleaving module 2420 and send the control signals to routing switches 2424 via classical signal path 2434. As described above, in some embodiments routing switches 2424 can route qubits from RSI 2422 to either a local path 2444a or a network path 2444b. (For instance, routing switches 2424 can include routing switches 2121-2124, 2131- 2134, and 2151-2158 of FIG. 21). Local path 2444a and network path 2444b transfer the qubits to reconfigurable fusion circuits 2426. As described above, local path 2444a connects to reconfigurable fusion circuit 2426 in the same interleaving module 2420 while network path 2444b connects to reconfigurable fusion circuit 2426 in a different interleaving module 2420. For clarity of illustration, FIG. 24 shows one local path and one network path;however, it should be understood that multiple paths of either type can be provided and that the routing paths for different qubits of a given resource state can be selected independently of each other. In some embodiments, classical control logic 2410 can select routing paths and corresponding control signals for routing switches 2424 for each RSI cycle based on a fusion graph representation of a quantum computation.

[0168] In some embodiments, the set of all routing switches 2424 across all instances of interleaving module 2420 can provide a fusion network router 2450. In some embodiments, fusion network router 2450 can be a reconfigurable fusion network router that supports different layer topologies, including examples described above, without requiring changes to the underlying hardware. For instance, alternative network routing paths 2444b can be provided between a routing switch 2424 in one interleaving module 2420 and reconfigurable fusion circuits 2426 in each of two or more other interleaving modules 2420. In various embodiments, network routing paths can be provided between any routing switch and any reconfigurable fusion circuit. In an extreme case, every routing switch can be connected to every reconfigurable fusion circuit; however, for fusion graphs having regular lattice structures (as in examples described above), not all possible connections are useful connections, and the set of local paths 2444a and network paths 2444b in a given implementation can be based on the fusion graph topologies that system 2400 is intended to support.

[0169] Classical control logic 2410 can also generate control signals for reconfigurable fusion circuits 2426 in each instance of interleaving module 2420 and send the control signals to reconfigurable fusion circuits 2426 via classical signal path 2436. As described above, in some embodiments each reconfigurable fusion circuit 2426 can be an implementation of reconfigurable three-way fusion circuit 2040 of FIG. 20 that operates on three input qubits. Reconfigurable three-way fusion circuit 2040 can be controlled by providing classical control signals to select the state of switches 2050, 2060, 2070, which has the effect of routing the three input qubits to the desired measurement operation(s), which can include either a three- qubit joint measurement operation or individual qubit measurements (e.g., in a particular Pauli basis) on each of the input qubits. More generally, reconfigurable fusion circuits 2426 can receive a number n of qubits (where n > 2) and selectably perform an / / -way fusion operation on all of the input qubits, a single-qubit measurement on each input qubit and / or other combinations of measurement operations on one or more of the input qubits (e.g., a two-way type II fusion on two of the qubits plus a single-qubit measurement on a third qubit).In some embodiments, classical control logic can select the desired measurement operations for each RSI cycle based on a fusion graph representation (or other representation) of a quantum computation.

[0170] Measurement outcome data (also referred to as “measurement results”) generated by reconfigurable fusion circuit 2426 can be provided to classical control logic 2410 via classical signal path 2437. As described above, in some embodiments, the measurement outcome data can include photon counts (or a binary-valued signal indicating presence or absence of a photon) for each detector in the reconfigurable fusion circuit or for the detector(s) on the active path(s) in a given cycle.

[0171] Classical control logic 2410 can decode the measurement outcome data received via classical signal path 2437 to determine a result of the quantum computation. In some embodiments, classical control logic 2410 can also incorporate the heralding signals received via signal paths 2433 into the decoding. Further description of decoding operations that can be implemented in classical control logic 2410 can be found in above-referenced WO 2021 / 155289.

[0172] FIG. 25 is a flow diagram of a process 2500 for operating an array of interleaving modules (e.g., interleaving module network 2412 of FIG. 24) according to some embodiments. Process 2500 can be implemented, e.g., in classical control logic 2410 coupled to interleaving module network 2412. At block 2502, the classical control logic can obtain a machine-readable representation of a fusion graph corresponding to a quantum computation (or other operation on logical qubits) to be executed. At block 2504, the classical control logic can determine an interleaving coordinate for a current RSI cycle based on a cycle counter. For example, the cycle counter can be initialized (e.g., to 1) for the first cycle and incremented by 1 for each cycle. An interleaving coordinate can be determined for each interleaving module. For instance, if each interleaving module is assigned a patch of size P P unit cells (this is sometimes referred to as an “interleaving length” of P), the interleaving coordinate for a particular interleaving module can define a location within the patch assigned to that interleaving module. At block 2506, the classical control logic can select routing switch settings and measurement operations for the reconfigurable fusion circuits based on the interleaving coordinate and the fusion graph. For instance, routing switch settings can be determined based on the location within the patch, as identified by the interleaving coordinate. In some embodiments using a network of interleaving modules2100, routing switch settings can be set to select local routing paths except at patch boundaries, where network paths can be selected. Measurement operations for the reconfigurable fusion circuits can be determined by consulting the fusion graph, which indicates an operation for each interleaving coordinate. (Since different interleaving modules are executing different patches, the measurement operations are selected separately for each reconfigurable fusion circuit in each interleaving module.) At block 2508, classical control logic can initiate an RSI cycle, e.g., by sending control signals based on the routing switch settings and measurement operations to the interleaving modules; the RSI cycle can correspond to process 2300 described above. At block 2510, classical control logic can receive the measurement outcome data from the interleaving modules. The measurement outcome data can include data produced by the reconfigurable fusion circuits, as well as other ancillary data (e g., data indicative of whether the resource states were successfully generated during the RSI cycle).

[0173] At block 2512, the classical control logic can determine whether the quantum computation has been completed, e g., whether the entire fusion graph has been executed. If not, then at block 2514, the RSI cycle counter can be incremented, and process 2500 can return to block 2504 to determine the next interleaving coordinate and process the next RSI cycle. Process 2500 can continue to iterate until the computation is completed, ending at block 2530. It should be understood that all interleaving modules can be operated in parallel, with photons propagating between different interleaving modules based on the settings of the routing switches. Delay lines within (or between) interleaving modules can be provided so that qubits from different resource states arrive at the reconfigurable fusion circuits with the correct relative timing to execute the fusion graph.

[0174] System 2400 of FIG. 24 and process 2500 of FIG. 25 are illustrative, and variations and modifications are possible. Blocks shown separately can be combined, or a single block can be implemented using multiple distinct components or operators. Order of operations can be varied to the extent that logic permits, and operations described as sequential can be performed concurrently. Interleaving modules 2420 can be implemented according any of the interleaving modules described above or variations or modifications thereof.

[0175] System 2400 is just one example of a quantum computer system that can incorporate interleaving modules as described herein to perform operations on logical qubits or other operations that can be defined using fusion graphs, including operations related toquantum computation, quantum communication, and other applications. Those skilled in the art with access to this disclosure will appreciate that many different systems can be implemented. Further, determination of interleaving coordinates and operations to be performed for a given interleaving coordinate can be based on a fusion graph or any other input that specifies a set of operations to be performed on a set of resource states.Accordingly, interleaving modules can be used in a variety of applications, including but not limited to FBQC.5. Additional Embodiments

[0176] All embodiments described herein are illustrative, and many modifications are possible. Photonic qubits can be implemented using dual-rail encodings as described above, other spatio-temporal encodings, polarization encodings, GKP qubit encoding, or any other encoding that provides the state behavior of a physical qubit. Further, while photonic qubits are particularly well suited for interleaving architectures (due in part to the relative ease of moving photonic qubits between locations), fusion networks of the kind described herein are not limited to photonic qubits; any physical system that can be used as a qubit can be used to implement a fusion network architecture of the kind described herein, provided that three- way fusion operations can be implemented.

[0177] The following are example embodiments:

[0178] Example 1 : A method comprising: providing, to a system having a plurality of reconfigurable fusion circuits, a plurality of resource states, each resource state including a plurality of qubits that are entangled with each other, wherein qubits of different resource states are not entangled with each other; routing qubits of the resource states to the reconfigurable fusion circuits such that each fusion circuit receives a qubit from each of three or more of the resource states; and performing, by each reconfigurable fusion circuit, a selected one of a plurality of measurement operations on the received qubits to produce measurement data, wherein at least some of the selected measurement operations are three- way fusion operations.

[0179] Example 2: The method of Example 1 wherein the qubits are routed to the reconfigurable fusion circuits according to a three-dimensional cell complex having vertices corresponding to resource states, edges corresponding to fusion measurements, and faces corresponding to check operators.

[0180] Example 3: The method of Example 2 further comprising: selecting, for each reconfigurable fusion circuit, one of the plurality of measurement operations, the selection being based on the three-dimensional cell complex.

[0181] Example 4: The method of Example 2 wherein the three-dimensional cell complex comprises a unit cell in a repeating pattern.

[0182] Example 5: The method of Example 4 wherein the unit cell corresponds to a cuboctahedral polyhedron.

[0183] Example 6: The method of any one of Examples 1 through 5 further comprising generating the resource states.

[0184] Example 7: The method of Example 6 wherein generating the resource states includes: generating a first subsystem of qubits in a 3-GHZ state and a second subsystem of qubits in a 3-GHZ state; and performing a two-way fusion operation on a qubit of the first subsystem and a qubit of the second subsystem.

[0185] Example 8: The method of Example 6 wherein generating the resource states includes: generating a plurality of Bell pairs of qubits; and performing fusion operations on qubits of different ones of the Bell pairs, wherein at least one of the fusion operations is a three-way fusion operation.

[0186] Example 9: The method of any one of Examples 1 through 8 wherein the acts of providing, routing, and performing are repeated for each of a plurality of cycles including a first cycle.

[0187] Example 10: The method of Example 9 wherein the act of routing the qubits includes: storing a qubit from one of the resource states generated during the first cycle in a memory; and providing the stored qubit to one of the reconfigurable fusion circuits in a subsequent one of the plurality of cycles.

[0188] Example 11 : The method of Example 10 wherein the memory includes an optical fiber that delays delivery of the qubit to the one of the reconfigurable fusion circuits for a fixed number of cycles.

[0189] Example 12: The method of Example 9 further comprising: for each cycle, generating control signals to the reconfigurable fusion circuits to select the measurement operations to be performed by the reconfigurable fusion circuits.

[0190] Example 13: A circuit comprising: a plurality of resource state interconnects, each resource state interconnect having a plurality of output paths to output a resource state during each of a plurality of operating cycles, wherein each resource state is a quantum system of multiple entangled qubits, wherein different qubits of the resource state are output on a different ones of the output paths, and wherein different resource state interconnects are associated with different directions in a multi-dimensional entanglement space; a plurality of first routing switches, each of the first routing switches having an input path coupled to a different one of the output paths of the resource state interconnects and a plurality of output paths, wherein each of the first routing switches is configured to receive a different qubit of one of the resource states on the input path and to selectably route the received qubit to one of the plurality of output paths; a plurality of routing paths including a plurality of local routing paths, a plurality of outbound network routing paths, and a plurality of inbound network routing paths, wherein each of the outbound network routing paths exits the circuit, and wherein each of the inbound network routing paths enters the circuit from an external source, wherein at least some of the routing paths include delay lines having different delay lengths; a plurality of reconfigurable fusion circuits, each of the plurality of reconfigurable fusion circuits being configured to receive three input qubits on three input paths and to selectably perform one of a plurality of measurement operations on the input qubits, thereby producing measurement outcome data, wherein the plurality of measurement operations including a projective entangling measurement between three input qubits and a plurality of single-qubit measurements on each of the three input qubits, wherein the local routing paths are coupled between the routing switches and the reconfigurable fusion circuits such that each of the routing switches is coupled to at least one of the reconfigurable fusion circuits; and a plurality of second routing switches, each of the second routing switches having a first input path coupled to one of the local routing paths, a second input path coupled to one of the inbound network routing paths, and an output path coupled to one of the input paths of one of the reconfigurable fusion circuits.

[0191] Example 14: The circuit of Example 13 wherein each resource state comprises four qubits.

[0192] Example 15: The circuit of Example 13 or Example 14 wherein each one of the reconfigurable fusion circuits receives qubits associated with three different orthogonal directions in the entanglement space.

[0193] Example 16: A system comprising: a network of interleaving modules, wherein each interleaving module includes: a plurality of resource state interconnects, each resource state interconnect having a plurality of output paths to output a resource state during each of a plurality of operating cycles, wherein each resource state is a quantum system of multiple entangled qubits, wherein different qubits of the resource state are output on a different ones of the output paths, and wherein different resource state interconnects are associated with different directions in a multi-dimensional entanglement space; a plurality of first routing switches, each of the first routing switches having an input path coupled to a different one of the output paths of the resource state interconnects and a plurality of output paths, wherein each of the first routing switches is configured to receive a different qubit of one of the resource states on the input path and to selectably route the received qubit to one of the plurality of output paths; a plurality of routing paths including a plurality of local routing paths, a plurality of outbound network routing paths, and a plurality of inbound network routing paths, wherein each of the outbound network routing paths exits the interleaving module, and wherein each of the inbound network routing paths enters the interleaving module from another interleaving module, wherein at least some of the routing paths include delay lines having different delay lengths; a plurality of reconfigurable fusion circuits, each of the plurality of reconfigurable fusion circuits being configured to receive three input qubits on three input paths and to selectably perform one of a plurality of measurement operations on the input qubits, thereby producing measurement outcome data, wherein the plurality of measurement operations including a projective entangling measurement between three input qubits and a plurality of single-qubit measurements on each of the three input qubits, wherein the local routing paths are coupled between the routing switches and the reconfigurable fusion circuits such that each of the routing switches is coupled to at least one of the reconfigurable fusion circuits; and a plurality of second routing switches, each of the second routing switches having a first input path coupled to one of the local routing paths, a second input path coupled to one of the inbound network routing paths, and an output path coupled to one of the input paths of one of the reconfigurable fusion circuits; and classical control logic coupled to the network of interleaving modules and configured to control the first and second routing switches and the reconfigurable fusion circuits and to receive classical data signals representing the measurement outcome data from the reconfigurable fusion circuits.

[0194] Example 17: The system of Example 16 further comprising: a plurality of resource state generator circuits coupled to the resource state interconnects and configured to generate resource states.

[0195] Example 18: The system of Example 17 wherein each resource state generator circuit comprises: a plurality of 3 -GHZ generator circuits configured to generate a plurality of 3 -GHZ states from a plurality of single photons, each 3 -GHZ state including three qubits; a two-way fusion circuit configured to perform a fusion operation on one qubit from each of two of the 3-GHZ states; a Hadamard gate configured to apply a basis rotation on one input of the two-way fusion circuit; a plurality of output paths connected to the inputs of one of the resource state interconnects and configured to provide the other two qubits from each of the 3-GHZ generator circuits to the resource state interconnect; and a plurality of photon sources configured to provide photons to the 3-GHZ generator circuits.

[0196] Example 19: The system of Example 17 wherein each resource state generator circuit comprises: a plurality of Bell state generator circuits configured to generate a plurality of Bell states from a plurality of single photons, each Bell state including two qubits; a plurality of fusion circuits configured to perform fusion operations on qubits of different ones of the Bell states, wherein at least one of the fusion operations is a three-way fusion operation; a plurality of output paths connected to the inputs of one of the resource state interconnects and configured to provide qubits from different ones of the Bell state generator circuits to the resource state interconnect; and a plurality of photon sources configured to provide photons to the Bell state generator circuits.

[0197] Circuits are described in examples herein as operating on single qubits, e.g., dualrail encoded qubits propagating on a pair of waveguides. In some embodiments, it may be desirable to implement a fusion network such as fusion network 1400 using encoded qubits; that is, each qubit (edge) in fusion network 1400 can be implemented as an entangled system of multiple physical qubits.

[0198] The use of directional labels (e.g., N, E, W, S, U, D) is for convenience of description and should be understood as referring to entanglement space, not as requiring or implying a particular physical arrangement of components or physical qubits. All numerical examples are for purposes of illustration and can be modified. In addition, while layers and patches are described with reference to square numbers, it should be understood that nonsquare layers and / or non-square patches can also be used. For example, patches or layers canbe rectangular. Triangular patches or layers (or patches or layers having other shapes) can also be generated, e.g., by varying the number of resource states per row. Further, while examples described above assume that all instances of a resource state have the same entanglement pattern, such uniformity is not required. For instance, in some embodiments, resource states having different entanglement patterns can be provided to a particular RSIs at various times. In addition, there may be stochastic variation among resource states, e g., due to the non-deterministic nature of resource state generation. To increase the probability of delivering a desired resource state to each RSI in a given cycle, some embodiments can provide a number (R) of resource state generator circuits. If is the total number of interleaving modules and each interleaving module includes three RSIs, then R can be greater than 3M, and R can be chosen to provide a sufficiently high probability that at least 3M resource states will be generated during a given cycle. (“Sufficiently high probability” in a given implementation can be determined based on the particular implementation of fault tolerance.) Active multiplexing techniques, examples of which are known in the art, can be used to select 3Mof the R resource state generators on each clock cycle to deliver resource states to the resource state interconnects of the A- / interleaving modules.

[0199] Some embodiments described above provide examples of implementing FBQC by providing resource states and performing appropriate measurements on qubits of different resource states. The particular size (number of qubits) and entanglement pattern of the resource states can be varied as appropriate for a particular use case. In addition or instead, the number of resource states and entanglement geometry between resource states can be varied according to the particular use-case. In some embodiments, resource states having different sizes and / or entanglement patterns can be used at different vertex positions within a fusion graph, and position-dependent selection of resource state configurations can be used to implement logical operations. Further, embodiments are not limited to FBQC and may be used in a variety of contexts, including measurement-base quantum computing (MBQC), other quantum computing systems, quantum communication systems, and any other context where it is desirable to perform measurements on a system involving a large number of physical qubits having a specified entanglement structure. The particular size (number of qubits) and entanglement pattern of the resource states can be varied as appropriate for a particular use case. In addition or instead, the number of resource states and entanglement geometry between resource states can be varied according to the particular use-case. For instance, while the foregoing description uses examples of fusion networks having three-dimensional geometry, fusion networks having more or fewer dimensions can be executed by providing an appropriate source of resource states and a suitably connected network of interleaving modules.

[0200] Further, while examples herein refer to surface codes, those skilled in the art with the benefit of this disclosure will appreciate that surface codes are one category of topological codes that can be used to provide quantum error correction by defining and operating on logical qubits and that systems and methods described herein can be applied to any topological code, including surface codes, color codes, and so on. The particular stabilizers implemented are a matter of design choice.

[0201] Further, embodiments described above include references to specific materials and structures (e.g., optical fibers), but other materials and structures capable of producing, propagating, and operating on photons can be substituted. As noted above, resource states can be generated using photonic circuits, or a resource state can be created using matterbased qubits, after which an appropriate transducer technology can be applied to swap the state of the matter-based qubits onto a photonic state. Interleaving as described herein exploits the propagation of photonic qubits, and similar techniques may be applicable to systems of physical qubits that are realized using entities that propagate along well-defined hardware paths.

[0202] Classical control logic can be implemented on-chip with the waveguides, beam splitters, detectors and / or and other photonic circuit components or off-chip as desired.

[0203] It should be understood that all numerical values used herein are for purposes of illustration and may be varied. In some instances ranges are specified to provide a sense of scale, but numerical values outside a disclosed range are not precluded.

[0204] It should also be understood that all diagrams herein are intended as schematic. Unless specifically indicated otherwise, the drawings are not intended to imply any particular physical arrangement of the elements shown therein, or that all elements shown are necessary. Those skilled in the art with access to this disclosure will understand that elements shown in drawings or otherwise described in this disclosure can be modified or omitted and that other elements not shown or described can be added.

[0205] This disclosure provides a description of the claimed invention with reference to specific embodiments. Those skilled in the art with access to this disclosure will appreciatethat the embodiments are not exhaustive of the scope of the claimed invention, which extends to all variations, modifications, and equivalents.

Claims

WHAT IS CLAIMED IS:

1. A method comprising: providing, to a system having a plurality of reconfigurable fusion circuits, a plurality of resource states, each resource state including a plurality of qubits that are entangled with each other, wherein qubits of different resource states are not entangled with each other; routing qubits of the resource states to the reconfigurable fusion circuits such that each fusion circuit receives a qubit from each of three or more of the resource states; and performing, by each reconfigurable fusion circuit, a selected one of a plurality of measurement operations on the received qubits to produce measurement data, wherein at least some of the selected measurement operations are three-way fusion operations.

2. The method of claim 1 wherein the qubits are routed to the reconfigurable fusion circuits according to a three-dimensional cell complex having vertices corresponding to resource states, edges corresponding to fusion measurements, and faces corresponding to check operators.

3. The method of claim 2 further comprising: selecting, for each reconfigurable fusion circuit, one of the plurality of measurement operations, the selection being based on the three-dimensional cell complex.

4. The method of claim 2 wherein the three-dimensional cell complex comprises a unit cell in a repeating pattern.

5. The method of claim 4 wherein the unit cell corresponds to a cuboctahedral polyhedron.

6. The method of claim 1 further comprising generating the resource states.

7. The method of claim 6 wherein generating the resource states includes: generating a first subsystem of qubits in a 3-GHZ state and a second subsystem of qubits in a 3-GHZ state; and performing a two-way fusion operation on a qubit of the first subsystem and a qubit of the second subsystem.

8. The method of claim 6 wherein generating the resource states includes: generating a plurality of Bell pairs of qubits; and performing fusion operations on qubits of different ones of the Bell pairs, wherein at least one of the fusion operations is a three-way fusion operation.

9. The method of claim 1 wherein the acts of providing, routing, and performing are repeated for each of a plurality of cycles including a first cycle.

10. The method of claim 9 wherein the act of routing the qubits includes: storing a qubit from one of the resource states generated during the first cycle in a memory; and providing the stored qubit to one of the reconfigurable fusion circuits in a subsequent one of the plurality of cycles.

11. The method of claim 10 wherein the memory includes an optical fiber that delays delivery of the qubit to the one of the reconfigurable fusion circuits for a fixed number of cycles.

12. The method of claim 9 further comprising: for each cycle, generating control signals to the reconfigurable fusion circuits to select the measurement operations to be performed by the reconfigurable fusion circuits.

13. A circuit compri sing : a plurality of resource state interconnects, each resource state interconnect having a plurality of output paths to output a resource state during each of a plurality of operating cycles, wherein each resource state is a quantum system of multiple entangled qubits, wherein different qubits of the resource state are output on a different ones of the output paths, and wherein different resource state interconnects are associated with different directions in a multi-dimensional entanglement space; a plurality of first routing switches, each of the first routing switches having an input path coupled to a different one of the output paths of the resource state interconnects and a plurality of output paths, wherein each of the first routing switches is configured to receive a different qubit of one of the resource states on the input path and to selectably route the received qubit to one of the plurality of output paths; a plurality of routing paths including a plurality of local routing paths, a plurality of outbound network routing paths, and a plurality of inbound network routingpaths, wherein each of the outbound network routing paths exits the circuit, and wherein each of the inbound network routing paths enters the circuit from an external source, wherein at least some of the routing paths include delay lines having different delay lengths; a plurality of reconfigurable fusion circuits, each of the plurality of reconfigurable fusion circuits being configured to receive three input qubits on three input paths and to selectably perform one of a plurality of measurement operations on the input qubits, thereby producing measurement outcome data, wherein the plurality of measurement operations including a projective entangling measurement between three input qubits and a plurality of single-qubit measurements on each of the three input qubits, wherein the local routing paths are coupled between the routing switches and the reconfigurable fusion circuits such that each of the routing switches is coupled to at least one of the reconfigurable fusion circuits; and a plurality of second routing switches, each of the second routing switches having a first input path coupled to one of the local routing paths, a second input path coupled to one of the inbound network routing paths, and an output path coupled to one of the input paths of one of the reconfigurable fusion circuits.

14. The circuit of claim 13 wherein each resource state comprises four qubits.

15. The circuit of claim 13 wherein each one of the reconfigurable fusion circuits receives qubits associated with three different orthogonal directions in the entanglement space.

16. A system comprising: a network of interleaving modules, wherein each interleaving module includes: a plurality of resource state interconnects, each resource state interconnect having a plurality of output paths to output a resource state during each of a plurality of operating cycles, wherein each resource state is a quantum system of multiple entangled qubits, wherein different qubits of the resource state are output on a different ones of the output paths, and wherein different resource state interconnects are associated with different directions in a multi-dimensional entanglement space;a plurality of first routing switches, each of the first routing switches having an input path coupled to a different one of the output paths of the resource state interconnects and a plurality of output paths, wherein each of the first routing switches is configured to receive a different qubit of one of the resource states on the input path and to selectably route the received qubit to one of the plurality of output paths; a plurality of routing paths including a plurality of local routing paths, a plurality of outbound network routing paths, and a plurality of inbound network routing paths, wherein each of the outbound network routing paths exits the interleaving module, and wherein each of the inbound network routing paths enters the interleaving module from another interleaving module, wherein at least some of the routing paths include delay lines having different delay lengths; a plurality of reconfigurable fusion circuits, each of the plurality of reconfigurable fusion circuits being configured to receive three input qubits on three input paths and to selectably perform one of a plurality of measurement operations on the input qubits, thereby producing measurement outcome data, wherein the plurality of measurement operations including a projective entangling measurement between three input qubits and a plurality of single-qubit measurements on each of the three input qubits, wherein the local routing paths are coupled between the routing switches and the reconfigurable fusion circuits such that each of the routing switches is coupled to at least one of the reconfigurable fusion circuits; and a plurality of second routing switches, each of the second routing switches having a first input path coupled to one of the local routing paths, a second input path coupled to one of the inbound network routing paths, and an output path coupled to one of the input paths of one of the reconfigurable fusion circuits; and classical control logic coupled to the network of interleaving modules and configured to control the first and second routing switches and the reconfigurable fusion circuits and to receive classical data signals representing the measurement outcome data from the reconfigurable fusion circuits.

17. The system of claim 16 further comprising:a plurality of resource state generator circuits coupled to the resource state interconnects and configured to generate resource states.

18. The system of claim 17 wherein each resource state generator circuit comprises: a plurality of 3 -GHZ generator circuits configured to generate a plurality of 3- GHZ states from a plurality of single photons, each 3 -GHZ state including three qubits; a two-way fusion circuit configured to perform a fusion operation on one qubit from each of two of the 3-GHZ states; a Hadamard gate configured to apply a basis rotation on one input of the two- way fusion circuit; a plurality of output paths connected to the inputs of one of the resource state interconnects and configured to provide the other two qubits from each of the 3-GHZ generator circuits to the resource state interconnect; and a plurality of photon sources configured to provide photons to the 3-GHZ generator circuits.

19. The system of claim 17 wherein each resource state generator circuit comprises: a plurality of Bell state generator circuits configured to generate a plurality of Bell states from a plurality of single photons, each Bell state including two qubits; a plurality of fusion circuits configured to perform fusion operations on qubits of different ones of the Bell states, wherein at least one of the fusion operations is a three-way fusion operation; a plurality of output paths connected to the inputs of one of the resource state interconnects and configured to provide qubits from different ones of the Bell state generator circuits to the resource state interconnect; and a plurality of photon sources configured to provide photons to the Bell state generator circuits.

Images