Fusion tiling generator based quantum information system

Abstract

A fusion tiler system can generate a plurality of fusion complexes using a tiling scheme. The tiling scheme can generate candidate cell complexes such that one or more of the candidates have cells or chambers having two faces joined at the boundaries to form a volume. Valid cell complexes can be identified from the candidates based on fusion complex criteria, including quantity incident faces per edge. A valid fusion complex be implemented to perform measurements and decoding according to the fusion complex.

Description

6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) FUSION TILING GENERATOR BASED QUANTUM INFORMATION SYSTEM CROSS-APPLICATION REFERENCE

[0001] This application claims priority to and the benefit of U.S. Provisional Patent Application No. 63 / 531,807, filed on August 10, 2023 the contents of which are incorporated herein by reference in its entirety. TECHNICAL FIELD

[0002] Embodiments of the disclosure relate generally to quantum systems and, more specifically, to data structures for quantum systems. BACKGROUND

[0003] Quantum computing systems implement qubits for information processing tasks. Whereas bits in conventional non-quantum computing systems (e.g., electronic computing systems, microprocessors) can represent a first state (e.g., binary ^1,^ a high state) or a second stage (e.g., binary ^0,^ a low state), qubits of quantum computing system can represent a first state and a second state (e.g., 1, 0, respectively), but also a superposition of the first and second state where the qubits can have complex valued (e.g., imaginary number values) representations that can be entangled with one another in a complex space (e.g., Hilbert space, configuration space), which can make it difficult to reliability implement data structures.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) BRIEF DESCRIPTION OF THE DRAWINGS

[0004] The present disclosure will be understood more fully from the detailed description given below and from the accompanying drawings of various embodiments of the disclosure.

[0005] FIGs. 1A-1J show examples of a quantum information processing system, in accordance with some example embodiments.

[0006] FIGs. 2A-2C show examples of resource states, fusion complexes, and surface codes, in accordance with some example embodiments.

[0007] FIGs. 3A-3C show examples of different fusion complexes, in accordance with some example embodiments.

[0008] FIG.4 shows a flow diagram of a method for implementing a fusion tiling scheme, in accordance with some example embodiments.

[0009] FIG. 5 shows a flow diagram of a method for generating and processing fusion complexes, in accordance with some example embodiments.

[0010] FIG. 6 shows an example flow diagram of a method for generating candidate cell complexes and fusion complexes, in accordance with some example embodiments.

[0011] FIG.7 shows an example computer system that can execute instructions, in accordance with some example embodiments.

[0012] Descriptions of certain details and implementations follow, including a description of the figures, which may depict some or all of the embodiments described below, as well as discussing other potential embodiments or implementations of the inventive concepts presented herein. An overview of embodiments of the disclosure is provided below, followed by a more detailed description with reference to the drawings. DETAILED DESCRIPTION

[0013] In the following description, for the purposes of explanation, numerous specific details are set forth to provide an understanding of various embodiments of the present disclosure. It will be evident, however, to those skilled in the art, that embodiments of the inventive subject matter may be practiced without these6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) specific details. In general, well-known instruction instances, structures, and techniques are not necessarily shown in detail.

[0014] A quantum information processing system (e.g., quantum logic device, quantum computer) can utilize entanglement between qubits to more efficiently solve problems that are intractable for non-quantum information processing systems (e.g., classical computers, electronic microprocessors, and so on). One approach for error or fault tolerant quantum computation includes measurement- based quantum computation in which a large quantum state must be generated and maintained. An alternative approach to MBQC is fusion-based quantum computation (FBQC) in which resource states are processed by projective fusion measurements to implement processing. In some example embodiments, fusion networks are high dimensional complex space objects and it can be difficult to generate networks that function in a fusion-based computation scheme.

[0015] To address the foregoing, in some example embodiments, a degenerate tiling generation application is configured to generate, using one or more processors (e.g., classical processors, electrical based processors) a plurality of candidate cell complexes where the tiling generation application is configured to implement a generation of degenerate cells, such as polyhedrons with only two faces (e.g., two faces joined at their boundaries that form a volume, resembling ravioli). In some example embodiments, the plurality of candidate cell complexes is filtered to filter out cell complexes having edges that have a number of incident faces other than four. That is, each cell complex is analyzed to determine whether each of the plurality of edges is incident to exactly four faces of the cell complex and, if each edge is incident to exactly four faces, the cell complex is kept while others are filtered out. In some example embodiments, each of the non-filtered-out cell complexes are then stored in memory as fusion-based cell complex networks. In some example embodiments, the fusion-based cell complex networks are implemented to perform fusion-based qubit information processing on a quantum information processing system. In some example embodiments, the plurality of fusion-based cell complex networks are implemented in simulation and benchmarking schemes to identify a selected fusion-based cell complex network that exhibits desired threshold values for implementation in a particular error-6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) tolerant quantum information processing system (e.g., fault tolerant quantum information processing system, quantum computer). For example, a first fusion- based cell complex network can be selected for implementation on a first type of quantum computer having a first set of error rates, and a second, different fusion cell complex can be selected for implementation on a second type of quantum computer having different error rates. In this manner, different fusion cell complexes having different characteristics can be implemented in different types and designs of quantum computing systems.

[0016] In some example embodiments, each fusion cell complex corresponds to a plurality of entangled resource states and fusion operations to perform on qubits of different entangled resource sources, where the resource states and the fusion operations are arranged in a three-dimensional cellular form or lattice. In some example embodiments, a resource state of the plurality of resource states comprises a plurality of entangled qubits and the fusion operations are fusions performed on sets of qubits from respective resource states.

[0017] In some example embodiments, a quantum information system (e.g., a photonic quantum information processing system, superconducting qubit information processing system) is configured to manage sets of qubits (e.g., parts of resource states) and apply fusion operations on qubits from different sets of qubits according to one of the plurality of fusion cell complexes (e.g., the selected fusion cell complex). In some example embodiments, the fusion operations generate readout data (e.g., detector data), which is stored in classical memory (e.g., random access memory) as measurement data. In some example embodiments, the measurement data is processing, using the one or more classical processors, to generate syndrome data. In some example embodiments, the syndrome data is generated using check operations that correspond to faces of a selected fusion cell complex. The faces can correspond to, for example, topological surface code operations of a surface code (e.g., bit flip check operations). Examples of syndrome data comprise array data that can be input into different quantum error correction decoders, such as the union find decoder, graph matching based schemes (e.g., minimum weight perfect matching), neural network-based decoders, or other types of decoders to generate state data. In some example6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) embodiments, the state data includes qubit corrective actions to perform on qubits to mitigate errors in the quantum information processing system (e.g., quantum corrections on specific physical qubits, stabilizer operations, logical qubit operations involving collections of qubits). In some example embodiments, the state data comprises gate information of qubit gates to perform according to a selected (e.g., runtime) quantum algorithm where the selected gates are algorithmically chosen according to the given runtime quantum algorithm to avoid further errors and / or termination of the quantum computing task due to the further errors.

[0018] The following FIGs. 1A-1J discuss example configurations for quantum information processing systems, in accordance with some embodiments. Qubits: 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 mechanics. As used herein, a ^qubit^ (or quantum bit) is a quantum system with an associated quantum state that may be used to encode information. A quantum state may be used to encode one bit of information if the quantum state space can be modeled as a (complex) two-dimensional vector space, with one dimension in the vector space being mapped to logical value 0 and the other to logical value 1. In contrast to classical bits, a qubit may have a state that is a superposition of logical values 0 and 1. More generally, a ^qudit^ describes any quantum system having a quantum state space that may be modeled as a (complex) n-dimensional vector space (for any integer n), which may be used to encode n bits of information. For the sake of clarity of description, the term ^qubit^ is used herein, although in some embodiments the system may also employ quantum information carriers that encode information in a manner that is not necessarily associated with a binary bit, such as a qudit.

[0019] Qubits (or qudits) may be implemented in a variety of quantum systems. Examples of qubits include polarization states of photons; presence of photons in waveguides; or energy states of molecules, atoms, ions, nuclei, or photons. Other examples include other engineered quantum systems such as flux qubits, phase qubits, or charge qubits (e.g., formed from a superconducting Josephson junction);6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) topological qubits (e.g., Majorana fermions); or spin qubits formed from vacancy centers (e.g., nitrogen vacancies in diamond).

[0020] As used herein, a distinction is made between a ^physical qubit,^ which is a physical quantum system such as a molecule, atom, photon, etc. that exists in a 2-level quantum state, and a ^logical qubit,^ which includes a plurality of physical qubits encoded (e.g., entangled) together according to a quantum error correcting code (such as a surface code) to encode logical quantum information. These terms are described in greater detail below.

[0021] Example Physical implementations: Qubits (and operations on qubits) may be implemented using a variety of physical systems. 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 (alsoreferred to as a vacuum mode) may correspond to the |0 L state of a photonicqubit. Alternatively, a state with a photon in the second waveguide and no photonin the first waveguide may correspond to the |1 L state of the photonic qubit. Toprepare a photonic qubit in 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 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. The waveguides in such a system need not have any spatial relationship to each other. For instance, they may be (but need not be) arranged in parallel.

[0022] Some embodiments described below relate to physical implementations of unitary operations that couple modes of a quantum system, which may be understood as transforming the quantum state of the system. For instance, if the initial state of the quantum system (prior to mode coupling) is one in which one mode is occupied with probability 1 and another mode is unoccupied with6224.007WO1 ^ PSIQuantum (PSIQ-548WO1)probability 1 (e.g., a state | 10 in the Fock notation introduced above), modecoupling may result in a state in which both modes have a nonzero probability ofbeing occupied, e.g., a state a_1 | 10 +a_2 | 01 , where |a_1 |^2+|a_2 |^2=1. Insome embodiments, operations of this kind may be implemented by using beam splitters to couple modes together and variable phase shifters to apply phase shifts to one or more modes. The amplitudes a1 and a2 depend on the reflectivity (or transmissivity) of the beam splitters and on any phase shifts that are introduced.

[0023] FIGs. 1A-J ^ Surface Codes for Constructing Fault-tolerant Logical Qubits: A single physical qubit (e.g., such as the 2-level physical qubit illustrated =a_1 | 0 +a_2 | 1 ) may in principle be usedfor quantum computation. However, individual physical qubits are generally highly susceptible to noise and decoherence. Fault-tolerant quantum computing utilizes a plurality of entangled physical qubits to encode a single logical qubit to mitigate the frailty and / or short coherence times of individual physical qubits. In fault-tolerant quantum computing schemes, a plurality of physical qubits such as those illustrated in FIG. 1B are entangled together according to a specific error correcting code to produce a single logical qubit that is less susceptible to noise and decoherence. Encoding qubits in this manner causes the resultant logical qubit to be less sensitive to error and noise, and resultant errors may be fixed via quantum error correction. Encoding a logical qubit may itself be vulnerable to errors.

[0024] In some quantum computing methodologies, such as fusion-based quantum computing and circuit-based quantum computing, a logical qubit is encoded from a plurality of physical qubits using a sequence of specific measurements (e.g., stabilizer measurements). The measurement sequence may be constructed where a subset of the physical qubits is measured (e.g., collapsing the quantum state and producing classical information, e.g., the measurement result) in such a way that the remaining unmeasured / un-collapsed degrees of freedom (e.g., a two-dimensional subspace which has support over all the physical qubits) form the desired encoded logical qubit. Accordingly, the processes of performing stabilizer measurements and / or encoding a fault-tolerant logical qubit may receive a plurality of physical qubits as input and as output may produce both6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) the encoded logical qubit and classical information (e.g., syndrome graph data) resulting from the measurement sequence.

[0025] 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 an initial state and measured according to a predetermined measurement sequence, it may also be pre-determined how the syndrome appears in the absence of any errors involving the physical qubits during the measurement sequence (e.g., Pauli or erasure errors). Accordingly, any deviation of the syndrome graph 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 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.

[0026] 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. A series of measurements (e.g., stabilizer measurements) are applied to the physical qubits of the error correcting code containing the encoded logical information, producing measurement outcomes as classical information. As described in further detail below, based on the knowledge of the particular geometry of the cluster state / error correcting code, these measurement outcomes may be used to determine classical data referred to herein as the ^syndrome graph data.^ The syndrome graph data may further include correction operators for the syndrome graph output by a decoder.

[0027] Errors that occur during operations on an encoded logical qubit may have varying degrees of severity. For example, errors in a fault-tolerant logical6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) qubit may cause logical failure if they link up in a way that spans the syndrome graph of the logical qubit. Conversely, localized errors that do not span the syndrome graph may be identifiable and correctable via quantum error correction.

[0028] FIG. 1B shows an arrangement of physical qubits that can be used to encode a fault-tolerant logical qubit using a surface code, according to one or more embodiments. In FIG. 1B, the solid grid lines are guides to the eye and form an array of squares, also referred to herein as a ^surface code,^ with physical ^data qubits^ disposed on the four vertices of each square and physical ^measure qubits^ disposed on the face of each square. As used herein, measure qubits are the physical qubits which are measured to perform measurement checks on adjacent data qubits without directly measuring the data qubits and collapsing the quantum information. In this example, the surface code has a length (e.g., a code distance) d of 12, but any length can be employed. In some embodiments, the buffer size to be added to the edge regions to suppress errors depends on the size of the code distance (e.g., which can depend on error performance of a given quantum computer and / or the quantum algorithm or application to be computed). In some embodiments, the buffer size added to the edge regions is dynamically adjusted by a decoder system 188 based on a given code distance (e.g., current code distance of a current quantum processing system set-up).

[0029] Continuing, in the example of FIG. 1B, the surface code arrangement of qubits also includes four lines of boundary measure qubits disposed adjacent to the outermost lines of data qubits. Each square is referred to herein as a plaquette. Within the bulk of the surface code (e.g., the plaquettes which do not form the outer boundary of the code), each data qubit may be coupled, via four 2- qubit gates, to its four nearest neighbor measure qubits (each on four different plaquettes) and likewise, each measure qubit may be coupled, via four 2-qubit gates to its four nearest neighbor data qubits. On the boundaries of the code, each boundary measure qubit may be coupled, via two 2-qubit gates to its nearest adjacent data qubits. According to one or more embodiments, the 2-qubit gates can be CNOT gates, CZ gates, and the like.

[0030] In order to operate the collection of data and measure qubits as a logical qubit that is protected against errors, the following set of measurements may be6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) repetitively performed on the system. For each plaquette within the bulk of the surface code, 4-qubit stabilizers are measured. For example, as shown in FIG. 1D, if the data qubits of a given plaquette are labeled 1, 2, 3 ,4 (e.g., data qubits 105, 107, 109, and 111) and the measure qubit is labeled a (e.g., measure qubit 103), the stabilizer to be measured can be X1Z2Z3X4. The ^quantum circuit^ (which is a term that refers to the sequence of gates and measurement operations to be performed on physical qubits) used to implement this stabilizer measurement is also shown in FIG. 1D and includes first initializing the measure qubit a in the |+> state, then performing the following gates: a CNOT gate between the measure qubit a and data qubit 1, respective CZ gates between the measure qubit a and qubit 2 and qubit 3, and a CNOT gate between the measure qubit a and qubit 4; followed by an x-basis measurement Mx of measure qubit a. The resulting measurement outcome (which takes the form of a classical bit, e.g., 0 or 1 or -1 or 1, depending on the choice of conventions) is equal to the outcome of the measurement of the parity check stabilizer X1Z2Z3X4 and becomes part of the syndrome graph. For the plaquettes found at the boundary of the surface code, and shown in FIG. 1E, a 2-qubit stabilizer of the form Z1X2 is measured. The quantum circuit used to implement this 2-qubit stabilizer measurement is also shown in FIG. 1E and includes first initializing the boundary measure qubit a in the |+> state then performing the following gates: a CZ gate between the measure qubit a and qubit 1 and CNOT gate between the measure qubit and qubit 2; followed by an x-basis measurement Mx of measure qubit a. In the example shown in FIG. 1C, there are two different types of boundaries depending on whether the boundary includes shaded plaquettes or unshaded plaquettes. A boundary surface that includes shaded plaquettes is referred to as a ^dual boundary surface^ and measurements including measure qubits within the dual boundary surface contribute to the ^dual syndrome graph.^ Similarly, a boundary surface that includes unshaded plaquettes is referred to herein as a ^primal boundary surface,^ and measurements including measure qubits within a primal boundary surface contribute to the ^primal syndrome graph.^

[0031] To implement the surface code scheme shown in FIG. 1C-1E, the plaquette measurements may be broken into two groups of measurements: a first6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) group of measurements that measures the stabilizers associated with the shaded plaquettes during a first duration of time and a second group of measurements that measures the stabilizers with the unshaded plaquettes during a second duration of time. These two sets of measurements are performed at different times to ensure that each qubit only participates in one quantum gate at a time. One of ordinary skill in the art will appreciate that any gates that can commute with one another may be performed in the same time step, or even simultaneously, if desired. The classical data generated by each one of these measurements, referred to herein as ^syndrome graph data,^ is then passed to a decoder for quantum error correction according to known methods, e.g., using union find decoding, minimum weight perfect matching, or any other decoding process.

[0032] One of ordinary skill will appreciate that the example shown in FIG. 1C is using a particular choice of local basis for the surface code and that other choices for the basis may be employed. For example, in some contexts, taking certain assumptions from the errors that may occur on the underlying data and measure qubits, one may apply a single qubit gate to each data qubit to obtain a modified surface code. One may modify the basis for each check to obtain a scheme for the modified code. One example includes the CSS (Calderbank, Shor, Steane) version, where stabilizer measurements are either x-type or z-type. To obtain this version of the surface code, the stabilizers are conjugated by a Hadamard H: X-> Z, Z->X on half the data qubits in a bipartition, thereby resulting in the CSS surface code. Note that the measurement schedule described above remains the same, but the new stabilizers are given by that summarized in FIG. 1F and FIG. 1G.

[0033] 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. Viewed yet another way, this process operates as a fault-tolerant logical channel.

[0034] FIG. 1H illustrates a three-dimensional graphical depiction of such a fault-tolerant logical identity gate. The surface labeled 114 is the input to the gate and includes a logical state encoded in a surface code, represented as the input6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) checkerboard surface. Likewise, the surface labeled 118 identifies the output qubits after the identity gate I has been applied to it. The input and output surfaces, which may be associated with the physical two-dimensional arrangement of data and measure qubits described above, are connected to each other via an intervening volume that represents the unique set of measurements to be applied over time (e.g., a slice or frame 116 corresponds to measurements applied at a given instant in time). Accordingly, in FIG. 1H, time flows from left to right and the lighter shaded (front and back) and darker shaded (top and bottom) sides of the boundaries of the volume depict whether the primal or dual plaquettes are disposed on that boundary as described above in reference to FIG. 1C-1E. FIG. 1I represents the same concept but written in a more familiar quantum circuit notation illustrating the analogy between the more familiar quantum circuit.

[0035] While FIG. 1H shows the logical identity gate, any gate can be depicted in this manner and such a depiction is one example of a ^logical block^ that specifies a set of instructions to be performed on the underlying surface code qubits to perform a logical operation (the Identity gate in this example) on the logical qubit that is encoded by surface code. Other examples of such gates are the S gate, the Hadamard gate, and the CX gate, among other possibilities.

[0036] The sequence of measurements performed over the flow of time illustrated in FIG. 1H (e.g., a sequence of measurements including the circuit measurements shown in FIGs.1C-1E) may include a subset of measurements that incur a logical error (e.g., a Pauli error) or an erasure error. To identify errors in the measurement outcomes, syndrome graph data may be generated from the collection of measurement outcomes resulting from the measurements of the physical qubits. For example, the bit values associated with a plurality of edge qubits may be combined to create a syndrome value associated with an adjacent vertex that results from the intersection of the respective edges, e.g., the result of the measurements shown in FIGs. 1D and 1E. A set of syndrome values (or ^syndromes^), also referred to herein as parity checks, may be associated with each vertex of the syndrome graph. The parity check values may be found by computing the parity of the bit values associated with each edge of the syndrome graph6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) incident to the vertex. In some embodiments, a parity computation entails determining whether the sum of the edge values is an even or odd integer, with the parity result being the result of the sum modulo 2. If no errors have occurred in the quantum state or in the qubit measurements, then all syndrome values should be even (or 0). On the contrary, if an error occurs, it may result in some odd (or 1) syndrome values.

[0037] In some embodiments, half of the bit values from the qubit measurements are associated with the primal boundary surfaces, and this syndrome graph is referred to herein as the ^primal graph.^ The syndrome graph resulting from measurements on the dual boundary surfaces is referred to as the ^dual graph.^ There is generally an equivalent decoding problem on the syndrome values of the primal and dual graphs.

[0038] FIG. 1J illustrates a quantum information processing system 191, in accordance with some embodiments. As illustrated, the system includes a classical processing system 193 coupled to a quantum object processing system 195 over a classical channel 112 (e.g., application programming interface (API)). The classical channel may relay classical information between the classical and quantum computing systems.

[0039] In some embodiments, the classical processing system 193 includes memory 194 (e.g., one or more non-transitory computer-readable memory media), one or more central processing units (CPUs) or processor(s) 192, a power supply, an input / output (I / O) subsystem, and a communication bus interconnecting these components. The processor(s) 192 may execute modules, programs, and / or instructions stored in memory 194 and thereby perform processing operations. The processor may comprise a dedicated processor, or it may be a field programmable gate array (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 194 stores one or more programs (e.g., sets of instructions) and / or data structures and is coupled to the processor(s).

[0040] Further, in some embodiments, the classical processing system 193 comprises a fusion tiler system 177 to generate fusion cell complexes, as discussed in further detail below. Further, in some embodiments, the classical processing6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) system 193 comprises a decoder system 188 that implements a decoding scheme (e.g., union find, matching (e.g., minimum weight matching), neural network- based decoding) to process syndrome data.

[0041] 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 194 and executed by the classical (e.g., digital) processor 192 of the classical computer. The memory 194 is classical in the sense that it stores data and / or program instructions in a storage medium in the form of bits (rather than as qubits containing quantum information), which have a single definite binary state at any point in time. The processor may read instructions from the computer program in the memory 194 and / or write data into memory, and may optionally receive input data from a source external to the computer, such as from a user input device such as a mouse, keyboard, or any other input device. The processor 192 may execute program instructions that have been read from the memory 194 to perform computations on data read from the memory 194 and / or input from the quantum computing system and generate output from those instructions. The processor 192 may store that output back into the memory 194.

[0042] The quantum object processing system 195 may include a plurality of qubits 199 and a controller 196 configured to interface with the plurality of qubits 199 to control, direct, and / or measure the qubits within the quantum circuit. The qubits may be configured to evolve in time under the directed influence of the controller 196, and a measurement system 198 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 controller 196 during a quantum computation, and they may additionally include classical results of the quantum computation. The measurement results may be communicated to the classical computing system and / or the controller 196, and further the classical computing system may provide directions and / or instructions to the controller 196 and the6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) measurement system 198 to guide the behavior of the quantum computing system to perform a quantum computation. For example, the classical processing system 193 may provide classical data signals used for quantum state preparation within the quantum object processing system 195, in response to which the controller may prepare the states of the qubits 199 into a desired initial state for a particular quantum computation.

[0043] In some embodiments, qubits 199 (e.g., physical qubits) are provided to the measurement system 198 and controller 196, where the measurement system and the controller function as a logical qubit encoder that performs a sequence of measurements on the physical qubits to produce a logical qubit (e.g., a logical qubit prepared in a magic state, or another type of fault-tolerant encoded logical qubit). For example, the measurement system and controller may perform a sequence of measurements on the physical qubits to entangle them in such a way as to produce a logical qubit. Encoding the logical qubit will also produce syndrome graph data for the logical qubit as classical information, which is output to the classical processing system 193 via the classical channel 112.

[0044] In some example embodiments, a cell complex (e.g., three dimensional cell complex, CW complex) comprises a plurality of cells, faces, vertices, and edges. In some example embodiments, a fusion complex is a three-dimensional cell complex in which the vertices correspond to resource states and the edges correspond to fusion projective measurements of a fusion network. The following is a summary of fusion measurements and resource states that can be implemented in fusion-based quantum computing schemes. In some example embodiments, fusion (projection) measurements are projective measurements on multiple qubits (e.g., joint measurement on two qubits). For example, for two qubits, a fusion operation can apply a particular two-particle projective measurement (e.g., a Bell projection) which, depending on the Bell basis chosen, can project the two qubits onto one of the four Bell states. The projective measurements produce two measurement outcomes that correspond to the eigenvalues of the corresponding pair of observables that are measured in the chosen basis. For example, an XX measurement (e.g., XX fusion) is a Bell projection that measures the XX and ZZ observables, each of which could have a6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) convention used). Similarly, a XZ measurement (e.g., XZ fusion) is a Bell projection that measures the XZ and ZX observables, and so on. Although two qubit joint projective measurements are discussed here as examples, it is appreciated that the joint measurements can be implemented jointly on three or more qubits.

[0045] In some example embodiments, resource states are small entangled quantum states of a fixed size that undergo fusions according to the fusion network. Examples of resource states can include bell pairs, 3GHZ states, or larger entangled states (e.g., 26-qubit entangled state, 26-photon entangled state). Resource states can be generated repeatedly and consumed during the computation process such that the generation of resource states is independent of the computation^s runtime or the code distance used for error correction.

[0046] In some example embodiments, a fusion network can be generated from patterning a cell complex (e.g., unit cell). In some example embodiments, a cell complex that has four faces incident at every edge in the bulk then it defines a surface-code fault tolerance scheme. In this way, the measurements, the structure of resource states, and the check graph are captured by a single geometric structure: a three-dimensional cell complex. In some example embodiments, checks are the products of measurement outcomes that give deterministic values in the absence of errors (e.g., detectors, detections).

[0047] FIGs. 2A-2C show examples structures from surface code fusion complexes, in accordance with some example embodiments. At a high level, FIG. 2A shows a surface protocol from which slices correspond to surface codes. Further FIG.2B shows a fusion cell complex (e.g., surface code fault tolerant complex) that describes relations between resource states, fusion measurement operations (e.g., of a fusion network graph G), and check operator locations (e.g., of a syndrome graph, decoding graph). Further, FIG. 2C shows an example resource state that corresponds to the cubic cell complex of FIG. 2B. In some example embodiments, in fusion-based quantum computing, fault tolerance can be implemented from a fusion network that defines a set of measurements (e.g., fusions, projective measurements) made on a collection of constant-sized entangled states (e.g.,6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) resource states). Resource states are stabilizer states, which can be described by a stabilizer group, R, and the fusion measurements by the fusion group, F. In some example embodiments of fusion networks, a fusion network can be conveniently represented as a graph where the vertices represent resource states and the edges represent fusion measurements between them. The following is a description of a surface code (e.g., a two-dimensional surface code), in accordance with some example embodiments (Definition 1): a stabilizer code, S, is a surface code if there exists a two-dimensional cell complex = {F, E, V} where vertices are 4-valent and faces are 2-colorable, such that S = Xf_a , Zf_b | Fa , fb Fb , Xf_a is a product of Pauli-X operators on all vertices belonging to the face fa , and Zf_b is a product of Pauli-Z operators on all vertices belonging to the face fband Faand Fbdenote the set of faces of each color.

[0048] In some example embodiments, the above surface code description corresponds to a a ^plaquette^ version of the surface code. In some example embodiments, the 4-valence condition implies a local bi-colorability of the faces, and for certain complexes, also a global bi-colorability. In some example embodiments, errors in a surface code fusion network can be represented by a syndrome graph. In some example embodiments, any stabilizer code equivalent to a surface code under local Clifford operations is also considered a surface code (e.g., a Wen plaquette model of surface codes).

[0049] In the following, the examples are surface code fusion networks where all fusions are 2-qubit Bell-basis measurements f = {XX, ZZ}, such that the fusion network can be represented by a graph G = (GV, GE) , whose vertices GV correspond to resource states and edges GE to fusions. It is noted that each edge supports two qubits, one from each neighboring resource state. The following is a description of surface-code fusion networks (Definition 2): A fusion network is a surface-code fusion network if, after performing fusions on any region A GV , the unmeasured qubits on the boundary of the region A are described by a surface code (up to signs of the stabilizers).

[0050] In some example embodiments, performing fusions on a region A GV means performing fusions between resource states contained within that region. The boundary refers to all un-fused qubits of resource states within A. This6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) implies that all resource states must be surface codes themselves (corresponding to the region of a single vertex). One property of surface-code fusion networks is that un-fused qubits in the network at any moment in time (e.g., a time-slice) during a quantum computation will always be in a surface-code state (up to signs of the stabilizers). An illustration of this is discussed below with reference to FIG. 2A.

[0051] FIG. 2A shows a region, A, of a fusion network shown with a boundary . In some example embodiments, when the fault tolerance protocol inside A has been performed, a surface code state is left on the remaining boundary qubits. FIG. 2B shows an example fusion complex (e.g., 3D cell complex, cubic cell complex) that can be implemented with the surface code, in accordance with some example embodiments. FIG. 2C shows an example of a resource state associated with a vertex as an inflated volume that corresponds to the cubic fusion complex of FIG. 2B. In the illustrated example of FIGs. 2A-2C, state is represented using the surface code (e.g., in the plaquette representation) on the surface of an octahedron. Though FIGs. 2A to 2C show different layers using a same type of code, it is appreciated, that the surface code need not be the same from one layer to the next. The slices could also pass through the network in a spatial dimension or around a closed surface, but regardless of the orientation, inspecting the state of the qubits along that slice show they are in a surface code state (e.g., any slice through the A leaves, at the slice boundary, a surface code) , in accordance with some example embodiments.

[0052] The following is a discussion of fusion complexes and corresponding check structures, which can be used to perform checks (e.g., check structures, syndrome data). For brevity, the following discusses fusions which are implemented as 2-qubit Bell basis measurements as an illustrative example. In some example embodiments, a fusion complex is a three-dimensional cell complex with a constraint that every edge has exactly four incident faces. In some example embodiments, a three-dimensional cell complex, = {C, F, E, V} , is a surface-code fusion complex if every edge has exactly four incident faces (Definition 3).

[0053] In some example embodiments, each surface-code fusion complex defines, or otherwise can be used to generate, a surface-code fusion network6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) dataset. For example, for any region A GV, the state on the qubits after performing fusions within A can be obtained (up to sign) by restricting the check operators to ; which is a surface code state, as every qubit belongs to two X- check and two Z-check generators, thereby meeting the conditions of Definition 1, following from the fact that every edge of the fusion complex belongs to four cells.

[0054] The following is a correspondence between surface-code fusion complexes and objects of a fusion network: (1) Each vertex, v V, corresponds to a resource state containing one qubit for each edge incident to v ; (2) Each edge, e E, corresponds to a fusion measurement on multiple qubits between the two resource states associated with the vertices at the endpoints of the edge; thus (3) the 1-skeleton of the fusion complex, e.g., cubical fusion complex as in FIG. 2B) describes a fusion network graph, G (measurement graph). In some example embodiments, a given fusion complex contains additional information in that each cell, c C, corresponds to a check operator (e.g., syndrome graph, decoding graph). In the fusion-based quantum computing approach, a check operator corresponds to a Pauli operator that is a member of both the resource group, R, and the group generated by all the fusion measurements, F. That is, check operators are members of R F . As used here, the term "cell" refers to a three-dimensional cell.

[0055] In some example embodiments, fusion measurements correspond to Bell state measurements of XX and ZZ , and thus the check operators are either XX or ZZ measurements. In some example embodiments, mixed type measurements and checks can be implemented using Hadamards (or other local Cliffords) in the protocol. In some example embodiments, the terminology ^X-type^ and ^Z-type^ is used to refer to the different types of checks, or simply ^X checks^ and ^Z checks^. In some example embodiments, the cells are always bi-colorable. The two colors of cells correspond to the X-type and Z-type check operators. For each X cell (or Z cell), a check exists that takes the product of the XX (or ZZ) outcomes from the fusion on each edge contained in the cell. As each edge has exactly four incident faces, there are also exactly four cells meeting at each edge. In this way, each fusion outcome (either XX or ZZ) is supported in two check operators.

[0056] In some example embodiments, the entangled resource state at any given vertex v is a stabilizer state defined by the structure of the surrounding6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) cells and edges. The resource state can be represented as a surface code on the surface of a topological sphere. For example, with reference to the cubical cell complex of FIG. 2C, the resource state is defined on the surface of an octahedron. In one approach, to visualize the resource state, one can ^inflate^ the vertex in the fusion complex to create a volume. This volume has qubits on its vertices and bi- colorable faces that define the resource state stabilizers. Where the inflated volume intersects an X-type cell, the resource state has an X-type plaquette for its face. Likewise, where the inflated volume intersects a Z-type cell, the resource state has a Z-type plaquette for its face. Since the code is on the surface of a sphere, this representation of the stabilizers gives an overcomplete-generating set, in accordance with some example embodiments. For example, if there are n qubits in the resource state, then there are n + 2 plaquette stabilizers on the sphere (e.g., with one X-type and one Z-type plaquette being a product of the others).

[0057] In some example embodiments, a check operator exists for each cell of the fusion complex. In some example embodiments implementing FBQC-base schemes, the check group is defined as C = R F, where R is the stabilizer group of all resource states, and F is the fusion group. Further, a check exists wherever there is an element of R that exists in F. It can be verified that for each cell, c, of the complex, there exists an element of the resource group rc in R that is identical to an element of the fusion group, fc in F. If, for example, the cell is X-type, then rcis the product of the X-type stabilizer generator from each vertex of the cell, and fc is the product of the XX fusion measurement operator for each edge of the cell (and similarly for the Z-type checks).

[0058] In some example embodiments, since there are always four cells meeting at an edge, each fusion is part of exactly four checks, and since the cells are bi- colorable, there must always be two X-checks and two Z-checks. In this way, an error flipping a fusion measurement outcome will, therefore, result in exactly two flipped checks, the property of a surface code under a single qubit Pauli-X or Z error.

[0059] FIGs.3A-3C show different example representations of fusion complexes and resource states in formats, in accordance with some example embodiments. In some example embodiments, a tiling scheme can generate 3D cell complexes as D-6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) symbols and the D-symbols can be rendered suing various tools (e.g., Gavrog), and / or analyzed using other representations, such as ZX notation. At a high level, FIG. 3A shows fusion complexes 300 in cell complex format (fusion complexes 1 to 6; 1 is cubic, 2 is alternative cubic, etc.); FIG. 3B shows resource states 325 for the fusion complexes 1 to 6 in plaquette format; and FIG. 3C shows resource states 350 for each of the fusion complexes 1 to 6 in ZX format. Each fusion complex and resource state corresponds to the following dataset: 1. Cubic D-symbol: <13:1,1,1,1:4,3,4> Syndrome check degree, C: 12 Resource state size, R: 6 Pauli threshold, perasure; 12% Pauli threshold, pflip; 1% 2. Alternated-cubic D-symbol: <23:12,12,12,2:33,34,4> Syndrome check degree, C: 2x6 + 12 Resource state size, R: 12 Pauli threshold, perasure; 12% , 25% Pauli threshold, pflip; 1%, 2.9% 3. Uniform-10 D-symbol: <53:135,2345,145,123 5:34,34,44> Syndrome check degree, C: 2x10 Resource state size, R: 10 Pauli threshold, perasure; 14.5% Pauli threshold, pflip; 1.4% 4. 4-star D-symbol: <23 : 2,12,12,2 : 6, 24 ,4> Syndrome check degree, C: 24 Resource state size, R: 3x4 Pauli threshold, perasure; 6.9% Pauli threshold, pflip; 0.75% 5. Pyrochlore D-symbol: <43:1234,124,134,24:336,33,4> Syndrome check degree, C: 2x6 + 2x18 Resource state size, R: 4x6 Pauli threshold, perasure; 10% Pauli threshold, pflip; 0.78% 6. Cuboctahedral D-symbol: <83:12345 67 8,1234 578,24 68,358 7:333 33 43,244,4> Syndrome check degree, C: 4x3 + 12 + 24 Resource state size, R: 3x8 Pauli threshold, perasure; 12.7% Pauli threshold, pflip; 1.3%6224.007WO1 ^ PSIQuantum (PSIQ-548WO1)

[0060] In the above list of fusion complexes 1 to 6, their names are followed by their Delani-Dress Symbols (D-Symbols), which is followed the syndrome-graph check degrees C, which is followed by the resource-state size (number of qubits) R, which is followed by the corresponding Pauli thresholds, erasure and flip. Generally, the first two integers in the D-symbol denote the size and the dimension, respectively, where the dimension is always 3 in accordance with some example embodiments. Further, the n1× d1+ (. . .), notation means there are n1checks / resource-states with degree / size d1, etc., in the unit cell. Further, as illustrated by the alternated cubic fusion complex, the XX and ZZ syndrome graphs can have different thresholds. In some example embodiments, for each generated fusion complex, the complex can be characterized via bench-marking (e.g., Monte Carlo based error simulations) to generate the threshold data (e.g., perasure,pflip), which can be used to select a given fusion complex to be implemented on a given set of hardware (e.g., based on the given hardware^s operating characteristics, such as noise).

[0061] In FIG. 3A, the fusion complexes 1 to 6 are generated as cellular format, where vertices correspond to resource states, edges correspond to fusions, and cells correspond to check operators. The colors illustrated in FIG. 3A are for visual aid purposes and do not indicate of the cell being X or Z type (each cell in FIG. 3A is generated by the fusion tiler system 177 with additional data that can be utilized to identify vertices, edges, and cells, and the color is merely a visual aid generated by the fusion tiler system 177).

[0062] FIG. 3B shows depictions of the resource states for each of the fusion complexes 1 to 6, in accordance with some example embodiments. In FIG.3B, each resource state is depicted as a surface code on a closed surface in the plaquette representation. In FIG.3B, each vertex represents a qubit and each face of a given polyhedral (e.g., a topological space, a three-dimensional shape) represents a stabilizer of the resource state. The colors in FIG. 3B indicate two stabilizers, namely an X-type stabilizer or a Z-type stabilizer.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1)

[0063] FIG.3C shows ZX-diagram notation of the resource states for each of the fusion complexes 1 to 6 to further aid understanding of the fusion complexes and resource states. In the ZX format, quantum operations are represented as a network of green and red spiders (e.g., a green spider (Z-type) and red spider (X- type). In accordance with some embodiments, the spiders correspond to stabilizer state projections, with each n-port spider describing stabilizer generators on n qubits. In some example embodiments, the stabilizer generators described by Z (X)spiders are X n (Z n) and all pairwise Z 2 (X 2) operators. Multiple spiders canbe connected to form composite ZX diagrams. Each connection between two spiders corresponds to a bell state projection. For example, the two qubits i and j correspond to the two ports, which are identified via the projection ZiZj= XiXj= +1. In the example of FIG. 3C, the outer legs of the fusion complex shapes (unterminated legs) correspond to qubits of the state (e.g., resource state). It is appreciated that each ZX diagram is not unique; rather, there are many possible ZX diagrams that correspond to each resource state for the fusion complexes 1 to 6, and in FIG. 3C only one illustrative example is shown for each to aid understanding. Further note that with reference to the the 4-star, the full 4-star fusion network is made up of both green and red spiders, but only one is shown in FIG. 3C for brevity.

[0064] In some example embodiments, fusion complexes can be generated by a fusion tiling scheme, which uses a tiling generator (e.g., combinatorial tiling generator scheme), in a specific way to more efficiently generate fusion complexes for implementation in computing systems (e.g., fault tolerant quantum computing systems). Generally, a tiling is a regular Closure Finite Weak Topology-complex (CW complex), which realizes a simply connected d-dimensional manifold. Its hisms that respect the induced CW- filtration. In some embodiments, a tiling can be thought of as a partition of space, where each tile is glued face-to-face in such a way that the entire space is filled. This partition determines a group of symmetries that respect the tiling, such as reflections, rotations, and translations that map the tiles back to themselves, ensuring the new tiling is the same as the old one. In some embodiments,6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) equivalence of tilings depends on both their topological structure and their symmetry group.

[0065] In some example embodiments, a tiling generator can be implemented in the following method 400 of FIG. 4, in accordance with some example embodiments. At operation 405, the fusion tiler system 177 generates a valid regular D-symbol according to D-symbol criteria, such as locality and duality, or varied in a search (e.g., vary the symbol size or other parameters). For example, the symbol can be selected such that that the corresponding state (if it exists) has desirable properties like locality or self-duality. In some example embodiments, only the maximum symbol size is input and the other parameters are iterated in a search, as discussed in further detail below.

[0066] At operation 410, the fusion tiler system 177 checks whether the generated symbol of operation 405 is valid by determining if the symbol is locally Euclidean. If yes, the method proceeds to operation 415; otherwise, method 400 returns to operation 405 for new D-symbol generation. In some example embodiments, the fusion tiler system 177 determines whether the D-symbol isspherical, Euclidean, or hyperbolic (e.g., local flatness) as follows: Let ( , ) be aD-symbol with index set = { , , }. A function is then defined: byand setnumber ( , ) is calledthe curvature of the D-symbol ( , ). It is closely related to the orbifold Eulercharacteristic in the sense of [Th2], which for manifolds coincides with the usualEuler characteristic. It is noted that the sign of ( , ) distinguishes thecurvature of the metric for a realization of ( , ). More precisely: a D-symbol( , ) ishyperbolic, iff ( , ) < 0 holds,Euclidean, iff ( , ) = 0 holds,spherical, iff ( , ) > 0 holds and, additionally, for all ,and all the quantity6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) is a natural number and non-semi-good otherwise.

[0067] At operation 415, the fusion tiler system 177 constructs an orbifold and orbifold invariants to determine space group. If it corresponds to one of the known space groups (219 known space groups as of the year 2024), the method proceeds to operation 420; otherwise, the method returns to operation 405.

[0068] At operation 420, the fusion tiler system 177 enumerates over finite index subgroups using a fundamental group constructed according to Group Criteria (GC), which comprises GC Parameter 1 and GC Parameter 2.

[0069] Parameter 1: Let be an extended Schläfli symbol with basepoint . Up to isomorphism, there is a unique symbol with basepoint and a covering : with ( ) = such that for any other covering : and any ( ),there is a unique : through which factors: = with ( ) = .We call the universal cover of .

[0070] Parameter 2: Let be an extended Schläfli symbol with basepoint , and let be its universal cover with covering map . Then the fundamental group of is given by the subgroup of automorphisms of annihilated by , (, ): = { Aut ( ): = }.

[0071] Following the construction of the universal cover, the fundamentalgroup can be constructed explicitly as ( , ) = Stab ( ) / with a finitepresentation. If this symbol is covered by a tiling, then the fundamental group corresponds to a subgroup of the symmetries of that tiling. For example, only reflections that map between symmetry equivalent flags are themselves symmetries of the tiling. Correspondingly, these are exactly the reflections thatbelong to Stab ( ), in accordance with some example embodiments.

[0072] Continuing, at operation 425, the fusion tiler system 177 determines whether the next group is isomorphic to Z3(if not, the method fails and returns to operation 405).

[0073] At operation 430, the fusion tiler system 177 generates a cover (e.g., toroidal cover) of the symbol as quotient of universal cover by that subgroup. Generally, a covering of a -dimensional symbol by another -dimensional6224.007WO1 ^ PSIQuantum (PSIQ-548WO1)symbol is an -equivariant map : satisfyingfor all. The covering is an isomorphism if is bijective. In some embodiments, coverings correspond to maps between symbols that preserve the structure of the symbol, namely the labeled adjacency relations and the vertex labels.

[0074] At operation 435, the fusion tiler system 177 determines whether a valid symbol is generated. For example, the symbol can be determined to be valid based on whether the symbol is a valid fusion complex. In some example embodiments, if the symbol is valid then it is stored as a valid symbol (e.g., valid fusion complex). In some example embodiments, if the symbol is not valid the method 400 returns to operation to operation 420 (e.g., if there are no more subgroups isomorphic to Z3, then the method 400 is complete). Although the method 400 is discussed here as an example of a tiling scheme to generate candidate cell complexes, in some example embodiments other types of tiling schemes can likewise be implemented to form cell complexes that can include valid fusion complexes, so long as the conditions above are satisfied (e.g., each face being incident to four faces, definition 3). In some example embodiments, the tiling scheme is implemented to output a set of cell complexes as a set of D-symbols), which then can be analyzed and processed directly as data items themselves (e.g., filtering based on items in sections of a given D-symbol), and optionally be rendered using different rendering approaches and software that can render D-symbols and cell complexes (e.g., Gavrog software, Mathematica).

[0075] FIGA. 5 shows a flow diagram for a method 500 for automated generation of fusion complexes using the fusion tiler system 177, in accordance with some example embodiments. At operation 505, the fusion tiler system 177 receives a query (e.g., candidate cell complex query) describing search parameters for fusion tiling search, using a fusion tiler scheme, as discussed above (FIG. 4). In some example embodiments, the search parameters include a maximum D- symbol size (e.g., maximum Delany-Dress symbol size), such that the fusion tiler system 177 generates different candidate symbols (e.g., different cell complexes) up to the specified size. At operation 510, the fusion tiler system 177 generates cell complexes that correspond to valid fusion complexes (e.g., error tolerant quantum cell complex) . Further details of cell complex generation are described in further6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) detail with reference to FIG.6. At operation 515, the fusion tiler system 177 stores the valid fusion complex cell complex data in memory.

[0076] At operation 520, the fusion tiler system 177 generates threshold data for each of the fusion complexes. For example, the fusion tiler system 177 performs simulations (e.g., Monte Carlo simulations, Fault Tolerant Simulation based on error models) to identify thresholds for each fusion complex (e.g., perasure,pflip). At operation 525, the fusion tiler system 177 selects one of the valid fusion complexes, based on the threshold data generated at operation 520. The selected fusion complex can, for example, be based upon the physical error characteristics of the quantum information processing system (quantum object processing system 195) such that the thresholds and operating characteristics of the selected fusion complex are congruent with the physical characteristics (e.g., high enough thresholds to ensure errors from the physical systems are manageable). In some example embodiments, the error rates and physical characteristics of the physical system can be ascertained via testing and then a fusion complex having a high enough threshold can be selected based on the error rates and physical characteristics.

[0077] As discussed above, a given valid fusion cell complex can be implemented to perform checks and decoding operations (e.g., generate check structures or graphs, and decoding structures or graphs). In particular, one color of a cell corresponds to an X-type check operator, where the other color corresponds to a Z- type check operator. Further, (1) for each X cell a check exists that takes the product of the XX outcomes from the fusion on each edge contained in the cell, and likewise, (2) for each Z cell, a check exists that takes the product of the ZZ outcomes from the fusion on each edge contained in the cell.

[0078] Continuing, at operation 530, the quantum object processing system 195 generates measurements according to the selected fusion complex. Further, at operation 535, the quantum object processing system 195 (e.g., decoder system 188) performs decoding using the selected fusion complex. In some example embodiments, the decoding scheme can be varied, and different options can be implemented, such as Union Find (UF) based decoding, Minimum Weight Perfect Matching (MWPM), Belief Propagation (BP) based decoding, and so on.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1)

[0079] FIG. 6 shows a flow diagram of a method 600 for generating fusion complexes, in accordance with some example embodiments. The method 600 can be implemented as a sub-routine of operation 510 in which fusion complexes are generated. In the following, Delany-Dress symbols (e.g., Delaney symbols, D- symbols) describe the cell complexes (e.g., cells, multi-cell structures) that can be generated by a tiling scheme (e.g. method 400, FIG.4). Generally, a D-symbol (e.g., an extended Schläfli symbol) is a list of numbers enclosed by brackets, such as <1.1:23:12,12,12,2:33,34,4> , where the D-symbol fully specifies a periodic tiling using reflection symmetries. (It is noted that that D-symbol brackets are only used as brackets, or delimiters, and are not bras nor kets as discussed elsewhere). In D- symbols, there are four sections separated by colons (e.g., <1.1:23:12,12,12,2:3 3,34,4> includes the following sections: (A) '1.1', (B) '23', (C) '12,12,12,2' and (D) '3 3,3 4,4^), where each section can have one or more items (e.g., numbers) delimited by commas. Generally, the first section is a reference number section (e.g., for tracking purposes). The second section comprises size data. For instance, the first number in the second section is the size of the given D-symbol (^2^ in the example above). Further, the third section comprises chamber data. Further, the fourth section mij data (e.g., orbit data). In some example embodiments, the fusion tiler system 177 performs a search, up to a given input D-symbol size, by implementing method 400 with varied values (e.g., varying different values in the second, third, or fourth sections, up to the given input D-symbol size), and the fourth section can be used to identify valid fusion complexes from a set of generated candidate symbols as further discussed below.

[0080] At operation 605, the fusion tiler system 177 generates a plurality of candidate cell complexes (e.g., each cell complex denoted by a D-symbol, optionally with other data describing the cell complex, optional tracking, reference data, render data, time stamps, etc.). In some example embodiments, the tiling scheme implemented to generate the cell complexes can include the method 400, discussed above with reference to FIG. 4. In some example embodiments, in generating the candidate cell complexes, the tiling scheme is configured such that degenerate cells (e.g., raviolis) are enabled and not excluded from the varied outputs of the candidate cell complexes (e.g., different iterations and combinations to form the6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) various candidate cell complexes). As discussed above, degenerate cells comprise two faces (e.g., identical faces) having boundaries that are joined to form a volume (e.g., chamber). In some example example embodiments, the fusion tiler system 177 generates candidate cell complexes up to a limit of the D-symbol size specified at operation 505.

[0081] At operation 610, the fusion tiler system 177 filters the candidate set to identify valid fusion complexes. In some example embodiments, the candidate set is filtered to remove results that have edges with incident faces, f, where f is not four. That is, the cell complexes having edges with each edge having four incident faces are kept and the other cell complexes are filtered out (e.g., those with edges having three incident faces). In some example embodiments, the fusion tiler system 177 can filter the candidate cell complexes based on their D-symbols. For example, the last section of the D-symbol can be filtered to only keep those with edges having four incident faces. For example, the last item in the last section of the D-symbol can be one or more fours (e.g., ^ ... : , 4 >; or ^ ... : 4 ^ , 44>; or ^ ... : 4, 3, 4>^ to not be filtered out (e.g., the last item is one or more 4^s, shown in bold italic underline for emphasis). Further examples are included below.

[0082] At operation 615, the fusion tiler system 177 then stores the remaining cell complexes as valid fusion complexes.

[0083] The following are examples of inputs and outputs for the fusion tiling scheme: for maximum size of D=2, the following valid fusion complexes are generated: <1.1 : 13 : 1,1,1,1 : 4,3,4> <1.1 : 23 : 12,12,12,2 : 33,34,4> <1.1 : 23 : 2,12,12,2 : 6,24,4>

[0084] Further, for a maximum size of D=3, the following (non-exhaustive) valid fusion complexes can be generated: <1.1: 43 : 1234,1234,13 4,24 : 2233,264,4> <1.1: 43 : 1234,1234,13 4,24 : 2244,263,4> <1.1: 43 : 1234,1234,13 4,24 : 2233,283,4>6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) <1.1: 43 : 1234,1234,13 4,24 : 2244,443,4> <1.1: 43 : 1234,1234,13 4,24 : 2244,462,4> <1.1: 43 : 1234,1234,13 4,24 : 3344,442,4>

[0085] The above are illustrative examples, and it is appreciated that further fusion complexes can be generated in a similar manner. Further, in some example embodiments, the cell complex cells can be processed and analyzed using other example representations to identify valid fusion complexes. For example, valid fusion complexes can be analyzed using a dual picture, where cells are replaced with vertices and faces are replaced with edges. In this approach, in order to be a valid fusion complex (e.g., valid surface-code fusion complex), all faces in the dual cell complex must have four edges (or, equivalently, four vertices). In some example embodiments, the fusion tiler system 177 identifies the valid fusion complex by first looping over all cells in a unit cell of the dual complex, followed by looping over all faces of the cell, and only keeping the candidate cell complex if all dual faces have four vertices, in accordance with some example embodiments.

[0086] FIG. 7 illustrates a diagrammatic representation of a machine 700 (e.g., classical processing system 193) in the form of a computer system within which a set of instructions may be executed for causing the machine 700 to perform any one or more of the methodologies discussed herein, according to an example embodiment. Specifically, FIG. 7 shows a diagrammatic representation of the machine 700 in the example form of a computer system, within which instructions 716 (e.g., software, a program, an application, an applet, an app, or other executable code) for causing the machine 700 to perform any one or more of the methodologies discussed herein may be executed. For example, the instructions 716 may cause the machine 700 to execute operations of methods 400, 500, and 600. In this way, the instructions 716 transform a general, non-programmed machine into a particular machine 700 that is specially configured to carry out any one of the described and illustrated functions in the manner described herein.

[0087] In alternative embodiments, the machine 700 operates as a standalone device or may be coupled (e.g., networked) to other machines. In a networked deployment, the machine 700 may operate in the capacity of a server machine or6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) a client machine in a server-client network environment, or as a peer machine in a peer-to-peer (or distributed) network environment. The machine 700 may comprise, but not be limited to, a server computer, a client computer, a personal computer (PC), a tablet computer, a laptop computer, a netbook, a smart phone, a mobile device, a network router, a network switch, a network bridge, or any machine capable of executing the instructions 716, sequentially or otherwise, that specify actions to be taken by the machine 700. Further, while only a single machine 700 is illustrated, the term ^machine^ shall also be taken to include a collection of machines 700 that individually or jointly execute the instructions 716 to perform any one or more of the methodologies discussed herein.

[0088] The machine 700 includes processors 710, memory 730, and input / output (I / O) components 750 configured to communicate with each other such as via a bus 702. In an example embodiment, the processors 710 (e.g., a CPU, a reduced instruction set computing (RISC) processor, a complex instruction set computing (CISC) processor, a graphics processing unit (GPU), a digital signal processor (DSP), an ASIC), a radio-frequency integrated circuit (RFIC), another processor, or any suitable combination thereof) may include, for example, a processor 712 and a processor 714 that may execute the instructions 716. The term ^processor^ is intended to include multi-core processors 710 that may comprise two or more independent processors (sometimes referred to as ^cores^) that may execute instructions 716 contemporaneously. Although FIG. 7 shows multiple processors 710, the machine 700 may include a single processor with a single core, a single processor with multiple cores (e.g., a multi-core processor), multiple processors with a single core, multiple processors with multiple cores, or any combination thereof.

[0089] The memory 730 may include a main memory 732, a static memory 734, and a storage unit 736, all accessible to the processors 710 such as via the bus 702. The main memory 732, the static memory 734, and the storage unit 736 store the instructions 716 embodying any one or more of the methodologies or functions described herein. The instructions 716 may also reside, completely or partially, within the main memory 732, within the static memory 734, within machine storage medium 738 of the storage unit 736, within at least one of the processors6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) 710 (e.g., within the processor^s cache memory), or any suitable combination thereof, during execution thereof by the machine 700.

[0090] The I / O components 750 include components to receive input, provide output, produce output, transmit information, exchange information, capture measurements, and so on. The specific I / O components 750 that are included in a particular machine 700 will depend on the type of machine. For example, portable machines such as mobile phones will likely include a touch input device or other such input mechanisms, while a headless server machine will likely not include such a touch input device. It will be appreciated that the I / O components 750 may include many other components that are not shown in FIG.7. The I / O components 750 are grouped according to functionality merely for simplifying the following discussion and the grouping is in no way limiting. In various example embodiments, the I / O components 750 may include output components 752 and input components 754. The output components 752 may include visual components (e.g., a display such as a plasma display panel (PDP), a light emitting diode (LED) display, a liquid crystal display (LCD), a projector, or a cathode ray tube (CRT)), acoustic components (e.g., speakers), other signal generators, and so forth. The input components 754 may include alphanumeric input components (e.g., a keyboard, a touch screen configured to receive alphanumeric input, a photo- optical keyboard, or other alphanumeric input components), point-based input components (e.g., a mouse, a touchpad, a trackball, a joystick, a motion sensor, or another pointing instrument), tactile input components (e.g., a physical button, a touch screen that provides location and / or force of touches or touch gestures, or other tactile input components), audio input components (e.g., a microphone), and the like.

[0091] Communication may be implemented using a wide variety of technologies. The I / O components 750 may include communication components 764 operable to couple the machine 700 to a network 780 or devices 770 via a coupling 782 and a coupling 772, respectively. For example, the communication components 764 may include a network interface component or another suitable device to interface with the network 780. In further examples, the communication components 764 may include wired communication components, wireless6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) communication components, cellular communication components, and other communication components to provide communication via other modalities. The devices 770 may be another machine or any of a wide variety of peripheral devices (e.g., a peripheral device coupled via a universal serial bus (USB)). The various memories (e.g., 730, 732, 734, and / or memory of the processor(s) 710 and / or the storage unit 736) may store one or more sets of instructions 716 and data structures (e.g., software) embodying or utilized by any one or more of the methodologies or functions described herein. These instructions 716, when executed by the processor(s) 710, cause various operations to implement the disclosed embodiments.

[0092] Described implementations of the subject matter can include one or more features, alone or in combination as illustrated below by way of examples.

[0093] Example 1 is a method comprising: receiving, using one or more classical processors, a candidate cell complex query; generating a plurality of candidate cell complexes using a tiling scheme; generating an error tolerant quantum cell complex from the plurality of candidate cell complexes, the error tolerant quantum cell complex comprising a plurality of edges, each edge being incident to four faces; and storing the error tolerant quantum cell complex.

[0094] In Example 2, the subject matter of Example 1 includes, generating a plurality of error tolerant quantum cell complexes from the plurality of candidate cell complexes, wherein each edge in each error tolerant quantum cell complex has four incident faces.

[0095] In Example 3, the subject matter of Examples 1^2 includes, wherein the error tolerant quantum cell complex comprises a fusion complex.

[0096] In Example 4, the subject matter of Examples 1^3 includes, wherein the error tolerant quantum cell complex comprises a three dimensional cell complex comprising cells, vertices and edges, wherein the vertices correspond to resource state entanglements, and wherein the edges correspond to fusion measurements.

[0097] In Example 5, the subject matter of Examples 1^4 includes, wherein one or more of the plurality of candidate cell complexes comprise degenerate cells.

[0098] In Example 6, the subject matter of Example 5 includes, wherein a degenerate cell comprises a polyhedra having a first face and a second face,6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) wherein the first face has a plurality of first boundaries, wherein the second face has a plurality of second boundaries, and wherein the plurality of first boundaries are joined to the plurality of second boundaries.

[0099] In Example 7, the subject matter of Examples 1^6 includes, wherein generating the plurality of candidate cell complexes comprises: generating initial cell complexes; and filtering initial cell complexes to form the plurality of candidate cell complexes, wherein the initial cell complexes is filtered out such that only initial cell complexes having two faces having joined boundaries remain. [000100] In Example 8, the subject matter of Examples 1^7 includes, wherein the candidate cell complex query comprises a maximum size limit. [000101] In Example 9, the subject matter of Example 8 includes, wherein the maximum size limit comprises a Delaney symbol size limit. [000102] In Example 10, the subject matter of Example 9 includes, wherein the Delaney symbol size limit comprises a quantity of chamber classes in Barycentric based subdivisions. [000103] In Example 11, the subject matter of Examples 1^10 includes, wherein the tiling scheme comprises a combinatorial tiling scheme. [000104] In Example 12, the subject matter of Examples 1^11 includes, generating threshold data based on simulations using the error tolerant quantum cell complex. [000105] In Example 13, the subject matter of Example 12 includes, wherein the simulations comprise Monte Carlo based simulations. [000106] In Example 14, the subject matter of Examples 12^13 includes, wherein the error tolerant quantum cell complex is selected from a plurality of error tolerant quantum cell complexes based on the threshold data of the error tolerant quantum cell complex being congruent with physical hardware of a quantum information processing system. [000107] Example 18 is a tangible computer-readable medium storing a set of instructions, the set of instructions comprising one or more instructions that, when executed by one or more classical processors (e.g., hardware processors) of a device, cause the device to perform operations to implement of any of Examples 1^ 14.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) [000108] Example 19 is a system comprising one or more classical processors (e.g., hardware processors); and a memory storing instructions that when executed by the one or more processors cause the system to perform operations to implement of any of Examples 1^14. [000109] The foregoing disclosure provides illustration and description but is not intended to be exhaustive or to limit the implementations to the precise form disclosed. Modifications may be made in light of the above disclosure or may be acquired from practice of the implementations. As used herein, the term ^component^ is intended to be broadly construed as hardware, firmware, or a combination of hardware and software. It will be apparent that systems and / or methods described herein may be implemented in different forms of hardware, firmware, and / or a combination of hardware and software. The actual specialized control hardware or software code used to implement these systems and / or methods is not limiting of the implementations. Thus, the operation and behavior of the systems and / or methods are described herein without reference to specific software code - it being understood that software and hardware can be used to implement the systems and / or methods based on the description herein. As used herein, satisfying a threshold may, depending on the context, refer to a value being greater than the threshold, greater than or equal to the threshold, less than the threshold, less than or equal to the threshold, equal to the threshold, and / or the like, depending on the context. Although particular combinations of features are recited in the claims and / or disclosed in the specification, these combinations are not intended to limit the disclosure of various implementations. In fact, many of these features may be combined in ways not specifically recited in the claims and / or disclosed in the specification. [000110] Although each dependent claim listed below may directly depend on only one claim, the disclosure of various implementations includes each dependent claim in combination with every other claim in the claim set. No element, act, or instruction used herein should be construed as critical or essential unless explicitly described as such. Also, as used herein, the articles ^a^ and ^an^ are intended to include one or more items and may be used interchangeably with ^one or more.^ Further, as used herein, the article ^the^ is intended to include one or6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) more items referenced in connection with the article ^the^ and may be used interchangeably with ^the one or more.^ Furthermore, as used herein, the term ^set^ is intended to include one or more items (e.g., related items, unrelated items, a combination of related and unrelated items, and / or the like), and may be used interchangeably with ^one or more.^ Where only one item is intended, the phrase ^only one^ or similar language is used. Also, as used herein, the terms ^has,^ ^have,^ ^having,^ or the like are intended to be open-ended terms. Further, the phrase ^based on^ is intended to mean ^based, at least in part, on^ unless explicitly stated otherwise. Also, as used herein, the term ^or^ is intended to be inclusive when used in a series and may be used interchangeably with ^and / or,^ unless explicitly stated otherwise (e.g., if used in combination with ^either^ or ^only one of^).

Claims

6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) CLAIMS What is claimed is:

1. A method comprising: receiving, using one or more classical processors, a candidate cell complex query; generating a plurality of candidate cell complexes using a tiling scheme; generating an error tolerant quantum cell complex from the plurality of candidate cell complexes, the error tolerant quantum cell complex comprising a plurality of edges, each edge being incident to four faces; and storing the error tolerant quantum cell complex.

2. The method of claim 1, further comprising: generating a plurality of error tolerant quantum cell complexes from the plurality of candidate cell complexes, wherein each edge in each error tolerant quantum cell complex has four incident faces.

3. The method of claim 1, wherein the error tolerant quantum cell complex comprises a fusion complex.

4. The method of claim 1, wherein the error tolerant quantum cell complex comprises a three dimensional cell complex comprising cells, vertices and edges, wherein the vertices correspond to resource state entanglements, and wherein the edges correspond to fusion measurements.

5. The method of claim 1, wherein one or more of the plurality of candidate cell complexes comprise degenerate cells.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) 6. The method of claim 5, wherein a degenerate cell comprises a polyhedra having a first face and a second face, wherein the first face has a plurality of first boundaries, wherein the second face has a plurality of second boundaries, and wherein the plurality of first boundaries are joined to the plurality of second boundaries.

7. The method of claim 1, wherein generating the plurality of candidate cell complexes comprises: generating initial cell complexes; and filtering initial cell complexes to form the plurality of candidate cell complexes, wherein the initial cell complexes is filtered out such that only initial cell complexes having two faces having joined boundaries remain.

8. The method of claim 1, wherein the candidate cell complex query comprises a maximum size limit.

9. The method of claim 8, wherein the maximum size limit comprises a Delaney symbol size limit.

10. The method of claim 9, wherein the Delaney symbol size limit comprises a quantity of chamber classes in Barycentric based subdivisions.

11. The method of claim 1, wherein the tiling scheme comprises a combinatorial tiling scheme.

12. The method of claim 1, further comprising: generating threshold data based on simulations using the error tolerant quantum cell complex.

13. The method of claim 12, wherein the simulations comprise Monte Carlo based simulations.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) 14. The method of claim 12, wherein the error tolerant quantum cell complex is selected from a plurality of error tolerant quantum cell complexes based on the threshold data of the error tolerant quantum cell complex being congruent with physical hardware of a quantum information processing system.

15. A system comprising: one or more classical processors; and a memory storing instructions that when executed by the one or more classical processors cause the system to perform operations comprising: receiving a candidate cell complex query; generating a plurality of candidate cell complexes using a tiling scheme; generating an error tolerant quantum cell complex from the plurality of candidate cell complexes, the error tolerant quantum cell complex comprising a plurality of edges, each edge being incident to four faces; and storing the error tolerant quantum cell complex.

16. The system of claim 15, wherein the operations further comprise: generating a plurality of error tolerant quantum cell complexes from the plurality of candidate cell complexes, wherein each edge in each error tolerant quantum cell complex has four incident faces.

17. The system of claim 15, wherein the error tolerant quantum cell complex comprises a fusion complex.

18. The system of claim 15, wherein the error tolerant quantum cell complex comprises a three dimensional cell complex comprising cells, vertices and edges, wherein the vertices correspond to resource state entanglements, and wherein the edges correspond to fusion measurements.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) 19. The system of claim 15, wherein one or more of the plurality of candidate cell complexes comprise degenerate cells.

20. The system of claim 19, wherein a degenerate cell comprises a polyhedra having a first face and a second face, wherein the first face has a plurality of first boundaries, wherein the second face has a plurality of second boundaries, and wherein the plurality of first boundaries are joined to the plurality of second boundaries.

21. The system of claim 15, wherein generating the plurality of candidate cell complexes comprises: generating initial cell complexes; and filtering initial cell complexes to form the plurality of candidate cell complexes, wherein the initial cell complexes is filtered out such that only initial cell complexes having two faces having joined boundaries remain.

22. The system of claim 15, wherein the candidate cell complex query comprises a maximum size limit.

23. The system of claim 22, wherein the maximum size limit comprises a Delaney symbol size limit.

24. The system of claim 23, wherein the Delaney symbol size limit comprises a quantity of chamber classes in Barycentric based subdivisions.

25. The system of claim 15, wherein the tiling scheme comprises a combinatorial tiling scheme.

26. The system of claim 15, wherein the operations further comprise: generating threshold data based on simulations using the error tolerant quantum cell complex.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) 27. The system of claim 26, wherein the simulations comprise Monte Carlo based simulations.

28. The system of claim 26, wherein the error tolerant quantum cell complex is selected from a plurality of error tolerant quantum cell complexes based on the threshold data of the error tolerant quantum cell complex being congruent with physical hardware of a quantum information processing system.

29. A tangible computer-readable medium storing a set of instructions, the set of instructions comprising one or more instructions that, when executed by one or more classical processors of a device, cause the device to perform operations comprising: receiving a candidate cell complex query; generating a plurality of candidate cell complexes using a tiling scheme; generating an error tolerant quantum cell complex from the plurality of candidate cell complexes, the error tolerant quantum cell complex comprising a plurality of edges, each edge being incident to four faces; and storing the error tolerant quantum cell complex.

30. The tangible computer-readable medium of claim 29, wherein the operations further comprise: generating a plurality of error tolerant quantum cell complexes from the plurality of candidate cell complexes, wherein each edge in each error tolerant quantum cell complex has four incident faces.

31. The tangible computer-readable medium of claim 29, wherein the error tolerant quantum cell complex comprises a fusion complex.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) 32. The tangible computer-readable medium of claim 29, wherein the error tolerant quantum cell complex comprises a three dimensional cell complex comprising cells, vertices and edges, wherein the vertices correspond to resource state entanglements, and wherein the edges correspond to fusion measurements.

33. The tangible computer-readable medium of claim 29, wherein one or more of the plurality of candidate cell complexes comprise degenerate cells.

34. The tangible computer-readable medium of claim 33, wherein a degenerate cell comprises a polyhedra having a first face and a second face, wherein the first face has a plurality of first boundaries, wherein the second face has a plurality of second boundaries, and wherein the plurality of first boundaries are joined to the plurality of second boundaries.

35. The tangible computer-readable medium of claim 29, wherein generating the plurality of candidate cell complexes comprises: generating initial cell complexes; and filtering initial cell complexes to form the plurality of candidate cell complexes, wherein the initial cell complexes is filtered out such that only initial cell complexes having two faces having joined boundaries remain.

36. The tangible computer-readable medium of claim 29, wherein the candidate cell complex query comprises a maximum size limit.

37. The tangible computer-readable medium of claim 36, wherein the maximum size limit comprises a Delaney symbol size limit.

38. The tangible computer-readable medium of claim 37, wherein the Delaney symbol size limit comprises a quantity of chamber classes in Barycentric based subdivisions.6224.007WO1 ^ PSIQuantum (PSIQ-548WO1) 39. The tangible computer-readable medium of claim 29, wherein the tiling scheme comprises a combinatorial tiling scheme.

40. The tangible computer-readable medium of claim 29, wherein the operations further comprise: generating threshold data based on simulations using the error tolerant quantum cell complex.

41. The tangible computer-readable medium of claim 40, wherein the simulations comprise Monte Carlo based simulations.

42. The tangible computer-readable medium of claim 40, wherein the error tolerant quantum cell complex is selected from a plurality of error tolerant quantum cell complexes based on the threshold data of the error tolerant quantum cell complex being congruent with physical hardware of a quantum information processing system.

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