Frozen gap scoring

Abstract

Systems, devices and methods for performing determining a quality metric of a logical qubit or a logical block. A syndrome graph is received for a brick of a target fusion network or another fault-tolerant protocol that identifies syndromes on vertices of the syndrome graph. At least one syndrome is removed from the syndrome graph to obtain a modified syndrome graph. A quality metric is determined based on the modified syndrome graph by determining a logical gap magnitude of the modified syndrome graph without the at least one removed syndrome. The quality metric is utilized to determine routing for the brick.

Description

7246-02301 PsiQ-603WO1 Frozen Gap Scoring Technical Field

[0001] Embodiments herein relate generally to quantum computational methods, systems and 5 devices, such as photonic devices (or hybrid electronic / photonic devices), semiconducting or superconducting quantum computing devices, or topological quantum computers for preparing fault-tolerant logical qubits in a quantum computer. Background 10

[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 conventionalbra / ket notation) or in a superposition of the two states (e.g., |0 |1 . By operating ona system (or ensemble) of qubits, a quantum computer may quickly perform certain 15 categories of computations that would require impractical amounts of time in a classical computer.

[0003] Because quantum computing utilizes quantum states as computational units, quantum computing systems are typically very sensitive to environmental noise, degradation and decoherence. Accordingly, there is a robust field of research into developing effective and20 efficient fault tolerance and error correction into quantum computing systems. In a fault- tolerant quantum computing scheme, multiple physical qubits may be entangled together to represent a single logical qubit, to make the logical qubit less susceptible to error. This process is time and resource intensive, and improvements in the field of fault-tolerant quantum computing are desired to increase the efficiency and fault tolerance of logical qubit 25 preparation. Summary

[0004] Some embodiments described herein include quantum computing devices, systems and methods for removing syndromes from syndrome graph data prior to determining a 30 quality metric for a brick. 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. - 1 -7246-02301 PsiQ-603WO1

[0005] In some embodiments, a plurality of copies is received for each brick of a plurality of bricks of a target fusion network. In some embodiments, a classical processor determines a quality metric for each copy of each brick.

[0006] In some embodiments, determining each quality metric includes receiving syndrome 5 graph data for the copy that identifies syndromes on vertices of a syndrome graph. In some embodiments, at least one syndrome is removed from the syndrome graph data to obtain modified syndrome graph data. The syndrome(s) may be selected to be removed based on their proximity to a port pseudo-vertex that has been added to the syndrome graph. In some embodiments, the quality metric is determined based on determining a logical gap magnitude 10 of the modified syndrome graph data without the removed syndrome(s).

[0007] In some embodiments, the quality metric is utilized to determine routing for the copies of the bricks. In some embodiments, for each brick of the plurality of bricks, a first copy is selected to fuse into an aggregate brick based on the quality metrics. In some embodiments, a fusion controller fuses the first copies for each brick together to produce the 15 aggregate brick.

[0008] 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, hybrid quantum / classical computing systems, and any of various other quantum computing systems. 20

[0009] This Summary is intended to provide a brief overview of some of the subject matter described in this document. Accordingly, it will be appreciated that the above-described features are merely examples and should not be construed to narrow the scope or spirit of the subject matter described herein in any way. Other features, aspects, and advantages of the subject matter described herein will become apparent from the following Detailed Description, 25 Figures, and Claims. Brief Description of the Drawings

[0010] 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 30 like reference numerals refer to corresponding parts throughout the Figures.

[0011] Figures 1A-J illustrate the utilization of surface codes to constructed an error-corrected fault-tolerant logical qubit, according to some embodiments; - 2 -7246-02301 PsiQ-603WO1

[0012] Figure 1K illustrates an example of a qubit fusion system interfacing with fusion sites, in accordance with some embodiments;

[0013] Figure 1L illustrates a qubit fusion system interfacing with a classical computing system, according to some embodiments; 5

[0014] Figure 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;

[0015] Figure 2B is a diagram of a controller, according to some embodiments;

[0016] Figures 3A-B illustrate a 2D toy model of brick encoding, according to some 10 embodiments;

[0017] Figures 4A-C illustrate a 3D 6-ring network for hierarchical multiplexing in logical block encoding, according to some embodiments;

[0018] Figure 5 is a flowchart illustrating a method for performing hierarchical multiplexing while encoding bricks, according to some embodiments; 15

[0019] Figure 6A illustrates two 6-qubit resource states that undergo a fusion to produce a single 10-qubit entangled resource state, according to some embodiments;

[0020] Figure 6B illustrates integration of two copies of a 10-qubit resource state into a multiplexing circuit, according to some embodiments;

[0021] Figure 6C is a legend that defines circuit elements shown in Figure 6D, according to 20 some embodiments;

[0022] Figure 6D illustrates an interleaving module configured to utilize the fault-tolerant 10- qubit resource state, according to some embodiments;

[0023] Figure 6E illustrates how four interleaving modules may be interconnected and incorporated into a larger quantum circuit, according to some embodiments; 25

[0024] Figure 7A illustrates performing two fusion measurements on two 10-qubit resource states to obtain a 16-qubit resource state, according to some embodiments;

[0025] Figure 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;

[0026] Figure 7C illustrates incorporating a multiplexed 16-qubit resource state into an 30 interleaving module, according to some embodiments;

[0027] Figure 8A illustrates an ordered sequence of fusion measurements, according to some embodiments; - 3 -7246-02301 PsiQ-603WO1

[0028] Figure 8B illustrates two stages of fusion measurements in a fusion network, according to some embodiments;

[0029] Figure 9A is a legend illustrating various circuit components, according to some embodiments; 5

[0030] 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;

[0031] Figure 9C is a circuit diagram illustrating routing of qubits to 1stand 2ndstage fusion measurements, according to some embodiments;

[0032] Figure 10 are circuit diagrams illustrating network router switches that route qubits 10 either locally or to another resource state generator in the fusion network, according to some embodiments;

[0033] Figure 11A is a circuit diagram illustrating a circuit utilizing network switches to create multiple copies of a brick, according to some embodiments;

[0034] Figure 11B is a circuit diagram illustrating multiple interconnected circuits, each with 15 multiple copies of a brick, according to some embodiments;

[0035] Figures 12A-C illustrates the construction of subsequent hierarchical brick layers using a circuit-based quantum computing, according to some embodiments;

[0036] Figures 13A-J illustrate quantum circuits for performing hierarchical multiplexing using circuit-based quantum computing, according to some embodiments; 20

[0037] Figures 14A-D illustrate two alternate corrections of a 2-dimensional syndrome graph, according to some embodiments;

[0038] Figures 15A-D illustrate connected components of primal and dual syndrome graphs in two dimensions in accordance with some embodiments;

[0039] Figure 16 illustrates gap-boundary and port pseudo-vertices for an identity logical 25 block, according to some embodiments;

[0040] Figures 17A-E illustrate an example method for freezing out syndromes that are matched to port pseudo-vertices in a syndrome graph, according to some embodiments;

[0041] Figure 18 is a flowchart that illustrates a method for determining a quality metric for a brick, according to some embodiments; 30

[0042] Figure 19 illustrates a fusion network divided into four 4x4 sub-bricks, according to some embodiments;

[0043] Figure 20 illustrates two different identifications of gap-boundary and port pseudo- vertices for a syndrome graph, according to some embodiments; - 4 -7246-02301 PsiQ-603WO1

[0044] Figure 21 illustrates gap-boundary and port pseudo-vertices for 3D Kagome-6 a syndrome graph, according to some embodiments;

[0045] Figure 22A-D illustrate the identification of error clusters in a Tanner graph, according to some embodiments; 5

[0046] Figure 23 illustrates a method for using union find decoding to freeze out syndromes, according to some embodiments;

[0047] Figure 24A illustrates a logical mask on a syndrome graph with periodic boundary conditions, according to some embodiments;

[0048] Figure 24B illustrates conversion of a syndrome graph with a logical mask to a modified 10 syndrome graph with pseudo-check nodes and hyperedges, according to some embodiments;

[0049] Figure 25A illustrates a Tanner graph with a logical mask, according to some embodiments; and

[0050] Figure 25B illustrates conversion of a Tanner graph with a logical mask to a Tanner graph with an additional pseudo-check node, according to some embodiments. 15

[0051] 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 20 within the spirit and scope of the subject matter as defined by the appended claims. DETAILED DESCRIPTION

[0052] Disclosed herein are examples (also referred to as “embodiments”) of systems and methods for performing fault-tolerant post-selection using various quantum computing 25 systems.

[0053] 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 30 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. - 5 -7246-02301 PsiQ-603WO1 Overview of Quantum Computing

[0054] 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 5 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” 10 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 15 as a qudit.

[0055] 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 20 superconducting Josephson junction); topological qubits (e.g., Majorana fermions); or spin qubits formed from vacancy centers (e.g., nitrogen vacancies in diamond).

[0056] 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) 25 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. Figures 1A-L – Surface Codes and Physical implementations

[0057] Qubits (and operations on qubits) may be implemented using a variety of physical systems. In some embodiments, qubits are provided in an integrated photonic system 30 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 - 6 -7246-02301 PsiQ-603WO1 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, 5 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. 10

[0058] 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) may correspond to the |0 state of a photonic qubit.Alternatively, a state with a photon in the second waveguide and no photon in the first15 waveguide may correspond to the |1 state of the photonic qubit. To prepare a photonic qubitin a known 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 20 having qubits whose logical states map to 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 25 be but need not be arranged in parallel.

[0059] 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 mode30 is unoccupied with probability 1 (e.g., a state |10 in Fock notation), mode coupling mayresult in a state in which both modes have a nonzero probability of being occupied, e.g., a state |10 |01 , where | | | | 1. In some embodiments, operations of thiskind may be implemented by using beam splitters to couple modes together and variable - 7 -7246-02301 PsiQ-603WO1 phase shifters to apply phase shifts to one or more modes. The amplitudes a1 and a2 depend on the reflectivity (or transmissivity) of the beam splitters and on any phase shifts that are introduced.

[0060] A single physical qubit (e.g., such as the 2-level physical qubit illustrated in Figure 51A with a quantum state | |0 |1 ) may be used for quantum computation inprinciple. 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 10 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, such as is shown in Figure 1B.

[0061] Figure 1B illustrates one example for constructing a fault-tolerant logical qubit using a circuit-based approach. In the illustrated example, the light shaded circles are data qubits 15 (e.g., qubits 125-131) that encode quantum information. The data qubits are entangled with adjacent measure qubits, illustrated as dark shaded circles (such as measure qubit 123). The measure qubits may be measured to determine aspects of the quantum information encoded in the data qubits. The example illustrated in Figure 1B has a code length of 12. Fusion- based approaches to encoding fault-tolerant logical qubits may also be used for embodiments 20 described herein. 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, which may likewise be corrected and / or tolerated.

[0062] Embodiments herein address these and other issues by implementing hierarchical 25 multiplexing in fault-tolerant codes and channels to improve the accuracy and fidelity of encoded logical qubits. 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 30 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 - 8 -7246-02301 PsiQ-603WO1 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 5 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. 10

[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 15 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 data 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 20 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. 25

[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 30 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 - 9 -7246-02301 PsiQ-603WO1 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 hierarchical multiplexing by determining an error metric based on the syndrome graph data, and utilizing 5 the error metric to determine how to utilize / route a plurality of multiplexed copies of a logical block within a quantum circuit. For example, multiplexing may be employed whereby multiple copies of each logical qubit are produced and the higher fidelity logical qubits are kept and used in a quantum computation, whereas the lower fidelity logical qubits are discarded, increasing the fidelity of the computation. As described in greater detail below, 10 exemplary embodiments employ hierarchical multiplexing, where multiplexing is iterated for bricks at multiple hierarchical scales.

[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 15 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 resource states such as those illustrated in Figure 1I 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 20 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 25 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 30 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 - 10 -7246-02301 PsiQ-603WO1 states (e.g., the 10 qubit 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. 5

[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 10 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 quantum 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. 15 Viewed yet another way, this process operates as a fault-tolerant logical channel.

[0071] Figure 1C 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 20 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 1C, 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 25 dual plaquettes are disposed on that boundary. Figure 1D represents the same concept but written in a more familiar quantum circuit notation illustrating the analogy between the more familiar quantum circuit. While Figure 1C 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 30 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. - 11 -7246-02301 PsiQ-603WO1

[0072] The protocol for preparing an encoded logical state may contain two parameters, and . Here is referred to as the “distance” of the scheme, which corresponds to the length and width of the cross section shown in Figure 1C – it determines the code distance of the surface code state being prepared. In some embodiments, L may be separated into two 5 parameters, Lx and 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 10 number of layers of resource states in FBQC. 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. 15

[0073] The sequence of measurements performed over the flow of time illustrated in Figure 1C (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 20 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 fusion 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 1E illustrates an example 2D 25 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 30 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. - 12 -7246-02301 PsiQ-603WO1

[0074] 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 5 on the syndrome values of the primal and dual graphs.

[0075] 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 10 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 1C). This error metric may then be used for 15 hierarchical multiplexing, to determine how and / or where to route each of a plurality of multiplexed copies of a brick, in some embodiments.

[0076] In some embodiments, hierarchial multiplexing 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 20 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 25 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.

[0077] Figures 1F and 1G 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 30 the rectangular sheet 120 in the top half of Figure 1G 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 1G, the vertical direction represents the depth of the logical qubit (i.e., time), which is a sequence of nine entangling - 13 -7246-02301 PsiQ-603WO1 measurements performed on the 9x9 grid of physical qubits representing each of the qubits q1-q4as well as a portion of the auxiliary qubits 121.

[0078] 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 5 of the surface code state being prepared. In some embodiments, L may be separated into two parameters, Lx and 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 10 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. 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 15 errors on the output state.

[0079] Figure 1H 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 20 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.

[0080] 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 25 “sub-bricks.” Figure 1H shows multiple sub-bricks stitched together into a single brick. As illustrated in Figure 1H, 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 1H delineates (potentially together with corresponding boxes in subsequent sheets) a respective sub-brick. For example, as can be 30 seen by comparison of Figure 1H with Figures 1F and 1G, the sub-brick 126 encodes a portion of the logical qubit q2,the sub-brick 124 encodes portions of both q1and q2, the sub- brick 128 encodes a portion of q1and 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 - 14 -7246-02301 PsiQ-603WO1 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 1H, which performs the Z2Z3two-qubit measurement. In performing hierarchical multiplexing on sub-bricks, multiple copies of each of the sub-bricks may be first created 5 separately (i.e., not yet stitched together), hierarchical multiplexing may 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.

[0081] Figure 1I 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 10 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 15 operations, by a graph G using a graph state representation. The graph state is defined as the quantum state | obtained by putting qubits in the | state at each vertex and performing acontrolled-Z gate between qubits for which 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 20 some embodiments, a resource state such as is shown in Figure 1I may be used to perform hierarchical multiplexing according to the circuit diagrams shown in Figures 6-11.

[0082] Figure 1J 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 1H, as one example. 25

[0083] 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 may be generally applied to various types of logical blocks, logical qubits, and / or components thereof, in various embodiments. 30

[0084] Figure 1K 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 computing system 201 shown in Figure 2A. - 15 -7246-02301 PsiQ-603WO1

[0085] 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 5 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 10 measurement outcomes in the form of classical data are output and then provided to a decoder system.

[0086] Figure 1L 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 15 may be an element of fusion array 138 (shown in Figure 1K), 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.

[0087] The qubit fusion system 1500 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 20 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 25 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 fusion 30 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 - 16 -7246-02301 PsiQ-603WO1 used to determine a quality metric to be used for brick selection in hierarchical multiplexing, in some embodiments.

[0088] Figure 1L 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 5 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 10 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 15 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 1534 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 20 1L, 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 25 otherwise controlling the operation of the detectors. Each of the detectors 1503, 1505, 1507, 1509 provides one bit of information (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. 30 Figures 2A-B – Quantum and Classical Computing Systems

[0089] Figure 2A is a system diagram of a quantum computing system 201 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 - 17 -7246-02301 PsiQ-603WO1 (QPU) 205 over a classical channel 212. The classical channel may relay classical information between the classical computing system and the QPU.

[0090] 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 5 (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(s) may additionally or alternatively perform operations based on information and / or instructions received from the QPU 205 over the channel 212. 10 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). 15

[0091] In some embodiments, the QPU 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 20 resources with other control aspects of the classical computing system.

[0092] 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 25 of the classical computer. The memory 204 is classical in the sense that it stores data and / or program instructions in a non-transitory 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 30 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 - 18 -7246-02301 PsiQ-603WO1 processor 202 may store that output back into the memory 204 and / or provide the output to the QPU over the channel 212.

[0093] The QPU 205 may include a plurality of qubits 210 and a controller 206 configured to interface with the plurality of qubits. In some embodiments, the qubits are divided into one or 5 more independent qubit modules, where each qubit module includes a self-contained plurality of fault-tolerant qubits, and different qubit modules may be interchangeably used for various steps within a quantum computation. The controller 206 may include physical hardware to interact with and / or perform operations on the qubits, e.g., to apply quantum gates or perform other operations. In some embodiments, the controller further includes a classical processor, 10 potentially coupled to its own dedicated non-transitory (classical) memory, that is configured to direct the physical hardware to interact with the qubits and communicate with the processor of the classical computing system 203 over the channel 212. Alternatively, the classical processor of the classical computing system 203 may directly communicate with the hardware of the controller to provide instructions for interacting with and manipulating the 15 qubits. 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 20 quantum controller 206 during a quantum computation, and they may additionally include classical results of the quantum computation. In some embodiments, the QPU further includes one or more decoders configured to receive and decode the classical measurement results, and the decoded measurement results may be provided to the classical computing system for processing. The measurement results may be communicated to the classical 25 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 QPU while performing a quantum computation.

[0094] 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 30 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 or brick. Encoding the - 19 -7246-02301 PsiQ-603WO1 logical qubit or brick will also produce syndrome graph data as classical information, which is output to the classical computing system 203 via the classical channel 212. The classical computing system may analyze the syndrome graph data to determine an error metric for the logical qubit or brick. Depending on the error metric, the classical computing system outputs 5 instructions back to the QPU 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 (or brick) in the quantum computation.

[0095] 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 10 more processors or central processing units (CPUs) 232, and one or more input / output ports to communicate with other elements of the QPU 205 and / or the classical computing system 201. Hierarchical Multiplexing for Logical Block Encoding

[0096] As described according to embodiments herein, hierarchical multiplexing performs 15 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. 20

[0097] 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.

[0098] As used herein, “multiplexing” refers to the construction of redundant copies of a 25 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. 30

[0099] 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 - 20 -7246-02301 PsiQ-603WO1 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, 5 and may be configured to construct a fusion network.

[0100] 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 FBQC, e.g., in 10 circuit-based quantum computation.

[0101] 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 as shown in a toy model in Figure 3A. For each resource state in the 15 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.

[0102] 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 20 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).

[0103] Embodiments herein may employ a “dicing scheme”, by which is meant a method for 25 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 30 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.

[0104] 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 - 21 -7246-02301 PsiQ-603WO1 be determined, where each brick is assigned 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 quality metric of a brick may be determined based on a logical gap. In some embodiments, one or 5 more syndromes may be frozen out of the logical gap calculation, based on their proximity to port vertices in the syndrome graph, as described in greater detail below.

[0105] 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 state represented as a node and containing four qubit 10 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 15 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.

[0106] 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 20 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 25 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.

[0107] Hierarchical multiplexing may be utilized to reduce both erasures and Pauli errors in 30 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 - 22 -7246-02301 PsiQ-603WO1 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 without syndromes, Pauli errors may be 5 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 10 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 15 ~12% and an error threshold of ~1% while the 2D limit has an erasure threshold of ~50% and an error threshold of ~11%.

[0108] 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 20 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.

[0109] In some embodiments, these competing considerations may be quantitatively investigated to optimize the dicing scheme and the allocation of multiplexing overhead at each stage. 25 Figure 5 – Flowchart for Hierarchical Multiplexing of Logical Blocks

[0110] 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 30 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 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 - 23 -7246-02301 PsiQ-603WO1 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, 5 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. 10

[0111] 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 15 other types of fault tolerant quantum networks.

[0112] 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 Kagome 6 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 20 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 25 stage 1.

[0113] 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 1I), as prescribed by the error-correcting code and desired 30 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 - 24 -7246-02301 PsiQ-603WO1 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 1H), the 5 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 1H).

[0114] 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 10 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 15 high-quality output bricks.

[0115] 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.

[0116] At 504, a respective first quality metric is determined for each layer one copy of each 20 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 the 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 25 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. 30

[0117] 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 - 25 -7246-02301 PsiQ-603WO1 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 1E. 5

[0118] 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 Nsis the number of syndromes in the syndrome graph data, Neis the number of erasures in the syndrome graph data, and and are tunable parameters. In some embodiments, the first quality metric is a 10 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 Ns and erasures Ne in 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 15 from the measurement system), or based on any other observable information communicated from the hardware modules that may be used to assess the reliability of measurement outcomes from the measurement system.

[0119] 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 20 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 monotonic25 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). When the logical gap magnitude is large, there is a larger difference in the number of correction operators used for the two distinct correction classes than there is when the logical gap magnitude is small. Accordingly, it is relatively more likely that the correction class with the smaller weight is the proper 30 correction to fix the underlying erasure errors, so that this error can be corrected with a higher statistical likelihood than if the logical gap magnitude is small. A larger logical gap magnitude may therefore be associated with a higher quality metric. - 26 -7246-02301 PsiQ-603WO1

[0120] In some embodiments, the logical gap is calculated using a frozen gap methodology, where a subset of the syndromes are “frozen out”, or removed, from the syndrome graph data based on their proximity to port vertices in the syndrome graph, prior to calculating the logical gap. The frozen gap methodology is described in greater detail below in reference to 5 Figure 16-23. The method steps described in reference to Figure 18 may be used in conjunction with the methods described in reference to Figure 5, as a particular methodology for determining a quality metric.

[0121] 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 10 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.

[0122] In some embodiments, aggregate quality metrics are determined based at least in part 15 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 20 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 25 and / or syndromes happen to line up in an adverse way), but may have a low likelihood 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 30 with a particular set of copies of other bricks in the target fusion network.

[0123] 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 - 27 -7246-02301 PsiQ-603WO1 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. 5

[0124] 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.

[0125] In some embodiments, the layer one copies for each brick are rank ordered based on 10 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 15 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.)

[0126] 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 20 instructions over a classical channel to the quantum computing system to discard these copies. In some embodiments, these copies may be rerouted in the quantum circuit, and kept for use in some other aspect of the quantum computation.

[0127] 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 25 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 30 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 - 28 -7246-02301 PsiQ-603WO1 plurality of sub-bricks; and the first layer two copies are fused together to produce the respective layer one copy.

[0128] 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 5 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- 10 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. 15

[0129] Using second quality metrics as inputs to the first quality metrics may enable more efficient and faster computation. In some embodiments, an 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 20 data structure that is conveniently and quickly updated and accessed. Additional Technical Detail

[0130] The following numbered paragraphs provide additional technical detail and description regarding embodiments herein. Figures 6-7 – Implementing Hierarchical Multiplexing with Interleaving Circuits 25

[0131] 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.

[0132] 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 30 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 - 29 -7246-02301 PsiQ-603WO1 measurement result), which may be output to a classical processor for determining a quality metric.

[0133] 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 5 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 10 two input qubits to route to the output, resulting in a fault-tolerant 10 qubit resource state 612. To preserve entanglement, the 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 15 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. 20

[0134] 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 25 duration of x clock cycles.

[0135] 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 1G. 30

[0136] 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. - 30 -7246-02301 PsiQ-603WO1

[0137] 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 5 bricks 406.

[0138] 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. Classical measurement results 705 are produced during the fusion measurements, which may be output to a controller to determine a quality metric and 10 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.

[0139] 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 15 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 20 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.

[0140] Figure 7C illustrates how the multiplexed 16-qubit resource state produced in Figure 25 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 1G. Figures 8-11 – Raster Scanning to Implement Fusion Network

[0141] 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 30 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 - 31 -7246-02301 PsiQ-603WO1 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 create 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 5 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.

[0142] Figure 8A illustrates an ordered sequence of fusion measurements, according to some 10 embodiments. The letters (x,y,z) indicate 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. In each fusion measurement, one of the qubits is labelled, and the other is labelled “ ”. Figure 8B illustrates two stages of fusion measurements in a fusion network, 15 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.

[0143] Figure 9A is a legend illustrating various circuit components used in subsequent 20 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” 25 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 the three 30 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 - 32 -7246-02301 PsiQ-603WO1 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 5 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. 10

[0144] 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 15 two input qubits, which his routed to a fusion circuit.

[0145] Figure 9C is a circuit diagram illustrating routing of qubits to 1stand 2ndstage fusionmeasurements, 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. 20

[0146] 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.

[0147] Figure 11A is a circuit diagram illustrating a circuit utilizing network switches to 25 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 30 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 - 33 -7246-02301 PsiQ-603WO1

[0148] 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 5 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.

[0149] Figure 12A illustrates a brick illustrated as a grid of entangled CBQC qubits that are 10 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. 15

[0150] 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 white 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 20 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 25 blocks may be utilized as either state preparations or projections.

[0151] 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 30 codes, which are illustrated in Figure 13E.

[0152] 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 - 34 -7246-02301 PsiQ-603WO1 "corresponding pairs" of qubits. Not all pairs are shown, and the measurements may take place on all qubits of all surface codes.

[0153] Figure 13G illustrates a quantum circuit that may be used to implement these measurements. The operation includes performing measurements of {XXXX, ZZII, IZZI, 5 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.

[0154] 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 10 Figure 13H. Each of the shaded nodes corresponds to either a 4GHZ preparation or projection. To turn it into a circuit, the 3D network is extrapolated into a linear one (see Figures 13I-J), with long-range connections.

[0155] 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 15 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.

[0156] Figure 13J illustrates one example of the connections between the upper and lower 20 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

[0157] The following paragraphs define methods for determining quality metrics based on a 25 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.

[0158] For the 6-ring fusion network, there are two distinct syndrome graphs termed the 30 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 - 35 -7246-02301 PsiQ-603WO1 independent logical error classes that generate all possible logical correlations from input to output. The set of distinct logical sectors is denoted herein by .

[0159] The logical gap rule , uses a metric that utilizes the fact that, below anerror correction threshold, logical errors may be suppressed due to distinguishability between 5 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 10 sectors.

[0160] 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 a configuration and possible corrections , such that composing the correction and error yields a logicaloperator on the code space—namely and respectively. The signed logical gap is defined 15 as

[0162] where denotes the log-likelihood weight of the correction for the sector given by a choice of decoder, defined as follows: an edge has weightwhereis the (marginal) probability of Pauli error on that edge, edges supporting erasures have 20 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 logical gap) may be known in this circumstance, which is denoted herein as | |(below we will drop the dependence on for brevity). 25

[0163] 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. If 0, the decoder will fail and a logical error will be introduced. If 0, the decoder will succeed in correcting the error and if 0, the 30 decoder will succeed / fail half of the time. Therefore, the EER for the brick becomesis the distribution of logical gaps of logical error classes for a fixed brick size and error rate. In more complex logical blocks (i.e., surface code - 36 -7246-02301 PsiQ-603WO1 protocols / channels), there may be many logical error classes and so one may compute a vector of logical gaps as the information of interest.

[0165] where recall, is the set of distinct logical error classes. A combined score may be 5 created for the brick to be thresholded by the policy as

[0166] ,(3)

[0167] ; (4)

[0168] where represent tunable linear weights to weight the addition of the scores for all logical error classes. 10

[0169] 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 15 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 20 multi-qubit Pauli operators (in the syndrome graph, this corresponds to adding additional edges with appropriate weight contributions).

[0170] 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 25 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 the value of the logical qubit, and the decoder may determine the most likely correction satisfying this 30 constraint.

[0171] Said another way, if we denote C as the correction and E as the true error that occurred (which the decoder does not know a priori), without the constraint (i.e., for the first correction) the decoder will determine the correction that has the highest probability of giving - 37 -7246-02301 PsiQ-603WO1 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 5 other words, the correction plus the error should flip the logical sector.

[0172] 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 10 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 15 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.

[0173] In some embodiments, the overall weight of each correction may be determined as a 20 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 respective corrected edges, as shown in the expression ln ,where is the (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 25 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 30 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 - 38 -7246-02301 PsiQ-603WO1 edges may have different error probabilities and may vary between different edges in the logical block.

[0174] 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 5 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 10 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. 15

[0175] 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 weight contributions based on a graph distance from the first physical 20 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 1C 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. 25

[0176] 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. 30

[0177] 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.

[0178] 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 - 39 -7246-02301 PsiQ-603WO1 Pauli errors), it may not be known a priori which correction will correct the error. Accordingly, calculating an effective quality metric is complicated by the fact that a syndrome does not definitively indicate a unique underlying error configuration. However, corrections with larger weights are generally less likely to be correct. All else being equal, the 5 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. This is because, in at least some cases, the likelihood of an error profile decreases exponentially with the number edge flips involved.

[0179] Accordingly, a large magnitude of the logical gap (i.e., a large magnitude in the 10 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 15 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 to 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 20 the decoder, where syndrome graph data with a larger magnitude logical gap are identified as corresponding to higher fidelity logical qubits.

[0180] 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 25 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 30 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 Graphs - 40 -7246-02301 PsiQ-603WO1

[0181] 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 5 the present disclosure.

[0182] 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. 10

[0183] FIG.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, 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 15 measurements can be implemented as shown above in FIGS.1E-1F or 1G. 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 FIG.14B in the context of the surface code example of FIGS.1E-1F is that any “syndromes” present in the syndrome graph data, are the 20 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 25 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. 30

[0184] Figure 14C illustrates an example correction that a decoder would produce when provided the syndrome graph data of FIG.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 - 41 -7246-02301 PsiQ-603WO1 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 5 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”. Note 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 10 of this correction is 8, because it identifies 8 Pauli operators on the underlying physical data qubits (shown by the 8 edges that are traversed by a 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 15 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 graphto the right edge of the syndrome graph (or from the top to the bottom).

[0185] 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 20 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 of the logical25 gap of |9 8| 1.

[0186] 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 30 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 - 42 -7246-02301 PsiQ-603WO1 14A-C). In addition, the syndrome graph data may include a second related matrix that includes erasure errors, where, e.g., in the case of the surface code shown in in FIGS.1E-1F, each entry of the matrix is mapped to an edge in the syndrome graph. Figures 15A-D - Connected Components – 2D Example 5

[0187] 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. 10 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, 4D, or other higher dimensional syndrome graph.

[0188] Figure 15A illustrates an overlay of an example 2D primal syndrome graph (dotted lines) and a 2D dual syndrome graph (dashed lines). Solid black circles connected to dotted 15 lines are syndrome values of the primal graph, whereas solid white 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 indicated 20 by double-sided arrows 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.

[0189] 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 25 measurement.

[0190] 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 30 dotted and dashed lines for the primal and dual edges, respectively.

[0191] 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 - 43 -7246-02301 PsiQ-603WO1 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 5 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.

[0192] 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 10 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.

[0193] In the example shown in Figure 15C, the two ends of the primal edge involved in the 15 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. It may therefore be desirable to select the basis for this fusion measurement to risk erasure of the primal edge in the event of a failure outcome, in some embodiments.

[0194] Figure 15D illustrates an alternative scenario, similar to Figure 15C but with a different 20 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 dotted and dashed lines for the primal and dual edges, respectively. The two edges involved in the next fusion measurement are indicated with the double-headed arrow. 25

[0195] 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.

[0196] 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 30 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 - 44 -7246-02301 PsiQ-603WO1 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. 5 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 (for the 3 unmeasured edges adjacent to this connected component), whereas the right connected component adjacent to the indicated edge of the primal graph in Figure 15D has an exposure of 6 (for the 6 unmeasured edges adjacent to this connected component), leading to an overall exposure of the indicated 10 edge of the primal graph of 3*6=18.

[0197] 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, 15 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. In this example, it may be desirable to select the basis for this fusion measurement to risk erasure of the dual edge in the event of a failure outcome, since it has a smaller exposure, in at least some embodiments. 20

[0198] 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 metric used to determine which copy of the two sub-bricks to select to fuse together may be determined based on the overall exposure for 25 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 30 each set of two or more 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 - 45 -7246-02301 PsiQ-603WO1 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. Removing Syndromes when Determining the Logical Gap

[0199] In some embodiments, one or more syndromes may be removed from the syndrome 5 graph of a brick prior to determining a logical gap for the syndrome graph. In some cases, a brick may be partially measured, where some of the edges of its fusion graph have been measured, and some of edges have not. Often, edges within the bulk of a brick will have been measured, while edges along the perimeter of the brick will have not yet been measured (e.g., when a brick at a particular layer is constructed, perimeter edges of the brick have yet to be 10 fused with another brick to form a higher layer aggregate brick). Incomplete checks are not useable to determine syndrome information, as the requisite measurements have not yet been performed. Accordingly, errors that occur on edges that are part of both complete checks and incomplete checks give less syndrome information than if all the checks were complete. As such, when a syndrome occurs in a fusion graph with incomplete measurements, the observed 15 syndrome may be indicative of an underlying error profile that neighbors both complete and incomplete checks. When a syndrome occurs close to an unmeasured edge (i.e., when it neighbors an incomplete check), it may be more likely than not that the syndrome is indicative of an underlying error profile that involves the unmeasured edge, since that error profile will involve fewer edge flips (e.g., the number of flips may be proportional to the 20 graph distance between the syndrome and the unmeasured edge). Conversely, a syndrome in the bulk of the brick and relatively far away from the unmeasured edges may be more likely to be indicative of an error profile that involves only complete checks in the bulk. If a syndrome is caused by an error profile that is adjacent to an unmeasured edge and is considered when determining a quality metric of the syndrome graph, it may adversely affect 25 the accuracy of the quality metric. Embodiments herein address these and other concerns by removing, or “freezing out” one or more syndromes based on their proximity to unmeasured edges of a syndrome graph.

[0200] Figure 16 illustrates a logical block encoding the identity gate at 1602. Figure 16 further illustrates at 1604 a 2D slice of a syndrome graph of the logical block. The logical 30 block has port faces on the left and right sides of the block 1602 (which correspond to the left and right sides of the syndrome graph 1604), and primal and dual boundaries on the front and back sides of the block 1602 (which correspond to the bottom and top sides of the syndrome graph 1604, respectively). The primal and dual boundaries are realized by certain modified - 46 -7246-02301 PsiQ-603WO1 measurement patterns, leading to a modified check structure in their vicinity. The ports consist of sets of un-measured qubits, which can be considered as inputs and outputs in the time domain of a logical block forming part of a computation. A chain of errors spanning between two opposite primal boundaries (or two opposite dual boundaries) of the illustrated 5 logical block will result in a logical error. In the illustrated syndrome graph 1604, a first set of opposing pseudo-vertices are introduced at the top and bottom, and the measurements on each boundary face that are only involved in a single check are connected to these pseudo- vertices. These added vertices are called “pseudo-vertices” because they are not actually a part of the syndrome graph (e.g., they are not derived from classical measurement results of 10 fusion measurements), but rather they are introduced as a computational tool to assist in computing the quality metric. A second set of pseudo-vertices is introduced on the left and right sides of the syndrome graph 1604, and the unmeasured edges along the left and right port faces are connected to these pseudo-vertices.

[0201] Figures 17A-E illustrate an example method for removing a subset of the syndromes 15 from a syndrome graph when calculating a logical gap magnitude. Figure 17A illustrates an underlying error configuration of a syndrome graph (which is not typically known, a priori). The nodes of the graph indicate parity checks, solid lines indicate outcomes of fusion measurements, gray lines indicate unmeasured outcomes, and gray highlights superimposed over a line indicate the presence of a Pauli error. 20

[0202] Figure 17B illustrates an augmented syndrome graph that would result from the underlying error configuration shown in Figure 17A. Syndromes are indicated by larger gray circles, and are present when a parity check is connected to an odd number of Pauli errors. The syndrome graph is augmented with gap-boundary and port pseudo-vertices, which are connected to the unmeasured edges along the perimeter of the syndrome graph. 25

[0203] Figure 17C illustrates matching of syndromes on the augmented syndrome graph. As illustrated, each syndrome is matched to either a pseudo-vertex or another syndrome. To determine the matching, a decoding algorithm such as minimum weight perfect matching or another decoding algorithm may be utilized, which finds a pairing of syndromes with combined minimal weight. Often this may lead to syndromes in close proximity to a gap- 30 boundary or port matching with the associated pseudo-vertex. For example, the syndrome 1702 is close to and matched to the left pseudo-vertex 1704, which is a port vertex. The syndrome 1706 is closest to and matched to the top pseudo-vertex 1708, which is a gap- boundary vertex. The syndrome 1710 is closest to and matched to a neighboring syndrome - 47 -7246-02301 PsiQ-603WO1 1712. Figure 17D illustrates how the two syndromes 1702 and 1714, which are matched to port pseudo-vertices, are removed from the syndrome graph. The logical gap may then be calculated, as shown in Figure 17E. The left image shows a first correction class, which has a weight of 3 due to the three flipped edges in the correction. The right image shown a second 5 correction class, which has a weight of 7 due to the seven flipped edges in the correction. The magnitude of the logical gap is therefore |3 7| 4. Note that, according to the initialunderlying error configuration, the left correction will not result in a logical error, while the right correction will. Figure 18 - Flowchart for Frozen Gap Scoring 10

[0204] Figure 18 is a flowchart that illustrates a method for determining a quality metric for a brick, which may be a portion of a logical qubit or logical block, according to some embodiments. The method shown in Figure 18 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 18 may be performed by a quantum computing device or system 15 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 20 instructions stored on a non-transitory computer-readable memory medium. In some embodiments the methods described in Figure 18 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 25 within the scope of the embodiments described herein. As illustrated, the method shown in Figure 18 may proceed as follows.

[0205] At 1802, syndrome graph data is received for a brick of a plurality of bricks of a target fusion network. The syndrome graph data identifies at least one syndrome on one or more respective vertices of a syndrome graph. The syndrome graph may include both a primal 30 syndrome graph and a dual syndrome graph, in some embodiments. For simplicity, some of the following method steps are described as being performed simply on a “syndrome graph”, without specifying whether the steps are performed on a primal or dual syndrome graph. In general, the described method steps may be equivalently performed on either the primal or - 48 -7246-02301 PsiQ-603WO1 dual syndrome graphs, and in many cases the method steps are separately performed for both the primal and dual syndrome graphs, and the results (i.e., the quality metrics) are combined to obtain an overall quality metric. Figures 17B shows a syndrome graph for a simple 2D brick. Each syndrome indicates a parity check error for the adjacent fusion measurements. In 5 some embodiments, the brick is a contiguous group of fused resource states with a perimeter of unmeasured edges.

[0206] At 1804, zero, one, or more syndromes of the syndrome graph data are removed from the syndrome graph data to obtain modified syndrome graph data. The syndromes may be selected for removal by a decoder, and the selection of syndromes to remove may be 10 determined based on a determined probability that the syndrome is indicative of an underlying error configuration in the syndrome graph that is connected to an unmeasured edge of the brick.

[0207] In some embodiments, the fusion graph data may be augmented with gap-boundary and port pseudo-vertices. Each unmeasured edge may be connected to either a gap-boundary 15 or a port pseudo-vertex that is added to the fusion graph data. In some embodiments, the syndrome(s) are selected for removal from the syndrome graph data based on their proximity to a port pseudo-vertex of the fusion graph data.

[0208] In some embodiments, the proximity of the removed syndromes from their connected port pseudo-vertices is stored and used to determine the quality metric. For example, a 20 separate quality metric may be determined when a particular syndrome is both removed and not removed. An overall quality metric for the brick may be determined based on a weighted average of these two quality metrics. The average may be weighted based on the graph distance of the removed syndrome from its connected port pseudo-vertex, where the quality metric calculated with the syndrome removed receives a weight that decreases with its graph 25 distance from the port pseudo-vertex. Advantageously, this weighted average procedure provides that the contribution of the quality metric with the frozen syndrome to the overall quality metric is proportional to the likelihood that the frozen syndrome is indicative of an underlying error profile that is connected to an unmeasured edge.

[0209] At 1808, a quality metric is determined from the modified syndrome graph data. In 30 some embodiments, determining the quality metric includes determining a logical gap magnitude of the modified syndrome graph data without the removed syndrome(s).

[0210] In some embodiments, the logical gap magnitude is a magnitude of a difference in a first weight of a first correction of the modified syndrome graph data and a second weight of - 49 -7246-02301 PsiQ-603WO1 a second correction of the modified syndrome graph data. The two corrections may correspond to two different correction classes, as described above in reference to Figures 17A-E. The first and second weights may be log-likelihood weights, in at least some embodiments. 5

[0211] In some embodiments, the logical gap magnitude is associated with a first logical error class of a set of distinct logical error classes of the syndrome graph. The method may further include determining a second logical gap magnitude of the modified syndrome graph data without the removed syndrome(s), wherein the second logical gap magnitude is associated with a second logical error class of the set of distinct logical error classes of the 10 syndrome graph. The quality metric may then be determined further based on the second logical gap magnitude. For example, in some embodiments the quality metric is a summation of a decaying exponential function of the first logical gap magnitude and a decaying exponential function of the second logical gap magnitude. In some embodiments, the first logical error class is a primal graph logical error, and the second logical error class is a dual 15 graph logical error.

[0212] In some embodiments, the method includes adding a plurality of pseudo-vertices to the syndrome graph data, where each pseudo-vertex connects to one or more unmeasured edges in the syndrome graph. A first subset of the plurality of pseudo-vertices may be identified as gap-boundary vertices, and a second subset of the plurality of pseudo-vertices 20 may be identified as port vertices. In some embodiments, the syndrome(s) are selected for removal from the syndrome graph data based on their proximity to port vertices in the identified second subset of the plurality of pseudo-vertices. For example, if the decoder determines that one or more syndromes are matched to a port pseudo-vertex with a higher probability than they are matched to a gap-boundary pseudo-vertex or another syndrome, 25 they may be selected for removal before calculating the logical gap. If none of the syndromes meet this criterion, none of the syndromes may be removed.

[0213] In some embodiments, a decoder may be utilized to match each syndrome in the syndrome graph to either a port pseudo-vertex, a gap-boundary pseudo-vertex, or another syndrome. The decoder may perform this matching based on graph distances between the 30 syndromes and the port pseudo-vertices, gap-boundary pseudo-vertices, and other syndromes, where the syndromes are matched based on the shortest graph distance, in some embodiments. The syndromes that are matched to port pseudo-vertices may be removed in the modified syndrome graph, while the syndromes that are matched to gap-boundary - 50 -7246-02301 PsiQ-603WO1 pseudo-vertices or other syndromes may remain in the modified syndrome graph. In some embodiments, the graph distance between the removed syndrome(s) and the matched port pseudo-vertex may be used to determine a weight for the contribution of the calculated logical gap to the quality metric. In some embodiments, a difference between the graph 5 distance between the removed syndrome(s) and the matched port pseudo-vertex and the graph distance between the removed syndrome(s) and the nearest gap-boundary pseudo-vertex or nearest neighbor syndrome may be used to determine a weight for the contribution of the calculated logical gap to the quality metric.

[0214] Determining the logical gap magnitude may be performed based on a difference 10 between two corrections of an underlying error configuration that spans two gap-boundary pseudo-vertices of the identified first subset of the plurality of pseudo-vertices.

[0215] In some embodiments, determining the quality metric further includes reidentifying the first subset of the plurality of pseudo-vertices as port vertices, and reidentifying at least a portion of the second subset of the plurality of pseudo-vertices as gap-boundary vertices. 15 Zero, one or more second syndromes are selected for removal from the syndrome graph data to obtain second modified syndrome graph data. The zero, one, or more second syndromes are selected for removal based on their proximity to a port vertex in the reidentified first subset of the plurality of pseudo-vertices. A second logical gap magnitude of the second modified syndrome graph data is determined without the zero, one or more removed second 20 syndrome(s). Determining the second logical gap magnitude is performed based at least in part on a difference between two corrections of a logical error that spans two gap-boundary vertices of the reidentified portion of the second subset of the plurality of pseudo-vertices. The quality metric is then determined as a summation of a decaying exponential function of the logical gap magnitude and a decaying exponential function of the second logical gap 25 magnitude. Figure 20 illustrates an example of these embodiments, where for a 2D brick, a contribution to the quality metric is determined from a first logical gap calculated with the pseudo-vertex identification shown at 2004, and a second logical gap is calculated with the pseudo-vertex identification shown at 2006. This process may be repeated for both the primal and dual syndrome graphs, and the quality metric may be determined as a sum of all four 30 decaying exponential functions of the four logical gap magnitudes.

[0216] At 1810, the quality metric is stored in a non-transitory computer-readable memory medium. The quality metric may be used by a controller to decide where to route the brick in a quantum computing method, e.g., for fault-tolerant post-selection, or for other applications. - 51 -7246-02301 PsiQ-603WO1 In some embodiments, the quality metric may be used to inform block selection for a hierarchical multiplexing procedure, e.g., as described in reference to Figure 5. For example, in some embodiments, it may be determined whether to fuse the brick with one or more second bricks to produce an aggregate brick based on the quality metric. 5

[0217] In some embodiments, second syndrome graph data that identifies one or more second syndromes is received for a second copy of the brick. Zero, one, or more second syndromes are removed from the second syndrome graph data to obtain modified second syndrome graph data, and a second quality metric is determined based on the modified second syndrome graph data. The second quality metric may be determined by determining a second 10 logical gap magnitude of the modified second syndrome graph data without the zero, one or more removed second syndromes. Either the brick or the copy of the brick may then be selected to be fused into an aggregate brick based on a comparison of the first and second quality metrics (e.g., the higher quality metric may be selected. A fusion controller may then fuse the selected brick or the selected copy of the brick to one or more second bricks to 15 produce the aggregate brick.

[0218] In some embodiments, an overall quality metric is determined based on a weighted average of a first quality metric determined from an unmodified syndrome graph and one or more second quality metrics determined from one or more respective modified syndrome graphs (i.e., modified by removing one or more syndromes). Advantageously, the weighted 20 average may be weighted based on the likelihood that a removed syndrome is indicative of an error profile that involves measured or unmeasured edges. For example, in some embodiments the weighted average is weighted based on graph distances of the removed syndromes from port vertices of the brick, and / or based on graph distances of unremoved syndromes from the port vertices of the brick. This may be particularly advantageous in cases 25 where a syndrome has equal weight (or close to equal weight) in its graph distance to be matched to both a port pseudo-vertex and a gap-boundary pseudo-vertex of another syndrome. In this case, a logical gap may be determined with this syndrome both removed and not removed, and the quality metric may be determined from an average of the two with equal weight (or close to equal weight). 30

[0219] More specifically, a first quality metric may be determined based on the received, unmodified syndrome graph by determining a logical gap magnitude from the syndrome graph. At least one modified syndrome graph may be constructed from the syndrome graph by removing one or more syndromes from the syndrome graph. For each modified syndrome - 52 -7246-02301 PsiQ-603WO1 graph, a respective second quality metric is determined by determining a logical gap magnitude of the respective modified syndrome graph without the one or more removed syndromes. An overall quality metric may be determined as a weighted average of the first quality metric and the second quality metric(s), and the overall quality metric may be stored 5 in a non-transitory computer-readable memory medium.

[0220] In some embodiments, constructing the modified syndrome graphs includes constructing a separate modified syndrome graph that removes each distinct subset of the syndromes of the syndrome graph. For example, a separate modified syndrome graph may be constructed with every distinct subset of the syndromes removed. In these embodiments, the 10 weighted average weighs the separate modified syndrome graphs based on graph distances of syndromes in the distinct subsets from port vertices of the brick. Frozen Gap Scoring for Hierarchical Multiplexing

[0221] In a typical hierarchical multiplexing setting, a brick includes resource states fused together in a region (e.g. a brick may include a 2x2x2 set of 6-rings fused together), with 15 some unfused resources on the boundary of this brick. In some embodiments, these unfused resources may be characterized as “gap-boundaries” and “ports”. Both gap-boundaries and ports are types of boundary vertices where qubits of resource states are not fused, and thus checks near a gap-boundary or port are not yet available and called “incomplete”. Gap- boundaries refer to a pair of edge vertices of the brick across which a continuous train of 20 errors is more likely to cause a logical error. For a complete logical block, a train of errors that spans across two opposing gap-boundaries will cause a logical error. However, for a brick that is at a lower layer and is only a portion of a full logical block, a train of errors that spans across two opposing gap-boundaries is simply more likely to cause a logical error, once the brick is combined with other bricks to form a full logical block. Port boundaries refer to 25 all edge vertices that are not identified as gap-boundaries, and which are used to identify syndromes to be “frozen out” (i.e., removed) from the logical gap calculation. In some cases, the port-boundaries are the input / output surfaces of a logical block.

[0222] Bricks are often constructed in a 3-dimensional arrangement, but some of the concepts are simpler to illustrate in a 2D context. Figure 19 illustrates a fusion network 30 divided into four 4x4 sub-bricks, according to some embodiments. As shown, both a primal and a dual syndrome graph can be constructed from a single 4x4 sub-brick. Before the four sub-bricks are fused together, the primal and dual graphs both have unmeasured port vertices around their boundary.. - 53 -7246-02301 PsiQ-603WO1

[0223] In some embodiments, such as the bricks shown in Figure 19, a brick has ports on all sides (e.g., also in a topological sphere). This type of hierarchical multiplexing, where there are no primal / dual boundaries on a brick, is referred to as “Spherical hierarchical multiplexing”. In this case, all unmeasured boundary vertices are port vertices, and there is no 5 preferred opposing pair of sides that should be identified as gap boundaries. To address this, in this case a separate quality metric may be determined when each opposing pair of faces (which are sets of unmeasured edges in the syndrome graph) are considered gap-boundaries, and the remaining opposing pairs of faces (which are also sets of unmeasured edges in the syndrome graph) are considered port edges. In each case, the gap-boundaries are connected to 10 gap-boundary pseudo-vertices and the ports are connected to port pseudo-vertices. The contributions from each separate quality metric (where each opposing set of unmeasured edges are connected to a gap-boundary vertex) may then be summed to obtain an overall quality metric.

[0224] An example of this for a simple 2D brick is shown in Figure 20. As illustrated, a first 15 quality metric (upper) may be determined where pseudo-vertices are added on either side of the brick along the x-axis and identified as port vertices, and pseudo-vertices are added on either side of the brick along the y-axis and identified as gap-boundary vertices (2004). A second quality metric (lower) is determined for the inverse case, where pseudo-vertices are added on either side of the brick along the y-axis and identified as port vertices, and pseudo- 20 vertices are added on either side of the brick along the x-axis and identified as gap-boundary vertices (2006). In some embodiments contributions from each of the four logical gaps may be combined to determine an overall quality metric (one for each axis identification and for each of the primal and dual syndrome graphs).

[0225] Figure 21 illustrates the identification of port and gap-boundary pseudo-vertices for a 25 fusion network composed of a 3D cubical brick, according to some embodiments. For a 3D cubical brick, there are three distinct axes that define three pairs of opposing faces. The long- dashed, short-dashed, and dotted lines illustrate the connection of unmeasured edges to these six pseudo-vertices along the three respective axes. In this example, a logical gap may be determined when one of the three pairs of opposing pseudo-vertices (e.g., the pair of pseudo- 30 vertices along the x-axis) is identified as a pair of gap-boundary vertices, and the remaining two pairs of pseudo-vertices (e.g., the pairs of pseudo-vertices along the y- and z-axes) are identified as port vertices. This process may be repeated to determine two additional logical gaps where the pairs along the y-axis, and then the pairs along the z-axis, are identified as - 54 -7246-02301 PsiQ-603WO1 gap-boundary vertices, and the other pseudo-vertices are identified as port vertices in each case. This process may be repeated for the other syndrome graph (i.e., it may be performed separately for both the primal and dual syndrome graphs), giving three more logical gaps. The overall quality metric may be determined based on a sum of contributions from each of 5 these six determined logical gaps.

[0226] Note that in the above examples, the pseudo-vertices are placed along the x-axis or the y-axis (or the z-axis). More generally, pseudo-vertices may be placed in any of a variety of locations around the perimeter of the brick, and various subsets of these pseudo-vertices may be identified as port or gap-boundary vertices. 10

[0227] In some embodiments, a quality metric, M, may be calculated according to the following expression:

[0229] where the summation is over contributions from the primal and dual syndrome graphs, denotes the magnitude of the logical gap, and are adjustable numerical 15 parameters, and w is a matching weight used to freeze out any syndromes during the initial step. Said another way, w is the weight of the path used to freeze out any syndromes by matching them to ports. For example, in the example shown in Figure 17D, the weight is 2 for each of the two syndromes that are matched to port pseudo-vertices (assuming all edge weights are 1). 20 Figures 22A-D - Frozen Gap Scoring with other Decoders and Topological Codes

[0230] In various embodiments, different types of decoders may be used to calculate the logical gap, and this may be done using any of a variety of topological protocols. In some embodiments, the checks and decoding problem for a topological protocol may be represented by a Tanner graph. A Tanner graph has check nodes and error nodes. In fusion- 25 based quantum computing (FBQC), the error nodes occur at fusion outcomes, and check nodes correspond to parity checks. An example of this is shown in Figure 22A, where each fusion outcome is associated with four checks, and checks are weight four. In Figures 22A-D, check nodes are represented by squares, and error nodes by circles. The dark shaded parity checks in Figures 22B-D indicate syndromes. 30

[0231] Embodiments herein describe methods to calculate a logical gap after removing some of the syndromes from the syndrome graph. The notion of a cluster of errors may be used to assist in determining which syndromes to remove. As used herein, for the code layout shown in Figures 22A-D, a set of errors forms a “cluster” when every error in the set is distance two - 55 -7246-02301 PsiQ-603WO1 (in the Tanner graph) to another error in the set. Said another way, a set of errors forms a cluster when there is at least one check node that connects any two errors in the cluster. Figures 22B-D illustrate three examples of clusters of errors. Port check nodes are defined as check nodes where not all of the neighboring error nodes have been measured. Examples of 5 port check nodes and clusters are shown in Figures 22B-D. Port check nodes are indicated by light-gray shaded squares.

[0232] The selection of syndromes to remove for any code or decoder may proceed as follows. Given the set of syndromes, the decoder will determine a recovery. The recovery is divided into one or more clusters of flipped errors. If a cluster is connected to a port (i.e. an 10 error in that cluster is distance two in the Tanner graph to a port node), then that cluster and the associated syndromes are selected to be frozen out (i.e., removed) prior to calculating the logical gap. The rest of the gap calculation may proceed as described above in reference to Figure 16. For example, a recovery for each class may be produced, and the absolute value of the difference of the weights is computed. The logical gap may then be calculated by 15 subtracting off a value proportional to the weight of the initial frozen clusters. Figure 23 - Union Find Decoding

[0233] Union Find decoding works by expanding erased clusters around each syndrome until they have an even number of syndromes in them, after which any correction explaining the syndrome in the cluster can be applied. Figure 23 illustrates an example of some syndromes 20 (large gray shaded circles), the erased clusters that are produced (light gray highlighted edges), and the correction that is applied (dark gray highlighted edges). In the illustrated example, the leftmost syndrome is matched to a port pseudo-vertex, and this syndrome would be frozen out. In this way, any other decoder can be used to perform the initial freeze, and to calculate the weights used for the logical gap calculation. 25 Soft-Output Decoder

[0234] In some embodiments, a logical gap calculated with port-syndromes frozen out can be considered more generally as an approach to give “soft output” for a decoder of topological codes. As used herein, a “soft output decoder” refers to a decoder that not only returns a correction, but returns a likelihood that the correction is correct. In other words, soft-output 30 decoding gives a probability distribution over all possible corrections (or a chosen subset of corrections), which may be useful in the setting of post-selection, concatenated codes, and / or hierarchical multiplexing. Importantly, the logical gap may provide soft output even with - 56 -7246-02301 PsiQ-603WO1 only a subset of check operators (i.e., before all relevant fusion measurements have been performed). For example, this may be advantageous for modular decoding.

[0235] Given a recovery belonging to class L, the soft output may be set as , whereis the gap corresponding to the class L, and where A is a normalization factor. 5 Calculating Gaps with Other Boundary Conditions.

[0236] In some embodiments, quantum computing systems and methods are described to calculate logical gaps (without or without frozen syndromes) and their associated quality metrics for general syndrome graphs or tanner graphs. In some embodiments, pseudo- syndromes are added to a syndrome graph and connected to boundaries of a brick. The parity 10 on the pseudo-vertices can be chosen to enable the decoder to return corrections for different logical equivalence classes for computing the gap and subsequent quality metrics.

[0237] In general, it may be desirable to compute a logical gap in situations where there are no open boundary conditions. For example, the boundary conditions of a brick may be periodic, so that there is not a perimeter of unmeasured edges. As another example, the code 15 that the gap will be computed for may be described by a Tanner graph and without a clear definition of boundaries to choose from. The following paragraphs describe embodiments to enable gap calculation in these types of situations (and other situations). First, we give an example for the syndrome-graph case, and then expand to the Tanner graph case. Periodic Syndrome Graph Example 20

[0238] Consider a syndrome graph and a logical mask. A logical mask is a set of edges on the syndrome graph which is used to determine if a logical failure has occurred. It corresponds to the set of measurements making up a logical membrane (also referred to as a logical correlator). In some embodiments, the logical membrane may be heuristically defined, if one wants to assess the probability of a set of errors that span a syndrome graph normal to the 25 logical mask. An example of a logical mask corresponding to a logical correlator is shown in Figure 24A, which shows a logical mask on a syndrome graph with periodic boundary conditions.

[0239] Figure 24B illustrates conversion of a syndrome graph with a logical mask to a modified syndrome graph with pseudo-check nodes and hyperedges, according to some30 embodiments. To compute the logical gap and its variants, in some embodiments a pseudo- vertex is added to the syndrome graph corresponding to this logical mask. The edges in the logical mask become hyper-edges, connecting the two vertices that they originally connected to, as well as the added pseudo-vertex. This now gives a syndrome hypergraph, whereby - 57 -7246-02301 PsiQ-603WO1 vertices correspond to checks (or pseudo-vertices), and hyperedges correspond to fusion outcomes that may be supported in one, two or three vertices. By setting the parity of the pseudo-vertex, the logical class a decoder will recover to may be determined.

[0240] In particular, similar to the case for open boundary conditions, the two recoveries for 5 the gap may be computed as follows.

[0241] First, the syndrome for the pseudo-vertex may be set to zero, and the decoder may be applied to obtain a first recovery r1. The first recovery may be determined by a decoder, and may describe a potential correction for the underlying error configuration that resulted in the set of syndromes in the syndrome graph. Then, the syndrome for the pseudo-vertex may be 10 set to 1, and the decoder may be applied to obtain a second recovery r2. The gap can be computed by taking the different in weights of the two recoveries, | |. Ingeneral, a modified decoder may be used in this case, as a minimum-weight perfect matching decoder may not be able to deal with hyperedges by default. In some embodiments, a decoder such as belief propagation, ordered statistics decoding (BP-OSD) may be used to compute the 15 recovery.

[0242] In some embodiments, as is the case for the open boundary conditions, ports corresponding to incomplete checks may be identified, and some of the syndromes may be frozen out prior to computing the gap. Additionally or alternatively, the quality metric may be updated by considering a weighted combination of the gap and the weight of the recovery 20 used to freeze out syndromes. For example, the methods described in reference to Figures 24A-B may be used in combination with the methods described in reference to Figure 18.

[0243] Note that this procedure may be applied to multiple logical masks, by adding the corresponding pseudo-vertices and updating the set of vertices each hyper-edge connects to. Figures 25A-B - Tanner Graph Example 25

[0244] Figures 25A-B illustrate conversion of a Tanner graph with a logical mask to a Tanner graph with an additional pseudo-check node, according to some embodiments. In some embodiments, the Tanner graph is updated by considering a logical mask and adding additional pseudo-check nodes. Here, a logical mask is represented by a collection of error nodes (circles in the figure). In Figure 25B, one pseudo-check node is added for the logical 30 mask, and connected to the error nodes in the support of the logical mask. The logical gap may be computed by modifying the value (0, or 1) on the pseudo-check node, as is done in the syndrome graph case described in reference to Figures 24A-B. Additional Embodiments - 58 -7246-02301 PsiQ-603WO1

[0245] The following numbered paragraphs describe additional embodiments.

[0246] In some embodiments, a method includes identifying syndrome data using one or more processors of a machine. A first set of decoding data may be generated using a decoding scheme. Syndromes are removed from the first set of decoding data to form a 5 modified set of decoding data. The removed syndromes correspond to syndrome data at sides of a surface code. Additional sets of decoding data are generated by applying the decoding scheme on the modified set of decoding data. The additional sets of decoding data are generated at least in part on modifying weights in the additional sets of decoding data. One or more components of a quantum state are assigned as successful components based on the 10 additional sets of decoded data. The successful components are selected for further processing.

[0247] In various embodiments, the decoding scheme used to select syndromes to freeze out (i.e., remove) obtain relevant weights for correcting each sector, and / or calculating the logical gap may be a minimum weight perfect matching (MWPM) decoding scheme, a union find 15 (UF) decoding scheme, or any other desired decoding scheme.

[0248] 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.

[0249] It should also be understood that all diagrams herein are intended as schematic. 20 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. 25

[0250] This disclosure provides a description of the claimed invention with reference to specific embodiments. Those skilled in the art with access to this disclosure will appreciate that the embodiments are not exhaustive of the scope of the claimed invention, which extends to all variations, modifications, and equivalents.

[0251] The terminology used in the description of the various described embodiments herein 30 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 - 59 -7246-02301 PsiQ-603WO1 “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 5 not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0252] 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, 10 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.

[0253] As used herein, the term “if” is, optionally, construed to mean “when” or “upon” or 15 “in response to determining” or “in response to detecting” or “in accordance with a determination that,” depending on the context.

[0254] 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. 20 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 practical 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. - 60 -

Claims

7246-02301 PsiQ-603WO1 Claims What is claimed is:

1. A method, comprising: 5 for each brick of a plurality of bricks of a target fusion network, receiving a plurality of respective copies; by a classical processor: determining a respective quality metric for each copy of each brick of the plurality of bricks, wherein determining the quality metrics comprises, for each respective 10 copy of each brick of the plurality of bricks: receiving syndrome graph data for the respective copy, wherein the syndrome graph data identifies at least one syndrome on one or more respective vertices of a syndrome graph; removing zero, one, or more syndromes of the syndrome graph data 15 from the syndrome graph data to obtain modified syndrome graph data; and determining the respective quality metric based on the modified syndrome graph data, wherein determining the respective quality metric comprises determining a logical gap magnitude of the modified syndrome graph data without the zero, one or more removed syndromes; 20 based at least in part on the quality metrics, selecting a first copy of the respective copies for each brick of the plurality of bricks; and by a fusion controller, fusing the first copies for each brick of the plurality of bricks together to produce an aggregate brick. 25 2. The method of claim 1, wherein the logical gap magnitude comprises a magnitude of a difference in a first weight of a first correction of the modified syndrome graph data and a second weight of a second correction of the modified syndrome graph data. 30 3. The method of claim 2, wherein the first and second weights comprise log-likelihood weights.

4. The method of claim 1, - 61 -7246-02301 PsiQ-603WO1 wherein the logical gap magnitude is associated with a first logical error class of a set of distinct logical error classes of the syndrome graph, wherein the method further comprises determining a second logical gap magnitude of the modified syndrome graph data without the zero, one or more removed syndromes, 5 wherein the second logical gap magnitude is associated with a second logical error class of the set of distinct logical error classes of the syndrome graph, and wherein the quality metric is determined further based on the second logical gap magnitude. 10 5. The method of claim 4, wherein the first logical error class comprises a primal graph logical error, and wherein the second logical error class comprises a dual graph logical error.

6. The method of claim 4, 15 wherein the quality metric comprises a summation of a decaying exponential function of the first logical gap magnitude and a decaying exponential function of the second logical gap magnitude.

7. The method of claim 1, 20 wherein selecting the first copy of each brick of the plurality of bricks comprises selecting a copy with a highest quality metric.

8. The method of claim 1, further comprising: for each brick of the plurality of bricks, rank ordering the copies based on their 25 respective quality metrics; fusing respective copies from each brick of the plurality of bricks together based at least in part on the rank ordering to produce a plurality of aggregate bricks.

9. The method of claim 1, 30 wherein the plurality of bricks comprise a plurality of Kagome-6 bricks.

10. A method, comprising: - 62 -7246-02301 PsiQ-603WO1 receiving syndrome graph data for a brick of a plurality of bricks of a target fusion network, wherein the syndrome graph data identifies at least one syndrome on one or more respective vertices of a syndrome graph; removing zero, one or more syndromes of the syndrome graph data from the 5 syndrome graph data to obtain modified syndrome graph data; determining a quality metric based at least in part on the modified syndrome graph data, wherein determining the quality metric comprises determining a logical gap magnitude of the modified syndrome graph data without the zero, one, or more removed syndromes; and storing the quality metric in a non-transitory computer-readable memory medium. 10 11. The method of claim 10, further comprising: based at least in part on the quality metric, determining whether to fuse the brick with one or more second bricks to produce an aggregate brick. 15 12. The method of claim 10, further comprising: selecting the zero, one, or more syndromes for removal from the syndrome graph data based at least in part on proximity of the zero, one, or more syndromes to an unmeasured edge of the syndrome graph. 20 13. The method of claim 10, further comprising: adding a plurality of pseudo-vertices to the syndrome graph data, wherein each pseudo-vertex of the plurality of pseudo-vertices connects to one or more unmeasured edges in the syndrome graph; identifying a first subset of the plurality of pseudo-vertices as gap-boundary vertices; 25 identifying a second subset of the plurality of pseudo-vertices as port vertices; selecting the zero, one, or more syndromes for removal from the syndrome graph data based at least in part on their proximity to port vertices in the identified second subset of the plurality of pseudo-vertices, and wherein determining the logical gap magnitude is performed based at least in part on a 30 difference between two corrections of a logical error that spans two gap-boundary vertices of the identified first subset of the plurality of pseudo-vertices.

14. The method of claim 13, - 63 -7246-02301 PsiQ-603WO1 wherein determining the quality metric further comprises: reidentifying the first subset of the plurality of pseudo-vertices as port vertices; reidentifying at least a portion of the second subset of the plurality of pseudo- 5 vertices as gap-boundary vertices; selecting zero, one, or more second syndromes for removal from the syndrome graph data to obtain second modified syndrome graph data, wherein the zero, one, or more second syndromes are selected for removal based at least in part on their proximity to at least one port vertex in the reidentified first subset of the plurality of pseudo-vertices; 10 determining a second logical gap magnitude of the second modified syndrome graph data without the zero, one, or more removed second syndromes, wherein determining the second logical gap magnitude is performed based at least in part on a difference between two corrections of a logical error that spans two gap-boundary vertices of the reidentified portion of the second subset of the plurality of pseudo-vertices; and 15 determining the quality metric as a summation of a decaying exponential function of the logical gap magnitude and a decaying exponential function of the second logical gap magnitude.

15. The method of claim 10, 20 wherein the logical gap magnitude comprises a magnitude of a difference in a first weight of a first correction of the modified syndrome graph data and a second weight of a second correction of the modified syndrome graph data.

16. The method of claim 15, 25 wherein the first and second weights comprise log-likelihood weights.

17. The method of claim 10, wherein the logical gap magnitude is associated with a first logical error class of a set of distinct logical error classes of the syndrome graph, 30 wherein the method further comprises determining a second logical gap magnitude of the modified syndrome graph data without the zero, one, or more removed syndromes, wherein the second logical gap magnitude is associated with a second logical error class of the set of distinct logical error classes of the syndrome graph, and - 64 -7246-02301 PsiQ-603WO1 wherein the quality metric is determined further based on the second logical gap magnitude.

18. The method of claim 17, 5 wherein the first logical error class comprises a primal graph logical error, and wherein the second logical error class comprises a dual graph logical error.

19. The method of claim 17, wherein the quality metric comprises a summation of a decaying exponential function 10 of the first logical gap magnitude and a decaying exponential function of the second logical gap magnitude.

20. The method of claim 10, further comprising: receiving second syndrome graph data for a copy of the brick, wherein the second 15 syndrome graph data identifies one or more second syndromes; removing zero, one, or more second syndromes from the second syndrome graph data to obtain modified second syndrome graph data; determining a second quality metric based on the modified second syndrome graph data, wherein determining the second quality metric comprises determining a second logical 20 gap magnitude of the modified second syndrome graph data without the zero, one, or more removed second syndromes; and based at least in part on the first and second quality metrics, selecting the brick or the copy of the brick to be fused into an aggregate brick; and by a fusion controller, fusing the selected brick or the selected copy of the brick to 25 one or more second bricks to produce the aggregate brick.

21. The method of claim 10, further comprising: determining, by a decoder and based on the modified syndrome graph, a correction to potentially correct an underlying error configuration of the brick; 30 determining a probability that the correction will correct the underlying error configuration, wherein the probability is determined based at least in part on the logical gap magnitude; and storing the probability in the non-transitory computer-readable memory medium. - 65 -7246-02301 PsiQ-603WO1 22. A method, comprising: receiving a syndrome graph for a brick of a plurality of bricks of a target fusion network, wherein the syndrome graph comprises at least one syndrome on one or more 5 respective vertices; determining a first quality metric based on the syndrome graph, wherein determining the first quality metric comprises determining a logical gap magnitude from the syndrome graph; constructing at least one modified syndrome graph from the syndrome graph, wherein 10 constructing the at least one modified syndrome graph comprises removing one or more syndromes from the syndrome graph; for each of the at least one modified syndrome graphs: determining a respective second quality metric, wherein determining the second quality metric comprises determining a logical gap magnitude of the respective 15 modified syndrome graph without the one or more removed syndromes; and determining an overall quality metric as a weighted average of the first quality metric and the at least one second quality metric; and storing the overall quality metric in a non-transitory computer-readable memory medium. 20 23. The method of claim 22, wherein the weighted average is weighted based at least in part on graph distances of the removed syndromes from port vertices of the brick. 25 24. The method of claim 23, wherein the weighted average is further weighted based on graph distances of unremoved syndromes from the port vertices of the brick.

25. The method of claim 22, 30 wherein constructing the at least one modified syndrome graph comprises constructing a separate modified syndrome graph that removes each distinct subset of the one or more syndromes of the syndrome graph, and - 66 -7246-02301 PsiQ-603WO1 wherein the weighted average weighs the separate modified syndrome graphs based on graph distances of syndromes in the distinct subsets from port vertices of the brick.

26. The method of claim 22, further comprising: 5 based at least in part on the overall quality metric, determining whether to fuse the brick with one or more second bricks to produce an aggregate brick.

27. A method, comprising: receiving syndrome graph data for a brick of a plurality of bricks of a target fusion 10 network, wherein the syndrome graph data identifies at least one syndrome on one or more respective vertices of a syndrome graph, wherein the syndrome graph exhibits periodic boundary conditions, and wherein the syndrome graph comprises a logical mask; adding a pseudo-vertex to the syndrome graph data, wherein the pseudo-vertex connects to pairs of vertices in the logical mask 15 determining a first recovery for the syndrome graph data when the pseudo-vertex is set to have a syndrome value of 1; determining a second recovery for the syndrome graph data when the pseudo-vertex is set to have a syndrome value of 0; determining a logical gap magnitude to be a magnitude of a difference between a first 20 weight of the first recovery and a second weight of the second recovery; determining a quality metric based at least in part on the logical gap magnitude; and storing the quality metric in a non-transitory computer-readable memory medium.

28. The method of claim 27, 25 wherein the first and second recoveries are determined by a belief-propagation, ordered statistics decoding (BP-OSD) decoder.

29. A method, comprising: receiving syndrome graph data for a brick of a plurality of bricks of a target fusion 30 network, wherein the syndrome graph data identifies at least one syndrome on one or more respective error nodes of a Tanner graph, and wherein the syndrome graph comprises a logical mask; - 67 -7246-02301 PsiQ-603WO1 adding a pseudo-check node to the Tanner graph, wherein the pseudo-vertex connects to one or more error nodes in the logical mask; determining a first recovery for the syndrome graph data when the pseudo-check node is set to have a syndrome value of 1; 5 determining a second recovery for the syndrome graph data when the pseudo-check node is set to have a syndrome value of 0; determining a logical gap magnitude to be a magnitude of a difference between a first weight of the first recovery and a second weight of the second recovery; determining a quality metric based at least in part on the logical gap magnitude; and 10 storing the quality metric in a non-transitory computer-readable memory medium.

30. The method of claim 29, wherein the first and second recoveries are determined by a belief-propagation, ordered statistics decoding (BP-OSD) decoder. 15 31. A non-transitory computer-readable memory medium storing program instructions which, when executed by a processor, direct the method steps of any of claims 1-30.

32. A controller, comprising: 20 a non-transitory computer-readable memory medium; a switching circuit coupled to a logical qubit generator; a fusion controller; one or more processors coupled to the memory medium, wherein the processor is configured to execute program instructions to direct the method steps of any of claim 1-30. - 68 -

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