Non-Abelian Topological Order and Anyon Creation in Ion Trap Processors

Abstract

The ground state of a non-abelian topological order is prepared using physical qubits logically organized on a lattice formed from multiple sublattices. The physical qubits assigned to each vertex of each sublattice in a first subset of sublattices are entangled with the bracket of each sublattice. The bracket of each sublattice in the first subset is measured. The physical qubits assigned to each vertex of each sublattice in a second subset of sublattices are entangled with the bracket of each sublattice. The bracket of each sublattice in the second subset is measured. Based on the measurements of the brackets, the controller determines whether any of the brackets host an abelian topological order. In response to the determination that a bracket hosts an abelian topological order, the controller triggers a feedforward action on the bracket to generate the ground state of a non-abelian topological order.

Description

[Technical Field]

[0001] Cross-reference of related applications This application claims priority to U.S. application No. 18 / 635,403, filed on April 15, 2024, which in turn claims priority to U.S. application No. 63 / 496,547, filed on April 17, 2023, the contents of which are incorporated herein by reference in their entirety.

[0002] Various embodiments relate to the preparation and / or generation of ground states of non-abelian topological order using entangled quantum objects. For example, various embodiments relate to the generation or preparation of ground states of non-abelian topological order using physical qubits of quantum charge-coupled device (QCCD) based quantum processors. [Background technology]

[0003] Complex quantum computations require levels of precision unattainable by conventional quantum computers. For example, conventional quantum computers are limited by noise, for instance, due to imperfect control and noise in the gate operations between data qubits. Through effort, ingenuity, and innovation, many of the shortcomings of conventional systems have been overcome by developing solutions constructed according to embodiments of the present invention, many of which are described in detail herein. [Prior art documents] [Patent Documents]

[0004] [Patent Document 1] U.S. Patent Application No. 17 / 810,082 [Patent Document 2] U.S. Patent No. 11,037,776 [Patent Document 3] U.S. Patent Application No. 17 / 533,587 [Patent Document 4] U.S. Patent Application No. 63 / 500,710 [Overview of the project] [Means for solving the problem]

[0005] Various embodiments provide methods, systems, system controllers, etc., for preparing and / or generating ground states of non-abelian topological order using entangled quantum objects. For example, various embodiments provide methods for preparing and / or generating ground states of non-abelian topological order using quantum charge-coupled device (QCCD) based quantum processors.

[0006] According to a first embodiment, a method is provided for generating a ground state of a non-abelian topological order. In an exemplary embodiment, the method includes the step of confining a plurality of physical qubits within a confinement device. The plurality of physical qubits are logically organized on a lattice comprising a plurality of vertices and a plurality of plaquettes connected by edges. The lattice is formed from a plurality of sublattices. Each sublattice comprises three or more vertices of a plurality of vertices and one or more plaquettes of a plurality of plaquettes. The three or more vertices and one or more plaquettes of each sublattice in the plurality of sublattices are connected by edges to form a sublattice. The method further includes the step of causing entanglement between one or more plaque qubits of the plurality of physical qubits and one or more vertices of the plurality of physical qubits allocated to one or more vertices of each sublattice in the first subset of sublattices, and the step of causing measurement of one or more plaque qubits allocated to one or more plaquettes of each sublattice in the first subset of sublattices. The method further includes the steps of causing entanglement between one or more bracket qubits of a plurality of physical qubits assigned to one or more brackets of each sublattice in each sublattice of the second subset of the sublattice and three or more vertices of each sublattice of the second subset of the sublattice, and causing a measurement of one or more bracket qubits of each sublattice of the second subset of the sublattice.The method further includes the steps of determining whether any of the brackets in a plurality of sublattices hosts an abelian topological order based on measurements of one or more bracket qubits in each sublattice of a first subset and measurements of one or more bracket qubits in each sublattice of a second subset, and initiating a feedforward action to be performed on the pair of brackets to generate a ground state having a non-abelian topological order in response to the determination that at least a plurality of pairs of brackets in sublattices hosts an abelian topological order.

[0007] In an exemplary embodiment, one or more bracket qubits of each sublattice are in a product state with one or more bracket qubits of each sublattice of the first subset of sublattices, prior to the entanglement of the vertex qubits of each sublattice of the first subset of sublattices.

[0008] In an exemplary embodiment, the step of causing entanglement between one or more bracket qubits of each sublattice of a first subset of the sublattice and the vertex qubits of each sublattice of a first subset of the sublattice includes causing entanglement between a particular bracket qubit assigned to a particular bracket of each of the one or more brackets and each of three vertex qubits assigned to a vertex of each of three or more vertices connected to the particular bracket via an edge of the sublattice, and causing entanglement between another particular bracket qubit assigned to another particular bracket of each of the one or more brackets and three vertex qubits assigned to a vertex of each of three or more vertices connected to the particular bracket via an edge of the sublattice, wherein the three vertex qubits are also connected to another particular bracket via an edge of the sublattice.

[0009] In an exemplary embodiment, the steps of causing entanglement between a specific bracket qubit assigned to a particular bracket from among the one or more brackets and each of the three vertex qubits assigned to each of the three or more vertices and connected to the particular bracket, and causing entanglement between another specific bracket qubit assigned to another specific bracket from among the one or more brackets and each of the three vertex qubits assigned to each of the three or more vertices and connected to another specific bracket, are carried out as four 2-qubit gates.

[0010] In an exemplary embodiment, the method further includes the step of inducing entanglement of multiple brackets using a pair of three-operator non-Clifford interactions, prior to measuring one or more bracket qubits of each sublattice in a first subset of the sublattices.

[0011] In an exemplary embodiment, a pair of three-operator non-Clifford interactions is implemented as four two-qubit gates.

[0012] In an exemplary embodiment, the grid is a kagome grid.

[0013] In an exemplary embodiment, the step of causing a measurement of one or more bracket qubits among the bracket qubits of each sublattice includes the step of determining the quantum state of the bracket qubit.

[0014] In an exemplary embodiment, the step of causing a feedforward action to be performed on a pair of brackets includes the step of causing a conditional Z gate to be performed on the pair of brackets.

[0015] In an exemplary embodiment, the grid has periodic boundary conditions.

[0016] In another embodiment, a system is provided configured to generate a ground state having a non-abelian topological order. In an exemplary embodiment, the system includes a confinement device configured to confine a plurality of physical qubits, one or more manipulation sources configured to generate respective manipulation signals for the interaction of the plurality of physical qubits with each other, and a controller configured to control the operation of the confinement device and one or more manipulation sources. The controller is configured to control the operation of the confinement device and / or one or more manipulation sources so that the plurality of physical qubits are confined within the confinement device. The plurality of physical qubits are logically organized on a lattice including a plurality of vertices and a plurality of brackets connected by edges. The lattice is formed from a plurality of sublattices. Each sublattice includes three or more vertices of a plurality of vertices and one or more brackets of a plurality of brackets. The three or more vertices of each sublattice and the one or more brackets of each sublattice are connected by edges to form a sublattice. The controller is further configured to cause entanglement between one or more bracket qubits of a plurality of physical qubits assigned to one or more brackets of each sublattice in each sublattice of the first subset of the sublattice and vertex qubits of a plurality of physical qubits assigned to three or more vertices of each sublattice in the first subset of the sublattice, and to cause measurement of one or more bracket qubits assigned to one or more brackets of each sublattice in the first subset of the sublattice.The controller is further configured to cause entanglement between one or more bracket qubits of a plurality of physical qubits assigned to one or more brackets of each sublattice in each sublattice of the second subset of sublattices and between physical qubits of a plurality of physical qubits assigned to three or more vertices of each sublattice in the second subset of sublattices, and to cause measurements of one or more bracket qubits of each sublattice in the second subset of sublattices. The controller is further configured to determine, based on the measurements of one or more bracket qubits of each sublattice in the first subset and the measurements of one or more bracket qubits of each sublattice in the second subset, whether any of the brackets of the plurality of sublattices host abelian topological order, and in response to the determination that at least a pair of brackets of sublattices host abelian topological order, to cause the execution of a feedforward action performed on the pair of brackets to generate a ground state having non-abelian topological order.

[0017] In an exemplary embodiment, one or more bracket qubits of each sublattice are in a product state with one or more bracket qubits of each sublattice of the first subset of sublattices, prior to the entanglement of the vertex qubits of each sublattice of the first subset of sublattices.

[0018] In an exemplary embodiment, causing entanglement of the vertex qubits of each sub-lattice of the first subset of sub-lattices with one or more plaquette qubits of each sub-lattice of the first subset of sub-lattices includes causing entanglement between a particular plaquette qubit assigned to a particular plaquette of each of the one or more plaquettes and each of three or more vertex qubits assigned to vertices among the vertices each connected to the particular plaquette via a sub-lattice edge, and causing entanglement between another particular plaquette qubit assigned to another particular plaquette of each of the one or more plaquettes and each of three vertex qubits assigned to vertices among the vertices each connected to the particular plaquette via a sub-lattice edge, where the three vertex qubits are also connected via a sub-lattice edge to another particular plaquette.

[0019] In an exemplary embodiment, causing entanglement between a particular plaquette qubit assigned to a particular plaquette of each of the one or more plaquettes and each of three vertex qubits each assigned to a respective vertex of three or more vertices connected to the particular plaquette, and causing entanglement between another particular plaquette qubit assigned to another particular plaquette of each of the one or more plaquettes and each of three vertex qubits each assigned to a respective vertex of three or more vertices connected to the another particular plaquette is implemented as four two-qubit gates.

[0020] In an exemplary embodiment, the controller is further configured to execute causing entanglement of a plurality of plaquettes using a pair of three-operator non-Clifford interactions prior to measurement of one or more plaquette qubits of each sub-lattice of the first subset of sub-lattices.

[0021] In an exemplary embodiment, a pair of three-operator non-Clifford interactions is implemented as four two-qubit gates.

[0022] In an exemplary embodiment, the lattice is a kagome lattice.

[0023] In an exemplary embodiment, causing the measurement of a plaquette qubit among one or more plaquette qubits of each sublattice includes determining the quantum state of the plaquette qubit.

[0024] In an exemplary embodiment, causing the execution of a feed-forward action on a pair of plaquettes includes causing the execution of a conditional Z gate on the pair of plaquettes.

[0025] In an exemplary embodiment, the lattice has periodic boundary conditions.

[0026] In another embodiment, a controller is provided configured to control one or more components of a system configured to generate a ground state having a non-abelian topological order. In an exemplary embodiment, the controller includes at least one processing device and at least one memory. The at least one memory stores executable instructions configured, when executed by the at least one processing device, to cause the controller to control one or more components of the system to cause a confinement device of the system to confine a plurality of physical qubits by the confinement device. The plurality of physical qubits are logically organized on a lattice including a plurality of vertices and a plurality of brackets connected by edges. The lattice is formed from a plurality of sublattices. Each sublattice includes three or more vertices of a plurality of vertices and one or more brackets of a plurality of brackets. The three or more vertices of each sublattice and the one or more brackets of each sublattice are connected by edges to form a sublattice. The executable instructions are further configured to cause the controller to control one or more components of the system to cause entanglement between one or more bracket qubits of a plurality of physical qubits assigned to one or more brackets of each sublattice of each sublattice of the first subset of sublattices, and between one or more bracket qubits of a plurality of physical qubits assigned to three or more vertices of each sublattice of the first subset of sublattices, when executed by at least one processing device, and to cause measurement of one or more bracket qubits assigned to one or more brackets of each sublattice of the first subset of sublattices.The executable instruction is further configured, when executed by at least one processing device, to cause entanglement of one or more bracket qubits of a plurality of physical qubits assigned to one or more brackets of each sublattice in each sublattice of the second subset of sublattices, and of physical qubits of a plurality of physical qubits assigned to three or more vertices of each sublattice in the second subset of sublattices, and to cause a measurement of one or more bracket qubits of each sublattice in the second subset of sublattices. The executable instruction is further configured, when executed by at least one processing device, to cause a controller to determine, based on the measurement of one or more bracket qubits of each sublattice in the first subset and the measurement of one or more bracket qubits of each sublattice in the second subset, whether any of the brackets of the plurality of sublattices host an abelian topological order, and in response to the determination that at least one pair of brackets of sublattices hosts an abelian topological order, to cause the execution of a feedforward action to be performed on the pair of brackets to generate a ground state having a non-abelian topological order.

[0027] In an exemplary embodiment, one or more bracket qubits of each sublattice are in a product state with one or more bracket qubits of each sublattice of the first subset of sublattices, prior to the entanglement of the vertex qubits of each sublattice of the first subset of sublattices.

[0028] In an exemplary embodiment, causing entanglement between one or more bracket qubits of each sublattice of a first subset of the sublattice and the vertex qubits of each sublattice of a first subset of the sublattice includes causing entanglement between a specific bracket qubit assigned to a particular bracket among the one or more brackets and each of three vertex qubits assigned to a vertex of each of three or more vertices connected to the specific bracket via the edge of the sublattice, and causing entanglement between another specific bracket qubit assigned to another specific bracket among the one or more brackets and three vertex qubits assigned to a vertex of each of three or more vertices connected to the specific bracket via the edge of the sublattice, wherein the three vertex qubits are also connected to another specific bracket via the edge of the sublattice.

[0029] In an exemplary embodiment, causing entanglement between a specific bracket qubit assigned to a particular bracket among the one or more brackets and each of the three vertex qubits, each assigned to each of the three or more vertices and connected to the particular bracket, and causing entanglement between another specific bracket qubit assigned to another specific bracket among the one or more brackets and each of the three vertex qubits, each assigned to each of the three or more vertices and connected to another specific bracket, is carried out as four 2-qubit gates.

[0030] In an exemplary embodiment, the executable instruction is further configured to induce multiple bracket entanglement using a pair of three-operator non-Clifford interactions before the measurement of one or more bracket qubits of each sublattice of a first subset of the sublattice, when executed by at least one processing device.

[0031] In an exemplary embodiment, a pair of three-operator non-Clifford interactions is implemented as four two-qubit gates.

[0032] In an exemplary embodiment, the grid is a kagome grid.

[0033] In an exemplary embodiment, causing a measurement of one or more bracket qubits in each sublattice includes determining the quantum state of the bracket qubit.

[0034] In an exemplary embodiment, triggering the execution of a feedforward action on a pair of brackets includes causing a conditional Z-gate to be executed on the pair of brackets.

[0035] In an exemplary embodiment, the grid has periodic boundary conditions.

[0036] In yet another embodiment, a computer program product is provided. The computer program product includes at least one non-temporary memory medium for storing executable instructions. The executable instructions are configured to confine the plurality of physical qubits by the confinement device when executed by a processing device of a system controller which includes a confinement device configured to confine the plurality of physical qubits. The plurality of physical qubits are logically organized on a lattice which includes a plurality of vertices and a plurality of brackets connected by edges. The lattice is formed from a plurality of sublattices. Each sublattice includes three or more vertices from each of the plurality of vertices and one or more brackets from each of the plurality of brackets. The three or more vertices and one or more brackets of each of the sublattices in the plurality of sublattices are connected by edges to form the sublattice. The executable instructions are further configured to cause the controller, when executed by the processing device, to control the operation of the system's confinement device and / or one or more manipulators to cause entanglement between one or more bracket qubits of a plurality of physical qubits assigned to one or more brackets of each sublattice of the first subset of the sublattice and between vertex qubits of a plurality of physical qubits assigned to three or more vertices of each sublattice of the first subset of the sublattice, and to cause measurement of one or more bracket qubits assigned to one or more brackets of each sublattice of the first subset of the sublattice.The executable instructions are further configured to cause the controller, when executed by the processing device, to control the operation of the system's confinement device and / or one or more manipulators to cause entanglement of one or more bracket qubits among a plurality of physical qubits assigned to three or more vertices in each of the sublattices of the second subset of the sublattices, with respect to one or more bracket qubits among a plurality of physical qubits assigned to one or more bracket qubits in each of the sublattices of the second subset of the sublattices, and to cause measurement of one or more bracket qubits in each of the sublattices of the second subset of the sublattices. The executable instructions are further configured to cause a controller, when executed by at least one processing device, to determine whether any of the brackets in a plurality of sublattices host an abelian topological order, based on measurements of one or more bracket qubits in each sublattice of a first subset and one or more bracket qubits in each sublattice of a second subset, and to control the operation of the system's confinement device and / or one or more manipulators to cause the execution of a feedforward action performed on the pair of brackets to generate a ground state having a non-abelian topological order, in response to the determination that at least a plurality of pairs of brackets in sublattices host an abelian topological order.

[0037] In an exemplary embodiment, one or more bracket qubits of each sublattice are in a product state with one or more bracket qubits of each sublattice of the first subset of sublattices, prior to the entanglement of the vertex qubits of each sublattice of the first subset of sublattices.

[0038] In an exemplary embodiment, causing entanglement between one or more bracket qubits of each sublattice of a first subset of the sublattice and the vertex qubits of each sublattice of a first subset of the sublattice includes causing entanglement between a specific bracket qubit assigned to a particular bracket among the one or more brackets and each of three vertex qubits assigned to a vertex of each of three or more vertices connected to the specific bracket via the edge of the sublattice, and causing entanglement between another specific bracket qubit assigned to another specific bracket among the one or more brackets and three vertex qubits assigned to a vertex of each of three or more vertices connected to the specific bracket via the edge of the sublattice, wherein the three vertex qubits are also connected to another specific bracket via the edge of the sublattice.

[0039] In an exemplary embodiment, causing entanglement between a specific bracket qubit assigned to a particular bracket among the one or more brackets and each of the three vertex qubits, each assigned to each of the three or more vertices and connected to the particular bracket, and causing entanglement between another specific bracket qubit assigned to another specific bracket among the one or more brackets and each of the three vertex qubits, each assigned to each of the three or more vertices and connected to another specific bracket, is carried out as four 2-qubit gates.

[0040] In an exemplary embodiment, the executable instruction is further configured to cause the controller, when executed by the processing device, to control the operation of the system's confinement device and / or one or more operating sources to cause entanglement of multiple brackets using a pair of three-operator non-Clifford interactions, prior to the measurement of one or more bracket qubits of each sublattice of a first subset of the sublattice.

[0041] In an exemplary embodiment, a pair of three-operator non-Clifford interactions is implemented as four two-qubit gates.

[0042] In an exemplary embodiment, the grid is a kagome grid.

[0043] In an exemplary embodiment, causing a measurement of one or more bracket qubits in each sublattice includes determining the quantum state of the bracket qubit.

[0044] In an exemplary embodiment, triggering the execution of a feedforward action on a pair of brackets includes causing a conditional Z-gate to be executed on the pair of brackets.

[0045] In an exemplary embodiment, the grid has periodic boundary conditions.

[0046] Having given a general overview of the present invention, the attached drawings, which are not necessarily drawn to the correct scale, will be referenced from here on. [Brief explanation of the drawing]

[0047] [Figure 1] This is a block diagram of an exemplary system configured to generate a ground state of a non-abelian topological order, according to an exemplary embodiment. [Figure 2] This flowchart illustrates, for example, the processes, procedures, and / or actions performed by the system's controller to generate a ground state of a non-abelian topological order, according to an exemplary embodiment. [Figure 3] This figure shows a grid formed from multiple subgrids and the operation defined on the grid, according to an exemplary embodiment. [Figure 4] This figure schematically illustrates various processes and / or procedures for generating a ground state of a non-abelian topological order based on subsets of multiple sublattices, according to exemplary embodiments. [Figure 5] This is a schematic diagram of a gate reduction used to perform entanglement of physical qubits assigned to each vertex of each sublattice in a subset of sublattices, with respect to each sublattice bracket, according to an exemplary embodiment. [Figure 6] This figure shows the result of generating a ground state of a non-abelian topological order according to an exemplary embodiment. [Figure 7] This is a schematic diagram of an exemplary controller for a system that includes a quantum object confinement device configured to confine a quantum object therein, according to an exemplary embodiment. [Figure 8] This is a schematic diagram of an exemplary computing entity of a system including a quantum object confinement device that may be used in exemplary embodiments. [Modes for carrying out the invention]

[0048] The present invention will be more fully described below with reference to the accompanying drawings showing embodiments that are part of but not all of the present invention. Furthermore, the present invention may be carried out in many different forms and should not be construed as being limited to the embodiments described herein, but rather these embodiments are provided to satisfy any legal requirements that this disclosure may have. The term “or” (also written as “ / ”) is used herein in both disjunctive and conjunctive senses unless otherwise indicated. The terms “illustrative” and “exemplary” are used to mean examples and do not indicate a level of quality. The terms “generally” and “approximately” mean, unless otherwise indicated, within applicable engineering and / or manufacturing tolerances, and / or within the user’s measurement capabilities. Throughout, similar numbers refer to similar elements.

[0049] In various scenarios, quantum objects are confined by quantum object confinement devices (also referred to herein as confinement devices). In various embodiments, quantum objects include ions, atoms, ionic molecules, molecular molecules, and / or multipolar molecules, quantum dots, quantum particles, groups thereof, crystals, and / or combinations thereof (e.g., an ionic crystal containing two or more ions). In exemplary embodiments where the quantum object is an ion and / or an ionic crystal, the confinement device is an ion trap, such as a surface ion trap or a Paul ion trap. In various other embodiments, the confinement device is a device configured to confine a quantum object.

[0050] In various embodiments, quantum objects confined by a confinement device are used to perform experiments, controlled quantum state evolution, quantum computation, and the like. In various embodiments, quantum objects are transported between various locations defined at least partially by the confinement device and / or the system including the confinement device. For example, a confined quantum object is a physical qubit of a quantum processor including a confinement device.

[0051] In various embodiments, physical qubits are logically organized based on a lattice formed from multiple sublattices. In various embodiments, the sublattices are separable from each other with respect to at least some operations performed on the lattice. For example, with respect to at least one operation defined on the lattice, at least one operation acts only on physical qubits assigned to one site of a single sublattice, and thus the sublattices are separable with respect to at least one operation. Therefore, with respect to the portion of a circuit performed on the lattice that includes only at least one operation for which the sublattices are separable, the sublattices may be treated as independent lattices. In an exemplary embodiment, the multiple sublattices include three sublattices.

[0052] In various embodiments, the grid is a Kagome grid. The Kagome grid is related to trihexagonal tiling, one of eleven uniform tilings of regular polygons in the Euclidean plane. The trihexagonal tiling consists of equilateral triangles and regular hexagons arranged such that each hexagon is surrounded by a triangle and vice versa. For example, two hexagons and two triangles alternately surround each vertex, and their edges form an infinite arrangement of lines. The Kagome grid consists of the vertices and edges of the trihexagonal tiling. For example, the grid sites of the Kagome grid are the vertices of the trihexagonal tiling, and these vertices are connected to each other by the edges of the trihexagonal tiling. In various embodiments where the grid is a Kagome grid, the sub-grid is a triangular grid.

[0053] In this specification, the term “logically organized” is used to indicate that the operational relationships between multiple physical qubits are determined based on a lattice and behavior defined on the lattice, and / or configured to conform to the lattice and behavior defined on the lattice. In at least one embodiment, physical qubits among multiple physical qubits forming a lattice may move and / or transport independently of one another. Thus, in this specification, the term “logically organized” is used to clarify that the multiple physical qubits forming a lattice do not need to be physically organized based on the lattice. In particular, the interactions between physical qubits in a lattice are governed, organized, and / or determined based on the lattice and behavior defined on the lattice. In various embodiments, the mobility of physical qubits allows for any physical rearrangement of physical qubits so that the interaction of any physical qubits is feasible.

[0054] Complex quantum computations require levels of precision unattainable in conventional quantum computers due to imperfect control and noise in gate operations between data qubits, for example. Proposed methods of fault-tolerant quantum computing involve performing quantum computations on logic qubits logically organized based on selected quantum error correction (QEC) codes. Conventional quantum error correction generally involves extracting syndromes that include the interaction of ancilla qubits with data qubits, defined by the QEC codes. However, if not performed carefully, such interactions between ancilla qubits and data qubits can catastrophically spread errors, leading to logic errors that would otherwise have been correctable given their initial weights. Therefore, there is a technical problem regarding how to perform quantum computations at a level of precision sufficient to perform complex calculations.

[0055] Various embodiments provide technical solutions to such technical problems. For example, various embodiments provide the implementation of fault-tolerant quantum computing and / or fault-tolerant quantum error correction using topological quantum computing. Topological quantum computing performs quantum computation using a phase of matter called topological order. Topological order is a manifestation of long-range quantum entanglement of multiple quantum objects, such as physical qubits. For example, a concentration of entanglement of underlying physical qubits forms a quasiparticle called an anyon. For example, an anyon is an excitation of topological order (similar to how phonons are excitations of motional modes of matter, for example).

[0056] Various embodiments provide the preparation and / or generation of ground states of non-abelian topological orders. A non-abelian topological order is a type of topological order that has non-abelian (e.g., non-commutative) properties. As a result of the non-abelian nature of such states of matter, non-abelian anyons "remember" their respective histories. For example, the non-commutative nature of a non-abelian topological order means that performing operation A on a non-abelian anyon and then operation B will result in a different outcome than performing operation B on a non-abelian anyon and then operation A. These features of non-abelian topological orders are expected to enable the execution of fault-tolerant quantum computing using non-abelian anyons and / or various states of non-abelian topological orders.

[0057] Classical simulations cannot simulate systems and / or matter exhibiting non-abelian topological order. Therefore, in order to test whether these expectations of fault-tolerant computing using non-abelian topological order are realized, non-abelian topological order states must be generated and experimentally investigated. However, there are no conventional experimentally successful techniques in the art for preparing and / or generating non-abelian topological order states. Thus, technical problems exist regarding the preparation and / or generation of non-abelian topological order states. Furthermore, technical problems exist regarding determining whether and / or how topologically protected quantum computing can provide more accurate quantum computation.

[0058] Various embodiments provide technical solutions to these technical problems. For example, various embodiments provide methods, systems, controllers for systems, and computer program products for configuring controllers for systems for preparing and / or generating states of non-abelian topological order. For example, in various embodiments, the ground state of the non-abelian topological order may be generated using multiple physical qubits of a quantum processor, such as a QCCD-based processor. The non-abelian topological order may then be manipulated and / or interacted to provide various excited states and / or various (other) ground states of the non-abelian topological order.

[0059] For example, due to the limited number of physical qubits available to a typical quantum processor (e.g., generally less than 50), various embodiments implement methods for generating ground states of non-abelian topological order using a reduced number of physical qubits. For example, conventional theoretical techniques for generating ground states of non-abelian topological order may require the use of more physical qubits than are available to a typical quantum processor. Various embodiments reduce the number of physical qubits required to prepare and / or generate ground states of non-abelian topological order by performing a sequence of entanglement and measurement operations on a first subset of sublattices, and then reinitializing and reusing one or more of the physical qubits used to perform the sequence of entanglement and measurement operations on the first subset of sublattices in order to perform the sequence of entanglement and measurement operations on a second subset of sublattices. In some embodiments, three or more subsets of a sublattice are used (for example, three or more subsets of a sublattice are used, and the maximum number of subsets of a sublattice is set by the number of sublattices such that none of the subsets of the sublattice are empty and the intersection of each pair of subsets of the sublattice is empty).

[0060] In another example, various embodiments reduce the number of gates executed to prepare and / or generate the ground state of a non-abelian topological order, thereby reducing the depth and time required to execute the quantum circuit that prepares and / or generates the ground state of the non-abelian topological order. For example, physical qubits have a finite coherence time. Therefore, having very deep quantum circuits (e.g., quantum circuits containing a large number of gates) can negatively impact the fidelity to which the ground state of a non-abelian topological order can be prepared and / or generated. For example, the usual theoretical technique for generating the ground state of a non-abelian topological order involves executing 108 2-qubit gates. However, according to an exemplary embodiment, the ground state of a non-abelian topological order is generated using only 78 2-qubit gates.

[0061] Therefore, various embodiments provide technical solutions to technical problems relating to the preparation and generation of ground states of non-abelian topological orders. Thus, various embodiments provide technical improvements to the field of fault-tolerant and / or topologically protected quantum computing.

[0062] Exemplary systems configured to generate non-abelian topological order states Various embodiments provide systems configured to generate non-abelian topological order states (e.g., one or more ground states which may be manipulated and / or interacted to result in one or more excited states). In various embodiments, the system includes a plurality of physical qubits. For example, in various embodiments, the system includes a confinement device that confines a plurality of quantum objects used as physical qubits. For example, in various embodiments, the system is a QCCD-based quantum computing system, such as a QCCD-based processor. These systems may be configured for use in performing fault-tolerant and / or topologically protected quantum computing, etc. In various embodiments, the system is configured to create and / or utilize non-abelian topological order and anyons. An exemplary QCCD-based quantum computing system is then disclosed.

[0063] Various embodiments provide a system 100 including a quantum object confinement device 50 (also referred to herein as a confinement operation), as shown in Figure 1. The confinement device 50 is configured to confine a plurality of quantum objects such that each quantum state of the quantum objects may be manipulated, evolved in a controlled manner (for example, according to a quantum circuit).

[0064] For example, quantum operation functions (such as one-qubit quantum gates, two-qubit quantum gates, initialization, read, and / or measurement operations) may be performed on quantum objects located within quantum operation locations defined by the confinement device 50 and / or the system 100 including the confinement device. For example, the confinement device 50 is configured to maintain one or more quantum objects in the quantum operation locations so that quantum operations may be performed on one or more quantum objects. For example, quantum objects confined by the confinement device 50 are used as physical qubits in the system 100.

[0065] In various embodiments, the system 100 including the confinement device 50 includes one or more operating sources 64 (e.g., 64A, 64B, 64C) configured to provide operating signals (e.g., laser beams and / or pulses, microwave signals, etc.) so that they interact with one or more quantum objects located at each quantum operation site. In various embodiments, the system 100 including the confinement device 50 includes one or more magnetic field sources 70 (e.g., 70A, 70B) configured to provide a controlled magnetic field and / or magnetic field gradient at the quantum operation site for use when performing one or more quantum operations on one or more quantum objects located at the quantum operation site. In various embodiments, the system 100 including the confinement device 50 includes an optical collection system 80 configured to collect and / or detect light and / or photons emitted (e.g., fluorescent) by one or more quantum objects located at each quantum operation site.

[0066] In exemplary embodiments, the system 100, including the confinement device 50, is a quantum charge-coupled device (QCCD) based quantum computer and / or includes a QCCD-based quantum computer. For example, one or more of the quantum objects confined by the confinement device 50 may be used as physical qubits in the quantum computer.

[0067] In various embodiments, system 100 includes a computing entity 10 and a quantum computer 110. In various embodiments, the quantum computer 110 includes a controller 30 and a quantum processor 115. In various embodiments, the quantum processor 115 includes a cryogenic and / or vacuum chamber 40 surrounding a confinement device 50, one or more operating sources 64 (e.g., 64A, 64B, 64C), one or more voltage sources 90, one or more magnetic field sources 70 (e.g., 70A, 70B), an optical collection system 80, and the like. In various embodiments, the controller 30 is configured to control the operation of the operating sources 64, voltage sources 90, magnetic field sources 70, vacuum system and / or cryogenic cooling system (not shown), etc. (e.g., to control one or more drivers configured to cause the operation). In various embodiments, the controller 30 is configured to receive signals (e.g., electrical signals) generated and provided by the optical collection system 80.

[0068] In exemplary embodiments, one or more manipulators 64 may include one or more lasers (e.g., optical lasers, microwave sources and / or masers) or other manipulators. In various embodiments, one or more manipulators 64 are configured to manipulate and / or induce the development of controlled quantum states of one or more quantum objects confined by the confinement device 50. For example, a first manipulator 64A is configured to generate and / or provide a first manipulator signal, a second manipulator 64B is configured to generate and / or provide a second manipulator signal, and the first and second manipulator signals are configured to perform one or more quantum operations (e.g., a one-qubit gate, a two-qubit gate, cooling, initialization, read / measurement) on the quantum objects confined by the confinement device 50.

[0069] In an exemplary embodiment, one or more operating sources 64 each provide operating signals (e.g., laser beams, etc.) to one or more portions of the confinement device 50 (e.g., quantum operating locations) via corresponding beampathing systems 66 (e.g., 66A, 66B, 66C). In various embodiments, at least one beampathing system 66 includes modulators configured to modulate the operating signals provided to the confinement device 50 via the beampathing system 66. In various embodiments, the operating sources 64 of the quantum computer 110, the active components of the beampathing systems 66 (e.g., modulators, etc.), and / or other components are controlled by a controller 30.

[0070] In various embodiments, the confinement device 50 is an ion trap such as a surface ion trap or a Paul ion trap. In various embodiments, the quantum object is an ion, an atom, an ionic crystal and / or group, an atomic crystal and / or group, an ionic molecule, a molecular molecule, and / or a multipolar molecule, a quantum dot, a quantum particle, a group thereof, a crystal, and / or combination thereof (e.g., an ionic crystal). In various embodiments, the confinement device 50 is a suitable confinement device for confining the quantum object of the embodiment. In various embodiments, the confinement device 50 is similar to the confinement devices and / or ion traps disclosed in U.S. Patent Application No. 17 / 810,082 filed June 30, 2022, U.S. Patent No. 11,037,776 issued June 15, 2021, U.S. Patent Application No. 17 / 533,587 filed November 23, 2021, and / or U.S. Patent Application No. 63 / 500,710 filed May 8, 2023.

[0071] In various embodiments, the quantum computer 110 includes one or more voltage sources 90. For example, the voltage sources may be arbitrary waveform generators (AWGs), digital-to-analog converters (DACs), direct digital synthesizers (DDSs), and / or other voltage signal generators. For example, the voltage sources 90 may include a plurality of control voltage drivers and / or voltage sources, and / or at least one RF driver and / or voltage source. In exemplary embodiments, the voltage sources 90 may be electrically coupled to corresponding potential generating elements (e.g., control electrodes and / or RF electrodes) of the confinement device 50.

[0072] In various embodiments, the voltage signal generated by the voltage source 90 is filtered before being applied to the potential generating elements of the confinement device 50 (e.g., control electrodes and / or RF electrodes). In an exemplary embodiment, the system 100 includes a filter configured to filter the voltage signal generated by the voltage source 90 and applied to the electrodes of the confinement device 50.

[0073] In various embodiments, the quantum computer 110 includes one or more magnetic field sources 70 (e.g., 70A, 70B). For example, the magnetic field sources may be an internal magnetic field source 70A located inside the cryogenic and / or vacuum chamber 40, and / or an external magnetic field source 70B located outside the cryogenic and / or vacuum chamber 40. In various embodiments, the magnetic field sources 70 include permanent magnets, Helmholtz coils, electromagnets, and the like. In various embodiments, the magnetic field sources 70 are configured to generate a magnetic field and / or magnetic field gradient having a specific magnitude and a specific magnetic field direction at one or more locations determined by the confinement device 50.

[0074] In various embodiments, the quantum computer 110 includes an optical collection system 80 configured to collect and / or detect photons (e.g., stimulated emission) generated by quantum objects placed at each quantum operation location (e.g., during read / measure operations). The optical collection system 80 may include one or more optical elements (e.g., lenses, mirrors, waveguides, optical fiber cables, etc.) and one or more photodetectors. In various embodiments, the photodetectors may be photodiodes, photomultiplier tubes, charge-coupled device (CCD) sensors, complementary metal-oxide-semiconductor (CMOS) sensors, micro-electromechanical system (MEMS) sensors, and / or other photodetectors that are highly sensitive to light of the expected fluorescence wavelength of the quantum objects. In various embodiments, the detectors may communicate electronically with the controller 30 via one or more A / D converters 725 (see Figure 7), etc.

[0075] In various embodiments, the computing entity 10 is configured to allow a user to provide input to the quantum computer 110 (for example, through the user interface of the computing entity 10) and to receive, view, etc., outputs from the quantum computer 110. For example, a user may operate the (classical and / or semiconductor-based) computing entity 10 to generate quantum programs and / or circuits, and the computing entity 10 may provide quantum programs and / or circuits and / or compiled versions thereof to the controller 30 of the quantum computer 110. The computing entity 10 may communicate with the controller 30 of the quantum computer 110 via one or more wired or wireless networks 20 and / or via direct wired and / or wireless communication. In exemplary embodiments, the computing entity 10 may translate, configure, format, etc., information / data, quantum computing algorithms (e.g., quantum circuits), etc., into a computing language, executable instructions, command set, etc., that the controller 30 can understand, execute, and / or implement.

[0076] In various embodiments, the controller 30 is configured to control the operation of a voltage source 90, a magnetic field source 70, a cryogenic system and / or vacuum system that controls the temperature and / or pressure in the cryogenic and / or vacuum chamber 40, an operating source 64, and / or other systems that control various environmental conditions (e.g., temperature, pressure, etc.) in the cryogenic and / or vacuum chamber 40, which are configured to manipulate and / or induce controlled development of quantum states of one or more quantum objects in the confinement device 50, and / or to read and / or measure the quantum (e.g., qubit) states of one or more quantum objects in the confinement device 50. For example, the controller 30 may induce controlled development of quantum states of one or more quantum objects confined by the confinement device 50 in order to execute quantum circuits and / or algorithms. For example, the controller 30 may induce entanglement of one or more quantum objects confined by the confinement device 50 in order to prepare and / or generate ground states or excited states of a non-abelian topological order. For example, the controller 30 may read and / or detect the quantum state of one or more quantum objects in the confinement device 50 at one or more points in time during the execution of the quantum circuit. In various embodiments, the quantum objects confined by the confinement device are used as qubits in the quantum computer 110.

[0077] Exemplary preparation and / or generation of ground states of non-abelian topological order In various embodiments, a lattice is defined that includes multiple vertices connected by their respective edges. In an exemplary embodiment, the lattice is a Kagome lattice. The physical qubits of the quantum processor 115 are assigned to each vertex of the lattice to logically organize the physical qubits based on the lattice. The vertices of the lattice are colored to define multiple sublattices that are separable with respect to at least one operation defined on the lattice. For example, in an exemplary embodiment, the lattice is colored in three colors.

[0078] In various embodiments, a lattice defines multiple brackets. Generally, a bracket is the smallest closed loop enclosing the region between four lattice sites. For example, when the lattice is a Kagome lattice, physical qubits assigned lattice sites at the centers of the lattice's hexagons represent the brackets of the lattice. Each of the brackets is made into a product state. A product state is a multi-qubit state that can be represented as a simple combination of various independent states of a number of 1-qubit systems. For example, a product state is a multi-qubit state that can be decomposed into a tensor product of the individual qubit states.

[0079] The vertex qubits of a first sublattice (e.g., the physical qubits assigned to each vertex) are also entangled with the brackets of the first sublattice (e.g., the physical qubits representing the brackets). The first sublattice is an element of the first subset of sublattices of the lattice. In various embodiments, the vertex qubits of each sublattice in the first subset of sublattices (e.g., the physical qubits assigned to each vertex) are also entangled with the brackets of the same sublattice (e.g., the physical qubits representing the brackets). In an exemplary embodiment, for each sublattice in the first subset of sublattices, the vertex qubits of the sublattice in the first subset of sublattices (e.g., the physical qubits assigned to each vertex) are also entangled with the brackets of the same sublattice (e.g., the physical qubits representing the brackets). The qubits representing the brackets may then be entangled using a pair of three-operator non-Clifford interactions. The three-operator non-Clifford interactions act on each of the sublattices (e.g., sublattices in both the first subset and the second subset of sublattices). Then, the bracket of the sublattice of the first subset of the sublattice is measured.

[0080] In various embodiments, after a measurement of the sublattice brackets of a first subset of the sublattice (for example, after performing a measurement operation on the physical qubits assigned to the sublattice brackets of the first subset of the sublattice), some of the physical qubits assigned to the sublattice brackets of the first subset of the sublattice may be reinitialized (for example, the quantum state of those physical qubits may be returned to a known initial quantum state such as |0>), reallocated to each vertex of the sublattice of a second subset of the sublattice, and / or used as auxiliary qubits.

[0081] In various embodiments, the first subset of the sublattice and the second subset of the sublattice do not overlap. In other words, the intersection of the first subset of the sublattice and the second subset of the sublattice is empty. In the embodiment shown in Figure 5, the first subset of the sublattice includes the green sublattice and the blue sublattice (for example, the sublattice colored by the first and second colors), and the second subset of the sublattice includes the red sublattice (for example, the remaining sublattice colored by the third color).

[0082] In various embodiments, the vertex qubits of each sublattice in a second subset of sublattices (e.g., the physical qubits assigned to each vertex) are also entangled with the brackets of the same sublattice (e.g., the physical qubits representing the brackets). In an exemplary embodiment, for each sublattice in a second subset of sublattices, the vertex qubits of that sublattice (e.g., the physical qubits assigned to each vertex) are also entangled with the brackets of the same sublattice (e.g., the physical qubits representing the brackets). The qubits representing the brackets may then be entangled using a pair of three-operator non-Clifford interactions. The brackets of the sublattices in a second subset of sublattices are then measured.

[0083] Based on measurements of the brackets, it is determined whether any of the brackets host an abelian anyon. For example, entanglement of physical qubits may result in topological order, and measurements may provide indication, for each bracket, whether the topological order present in that bracket is abelian or non-abelian. For instance, a bracket may be identified as hosting an abelian anyon and / or having an abelian topological order present therein.

[0084] For brackets determined to host and / or have an abelian topological order, a feedforward action is performed to make the topological order present in those brackets non-abelian. In various embodiments, the feedforward action is performed on one or more pairs of brackets determined to host and / or have an abelian topological order. In an exemplary embodiment, the brackets of the pair of brackets are sublattices. In an exemplary embodiment, performing a feedforward action on a pair of brackets includes performing a controlled Z gate on the pair of brackets. In an exemplary embodiment, the ground state resulting from the non-abelian topological order is a D4 topological order.

[0085] Figure 2 provides a flowchart illustrating the various processes, procedures, etc., performed by the controller 30 to cause the preparation and / or generation of a non-abelian topological order using the physical qubits of the quantum processor 115.

[0086] Starting from step 202, the grid is defined. In various embodiments, the grid is a Kagome grid. In various embodiments, boundary conditions for the grid are defined. For example, in various embodiments, the grid has periodic boundary conditions. For example, the grid may be formed on a torus. For example, the grid may be a two-dimensional grid housed and / or formed on a toroidal manifold.

[0087] In various embodiments, the grid may be colored to define a set of subgrids. For example, each site of the grid (e.g., each vertex and each bracket of the grid) may be assigned a color. In various embodiments, the intersection of each subgrid with each of the other subgrids is empty, and the union of all subgrids is the grid. In an exemplary embodiment, the grid is colored with three colors so that three subgrids are defined. In various other embodiments, fewer or more colors may be used, and / or fewer or more subgrids may be defined. In an exemplary embodiment where the grid is a Kagome grid, each subgrid is a triangular grid.

[0088] In exemplary embodiments, a first subset and a second subset of the sublattice are defined. For example, in the embodiment shown in Figure 4, the first subset of the sublattice includes a first sublattice of a first color (e.g., blue) and a second sublattice of a second color (e.g., green), and the second subset of the sublattice includes a third sublattice of a third color (e.g., red). In various embodiments, the intersection of the first subset of the sublattice and the second subset of the sublattice is empty, and the union of the subsets of the sublattice (for example, the union of the first subset and the second subset when only two subsets exist) is the lattice. In various embodiments, two or more non-empty subsets of the sublattice may be defined.

[0089] Figure 3 shows an exemplary grid 300 including multiple grid sites 302 colored with a first color (e.g., blue site 302B), a second color (e.g., green site 302G), and a third color (e.g., red site 302R). The grid sites 302 are connected by edges 304 to define the grid. Each edge 304 connects two grid sites 302 having different colors. Two grid sites 302 that are endpoints of a particular edge 304 are said to be directly connected to each other through the grid. In various embodiments, the grid has a periodic boundary condition such that when the grid is folded to align its boundaries, a boundary site 306A aligns with and / or overlaps with, for example, a boundary site 306B, as an exemplary scenario in which a portion of the grid enclosed by a dashed line 308 is implemented. In other words, as a result of the periodic boundary condition, boundary site 306A is connected (e.g., via edges 304 of the grid) to the blue and green sites to which boundary site 306B is connected.

[0090] As shown in Figure 3, various operations are defined on the grid 300. Each bracket supports one star operator 310 and two triangular operators 315A and 315B. For example, the star operator 310 is an operator that works on 12 sites. As shown, the star operator 310 is an operator

number

number

number

number

[0091] The Hamiltonian of qubits arranged on a periodic kagome lattice that produces a non-abelian topological order is:

number

number

number

[0092] Furthermore, for each color of the lattice and each direction along the torus (e.g., vertical, horizontal), two logical string operators are defined. The logical Z operator is the product of local Pauli Z gates that act on all lattice sites of each color in the selected direction (e.g., physical qubits assigned to the lattice sites). For example, Figure 3 shows the logical horizontal green Z operator. GH And the logical vertical blue Z operator Z BV This is shown. Figure 3 shows the logical vertical blue X operator χ. BV The logical X operator shown is the product of a local Pauli X gate and a series of controlled Z gates that connect every green vertex to the previous red vertex on a path in the selected direction.

[0093] Continuing with Figure 2, in step 204, the brackets are prepared in a product state. For example, the controller 30 assigns physical qubits to each of the brackets and to the vertices of the sublattice of at least the first subset of the sublattice. Notably, the physical qubits are logically organized into the brackets and vertices of the lattice and do not need to be physically organized within the confinement device to physically form the lattice. In other words, the lattice defines the functional and / or operational relationships between the physical qubits, rather than the physical relationships between the physical qubits.

[0094] In various embodiments, physical qubits are assigned to each of the brackets. For example, each physical qubit is assigned to each bracket of each sublattice (including sublattices of the first subset of the sublattice and sublattices of the second subset of the sublattice). Furthermore, physical qubits are assigned to the vertices of the sublattices of the first subset of the sublattice.

[0095] Each physical qubit assigned to a bracket is prepared in a product state. A product state is a multi-qubit state that can be represented as a simple combination of various independent states of a large number of single-qubit systems. For example, a product state is a multi-qubit state that can be decomposed into a tensor product of the individual qubit states. For example, a bracket is:

number

number

[0096] In various embodiments, the physical qubits assigned to the vertices of the sublattice in a first subset of the sublattice are prepared in a known initial state. For example, in an exemplary embodiment, each of the physical qubits assigned to the vertices of the sublattice in a first subset of the sublattice is prepared in a known initial state such as |0>.

[0097] In step 206, the controller causes entanglement of the vertex qubits of a first subset of the sublattice with the bracket qubits. In the exemplary embodiment shown in Figure 4, the first subset of the sublattice includes a first sublattice (e.g., the blue sublattice) and a second sublattice (e.g., the green sublattice). With respect to each sublattice in the first subset of the sublattice, the vertices of the sublattice are also entangled with the brackets of the sublattice. In various embodiments, the vertices of the sublattice are also entangled with the brackets of the sublattice that are (directly) connected to the vertices of the sublattice via the edges of the sublattice.

[0098] For example, as shown in panel 402 of Figure 4, with respect to a first sublattice (e.g., the blue sublattice), physical qubits are assigned to the vertices of the first sublattice (e.g., v1-v9). Physical qubits are also assigned to the brackets of the first sublattice (e.g., p1-p3). The physical qubits assigned to the brackets of the first sublattice (also referred to herein as the first bracket qubits) are entangled by an entanglement gate with the physical qubits assigned to the vertices of the first sublattice (also referred to herein as the first vertex qubits). In an exemplary embodiment, the entanglement gate is a controlled NOT gate (e.g., a CNOT gate). In another exemplary embodiment, the entanglement gate is a controlled Z gate (e.g., a CZ gate). For example, as shown in panel 402 of Figure 4, the first bracket qubit p1 is coupled and / or directly connected to the first vertex qubits v1, v2, v3, v7, v8, and v9 via each edge of the first sublattice. Thus, as shown in Figure 5, an entanglement gate is performed between the first bracket qubit and each of the first vertex qubits v1, v2, v3, v7, v8, and v9.

[0099] Figure 5 shows a simplified notation of circuit 505, which demonstrates the execution of entanglement gates between the first bracket qubits and each of the first vertex qubits v1, v2, v3, v7, v8, and v9 after executing a one-qubit Hadamard gate on each of the first bracket qubits and each of the first vertex qubits v1, v2, v3, v7, v8, and v9, the execution of entanglement gates between the second bracket qubits and each of the first vertex qubits (e.g., the first vertex qubits v1, v2, v3, v4, v5, and v6) that are (directly) connected to the second bracket qubits via the respective edges of the first sublattice, and the execution of entanglement gates between the third bracket qubits (e.g., the first vertex qubits v4, v5, v6, v7, v8, and v9) after executing a one-qubit Hadamard gate on each of the first bracket qubits and each of the first vertex qubits v1, v2, v3, v7, v8, and v9 v9, the execution of entanglement gates between the second bracket qubits and each of the first vertex qubits (e.g., the first vertex qubits v1, v2, v3, v4, v5, and v6) that are (directly) connected to the third bracket qubit via the respective edges of the first sublattice.

[0100] Figure 5 also shows the circuit 510 in expanded notation which is equivalent to the circuit 505 in reduced notation. Furthermore, due to the fact that each of the bracket qubits (including the first bracket qubit) is prepared in a product state (e.g., |+>product state) and each of the first vertex qubits is prepared in a known initial quantum state (e.g., |0>1 qubit state), the execution of the set of entanglement gates between the first bracket qubit p1 and the first vertex qubits v1, v2, v3, v7, v8, and v9 is equivalent to the execution of the entanglement gates between the first bracket qubit p1 and the first vertex qubits v1, v2, and v3, as well as the entanglement gates between the first bracket qubit p1 and the first bracket qubits p3 (other first bracket qubits that are (directly) connected to the first vertex qubits v7, v8, and v9 via the edges of the sublattice). This is shown in the reduced circuit 515 in Figure 5. Therefore, instead of executing six 2-qubit gates for each bracket in the sublattice to entangle the first vertex qubit with the first bracket qubit, only four 2-qubit gates need to be executed for each bracket in the sublattice. For each sublattice, this results in a reduction of six fewer 2-qubit gates per sublattice.

[0101] In an exemplary embodiment, the first subset of the sublattice includes a second sublattice (e.g., a green sublattice). In such an embodiment, physical qubits assigned to the vertices of the second sublattice (which are elements of the first subset of the sublattice) are entangled with physical qubits assigned to the brackets of the second sublattice. The physical qubits assigned to the brackets of the second sublattice (also referred herein as second bracket qubits) are entangled by an entanglement gate with physical qubits assigned to the vertices of the second sublattice (also referred herein as second vertex qubits). In an exemplary embodiment, the entanglement gate is a controlled NOT gate (e.g., a CNOT gate). In another exemplary embodiment, the entanglement gate is a controlled Z gate (e.g., a CZ gate). For example, as shown in panel 404 of Figure 4, a second bracket qubit p1 is connected and / or directly to the second vertex qubits v1, v2, v3, v7, v8, and v9 via each edge of the second sublattice; a second bracket qubit p2 is connected and / or directly to the second vertex qubits v1, v2, v3, v4, v5, and v6 via each edge of the second sublattice; and a second bracket qubit p3 is connected and / or directly to the second vertex qubits v4, v5, v6, v7, v8, and v9 via each edge of the second sublattice. In various embodiments, an entanglement gate is performed to entangle the second bracket qubits with the second vertex qubits that are connected and / or directly to the second bracket qubits (for example, via the edges of the second sublattice). In various embodiments, the entanglement gate may be executed according to the circuit 505 in reduced notation / circuit 510 in expanded notation, or according to the reduced circuit 515.For example, when the entanglement of the second bracket qubit with the second vertex qubit is performed according to circuit 515, instead of performing six 2-qubit gates for each bracket of the second sublattice to entangle the second vertex qubit with the second bracket qubit, only four 2-qubit gates need to be performed for each bracket of the second sublattice.

[0102] In various embodiments, entanglement of two physical qubits is performed by transporting the physical qubits closer to each other and causing an interaction between the two physical qubits. For example, in an exemplary embodiment where the physical qubits are ions confined in a confinement device such as a surface ion trap, a voltage source 90 configured to generate and provide a voltage signal to be applied to the electrodes of the surface ion trap is operated to generate and provide a voltage signal that transports the two physical qubits to a common location and places them in a common potential well. One or more operating limits 64 are operated to generate and provide their respective operating signals so that the operating signals are incident on the two physical qubits and cause an interaction between the two physical qubits resulting in the execution of the entanglement gate. In various embodiments, the operating signals may be a laser beam / pulse, a microwave field / pulse, a magnetic field gradient, etc. After the execution of the entanglement gate, the two physical qubits may be transported away from each other.

[0103] For example, the controller 30 may control the operation of the confinement device 50 and the operation of the operation source 64, such that with respect to each sublattice of the first subset of the sublattice, an entanglement gate is performed on two physical qubits according to the reduced notation circuit 505 / expanded notation circuit 510 or the reduced circuit 515 (for example, by controlling the operation of the voltage source 90 to generate and provide voltage signals to be applied to the electrodes of the confinement device in an exemplary embodiment).

[0104] Returning to Figure 2, in step 208, the controller 30 causes entanglement of the lattice's bracket qubits. For example, in various embodiments, the brackets are entangled with each other by a three-operator non-Clifford interaction. In exemplary embodiments, the number of operators in the non-Clifford interaction may be selected based on the shape of the lattice and / or the number of sublattices forming the lattice.

[0105] For example, as shown in panel 410 of Figure 4, a three-operator non-Clifford interaction is performed on each triangle of the bracket qubits containing the brackets of each sublattice. In various embodiments, the three-operator non-Clifford interaction is performed as shown in panel 412 of Figure 4,

number

[0106] In various embodiments, the entanglement of two triangles of a bracket qubit is performed as a parallelogram entanglement of a bracket qubit, as shown in panel 416. The parallelogram entanglement of a bracket qubit is performed using four 2-qubit gates, as shown by circuit portion 418. The execution of circuit portion 418 has the same effect as the execution of circuit portion 414, but requires two fewer 2-qubit gates. For example, in an exemplary embodiment, circuit portion 418 is performed for each of the nine parallelograms shown in panel 420. Thus, bracket qubit entanglement using circuit portion 418 results in a reduction of 18 2-qubit gates from the number of 2-qubit gates required to prepare the ground state of the non-abelian topological order compared to when circuit portion 414 is used.

[0107] As described above, in various embodiments, the controller 30 controls the operation of the confinement device (for example, by controlling the operation of a voltage source 90 that generates and provides a voltage signal to be applied to the electrodes of the confinement device) and the operation of the operation source 64, causing two physical qubits to be transported together in close proximity to each other (for example, into a common potential well that confines both qubits), and causing one or more operation signals to be injected into the two qubits to execute a two-qubit gate. For example, the controller 30 may control the operation and operation signals of the confinement device 50 to transport a suitable pair of physical qubits in close proximity to each other and cause a suitable operation signal to be injected into that pair to execute a two-qubit gate, for example, according to a sequence in circuit section 418.

[0108] Continuing with Figure 2, in step 210, the controller causes the bracket qubits of the sublattices of the first subset of the sublattice to be measured. For example, in the example shown where the first subset of the sublattice includes a first sublattice (e.g., the blue sublattice) and a second sublattice (e.g., the green sublattice), the first and second bracket qubits are measured. For example, the qubit state of each of the bracket qubits of the sublattices that are elements of the first subset of the sublattice is determined by performing a measurement operation on those bracket qubits. In an exemplary embodiment, the measurement of the bracket qubits is performed on the X basis and results in a |+> state determination (e.g., a measurement result of +1) or a |-> state determination (e.g., a measurement result of -1). In an exemplary embodiment, a one-qubit Hadamard gate is executed on the qubits before the measurement operation is performed in order to perform the measurement of the bracket qubits on the X basis. For example, panel 430 of Figure 4 shows the measurement of the bracket qubits of the sublattice of the first subset of the sublattice.

[0109] In various embodiments, performing a measurement operation on a physical qubit (e.g., a physical qubit of a QCCD-based quantum processor 115) involves injecting a read operation signal into the physical qubit that resonates with or is near resonant to a particular transition of the physical qubit. When the wave function of the physical qubit collapses into a first qubit state of the physical qubit, the qubit fluoresces in response to the read operation signal injected into it, and when the wave function of the physical qubit collapses into a second qubit state of the physical qubit, the qubit does not fluoresce in response to the read operation signal injected into it. An optical acquisition system 80 detects and / or captures all fluorescence generated in response to the read operation signal injected into the physical qubit. The optical acquisition system 80 provides the sensor signal to a controller 30 (e.g., via an A / D converter 725). The controller 30 processes the sensor signal to determine which qubit state the wave function of the physical qubit has collapsed into.

[0110] In various embodiments, after measurement of the bracket qubits of a sublattice in a first subset of the sublattice, some of the physical qubits previously assigned to one or more brackets of the sublattice in the first subset of the sublattice are reinitialized (to a known initial state, e.g., |0>) and reallocated to the respective vertices of the sublattice in a second subset of the sublattice. In the exemplary embodiment shown in Figure 4, the second subset of the sublattice includes a third sublattice (e.g., a red sublattice). For example, in the exemplary embodiment, one or more of the physical qubits used as the first or second bracket qubits are reinitialized and reallocated as third vertex qubits (also referred herein as physical qubits assigned to the vertices of the third sublattice) and / or auxiliary qubits.

[0111] In step 212 of Figure 2, the physical qubits assigned to the vertices of the sublattice in the second subset of the sublattice are entangled with the bracket qubits of each sublattice.

[0112] Continuing with the exemplary embodiment shown in Figure 4, the second subset of the sublattice includes a third sublattice (e.g., the red sublattice). As shown in Panel 406, the physical qubits assigned to the vertices of the third sublattice (which are elements of the second subset of the sublattice) are entangled with the physical qubits assigned to the brackets of the third sublattice. The physical qubits assigned to the brackets of the third sublattice (also referred to herein as the third bracket qubits) are entangled by an entanglement gate with the physical qubits assigned to the vertices of the third sublattice (also referred to herein as the third vertex qubits). In an exemplary embodiment, the entanglement gate is a controlled NOT gate (e.g., a CNOT gate). In another exemplary embodiment, the entanglement gate is a controlled Z gate (e.g., a CZ gate).

[0113] For example, a third bracket qubit p1 is connected and / or directly to the third vertex qubits v1, v2, v3, v7, v8, and v9 via each edge of the third sublattice; a third bracket qubit p2 is connected and / or directly to the third vertex qubits v1, v2, v3, v4, v5, and v6 via each edge of the third sublattice; and a third bracket qubit p3 is connected and / or directly to the third vertex qubits v4, v5, v6, v7, v8, and v9 via each edge of the third sublattice. In various embodiments, an entanglement gate is performed to entangle the third bracket qubits with the third vertex qubits that are connected and / or directly to the third bracket qubits (for example, via the edges of the third sublattice). In various embodiments, the entanglement gate may be performed according to the reduced notation circuit 505 / expanded notation circuit 510, or according to the reduced circuit 515. For example, when the entanglement of a third bracket qubit with a third vertex qubit is performed according to the reduced circuit 515, only four two-qubit gates need to be performed for each bracket of the third sublattice, instead of performing six two-qubit gates for each bracket of the third sublattice to entangle the third vertex qubit with the third bracket qubit.

[0114] For example, the controller 30 may control the operation of the confinement device 50 and the operation of the operation source 64, so as to cause the execution of an entanglement gate on a pair of physical qubits with respect to each sublattice of a second subset of the sublattice, according to the circuit 505 in reduced notation / circuit 510 in expanded notation, or according to the reduced circuit 515 (for example, by controlling the operation of the voltage source 90 to generate and provide voltage signals to be applied to the electrodes of the confinement device in an exemplary embodiment).

[0115] Continuing with Figure 2, in step 214, the controller causes the bracket qubits of the sublattice of the second subset of the sublattice to be measured. For example, in the example shown, where the second subset of the sublattice includes a third sublattice (e.g., the red sublattice), the third bracket qubit is measured. For example, the qubit state of each of the bracket qubits of the sublattice that are elements of the second subset of the sublattice is determined by performing a measurement operation on those bracket qubits. In an exemplary embodiment, the measurement of the bracket qubits is performed on an X basis and results in a |+> state determination (e.g., a measurement result of +1) or a |-> state determination (e.g., a measurement result of -1). For example, panel 440 of Figure 4 shows the measurement of the bracket qubits of the sublattice of the second subset of the sublattice.

[0116] In various embodiments, measuring a physical qubit (for example, a physical qubit of a QCCD-based quantum processor 115) involves injecting a read operation signal into the physical qubit that resonates with or near-resonates with a particular transition of the physical qubit. When the wave function of the physical qubit collapses into a first qubit state of the physical qubit, the qubit fluoresces in response to the read operation signal injected into it, and when the wave function of the physical qubit collapses into a second qubit state of the physical qubit, the qubit does not fluoresce in response to the read operation signal injected into it. An optical acquisition system 80 detects and / or captures all fluorescence generated in response to the read operation signal injected into the physical qubit. The optical acquisition system 80 provides the sensor signal to a controller 30 (for example, via an A / D converter 725). The controller 30 processes the sensor signal to determine which qubit state the wave function of the physical qubit collapsed into.

[0117] In various embodiments, the controller 30 processes sensor signals to determine measurement results for each of the lattice's bracket qubits. Based on the measurement results, the controller 30 determines whether any of the brackets host abelian topological order. For example, the controller 30 determines whether any of the brackets in a plurality of sublattices host abelian topological order based on measurements of the lattice's brackets (e.g., the brackets of a sublattice in a first subset of the sublattice and the brackets of a sublattice in a second subset). For example, in an exemplary embodiment, a bracket is determined to host non-abelian topological order when its measurement yields a +1 result, and a bracket is determined to host abelian topological order when its measurement yields a -1 result.

[0118] In an exemplary embodiment, after measurement of the third bracket qubits, one or more of the third bracket qubits are reinitialized to be used as auxiliary qubits.

[0119] In step 216, in response to the determination that at least one pair of brackets hosts an abelian topological order, the controller 30 triggers the execution of a feedforward action performed on the pair of brackets to generate a ground state having a non-abelian topological order. For example, processing a sensor signal, the controller 30 may determine that a pair of brackets in a particular sublattice hosts a phase of material that is abelian topological order. In this case, the controller 30 may trigger the execution of a feedforward action on the pair of brackets to cause each of the brackets in the pair to host a phase of material that is non-abelian topological order. In various embodiments, the feedforward action includes the execution of one or more conditional Z gates on physical qubits assigned to the pair of brackets determined to host an abelian topological order. In various embodiments, the brackets of the pair of brackets are from the same sublattice (for example, both are from a first sublattice, both are from a second sublattice, or both are from a third sublattice).

[0120] As shown in panel 450 of Figure 4, in one or more of the sublattices, a number of pairs of brackets hosting abelian topological order may be identified, and a feedforward action such as a conditional Z gate may be performed on each pair to host a phase of non-abelian topological order in the entire lattice.

[0121] For example, the controller 30 may control the operation of the confinement device 50 (for example, by generating and providing voltage signals to be applied to the electrodes of the confinement device, and controlling the operation of the operating source 90) and the operation of the operating source 64, at least in part, based on the results of processing measurements of the bracket qubits, so as to trigger the execution of a conditional Z gate on the pair of physical qubits assigned to each pair of brackets.

[0122] In various embodiments, non-Abelian topological orders are prepared and / or generated in their ground states. Notably, non-Abelian topological orders may have multiple degenerate distinguishable ground states (e.g., distinguishable based on the results of their actions by various operators defined on a lattice). Non-Abelian topological orders may be manipulated to cause the execution of various quantum computations. For example, entanglements (e.g., anyons) may be generated and / or manipulated (e.g., braided) to cause the execution of (fault-tolerant) quantum computations. For example, non-Abelian anyons may be used as logical qubits of a quantum processor in which a non-Abelian topological order is prepared.

[0123] FIG. 6 provides the results of experimental measurements of the prepared ground states of the non-Abelian topological order shown in FIG. 4. For example, with respect to lattice 600, the expectation values of star operator 310(A S ) and each triangular operator 315A, 315B(B T ) and logical Z operator for each of the plaquettes 605 (e.g., 605A, 605B,..., 605N) are shown. As shown in FIG. 6, the ground state of the non-Abelian topological order is deterministically prepared with A S ≈ B T ≈ 1. Further, the ground state of the non-Abelian topological order is deterministically prepared using significantly fewer two qubits (e.g., without requiring post selection) compared to naive preparation of the non-Abelian topological order.

[0124] Technical Advantages Complex quantum computations require levels of precision unattainable in conventional quantum computers due to imperfect control and noise in gate operations between data qubits, for example. Proposed methods of fault-tolerant quantum computing involve performing quantum computations on logic qubits logically organized based on selected quantum error correction (QEC) codes. Conventional quantum error correction generally involves extracting syndromes that include the interaction of auxiliary qubits with data qubits, defined by the QEC codes. However, if not performed carefully, such interactions between auxiliary and data qubits can catastrophically spread errors, leading to logic errors that would otherwise have been correctable given their initial weights. Therefore, there is a technical problem regarding how to perform quantum computations at a level of precision sufficient to perform complex calculations.

[0125] Various embodiments provide technical solutions to such technical problems. For example, various embodiments provide the implementation of fault-tolerant quantum computing and / or fault-tolerant quantum error correction using topological quantum computing. Topological quantum computing performs quantum computation using topological order. Topological order is a manifestation of long-range quantum entanglement of multiple quantum objects, such as physical qubits. For example, a concentration of entanglement of underlying physical qubits forms a quasiparticle called an anyon. For example, an anyon is an excitation of topological order (similar to how phonons are excitations of the motion modes of matter, for example).

[0126] Various embodiments provide the preparation and / or generation of ground states of non-abelian topological orders. A non-abelian topological order is a type of topological order that has non-abelian (e.g., non-commutative) properties. As a result of the non-abelian nature of such states of matter, non-abelian anyons "remember" their respective histories. For example, the non-commutative nature of a non-abelian topological order means that performing operation A on a non-abelian anyon and then operation B will result in a different outcome than performing operation B on a non-abelian anyon and then operation A. These features of non-abelian topological orders are expected to enable the execution of fault-tolerant quantum computing using non-abelian anyons and / or various states of non-abelian topological orders.

[0127] Classical simulations cannot simulate systems and / or matter exhibiting non-abelian topological order. Therefore, in order to test whether these expectations of fault-tolerant computing using non-abelian topological order are realized, non-abelian topological order states must be generated and experimentally investigated. However, there are no conventional experimentally successful techniques in the art for preparing and / or generating non-abelian topological order states. Thus, technical problems exist regarding the preparation and / or generation of non-abelian topological order states. Furthermore, technical problems exist regarding determining whether and / or how topologically protected quantum computing can provide more accurate quantum computation.

[0128] Various embodiments provide technical solutions to these technical problems. For example, various embodiments provide methods, systems, controllers for systems, and computer program products for configuring controllers for systems for preparing and / or generating states of non-abelian topological order. For example, in various embodiments, the ground state of a non-abelian topological order may be generated using multiple physical qubits of a quantum processor, such as a QCCD-based processor. Figure 6 shows a measurement result of an exemplary successful generation of a ground state of a non-abelian topological order prepared by an exemplary embodiment. Once prepared, the non-abelian topological order may be manipulated and / or interacted to provide various excited states and / or various (other) ground states of the non-abelian topological order.

[0129] For example, due to the limited number of physical qubits available to a typical quantum processor (e.g., generally less than 50), various embodiments implement methods for generating ground states of non-abelian topological order using a reduced number of physical qubits. For example, conventional theoretical techniques for generating ground states of non-abelian topological order may require the use of more physical qubits than are available to a typical quantum processor. Various embodiments reduce the number of physical qubits required to prepare and / or generate ground states of non-abelian topological order by performing a sequence of entanglement and measurement operations on a first subset of sublattices, and then reinitializing and reusing one or more of the physical qubits used to perform the sequence of entanglement and measurement operations on the first subset of sublattices in order to perform the sequence of entanglement and measurement operations on a second subset of sublattices.

[0130] In another example, various embodiments reduce the number of gates executed to prepare and / or generate the ground state of a non-abelian topological order, thereby reducing the depth and time required to execute the quantum circuit that prepares and / or generates the ground state of the non-abelian topological order. For example, physical qubits have a finite coherence time. Therefore, having very deep quantum circuits (e.g., quantum circuits containing a large number of gates) can negatively impact the fidelity to which the ground state of a non-abelian topological order can be prepared and / or generated. For example, the usual theoretical technique for generating the ground state of a non-abelian topological order involves executing 108 2-qubit gates. However, according to an exemplary embodiment, the ground state of a non-abelian topological order is generated using only 78 2-qubit gates.

[0131] Therefore, various embodiments provide technical solutions to technical problems relating to the preparation and generation of ground states of non-abelian topological orders. Thus, various embodiments provide technical improvements to the field of fault-tolerant and / or topologically protected quantum computing.

[0132] Example Controller Various embodiments provide systems including a confinement device 50, systems including a confinement device 50, and / or methods for using them. In exemplary embodiments, the system is a quantum charge-coupled device (QCCD) based quantum computer 110 or other quantum computer. In various embodiments, the system (e.g., quantum computer 110) further includes a controller 30 configured to control various elements of the system. For example, the controller 30 is configured to control various elements of the system to prepare and / or generate a ground state of a non-abelian topological order, and / or to perform quantum computation using the non-abelian topological order and its anyons.

[0133] For example, the controller 30 may be configured to control a voltage source 90 configured to manipulate and / or induce a controlled evolution of the quantum state of one or more quantum objects (e.g., physical qubits) confined by the confinement device 50, and / or to read and / or measure the quantum state of one or more quantum objects confined by the confinement device, a cryogenic system and / or vacuum system for controlling the temperature and pressure in the cryogenic and / or vacuum chamber 40, an operating source 64 (e.g., 64A, 64B, 64C), active components of the beampath system 66 (e.g., 64A, 64B, 64C), a magnetic field source 70 (e.g., 70A, 70B), and / or other systems for controlling environmental conditions (e.g., temperature, humidity, pressure, magnetic field gradient, etc.) in the cryogenic and / or vacuum chamber 40.

[0134] As shown in Figure 7, in various embodiments, the controller 30 may include various controller elements, including one or more processing devices 705, memory 710, driver controller elements 715, communication interfaces 720, analog-to-digital converter elements 725, and so on. For example, one or more processing devices 705 may include one or more processing elements such as programmable logic devices (CPLDs), microprocessors, coprocessing entities, application-specific instruction-set processors (ASIPs), integrated circuits, application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), programmable logic arrays (PLAs), hardware accelerators, and other processing devices and / or circuits. The term "circuit" may refer to an entirely hardware embodiment or a combination of hardware and computer program products. In exemplary embodiments, one or more processing devices 705 of the controller 30 include and / or communicate with a clock. In various embodiments, this clock defines the system's clock cycle.

[0135] For example, memory 710 may include non-temporary memory such as volatile and / or non-volatile memory storage, such as one or more of the following: hard disk, ROM, PROM, EPROM, EEPROM, flash memory, MMC, SD memory card, memory stick, CBRAM, PRAM, FeRAM, RRAM, SONOS, racetrack memory, RAM, DRAM, SRAM, FPM DRAM, EDO DRAM, SDRAM, DDR SDRAM, DDR2 SDRAM, DDR3 SDRAM, RDRAM, RIMM, DIMM, SIMM, VRAM, cache memory, register memory, etc. In various embodiments, memory 710 may store qubit records corresponding to the logical and / or physical qubits of a quantum computer (e.g., in a qubit record datastore, qubit record database, qubit record table, etc.), calibration tables, executable queues, computer program code or executable instructions (e.g., one or more computer languages, special controller languages, etc.). In an exemplary embodiment, the execution of at least a portion of computer program code or executable instructions stored in memory 710 (for example, by processing device 705) causes the controller 30 to perform one or more steps, operations, processes, procedures, etc. described herein for controlling one or more components of the quantum computer 110 (e.g., voltage source 90, operation source 64, magnetic field source 70, etc.) to measure and / or read the quantum state of one or more quantum objects, causing a controlled evolution of the quantum state of one or more quantum objects. For example, the execution of at least a portion of computer program code or executable instructions stored in memory 710 (for example, by processing device 705) causes the controller 30 to perform one or more steps, operations, processes, procedures, etc. in the flowchart of Figure 2 in an exemplary embodiment.

[0136] In various embodiments, the driver controller element 715 may include one or more drivers and / or controller elements, each configured to control one or more drivers. In various embodiments, the driver controller element 715 may include drivers and / or driver controllers. For example, a driver controller may be configured to operate one or more corresponding drivers according to executable instructions, commands, etc., scheduled and executed by the controller 30 (e.g., by the processing device 705). In various embodiments, the driver controller element 715 may enable the controller 30 to operate the operating source 64. In various embodiments, drivers may be laser drivers, vacuum component drivers, drivers for controlling current and / or voltages applied to RF, control, and / or other electrodes (e.g., shim electrodes) used to maintain and / or control the confinement potential of the confinement device (and / or other drivers for providing driver action sequences and / or control signals to the potential generating elements of the confinement device), cryogenic and / or vacuum system component drivers, etc. For example, the driver may control and / or include a control and / or RF voltage driver and / or voltage source that provides voltage and / or electrical signals to the control electrodes and / or RF electrodes of the confinement device 50.

[0137] In various embodiments, the controller 30 includes means for transmitting and / or receiving signals from one or more detectors, such as photodetector components of the optical acquisition system 80 (e.g., cameras, MEMS cameras, CCD cameras, photodiodes, photomultiplier tubes, etc.). For example, the controller 30 may include one or more analog-to-digital converter elements 725 configured to receive signals from one or more detectors, photodetector components, calibration sensors, etc.

[0138] In various embodiments, the controller 30 may include a communication interface 720 for interfaceing with and / or communicating with one or more computing entities 10. For example, the controller 30 may include a communication interface 720 for receiving executable instructions, command sets, etc., from the computing entities 10 and providing the computing entities 10 with outputs received from the quantum processor 115 (e.g., via the optical collection system 80) and / or the results of processing the outputs (received from the quantum processor 115). In various embodiments, the computing entities 10 and the controller 30 may communicate directly via wired and / or wireless connections and / or via one or more wired and / or wireless networks 20.

[0139] Exemplary Computing Entity Figure 8 provides a descriptive schematic diagram representing an exemplary computing entity 10 that may be used in conjunction with embodiments of the present invention. In various embodiments, the computing entity 10 is configured to allow a user to provide input to a quantum computer 110 (for example, through a user interface of the computing entity 10) and to receive, display, analyze, and so on, outputs from the quantum computer 110.

[0140] As shown in Figure 8, the computing entity 10 may include an antenna 812, a (e.g., wireless) transmitter 804, a (e.g., wireless) receiver 806, and a processing device 808 that provides a signal to the transmitter 804 and receives a signal from the receiver 806, respectively.

[0141] The signals provided to the transmitter 804 and the signals received by the receiver 806 may include signaling information / data according to applicable wireless system radio interface standards for communication with various entities such as the controller 30 and other computing entities 10. In this regard, computing entity 10 may operate with one or more radio interface standards, communication protocols, modulation types, and access types. For example, computing entity 10 may be configured to receive and / or provide communications using wired data transmission protocols such as Fiber Optic Distributed Data Interface (FDDI), Digital Subscriber Line (DSL), Ethernet, Asynchronous Transfer Mode (ATM), Frame Relay, Data Over Cable Service Interface Specification (DOCSIS), or any other wired transmission protocol.Similarly, Computing Entity 10 supports General-Purpose Packet Radio Services (GPRS), Universal Mobile Communications System (UMTS), Code Division Multiple Access 2000 (CDMA2000), CDMA2000 1X (1xRTT), Wideband Code Division Multiple Access (WCDMA), Global System for Mobile Communications (GSM), Enhanced Data rates for GSM Evolution (EDGE), Time Division Synchronous Code Division Multiple Access (TD-SCDMA), Long-Term Evolution (LTE), Evolutionary Universal Terrestrial Radio Access Network (E-UTRAN), Evolution-Data Optimized (EVDO), High-Speed ​​Packet Access (HSPA), High-Speed ​​Downlink Packet Access (HSDPA), IEEE 802.11 (Wi-Fi), and Wi-Fi. It may be configured to communicate over a wireless external communication network using any of the following protocols: Direct, 802.16 (WiMAX), Ultra Wideband (UWB), Infrared (IR) protocol, Near Field Communication (NFC) protocol, Wibree, Bluetooth protocol, Wireless Universal Serial Bus (USB) protocol, and / or any other wireless protocol.Computing entity 10 may communicate using such protocols and standards, including Border Gateway Protocol (BGP), Dynamic Host Configuration Protocol (DHCP), Domain Name System (DNS), File Transfer Protocol (FTP), Hypertext Transfer Protocol (HTTP), HTTP, HTTP over TLS / SSL / Secure, Internet Message Access Protocol (IMAP), Network Time Protocol (NTP), Simple Mail Transfer Protocol (SMTP), Telnet, Transport Layer Security (TLS), Secure Sockets Layer (SSL), Internet Protocol (IP), Transmission Control Protocol (TCP), User Datagram Protocol (UDP), Datagram Congestion Control Protocol (DCCP), Stream Control Transmission Protocol (SCTP), and Hypertext Markup Language (HTML).

[0142] These communication standards and protocols allow the computing entity 10 to communicate with various other entities using concepts such as Unstructured Supplementary Service information / data (USSD), Short Message Service (SMS), Multimedia Messaging Service (MMS), Dual-Tone Multi-Frequency Signaling (DTMF), and / or Subscriber Identification Module Dialer (SIM Dialer). The computing entity 10 can also download changes, add-ons, and updates to its firmware, software (including executable instructions, applications, and program modules), and operating system, for example. In various embodiments, the computing entity 10 further includes one or more network interfaces 820 configured to communicate over one or more wired and / or wireless networks 20.

[0143] The computing entity 10 may also include user interface devices that include one or more user input / output interfaces (for example, a display 816 and / or speaker / speaker driver coupled to the processing device 808, as well as a touchscreen, keyboard, mouse, and / or microphone coupled to the processing device 808). For example, a user output interface may be configured to provide applications, browsers, user interfaces, interfaces, dashboards, screens, web pages, pages, and / or similar words used herein to be interchangeable, for the purpose of causing the display or audible presentation of information / data, and for interaction with them via one or more user input interfaces. A user input interface may include any of many devices that enable the computing entity 10 to receive data, such as a keypad 818 (hard or soft), a touch display, a voice / speech or motion interface, a scanner, a reader, or other input device. In embodiments including a keypad 818, the keypad 818 may include (or trigger the display of) regular numeric keys (0-9) and associated keys (#, *), as well as other keys used to operate the computing entity 10, and may include a set of keys that can be operated to provide a complete set of alphabet keys or a complete set of alphanumeric keys. In addition to providing input, the user input interface may be used to activate or deactivate certain functions, such as a screen saver and / or sleep mode. Through such input, the computing entity 10 can collect information / data, user interactions / input, etc.

[0144] Computing entity 10 may also include volatile storage or memory 822 and / or non-volatile storage or memory 824, which may be embedded and / or removable. For example, non-volatile memory may be ROM, PROM, EPROM, EEPROM, flash memory, MMC, SD memory card, memory stick, CBRAM, PRAM, FeRAM, RRAM, SONOS, racetrack memory, etc. Volatile memory may be RAM, DRAM, SRAM, FPM DRAM, EDO DRAM, SDRAM, DDR SDRAM, DDR2 SDRAM, DDR3 SDRAM, RDRAM, RIMM, DIMM, SIMM, VRAM, cache memory, register memory, etc. Volatile and non-volatile storage or memory may store databases, database instances, database management system entities, data, applications, programs, program modules, scripts, source code, object code, bytecode, compiled code, interpreted code, machine code, executable instructions, etc., for implementing the functions of computing entity 10.

[0145] conclusion Many modifications and other embodiments of the invention described herein will come to mind to those skilled in the art who benefit from the teachings presented in the above description and the accompanying drawings. It should be understood that the invention should not be limited to any particular embodiment disclosed, and that modifications and other embodiments are intended to be included within the scope of the appended claims. While specific terminology is used herein, it is used only in a general and descriptive sense and not for limiting purposes. [Explanation of Symbols]

[0146] 10 Computing Entities 20. Wired or wireless network 30 controllers 40 Cryogenic and / or vacuum chamber 50 Quantum Object Confinement Device 64, 64A, 64B, 64C operation source 66, 66A, 66B, 66C Beampathing Systems 70 Magnetic field source 70A magnetic field source, internal magnetic field source 70B Magnetic field source, external magnetic field source 80 Optical Acquisition System 90 Voltage source 100 Systems 110 Quantum Computers 115 Quantum Processors 300 grid 302 Grid Site 302B Blue Site, First Color 302G Green site, second color 302R Red Sight, Third Color 304 sides 306A Boundary Site 306B Boundary Site 308 Dashed line 310 Star-shaped operator 315, 315A, 315B Triangle Operator 402 Panel 404 Panel 406 Panel 410 panels 412 panels 414 Circuit part 416 panels 418 Circuit part 420 panels 430 panels 440 panels 450 panels 505 Circuits with simplified notation 510 Circuit diagram with expanded notation 515 circuits reduced 600 grid 605, 605A, 605B, ..., 605N brackets 705 Processing Devices 710 memory 715 Driver Controller Element 720 Communication Interfaces 725 A / D converter, analog-to-digital converter element 804 Transmitter 806 Receiver 808 Processing Devices 812 Antenna 816 displays 818 Keypad 820 Network Interfaces 822 Volatile storage or memory 824 Non-volatile storage or memory

Claims

1. A method for generating a ground state having a non-abelian topological order, A step of confining a plurality of physical qubits with a confinement device, wherein the plurality of physical qubits are logically organized on a lattice comprising a plurality of vertices and a plurality of brackets connected by edges, the lattice is formed from a plurality of sublattices, each sublattice comprising three or more vertices of the plurality of vertices and one or more brackets of the plurality of brackets, and the three or more vertices and one or more brackets of each sublattice among the plurality of sublattices are connected by edges to form the sublattice, A step of causing entanglement between one or more bracket qubits of the plurality of physical qubits, which are assigned to each of the one or more brackets of each sublattice in the first subset of the sublattice, and vertex qubits of the plurality of physical qubits, which are assigned to each of the three or more vertices of each sublattice in the first subset of the sublattice, A step of causing a measurement of the one or more bracket qubits assigned to each of the one or more brackets of each of the sublattices in the first subset of the sublattices, A step of causing entanglement between one or more bracket qubits of the plurality of physical qubits, which are assigned to each of the one or more brackets of each sublattice in the second subset of the sublattice, and the physical qubits of the plurality of physical qubits, which are assigned to each of the three or more vertices of each sublattice in the second subset of the sublattice, A step of causing a measurement of one or more bracket qubits of each of the sublattices of the second subset of the sublattices, A step of determining whether any of the brackets in the plurality of sublattices host an abelian topological order, based on the measurements of one or more bracket qubits in each of the sublattices in the first subset and the measurements of one or more bracket qubits in each of the sublattices in the second subset, The steps include: triggering the execution of a feedforward action performed on the pair of brackets to generate the ground state having a non-abelian topological order, in response to the determination that at least the pair of brackets of the plurality of sublattices hosts an abelian topological order; Methods that include...

2. The method according to claim 1, wherein one or more bracket qubits of each sublattice are in a product state with one or more bracket qubits of each sublattice of the first subset of the sublattice before the entanglement of the vertex qubits of each sublattice of the first subset of the sublattice.

3. The method according to claim 1 or 2, wherein the step of causing entanglement between the one or more bracket qubits of each of the sublattices of the first subset of the sublattices and the vertex qubits of each of the sublattices of the first subset of the sublattices includes causing entanglement between a specific bracket qubit assigned to a particular bracket among the one or more brackets and each of three vertex qubits assigned to a vertex among the three or more vertices connected to the specific bracket via an edge of the sublattices, and the step of causing entanglement between another specific bracket qubit assigned to another specific bracket among the one or more brackets and the three vertex qubits assigned to a vertex among the three or more vertices connected to the specific bracket via an edge of the sublattices, the three vertex qubits are also connected to the other specific bracket via an edge of the sublattices.

4. The method according to claim 3, wherein the steps of causing entanglement between a specific bracket qubit assigned to a specific bracket among the one or more brackets and each of three vertex qubits assigned to each of the three or more vertices and connected to the specific bracket, and causing entanglement between another specific bracket qubit assigned to another specific bracket among the one or more brackets and each of the three vertex qubits assigned to each of the three or more vertices and connected to the other specific bracket, are performed as four 2-qubit gates.

5. The method according to any one of claims 1 to 4, further comprising the step of causing entanglement of the plurality of brackets using a pair of three-operator non-Clifford interactions prior to the measurement of the one or more bracket qubits of each of the sublattices of the first subset of the sublattices.

6. The method according to claim 5, wherein a pair of three-operator non-Clifford interactions is implemented as four two-qubit gates.

7. The method according to any one of claims 1 to 6, wherein the grid is a Kagome grid.

8. The method according to any one of claims 1 to 7, wherein the step of causing the measurement of a bracket qubit among the one or more bracket qubits of each sublattice includes the step of determining the quantum state of the bracket qubit.

9. The method according to any one of claims 1 to 8, wherein the step of causing the feedforward action to be performed on the pair of brackets includes the step of causing a conditional Z gate to be performed on the pair of brackets.

10. The method according to any one of claims 1 to 9, wherein the grid has periodic boundary conditions.

11. A system configured to generate a ground state having a non-abelian topological order, A confinement device configured to confine multiple physical qubits, One or more manipulation sources configured to generate respective manipulation signals for interaction with each of the multiple physical qubits, A controller configured to control the operation of the confinement device and the one or more operating sources, Confining the plurality of physical qubits by the confinement device, wherein the plurality of physical qubits are logically organized on a lattice including a plurality of vertices and a plurality of brackets connected by edges, the lattice is formed from a plurality of sublattices, each sublattice including three or more vertices of the plurality of vertices and one or more brackets of the plurality of brackets, and the three or more vertices and one or more brackets of each sublattice among the plurality of sublattices are connected by edges to form the sublattice, thereby confining the qubits. To cause entanglement between one or more bracket qubits of the plurality of physical qubits, which are assigned to one or more brackets of each sublattice in the first subset of the sublattice, and vertex qubits of the plurality of physical qubits, which are assigned to three or more vertices of each sublattice in the first subset of the sublattice, To cause a measurement of the one or more bracket qubits assigned to each of the one or more brackets of each of the sublattices in the first subset of the sublattices, To cause entanglement between one or more bracket qubits of the plurality of physical qubits, which are assigned to each of the one or more brackets of each sublattice in the second subset of the sublattice, and the physical qubits of the plurality of physical qubits, which are assigned to each of the three or more vertices of each sublattice in the second subset of the sublattice, To cause a measurement of one or more bracket qubits of each of the sublattices of the second subset of the sublattices, Based on the measurements of one or more bracket qubits in each of the sublattices of the first subset, and the measurements of one or more bracket qubits in each of the sublattices of the second subset, it is determined whether any of the brackets in the plurality of sublattices host an abelian topological order, and A controller configured to perform, in response to a determination that at least the pair of brackets of the plurality of sublattices hosts an abelian topological order, trigger the execution of a feedforward action performed on the pair of brackets to generate the ground state having a non-abelian topological order, and A system that includes this.

12. The system according to claim 11, wherein one or more bracket qubits of each sublattice are in a product state with one or more bracket qubits of each sublattice of the first subset of the sublattice before the entanglement of the vertex qubits of each sublattice of the first subset of the sublattice.

13. The system according to claim 11 or 12, wherein causing entanglement between the vertex qubits of each of the sublattices of the first subset of the sublattices and the one or more bracket qubits of each of the sublattices of the first subset of the sublattices causes entanglement between a specific bracket qubit assigned to a specific bracket among the one or more brackets and each of the three vertex qubits assigned to a vertex among the three or more vertices connected to the specific bracket via an edge of the sublattices, and causes entanglement between another specific bracket qubit assigned to another specific bracket among the one or more brackets and the three vertex qubits assigned to a vertex among the three or more vertices connected to the specific bracket via an edge of the sublattices, the three vertex qubits are also connected to the other specific bracket via an edge of the sublattices.

14. The system according to claim 13, wherein the following is performed as four 2-qubit gates: causing an entanglement between a specific bracket qubit assigned to a specific bracket among the one or more brackets and each of three vertex qubits assigned to each of the three or more vertices and connected to the specific bracket, and causing an entanglement between another specific bracket qubit assigned to another specific bracket among the one or more brackets and each of the three vertex qubits assigned to each of the three or more vertices and connected to the other specific bracket.

15. The system according to any one of claims 11 to 14, wherein the controller is further configured to perform the entanglement of the plurality of brackets using a pair of three-operator non-Clifford interactions before the measurement of the one or more bracket qubits of each of the sublattices of the first subset of the sublattices.

16. The system according to claim 15, wherein pairs of three-operator non-Clifford interactions are implemented as four two-qubit gates.

17. The system according to any one of claims 11 to 16, wherein the grid is a Kagome grid.

18. The system according to any one of claims 11 to 17, wherein causing the measurement of a bracket qubit among the one or more bracket qubits of each of the sublattices includes determining the quantum state of the bracket qubit.

19. The system according to any one of claims 11 to 18, wherein causing the execution of the feedforward action on the pair of brackets includes causing a conditional Z gate to be executed on the pair of brackets.

20. The system according to any one of claims 11 to 19, wherein the grid has periodic boundary conditions.

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