Entanglement recycling in state generation

Abstract

Resource state generation systems and methods in which states can be evaluated and used in further entangling measurements for generation of a target resource state. Multi-qubit measurements can be performed to generate a plurality of entangled states of different types. Quality metrics for the entangled states can be generated and used to select entanglements for further entangling measurements to form one or more target resource state entanglements.

Description

Attorney Docket No.7246-02501 Entanglement Recycling in State Generation Technical Field

[0001] Embodiments herein relate generally to approaches for quantum information processing systems, and more particularly to entangled state generation and processing. Background

[0002] Quantum computing can be 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 may exist in one of two orthogonal states(denoted as |0 and |1 in the conventional bra / ket notation) or in a superpositionof the two states (e.g., |0 |1 . By operating on a system (or ensemble) ofqubits, a quantum computer may quickly perform certain categories of computations that would otherwise require an impractical amount of resources (e.g., time, energy, classical compute resources, electricity, cooling power, etc.). Entanglement may be used as a resource in quantum information processing systems. However, generating desired entanglements can be difficult. Brief Description of the Drawings

[0003] For a better understanding of the various described embodiments, reference should be made to the Detailed Description below, in conjunction with the following drawings, in which like reference numerals refer to corresponding parts throughout the Figures.

[0004] FIGs. 1A-I illustrate the utilization of surface codes to construct an error- corrected fault-tolerant logical qubit, according to some embodiments;

[0005] FIG. 1J illustrates an example of a qubit fusion system interfacing with fusion sites, in accordance with some embodiments;

[0006] FIG. 1K illustrates a qubit fusion system interfacing with a classical computing system, according to some embodiments;

[0007] FIG. 2A is a system diagram of a quantum computing system that may be utilized to implement fault-tolerant post-selection of logical encoded qubits, according to some embodiments;Attorney Docket No.7246-02501

[0008] FIG. 2B is a diagram of a controller, according to some embodiments;

[0009] FIGs. 3A-B illustrate a 2D model of brick encoding, according to some embodiments;

[0010] FIGs. 4A-C illustrate a 3D 6-ring network for hierarchical multiplexing in logical block encoding, according to some embodiments;

[0011] FIG. 5 is a flowchart illustrating a method for performing hierarchical multiplexing while encoding bricks, according to some embodiments;

[0012] FIG. 6A illustrates two 6-qubit resource states that undergo a fusion to produce a single 10-qubit entangled resource state, according to some embodiments;

[0013] FIG.6B illustrates integration of two copies of a 10-qubit resource state into a multiplexing circuit, according to some embodiments;

[0014] FIG. 6C is a legend that defines circuit elements shown in Figure 6D, according to some embodiments;

[0015] FIG. 6D illustrates an interleaving module configured to utilize the fault- tolerant 10-qubit resource state, according to some embodiments;

[0016] FIG. 6E illustrates how four interleaving modules may be interconnected and incorporated into a larger quantum circuit, according to some embodiments;

[0017] FIG. 7A illustrates performing two fusion measurements on two 10-qubit resource states to obtain a 16-qubit resource state, according to some embodiments;

[0018] FIG. 7B illustrates a multiplexing circuit that receives two 16-qubit resource states and produces a single multiplexed 16-qubit resource state, according to some embodiments;

[0019] FIG.7C illustrates incorporating a multiplexed 16-qubit resource state into an interleaving module, according to some embodiments;

[0020] FIG. 8A illustrates an ordered sequence of fusion measurements, according to some embodiments;

[0021] FIG. 8B illustrates two stages of fusion measurements in a fusion network, according to some embodiments;

[0022] FIG. 9A is a legend illustrating various circuit components, according to some embodiments;Attorney Docket No.7246-02501

[0023] FIG. 9B is a circuit diagram illustrating a 2-to-1 multiplexing scheme for using rasterized resource state generators to construct a brick, according to some embodiments;

[0024] FIG. 9C is a circuit diagram illustrating routing of qubits to 1stand 2ndstage fusion measurements, according to some embodiments;

[0025] FIG. 10 are circuit diagrams illustrating network router switches that route qubits either locally or to another resource state generator in the fusion network, according to some embodiments;

[0026] FIG. 11A is a circuit diagram illustrating a circuit utilizing network switches to create multiple copies of a brick, according to some embodiments;

[0027] FIG. 11B is a circuit diagram illustrating multiple interconnected circuits, each with multiple copies of a brick, according to some embodiments;

[0028] FIGs. 12A-C illustrates the construction of subsequent hierarchical brick layers using a circuit-based quantum computing, according to some embodiments;

[0029] FIGs. 13A-J illustrate quantum circuits for performing hierarchical multiplexing using circuit-based quantum computing, according to some embodiments;

[0030] FIGs. 14A-D illustrate two alternate corrections of a 2-dimensional syndrome graph, according to some embodiments;

[0031] FIGs. 15A-D illustrate connected components of primal and dual syndrome graphs in two dimensions in accordance with some embodiments;

[0032] FIGs. 16A illustrates a quality distribution for generated resource states, according to some embodiments;

[0033] FIG. 16B illustrates adaptive fusion complex construction, according to some embodiments;

[0034] FIG. 17 illustrates permutations and translations of a resource state in a target fusion network, according to some embodiments;

[0035] FIG. 18 illustrates permutations of a 6-ring resource state according to different symmetry groups, according to some embodiments;

[0036] FIG. 19A illustrates a switching circuit configured to perform network fusion adaptivity, according to some embodiments;Attorney Docket No.7246-02501

[0037] FIG.19B illustrates a combination of a hierarchical multiplexing switching circuit and a network fusion adaptivity switching circuit, according to some embodiments;

[0038] FIG. 20 is a flowchart illustrating a method for performing fusion network adaptivity, according to some embodiments;

[0039] FIG.21 is a system diagram illustrating aspects of a modularized quantum computing system, according to some embodiments;

[0040] FIGs. 22A-22B illustrate a plurality of entangled states for implementing resource state generation using recycling-based adaptivity, according to some embodiments;

[0041] FIG. 23 illustrates encoded fusion measurements between qubit nodes on different clusters, according to some embodiments;

[0042] FIG. 24 illustrates a flow diagram for generation of a resource state from a plurality of entangled states, according to some embodiments;

[0043] FIGs. 25A-25D illustrate ZX diagrams, in accordance with some example embodiments;

[0044] FIGs. 26A-26C illustrate quantum information processing systems, in accordance with some example embodiments; and

[0045] FIG.27 shows a classical information processing system, in accordance with some example embodiments.

[0046] While the features described herein may be susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the drawings and detailed description thereto are not intended to be limiting to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the subject matter as defined by the appended claims. DETAILED DESCRIPTION

[0047] Disclosed herein are examples (also referred to as “embodiments”) of systems and methods for performing fault-tolerant post-selection using various quantum computing systems.Attorney Docket No.7246-02501

[0048] 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 may be practiced without these details. Reference will now be made in detail to embodiments, examples of which are illustrated in the accompanying drawings. In other instances, well-known methods, procedures, components, circuits, and networks have not been described in detail so as not to unnecessarily obscure aspects of the embodiments.

[0049] Some embodiments described herein include quantum computing devices, systems and methods for performing fusion network adaptivity while encoding a logical qubit or logical block. In some embodiments, a classical computing system including a classical processor coupled to a non-transitory memory medium may execute program instructions to direct the method steps of a quantum computing system.

[0050] In some embodiments, a fusion controller performs a plurality of first multi-qubit fusion measurements on qubits from different seed states of a plurality of seed states to produce a plurality of clusters.

[0051] In some embodiments, a classical processor determines quality metrics for each cluster of the plurality of clusters.

[0052] In some embodiments, the system can perform second multi-qubit fusion measurement on qubits from two or more different clusters of the plurality of clusters to produce a second-stage cluster. In some embodiments, the qubits from the different clusters are selected based on their respective quality metrics, such as average erasure, entanglement size, or other types of metrics such that a likelihood of outputting a target resource state is increased.

[0053] In some embodiments, after encoding the resource state, the resource state may be fused with a second resource state. The fusion may proceed adaptively, and may be performed on two different resource states with arbitrary encoding.

[0054] The techniques described herein may be implemented in and / or used with a number of different types of devices, including but not limited to photonic, superconductor, or semiconductor quantum computing devices and / or systems,Attorney Docket No.7246-02501 hybrid quantum / classical computing systems, and any of various other quantum computing systems. Qubits

[0055] 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. As used herein, a “qubit” (or quantum bit) is a quantum system with an associated quantum state that may be used to encode information. A quantum state may 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 may have a state that is a superposition of logical values 0 and 1. More generally, a “qudit” describes any quantum system having a quantum state space that may be modeled as a (complex) n-dimensional vector space (for any integer n), which may 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 may also employ quantum information carriers that encode information in a manner that is not necessarily associated with a binary bit, such as a qudit.

[0056] Qubits (or qudits) may 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).

[0057] As used herein, a distinction is made between a “physical qubit” which is a physical quantum system such as a molecule, atom, photon, etc. that exists in a 2-level quantum state, and a “logical qubit” which includes a plurality of physical qubits encoded (e.g., entangled) together according to a quantum error correcting code (such as a surface code) to encode logical quantum information. These terms are described in greater detail below.Attorney Docket No.7246-02501 Figures 1A-K – Surface Codes and Physical implementations

[0058] Qubits (and operations on qubits) may be implemented using a variety of physical systems. In some examples described herein, qubits are provided in an integrated photonic system employing waveguides, beam splitters, photonic switches, and single photon detectors, and the modes that may be occupied by photons are spatiotemporal modes that correspond to presence of a photon in a waveguide. Modes may be coupled using mode couplers, e.g., optical beam splitters, to implement transformation operations, and measurement operations may 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, may 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 may 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 may be any physical mechanism that couples two modes, e.g., a pulsed electromagnetic field that is tuned to couple two internal states of the atom / ion.

[0059] In some embodiments of a photonic quantum computing system using dual-rail encoding, a qubit may be implemented using a pair of waveguides. In some embodiments, a photon in a first waveguide of the pair and no photon in a second waveguide of the pair (also referred to as a vacuum mode) maycorrespond to the |0 state of a photonic qubit. Alternatively, a state with aphoton in the second waveguide and no photon in the first waveguide maycorrespond to the |1 state of the photonic qubit. To prepare a photonic qubit in aknown logical state, a photon source may be coupled to one end of one of the waveguides. The photon source may 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 toAttorney Docket No.7246-02501 different temporal modes of the photonic system may 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 may be created. It should be understood that the waveguides in such a system need not have any particular spatial relationship to each other. For instance, they may be but need not be arranged in parallel.

[0060] Some embodiments described below relate to physical implementations of unitary operations that couple modes of a quantum system, which may 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 withprobability 1 (e.g., a state |10 in Fock notation), mode coupling may result in astate in which both modes have a nonzero probability of being occupied, e.g., astate |10 |01 , where | | | | 1.some embodiments, operationsof this kind may 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 a1and a2depend on the reflectivity (or transmissivity) of the beam splitters and on any phase shifts that are introduced.

[0061] A single physical qubit (e.g., such as the 2-level physical qubit illustratedin Figure 1A with a quantum state|0 |1 ) may be used for quantumcomputation in principle. However, individual physical qubits are generally highly susceptible to noise and decoherence. Fault-tolerant quantum computing utilizes a plurality of entangled physical qubits to encode a single logical qubit to mitigate the frailty and / or short coherence times of individual physical qubits. In fault-tolerant quantum computing schemes, a plurality of physical qubits is entangled together according to a specific error-correcting code (e.g., using fusion measurements on resource states) to produce a single logical qubit that is less susceptible to noise and decoherence. Encoding qubits in this manner causes the resultant logical qubit to be less sensitive to error and noise, and resultant errors may be fixed via quantum error correction. Encoding a logical qubit may itself be vulnerable to errors.Attorney Docket No.7246-02501

[0062] Embodiments herein address these and other issues by implementing fusion network adaptivity in fault-tolerant codes and channels to improve the accuracy and fidelity of encoded logical qubits. The terms “fusion network adaptivity” and “fusion complex adaptivity” are used interchangeably throughout this disclosure. At a high level, various quality metrics may be employed to assess the fidelity of an encoded brick, to determine whether to discard the brick, keep it for use in the quantum computation, and / or reroute to another location within a quantum circuit. As used herein, a “brick” refers to two or more resource states that are entangled together, which may be used to encode a logical block. As described in greater detail below, bricks may be iteratively entangled together at subsequent hierarchical stages to encode a logical block.

[0063] In some quantum computing methodologies, such as fusion-based quantum computing, a logical qubit is encoded from a plurality of physical qubits using a sequence of specific measurements (e.g., stabilizer measurements). The measurement sequence may be constructed where a subset of the physical qubits is measured (e.g., collapsing the quantum state and producing classical information, i.e., the measurement result) in such a way that the remaining unmeasured / un-collapsed degrees of freedom (e.g., a 2-dimensional subspace which has support over all the physical qubits) form the desired encoded logical qubit. Accordingly, the processes of performing stabilizer measurements and / or encoding a fault-tolerant logical qubit may receive a plurality of physical qubits as input and as output may produce both the encoded logical qubit and classical information (e.g., syndrome graph data) resulting from the measurement sequence.

[0064] In some quantum computing implementations, the classical information takes the form of syndrome graph data, where the syndrome graph is a geometric representation of the outcomes of the measurement sequence. Because the input physical qubits are prepared in a known initial state and measured according to a predetermined measurement sequence, it may be determined (e.g., using classical computing) how the syndrome should appear in the absence of any errors involving the physical qubits during the measurement sequence (e.g., Pauli or erasure errors). Accordingly, any deviation of the syndrome graph dataAttorney Docket No.7246-02501 from the expected result may be indicative of one or more errors within the logical qubit. In general, these deviations may not indicate precisely which measurement(s) had an error, or which type of error has occurred, as there may be more than one type of error or combination of errors that is consistent with a given observed deviation from the anticipated error-free syndrome graph. For example, a syndrome graph may be determined as a grid of parity checks for adjacent nodes of the grid, whereby a parity error may indicate that one or more of the adjacent nodes had an error, but the parity error may not indicate precisely which adjacent node had an error, or which error occurred.

[0065] As used herein, the term “syndrome graph data” refers to a set of classical information (e.g., data represented by digital values such as ones and zeros) that specifies the location of one or more syndromes and / or one or more erasure errors within the syndrome graph of a logical block. Said another way, based on the knowledge of the particular geometry of the cluster state / error correcting code, measurement outcomes may be used to determine the syndrome graph data. In some embodiments, the syndrome graph data may further include correction operators for the syndrome graph output by a decoder.

[0066] Errors that occur during operations on an encoded logical qubit may have varying degrees of severity. For example, errors in a fault-tolerant logical qubit may cause logical failure if they link up in a way that spans the syndrome graph of the logical qubit. Conversely, localized errors that do not span the syndrome graph may be identifiable and correctable via quantum error correction. Embodiments herein perform fusion network adaptivity by determining a quality metric based on Tanner graph or syndrome graph data for different orientations and / or locations of a brick in a target fusion network, and utilizes the quality metric to determine how to route a brick into the fusion network.

[0067] In some embodiments, the brick may be a component of a quantum error- correcting code where an operation (for example, a quantum gate acting on a logical qubit) may be performed on encoded logical information. For example, a brick may include multiple resources states that are entangled with one another in a specific way. Resource states are defined as a plurality of physical qubits prepared in a specific entangled manner. In some embodiments, 6-qubit resourceAttorney Docket No.7246-02501 states such as those illustrated in Figure 1H may be used, or other types of resource states may be used. Depending on the layer or stage, a brick may include a portion of a logical block, logical qubit, or logical gate, an entire one of any of these, or a combination thereof.

[0068] As used herein, a “logical block” refers to an entangled arrangement of qubits with sufficient structure to perform a logical operation on one or more inputs and provide one or more outputs. For example, when a sufficient number of bricks of sufficient size and complexity are entangled together such that the resultant aggregate brick is configured to perform one more or logical operations, the aggregate brick may be considered to be a logical block. Logical qubits and logical gates are both examples of logical blocks. In some embodiments, a logical block may include one or more input ports and one or more output ports. The logical operation(s) performed by a logical block may be fault tolerant, in some embodiments. Logical blocks are described in greater detail in Héctor Bombín, Chris Dawson, Ryan V. Mishmash, Naomi Nickerson, et al., Logical Blocks for Fault-Tolerant Topological Quantum Computation, PRX Quantum 4, 020303 (2023). Note that in FBQC, resource states are entangled by performing fusion measurements on a subset of the qubits of the resource states. As one example, a brick that includes two entangled 6-qubit resource states (e.g., the brick 606 shown in Figure 6A) will have fewer remaining unmeasured qubits than were present in the 12 qubits contained in the original resource states (e.g., the 12 qubits in the two resource states 602 and f604), since two of the qubits (x0+ and x1- in Figure 6) are measured in a fusion measurement.

[0069] As used herein, “sub-brick” is used to describe the smaller bricks that make up a given brick, and “aggregate brick” is used to describe the larger brick that is obtained by fusing together a plurality of bricks. Note that the layers of a sub-brick and an aggregate brick are relative to the layer of the brick to which they refer, as the brick may be at different layers or stages (e.g., the brick may be any of the stages shown in Figure 4A, and the sub-brick and the aggregate brick are one stage lower and higher, respectively).

[0070] If the above-described surface code measurement schedule is applied for numerous time steps, the system effectively acts as a fault-tolerant quantumAttorney Docket No.7246-02501 memory for the logical qubit encoded by the underlying surface code or, viewed another way, as a fault-tolerant logical identity gate on the logical qubit that is encoded by the underlying surface code. Viewed yet another way, this process operates as a fault-tolerant logical channel.

[0071] Figure 1B illustrates a 3-dimensional graphical depiction of such a fault- tolerant logical identity gate. The surface labeled 114 is the input to the gate and includes an arbitrary logical state encoded in a surface code, represented as the input checkerboard surface. Likewise, the surface labeled 118 identifies the output qubits after the identity gate I has been applied to it. The input and output surfaces, which may be associated with either the physical or relational arrangement of qubits, are connected to each other via an intervening volume that represents the unique set of measurements to be applied over time. Accordingly, in Figure 1B, time flows from left to right and the lighter shaded (front and back) and darker shaded (top and bottom) sides of the boundaries of the volume depict whether the primal or dual plaquettes are disposed on that boundary. Figure 1C represents the same concept but written in a more familiar quantum circuit notation illustrating the analogy between the more familiar quantum circuit. While Figure 1B shows the logical identity gate, any gate can be depicted in this manner and such a depiction is one example of a logical block that specifies a set of instructions to be performed on the underlying surface code qubits to perform a logical operation (the identity gate in this example) on the logical qubit that is encoded by surface code. Other examples of such gates are the S gate, the Hadamard gate, and the CX gate, among other possibilities.

[0072] The sequence of measurements performed over the flow of time illustrated in Figure 1B (e.g., a sequence of measurements including selective fusion measurements) may include a subset of measurements that incur an error such as a Pauli error or an erasure error. To identify errors in the measurement outcomes, syndrome graph data may be generated from the collection of measurement outcomes resulting from the measurements of the physical qubits. For example, the bit values associated with a plurality of edge qubits may be combined to create a syndrome value associated with an adjacent vertex that results from the intersection of the respective edges, e.g., the result of fusionAttorney Docket No.7246-02501 measurements. A set of syndrome values (or “syndromes”), also referred to herein as parity checks, may be associated with each vertex of the syndrome graph. Figure 1D illustrates an example 2D representation of a syndrome graph including a plurality of interspersed syndromes and erasures. The parity check values may be found by computing the parity of the bit values associated with each edge of the syndrome graph incident to the vertex. In some embodiments, a parity computation entails determining whether the sum of the edge values is an even or odd integer, with the parity result being the result of the sum modulo 2. If no errors have occurred in the quantum state or in the qubit measurements, then all syndrome values should be even (or 0). On the contrary, if an error occurs, it may result in some odd (or 1) syndrome values.

[0073] In some embodiments, half of the bit values from the qubit measurements are associated with the primal boundary surfaces, and this syndrome graph is referred to herein as the “primal graph”. The syndrome graph resulting from measurements on the dual boundary surfaces is referred to as the “dual graph”. There is generally an equivalent decoding problem on the syndrome values of the primal and dual graphs.

[0074] Syndromes may be identified and appropriately removed via quantum error correction, via a process known as decoding. Decoding produces a recovery that is consistent with the syndrome, attempting to correct for the error. Decoding succeeds when the combined effect of the error and recovery does not give rise to a logical error. However, this process does not always succeed, and certain combinations of error and recovery may result in an error chain that spans the surface code and damages the logical information. In some embodiments, logical gap magnitudes for correcting syndromes may be extracted from syndrome graph data to determine an error metric associated with a logical encoded qubit (i.e., for the output surface code shown in Figure 1B). This error metric may then be used for hierarchical multiplexing, to determine how and / or where to route each of a plurality of multiplexed copies of a brick, and / or to determine where to route different qubits within the brick, in some embodiments.Attorney Docket No.7246-02501

[0075] In some embodiments, hierarchial multiplexing and / or fusion network adaptivity may utilize information metrics based on visible syndrome and erasure information. In some embodiments, different metrics may be employed for ranking the quality of bricks based on their respective configurations of syndromes and erasures. For example, some embodiments utilize a logical gap (and variants thereof) which determines an unsigned weight difference between inequivalent logical corrections as a metric for predicting logical error rates of bricks (also known as fault-tolerant channels) based on error-correcting codes. Advantagously, this metric is highly adaptable to various types of noise and decoders. In some embodiments, hierarchial multiplexing may be deployed to prepare low-error surface code logical qubits with low overheads under an i.i.d. model of Pauli error and erasure error rates. Hierarchial multiplexing strategies based on the logical gap may suppress the encoding error rate of a logical qubit.

[0076] Figures 1E and 1F illustrate an arrangement of physical qubits that may be used to perform a (Z2, Z3) measurement on four logical qubits q1-q4. The individual circles shown in the rectangular sheet 120 in the top half of Figure 1F represent individual physical qubits, and the lines connecting adjacent qubits indicate entanglement (e.g., via fusion measurements). In the stack of d = 9 layers shown at 122 of Figure 1F, the vertical direction represents the depth of the logical qubit (i.e., time), which is a sequence of nine entangling measurements performed on the 9x9 grid of physical qubits representing each of the qubits q1-q4 as well as a portion of the auxiliary qubits 121.

[0077] The protocol for preparing an encoded logical state may contain two parameters, and . Here is referred to as the “distance” of the scheme – it determines the code distance of the surface code state being prepared. In some embodiments, L may be separated into two parameters, Lxand Ly,i.e., the code distance may be different in the two spatial directions. This may be desirable, for example, when there is an asymmetery in the noise model or logical error rates in the X and Z directions, and the code distance may be separately tuned in the two spatial directions. is referred to as the “depth” of the scheme – it can be thought of as simulated time, i.e. the number of rounds of stabilizer measurements in CBQC, or the number of layers of resource states in FBQC.Attorney Docket No.7246-02501 may determine the number of stabilizer checks in the protocol from which information may be gathered for post-selection. A minimal depth of 2 may be chosen, however, longer depths may also be used (using more overhead) to allow for more information to be collected in order to better predict logical errors on the output state.

[0078] Figure 1G illustrates how the physical qubits illustrated in Figures IE-F may be encoded using 18 units of physical hardware (e.g., 18 interleaving circuits). As illustrated, each interleaving circuit encodes a set of 36 physical qubits in the illustrated sheet. As one example, the physical qubits may be encoded using a raster scan methodology where the 36 qubits are sequentially produced, mutually entangled, and preserved for 36 time cycles using variable fiber optic cable lengths until all 18 interleaving circuits have completed 36 cycles, whereupon the next sheet of physical qubits may be encoded.

[0079] In some embodiments, hierarchical multiplexing may be performed on a portion of one or more bricks, where the lower layer components of a brick are referred to herein as “sub-bricks.” Figure 1G shows multiple sub-bricks stitched together into a single brick. As illustrated in Figure 1G, a given hardware block such as an interleaving circuit may be configured to be encoded as multiple sub- bricks that each include only a portion of one or more logical qubits. Each dashed box in Figure 1G delineates (potentially together with corresponding boxes in subsequent sheets) a respective sub-brick. For example, as can be seen by comparison of Figure 1G with Figures 1E and 1F, the sub-brick 126 encodes a portion of the logical qubit q2,the sub-brick 124 encodes portions of both q1 and q2, the sub-brick 128 encodes a portion of q1 and is partially dormant, the sub- brick 130 is entirely dormant for the illustrated set of clock cycles, and the sub- brick 132 encodes a portion of a block of logical ancillary qubits. The sub-bricks in Figure IG as illustrated are stitched together (e.g., through fusion measurements in FBQC) to form the entire brick shown in Figure 1G, which performs the Z2Z3 two-qubit measurement. In performing hierarchical multiplexing on sub-bricks, multiple copies of each of the sub-bricks may be first created separately (i.e., not yet stitched together), hierarchical multiplexing mayAttorney Docket No.7246-02501 be performed to select high-quality copies of each sub-brick, and the selected sub- bricks may be then entangled to produce the desired larger brick.

[0080] Figure 1H illustrates a logical block composed of a plurality of 6-qubit resource states in a fusion network, according to some embodiments. In the illustrated example, each resource state is composed of 6 physical qubits in a specific entangled arrangement. The entanglement is illustrated with thin lines connecting the different numbered qubits 1-6 of the resource state. Bold lines are shown to indicate 2-qubit fusion measurements that are performed on one qubit from each of two different resource states to construct the logical block. In some embodiments stabilizer resource states may be utilized, which may be described, up to local Clifford operations, by a graph G using a graph state representation.The graph state is defined as the quantum state | obtained by putting qubits inthe | state at each vertex and performing a controlled-Z gate between qubits forwhich the corresponding vertices in the graph are neighbors. Stabilizer resource states are described in greater detail in Bartolucci, S., Birchall, P., Bombín, H. et al. Fusion-based quantum computation. Nat Commun 14, 912 (2023). In some embodiments, a resource state such as is shown in Figure 1H may be used to perform hierarchical multiplexing and / or fusion network adaptivity according to the circuit diagrams shown in Figures 6-11 and 18.

[0081] Figure 1I is a circuit diagram illustrating raster-scanned interleaving modules, according to some embodiments. The interleaving modules may be used to implement the qubits illustrated in the network shown in Figure 1G, as one example.

[0082] When the interleaving length l is larger than the code depth d (not illustrated), a sub-brick may include multiple logical qubits and / or portions of logical qubits. Methods described herein for performing hierarchical multiplexing and fusion network adaptivity may be generally applied to various types of logical blocks, logical qubits, and / or components thereof, in various embodiments.

[0083] Figure 1J shows one example of qubit fusion system 134 in accordance with some embodiments. In some embodiments, qubit fusion system 134 may be employed within a larger FBQC system such as the quantum processing unit 205 shown in Figure 2A.Attorney Docket No.7246-02501

[0084] Qubit fusion system 134 includes a fusion controller 140 that is coupled to a fusion array 138. Fusion controller 140 is configured to operate as described herein to direct the fusion sites to perform fusion measurements in a particular manner (e.g., in a particular basis). Fusion array 138 includes a collection of fusion sites that each receive two or more qubits from different resource states (not shown) and perform one or more fusion operations (e.g., Type II fusion) on selected qubits from the two or more resource states. The fusion operations performed on the qubits may be controlled by the fusion controller 140 via signals that are sent from the fusion controller 140 to each of the fusion gates via classical control channels 136a, 136b, etc. Based on the joint measurements performed at each fusion site, classical measurement outcomes in the form of classical data are output and then provided to a decoder system.

[0085] Figure 1K shows one possible example of a fusion site 1501 as configured to operate with a fusion controller 140 to provide measurement outcomes to a decoder for fault tolerant quantum computation in accordance with some embodiments. In this example, fusion site 1501 may be an element of fusion array 138 (shown in Figure 1J), and although only one instance is shown for purposes of illustration, the fusion array 138 may include any number of instances of fusion sites 1501.

[0086] The qubit fusion system 1505 may receive two or more qubits (qubit 1 and qubit 2) that are to be fused. Qubit 1 is one qubit that may be entangled with one or more other qubits (not shown) as part of a first resource state and qubit 2 is another qubit that may be entangled with one or more other qubits (not shown) as part of a second resource state (e.g., the resource states 602, 604, 606, 608, 610 and / or 612 illustrated in Figures 6A-B). The fusion operations that take place at the fusion sites are fully destructive joint measurements between qubit 1 and qubit 2 such that classical information remains after the measurement is performed representing the measurement outcomes on the detectors, e.g., detectors 1503, 1505, 1507, 1509. Quantum information contained within Qubits 1 and / or 2 may be transferred to the remaining (i.e., unmeasured) qubits of their respective resource states. The classical information is decoded by a decoder 146 and may be used in subsequent steps of the described embodiments. For example, the result of the fusion measurement may be used to determine whether the fusionAttorney Docket No.7246-02501 was successful (i.e., whether it resulted in a desired outcome), and / or whether the remaining unmeasured qubits of the resource states associated with qubits 1 and 2 are in a desired configuration, among other possibilities. More broadly, the fusion measurement results may be used to determine a quality metric to be used for fusion network adaptivity, in some embodiments.

[0087] Figure 1K shows an illustrative example for one way to implement a fusion site as part of a photonic quantum computer architecture, according to some embodiments. In this example, qubit 1 and qubit 2 are dual rail encoded photonic qubits. Accordingly, qubit 1 and qubit 2 are input on waveguides 1521, 1523 and 1525, 1527, respectively. An interferometer 1524, 1528 may be placed in line with each qubit, and within one arm of each interferometer 1524, 1528 a programmable phase shifter 1530, 1532 may be applied to affect the basis in which the fusion operation is applied, e.g., XX, XY, YY, ZZ, etc.). The programmable phase shifters 1530, 1532 may be coupled to the fusion controller 1519 via control line 1529 and 1531 such that signals from the fusion controller 1519 may be used to set the basis in which the fusion operation is applied to the qubits. For example, the programmable phase shifters may be programmable to either apply or not apply a Hadamard gate to their respective qubits, altering the basis (e.g., x vs. z) of the type II fusion measurement. In some embodiments the basis may be hard-coded within the fusion controller 1519, or in some embodiments the basis may be chosen based upon external inputs, e.g., instructions provided by the fusion pattern generator 144. Additional mode couplers, e.g., mode couplers 1533 and 1532 may be applied after the interferometers followed by single photon detectors 1503, 1505, 1507, 1509 to provide a readout mechanism for performing the joint measurement. In the example shown in Figure 1K, the fusion site implements an un-boosted Type II fusion operation on the incoming qubits. One of ordinary skill will appreciate that any type of fusion operation may be applied (and may be boosted or un-boosted) without departing from the scope of the present disclosure. In some embodiments, the fusion controller 1519 may also provide a control signal to the detectors 1503, 1505, 1507, 1509. A control signal may be used, e.g., for gating the detectors or for otherwise controlling the operation of the detectors. Each of the detectors 1503, 1505, 1507, 1509 provides one bit of informationAttorney Docket No.7246-02501 (representing a “photon detected” or “no photon detected” state of the detector), and these four bits may be preprocessed at the fusion site 1501 to determine a measurement outcome (e.g., fusion success or not) or passed directly to the decoder 146 for further processing. Figures 2A-B – Quantum and Classical Computing Systems

[0088] Figure 2A illustrates a quantum processing unit coupled to a classical computing system that may be utilized to implement method steps of embodiments described herein. As illustrated, the system includes a classical computing system 203 coupled to a quantum processing unit 205 over a classical channel 212. The classical channel may relay classical information between the classical computing system and quantum processing unit.

[0089] In some embodiments, the classical computing system 203 includes one or more non-transitory computer-readable memory media 204, one or more central processing units (CPUs) or processor(s) 202, a power supply, an input / output (I / O) subsystem, and a communication bus or interconnecting these components. The processor(s) 202 may execute modules, programs, and / or instructions stored in memory 204 and thereby perform processing operations. The processor may comprise a dedicated processor, or it may be a field programmable gate arrays (FPGA), an application specific integrated circuit (ASIC), or a “system on a chip” that includes classical processors and memory, among other possibilities. In some embodiments, memory 204 stores one or more programs (e.g., sets of instructions) and / or data structures and is coupled to the processor(s).

[0090] In some embodiments, the classical computing system may have installed thereon a dedicated module acting as a controller, also referred to as a multiplexing controller. In some embodiments, the controller may include its own dedicated memory medium and / or processor(s), which may be a dedicated processor, an FPGA, or an ASIC, among other possibilities. In some embodiments, the controller may be implemented as software and may share processing resources with other control aspects of the classical computing system.Attorney Docket No.7246-02501

[0091] The classical computing system may be classical in the sense that it operates computer code represented as a plurality of classical bits that may take a value of 1 or 0. Programs may be written in the form of ordered lists of instructions and stored within the classical (e.g., digital) memory 204 and executed by the classical (e.g., digital) processor 202 of the classical computer. The memory 204 is classical in the sense that it stores data and / or program instructions in a storage medium in the form of bits (rather than as qubits containing quantum information), which have a single definite binary state at any point in time. The processor may read instructions from the computer program in the memory 204 and / or write data into memory, and may optionally receive input data from a source external to the computer 203, such as from a user input device such as a mouse, keyboard, or any other input device. The processor 202 may execute program instructions that have been read from the memory 204 to perform computations on data read from the memory 204 and / or input from the quantum computing system, and generate output from those instructions. The processor 202 may store that output back into the memory 204.

[0092] The quantum processing unit 205 may include a plurality of qubits and a controller 206 configured to interface with the plurality of qubits 210 to control, direct and / or measure the qubits within the quantum circuit. The qubits may be configured to evolve in time under the directed influence of the controller, and a measurement system 208 may at times perform quantum measurements on all or a subset of the qubits to obtain quantum measurement results in the form of classical data bits (e.g., ones and zeros). The classical data from the measurement results may be intermediate results that inform behavior of the classical computing system and / or the quantum controller 206 during a quantum computation, and they may additionally include classical results of the quantum computation. The measurement results may be communicated to the classical computing system and / or the controller 206, and further the classical computing system may provide directions and / or instructions to the controller 206 and the measurement system 208 to guide the behavior of the quantum computing system to perform a quantum computation. For example, the classical computing system 203 may provide classical data signals used for quantum stateAttorney Docket No.7246-02501 preparation within the quantum processing unit 205, in response to which the controller may prepare the states of the qubits 210 into a desired initial state for a particular quantum computation.

[0093] In some embodiments, physical qubits 210 are provided to the measurement system 208 and controller 206, where the measurement system and the controller function as a logical qubit encoder that perform a sequence of measurements on the physical qubits to produce a logical qubit (e.g., a fault- tolerant encoded logical qubit). For example, the measurement system and controller may perform a sequence of measurements on the physical qubits to entangle them in such a way as to produce a logical qubit. Encoding the logical qubit will also produce syndrome graph data for the logical qubit as classical information, which is output to the controller of the classical computing system 203 via the classical channel 212. The controller analyzes the syndrome graph data to determine an error metric for the logical qubit. Depending on the error metric, the controller outputs instructions back to the quantum processing unit 205 along the classical channel 212 to discard the logical qubit, to flag the logical qubit as poor quality and / or reroute it to a different aspect of the quantum computation, or to keep the logical qubit in the quantum computation.

[0094] Figure 2B is an illustration of components of a controller 206, according to some embodiments, and includes a non-transitory computer-readable memory medium 230, one or more processors or central processing units (CPUs) 232, and one or more input / output ports to communicate with other elements of the quantum processing unit 205 and / or the classical computing system 201. Hierarchical Multiplexing for Logical Block Encoding

[0095] As described according to embodiments herein, hierarchical multiplexing performs post-selection and multiplexing at the fault-tolerance scale (e.g., the logical scale), making use of syndromes and erasures (or any other visible information) to select favorable parts of a topological error-correcting code. As used herein, “post-selection” refers to the analysis of one or more quality metrics of a brick, and routing the brick in a quantum circuit based on the quality metrics.Attorney Docket No.7246-02501

[0096] The quality metrics may vary depending on the scale (i.e., the layer or stage) of the brick. For example, binary quality metrics (e.g., that indicate that a brick is definitively either valid or invalid) may be used for smaller bricks, whereas bricks that are greater than a threshold size may utilize syndrome graph data to analyze their quality.

[0097] As used herein, “multiplexing” refers to the construction of redundant copies of a brick, where higher and lower quality copies may be dynamically rerouted in the quantum circuit (as one example, the highest quality copies may be kept and the lower quality copies may be discarded). In the context of FBQC, a target fusion network is divided into smaller pieces, each of which may be constructed multiple times in parallel, in order to create a fusion network that has fewer imperfections.

[0098] As used herein, a “fusion network” is a prescribed set of resource states and fusion events to be performed between these resource states, which produces a network of entangled resource states in a particular entangled arrangement. A fusion network may be represented by a graph (e.g., a fusion graph), where vertices represent resource states and edges represent fusions. A “target fusion network” is the desired fusion network in the absence of any hierarchical multiplexing, i.e., it is the intended fusion network to be constructed by a quantum computing system. A quantum computing system may include resource state generators (RSG), fusion routing networks, controllers, and a classical computing system, and may be configured to construct a fusion network.

[0099] Hierarchical multiplexing may also be used outside of FBQC. For example, in addition to the multiplexing that occurs in resource state generation, hierarchical multiplexing may utilize multiplexing in one or more higher levels of the fusion network. Hierarchical multiplexing may also be used for other models of computation besides than FBQC, e.g., circuit-based quantum computation.

[0100] In FBQC, the fusion network is built up by fusing individual resource states together. In hierarchical multiplexing, a larger fusion network is built incrementally from bricks, which may be ranked and dynamically selected for at each stage. At the zeroth level, each brick is a single resource state asAttorney Docket No.7246-02501 shown in a toy model in Figure 3A. For each resource state in the target fusion network, N0 copies may be constructed in the hierarchical multiplexing scheme. For the next, first level, a brick may include several resource states fused together.

[0101] In some embodiments, N0copies of level one bricks may be constructed and ranked according to one or more quality metrics, and only N1copies of these level one bricks may be kept. This process may be iterated, building up bricks of increasing size from lower level bricks, and post-selecting for the high quality ones. Each level is called a "stage", and the final stage may include ballistic fusions of the top-level bricks to complete the fusion network. The top-level ballistic fusions may occur without multiplexing (i.e., without multiple redundant copies of each brick).

[0102] Embodiments herein may employ a “dicing scheme”, by which is meant a method for dividing up the target fusion network into different stages. Each stage includes a set of disconnected bricks to be fused together, where each brick may be multiplexed. At the lowest level (the zeroth level), the bricks are resource states, whereas higher level bricks (i.e., at higher stages) are sets of resource states fused together. We may specify a dicing scheme by a set of stages (S0; S1;…; SK), where each stage Si is a set of fusions in the target fusion network that are to be performed. Each stage is performed many times in parallel and a subset of the bricks exhibiting the highest quality at each stage may be kept for subsequent stages.

[0103] Various quality metrics may be used to direct the post-selection strategy, to assess the quality of each brick and rank them. As a first example, a mixed erasure / syndrome score may be determined, where each brick is assigned a score , where Nsis the number of syndromes in the syndrome graph data, Ne is the number of erasures in the syndrome graph data, and and are tunable parameters. In some embodiments, the quality metric of a brick may be determined based on a logical gap, as described in greater detail below.

[0104] Figures 3A-B illustrate a simple 2D toy example, where the resource states are four qubit states, and the target fusion network is described by a 2D square lattice. For example, Figure 3A illustrates a single resource stateAttorney Docket No.7246-02501 represented as a node and containing four qubit states that may undergo two- qubit fusion measurements with neighboring resource states along the illustrated lines. Figure 3B illustrates a dicing strategy for the 2D network, according to some embodiments. As illustrated, Stage 0 bricks include single resource states. Stage 1 bricks include pairs of resource states fused together. Stage 2 bricks include fused pairs of stage 1 bricks. The final stage (not illustrated) involves ballistically fusing all stage 2 bricks together, completing the target fusion network. Note that at each stage multiple copies may be created of each brick, where the best copies are selected to complete the fusion network.

[0105] A more realistic example of a fusion network is the 6-ring network. In a 6-ring network, resource states are 6-qubit cluster states arranged on a ring. The fusion graph for this network is a cubic lattice. Embodiments herein utilize a multi-stage hierarchical multiplexing scheme, where at each stage pairs of lower- level bricks are fused together. Pairs are fused along the x, then y, and then z directions in sequence, and this process may then be repeated to get even larger bricks. Figure 4A illustrates an example dicing scheme for the 6-ring fusion network for the zeroth stage through the fourth stage. At each stage, multiplexing may be employed to select the best bricks for fusion at each stage. An example is shown in Figure 4B, where 3:1 multiplexing is employed for level- 3 bricks (following the dicing scheme of Fig. 4A), where the best level-3 bricks 412a-c and 414a-c are selected to create a level-4 brick 416.

[0106] Hierarchical multiplexing may be utilized to reduce both erasures and Pauli errors in the produced fusion network. Figure 4C shows an example of using hierarchical multiplexing in the 6-ring fusion network where fusions inside a brick are post-selected. With post-selection, bricks such as the delineated brick 418 may be selected that have fewer erasures on the dot-dashed and solid line fusions to selectively suppress erasures on these fusion measurements. Pauli errors on both measurements from the fusions with suppressed erasures and errors (dot-dashed lines) and measurements from the fusions with suppressed erasures and partially suppressed errors (solid lines) may be indicated by syndromes inside the post-selected bricks. By picking copies of bricks withoutAttorney Docket No.7246-02501 syndromes, Pauli errors may be suppressed as well. Long chains of Pauli errors that span the brick will not light up syndromes inside the brick and cannot be post-selected out, but these higher order Pauli errors occur with much lower probability than the 1storder Pauli errors that are caught. As a result, most of the errors are restricted to the ballistic fusions (dashed lines) that lie between the bricks and don't have any post-selection. As illustrated in Figure 4C, these ballistic fusions form a small fraction of the fusions and therefore most of the errors and erasures can be suppressed. With larger brick size, this suppression may be further increased but larger brick sizes may also require more post- selection. In the limit of extremely large brick size, errors are isolated into 2D planes which have a much higher threshold than the original 2D fusion network. As an example, the 3D 6-ring fusion network has an erasure threshold of ~12% and an error threshold of ~1% while the 2D limit has an erasure threshold of ~50% and an error threshold of ~11%.

[0107] To determine an appropriate multiplexing scale (i.e., a dicing strategy and an allocation of multiplexing overhead for each stage), two competing effects may be considered. For a fixed multiplexing budget, smaller bricks may be more effectively multiplexed because there are more choices out of the bricks that are created. However, larger bricks leave fewer ballistic fusions interfacing the bricks which cannot be multiplexed away.

[0108] In some embodiments, these competing considerations may be quantitatively investigated to optimize the dicing scheme and the allocation of multiplexing overhead at each stage. Figure 5 – Flowchart for Hierarchical Multiplexing of Logical Blocks

[0109] Figure 5 is a flowchart that illustrates a method for performing hierarchical multiplexing for encoding logical qubits or logical blocks, according to some embodiments. The method shown in Figure 5 may be used in conjunction with any of the computer systems or devices shown in the above Figures, among other devices. For example, the method shown in Figure 5 may be performed by a quantum computing device or system as illustrated in Figures 2A and 2B. The quantum computing system may further include a controller (e.g., the controllerAttorney Docket No.7246-02501 206 illustrated in Figure 2B) to direct the described method steps, and may be included in (or be coupled to) a classical computing system for processing classical information and directing operations of the quantum computing system. For example, the controller may include one or more processors configured to execute program instructions stored on a non-transitory computer-readable memory medium. In some embodiments the methods described in Figure 5 may be utilized in a quantum communication network, quantum internet, or more generally in any application where it is desired to encode high fidelity qubits. It is to be understood that this method may be used by any quantum computing architecture, and these other architectures should be considered within the scope of the embodiments described herein. As illustrated, the method shown in Figure 5 may proceed as follows.

[0110] At 502, a plurality of respective layer one copies are produced for each brick of a plurality of bricks of a target fusion network. The methods of Figure 5 describe utilizing a target fusion network in the context of FBQC. More generally, the embodiments described herein may be used in other types of quantum computing applications, such as circuit-based quantum computing. In these embodiments, the plurality of bricks may be components of other types of fault tolerant quantum networks.

[0111] In some embodiments, a controller such as the controller 206 illustrated in Figure 2B may direct a quantum circuit to produce the layer one copies. The target fusion network may be a 6-ring network, or another type of fusion network. The bricks may be at any fusion stage of the target fusion network. For example, in the 6-ring fusion network shown in Figures 4A-C, the bricks may be stage 0, 1, 2, 3 or 4 bricks. To take one example, if the plurality of bricks are stage 3 bricks, multiple copies of each stage 3 brick may be produced as shown in Figure 4B (where 3 copies of each of two stage 3 bricks are produced). Note that the term “layer one” as used herein is an identifier to distinguish from other hierarchical layers in the multiplexing scheme, and does not imply that the copies are necessarily made in stage 1.

[0112] In an FBQC implementation, the bricks are comprised of physical qubits within multi-qubit resource states. Each resource state includes aAttorney Docket No.7246-02501 plurality of physical qubits prepared in a specific entangled state. For example, in some embodiments a 6-qubit resource state is employed (e.g., as shown in Figure 1H), as prescribed by the error-correcting code and desired logical operation. At each time step, a subset of the qubits of each resource state are fused with respective qubits of other resource states (thus creating higher level bricks), and a remaining subset of the qubits of each resource state are propagated to a subsequent time step. The fusion measurement causes the unmeasured qubits of the resource states to become entangled with each other, and this entangled quantum information is carried forward until the end of the computation, whereupon all remaining resource state qubits may be measured to produce classical measurement results. As described herein, the logical block includes the resource states that are input into the block (e.g., the input surface 114 from Figure 1G), the classical information that results from the fusion measurements, and the qubits that are output at the back end of the block (e.g., the output surface 118 from Figure 1G).

[0113] In some embodiments, the size and number of bricks in the plurality of bricks is determined based at least in part on a number of layer one copies, a circuit switching complexity, and a complexity of the first quality metrics. For example, brick size and number may be selected to balance the overhead for producing more copies with the number of ballistic fusions that are performed (e.g., a smaller brick size will result in a larger number of bricks for a given aggregate brick, which will increase the number of ballistic fusions). Further, if one has access to a high degree of switching (e.g., if one can switch any n inputs to any n outputs) and high-accuracy quality metrics, fewer input copies may be used to achieve high-quality output bricks.

[0114] In some embodiments, the plurality of bricks of the target fusion network are mutually disjoint. For example, each brick may contribute to a separate and distinct portion of the target fusion network.

[0115] At 504, a respective first quality metric is determined for each layer one copy of each brick of the plurality of bricks. For example, a first quality metric may be determined for each of the six copies 412a-c and 414a-c shown in Figure 4B. In some embodiments, syndrome graph data for the copies of theAttorney Docket No.7246-02501 bricks is received, and the quality metric is determined based on the syndrome graph data. The bricks may be encoded blocks that each include a plurality of physical qubits, and the syndrome graph data may include classical information describing outcomes of the encoding process of each brick. The syndrome graph data may specify the locations of one or more syndromes (i.e., parity errors) and / or one or more erasure errors in the syndrome graph of the logical qubit. The syndrome graph data may also specify a respective set of corrected edges produced by a decoder for each of a first and second correction of the syndromes and / or erasure errors, as described in greater detail below.

[0116] The syndrome graph data may include one or more syndromes, which represent one or more Pauli errors in the syndrome graph. A Pauli error refers to, as one example, a qubit flip error where a qubit has flipped its value (e.g., in the specific case of a dual-rail encoded photonic qubit, the photon may have inadvertently moved to the other waveguide). The syndrome graph data may further contain one or more erasure errors, which represent locations where a physical qubit has escaped the circuit (e.g., a photon may have tunneled out of and escaped the waveguide). An illustration of syndromes and erasure errors within a syndrome graph is shown in Figure 1D.

[0117] In some embodiments, the quality metric may be determined based on the total number of syndromes and erasures in the syndrome graph data for the brick. For example, the quality metric may be determined as a score , where Ns is the number of syndromes in the syndrome graph data, Ne is the number of erasures in the syndrome graph data, and and are tunable parameters. In some embodiments, the first quality metric is a weighted summation of a number of syndromes and erasures contained within a respective syndrome graph of each layer one copy. For example, rather than simply considering the total number of syndromes Nsand erasures Nein determining the score , each syndrome and / or erasure may be weighted based on its location in the syndrome graph, based on the expected rate of error or erasure (as, for example, may be obtained from aggregate statistics from the measurement system), or based on any other observable informationAttorney Docket No.7246-02501 communicated from the hardware modules that may be used to assess the reliability of measurement outcomes from the measurement system.

[0118] In some embodiments, the first quality metric is a logical gap magnitude of the respective syndrome graph of each layer one copy. The logical gap may be determined from the syndrome graph data. The logical gap magnitude is a magnitude of the logical gap, where the logical gap is the difference in weights between first and second correction operators belonging to distinct classes of correction. In some embodiments, the first and second corrections correct for syndromes indicated by the syndrome graph data. In some embodiments, the weight of a correction operator is computed as a real-valued monotonic function of the number of syndrome graph edges that it corrects (i.e., the number of non-identity single-qubit Pauli operators in the correction).

[0119] In some embodiments, the first quality metric is a summation over the set of distinct logical error classes of a decaying exponential function of the respective magnitudes of the respective logical gaps. More generally, any monotonically decreasing function of the logical gap magnitudes may be used for the error metric, and the specific form of function may be determined empirically to improve performance metrics of the fault-tolerant post-selection procedure.

[0120] In some embodiments, aggregate quality metrics are determined based at least in part on the first quality metrics for the layer one copies of the plurality of bricks. For example, rather than analyzing the quality of the brick copies in isolation, an aggregate quality metric may be considered that looks at the quality of the fused combination of the plurality of bricks within an aggregate brick. As one example, it may be considered whether the errors in adjacent bricks line up to form a lattice-spanning error for the aggregate brick. More generally, the aggregate quality metric may consider how likely it is for a set of syndromes and erasures of a brick to result in an overall logical error of the target fusion network, given knowledge of the syndromes and erasures of the copies of the other bricks in the target fusion network. For example, a first copy of a first brick may have a high likelihood of a lattice-spanning logical error when fused to a first copy of a second brick (e.g., if their erasures and / or syndromes happen to line up in an adverse way), but may have a low likelihoodAttorney Docket No.7246-02501 of a lattice-spanning logical error when fused to a second copy of the second brick. Note that the aggregate quality metric may be qualitatively different from the first quality metrics considered in isolation. For example, a copy of a first brick with a relatively poor individual quality metric may have a better aggregate quality metric when considered in combination with a particular set of copies of other bricks in the target fusion network.

[0121] At 506, based at least in part on the first quality metrics, a first layer one copy of the respective layer one copies is selected for each brick of the plurality of bricks. For example, the layer one copy with the highest first quality metric may be selected. In Figure 4B, copies are selected for the left (412a-c) and right (414a-c) sets of bricks that have the fewest number of missing fusions, leading to a higher quality metric in the fused result 416. In some embodiments, aggregate quality metrics are used instead of the first quality metrics when performing selection of the layer one copies.

[0122] At 508, the first layer one copies are fused together to produce an aggregate brick. This is illustrated schematically in the center of Figure 4B at 416. For example, a controller such as the controller 206 illustrated in Figure 2B may direct the quantum circuit to fuse together the first layer one copies.

[0123] In some embodiments, the layer one copies for each brick are rank ordered based on their respective quality metrics, and respective layer one copies from each brick of the plurality of bricks are fused together based on the rank ordering to produce a plurality of logical qubits. In other words, rather than only keeping the highest quality copy for each brick, the copies may rank ordered and may be paired based on their rank ordering. For example, a subset of the copies may be kept and paired (e.g., 50% of copies with the highest quality), or all of the copies may be kept and paired according to the rank ordering (e.g., the two highest quality copies are fused together, the two second highest quality copies are fused together, etc.)

[0124] In some embodiments, for any copies that are determined to not be kept and fused, instructions may be provided to discard the copies. For example, the controller may provide instructions over a classical channel to the quantum computing system to discard these copies. In some embodiments, these copiesAttorney Docket No.7246-02501 may be rerouted in the quantum circuit, and kept for use in some other aspect of the quantum computation.

[0125] In some embodiments, the methods described in reference to steps 502-508 may be hierarchically iterated for one or more additional stages of the target fusion network. For example, producing the plurality of respective layer one copies of each brick of the plurality of bricks may itself include multiplexing to produce higher quality layer one copies. For example, for each layer one copy of each brick of the plurality of bricks, a plurality of respective layer two copies of each sub-brick of a plurality of sub-bricks of the respective brick may be produced. (Note that “layer two” is at a lower stage than “layer one”, e.g., if the layer one copies are for stage 3 bricks, the layer two copies are for stage 2 sub- bricks). A respective second quality metric may be determined for each layer two copy of each sub-brick of the plurality of sub-bricks. Based at least in part on the second quality metrics, a first layer two copy of the respective layer two copies is selected for each sub-brick of the plurality of sub-bricks; and the first layer two copies are fused together to produce the respective layer one copy.

[0126] In some embodiments, the second quality metrics for the layer two copies may be used to inform the determination of the first quality metrics of the layer one copies. In other words, the first quality metrics may be determined based at least in part on the second quality metrics. As one example, quality metrics (such as one based on counting erasure and syndromes) may be updated based on the performance of the fusions between two or more copies. For example, the second quality metrics may specify locations of erasures and / or syndromes in the layer two copies of the sub-bricks, and it may be determined, when the sub-bricks are aligned to be fused into bricks, whether these erasures and / or syndromes connect to one another across adjacent sub-bricks. The first quality metric may rate erasures and / or syndromes that line up across adjacent sub-bricks as lower quality than erasures and / or syndromes that do not line up across adjacent sub-bricks, as these may be less likely to cause a lattice-spanning (and hence uncorrectable) error.

[0127] Using second quality metrics as inputs to the first quality metrics may enable more efficient and faster computation. In some embodiments, anAttorney Docket No.7246-02501 exposure metric for the layer two copies may be used to determine the first quality metric. An exposure metric may be determined for each layer two copy (i.e., for each sub-brick), and the exposure metric may be updated for the fused layer one bricks. Exposure may be calculated and stored in a classical data structure that is conveniently and quickly updated and accessed. 3. Additional

[0128] The paragraphs provide additional technical detail and description regarding embodiments herein.Hierarchical

[0129] Figures 6A-E illustrate an example of how hierarchical multiplexing may be physically implemented when encoding a brick with a 6-ring network using interleaving circuits, according to some embodiments.

[0130] Figure 6A illustrates two 6-qubit resource states that undergo a fusion between x0+ of resource state 0 (602) and x1- of resource state 1 (604), leading to a single 10-qubit entangled resource state (606), according to some embodiments. Note that this process corresponds to the transition from stage 0 (402) to stage 1 (404) shown in Figure 4A. The fusion measurement shown in Figure 6A also produces classical information 605 (i.e., the fusion measurement result), which may be output to a classical processor for determining a quality metric.

[0131] Figure 6B illustrates how two copies (608 and 610) of the 10-qubit resource state of Figure 6A may be integrated into a multiplexing circuit, according to some embodiments. As illustrated, each of the 10 qubits from each of the two resource states is routed to a respective 2 to 1 multiplexer (MUX) 614. In some embodiments, rather than having a dedicated 2-to-1 MUX for each pair of qubits, a single larger MUX may receive two or more pairs of qubits. In these embodiments, each input port of each MUX will receive a unique qubit. The MUXes are each coupled to a controller 206 which is configured to instruct the MUXes which of their two input qubits to route to the output, resulting in a fault-tolerant 10 qubit resource state 612. To preserve entanglement, theAttorney Docket No.7246-02501 controller may direct each of the MUXes to select a qubit from one of the resource states 608 and 610, depending on which resource state has a higher quality metric. The controller 206 may receive classical measurement results of the fusion measurements used to produce the resource states and provide instructions to the MUXes to select the resource state with the higher quality metric. To avoid clutter, only a single classical input 617 to the controller from the resource state 610 and a single output 619 to one of the MUXes is illustrated in Figure 6B, but the controller may be connected to receive input from both resource states 608 and 610 (as well as other elements of the quantum circuit, potentially) and provide instructions to each of the illustrated MUXes.

[0132] Figure 6C is a legend that defines circuit elements shown in Figure 6D, according to some embodiments. A modified fusion gate is F’ is defined, which is controllable to receive two inputs and either perform a fusion measurement, separately measure the qubits in the X, Y or Z basis, or perform a phase shift followed by a Z-measurement (e.g., for magic state preparation). Figure 6C additionally defines the symbol used to denote a delay line with a duration of x clock cycles.

[0133] Figure 6D illustrates an interleaving module configured to utilize the fault-tolerant 10-qubit resource state 612, according to some embodiments. Note that the circuit shown in Figure 6D corresponds to two of the physical qubits shown as open circles in the circuit of Figure 1F.

[0134] Figure 6E illustrates how four of the interleaving modules illustrated in Figure 6D may be interconnected and incorporated into a larger quantum circuit, according to some embodiments.

[0135] Figures 7A-C illustrate a physical implementation of a subsequent level of hierarchical multiplexing, relative to that shown in Figure in Figures 6A- E, according to some embodiments. Said another way, Figures 7A-C illustrate how to implement in hardware a multiplexing process on the level 1 bricks 404 of Figure 4A to obtain multiplexed level 2 bricks 406.

[0136] Figure 7A illustrates how two fusion measurements are performed on two 10-qubit resource states (i.e., level 1 bricks) to obtain a 16-qubit resource state (i.e., a level 2 brick), according to some embodiments. ClassicalAttorney Docket No.7246-02501 measurement results 705 are produced during the fusion measurements, which may be output to a controller to determine a quality metric and perform hierarchical multiplexing. Note that the qubits y00+, y01+, y10- and y11- are measured (i.e., destroyed) during the two fusion measurements, hence reducing the total number of qubits from 20 to 16.

[0137] Figure 7B illustrates a multiplexing circuit that receives two of the 16-qubit resource states shown in Figure 7B, routes each qubit to a respective 2- to-1 MUXer, and produces a single multiplexed 16-qubit resource state, according to some embodiments. The controller 721 may be configured to receive classical measurement results from the resource states (shown at 717 for the resource state 710) and direct the MUXers which of the two input qubits to select (shown as output 719 for one of the MUXes in Figure 7B), depending on the quality metrics of the two resource states 708 and 710. For example, depending on the quality metrics associated with the resource states 708 and 710, the controller may direct each of the 16 MUXers to select and output the qubits from either the resource state 708 or 710. Note that the controller 721 may be the same as or a different controller from the controller 206 illustrated in Figure 6B.

[0138] Figure 7C illustrates how the multiplexed 16-qubit resource state produced in Figure 7B may be incorporated into an interleaving module. Note that the circuit shown in Figure 7C corresponds to four of the physical qubits shown as open circles in the circuit of Figure 1F. Figures 8-11 – Raster Scanning to Implement Fusion Network

[0139] Figures 8-11 illustrate an alternative implementation to what is shown in Figures 6-7 for performing hierarchical multiplexing when constructing a brick, according to some embodiments. The circuit diagrams shown in Figures 8-11 utilize fewer resource state generators (RSGs) than the schemes shown in Figure 6-7, at the cost of a longer time for constructing the brick. At a high level, the circuits shown in Figures 8-11 perform a raster scanning procedure to sequentially perform fusion measurements, rather than performing them concurrently as in the embodiments shown in Figure 6-7. For example, in the raster scanning methodology, a single resource state generator (RSG) may createAttorney Docket No.7246-02501 a physical qubit that is fused at a subsequent time step with a physical qubit created by the same RSG. Accordingly, a single RSG may perform the desired fusion measurement in two sequential time steps. In the embodiments shown in Figures 6-7, the corresponding fusion measurement may be performed in a single time step with qubits produced by two separate RSGs. Either of the two methodologies may be more desirable, depending on specifics of the hardware cost, efficiency, and / or other considerations of the quantum computer.

[0140] Figure 8A illustrates an ordered sequence of fusion measurements, according to some embodiments. The lettersindicate the direction separating the two qubits to be fused, and the subscripts 1 and 2 indicate the stage of the fusion measurement . As illustrated, an ordered sequence of fusion measurements is sequentially performed .each fusion measurement, one of the qubits is labelledand the other is labelled “- ”. Figure 8B illustrates two stages of fusion measurements in a fusion network, according to some embodiments. The left half of Figure 8B illustrates the stage 1 fusion measurements (illustrated with a subscript 1), which are performed sequentially according to the order shown in Figure 8A, followed by the stage 2 fusion measurements, which are also performed sequentially , between adjacent bricks in the fusion network.

[0141] Figure 9A is a legend illustrating various circuit components used in subsequent Figures, according to some embodiments. As illustrated, a 6-qubit resource state generator 902 is shown as a large positively-sloped hashed rectangle with 6 single lines emanating from it, which correspond to the 6 entangled qubits of the resource state. A network / local switch router 904 is illustrated as a square enclosing a capital “N”, with a single qubit input and two outputs to a local destination (solid) and a network destination (dashed). Note that “local” means that the qubit is routed to a location within the same interleaving circuit (e.g., potentially to be fused by another qubit created by the same RSG), whereas “network” means that the qubit is routed to another interleaving circuit in the quantum network. A delay line 906 is illustrated as a loop, and may be of various durations. A stage router 908 is illustrated as a negatively sloped hashed rectangle with one input and three outputs, where theAttorney Docket No.7246-02501 three outputs determine which of three sequential fusion stages to route the input qubit. Some of the stage routers in the subsequent Figures have two outputs (rather than 3). A brick multiplexer (“muxer”) 910 is illustrated as a positively-sloped hashed rectangle with two qubit inputs (solid lines on left), one qubit output (solid line on right) and a classical control input (dashed arrow). The brick muxer receives the two qubits and selects one to output based on the classical control input. Note that the brick muxer is distinguishable from the stage router by having a positive slope on its hash marks, whereas the stage router has negatively sloped hash marks. A multiplex (“mux”) controller 912 is illustrated as a dashed rectangle with internal hashing, which receives input from a classical processor and outputs instructions (e.g., to a brick muxer). Finally, a fusion circuit 914 is illustrated as a curved box with two qubit inputs (solid lines) and a classical instruction input (dashed arrow). The classical instruction input may be received from a controller and may determine which type of fusion measurement to perform.

[0142] Figure 9B is a circuit diagram illustrating a 2-to-1 multiplexing scheme for using rasterized resource state generators to construct a brick, according to some embodiments. Two 6-qubit RSGs create 6 qubits each, which are routed to respective stage routers. The stage routers direct the input qubits, after a time delay, to one of two brick muxers, along with a qubit from the other RSG. The mux controller directs the brick muxers to select one of the two input qubits, which his routed to a fusion circuit.

[0143] Figure 9C is a circuit diagram illustrating routing of qubits to 1stand 2ndstage fusion measurements, according to some embodiments. The top 6 brick muxers are routed to 1ststage fusion measurements shown in the left half of Figure 8B, and the bottom 6 brick muxers are routed to 2ndstage fusion measurements shown in the right half of Figure 8B.

[0144] Figure 10 are circuit diagrams illustrating network router switches that route qubits either locally or to another resource state generator in the fusion network, according to some embodiments. Prior to routing the qubits to stage routers, the qubits may first pass through network router switches to route the qubits either locally or to the network.Attorney Docket No.7246-02501

[0145] Figure 11A is a circuit diagram illustrating a circuit utilizing network switches to create multiple copies of a brick, according to some embodiments. Multiplexing may be performed on the multiple copies, e.g., to select the highest quality copies for routing to higher layers of the circuit. Figure 11B is a circuit diagram illustrating multiple interconnected circuits, each with multiple copies of a brick, according to some embodiments. Figures 11A-B illustrate an example implementation of the hierarchical aspect of the multiplexing scheme, as the multiplexed sub-bricks created via the circuits shown in Figures 9A-C are used to create bricks that are themselves redundantly created and multiplexed, in some embodiments. Figures 12-13 – Hierarchical Multiplexing for Circuit-Based Quantum Computing

[0146] Figures 12A-C and 13A-G illustrate quantum circuits and methods for performing hierarchical multiplexing using circuit-based quantum computing (CBQC). As opposed to the fusion-based implementations described for performing hierarchical multiplexing, Figures 12A-C and 13A-G illustrate circuit configurations for utilizing CBQC in a hierarchical multiplexing scheme. Note that while the physical implementations are quite different, the high-level logic for selecting copies of sub-bricks for constructing larger bricks in a fault tolerant code based on quality metrics may be similar for both CBQC and FBQC implementations.

[0147] Figure 12A illustrates a brick illustrated as a grid of entangled CBQC qubits that are produced with two different types of stabilizer measurements (illustrated as shaded and unshaded boxes). Figure 12B illustrates three copies of each of two bricks, A and B, where ancilla qubit measurements (physical brick measurements, BM) are performed for the second copy of brick A and the first copy of brick B. Figure 12C illustrates a subsequent stage in the hierarchical multiplexing scheme.

[0148] Figure 13A utilizes logical block notation to illustrate a 4 GHZ state. The spacetime diagram on the left of Figure 13A corresponds to a brick that outputs a 4GHZ state stabilized by {ZZZZ, XXII, IXXI, IIXX}. The four whiteAttorney Docket No.7246-02501 ports on top are the output surface codes, whose distance may be tuned by changing the dimensions of the brick. The light and dark shaded surfaces correspond to X and Z-type boundaries, respectively. Time goes from bottom to top. The right half of Figure 13A uses a shorthand "tensor network" notation to denote the brick. Each brick has a distance that may be tuned. Each brick may be built and multiplexed before the 4GHZ projection is applied. Figure 13B illustrates another type of logical block that is stabilized by {XXXX, ZZII, IZZI, IIZZ} and is denoted by a dark-shaded circle, as opposed to the light shaded circle of the logical block shown in Figure 13A. Both logical blocks may be utilized as either state preparations or projections.

[0149] Figure 13C illustrates how the logical block of Figure 13B may be used for performing a set of "transversal" measurements between 4 surface codes. Figure 13D illustrates the four logical blocks that are included in the transversal measurements shown in Figure 13C. These operations only take place on the inner and upper surfaces of the surface codes, which are illustrated in Figure 13E.

[0150] Figure 13F illustrates these four transversal measurements in an alternative representation, where the four bricks of Figure 13D are illustrated as 5x5 grids and the lines denote the couplings of the transversal measurements. Figure 13F depicts a few "corresponding pairs" of qubits. Not all pairs are shown, and the measurements may take place on all qubits of all surface codes.

[0151] Figure 13G illustrates a quantum circuit that may be used to implement these measurements. The operation includes performing measurements of {XXXX, ZZII, IZZI, IIZZ} between corresponding qubits of the four surface codes, as shown in the quantum circuit diagram. Note that there are many possible circuit for implementing this, in various embodiments. After the measurement is done, the surface code qubits may be measured out.

[0152] Figure 13H illustrates how each brick may be composed of a number of light and dark shaded tensors itself, in a 3D lattice-like structure, a portion of which is shown in the inset in Figure 13H. Each of the shaded nodes corresponds to either a 4GHZ preparation or projection. To turn it into a circuit,Attorney Docket No.7246-02501 the 3D network is extrapolated into a linear one (see Figures 13I-J), with long- range connections.

[0153] Figure 13I illustrates extrapolating the network into a linear ordering, with light-shaded tensors on the bottom, and dark-shaded ones on top. In principle, any ordering or geometry is possible, according to various embodiments. A 2D grid may be arranged, for instance. It may be desirable to keep nearest neighbours in the 3D network as close as possible in the linear arrangement. More generally, an arbitrary linear ordering may be used, or a raster ordering may be utilized for all tensors.

[0154] Figure 13J illustrates one example of the connections between the upper and lower tensors, where they are connected up according to their connectivity in 3D. This will introduce long-range connections. Note that these long-range connections correspond to some circuit that needs to be implemented on the corresponding surface-code qubits. Computing a Logical Gap for Hierarchical Multiplexing

[0155] The following paragraphs define methods for determining quality metrics based on a logical gap for hierarchical multiplication. The following description may be applied to any of a variety of types of bricks, encoded logical qubits, and configurations. Furthermore, while the following description focuses on FBQC with the 6-ring network, the techniques may generalize to other models and schemes.

[0156] For the 6-ring fusion network, there are two distinct syndrome graphs termed the primal / dual syndrome graphs, analogous to the planar surface code, with checks belonging to the two independent syndrome graphs. The dual syndrome graphs may be collectively represented as syndrome graph that may be used to determine an error metric for a logical qubit. For more general logical blocks encoding channels from to qubits, there are independent logical error classes that generate all possible logical correlations from input to output. The set of distinct logical sectors is denoted herein by .

[0157] The logical gap rule , uses a metric that utilizes the factthat, below an error correction threshold, logical errors may be suppressed due toAttorney Docket No.7246-02501 distinguishability between logical sectors. On the other hand, above an error correction threshold, logical errors are not suppressed due to a loss of distinguishability between distinct logical sectors. In other words, below the threshold, the decoder may reliably differentiate which logical sector of the code space to recover to (as the code distance increases). In this spirit, one may define the logical gap as the difference between the correction weights that return the system to different logical sectors.

[0158] For example, consider a simple case of a single logical operator in a surface code memory block (e.g. only the primal syndrome graph), with aconfiguration and possible corrections , such that composing thecorrection and error yields a logical operator on the code space—namely and respectively. The signed logical gap is defined asweight of the correction forthe sector given by a choice of decoder, defined as follows: an edge has weightwhere is the (marginal) probability of Pauli error on that edge,edges supporting erasures have weight 0, and the total weight of a correction is . Other types of weights may also be used. The error as part of may be unknown and therefore which correction is correct is unknown; hence only the unsigned logical gap (i.e., the magnitude of the logicalgap) may be known in this circumstance, which is denoted herein as | |(below we will drop the dependence on for brevity).

[0161] In general, any decoder may be used to compute an unsigned logical gap and biased noise may be accommodated by modifying the weights appropriately. If one chooses a minimum-weight perfect-matching (MWPM)decoder, then the decoder may always choose the minimum weight correction. If0, the decoder will fail and a logical error will be introduced. If 0, thedecoder will succeed in correcting the error and if 0, the decoder will succeed / fail half of the time. Therefore, the EER for the brick becomes , where is the distribution of logical gaps of logical error classes for a fixed brick size and error rate. In more complex logical blocks (i.e.,Attorney Docket No.7246-02501 surface code protocols / channels), there may be many logical error classes and so one may compute a vector of logical gaps as the information of interest.

[0162] , | |, ,

[0163] where recall, is the set of distinct logical error classes. A combined score may be created for the brick to be thresholded by the policy as

[0165] ;

[0166] where represent tunable linear weights to weight the addition of the scores for all logical error classes.

[0167] To determine the two correction weights whose difference is the logical gap, in some embodiments weight contributions are computed for each single-qubit Pauli operator for a respective correction on the code or fusion network (on the syndrome graph, this corresponds to assigning a weight contribution to each corrected edge of the syndrome graph), and the overall weight of each correction is obtained by combining the weight contributions for each corrected edge of the correction. In some embodiments, the weight contributions may be computed as the log-likelihood ratios of the error rate that each qubit or fusion outcome is subject to. Edges may have weight zero, as could be the case, for instance, if an erasure error is detected on that qubit / fusion outcome. In the event that one considers a correlated error model, one can compute the weight of the correction in terms of new weights assigned to multi- qubit Pauli operators (in the syndrome graph, this corresponds to adding additional edges with appropriate weight contributions).

[0168] In some embodiments, the first and second corrections may be determined by a decoder, and may correspond to two potential corrections to the syndrome graph that either preserve or flip the overall logical state of the logical qubit, in some embodiments. For example, the syndrome graph data may be provided to a decoder, and the decoder may determine the first correction as the most likely correction that does not alter the overall logical state of the logical qubit. To obtain the second correction, the syndrome graph may be provided to the decoder with the constraint that it is to return a correction that flips theAttorney Docket No.7246-02501 value of the logical qubit, and the decoder may determine the most likely correction satisfying this constraint.

[0169] Said another way, if we denote C as the correction and E as the true error that occurred (which the decoder does not know), without the constraint (i.e., for the first correction) the decoder will determine the correction that has the highest probability of giving C+E=I, where I denotes a logical identity. In other words, it tries to find the first correction C that fixes what it thinks is the true error such that nothing happens to the logical state of the brick. For the second correction, the decoder is constrained to determine a C with the highest probability of giving C+E = Flip, where Flip denotes a flip of the state of the logical qubit. In other words, the correction plus the error should flip the logical sector.

[0170] The decoder may add information related to the first and second corrections to the syndrome graph data, and this supplemented syndrome graph data may then be provided to the Controller. In some embodiments, the decoder may be comprised within the Controller, or alternatively it may be instantiated as separate circuitry (e.g., as a dedicated classical processor and memory coupled to the controller, which may be contained within the classical computing system 203). The information related to the first and second corrections specifies the modifications to the syndrome graph that are entailed by the respective corrections (i.e., the location and / or number of the edges that are flipped by the correction). These first and second corrections are then the two alternative corrections for which two respective weights are determined by the Controller from the syndrome graph data received from the decoder, and the magnitude of the difference between the two weights is the magnitude of the logical gap. A simple example of two alternative corrections to a 2D syndrome graph is shown in Figures 14C-D and described in greater detail below.

[0171] In some embodiments, the overall weight of each correction may be determined as a weighted summation over the weight contributions of each corrected edge of the respective correction. In some embodiments, the summation is weighted based on log-likelihood ratio (LLR) weights of the respectivecorrected edges, as shown in the expression ln , where is theAttorney Docket No.7246-02501 (marginal) probability of a Pauli error on that edge. In these embodiments, a corrected edge with a smaller error probability will have a larger weight than if the corrected edge had a larger error probability. In this manner, edges that are relatively more likely to have experienced an error will be granted a smaller weight. Accordingly, corrected edges that are more likely to have experienced an error will have a smaller weight contribution (as else being equal), where a smaller overall weight corresponds to a correction that is more likely to not result in a logical error. Said another way, edges with a large are relatively more likely to require correction, and the LLR weights promote these corrections. For some logical blocks, each edge may have the same error probability such that has a single value that is constant throughout the logical block. However, in some cases, different edges may have different error probabilities and may vary between different edges in the logical block.

[0172] For a pair of parity errors that is identified as a pair of syndromes to be corrected in the syndrome graph, there may be many different combinations of Pauli errors that may be used to attempt to correct the parity error (e.g., any sequence of flipped edges on the syndrome graph that share endpoints with the pair of parity errors). The decoder may provide two specific alternative corrections, as shown in Figures 14C and 14D. The weight of each correction may be generally understood to be an increasing function of the number of flipped edges in the correction, whereby (all else being equal, e.g., without accounting for any potential weighting of the flipped edges based on their log-likelihood ratios and / or their location in the syndrome graph) corrections that flip a larger number of edges will have a larger weight than corrections that flip fewer edges. In some embodiments, the set of corrected edges connects each pair of syndromes in the syndrome graph in a particular way according to a geometry determined by the decoder.

[0173] As used herein, the term “graph distance” refers to the separation between two nodes on a syndrome graph. For example, two adjacent nodes (i.e., parity checks) connected by a fusion measurement (i.e., and edge) in a fusion- based encoding scheme have a graph distance of one. Two parity checks with one intervening parity check have a graph distance of two, etc. Weighting the weightAttorney Docket No.7246-02501 contributions based on a graph distance from the first physical qubit (e.g., a center qubit of the logical block) may improve the effectiveness of a quality metric, as described below, in identifying high-fidelity logical states. For example, a set of errors that spans across the face 118 shown in Figure 1B will be uncorrectable via quantum error correction, whereas a set of errors that lies within the face without spanning across may be correctable via quantum error correction.

[0174] Weighting the summation of weight contributions of corrected edges based on the respective graph distances may include weighting each weight contribution with a factor comprising the respective graph distance raised to a power of a tunable parameter. The tunable parameter may be empirically adjusted to improve the effectiveness of the quality metric.

[0175] The logical gap magnitude may be determined by taking the magnitude of the difference of the overall weights of the two alternate corrections of the syndrome graph.

[0176] Because the parity error may not uniquely identify the specific set of Pauli errors that occurred (e.g., the observed parity error may be consistent with two or more potential sets of Pauli errors), it may not be known a priori which correction will correct the error. However, corrections with larger weights are generally less likely to be correct. All else being equal, the weight of a correction increases for corrections that involve flipping a larger number of edges, and errors of this type are relatively less common than simpler errors that involve flipping fewer edges.

[0177] Accordingly, a large magnitude of the logical gap (i.e., a large magnitude in the difference between the two weights of the two alternative corrections) may indicate that one correction is much more likely to be correct than the other one (e.g., the correction with a small weight may be more likely to be correct). In this case, the error is likely correctable since it may be determined with a high probability that one of the two alternate corrections is the correct one, and the error may be likely fixable using a decoder. Conversely, when the magnitude of the logical gap is small, both of the two corrections may be comparably likely to be correct so it may be less likely for the decoder toAttorney Docket No.7246-02501 implement the proper correction (e.g., the decoder may have close to a 50 / 50 chance of implementing the proper correction). Accordingly, the magnitude of the logical gap may serve as an effective quality metric to quantify how likely the errors indicated by the syndrome graph data are to be correctable by the decoder, where syndrome graph data with a larger magnitude logical gap are identified as corresponding to higher fidelity logical qubits.

[0178] In some embodiments, a respective logical gap magnitude of the syndrome graph data is determined for each class of logical error (e.g., for each logical error sector) of the syndrome graph data. For example, a syndrome graph may include a primal graph and a dual graph, each of the primal and dual graphs may have their own respective class of logical errors, and a respective logical gap magnitude may be determined for each of the primal and dual graphs. In some embodiments, the primal and / or the dual graph may themselves contain multiple classes of logical error, and a respective logical gap magnitude may be determined for each class of error. The logical gap magnitudes for each class of error may be combined in any of a variety of ways (e.g., summed) to obtain an overall error metric based on the logical gap magnitudes. Figure 14A-D – Correction of Syndrome

[0179] Figures 14A-D illustrate an example of a syndrome graph and syndrome graph data in accordance with one or more embodiments. In this example, a simplified 2D syndrome graph is used but one of ordinary skill will appreciate that any type of syndrome graph corresponding to any error correcting code may be used without departing from the scope of the present disclosure.

[0180] Figure 14A illustrates the geometry of the syndrome graph, a square lattice in this example. Such a square lattice syndrome graph is associated with the primal or dual syndrome graphs of the surface code implementation discussed above in reference to FIGS. 1A-F.

[0181] Figure 14B shows an example of syndrome graph data that is superimposed on the syndrome graph, where vertices correspond to the set of measurement outcomes of the various parity checks. In the illustrated example,Attorney Docket No.7246-02501 each vertex includes a 1 if the parity check measurement returned an odd parity measurement outcome (also referred to herein as a “syndrome”) and a zero if the parity check returned an even parity, where the parity check measurements can be implemented as shown above in FIGS.1D-1E or 1F. Depending on the type of error correcting code being employed, one of ordinary skill will appreciate that the parity measurement outcomes may be computed by any known method. A useful way of understanding the syndrome graph data shown in Figure 14B in the context of the surface code example of FIGS.1D-1E is that any “syndromes” present in the syndrome graph data, are the result of one or more errors on the underlying data qubits, which can be thought of as positioned on the edges of the superimposed syndrome graph. Vertices having an odd number of data qubit errors incident thereon will result in a syndrome being present at the vertex (i.e., the vertex is labeled 1) while vertices having zero or an even number of data qubit errors incident thereon will result in no syndrome being present on the vertex (i.e., the vertex is labeled 0). The situation for an FBQC implementation of the surface code is slightly more complicated because, rather than being directly computed from the measurement of the measure qubit located at each vertex, the syndromes are computed from the surrounding two-qubit fusion measurement outcomes that can be visualized as located on each incident edge.

[0182] Figure 14C illustrates an example correction that a decoder would produce when provided the syndrome graph data of Figure 14B. The correction can include the one or more Pauli operators representing the set of operations that could be applied (or alternatively, not applied, but instead tracked and accounted for in the system as the quantum computation progresses) on the underlying data qubits of the error correction code to correct errors on those qubits while still preserving the overall quantum state of the logical qubit. One of ordinary skill will appreciate that many different decoders are known and any could be deployed here without departing from the scope of the present disclosure. For example, a minimum-weight perfect matching decoder may be used to determine the one or more corrections to be used for fault-tolerant post selection processing. The correction shown in Figure 14C is an example of a minimum-weight correction also referred to herein as the “first correction”. NoteAttorney Docket No.7246-02501 that there can be many possible minimum weight corrections that could be computed by the decoder, and Figure 14C only illustrates one particular example. The weight of this correction is 8, because it identifies 8 Pauli operators on the underlying physical data qubits (shown by the 8 edges are traversed by the thick black line). In some embodiments, to calculate a logical gap, a second correction can be computed with the constraint that the correction should flip the overall logical value of the logical qubit with the assumption that the first correction properly identifies the underlying errors on the data qubits, i.e., the second correction can be computed such that it that differs from the first correction by a logical operator (e.g., a chain of Pauli operations that spans from the left edge of the syndrome graph to the right edge of the syndrome graph (or from the top to the bottom).

[0183] Figure 14D shows one example of a second correction represented as a light grey thick line. In this example, the second correction plus the first correction results in a logical error on the logical qubit because a chain of errors spans the lattice from left to right. Note, there are many possible candidate corrections for this second correction as well, as there was for the first correction. The weight of the second correction shown in Figure 14D is 9. The magnitude of the logical gap may then be obtained by taking the difference of the weights of the corrections shown in Figures 14C and 14D, which results in a magnitude ofthe logical gap of |9 8| 1.

[0184] In the example above, the decoder receives syndrome graph data that indicates the location(s) of syndromes. More generally as described in more detail below, the decoder can receive a set of data representing the visible error which includes the location of both the syndromes and underlying erasure errors if any. As used herein the term syndrome graph will be used synonymously with visible error and it is therefore understood that syndrome graph data includes both syndromes and erasures. In some examples, the syndrome graph data received by the decoder may take the form of a matrix of syndrome values, where each entry of the matrix is mapped to a vertex in the syndrome graph (e.g., the vertices shown in Figures 14A-C). In addition, the syndrome graph data may include a second related matrix that includes erasure errors, where, e.g., in theAttorney Docket No.7246-02501 case of the surface code shown in in FIGS. 1D-1E, each entry of the matrix is mapped to an edge in the syndrome graph. Figures 15A-D - Connected Components – 2D Example

[0185] Figure 15A-D are simplified diagrams of 2-dimensional syndrome graphs illustrating the relationship between the primal and dual graphs during a sequence of fusion measurements. Actual implementations of FBQC will typically employ 3-dimensional syndrome graphs. However, the essential concepts of primal / dual correspondence and connected components may be more easily visualized and explained in the context of a 2-dimensional syndrome graph. It may be appreciated by one of skill in the art that the concepts introduced herein for a connected component in the context of a 2D syndrome graph may be generalized to apply to a 3D syndrome graph.

[0186] Figure 15A illustrates an overlay of an example 2D primal syndrome graph (dotted lines) and a 2D dual syndrome graph (dashed lines). Solid circles connected to dotted lines are syndrome values of the primal graph, whereas solid circles connected to dashed lines are syndrome values of the dual graph. Syndrome values may be calculated by performing a parity check once all of the syndrome graph edges connected to the syndrome value have been measured. A single fusion measurement will measure both a single edge in the primal graph and a single corresponding edge in the dual graph, two examples of which are shown in Figure 15A. More generally, each primal edge and dual edge pair that intersects at their midpoint will have both edges measured by a single fusion measurement.

[0187] Figure 15B illustrates the same syndrome graphs as Figure 15A, but with the lattices separated for clarity. The arrows illustrate the correspondence between two sets of primal / dual edges, where the two edges indicated by an arrow will be measured by a single fusion measurement.

[0188] Figure 15C illustrates the same syndrome graphs as Figures 15A- 15B after a subset of the sequence of fusion measurements have been performed. As illustrated, bold solid black edges indicate that the fusion measurement resulted in an erasure, whereas non-bold solid edges indicate that the fusion measurement was successful. Unmeasured edges are indicated with dashed lines.Attorney Docket No.7246-02501 For clarity, primal and dual edges are not distinguished by line type in Figures 15C-D, rather, they are separated (left and right, respectively).

[0189] As used herein, the term “connected component” refers to any contiguous set of erased edges in either the primal or dual syndrome graph. For example, a connected component composed of seven erased edges is shown in the primal graph on the left side of Figure 15C, whereas the dual graph shown on the right side of Figure 15C has a larger number of smaller connected components (e.g., 4 connected components of size two, one, one, and one, indicated by the bold lines). An edge may be erased by either a loss measurement result or a failed measurement result. As used herein, any node that is connected to an edge of a connected component is considered to be “within” the connected component.

[0190] In the example shown in Figure 15C, the subsequent fusion measurement to be performed involves the two edges indicated by the double- headed arrow (one primal edge and one dual edge). A failure outcome will result in one of the primal or dual edges involved in the measurement being erased, and the basis for performing the fusion measurement may be selected to determine which of the involved primal or dual edge will be erased in the event of a failure outcome. A successful outcome will successfully measure both of the involved primal and dual edges, while a loss outcome of a fusion measurement will erase both edges.

[0191] In the example shown in Figure 15C, the two ends of the primal edge involved in the fusion measurement are part of the same connected component. Accordingly, an erasure of this primal edge will not increase the likelihood of an overall logical error.

[0192] Figure 15D illustrates an alternative scenario, similar to Figure 15C but with a different set of prior fusion measurement results. Similar to Figure 15C, bold solid black edges indicate that a fusion measurement of the edge resulted in an erasure, non-bold solid edges indicate that the fusion measurement was successful, and unmeasured edges are indicated with dashed lines. The two edges involved in the next fusion measurement are indicated with the double-headed arrow.Attorney Docket No.7246-02501

[0193] In the example illustrated in Figure 15D, an erasure of either of the involved primal or dual edges will increase the likelihood of a logical error. To quantify the risk of causing a logical error, an exposure may be calculated for each of the involved primal and dual edges.

[0194] The “exposure” of an edge is defined herein as follows. Each edge connects two nodes of either the primal or dual graph. Often, one or both of these nodes may be part of one or more respective connected components. For example, the left node connected to the indicated edge of the primal graph in Figure 15D is part of a 6-node connected component, and the right node connected to the indicated edge of the primal graph is part of a different 6-node connected component. The exposure of an edge is calculated as the product of the exposures of these two connected components. The exposure of a connected component, in turn, is defined as a sum (either weighted or unweighted) of the unmeasured edges and / or the measured edges adjacent to the connected component, not counting the edge for which the exposure is being calculated. As one example, the left connected component adjacent to the indicated edge of the primal graph in Figure 15D may be determined to have an exposure of 3, whereas the right connected component adjacent to the indicated edge of the primal graph in Figure 15D has an exposure of 6, leading to an overall exposure of the indicated edge of the primal graph of 3*6=18.

[0195] In some cases, the edge may be adjacent to a node that is not part of a connected component. For example, the indicated edge in the dual graph of Figure 15D is directly above a connected component of size 3, but directly below a region of unmeasured nodes that does not contain a connected component. For such nodes that are not part of a connected component, the exposure of the node is set equal to 3 to account for the three unmeasured edges (not counting the indicated edge) adjacent to this node. Accordingly, the indicated edge of the dual graph of Figure 15D has an exposure of 3*1=3.

[0196] When performing hierarchical multiplexing, the exposure of copies of sub-bricks may be considered when determining an aggregate quality metric based on the quality metrics for multiple sub-bricks. For example, the illustrated primal graph may include two sub-bricks 1502 and 1504, and the quality metricAttorney Docket No.7246-02501 used to determine which copy of the two sub-bricks to select to fuse together may be determined based on the overall exposure for each pair of copies. For example, for each copy of each of the two sub-bricks, an overall exposure may be calculated for the fusions that are to be performed between the two copies, and these overall exposure values may be used to determine an aggregate quality metric for each pair of copies (e.g., with higher overall exposure values corresponding to lower quality metrics). Accordingly, the quality metric may be a joint quality metric that is determined for each set of two or copies of different sub-bricks, rather than an individual quality metric that depends solely on the properties of the respective copy in isolation. Said another way, the aggregate quality metric may be based on the positions of errors in a copy of the sub-brick 1502 relative to the positions of errors in a copy of the sub-brick 1504, rather than being only based on the number of errors in the two copies. Fusion Network Adaptivity for Logical Block Construction

[0197] Embodiments herein describe systems and methods to perform fusion network adaptivity to provide improved error tolerance in a quantum computing architecture. The process of generating large, entangled resource states involves randomness through fusion failures and erasure errors due to loss. Embodiments herein utilize “fusion network adaptivity”, which as used herein refers to adaptively modifying an orientation and / or location of a block in a target fusion network, and / or adaptively modifying the target fusion network, based on classical information from previous fusion measurement results. As used herein, “orientation” refers to how the qubits of a brick are fused into a fusion network at a given location. Different orientations may correspond to rotations, reflections, and other permutations or rearrangements of the qubits of the brick, with respect to the location of their fusion sites in the fusion network. Figure 18 illustrates several examples of different orientations for a Kagome-6 (i.e., a 6-ring) brick.

[0198] In some embodiments, the target fusion network may be adaptively modified based on Tanner graph data of the previously constructed blocks. Advantageously, these methods make more effective use of resource states withAttorney Docket No.7246-02501 one or more errors, to maintain high fidelity thresholds and error tolerance. Embodiments herein provide for methods to construct a logical block with a given level of error tolerance using a lower quantity of computational resources. For example, resource states are generated with a random distribution of errors, resulting in a corresponding quality distribution of the generated resource states. An example quality distribution of generated resource state is shown in Figure 16A. Fusion network adaptivity may allow for effective utilization of a larger percentage of these resource states (i.e., the quality threshold may be moved left in Figure 16A, as resource states of lower quality may still be made effective by adapting their orientation and / or location in the fusion network). Figure 16B shows how a generated resource state may be adaptively placed in a fusion complex based on the error profile of the generated resource state.

[0199] In some embodiments, methods for network fusion adaptivity be used in conjunction with a hierarchical multiplexing scheme, to construct logical blocks that satisfy a given error tolerance while using a smaller number of copies. This increased efficiency may be obtained by adapting to and improving tolerance for the inherent randomness involved in entanglement generation.

[0200] Embodiments herein describe switching networks that can alter the way a resource state or block fuses to other resource states or blocks. By implementing these switching options, failures and erasures indicated by Tanner graph data may be more favorably placed in the fusion network. For example, fusion network adaptivity may enable a favorable alignment of errors from resource state generation with those of other resource states or fusions.

[0201] Figure 17 illustrates an example of how the location and orientation of a resource state in a target fusion network may be selected from a plurality of options through translation and / or permutation of the resource state. Before fusing two resource states together (in the illustrated example, two 6-ring resource states), the qubits of the resource state may be permutated (such as a rotation or reflection of the qubits in the ring). Additionally or alternatively, the resource state may be fused to a different location in the fusion network (translation).Attorney Docket No.7246-02501

[0202] In some embodiments, fusion network adaptivity is used in the context of a fixed background fusion network, where the adaptivity takes the form of determining where and how resource states are fused in a fixed target fusion network. These embodiments are referred to herein as bulk-structured fusion complex adaptivity (BS-FCA). BS-FCA assumes an ambient fusion complex, and adaptivity is determined by resource-state qubit permutations and translations that preserve the fusion complex.

[0203] Permutations of qubits in a fusion network may be identified with different symmetry classes. For example, a resource state generator (RSG)- preserving symmetry, SRSG, is associated with a qubit permutation that leaves the RSG network invariant. i.e., permuting the outer qubits and the seed-states they belong to leads to a permutation of the RSG fusions. A resource state (RS)- preserving symmetry, SRS, describes a qubit permutation that leaves the resource state invariant. A fusion network (FN)-preserving RS symmetry, SFN, is associated with a permutation of the resource state that is a RS-preserving symmetry up to single Clifford unitary operators (The idea behind is that the local Clifford(s) can be compensated by an adaptive fusion – but this depends on the local encoding and may not always be permissible.). Finally, a FN-preserving symmetry, S, is associated with a permutation of any qubits in a fusion network that leaves the fusion network invariant (in terms of checks and membranes). SRSGare syndrome-graph preserving operations, but in general elements of SRSand SFN may not preserve the syndrome graph. Figure 18 illustrates examples of qubit rotations and reflections for a 6-ring resource state according to the different symmetry classes. Note that for a cluster-state version of a 6-ring resource state, the 60-degree rotation depicted for SFN / SRSin Figure 18 becomes an element of SRS(i.e., the state is more symmetric than the CSS version). This symmetry swaps primal and dual outcomes.

[0204] In contrast to BS-FCA, in some embodiments fusion network adaptivity may dynamically modify the target fusion network. In particular, permutations and translations can be applied to a resource state to fuse it together with other fusion networks in a way that a final target fusion network is not set before run-time. In this way, the fusion locations, and subsequently theAttorney Docket No.7246-02501 checks of the fusion network, can be dynamically chosen in real time as the fusion network is constructed. Dynamic choice of fusion locations may be used to generate fusion networks that do not have a fixed check structure, but that is adaptively chosen based on the Tanner graph data (syndromes and erasures) and current resource-state samples, and is obtained during runtime. The location of the fusions to be performed (i.e., which pairs of qubits from which resource states are to be fused) at each step may be determined by considering a set of available fusions and ranking them according to a quality metric.

[0205] For instance, one may choose to scan all qubits which are available to be fused (or all viable fusion locations contained within an allowed neighborhood), and consider all possible fusions between pairs of qubits (in the case of 2-way fusions), including potentially all permutations of resource states. From these candidate fusions, one may consider a quality metric that is a function of one or more of: (i) the existing check weights and erasure configuration, (ii) the stabilizer weights on the currently unfused qubits, (iii) the expected updates to both of these items based on the candidate fusion, and / or (iv) the most recently sampled resource state. Based on the quality metric, the candidate fusion to perform may be selected.

[0206] For example, one may consider a quality metric for a candidatefusion that is given by | | 4 , , where w is the expectedweight of the stabilizers on the outer (unfused) qubits, after the candidate fusionhas been performed, expressed in a minimal basis, where . refers to a sampleaverage over all outer stabilizers, and where a and , are tunable parameters. A choice of 4 is natural if one wants to penalize stabilizers of weight other than 4, which is that of the surface code. By expected, we mean one can determine based on the prior probabilities of fusion outcomes, failures and losses, what the average stabilizer weight should be. Another example is where thequality metric is given by , 0 , where is a tunableparameter, and is the weight (number of fusion outcomes) of any checks that have support on the outer (unfused) qubits. One may also consider computing quality metrics that involve exposure or logical gaps, in some embodiments.Attorney Docket No.7246-02501

[0207] Figure 19A illustrates a switching circuit that may be utilized to implement network fusion adaptivity, according to some embodiments. As illustrated, the switch 1904 receives 6 qubits of a resource state 1902 through 6 respective inputs and may be controlled to adaptively direct the 6 qubits to 6 different outputs. Note that while the circuit shown in 19A illustrates a switch where each of the 6 inputs may be redirected to any of the 6 outputs, in some embodiments the switching circuitry may be simplified to accommodate the case where not all redirects of input qubits are desirable. For example, if the different orientations are limited to one or more of the examples shown in Figure 18, the switching circuitry may be modified to only accommodate this limited number of orientations.

[0208] Figure 19B shows a combination of the network fusion adaptivity switching circuit 1912 and a hierarchical multiplexing switching circuit 1910. As illustrated, fusion network adaptivity switch may be used in conjunction with the hierarchical multiplexing switch, where either the first or second copy of the illustrated resource state is selected by the hierarchical multiplexing switch (e.g., based on quality metrics of the two copies), and the fusion network adaptivity switch subsequently reroutes the six qubits of the selected copy based on the determined orientation (e.g., based on quality metrics of different orientations). Figure 19B illustrates the case of 2-to-1 multiplexing, but more generally N-to-1 multiplexing may be implemented for any positive integer N. Note that while Figure 19B places the hierarchical multiplexing switch before the fusion network adaptivity switch, in some embodiments this order may be reversed (e.g., so that an orientation is selected for each copy prior to selecting a copy by the hierarchical multiplexing switch). In these embodiments, a separate instance of the fusion network adaptivity switch 1912 may be implemented for each of the two (or more copies). Figure 19B illustrates the switches 1910 and 1912 as separate circuits, but in some embodiments the two types of switching functionality may be combined in a single switching fabric.

[0209] In some embodiments, an additional switching circuit may be added into the setup shown in Figure 19B to implement translations of the resource state in the target fusion network (not shown in the Figure). This switchingAttorney Docket No.7246-02501 circuit may receive the six qubits of the resource state as inputs and output them to different locations in the target fusion network. Figure 20 – Flowchart for Fusion Network Adaptivity of Logical Blocks

[0210] Figure 20 is a flowchart that illustrates a method for performing topological fusion network adaptivity for encoding logical qubits or logical blocks, according to some embodiments. The method shown in Figure 20 may be used in conjunction with any of the computer systems or devices shown in the above Figures, among other devices. For example, the method shown in Figure 20 may be performed by a quantum computing device or system as illustrated in Figures 2A and 2B. The quantum computing system may further include a controller (e.g., the controller 206 illustrated in Figure 2B) to direct the described method steps, and may be included in (or be coupled to) a classical computing system for processing classical information and directing operations of the quantum computing system. For example, the controller may include one or more processors configured to execute program instructions stored on a non-transitory computer-readable memory medium. In some embodiments the methods described in Figure 20 may be utilized in a quantum communication network, quantum internet, or more generally in any application where it is desired to encode high fidelity qubits.

[0211] In some embodiments, the methods described in reference to Figure 20 may be used in conjunction with hierarchical multiplexing, as described in reference to Figure 5. For example, the determination of quality metrics for the multiplexed copies of bricks, as described in reference to Figure 5, may be supplemented by considering quality metrics for multiple orientations and / or locations in the target fusion network of each of the copies (as described in greater detail below in reference to Figure 20). This may result in preparation of a more efficient and higher fidelity logical block, as a particular copy of a brick may have a relatively low quality metric in its default orientation, but a rotation and / or reflection of the copy may result in a higher quality metric. This copy may then be selected for fusion in the target fusion network in the rotated and / or reflected orientation.Attorney Docket No.7246-02501

[0212] It is to be understood that this method may be used by any quantum computing architecture, and these other architectures should be considered within the scope of the embodiments described herein. As illustrated, the method shown in Figure 20 may proceed as follows.

[0213] At 2002, a plurality of bricks of a target fusion network is received. The bricks may be individual resource states (e.g., 6-ring resource states or another resource state), or they may be multiple resource states fused together (e.g., a higher level brick). The methods of Figure 20 describe utilizing a target fusion network in the context of FBQC. More generally, the embodiments described herein may be used in other types of quantum computing applications, such as circuit-based quantum computing. In these embodiments, the plurality of bricks may be components of other types of fault tolerant quantum networks.

[0214] In some embodiments, a controller, such as the controller 206 illustrated in Figure 2B, may direct a quantum circuit to produce the plurality of bricks, which are then received by a quantum computing system. The target fusion network may be a 6-ring network, or another type of fusion network. The bricks may be at any fusion stage of the target fusion network. For example, in the 6-ring fusion (also known as Kagome-6) network shown in Figures 4A-C, the bricks may be stage 0, 1, 2, 3 or 4 bricks.

[0215] In an FBQC implementation, the bricks are comprised of physical qubits within multi-qubit resource states. Each resource state includes a plurality of physical qubits prepared in a specific entangled state. For example, in some embodiments a 6-qubit resource state is employed (e.g., as shown in Figure 1H), as prescribed by the error-correcting code and desired logical operation. At each time step, a subset of the qubits of each resource state are fused with respective qubits of other resource states (thus creating higher level bricks), and a remaining subset of the qubits of each resource state are propagated to a subsequent time step. The fusion measurement causes the unmeasured qubits of the resource states to become entangled with each other, and this entangled quantum information is carried forward until the end of the computation, whereupon all remaining resource state qubits may be measured to produce classical measurement results. The intermediate fusion measurementsAttorney Docket No.7246-02501 produce classical information, such as Tanner graph data, which may be used by a classical processor to determine a quality metric for the bricks. As described herein, the logical block includes the resource states that are input into the block (e.g., the input surface 114 from Figure 1G), the classical information that results from the fusion measurements, and the qubits that are output at the back end of the block (e.g., the output surface 118 from Figure 1G).

[0216] In some embodiments, the plurality of bricks of the target fusion network are mutually disjoint. For example, each brick may contribute to a separate and distinct portion of the target fusion network.

[0217] At 2004, a respective first quality metric is determined for each of a plurality of orientations in the target fusion network of a first brick of the plurality of bricks. The different orientations may be different rotations and / or reflections of the first brick. For example, for the case of a 6-ring resource state, there are 2 nontrivial rotations and 3 non-trivial reflections that may be performed. In some embodiments, Tanner graph data (e.g., syndrome graph data) is received for the plurality of bricks, and the quality metric for each orientation is determined based on the Tanner graph data of the first brick and / or of a second brick to which the first brick will be fused. The bricks may be encoded blocks that each include a plurality of physical qubits, and the Tanner graph data may include classical information describing outcomes of the encoding process of the first brick. The Tanner graph data may be syndrome graph data that specifies the locations of one or more syndromes (i.e., parity errors) and / or one or more erasure errors in the syndrome graph of the logical qubit. The syndrome graph data may also specify a respective set of corrected edges produced by a decoder for each of a first and second correction of the syndromes and / or erasure errors, as described in greater detail below.

[0218] The syndrome graph data may include one or more syndromes, which represent one or more Pauli errors in the syndrome graph. A Pauli error refers to, as one example, a qubit flip error where a qubit has flipped its value (e.g., in the specific case of a dual-rail encoded photonic qubit, the photon may have inadvertently moved to the other waveguide). The syndrome graph data may further contain one or more erasure errors, which represent locations whereAttorney Docket No.7246-02501 a physical qubit has escaped the circuit (e.g., a photon may have tunneled out of and escaped the waveguide). An illustration of syndromes and erasure errors within a syndrome graph is shown in Figure 1D.

[0219] In some embodiments, the Tanner graph data specifies locations of one or more erasures in both the first brick and a second brick to which the first brick will be fused. In this case, an orientation of the first brick that places a first erasure in the Tanner graph data of the first brick on a same multi-edge as a second erasure in the Tanner graph data of the second brick is determined to have a higher first quality metric than an orientation that does not place the first erasure on the same multi-edge as the second erasure. As used herein, for fusion networks that utilize syndrome graph data, a “multi-edge” is a set of two or more edges in a syndrome graph that have the same source and target vertices. When Tanner graph data is used, a multi-edge refers to a fusion with an orientation that places a first erasure in the first Tanner graph data on an already erased outcome node in the second Tanner graph data.

[0220] Advantageously, fusing two erasures together (from the first and second bricks) on a shared multi-edge doesn’t increase the likelihood of a lattice- spanning logical error. Accordingly, the overall quality of the aggregate brick may be improved by rotating and / or reflecting the first brick such that an erasure on the first brick is fused to an erasure on the second brick. In some embodiments, contributions to the quality metric may be determined from both the primal and dual graphs, and these contributions may be summed to obtain the overall quality metric. For example, the overall quality metric may be proportional to total the number of aligned erasures (aligned between the first and second bricks) in both the primal and dual graphs. Note in the case of a K6 RS, networked fusions may have support on a multi-edge with RSG fusions from RSs that are yet to be fused in.

[0221] In some embodiments, the first quality metrics are determined based on exposures calculated for a fusion of the first brick to the second brick according to different orientations. This may be implemented according to a variety of specific methodologies, in various embodiments. In some embodiments, a quantity referred to herein as the “cumulative exposure” may be calculatedAttorney Docket No.7246-02501 between the vertices of the first Tanner graph data (of the first brick whose quality metric is being determined), and the currently sampled (full) syndrome graph. In some embodiments, referred to herein as scoring approach A, let be defined as the primal sub-syndrome graph corresponding to the current resource state sample (the first brick) for a given candidate location / orientation. For every vertex , define

[0222] ,,

[0224] where is the size of the cluster in the RSG syndrome graph, and is the size of the w cluster in the full syndrome graph (note: not the true exposure) that v belongs to. The variablemay be selected to have various values, depending on the implementation. For example, (i),0 if , for an intra-RSG cluster, (ii),1 if v and w belong to the same cluster and (intra cluster exposure), and (iii),1 if v and w belongto distinct clusters and (inter cluster exposure). In some embodiments,1if is erasure free, otherwise 1 (RSG quality). is anadjustable parameter that may be set to +1 or -1. The dual cumulative exposure may be calculated similarly to the primal cumulative exposure, and the overall score calculated simply as the sum (note that larger scores are preferred).

[0225] In some example embodiments, exposure may be determined according to a second methodology, referred to herein as scoring approach B: For a configuration (e.g., an orientation of the brick or resource state), ,primalmay be determined as:

[0226] where is the size of the cluster . Alternatively, in some embodiments, is the exposure of the cluster c. Similarly, for :Attorney Docket No.7246-02501

[0227] In this approach, the scores are then as follows:

[0228] Score

[0229] In some embodiments, the scores between primal and dual may bealternatively combined, e.g. as Score exp b primal dual ) whereb is a tunable parameter.

[0230] In some example embodiments (e.g., where 2 ), the error model can be implemented as: (1) network fusion outcomes (e.g., fusions performed for a fusion network, by a fusion network projection device (FNP)) are erased with rate , (2) while RSG fusions (fusions performed to generate a resource state, bya resource state projection device (RSP), discussed below) are erased with rate / , where can correspond to the level of multiplexing that is performed in anRSG to generate resource states. In some example embodiments, the fusion complex adaptivity scoring can be implemented where the error model is based on current RSG fusions of the current stage (e.g., at rate x), and RSG fusions of a previous RSG stage (e.g., at rate x / r), as discussed in further detail below with reference to Figure 21.

[0231] In some embodiments, translation is performed on the first brick, where quality metrics are determined for different locations of the first brick in the target fusion network, and a location is selected with the highest quality metric. This may be performed in addition to or as an alternative to determining a quality metric for different orientations (e.g., a separate quality metric may be determined for each of a plurality of orientations at each of a plurality of different locations). In these embodiments, a first location may be selected to fuse the first brick in the target fusion network, and the second brick to which the first brick is fused may be selected based on the first location.

[0232] In some embodiments, the first quality metric is a logical gap magnitude of the respective syndrome graph of each layer one copy. The logical gap may be determined from the syndrome graph data. The logical gap magnitude is a magnitude of the logical gap, where the logical gap is the difference in weights between first and second correction operators belonging to distinct classes of correction. In some embodiments, the first and second corrections correct for syndromes indicated by the syndrome graph data. In someAttorney Docket No.7246-02501 embodiments, the weight of a correction operator is computed as a real-valued monotonic function of the number of syndrome graph edges that it corrects (i.e., the number of non-identity single-qubit Pauli operators in the correction).

[0233] In some embodiments, the first quality metric is a summation over the set of distinct logical error classes of a decaying exponential function of the respective magnitudes of the respective logical gaps. More generally, any monotonically decreasing function of the logical gap magnitudes may be used for the error metric, and the specific form of function may be determined empirically to improve performance metrics of the fault-tolerant post-selection procedure.

[0234] In some embodiments, second quality metrics are determined for different orientations in the target fusion network of the second brick. In these embodiments, a second orientation in the target fusion network of the second brick is selected based on the second quality metrics, and fusing the first brick to the second brick is performed with the second brick oriented according to the second orientation. In some embodiments, an aggregate quality metric for combinations of orientations of the first brick and orientations of the second brick in the target fusion network. For example, aggregate quality metrics may be determined that consider combinations of orientations of both the first and second brick, and the first orientation and the second orientation may be selected based on the aggregate quality metric.

[0235] Said another way, rather than analyzing the quality metrics of the orientations of the first and second bricks in isolation, an aggregate quality metric may be considered that looks at the quality of the fused combination of the first and second bricks according to their different respective orientations. As one example, it may be considered whether the errors in adjacent bricks line up to form a lattice-spanning error for the aggregate brick. More generally, the aggregate quality metric may consider how likely it is for the set of syndromes and erasures in the two bricks to result in an overall logical error of the target fusion network. For example, a first orientation of a first brick may have a high likelihood of a lattice-spanning logical error when fused to a second brick in a second orientation (e.g., if their erasures and / or syndromes happen to line up in an adverse way), but may have a low likelihood of a lattice-spanning logical errorAttorney Docket No.7246-02501 when fused to the second brick with one or both of the first and second bricks in a different orientation. Note that the aggregate quality metric may be qualitatively different from the quality metrics of each brick considered in isolation. For example, an orientation of a first brick with a relatively poor individual quality metric may have a better aggregate quality metric when considered in combination with a particular orientation of the second brick in the target fusion network.

[0236] At 2006, based at least in part on the first quality metrics, a first orientation in the target fusion network is selected for the first brick. For example, the orientation with the highest first quality metric may be selected.

[0237] At 2008, a controller such as the controller 206 illustrated in Figure 2B may direct the quantum circuit to fuse together the first brick to a second brick of the plurality of bricks to produce an aggregate brick. The fusion may be performed according to the determined first orientation. Fusing the first brick to the second brick according to the first orientation may be performed by routing a plurality of qubits of the first brick through a switching circuit that adjustably couples the plurality of qubits to specific qubits in the second brick according to the first orientation. An example switching circuit for a K6 resource state is shown in Figures 19A-B, in accordance with some embodiments.

[0238] In some embodiments, the methods described in Figure 20 may be used in conjunction with hierarchical multiplexing, as described in Figure 5. In these embodiments, the switching circuit may be the same switching circuit used to route bricks according to a hierarchical multiplexing scheme, as described in reference to Figure 5. For example, for a first brick with m qubits and n-fold multiplexing (where n copies are made of each brick, and one of the copies is selected), a switching circuit with m*n inputs and m outputs may be utilized to select one of the n copies and reroute the m qubits in the selected brick to the desired outputs, according to the selected orientation. Alternatively, in some embodiments the switching circuit for implementing the selected orientation (e.g., the switch 1904 in Figure 19A) may be in a separate switching layer from the switching circuit used for hierarchical multiplexing (e.g., the switching circuits 614 in Figure 6B or 712 in Figure 7B).Attorney Docket No.7246-02501

[0239] In some embodiments, the methods described in reference to steps 2002-2008 may be iterated for one or more additional stages of the target fusion network. For example, quality metrics may be determined for different orientations of the aggregate brick in the target fusion network, and an orientation of the aggregate brick may be selected for it fusion to a second aggregate brick. In general, the methods described in reference to Figure 20 may be performed at any stage of a multi-stage construction of a logical block. FIG. 21 – Quantum Computing System

[0240] Figure 21 shows an example of a quantum computing device 2100. The quantum computing device 2100 is similar in some respects to the quantum computing system 201 shown in Figure 2A, with additional detail provided on some modular internal components, in accordance with some example embodiments. At a high level, the quantum computing device 2100 shows two types of fusion measurement devices, (1) resource state projection devices (RSPs) 2106 (which may include one or more instances of a qubit fusion system) that generate resource state entanglements, and (2) fusion network projection devices (FNPs) 2108 (which may include one or more instances of a qubit fusion system). In some example embodiments, the fusion network adaptivity described above (e.g., in reference to FIG. 16A to FIG. 20) is performed by the fusion network switches and fusion measurement chips in the FNP(s) 2108. In some example embodiments, resource state generation can implement FCA-like adaptivity in the RSPs 2106, which is discussed in further detail below with reference to FIG. 22A to FIG. 24.

[0241] In the illustrated example of FIG. 21, an arrow with an intersecting diagonal segment and a nearby letter corresponds to one or more modes (e.g., fibers / waveguides for photonic qubits; shuttling paths for trap based qubits, etc.). In some example embodiments, the mode count can correspond to a number of physical pathways (e.g., fibers, integrated waveguides; or an amount of qubits being shuttled, and so on), and the mode count can have the value of the latter. In the following, photons are discussed for brevity, but it is appreciated that theAttorney Docket No.7246-02501 modularization can likewise be implemented with other qubit types, as discussed in FIGs. 26A-26C.

[0242] Continuing, input light 2102 (e.g., photons) can be input into 64 channels where each channel can be implemented as an integrated waveguide or fiber (e.g., 64 fibers). Further, in FIG.21, dotted-line arrows correspond to data lines (e.g., classical data, electrical data cables, optical fibers transmitting classically modulated light (e.g., pulse width amplification (PAM), quadrature amplitude modulation (QAM)). The data lines can be implemented to send and receive data (e.g., “D1”, “D2”, “D3”, “F”, “R”, and “O”) to different devices. For example, the controller 2112 (e.g., controller 206 (FIG.2B), a central processing unit (CPU), a microcontroller, a graphical processing unit (GPU), a field programmable gate array (FPGA), electrical logic circuitry) can receive readout data (e.g., D1, D2, and D3) that can be stored in memory of the controller 2112. Further, the controller 2112 can generate instructions to be performed by photonic and electrical components in the quantum computing device 2100 (e.g., instructions F, and R), and can further generate outcome data (O), which may be transferred to devices that are external to the quantum computing device 2100 (e.g., classical computing system 142, decoder 146, FIG.1K above).

[0243] In some example embodiments, the quantum computing device 2100 includes a non-cryogenic structure 2101 (e.g., service racks, towers) having a plurality of non-cryogenic photonic and electric systems. For example, the non- cryogenic structure 2101 can include a seed state generator (SSG) set 2104 having SSG units that generate seed state entanglements (e.g., Bell Pairs, 3- GHZs), a resource state projection device (RSP) set 2106 having RSP units that generate resource state entanglements, and a fusion network projection device (FNP) set 2108 having FNP units that perform projective measurements on the generated resource states (e.g., to perform task based fusion computation).

[0244] As further illustrated in FIG. 21, the quantum computing device 2100 comprises a cryogenic enclosure 2110 (e.g., a cryogenic structure, such as cryostat or cryo-cabinet) to manage cryogenic components. The cryogenic enclosure 2110 can be cooled to cryogenic temperatures (e.g., below 120 Kelvin (K), 2K, 4K, 77K) using circulation of cryogenic coolants, such as liquid nitrogenAttorney Docket No.7246-02501 and / or liquid helium. In some example embodiments, the connections between the components of the non-cryogenic structure 2101 and the cryogenic enclosure 2110 are implemented using a plurality of optical cables and electrical cables to couple items in and out of the cryogenic environment. For example, a plurality of optical fibers and electrical cables can connect the components of the non- cryogenic structure 2101 to the components in the cryogenic enclosure by passing through feed through (e.g., vacuum feed throughs, ports) of the cryogenic enclosure 2110.

[0245] In some example embodiments, the cryogenic enclosure 2110 can include a sealed environment under moderate or high vacuum in which cryogenic coolants (e.g., liquid helium, liquid nitrogen) are circulated to cool chips that function at cryogenic temperatures, such as detectors (DET) 2103, detectors 2105, and detectors 2107 (e.g., single photon detectors, photon number resolving detectors).

[0246] In some example embodiments, a plurality of photons (e.g, single photons, photonic states) is provided as the input light 2102 into the SSG set 2104. Each SSG unit comprises photonic circuits to generate seed state entanglements (e.g., Bell pairs, GHZ (Greenberger–Horne–Zeilinger) states)). Each SSG unit comprises a plurality of output channels, including “B” channels that are coupled to a plurality of detectors 2103. In some example embodiments, each SSG (e.g., an entanglement device) is coupled to other entanglement devices using entanglement device optical interconnects (e.g., fibers), such as “C” channels that are coupled to RSP units of the RSP set 2106, and “D” channels that are coupled to FNP units of the FNP set 2108. In some example embodiments, the plurality of seed state generators run at a seed state clock cycle (e.g., a seed state frequency, such as 0.5 to 2 GHz), where in a given cycle, each SSG receives its own set of photonic states (e.g., it receives its own set of qubits; three photons, six photons. etc.) for generation of seed entangled states.

[0247] The outputs on the B channels are measured by the detectors 2103 to output detection data D1 that is transmitted to the controller 2112. In some example embodiments, the controller 2112 determines which sets of B channels have success patterns and transmits resource state configuration settings “R” toAttorney Docket No.7246-02501 the RSP units of the RSP set 2106 (e.g., informing the RSP units which channels comprise seed states and settings to perform projective measurements). In some example embodiments, while the RSP set 2106 receives the resource state configuration settings and configures the RSP components (e.g., phase shifters), the light that is on the C channels is delayed (e.g., via optical delays, lengths of fiber).

[0248] The RSP set 2106 then receives the light on the C channels and performs interference on the light, where the interfered light (e.g., interfered qubits) is output on the “E” channels to the detectors 2105. The detectors 2105 generate detection data D2 (resource entangled state readout data) that is transmitted to the controller 2112. In some example embodiments, the controller 2112 determines from the detection data D2 which of the RSP units successfully performed projective measurements, thereby indicating which of the D channels comprise resource state entanglements (e.g., 4GHZ states). For instance, light on the C and D channels may be entangled, and a RSP unit can perform a two- particle projective measurement on qubits on the C channel such that their corresponding entangled states on the D channel are in a resource state, as discussed in further detail below. In some example embodiments, the RSP units have a clock rate that is longer than the seed state cycles (e.g., RSPs can operate at 1 to 2 MHz, whereas the seed state system 2115 (and pump light system) may run at approximately 200 MHz or above).

[0249] In some example embodiments, after determining which of the RSP units of the RSP set 2106 successfully performed their respective projective measurements, the controller 2112 generates fusion network configuration settings “F”, which is transmitted to the FNP units of the FNP system 2108. The fusion network configuration settings data “F’ can include data indicating which channels of D have resource states on them based on successful projection measurements being performed by the RSP units. As further discussed below, the settings data for the FNP system 2108 can include optical switch setting data to implement interferences for various measurements (e.g., XX, ZZ and the like; single particle measurements). Additionally, the settings data for FNP system 2108 can include shuttling paths data (e.g., for trap-based qubits), or similarAttorney Docket No.7246-02501 settings data for other types of qubits (e.g., cavity connectivity routing data for a superconducting circuit-based qubit). In some example embodiments, the light on the D channels is delayed (e.g., via an optical delay, length of fiber) while the FNP system 2108 receives the data “F” and configures the FNP units (e.g., configures phase shifters). In some example embodiments, the FNP units have a clock rate that is longer than the SSG and RSG units (e.g., lower than 1 MHz). In some example embodiments, in non-photonic qubit-based approaches, the qubits are managed so that they maintain state and do not decohere until they are selected for fusion measurements (e.g., no delay may be applied to some non- photonic qubit systems).

[0250] In some example embodiments, the FNP set 2108 then receives the light (e.g., qubit channels) on the D channels and performs interference of the light. In some example embodiments, the interfered light (e.g., interfered qubits) from the FNP system 2108 is then output on the “F” channels to the detectors 2107. The detectors 2107 generate detector data D3, which is received by the controller 2112 and transmitted as outcome data “O” to one or more external destinations.

[0251] In some example embodiments, fusion network adaptivity applied to the network fusion level (e.g., at the computation level), as discussed in FIGs.16- 20 above, is implemented entirely within the FNPs 2108 (e.g., based on data from the FNPs and data from the RSPs). In some example embodiments, fusion network adaptivity-like operations can be implemented at the RSG level, within RSPs 2106. For example, in accordance with some example embodiments, resource states are generated in stages by entangling different resource tree portions to one another within the RSP set 2106. In some example embodiments, the fusion network adaptivity-like operations for the resource states are implemented using scoring approaches such as quality metrics to select different entangled states for projective measurements by the RSPs 2106. FIGs. 22A-22B – Resource State Generation using Entanglement Recycling

[0252] In some example embodiments, FCA-like approaches can be implemented in the process of resource state generation, where resourceAttorney Docket No.7246-02501Attorney Docket No.7246-02501Attorney Docket No.7246-02501

[0259] Alternatively, other quality metrics can be implemented to select states for re-use and re-cycling in target resource state generation, such as evaluating the erasure probability of encoded outcomes when undergoing encoded fusion, which is further discussed in U.S. Pat. No 11,308,416 and U.S. Pat. No. 12,360,869, which are hereby incorporated by reference in their entirety.

[0260] In the following, to elucidate in detail one of the quality metrics that may be implemented for recycling-based resource state generation, the average erasure heuristic (AEH)-based approach is discussed and it is appreciated that the re-use / re-cycling of states can likewise be implemented using any of the above approaches (e.g., AEH, ranking by size and selecting largest entanglements for further use, recycling based on moment of inertia).

[0261] Continuing, an some example embodiments, a quality metric of the encoded qubit (or tree portion) is determined using the following average erasure heuristic: Assuming that every qubit in the encoded qubit has a loss probability of , the probabilities are computed that (a) the logical XX outcome is erased if all qubits undergo a single qubit X measurement and (b) the logical ZZ outcome is erased if all qubits undergo single qubit Z measurement.

[0262] In some example embodiments, the above probability is determined as follows: Assume that each of n physical qubits in an encoded qubit undergoes loss with probability l. The probability of a specific configuration of k erased single qubit measurements out of n isFor each erasure configuration, the logical operator is erased if and only if the rank of the parity check matrix constructed from the code stabilizers is less than the rank of the stacked matrix that consists of the code stabilizers stacked on top of the logical operator. In this way, the total probability that a logical X / Z outcome is erased may be determined.

[0263] In some example embodiments, the status of the RSG staging protocol at any stage may be shown as a snapshot of ZX diagrams (e.g., FIG. 22A at an earlier stage, and FIG. 22B at a later stage). In some example embodiments, at any stage of the protocol, a “cluster” is defined as a set ofAttorney Docket No.7246-02501 nodes / qubits that are entangled / connected by a set of edges. Clusters have “qubit” nodes that have qubits attached to them that have not been measured yet. For the qubit nodes of each cluster, the AEH is computed under the assumption that the qubit in the qubit node is connecting the rest of the state to the kernel / fundamental resource state and the rest of the qubits are the encoded qubit (note that the number of qubit nodes is smaller than the number of unmeasured qubits, which reduces the number of computations).

[0264] In some example embodiments, the qubit node with the lowest AEH is labeled as the center node. In some example embodiments, the value of the AEH at the center node is the erasure cost of the cluster. In some example embodiments, the clusters are sorted (e.g., ranked) according to their erasure cost (e.g., generated via scoring approach B) and the centers of f clusters with the lowest erasure costs are sent to a fusion, the centers of the next f lowest erasure costs are sent to the next fusion, and so on. In someexample embodiments, where there are c clusters, [ / ] fusions are thenperformed.

[0265] In some example embodiments, each fusion site is assigned as a black spider (e.g., a black node, or sometimes called a “red spider” in ZX notation). In some example embodiments, a black spider suppresses X erasure more than any of its inputs and does the opposite for Z erasure. To adaptively maintain a balance between X and Z, a Hadamard is added to an input whenever the X erasure for a cluster is smaller than the Z erasure, which reverses the colors (e.g., white to black, or vice versa) of the tree (or tree portion) up to Hadamards on the leaves (which can be corrected in subsequent operations).

[0266] In some example embodiments, starting with seed states, this procedure is repeated (e.g., looped) until there are fewer than clusters remaining. In some example embodiments, fusions measurements in RSPs are only performed between clusters which restrict the state to tree form. In some embodiments, the desired or target fundamental resource state is a GHZ of size, . In the final step, all nodes of degree are analyzed, and the nodes are split to obtain the lowest average erasure rate (e.g., based on theAttorney Docket No.7246-02501 heuristic) for the encoded qubits adjacent to a degree node, which is the average erasure rate for the state. In some embodiments, the state with the best average erasure rate is then selected.

[0267] FIGs. 22A and 22B show an example RSG stages in ZX notation in accordance with some example embodiments. At a high level, FIG. 22A shows a snapshot of entanglements for a set of physical devices managing qubits at a first time, and FIGs. 22B shows a snapshot at a subsequent time (subsequent resource state cycle). In the illustrated example of FIGs. 22A- 22B, the seed state size is 3 (s=3), fusion size (e.g., count of inputs into afusion) is 4 ( =4), and the target GHZ size is 4 ( =4).

[0268] In the example of FIG.22A, there are two types of nodes of the depicted ZX notation (e.g., white nodes (Z) and black nodes (X)). Further, in FIG. 22A and 22B, there are two types of white nodes (Z spiders): blank white nodes 2205, and white nodes with a cross, such as crossed white node 2210. Further, there are two types of black nodes (X spiders), blank black node 2215, and black node with a cross, such as crossed black node 2220. The crossed nodes (e.g., node 2210 and 2220) are qubit nodes and have additional qubits attached to them (stubs attached to the nodes) that are not depicted in the figures to reduce clutter, and the blank nodes (e.g., node 2205, node 2220) do not have further qubits (e.g., further trees) attached to them.

[0269] The example of Figure 22A shows the state 2200 of the RSG formation in all the RSP units (e.g., RSP set 2106) at a random intermediate step (e.g., random snapshot, an initial or first resource state cycle). As illustrated, clusters of various sizes have been constructed from previous rounds of fusion measurements (e.g., Bell Pairs, n-GHZ states, W states, non- GHZ states, line states, tree states). The clusters are of different sizes due to the fusion measurements being probabilistic and thus may result in either a fusion (success) or a single qubit measurement (failure). The cluster containing the qubits 2210, 2215 and 2220 is the largest cluster, and may likely be determined to have a lower AEH than the other clusters (e.g., the smaller cluster containing qubit 2205). Note that lower AEH values are desirable, as they reflect a lower probability of a logical erasure. In someAttorney Docket No.7246-02501perform an encoded fusion between two different codes, e.g., a measurement of logical XX and ZZ on two encoded qubits. Embodiments herein may fuse sets of two qubits or perform measurements on single qubits in an adaptive order.Attorney Docket No.7246-02501 Example scenarios depicted in FIG.23 provide a approaches for determining which operation should be performed at any stage based on previous outcomes.

[0273] FIG. 23 illustrates encoded fusion measurements between two

[0274] Initially, the stabilizers on both encoded qubits may be known. Furthermore, methods may be utilized for robustly measuring the X / Z logical qubit by performing fixed single qubit measurements on every physical qubit of both encoded qubits. In the case of any tree code, the logical X / Z operator can be measured robustly by performing single qubit X / Z measurements on every physical qubit.

[0275] During the procedure, the XX / ZZ erasure heuristic may be defined on the set of two encoded qubits as the probability of recovering the logical XX and logical ZZ by performing the single qubit measurements corresponding to logical X / Z measurements on each encoded qubit. The average erasure heuristic may be determined as the average of the XX and ZZ erasure heuristics.

[0276] The XX / ZZ erasure heuristic may be computed as follows: Assume that each of n physical qubits undergoes loss with probability l. The probability of a specific configuration of k erased single qubit measurements out of n total fusions is For each erasure configuration, the logical operator is erased if and only if the rank of the parity check matrix constructed from the code stabilizers is less than the rank of the stacked matrix that consists of the codeAttorney Docket No.7246-02501 stabilizers stacked on top of the logical operator. This allows us to compute the total probability that the logical XX / ZZ outcome is erased.

[0277] In some embodiments, the method (e.g., for scenarios B and C) proceeds as follows.

[0278] First, all possible operations that can be performed may be enumerated. A typical example is to consider, for all pairs of unmeasured qubits, a fusion measuring XX and ZZ, and failing in ZZ; a fusion measuring XX and ZZ, and failing in XX; a fusion measuring XZ and ZX, and failing in XZ; a fusion measuring XZ and ZX, and failing in ZX; and for all qubits, a single qubit X measurement and a single qubit Z measurement.

[0279] Second, for each of the enumerated possible operations, compute (a) the probabilities of different measurement outcomes (e.g., which measurements are obtained and which are erased) in the presence of a loss outcome, (b) the stabilizers after the measurements are performed, and (c) the average erasure heuristic for each of the post measurement states. It is noted that by weighing the quality metric (e.g., average erasure heuristic, node count, entanglement size, etc.) for the different measurement outcomes with their probabilities, the expected erasure heuristic is obtained for every possible operation. The operation may be performed based a ranked value. E.g., in AEH, the operation is performed based on which has lowest average erasure heuristic; in sized based ranking, the largest entanglement is selected. Subsequently, the actual measured outcomes may be observed, and the stabilizers and logical XX and ZZ operators may be updated according to the observed outcomes.

[0280] Third, the first and second steps may be repeated until no qubits are left.

[0281] Fourth, it may be determined whether logical XX / ZZ has been measured. If the measurement group described by the set of measured operators contains the logical XX / ZZ operator, it has been successfully measured, and otherwise it has been erased.

[0282] FIG. 24 shows a flow diagram 2400 for implementing resource stage generating using recycling of states based on quality metrics using a quantum information processing system and classical information processing system, inAttorney Docket No.7246-02501 accordance with some example embodiments. At operation 2405, a plurality of entanglements are generated. For example, at operation 2405, the entanglements depicted in FIG.22A are generated. In some example embodiments, the states of FIG. 22A can include Bell pairs, n-GHZs (3GHZ, 4GHZs), W states, tree encodings, clusters, and so on. In some example embodiments, a quantum information processing system (e.g., FIGs.26A-26C, or other type of entanglement system) generates the entanglements and data describing the entanglements is stored in memory for processing by a classical information processing system (e.g., FIG. 27).

[0283] At operation 2410, quality metrics data describing the entanglements of operation 2405 is generated. For example, at operation 2410, the classical information processing system generates the metrics data by evaluating average erasure heuristics (AEH), entanglement size, or other types of quality metrics (e.g., inertia based, erasure adaptivity based, and so on).

[0284] At operation 2415, additional entanglements are generated based on the quality metrics data. For example, at operation 2415 the quantum information processing system generates the entanglements as depicted in FIG. 22B using the quality metrics data (e.g., to select qubits from different entangled states and jointly measure them).

[0285] In some example embodiments, after operation 2415, the target resource state entanglement is output (e.g., along with data describing which physical devices correspond to qubits of the target resource state). In some example embodiments, the loop is repeated and additional quality metrics data sets are generated, followed by additional entangling measurements to yield new snapshots (e.g., additional updated and reduced versions of FIGs.22A and 22B) until a target entanglement is achieved at operation 2420 (e.g., a 4GHZ is output, with local encodings on it each of its four qubits). In some example embodiments, the generated resource states can then be used in fusion level computation measurements as discussed above (e.g., FIG. 1F, FIG. 1G; computation level measurements implemented a plurality of FNPs 2108 in FIG. 21).Attorney Docket No.7246-02501

[0286] In some example embodiments, what entanglements are in a given snapshot is constrained by the type of physical devices and / or qubits being managed. As an example, what qubits and entangled states are considered in FIG. 22A for the FCA-like recycling analysis is constrained by physical performance characteristics of what is reachable by the quantum information processing system. For example, what photonic qubits and photonic entangled states are considered in a given snapshot of FIG.22A can be constrained by switch bandwidth, optical loss, erasure, or other photonic parameters. While, in a matter qubit based quantum information based system (e.g., ion trap based qubit systems, neutral atom based systems), what qubits and states are considered in FIG. 22A is constrained by physical constraints such as the ability to shuttle qubits from entanglements to other entanglements (e.g., without errantly encountering other non-intended qubits / entanglements or the environment). In some example embodiments, the classical information processing system is agnostic to qubit type and routing instructions, and instead performs the method 2400 based on what is stored or otherwise input to the RSG FCA system (and routing is handling at a lower level of the classical control system). As an example, the method 2400 can implement the method 2400 on all state generations system, in which case the entanglements depicted in FIG.22A would encompass all entanglements generated across the entire quantum information processing system (e.g., FIGs.22A shows a snapshot of all entanglements in the entire quantum information processing system; involving all of the RSPs 2106). Alternatively, in some example embodiments, based on physical device constraints, the method 2400 can be implemented for only a subset of the entangled state generators (e.g., FIG.22A shows a snapshot of entanglements for a subset of physical devices generating entanglements (e.g., a subset of RSPs 2106 depicted in FIG.21; two of the example three RSPs depicted in FIG.21 generate entanglements which are depicted in the snapshot of FIG. 22A). ZX Notation

[0287] The following is a general description of ZX notation with reference to FIGs.25A to 25D. The In the following, different physical items of quantumAttorney Docket No.7246-02501 hardware 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., 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, which can be performed according to a list of tasks, such as a quantum algorithm.

[0288] FIG. 25A shows an example of nodes of a quantum node diagram, in accordance with some example embodiments. In the illustrated example, a scalar 2505 (e.g., a number; zeroth order tensor), a vector 2510, a matrix 2515, and a tensor 2520 are shown (e.g., spiders). At a high level, the scalar node 2500 (shown as a circle with no legs) corresponds a zeroth order tensor, the vector node 2510 (shown as a circle with one leg) corresponds to a list of numbers having one index (e.g., first order tensor), the matrix node 2515 (shown as a circle with 2 legs) corresponds to an array of numbers having with two indices (e.g., a second order tensor), and the tensor 2520 (shown as a circle or dot with more than two legs) corresponds to a set of numbers having N number of indices (e.g., an N-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 N-indexes where 0 N ).

[0289] Generally, in quantum information processing (e.g., quantum computing), and as used here 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— 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 (e.g., theAttorney Docket No.7246-02501 stabilizers of a green / red nodes are products of X / Z on all qubits and all pairwise Z / X operators.).

[0290] FIG. 25B shows an example of a white node 2530 (e.g., 0 ) and ablack node 2535 (e.g., + ) shown as states with legs extending up (e.g., quantumstates, sources, vectors). FIG.25B further shows their counterparts, white node 2540 and black node 2545 arranged as measurements (e.g., projections, tests) having downward facing legs.

[0291] FIG. 25C shows example quantum node diagrams with nodes having legs facing up and down. In FIG.25C, white node 2550 (e.g., Z spider) has three downward legs and two upward legs and corresponds to the following: 00 000 + 11 111 (in bra-ket notation); whereas black node 2555 (e.g., Xspider) also has three downward legs and two upward legs but corresponds to thefollowing: |++ +++| + |-- ---|.

[0292] FIG. 25D shows example node relations, in accordance with some example embodiments. In some example embodiments, two nodes of the same shape can be merged with one another (e.g., spider fusing, contraction) as shown in diagram 2550, read from left to right. Alternatively, a single node can be expanded to two nodes (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 2560 shows an additional illustrative example; in diagram 2560 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 2570, a white node and 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., Backens, arXiv:1602.04744, 2016).Attorney Docket No.7246-02501

[0293] FIGs.26A-26C show example quantum information processing systems (e.g., quantum processing unit 205, FIG. 2A), according to some example embodiments. The quantum information processing systems illustrated in FIGs. 26A-26C can implement different physical embodiments that correspond to quantum computing (e.g., photonic qubits, trap-based qubits (ion trap based, neutral atom based), charge qubits, flux qubits). Each of the example quantum information processing systems can implement different physical components to prepare different quantum states and implement unitary evolution of the qubit systems (e.g., collectively a quantum state) to perform different qubit based operations.

[0294] FIG. 26A shows a photonic quantum computing system 2600, in accordance with some example embodiments. In the example illustrated in FIG. 26A, qubits are implemented as photonic qubits 2612 which can be based on degrees of freedom of photonic systems, such as photon location (e.g., which way systems), photon polarization, and so on. The photonic quantum computing system 2600 further includes a photonic controller 2602 to prepare and control the photonic qubits 2612. For example, the photonic controller 2602 can include a photonic state system 2604 to prepare the photonic qubits (e.g., weak coherent laser, single photon sources, quantum dots sources that generate single photons or entangled photonic states (such as caterpillar states)), and further include photonic controls 2607 to perform unitary evolution of a quantum state or other types of processes corresponding to the photonic qubits 2612 (e.g., discarding, post selection, feed forward control). The photonic controls 2607 can include optical components to process the photonic qubits 2612, such as phase shifters, polarizers, and beam splitters to implement photonic qubit operations (e.g., gates, unitary operations). The photonic controls 2607 further comprise an optical measurement system to perform measurements on photons of the photonic qubits 2612. In some example embodiments, the measurements performed by the photonic quantum computing system 2600 can include readout measurements (e.g., outcomes) to generate output data or measurements as gates (e.g., projective measurements, fusions) to further process the photonic qubits 2612.Attorney Docket No.7246-02501

[0295] FIG. 26B shows an trap-based quantum computing system 2620, in accordance with some example embodiments. In the example illustrated in FIG. 26B, qubits are implemented as trap-based qubits 2629 (e.g., ion-trap based qubits, neutral atom-based qubits). As an example, ion-based qubits can be implemented based on degrees of freedom of ionic systems, such as hyperfine states. As a further example, neutral atom-based qubits can be implemented based on levels of hyperfine ground subspace and so on). The trap-based quantum computing system 2620 further includes an trap qubit controller system 2622 to prepare and control the trap-based qubits 2629. For example, the trap controller system 2622 can include an trapped qubit state system 2624 to prepare the trap-based qubits (e.g., for ion based qubits: a quadrupole trap, optical pumping system, cooling system; for neutral atom based qubits: tweezer, optical lattice), and further include trapped qubit controls 2626 to perform unitary evolution of the quantum state corresponding or other processing of the trap-based qubits 2629. The ion controller 2626 can include control components, such as a laser pulse system, to process the trap-based qubits 2629 to impart qubit operations (e.g., gates, unitary operations) according to a given quantum task (e.g., qubit gate, circuit comprising a plurality of gates in an order). The trap-based quantum computing system 2620 further comprises a trapped qubit measurement system 2627 to perform measurements on the trap-based qubits (e.g., measurements devices for trap-based qubits can include a photomultiplier tube, avalanche photodiode or CCD based imaging device to generate readout data or perform measurements for qubit processing).

[0296] FIG. 26C shows a current-based quantum computing system 2630, in accordance with some example embodiments. In the example illustrated in FIG. 26C, qubits are implemented as current-based qubits 2640 (charge-based, flux-based, transmons), which can be implemented based on degrees of freedom of charge or current state (e.g., in superconducting circuits). The current-based quantum computing system 2630 further includes a current controller 2632 to prepare and control the current-based qubits 2640. For example, the current controller 2632 can include a current state system 2634 to prepare the current- based qubits, and further includes current controls 2636 to perform unitaryAttorney Docket No.7246-02501 evolutions of the quantum state corresponding to the current-based qubits 2640. The current controller 2632 can include control components, such as flux couplers, microwave photons, to process the current-based qubits 2640 to impart qubit operations (e.g., gates, unitary operations) according to a given quantum task (e.g., qubit gate, circuit comprising a plurality of gates in an order). The system 2630 further comprises a current measurement system 2637 to perform measurements on the current-based qubits 2640, such a readout cavity (e.g., readout resonator) to generate readout data (outcomes) or perform measurements for qubit processing.

[0297] In some example embodiments, given a target quantum state for preparation any of the systems of FIGs.26A-26C can implement respective qubits via gates (e.g., gates in circuit-based quantum computing) and / or measurements (in measurement-based approaches, such as Measurement Based Quantum Computing (MBQC), and Fusion Based Quantum Computing (FBQC)) to implement the target quantum state.

[0298] FIG. 27 illustrates a diagrammatic representation of a machine 2700 in the form of a computer system (e.g., classical information processing system 201, FIG.2A) within which a set of instructions may be executed for causing the machine 2700 to perform any one or more of the methodologies discussed herein, according to an example embodiment. Specifically, FIG.27 shows a diagrammatic representation of the machine 2700 in the example form of a computer system, within which instructions 2716 (e.g., software, a program, an application, an applet, an app, or other executable code) for causing the machine 2700 to perform any one or more of the methodologies discussed herein may be executed. For example, the instructions 2716 may cause the machine 2700 to execute any one or more operations of FIGs.6 to 9. In this way, these instructions 2716 configure the classical processor and / or the controllers (e.g., controller 206) to implement the several-body hadronic Hamiltonian approaches discussed above.

[0299] In alternative embodiments, the machine 2700 operates as a standalone device or may be coupled (e.g., networked) to other machines. In a networked deployment, the machine 2700 may operate in the capacity of a serverAttorney Docket No.7246-02501 machine or a client machine in a server-client network environment, or as a peer machine in a peer-to-peer (or distributed) network environment. The machine 2700 may comprise, but not be limited to, a server computer, a client computer, a personal computer (PC), a tablet computer, a laptop computer, a netbook, a smart phone, a mobile device, a network router, a network switch, a network bridge, or any machine capable of executing the instructions 2716, sequentially or otherwise, that specify actions to be taken by the machine 2700. Further, while only a single machine 2700 is illustrated, the term “machine” shall also be taken to include a collection of machines 2700 that individually or jointly execute the instructions 2716 to perform any one or more of the methodologies discussed herein.

[0300] The machine 2700 includes processors 2710, memory 2730, and input / output (I / O) components 2750 configured to communicate with each other such as via a bus 2702. In an example embodiment, the processors 2710 (e.g., a central processing unit (CPU), a reduced instruction set computing (RISC) processor, a complex instruction set computing (CISC) processor, a graphics processing unit (GPU), a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a radio-frequency integrated circuit (RFIC), another processor, or any suitable combination thereof) may include, for example, a processor 2712 and a processor 2714 that may execute the instructions 2716. The term “processor” is intended to include multi-core processors 2710 that may comprise two or more independent processors (sometimes referred to as “cores”) that may execute instructions 2716 contemporaneously. Although FIG. 27 shows multiple processors 2710, the machine 2700 may include a single processor with a single core, a single processor with multiple cores (e.g., a multi-core processor), multiple processors with a single core, multiple processors with multiple cores, or any combination thereof.

[0301] The memory 2730 may include a main memory 2732, a static memory 2734, and a storage unit 2736, all accessible to the processors 2710 such as via the bus 2702. The main memory 2732, the static memory 2734, and the storage unit 2736 store the instructions 2716 embodying any one or more of the methodologies or functions described herein. The instructions 2716 may alsoAttorney Docket No.7246-02501 reside, completely or partially, within the main memory 2732, within the static memory 2734, within machine storage medium 2738 of the storage unit 2736, within at least one of the processors 2710 (e.g., within the processor’s cache memory), or any suitable combination thereof, during execution thereof by the machine 2700.

[0302] The I / O components 2750 include components to receive input, provide output, produce output, transmit information, exchange information, capture measurements, and so on. The specific I / O components 2750 that are included in a particular machine 2700 will depend on the type of machine. For example, portable machines such as mobile phones will likely include a touch input device or other such input mechanisms, while a headless server machine will likely not include such a touch input device. It will be appreciated that the I / O components 2750 may include many other components that are not shown in FIG. 27. The I / O components 2750 are grouped according to functionality merely for simplifying the following discussion and the grouping is in no way limiting. In various example embodiments, the I / O components 2750 may include output components 2752 and input components 2754. The output components 2752 may include visual components (e.g., a display such as a plasma display panel (PDP), a light emitting diode (LED) display, a liquid crystal display (LCD), a projector, or a cathode ray tube (CRT)), acoustic components (e.g., speakers), other signal generators, and so forth. The input components 2754 may include alphanumeric input components (e.g., a keyboard, a touch screen configured to receive alphanumeric input, a photo-optical keyboard, or other alphanumeric input components), point-based input components (e.g., a mouse, a touchpad, a trackball, a joystick, a motion sensor, or another pointing instrument), tactile input components (e.g., a physical button, a touch screen that provides location and / or force of touches or touch gestures, or other tactile input components), audio input components (e.g., a microphone), and the like.

[0303] Communication may be implemented using a wide variety of technologies. The I / O components 2750 may include communication components 2764 operable to couple the machine 2700 to a network 2780 or devices 2770 via a coupling 2782 and a coupling 2772, respectively. For example, theAttorney Docket No.7246-02501 communication components 2764 may include a network interface component or another suitable device to interface with the network 2780. In further examples, the communication components 2764 may include wired communication components, wireless communication components, cellular communication components, and other communication components to provide communication via other modalities. The devices 2770 may be another machine or any of a wide variety of peripheral devices (e.g., a peripheral device coupled via a universal serial bus (USB)).

[0304] The various memories (e.g., 2730, 2732, 2734, and / or memory of the processor(s) 2710 and / or the storage unit 2736) may store one or more sets of instructions 2716 and data structures (e.g., software) embodying or utilized by any one or more of the methodologies or functions described herein. These instructions 2716, when executed by the processor(s) 2710, cause various operations to implement the disclosed embodiments.

[0305] As used herein, the terms “machine-storage medium,” “device- storage medium,” and “computer-storage medium” mean the same thing and may be used interchangeably in this disclosure. The terms refer to a single or multiple tangible storage devices and / or tangible media (e.g., a centralized or distributed database, and / or associated caches and servers) that store executable instructions and / or data. The terms shall accordingly be taken to include, but not be limited to, solid-state memories, and optical and magnetic media, including memory internal or external to processors. Specific examples of machine-storage media, computer-storage media, and / or device-storage media include non-volatile memory, including by way of example semiconductor memory devices, e.g., erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), field-programmable gate arrays (FPGAs), and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROM and DVD-ROM disks. The terms “machine-storage media,” “computer-storage media,” and “device- storage media” specifically exclude carrier waves, modulated data signals, and other such media, at least some of which are covered under the term “signal medium” discussed below.Attorney Docket No.7246-02501

[0306] In various example embodiments, one or more portions of the network 2780 may be an ad hoc network, an intranet, an extranet, a virtual private network (VPN), a local-area network (LAN), a wireless LAN (WLAN), a wide-area network (WAN), a wireless WAN (WWAN), a metropolitan-area network (MAN), the Internet, a portion of the Internet, a portion of the public switched telephone network (PSTN), a plain old telephone service (POTS) network, a cellular telephone network, a wireless network, a Wi-Fi® network, another type of network, or a combination of two or more such networks. For example, the network 2780 or a portion of the network 2780 may include a wireless or cellular network, and the coupling 2782 may be a Code Division Multiple Access (CDMA) connection, a Global System for Mobile communications (GSM) connection, or another type of cellular or wireless coupling. In this example, the coupling 2782 may implement any of a variety of types of data transfer technology, such as Single Carrier Radio Transmission Technology (1xRTT), Evolution-Data Optimized (EVDO) technology, General Packet Radio Service (GPRS) technology, Enhanced Data rates for GSM Evolution (EDGE) technology, third Generation Partnership Project (3GPP) including 3G, fourth generation wireless (4G) networks, Universal Mobile Telecommunications System (UMTS), High-Speed Packet Access (HSPA), Worldwide Interoperability for Microwave Access (WiMAX), Long Term Evolution (LTE) standard, others defined by various standard-setting organizations, other long-range protocols, or other data transfer technology.

[0307] The instructions 2716 may be transmitted or received over the network 2780 using a transmission medium via a network interface device (e.g., a network interface component included in the communication components 2764) and utilizing any one of a number of well-known transfer protocols (e.g., hypertext transfer protocol (HTTP)). Similarly, the instructions 2716 may be transmitted or received using a transmission medium via the coupling 2772 (e.g., a peer-to-peer coupling) to the devices 2770. The terms “transmission medium” and “signal medium” mean the same thing and may be used interchangeably in this disclosure. The terms “transmission medium” and “signal medium” shall be taken to include any intangible medium that is capable of storing, encoding, orAttorney Docket No.7246-02501 carrying the instructions 2716 for execution by the machine 2700, and include digital or analog communications signals or other intangible media to facilitate communication of such software. Hence, the terms “transmission medium” and “signal medium” shall be taken to include any form of modulated data signal, carrier wave, and so forth. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal.

[0308] In the foregoing detailed description, the method and apparatus of the present inventive subject matter have been described with reference to specific exemplary embodiments thereof. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the present inventive subject matter. The present specification and figures are accordingly to be regarded as illustrative rather than restrictive.

[0309] The following are example embodiments:

[0310] Example 1: A method may include: performing, on a quantum information processing system, multi-qubit measurements on a first plurality of entangled states to form a second plurality of entangled states; determining, on a classical information processing system, quality metrics for entangled states in the second plurality of entangled states; and performing, based on the quality metrics, additional multi-qubit measurements on the second plurality of entangled states to form a resource entangled state, the resource entangled state being a first type of entangled state, the second plurality of entangled states may include a second type of entangled state that is different from the first type of entangled state, the performing the additional multi-qubit measurements based on the quality metrics increasing a likelihood of generation of the resource entangled state.

[0311] Example 2: The method as example 1 describes, where the first type of entangled states corresponds to a first cluster shape and where the second type of entangled state corresponds to a different cluster shape that is different than the first cluster shape.Attorney Docket No.7246-02501

[0312] Example 3: The method as either of examples 1 or 2 describe, where the first type of entangled state and second type of entangled state correspond to different types of ZX diagrams.

[0313] Example 4: The method as any of examples 1-3 describe, where the first type of entangled state may include a GHZ state and the second type of entangled state may include a non-GHZ state.

[0314] Example 5: The method as any of examples 1-4 describe, where the quantum information processing system may include a plurality of entanglement devices configured to probabalistically form entangled states, where the first type of entangled state corresponds to a target entangled state and where the second type of entangled state corresponds to entanglements that are not the target state.

[0315] Example 6: The method as any of examples 1-5 describe, where the quality metric may include ranking the second plurality of entangled states based on as average erasure scheme.

[0316] Example 7: The method as any of examples 1-6 describe, where the quality metric may include ranking the second plurality of entangled states by size from largest to smallest.

[0317] Example 8: The method as any of examples 1-7 describe, where the multi-qubit measurements may include joint projective measurement on a plurality of qubits.

[0318] Example 9: The method as any of examples 1-8 describe, where the plurality of qubits may include a first qubit from a first entangled state and a second qubit from a second entangled state, and where the joint projective measurement may include destructively measuring the first qubit and second qubit.

[0319] Example 10: The method as any of examples 1-9 describe, where the multi-qubit measurement may include a Type-II fusion measurement.

[0320] Example 11: The method as any of examples 1-10 describe, where the quantum information processing system may include one of: a photonic-based quantum information processing system, an ion-trap based quantum informationAttorney Docket No.7246-02501 processing system, a super-conducting circuit-based quantum information processing system.

[0321] Example 12: The method as any of examples 1-11 describe, where the classical information processing system may include one or more of: a central processing unit (CPU), a graphics processing unit (GPU), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA).

[0322] Example 13: A system may include: one or more processors; a non- transitory computer readable memory, the non-transitory computer readable memory storing instructions that when executed by the one or more processors perform operations may include: identify first multi-qubit measurement data, the first-multi-qubit measurement data generated from a quantum information processing system by performing a plurality of multi-qubit measurements on a first plurality of entangled states to form a second plurality of entangled states; determine quality metrics for entangled states in the second plurality of entangled states; and store the quality metrics.

[0323] Example 14: The system as example 13 describes, where the quantum information processing system performs additional multi-qubit measurements on the second plurality of entangled states based on the quality metrics to form a resource entangled state, the resource entangled state being a first type of entangled state, the second plurality of entangled states may include a second type of entangled state that is different from the first type of entangled state, the performing the additional multi-qubit measurements based on the quality metrics increasing a likelihood of generation of the resource entangled state.

[0324] Example 15: The system as either of examples 13 or 14 describe, where the first type of entangled states corresponds to a first cluster shape and where the second type of entangled state corresponds to a different cluster shape that is different than the first cluster shape.

[0325] Example 16: The system as any of examples 13-15 describe, where the quality metric may include ranking the second plurality of entangled states based on as average erasure scheme.Attorney Docket No.7246-02501

[0326] Example 17: The system as any of examples 13-16 describe, where the quality metric may include ranking the second plurality of entangled states by size from largest to smallest.

[0327] Example 18: The system as any of examples 13-17 describe, where the multi-qubit measurements may include joint projective measurement on a plurality of qubits.

[0328] Example 19: The system as any of examples 13-18 describe, where the quantum information processing system may include one of: a photonic based quantum information processing system, a ion-trap based quantum information processing system, a super-conducting circuit-based quantum information processing system.

[0329] Example 20: The system as any of examples 13-19 describe, where the classical information processing system may include one or more of: a central processing unit (CPU), a graphics processing unit (GPU), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA).

[0330] Example 21: A method, may include: receiving a first resource state and a second resource state; identify a first plurality of qubits of the first resource state that is to be fused to a second plurality of qubits in the second resource to obtain an encoded fusion outcome for a logical block; for each of a plurality of potential fusion measurements for the first and second pluralities of qubits: determine an average erasure heuristic for each of a plurality of outcomes of the potential fusion measurement; and determine a weighted average erasure heuristic based on the average erasure heuristics, where the weighted average erasure heuristic is weighted based on probabilities of the plurality of outcomes; and by a fusion controller, performing a first fusion measurement of the potential fusion measurements with a smallest weighted average erasure heuristic to produce the resource state.

[0331] Example 22: The method as example 21 describes, further may include: by the fusion controller, performing one or more second fusion measurements of the plurality of potential fusion measurements until all qubits in the first and second pluralities of qubits have been measured.Attorney Docket No.7246-02501

[0332] Example 23: A non-transitory computer-readable memory medium storing program instructions which, when executed by a classical processor, direct the method steps of any as either of examples 21 or 22.

[0333] Example 24: A controller, including: a non-transitory computer- readable memory medium; a switching circuit coupled to a logical qubit generator; a fusion controller; a classical processor coupled to the memory medium, where the classical processor is configured to execute program instructions to direct the methods of any of examples 21-23.

[0334] Example 25: A method may include: identifying, in memory, a resource state generator (RSG) protocol for a target RSG state to be generated by a plurality of projective measurement devices over a plurality of stages, where each stage may include joint measurements on multiple qubits to form entanglements, where the target RSG state may include qubits in a local encoding based on metric data from previous stages; generating, in a first resource state cycle, a first entanglement state may include first local encodings in a first arrangement based on first metric data of the first resource state cycle; generating, in a second resource state cycle, a second state may include second local encodings in a second arrangement based on second metric data of the second resource state cycle; performing, in a third resource state cycle, a joint projective measurement on qubits from the first entangled state and the second entangled state to form the target RSG state.

[0335] Example 26: The method as example 25 describes, where the first arrangement and second arrangement may include different tree encodings.

[0336] Example 27: The method as either of examples 25 or 26 describe, where metric data indicates a likelihood of erasure for a local encoding.

[0337] Example 28: The method as any of examples 25-27 describe, where metric data indicates a quantity of qubits in a local encoding.

[0338] Example 29: The method as any of examples 25-28 describe, where the second resource state cycle is subsequent in time to the first resource state cycle.Attorney Docket No.7246-02501

[0339] Example 30: The method as any of examples 25-29 describe, where qubits from the first entangled state are delayed for joint measurement with qubits from the second entangled state.

[0340] Example 31: The method of claim as any of examples 25-30 describe, where delaying qubits may include delaying photonic qubits using one or more of: an photonic integrated circuit based delay, an optical fiber based delay.

[0341] Example 32: The method as any of examples 25-31 describe, where delaying qubits may include maintaining qubits from the first entangled state until the qubits from the first entangled state can be joint measured with qubits of the second entangled state.

[0342] Example 33: The method as any of examples 25-32 describe, where maintaining the qubits may include maintaining one or more states of the qubits such that the qubits do not decohere.

[0343] Example 34: A method, may include: by a fusion controller, performing a plurality of first multi-qubit fusion measurements on qubits from different seed states of a plurality of seed states to produce a plurality of clusters; by a classical processor, determining a quality metric for each cluster of the plurality of clusters; and by the fusion controller, performing a second multi- qubit fusion measurement on qubits from two or more different clusters of the plurality of clusters to produce a second-stage cluster, where the qubits from the different clusters are selected based on their respective quality metrics.

[0344] Example 35: The method as example 34 describes, where the quality metric may include an average erasure heuristic, and where the average erasure heuristic may include an average of a first probability that a logical XX outcome is erased when all qubits of a cluster undergo single qubit X measurements and a second probability that a logical ZZ outcome is erased when all qubits of the cluster undergo single qubit Z measurements.

[0345] Example 36: The method as either of examples 34 or 35 describe, further may include: for each cluster of the plurality of clusters: determining a respective potential average erasure heuristic when each node of the cluster is identified as a center node for the cluster, where the average erasure heuristicsAttorney Docket No.7246-02501 are determined to be the lowest potential average erasure heuristics for each cluster.

[0346] Example 37: The method as any of examples 34-36 describe, further may include: sorting the plurality of clusters into a plurality of bins based on their average erasure heuristics; where the second multi-qubit fusion measurement is performed on qubits within clusters of a first bin with a lowest average erasure heuristic.

[0347] Example 38: The method as any of examples 34-37 describe, where quality metric may include entanglement size, and where larger sized entanglements are selected to produce the second-stage cluster.

[0348] Example 39: The method as any of examples 34-38 describe, where performing the second multi-qubit fusion measurement on qubits from different clusters of the plurality of clusters may include performing a multi-qubit fusion measurement on a respective qubit from the center node each of the two or more different clusters.

[0349] Example 40: The method as any of examples 34-39 describe, where the plurality of seed states may include 6-ring seed states.

[0350] 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.

[0351] 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 may be modified or omitted and that other elements not shown or described may be added.

[0352] This disclosure provides a description of the claimed invention with reference to specific embodiments. Those skilled in the art with access to this disclosure will appreciate that the embodiments are not exhaustive of the scopeAttorney Docket No.7246-02501 of the claimed invention, which extends to all variations, modifications, and equivalents.

[0353] The terminology used in the description of the various described embodiments herein is for the purpose of describing particular embodiments only and is not intended to be limiting. As used in the description of the various described embodiments and the appended claims, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will also be understood that the term “and / or” as used herein refers to and encompasses any and all possible combinations of one or more of the associated listed items. It will be further understood that the terms “includes,” “including,” “comprises,” and / or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0354] It will also be understood that, although the terms first, second, etc., are, in some instances, used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first switch could be termed a second switch, and, similarly, a second switch could be termed a first switch, without departing from the scope of the various described embodiments. The first switch and the second switch are both switches, but they are not the same switch unless explicitly stated as such.

[0355] As used herein, the term “if” is, optionally, construed to mean “when” or “upon” or “in response to determining” or “in response to detecting” or “in accordance with a determination that,” depending on the context.

[0356] The foregoing description, for purpose of explanation, has been described with reference to specific embodiments. However, the illustrative discussions above are not intended to be exhaustive or to limit the scope of the claims to the precise forms disclosed. Many modifications and variations are possible in view of the above teachings. The embodiments were chosen in order to best explain the principles underlying the claims and their practicalAttorney Docket No.7246-02501 applications, to thereby enable others skilled in the art to best use the embodiments with various modifications as are suited to the particular uses contemplated.

Claims

Attorney Docket No.7246-02501 Claims What is claimed is:

1. A method comprising: performing, on a quantum information processing system, multi-qubit measurements on a first plurality of entangled states to form a second plurality of entangled states; determining, on a classical information processing system, quality metrics for entangled states in the second plurality of entangled states; and performing, based on the quality metrics, additional multi-qubit measurements on the second plurality of entangled states to form a resource entangled state, the resource entangled state being a first type of entangled state, the second plurality of entangled states comprising a second type of entangled state that is different from the first type of entangled state, wherein performing the additional multi-qubit measurements based on the quality metrics increases a likelihood of generation of the resource entangled state.

2. The method of claim 1, wherein the first type of entangled states corresponds to a first cluster shape and wherein the second type of entangled state corresponds to a second cluster shape that is different from the first cluster shape.

3. The method of claim 1, wherein the first type of entangled state and second type of entangled state correspond to different types of ZX diagrams.

4. The method of claim 1, wherein the first type of entangled state comprises a GHZ state and the second type of entangled state comprises a non-GHZ state.

5. The method of claim 1, wherein the quantum information processing system comprises a plurality of entanglement devices configured to probabalistically form entangled states, wherein the first type of entangled stateAttorney Docket No.7246-02501 corresponds to a target entangled state and wherein the second type of entangled state corresponds to entanglements that are not the target entangled state.

6. The method of claim 1, wherein the quality metric comprises ranking the second plurality of entangled states based on an average erasure scheme.

7. The method of claim 1, wherein the quality metric comprises ranking the second plurality of entangled states by size from largest to smallest.

8. The method of claim 1, wherein the multi-qubit measurements comprise joint projective measurements on a plurality of qubits.

9. The method of claim 8, wherein the plurality of qubits comprises a first qubit from a first entangled state and a second qubit from a second entangled state, and wherein the joint projective measurement comprises destructively measuring the first qubit and second qubit.

10. The method of claim 1, wherein the multi-qubit measurement comprises a Type-II fusion measurement.

11. The method of claim 1, wherein the quantum information processing system comprises one of: a photonic based quantum information processing system, a ion-trap based quantum information processing system, or a super- conducting circuit based quantum information processing system.

12. The method of claim 1, wherein the classical information processing system comprises one or more of: a central processing unit (CPU), a graphics processing unit (GPU), an application specific integrated circuit (ASIC), and a field programmable gate array (FPGA).

13. A system comprising: one or more processors;Attorney Docket No.7246-02501 a non-transitory computer-readable memory, the non-transitory computer- readable memory storing instructions that when executed by the one or more processors cause the system to: identify first multi-qubit measurement data, the first-multi-qubit measurement data generated from a quantum information processing system by performing a plurality of multi-qubit measurements on a first plurality of entangled states to form a second plurality of entangled states; determine quality metrics for entangled states in the second plurality of entangled states; and store the quality metrics.

14. The system of claim 13, wherein the quantum information processing system performs additional multi-qubit measurements on the second plurality of entangled states based on the quality metrics to form a resource entangled state, the resource entangled state being a first type of entangled state, the second plurality of entangled states comprising a second type of entangled state that is different from the first type of entangled state, and wherein performing the additional multi-qubit measurements based on the quality metrics increases a likelihood of generation of the resource entangled state.

15. The system of claim 13, wherein the first type of entangled states corresponds to a first cluster shape and wherein the second type of entangled state corresponds to a second cluster shape that is different from the first cluster shape.

16. The system of claim 13, wherein the quality metric comprises ranking the second plurality of entangled states based on an average erasure scheme.

17. The system of claim 13, wherein the quality metric comprises ranking the second plurality of entangled states by size from largest to smallest.Attorney Docket No.7246-02501 18. The system of claim 13, wherein the multi-qubit measurements comprise joint projective measurements on a plurality of qubits.

19. The system of claim 13, wherein the quantum information processing system comprises one of: a photonic based quantum information processing system, a ion-trap based quantum information processing system, or a super- conducting circuit based quantum information processing system.

20. The system of claim 13, wherein the classical information processing system comprises one or more of: a central processing unit (CPU), a graphics processing unit (GPU), an application specific integrated circuit (ASIC), and a field programmable gate array (FPGA).

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