Logical Hadamard gate operation and gauge fixing in subsystem code

Abstract

Systems, computer-implemented methods, and computer program products are provided that facilitate logical Hadamard gate operations and gauge fixing in subsystem codes. According to one embodiment, the system can include a processor that executes computer-executable components stored in a memory. The computer-executable components can include a gauge-fixing component that applies a gauge-fixing operation to the encoded qubit subsystem code to generate a switched subsystem code. The computer-executable components can further include a transverse component that applies a transversal Hadamard operation to the switched subsystem code to generate a rotated subsystem code.

Description

Technical Field

[0001] The present disclosure relates to subsystem code, and more particularly, to logical Hadamard gate operations and gauge fixing in subsystem code.

Background Art

[0002] Quantum computing generally involves using quantum mechanical phenomena for the purpose of performing computing functions and information processing functions. Quantum computing can generally be seen as contrasting with classical computing, which performs operations on binary values using transistors. That is, while a classical computer can perform operations on bit values that are either 0 or 1, a quantum computer can perform operations on qubits (quantum bits) that include superpositions of both 0 and 1, can entangle multiple qubits, and can use interference.

[0003] Some existing quantum technologies utilize subsystem code to perform various quantum operations, such as a Controlled NOT (CNOT) logic gate. A problem with such existing quantum technologies is that they do not perform logical Hadamard gates in such subsystem code. Another problem with such existing quantum technologies is that they are unable to perform gauge fixing in such subsystem code.

Summary of the Invention

[0004] The following presents an overview that provides a basic understanding of one or more embodiments of the present invention. This overview is not intended to identify key or critically important elements, nor is it intended to limit the scope of a particular embodiment or the scope of the claims. Its sole purpose is to present concepts in a simplified form as a prelude to the more detailed description that follows. One or more embodiments described herein include a system, device, computer-implemented method, or computer program product, or a combination thereof, that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code.

[0005] According to one embodiment, a system can include a processor that executes computer-executable components stored in a memory. The computer-executable components can include a gauge fixing component that applies a gauge fixing operation to coded qubit subsystem code to generate switched subsystem code. The computer-executable components can further include a transverse component that applies a transversal Hadamard operation to the switched subsystem code to generate rotated subsystem code. An advantage of such a system is that it can be implemented to perform a fault tolerant logical Hadamard gate on a quantum code that has the property of reducing at least one of frequency collision or crosstalk error associated with a quantum device.

[0006] In some embodiments, this computer-executable component can further include a rotation component that generates code to perform a fault-tolerant logical Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with a quantum device by rotating the rotated subsystem code. The advantage of such a system is that the system can be implemented to perform a fault-tolerant logical Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with a quantum device.

[0007] According to another embodiment, a computer-implemented method can include applying a gauge fixing operation to a subsystem code of encoded qubits by a system operably coupled to a processor to generate a switched subsystem code. The computer-implemented method can further include applying a transversal Hadamard operation to the switched subsystem code by the system to generate a rotated subsystem code. The advantage of such a computer-implemented method is that the computer-implemented method can be implemented to perform a fault-tolerant logical Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with a quantum device.

[0008] In some embodiments, this computer-implemented method can further include generating, by the system, code for performing a fault-tolerant logical Hadamard gate on a quantum code that has the property of rotating the rotated subsystem code to reduce at least one of frequency collisions or cross-talk errors associated with the quantum device. The advantage of such a computer-implemented method is that the computer-implemented method can be implemented to perform a fault-tolerant logical Hadamard gate on a quantum code that has the property of reducing at least one of frequency collisions or cross-talk errors associated with the quantum device.

[0009] According to another embodiment, a computer program product includes a computer-readable storage medium having program instructions implemented thereon, the program instructions being executable by a processor to cause the processor to apply a gauge fixing operation to a subsystem code of encoded qubits to generate a switched subsystem code. The program instructions are further executable by the processor to cause the processor to apply a transversal Hadamard operation to the switched subsystem code to generate a rotated subsystem code. The advantage of such a computer program product is that the computer program product can be implemented to perform a fault-tolerant logical Hadamard gate on a quantum code that has the property of reducing at least one of frequency collisions or cross-talk errors associated with the quantum device.

[0010] In some embodiments, the processor is executable by the processor to further cause the program instructions to generate code that executes a fault-tolerant logic Hadamard gate on a quantum code that has the property of rotating the rotated subsystem code to reduce at least one of frequency collision or crosstalk errors associated with the quantum device. The advantage of such a computer program product is that the computer program product can be implemented to execute a fault-tolerant logic Hadamard gate on a quantum code that has the property of reducing at least one of frequency collision or crosstalk errors associated with the quantum device.

[0011] According to one embodiment, the system can include a processor that executes computer-executable components stored in a memory. The computer-executable components can include a gauge fixing component that applies a gauge fixing operation to a subsystem code of encoded qubits to generate a switched subsystem code. The computer-executable components can further include a lattice shift component that shifts a lattice of the switched subsystem code to generate a shifted and switched subsystem code. The advantage of such a system is that the system can be implemented to execute a fault-tolerant logic Hadamard gate on a quantum code that has the property of reducing at least one of frequency collision or crosstalk errors associated with the quantum device.

[0012] In some embodiments, this computer-executable component can further include a transversal component that applies a transversal Hadamard operation to the shifted and switched subsystem code to generate a rotated subsystem code. In these embodiments, this computer-executable component can further include a rotation component that rotates the rotated subsystem code to generate code that performs a fault-tolerant logical Hadamard gate on a quantum code having a property of reducing at least one of frequency collisions or crosstalk errors associated with the quantum device. The advantage of such a system is that the system can be implemented to perform a fault-tolerant logical Hadamard gate on a quantum code having a property of reducing at least one of frequency collisions or crosstalk errors associated with the quantum device.

[0013] According to another embodiment, a computer-implemented method can include applying a gauge fixing operation to a subsystem code of encoded qubits by a system operably coupled to a processor to generate a switched subsystem code. The computer-implemented method can further include shifting a lattice of the switched subsystem code by the system to generate a shifted and switched subsystem code. The advantage of such a computer-implemented method is that the computer-implemented method can be implemented to perform a fault-tolerant logical Hadamard gate on a quantum code having a property of reducing at least one of frequency collisions or crosstalk errors associated with the quantum device.

[0014] In some embodiments, the computer-implemented method can further include applying a transversal Hadamard operation by the system to the shifted and switched subsystem code to generate a rotated subsystem code. In these embodiments, the computer-implemented method can further include generating, by the system, code for performing a fault-tolerant logical Hadamard gate on a quantum code having a property of reducing at least one of frequency collision or crosstalk errors associated with the quantum device by rotating the rotated subsystem code. The advantage of such a computer-implemented method is that the computer-implemented method can be implemented to perform a fault-tolerant logical Hadamard gate on a quantum code having a property of reducing at least one of frequency collision or crosstalk errors associated with the quantum device.

Brief Description of the Drawings

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[0016] The following detailed description is for illustrative purposes only, and is not intended to limit the embodiments or the application or uses of the embodiments, or both, nor is it intended to be bound by any express or implied information presented in the Background or Summary or Detailed Description above.

[0017] One or more embodiments will now be described with reference to the drawings. The same reference numerals are used to refer to the same elements throughout. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a more thorough understanding of one or more embodiments. However, it will be apparent that in various cases one or more embodiments may be practiced without these specific details.

[0018] Considering the above problems related to existing quantum technologies, the present disclosure can be implemented to produce solutions to these problems in the form of a system, a computer-implemented method, a computer program product, or a combination thereof, that facilitates logical Hadamard gate operations and gauge fixing in subsystem codes by applying a gauge fixing operation to the subsystem code of encoded qubits to produce a switched subsystem code, or by applying a transversal Hadamard operation to the switched subsystem code to produce a rotated subsystem code, or both. The advantage of such a system, computer-implemented method, computer program product, or combination thereof is that they can be implemented to perform a fault-tolerant logical Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with a quantum device.

[0019] In some embodiments, the present disclosure can be implemented to produce solutions to the above problems in the form of a system, a computer-implemented method, a computer program product, or a combination thereof, that further facilitates logical Hadamard gate operations and gauge fixing in subsystem codes by generating a code that rotates the rotated subsystem code to perform a fault-tolerant logical Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with a quantum device. The advantage of such a system, computer-implemented method, computer program product, or combination thereof is that they can be implemented to perform a fault-tolerant logical Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with a quantum device.

[0020] As used herein, "entity" can include a human, client, user, computing device, software application, agent, machine learning (ML) model, artificial intelligence (AI) or another entity or a combination thereof. When an element is referred to herein as being "coupled" to another element, this can describe one or more different types of couplings including, but not limited to, chemical bonding, communicable coupling, electrical coupling, electromagnetic coupling, operable coupling, optical coupling, physical coupling, thermal coupling or another type of coupling or a combination thereof.

[0021] Figures 1 and 2 respectively show block diagrams of exemplary and non-limiting systems 100 and 200 that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. Systems 100 and 200 can each include a quantum gate operation system 102. The quantum gate operation system 102 of system 100 shown in FIG. 1 can include a memory 104, a processor 106, a gauge fixing component 108, a transverse component 110 or a bus 112 or a combination thereof. The quantum gate operation system 102 of system 200 shown in FIG. 2 can further include a rotation component 202 or a lattice shift component 204 or both.

[0022] The embodiments of the subject disclosure shown in the various figures disclosed herein are for illustrative purposes only, and thus it should be recognized that the architecture of such embodiments is not limited to the systems, devices or components shown in those figures or combinations thereof. For example, in some embodiments, system 100, system 200 or quantum gate computing system 102 or combinations thereof may further include various computer elements or computing-based elements or both described herein with respect to the operating environment 1000 of FIG. 10. In some embodiments, such computer elements or computing-based elements or both may be used with respect to one or more implementations of the systems, devices, components or computer-implemented operations or combinations thereof illustrated and described herein with respect to FIG. 1, FIG. 2 or other figures disclosed herein or combinations thereof.

[0023] Memory 104 is one or more components or instructions or both that can be read, written, executed, or a combination thereof, by a computer or machine or both, and that, when executed by a processor 106 (e.g., a classical processor, a quantum processor, or another type of processor or a combination thereof), can facilitate the execution of operations defined by executable components or instructions or both. For example, memory 104 can store components or instructions or both that can be read, written, executed, or a combination thereof, by a computer or machine or both, and that, when executed by processor 106, can facilitate the execution of various functions related to quantum gate operation system 102, gauge fixing component 108, transverse component 110, rotation component 202, lattice shift component 204, or another component related to quantum gate operation system 102 described herein with or without reference to the various figures of the subject disclosure, or a combination thereof.

[0024] Memory 104 can include volatile memory (e.g., random access memory (RAM), static RAM (SRAM), dynamic RAM (DRAM), or another type of volatile memory or combinations thereof) or non-volatile memory (e.g., read only memory (ROM), programmable ROM (PROM), erasable programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or another type of non-volatile memory or combinations thereof) or both, and those volatile memory or non-volatile memory or both can use one or more memory architectures. Additional examples of memory 104 are described below with respect to system memory 1016 of FIG. 10. Such examples of memory 104 can be used to implement any embodiment of the subject disclosure.

[0025] Processor 106 can include one or more components or instructions or both that can be read, written, executed, or combined by a computer or machine or both, and can implement one or more components or instructions or both that can be stored in memory 104. It can include one or more types of processors or electronic circuits or both (e.g., classical processors, quantum processors, and / or other types of processors and / or electronic circuits). For example, processor 106 can execute various operations specified by such components or instructions or both that can be read, written, executed, or combined by a computer or machine or both. These operations include, but are not limited to, logical, control, input / output (I / O), arithmetic operations, or other operations or combinations thereof. In some embodiments, processor 106 can include one or more central processing units, multi-core processors, microprocessors, dual microprocessors, microcontrollers, system-on-a-chip (SOC), array processors, vector processors, quantum processors, or other types of processors, or combinations thereof. Additional examples of processor 106 are described below with respect to processing unit 1014 of FIG. 10. Such examples of processor 106 can be used to implement any embodiment of the subject disclosure.

[0026] To perform the functions of system 100, system 200, quantum gate operation system 102 or components coupled thereto or combinations thereof, quantum gate operation system 102, memory 104, processor 106, gauge fixing component 108, transverse component 110, rotation component 202, lattice shift component 204, or another component of quantum gate operation system 102 described herein, or combinations thereof, can be communicatively, electrically, operably or optically or in combination thereof coupled to each other via bus 112. Bus 112 can include one or more memory buses, memory controllers, peripheral buses, external buses, local buses, quantum buses or another type of bus, or combinations thereof, and these buses can use various bus architectures. Additional examples of bus 112 are described below with respect to system bus 1018 of FIG. 10. Using such examples of bus 112, any embodiment of the subject disclosure can be implemented.

[0027] The quantum gate operation system 102 can include any type of component, machine, device, facility, apparatus, and / or instrument that includes a processor, and / or can communicate effectively and / or operably with a wired network and / or a wireless network. All such embodiments are contemplated. For example, the quantum gate operation system 102 can include a server device, a computing device, a general-purpose computer, a dedicated computer, a quantum computing device (e.g., a quantum computer), a tablet computing device, a handheld device, a server-class computing machine and / or a database, a laptop computer, a notebook computer, a desktop computer, a mobile phone, a smartphone, a consumer device and / or appliance, an industrial and / or commercial device, a digital assistant, an Internet-enabled multimedia telephone, a multimedia player, and / or other types of devices.

[0028] The quantum gate operation system 102 can be coupled (e.g., communicatively, electrically, operably, optically, or by another type of coupling, or a combination thereof) to one or more external systems, sources or devices, or combinations thereof (e.g., classical and / or quantum computing devices, communication devices, and / or another type of external system, source and / or device) using one or more wires and / or cables or both. For example, the quantum gate operation system 102 can be coupled (e.g., communicatively, electrically, operably, optically, or by another type of coupling, or a combination thereof) to one or more external systems, sources or devices, or combinations thereof (e.g., classical and / or quantum computing devices, communication devices, and / or another type of external system, source and / or device) using a data cable including, but not limited to, a High-Definition Multimedia Interface (HDMI) cable, a recommended standard (RS) 232 cable, an Ethernet cable or another data cable, or a combination thereof.

[0029] In some embodiments, the quantum gate operation system 102 can be coupled (e.g., communicatively, electrically, operably, optically, or by another type of coupling, or a combination thereof) via a network to one or more external systems, sources or devices, or combinations thereof (e.g., classical and / or quantum computing devices, communication devices, and / or another type of external system, source and / or device). For example, such a network can include, without limitation, a cellular network, a wide area network (WAN) (e.g., the Internet), a local area network (LAN), or another network, or a combination thereof, including wired networks, wireless networks, or both. The quantum gate operation system 102 can communicate with one or more external systems, sources or devices, such as computing devices, or combinations thereof, using substantially any desired wired technology, wireless technology, or both, which substantially any desired wired and wireless technologies include, without limitation, wireless fidelity (Wi-Fi), global system for mobile communications (GSM), universal mobile telecommunications system (UMTS), worldwide interoperability for microwave access (WiMAX), enhanced general packet radio service (Enhanced GPRS), third generation partnership project (3GPP) long termevolutions such as Long Term Evolution (LTE), Third Generation Partnership Project 2 (3GPP2) Ultra Mobile Broadband (UMB), High Speed Packet Access (HSPA), Zigbee and other 802.XX wireless technologies and / or legacy telecommunications technologies, BLUETOOTH(R), Session Initiation Protocol (SIP), ZIGBEE(R), RF4CE protocol, WirelessHART protocol, 6LoWPAN (IPv6 over Low power Wireless Area Networks), Z-Wave, ANT, ultra-wideband (UWB) standard protocols, and / or other proprietary and non-proprietary communication protocols. Thus, in some embodiments, the quantum gate operation system 102 can include hardware (e.g., a central processing unit (CPU), transceiver, decoder, quantum hardware, quantum processor or other hardware or combinations thereof), software (e.g., a set of threads, a set of processes, running software, quantum pulse schedule, quantum circuit, quantum gate or other software or combinations thereof), or a combination of hardware and software that can facilitate information transfer between the quantum gate operation system 102 and an external system, source or device or combinations thereof (e.g., a computing device, communication device and / or another type of external system, source and / or device).

[0030] The quantum gate operation system 102 can be one or more components or instructions or both that can be read, written, executed, or combined by a computer or machine or both, and when executed by a processor 106 (such as a classical processor, a quantum processor, or another type of processor or a combination thereof), can facilitate the execution of operations defined by such components or instructions or both, and can include one or more components or instructions or both. Further, in many embodiments, any component described herein, with or without reference to the various figures of the subject disclosure related to the quantum gate operation system 102, can be one or more components or instructions or both that can be read, written, executed, or combined by a computer or machine or both, and when executed by a processor 106, can facilitate the execution of operations defined by such components or instructions or both, and can include one or more components or instructions or both. For example, a gauge fixing component 108, a transverse component 110, a rotation component 202, a lattice shift component 204, or other components disclosed herein related to the quantum gate operation system 102, or a combination thereof (such as communicatively, electronically, operably, and / or optically coupled to the quantum gate operation system 102, and / or communicatively, electronically, operably, and / or optically used by the quantum gate operation system 102), can include such a plurality of components or instructions or both that can be read, written, executed, or combined by a computer or machine or both.As a result, according to a number of embodiments, the quantum gate operation system 102, or any component disclosed herein related to the quantum gate operation system 102, or both, can use the processor 106 to facilitate the execution of one or more operations described herein with respect to the quantum gate operation system 102, or any component related to the quantum gate operation system 102, or both, such that a computer, a machine, or both can read, write, execute, or a combination of these, and execute such components, instructions, or both.

[0031] The quantum gate operation system 102 is executed by a gauge fixing component 108, a transverse component 110, a rotation component 202, a lattice shift component 204, and / or another component disclosed herein related to the quantum gate operation system 102, and / or can facilitate the execution of operations (e.g., by the processor 106) related to the gauge fixing component 108, the transverse component 110, the rotation component 202, the lattice shift component 204, and / or another component disclosed herein related to the quantum gate operation system 102. For example, as will be described in detail later, the quantum gate operation system 102 can apply a gauge fixing operation to the subsystem code of the encoded qubits (e.g., by the processor 106) to generate a switched subsystem code, or apply a transverse Hadamard operation to the switched subsystem code to generate a rotated subsystem code, or facilitate both.In this example, as will be described in detail later, the quantum gate operation system 102 further has the property of rotating the rotated subsystem code (e.g., by the processor 106) to reduce at least one of the frequency collision or crosstalk errors associated with the quantum device, generating code for executing a fault-tolerant logical Hadamard gate on a quantum code, applying a gauge fixing operation to the subsystem code to exchange two-body gauge operators and four-body gauge operators in a subset of the subsystem code, applying a transversal Hadamard operation to the switched subsystem code to apply a Hadamard gate on the data qubits and exchange X-type gauge operators and Z-type gauge operators, X-type stabilizers and Z-type stabilizers, and X-type logical operators and Z-type logical operators, or performing two pairwise swap gate operations between qubits that are symmetric with respect to the vertical and diagonal axes of the rotated subsystem code, or facilitating combinations thereof.

[0032] In another example, as will be described in detail later, the quantum gate operation system 102 can apply a gauge fixing operation to the subsystem code of the encoded qubits (e.g., by the processor 106) to generate a switched subsystem code, or shift the lattice of the switched subsystem code to generate a shifted and switched subsystem code, or facilitate both. In this example, as will be described in detail later, the quantum gate operation system 102 can further apply a transversal Hadamard operation to the shifted and switched subsystem code (e.g., by the processor 106) to generate a rotated subsystem code, rotate the rotated subsystem code to generate a code that performs a fault-tolerant logic Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with the quantum device, apply a gauge fixing operation to the subsystem code to exchange two-body gauge operators and four-body gauge operators in a subset of the subsystem code, apply a transversal Hadamard operation to the shifted and switched subsystem code to apply a Hadamard gate on the data qubits and exchange X-type gauge operators and Z-type gauge operators, X-type stabilizers and Z-type stabilizers, and X-type logic operators and Z-type logic operators, or perform two pairwise swap gate operations between qubits that are symmetric with respect to the vertical and diagonal axes of the rotated subsystem code, or facilitate combinations thereof.

[0033] The gauge-fixing component 108 can apply a gauge-fixing operation to the subsystem code of the encoded qubits to generate a switched subsystem code. For example, referring to the exemplary and non-limiting FIG. 300 shown in FIG. 3, the gauge-fixing component 108 can apply a gauge-fixing operation to the subsystem code 302 of the encoded qubits (shown as C in FIG. 3) to generate a switched subsystem code 304 (shown as C' in FIG. 3).

[0034] FIG. 3 shows an exemplary and non-limiting FIG. 300 that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. For the sake of brevity, the description of the same elements or processes or both used in the corresponding respective embodiments and their repetitions is omitted.

[0035] As described above and shown in the exemplary and non-limiting FIG. 300 shown in FIG. 3, the gauge-fixing component 108 can apply a gauge-fixing operation to the subsystem code 302 of the encoded qubits to generate a switched subsystem code 304. For example, as will be described later with reference to FIGS. 4-6E, the gauge-fixing component 108 can apply a gauge-fixing operation to the subsystem code 302 to exchange two-body gauge operators and four-body gauge operators in a subset of the subsystem code 302. For example, the gauge-fixing component 108 can apply a gauge-fixing operation to the subsystem code 302 to exchange both two-body gauge operators and four-body gauge operators of both X-type and Z-type in the bulk of the subsystem code 302, and this exchange effectively switches the subsystem code 302 to the switched subsystem code 304 shown in FIG. 3.

[0036] In the exemplary and non - limiting FIG. 300 shown in FIG. 3, the transverse component 110 can apply a transverse Hadamard operation to the switched subsystem code 304 to generate a rotated subsystem code 306 (shown as C” in FIG. 3). For example, the transverse component 110 can apply a transverse Hadamard operation to the switched subsystem code 304 to apply a Hadamard gate on the data qubits and to swap X - type and Z - type gauge operators, X - type and Z - type stabilizers, and X - type and Z - type logical operators. That is, for example, the transverse component 110 can apply a transverse Hadamard operation to apply a Hadamard gate on all data qubits, which gate swaps X and Z - type gauge operators and stabilizers and effectively switches the switched subsystem code 304 to the rotated subsystem code 306.

[0037] In the exemplary and non - limiting FIG. 300 shown in FIG. 3, the rotation component 202 can generate code 308 that rotates the rotated subsystem code 306 to perform a fault - tolerant logic Hadamard gate on a quantum code having the property of reducing at least one of frequency collision or crosstalk errors related to a quantum device. In this exemplary embodiment, to rotate the rotated subsystem code 306 to generate code 308, the rotation component 202 can perform two pairwise swap - gate operations between qubits that are symmetric with respect to the vertical axis 310 and the diagonal axis 312 of the rotated subsystem code 306. For example, as shown in FIG. 3, to rotate the rotated subsystem code 306 to generate code 308, the rotation component 202 can use the formula (1) defined below to perform two pairwise swap - gate operations between qubits that are symmetric with respect to the vertical axis 310 and the diagonal axis 312, and such swap - gate operations effectively perform a 90 - degree (90°) rotation.

[0038] Formula (1) SWAP(n⇔d - n)·SWAP(m⇔n) In the above formula, m represents the row of the data - qubit position in the subsystem code, n represents the column of the data - qubit position in the subsystem code, d represents the subsystem - code distance of the subsystem code, and this subsystem - code distance is equal to the total number of rows and columns of the data - qubits in the subsystem code. According to many embodiments of the subject disclosure, such data - qubits are represented as points 314 in the various subsystem codes shown in the figures described herein. In the exemplary and non - limiting FIG. 300 shown in FIG. 3, by performing two pairwise swap - gate operations between such qubits as described above using the formula (1) defined above, the rotation component 202 can perform a qubit replacement, which effectively rotates the code by 90° and switches it back to the subsystem code 302.

[0039] In some embodiments, for example as will be described later with reference to FIGS. 7(A) and 7(B), the gauge fixing component 108 can apply a gauge fixing operation to the subsystem code of the encoded qubits to generate a switched subsystem code, and the lattice shift component 204 can shift the lattice of the switched subsystem code to generate a shifted and switched subsystem code. In these embodiments, the transverse component 110 can apply a transverse Hadamard operation to the shifted and switched subsystem code to generate a rotated subsystem code, and the rotation component 202 can rotate the rotated subsystem code to generate a code that performs a fault-tolerant logical Hadamard gate having a property of reducing at least one of frequency collision or crosstalk errors associated with the quantum device on the quantum code. In these embodiments, the gauge fixing component 108 can apply a gauge fixing operation to the subsystem code to exchange two-body gauge operators and four-body gauge operators in a subset of the subsystem code. In these embodiments, the transverse component 110 can apply a transverse Hadamard operation to the shifted and switched subsystem code to apply a Hadamard gate on the data qubits and to exchange X-type gauge operators and Z-type gauge operators, X-type stabilizers and Z-type stabilizers, and X-type logical operators and Z-type logical operators. In these embodiments, the rotation component 202 can perform two pairwise swap gate operations between qubits that are symmetric with respect to the vertical and diagonal axes of the rotated subsystem code.

[0040] FIG. 4 shows an exemplary and non-limiting FIG. 400 that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. For the sake of brevity, descriptions of the same elements or processes or both that are used in corresponding respective embodiments are omitted.

[0041] FIG. 400 shows gauge operators after gauge fixing to a heavy-hex code. The gauge group G’ is a gauge fixing of G when S(G)≦S(G’)≦G’≦G and k(G)=k(G’), where S(G) is the stabilizer of G and k(G) is the number of encoded qubits. In the exemplary FIG. 400 shown in FIG. 4, the subsystem code 302 and the switched subsystem code 304 are gauge fixings of the Bacon-Shor code shown as "C" in FIG. 4. BS

[0042] In the exemplary FIG. 400 shown in FIG. 4, the bulk of the subsystem code 302 includes 4-body X gauge operators (shown in the dark gray squares of the subsystem code 302) and vertical 2-body Z gauge operators (shown in the light gray vertical semi-circles of the subsystem code 302). In various embodiments of the subject disclosure, such a bulk of the subsystem code 302 can constitute a subset of the subsystem code 302. At the boundary of the subsystem code 302 shown in the exemplary FIG. 400 shown in FIG. 4, there are horizontal 2-body X gauge operators (shown in the dark gray horizontal semi-circles of the subsystem code 302). In various embodiments of the subject disclosure, the boundary of the subsystem code 302 can constitute a subset of the subsystem code 302.

[0043] In the exemplary FIG. 400 shown in FIG. 4, the bulk of the switched subsystem code 304 includes a 4-body Z gauge operator (shown in the light gray square of the switched subsystem code 304) and a horizontal 2-body X gauge operator (shown in the dark gray horizontal semi-circle of the switched subsystem code 304). In various embodiments of the subject disclosure, such a bulk of the switched subsystem code 304 may constitute a subset of the switched subsystem code 304. At the boundary of the switched subsystem code 304 shown in the exemplary FIG. 400 shown in FIG. 4, there is a vertical 2-body Z gauge operator (shown in the light gray vertical semi-circle of the switched subsystem code 304). In various embodiments of the subject disclosure, the boundary of the switched subsystem code 304 may constitute a subset of the switched subsystem code 304.

[0044] In the exemplary FIG. 400 shown in FIG. 4, before applying the gauge fixing operation, the gauge fixing component 108 can split the 4-body X gauge operator in the bulk into a pair of horizontal 2-body gauge operators. In the exemplary FIG. 400 shown in FIG. 4, before applying the gauge fixing operation, the gauge fixing component 108 can merge the pair of vertical 2-body Z gauge operators in the bulk into one 4-body gauge operator.

[0045] FIG. 5 shows an exemplary and non-limiting FIG. 500 that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. For brevity, the description of the same elements and / or processes or both that are used in the corresponding respective embodiments and their repetitions is omitted.

[0046] Figure 500 shows the heavy-hexagon gauge-fixed stabilizer described above. For example, in the exemplary Figure 500 shown in Figure 5, subsystem code 302 shows the stabilizer before the application of the gauge-fixing operation by the gauge-fixing component 108, and the switched subsystem code 304 shows the stabilizer after the application of the gauge-fixing operation by the gauge-fixing component 108.

[0047] In the bulk of the subsystem code 302 shown in the exemplary Figure 500 shown in Figure 5, the 4-body Z stabilizer is the product of a pair of vertical 2-body Z gauge operators (shown by the light gray vertical semi-circles in the subsystem code 302), and the X stabilizer is a 2-column vertical strip of Pauli X. At the boundary of the subsystem code 302 shown in the exemplary Figure 500 shown in Figure 5, there are vertical 2-body Z gauge operators (shown by the light gray vertical semi-circles in the subsystem code 302).

[0048] In the bulk of the switched subsystem code 304 shown in the exemplary Figure 500 shown in Figure 5, the 4-body X stabilizer is the product of a pair of horizontal 2-body X gauge operators (shown by the dark gray horizontal semi-circles in the switched subsystem code 304), and the Z stabilizer is a 2-row horizontal strip of Pauli Z. At the boundary of the switched subsystem code 304 shown in the exemplary Figure 500 shown in Figure 5, there are horizontal 2-body X gauge operators (shown by the dark gray horizontal semi-circles in the switched subsystem code 304).

[0049] According to one or more embodiments of the subject disclosure described herein, to facilitate applying a gauge fixing operation to a subsystem code of encoded qubits (e.g., subsystem code 302) to generate a switched subsystem code (e.g., switched subsystem code 304), the gauge fixing component 108 can use the gauge fixing protocol defined below.

[0050] Gauge fixing protocol (1) With respect to the bulk, the gauge fixing component 108 can measure all two-body X gauge operators and further infer the four-body X stabilizer eigenvalue (recorded as M, for example).

[0051] If M = +1, no correction by the gauge fixing component 108 is necessary.

[0052] If M = -1, the gauge fixing component 108 can apply Pauli Z to correct the stabilizer.

[0053] In this way, the original four-body X gauge operator in the subsystem code 302 is fixed to the stabilizer of the switched subsystem code 304.

[0054] In the absence of measurement noise, the gauge fixing component 108 can execute one cycle of the above operations.

[0055] In the presence of measurement noise, the gauge fixing component 108 can execute d cycles of measurement to decode and apply the Z correction.

[0056] (2) With respect to the boundary, when performing bulk measurements, the gauge fixing component 108 can simultaneously measure all two-body X boundary stabilizers (recorded as M, for example) and further apply corrections (e.g., in the same way as performed for the bulk described above).

[0057] Doing so fixes the original two-body X gauge operator in the subsystem code 302 to the boundary stabilizer in the switched subsystem code 304.

[0058] (3) If there is a measurement error, the gauge fixing component 108 further applies a measurement of the d cycles of the four-body Z gauge operator (in contrast to one cycle when there is no measurement error, for example).

[0059] Each cycle of the Z measurement can be applied by the gauge fixing component 108 after each cycle of the X measurement.

[0060] Therefore, the gauge fixing component 108 can apply an X correction after decoding the d cycles of the syndrome history.

[0061] Figures 6A and 6B respectively show exemplary and non-limiting FIGS. 600a and 600b that can facilitate logical Hadamard gate operations and gauge fixing in the subsystem code according to one or more embodiments described herein. For the sake of brevity, the description of the same elements and / or processes or both repetitions used in the corresponding respective embodiments is omitted.

[0062] The exemplary FIGS. 600a and 600b shown in FIGS. 6A and 6B respectively illustrate the CNOT gate scheduling that can be used by the gauge fixing component 108 to measure the X gauge operator described above in two consecutive cycles (e.g., X syndrome measurement), where FIG. 600a shows cycle 1 and FIG. 600b shows cycle 2. To facilitate such CNOT gate scheduling that can be used by the gauge fixing component 108 to measure the X gauge operator described above in two consecutive cycles (e.g., X syndrome measurement), the gauge fixing component 108 can use the corresponding circuit described below and shown in FIG. 6C.

[0063] FIG. 6C shows an exemplary and non-limiting FIG. 600c that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. For the sake of brevity, the description of the same elements and / or processes used in each corresponding embodiment, or both, is omitted.

[0064] To facilitate the CNOT gate scheduling described above and shown in FIGS. 600a and 600b that can be used by the gauge fixing component 108 to measure the X gauge operator described above in two consecutive cycles (e.g., X syndrome measurement (e.g., bulk)), the gauge fixing component 108 can use the corresponding circuit shown in FIG. 600c shown in FIG. 6C. The exemplary FIG. 600c shown in FIG. 6C can include a measurement circuit for a two-body X gauge operator in the bulk of the subsystem code (e.g., in the bulk of subsystem code 302 or switched subsystem code 304 or both). In some embodiments, the gauge fixing component 108 can measure the boundary two-body X gauge operator using a standard circuit including two CNOT gates.

[0065] FIG. 6D shows an exemplary and non-limiting FIG. 600d that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. For the sake of brevity, the description of the same elements or processes or both repetitions used in the corresponding respective embodiments is omitted.

[0066] The exemplary FIG. 600d shown in FIG. 6D shows CNOT gate scheduling that can be used by the gauge fixing component 108 to measure the Z gauge operator in the third cycle (e.g., Z syndrome measurement), and FIG. 600d shows cycle 3. To facilitate such CNOT gate scheduling that can be used by the gauge fixing component 108 to measure the Z gauge operator in cycle 3 (e.g., Z syndrome measurement), the gauge fixing component 108 can use the corresponding circuit described below shown in FIG. 6E.

[0067] FIG. 6E shows an exemplary and non-limiting FIG. 600e that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. For the sake of brevity, the description of the same elements or processes or both repetitions used in the corresponding respective embodiments is omitted.

[0068] To facilitate the CNOT gate scheduling described above shown in FIG. 600d that can be used by the gauge fixing component 108 to measure the Z gauge operator in cycle 3 (e.g., Z syndrome measurement (e.g., bulk)), the gauge fixing component 108 can use the corresponding circuit shown in FIG. 600e shown in FIG. 6E. The exemplary FIG. 600e shown in FIG. 6E can include a measurement circuit for two-body Z gauge operators in the bulk of the subsystem code (e.g., in the bulk of subsystem code 302 or switched subsystem code 304 or both).

[0069] Figures 7(A) and 7(B) each show exemplary and non-limiting FIGS. 700a and 700b that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. For the sake of brevity, the description of the same elements or processes or both that are used in the respective corresponding embodiments is omitted.

[0070] In some embodiments, the gauge fixing component 108 can apply a gauge fixing operation on a heavy hexagon lattice. In these embodiments, in order to perform a gauge fixing operation on a heavy hexagon lattice, the qubit lattice can be shifted by one lattice constant after the application of the gauge fixing operation. In these embodiments, in order to apply a gauge fixing operation on a heavy hexagon lattice, the gauge fixing component 108 can apply a gauge fixing operation on the subsystem code of the encoded qubits to generate a switched subsystem code, and the lattice shift component 204 can shift the lattice of the switched subsystem code to generate a shifted and switched subsystem code. For example, as shown in the exemplary FIGS. 700a and 700b shown in FIGS. 7(A) and 7(B) respectively, the gauge fixing component 108 can apply a gauge fixing operation to the subsystem code 702 to generate a switched subsystem code as described above with reference to the exemplary embodiments shown in FIGS. 1-6E, and the lattice shift component 204 can shift the qubit lattice of the switched subsystem code to generate a shifted and switched subsystem code 704. As shown in the exemplary FIGS. 700a and 700b shown in FIGS. 7(A) and 7(B) respectively, such a qubit lattice can include a plurality (e.g., 25) of qubits (represented by gray circles in FIGS. 700a and 700b).

[0071] To shift such a qubit lattice, the lattice shift component 204 can perform the following swap protocol to shift the qubit lattice by, for example, a lattice constant of 1. For example, as shown in the exemplary FIGS. 700a and 700b respectively shown in FIGS. 7(A) and 7(B), to shift the qubit lattice by a lattice constant of 1, the lattice shift component 204 can add a row of additional data qubits 706 to the bottom of the qubit lattice shown in FIG. 700a, and further add corresponding additional auxiliary qubits 708a, 708b (represented by white and black circles respectively in FIGS. 700a and 700b) to the qubit lattice. For clarity, all additional auxiliary qubits 708a, 708b in FIGS. 700a and 700b are not labeled. In one example, to shift the qubit lattice by a lattice constant of 1, the lattice shift component 204 applies the first step of the swap represented by the arrow 710 in FIG. 700a to all data qubits of the qubit lattice, and can move the quantum information of such data qubits to the additional auxiliary qubit 708a. In FIGS. 700a and 700b, such quantum information is represented by numbers 1 to 25. In this example, to complete such a qubit lattice shift operation, the lattice shift component 204 can apply the second step of the swap to move the quantum information to the additional data qubit 706 located one row below.

[0072] In the above example, by performing the gauge fixing operation and the swap protocol described above, the gauge fixing component 108 and the lattice shift component 204 can generate the shifted and switched subsystem code 704 shown in FIG. 700b shown in FIG. 7(B), and the shifted and switched subsystem code 704 can be generated by performing the two steps of the swap and the gauge fixing operation described above. The gauge fixing component 108 and the lattice shift component 204 each satisfy one or more hardware criteria of a quantum device that can be used to execute subsystem code (e.g., heavy hexagon code). It should be recognized that the gauge fixing operation and the swap protocol described above can be performed.

[0073] In embodiments where the gauge fixing component 108 and the lattice shift component 204 each perform the gauge fixing operation and the swap protocol described above, the gauge fixing component 108 applies the gauge fixing operation described above to the subsystem code 702 (as described above with reference to FIGS. 1-6E, for example), and can exchange two-body gauge operators and four-body gauge operators in a subset (e.g., bulk) of the subsystem code 702. In these embodiments, the transverse component 110 applies a transverse Hadamard operation to the shifted and switched subsystem code 704 (as described above with reference to FIGS. 1, 2, and 3, for example), and can generate a rotated subsystem code (not shown in FIGS. 7(A) or 7(B)). In these embodiments, the transverse component 110 applies a transverse Hadamard operation to the shifted and switched subsystem code 704 (as described above with reference to FIGS. 1, 2, and 3, for example), applies a Hadamard gate on data qubits (e.g., data qubits in a qubit lattice), and can exchange X-type gauge operators and Z-type gauge operators, X-type stabilizers and Z-type stabilizers, and X-type logical operators and Z-type logical operators. In these embodiments, the rotation component 202 can perform two pairwise swap gate operations between qubits that are symmetric with respect to the vertical and diagonal axes of the rotated subsystem code (as described above with reference to FIGS. 1, 2, and 3, for example). In these embodiments, the rotation component 202 rotates the rotated subsystem code (as described above with reference to FIGS. 1, 2, and 3, for example), and can generate a code (not shown in FIGS. 7(A) or 7(B)) that performs a fault-tolerant logical Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with the quantum device.

[0074] FIG. 8 shows a flow diagram of an exemplary and non - limiting computer - implemented method 800 that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. For the sake of brevity, descriptions of the same elements or processes or both that are used in the corresponding respective embodiments and are repeated are omitted.

[0075] At 802, the computer - implemented method 800 can include applying a gauge - fixing operation to subsystem code of encoded qubits (e.g., subsystem code 302) by a system operably coupled to a processor (e.g., processor 106) (e.g., via quantum gate operation system 102 or gauge - fixing component 108 or both) to generate switched subsystem code (e.g., switched subsystem code 304). For example, as described above with reference to FIGS. 1 - 6E, gauge - fixing component 108 can apply a gauge - fixing operation to subsystem code 302 to generate switched subsystem code 304.

[0076] At 804, the computer - implemented method 800 can include applying a transverse Hadamard operation to the switched subsystem code by this system (e.g., via quantum gate operation system 102 or transverse component 110 or both) to generate rotated subsystem code (e.g., rotated subsystem code 306). For example, as described above with reference to FIGS. 1, 2, and 3, transverse component 110 can apply a transverse Hadamard operation to switched subsystem code 304 to generate rotated subsystem code 306.

[0077] Although not shown in the exemplary embodiment shown in FIG. 8, in some embodiments, computer-implemented method 800 may further include, by this system (e.g., via quantum gate operation system 102 or rotation component 202 or both), rotating the rotated subsystem code (e.g., rotated subsystem code 306) to generate code (e.g., code 308) that performs a fault-tolerant logical Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with the quantum device. For example, as described above with reference to FIGS. 1, 2, and 3, rotation component 202 can rotate rotated subsystem code 306 to generate code 308 that performs a fault-tolerant logical Hadamard gate on a quantum code having the property of reducing at least one of frequency collisions or crosstalk errors associated with the quantum device.

[0078] FIG. 9 shows a flow diagram of an exemplary and non-limiting computer-implemented method 900 that can facilitate logical Hadamard gate operations and gauge fixing in subsystem code according to one or more embodiments described herein. For the sake of brevity, the description of the same elements or processes or both that are used in the corresponding respective embodiments and repeated is omitted.

[0079] At 902, the computer-implemented method 900 can include applying a gauge fixing operation to a subsystem code of encoded qubits (e.g., subsystem code 702) by a system operably coupled to a processor (e.g., processor 106), such as via a quantum gate operation system 102 or a gauge fixing component 108 or both, to generate a switched subsystem code (not shown). For example, as described above with reference to FIGS. 7(A) and 7(B), the gauge fixing component 108 can apply a gauge fixing operation to the subsystem code 702 to generate a switched subsystem code (not shown).

[0080] At 904, the computer-implemented method 900 can include shifting a lattice of the switched subsystem code (e.g., a qubit lattice) by this system, such as via a quantum gate operation system 102 or a lattice shift component 204 or both, to generate a shifted and switched subsystem code (e.g., shifted and switched subsystem code 704). For example, as described above with reference to FIGS. 7(A) and 7(B), the lattice shift component 204 can use the swap protocol described above to shift the qubit lattice of the switched subsystem code generated by the gauge fixing component 108, thereby generating the shifted and switched subsystem code 704.

[0081] Although not shown in the exemplary embodiment shown in FIG. 9, in some embodiments, computer-implemented method 900 may further include applying a transversal Hadamard operation to the shifted and switched subsystem code by this system (e.g., via quantum gate operation system 102 or transversal component 110 or both) to generate a rotated subsystem code (not shown). For example, as described above with reference to FIGS. 7(A) and 7(B), transversal component 110 may apply a transversal Hadamard operation to the shifted and switched subsystem code 704 (e.g., as described above with reference to FIGS. 1, 2, and 3) to generate a rotated subsystem code (not shown).

[0082] Although not shown in the exemplary embodiment shown in FIG. 9, in some embodiments, computer-implemented method 900 may further include rotating the rotated subsystem code by this system (e.g., via quantum gate operation system 102 or rotation component 202 or both) to generate code that executes a fault-tolerant logic Hadamard gate on a quantum code having a property of reducing at least one of frequency collisions or crosstalk errors associated with the quantum device. For example, as described above with reference to FIGS. 7(A) and 7(B), rotation component 202 may rotate the rotated subsystem code (e.g., as described above with reference to FIGS. 1, 2, and 3) to generate code (not shown) that executes a fault-tolerant logic Hadamard gate on a quantum code having a property of reducing at least one of frequency collisions or crosstalk errors associated with the quantum device.

[0083] The quantum gate operation system 102 can be associated with various technologies. For example, the quantum gate operation system 102 can be associated with quantum computing technology, quantum hardware and / or software technology, quantum gate scheduling technology, quantum algorithm technology, machine learning technology, artificial intelligence technology, cloud computing technology, and / or other technologies.

[0084] The quantum gate operation system 102 can provide technical improvements to systems, devices, components, operation steps or processing steps, or combinations thereof related to the various technologies identified above. For example, the quantum gate operation system 102 can apply a gauge fixing operation to the subsystem code of encoded qubits to generate a switched subsystem code, apply a transversal Hadamard operation to the switched subsystem code to generate a rotated subsystem code, or rotate the rotated subsystem code to generate a code that executes a fault-tolerant logic Hadamard gate on a quantum code having a property of reducing at least one of frequency collisions or crosstalk errors related to a quantum device (e.g., a quantum processor or a quantum computer or both that execute a code or a quantum code or both), or perform a combination thereof.

[0085] The quantum gate operation system 102 can provide technical improvements to a processing unit (such as processor 106, a quantum processor, or another processor, or a combination thereof) associated with the quantum gate operation system 102. For example, as described above, the quantum gate operation system 102 can apply a gauge fixing operation to the subsystem code of the encoded qubits to generate a switched subsystem code, apply a transversal Hadamard operation to the switched subsystem code to generate a rotated subsystem code, or rotate the rotated subsystem code to perform a fault-tolerant logic Hadamard gate on a quantum code having a property of reducing at least one of frequency collisions or crosstalk errors associated with a quantum device (such as a quantum processor or a quantum computer, or both, that executes such a code or such a quantum code, or both), or perform a combination thereof. In this example, the quantum gate operation system 102 can thereby reduce at least one of frequency collisions or crosstalk errors associated with a quantum device, such as a quantum processor that executes such a code or such a quantum code, or both. In this example, by reducing at least one of frequency collisions or crosstalk errors, such as at least one of frequency collisions or crosstalk errors associated with a quantum processor that executes such a code or such a quantum code, or both, the quantum gate operation system 102 can improve at least one of performance, accuracy, or fidelity, or a combination thereof, associated with the quantum processor, and reduce the computational cost associated with the quantum processor.

[0086] The practical use of the quantum gate operation system 102 is to use a quantum computing device (e.g., a quantum processor or a quantum computer or both) to generate code for executing a fault-tolerant logical Hadamard gate on a quantum code in order to calculate one or more solutions (e.g., discovery solutions) to various problems of different complexities (e.g., estimation problems, optimization problems or other problems or combinations thereof) in various fields (e.g., finance, chemistry, medicine or another field or combinations thereof). For example, the practical use of the quantum gate operation system 102 is one or more solutions (e.g., discovery solutions) to an estimation problem or an optimization problem or both in the field of chemistry, medicine or finance or combinations thereof, which can be used, for example, for the purpose of designing new compounds, new drug therapies or new option premiums or combinations thereof. It is possible to implement the quantum gate operation system 102 to generate code for executing a fault-tolerant logical Hadamard gate on a quantum code by using a quantum computing device (e.g., a quantum processor or a quantum computer or both).

[0087] It should be recognized that the quantum gate operation system 102 provides a new approach driven by relatively new quantum computing technology. For example, the quantum gate operation system 102 provides a new approach for generating code that executes a fault-tolerant logical Hadamard gate on a quantum code (e.g., a subsystem code such as, for example, a heavy hexagon code).

[0088] The quantum gate operation system 102 can solve problems that are highly technical in nature, not abstract, and cannot be performed as a set of mental acts by humans, using hardware or software. In some embodiments, one or more dedicated computers (e.g., dedicated processing units, dedicated classical computers, dedicated quantum computers or another type of dedicated computer or combinations thereof) can perform one or more of the processes described herein to perform defined tasks related to the various techniques identified above. The quantum gate operation system 102, or its components, or both, can be used to solve new problems arising from the use of the above-described technological advancements, quantum computing systems, cloud computing systems, computer architectures or other technologies or combinations thereof.

[0089] Since the various operations that can be performed by the quantum gate operation system 102 described herein, or its components, or both, are operations that exceed human intelligence, it should be recognized that the quantum gate operation system 102 can utilize various combinations of electrical components, mechanical components and circuits that cannot be replicated by human intelligence or performed by humans. For example, the amount of data processed by the quantum gate operation system 102 over a period of time, the speed at which such data is processed, or the type of data can be a greater amount, a faster speed, or a different type of data than can be processed by human intelligence over the same period of time.

[0090] According to some embodiments, the quantum gate operation system 102 can also be fully operable to perform one or more other functions (e.g., a fully power-on function, a fully execution function, or other functions or combinations thereof) while performing the various operations described herein. It should be recognized that such simultaneous multi-operation execution is beyond human intelligence. It should also be recognized that the quantum gate operation system 102 can contain information that is impossible for an entity such as a human user to manually obtain. The type, quantity, or diversity of information contained in the quantum gate operation system 102, the gauge fixing component 108, the transverse component 110, the rotation component 202, or the lattice shift component 204, or combinations thereof, can be more complex than the information manually obtained by a human user.

[0091] In some embodiments, the quantum gate operation system 102 can be associated with a cloud computing environment. For example, the quantum gate operation system 102 can be associated with a cloud computing environment 1150 described later with reference to FIG. 11, or one or more functional abstraction layers (e.g., the hardware and software layer 1260, the virtualization layer 1270, the management layer 1280, or the workload layer 1290, or combinations thereof) described later with reference to FIG. 12, or both.

[0092] The quantum gate operation system 102 or its components (e.g., the gauge fixing component 108, the transverse component 110, the rotation component 202, the lattice shift component 204, or another component or a combination thereof), or both, may use one or more computing resources of the cloud computing environment 1150, which will be described later with reference to FIG. 11, or one or more functional abstraction layers (e.g., quantum software), which will be described later with reference to FIG. 12, or both, to perform one or more operations based on one or more embodiments of the subject disclosure described herein. For example, for example, the cloud computing environment 1150, or such one or more functional abstraction layers, or a combination thereof, may be used by the quantum gate operation system 102 or its components or both to perform one or more operations based on one or more embodiments of the subject disclosure described herein, including one or more classical computing devices (e.g., classical computers, classical processors, virtual machines, servers, or another classical computing device or a combination thereof), quantum hardware, or quantum software (e.g., quantum computing devices, quantum computers, quantum processors, quantum circuit simulation software, superconducting circuits, and / or other quantum hardware and / or quantum software), or a combination thereof. For example, the quantum gate operation system 102 or its components or both may use such one or more classical and / or quantum: mathematical functions, calculations, and / or equations; computing and / or processing scripts; algorithms; models (e.g., artificial intelligence (AI) models, machine learning (ML) models, or another type of model or a combination thereof); and / or other operations based on one or more embodiments of the subject disclosure described herein, using such one or more classical or computing resources or both.

[0093] Although this disclosure includes a detailed description regarding cloud computing, it should be understood that the embodiments of the teachings described herein are not limited to only cloud computing environments. Rather, embodiments of the present invention can be implemented in connection with any other type of computing environment now known or later developed.

[0094] Cloud computing is a service delivery model that enables convenient on-demand network access to a shared pool of configurable computing resources (e.g., networks, network bandwidth, servers, processing, memory, storage, applications, virtual machines, and services) that can be rapidly provisioned and released with minimal management effort or interaction with the service provider. This cloud model can include at least five characteristics, at least three service models, and at least four deployment models.

[0095] The characteristics are as follows.

[0096] On-demand self-service: A cloud consumer can unidirectionally and automatically supply computing capabilities such as server time and network storage as needed without the need for human interaction with the service provider.

[0097] Broad network access: The capabilities are available over the network and are accessed through standard mechanisms that facilitate use by heterogeneous thin or thick client platforms (e.g., mobile phones, laptops, and PDAs).

[0098] Resource pooling: To provide services to a large number of consumers using a multi-tenant model, the provider's computing resources are pooled, and different physical and virtual resources are dynamically allocated and reallocated in response to requests. Consumers generally have a sense of location independence in that they cannot control or know the exact location of the provided resources, but can specify the location at a higher level of abstraction (e.g., country, state, or data center).

[0099] Rapid elasticity: The function can be quickly and elastically, and in some cases automatically, provisioned to scale out rapidly and released rapidly to scale in. To the consumer, the functions available for provisioning often appear to be infinite and can be purchased in any amount at any time.

[0100] Measured service: The cloud system automatically controls and optimizes resource usage by intervening in the metering function at an appropriate level of abstraction for the type of service (e.g., storage, processing, bandwidth, and active user accounts). Monitoring, controlling, and reporting resource usage can provide transparency to both the provider and the consumer of the services being utilized.

[0101] The service model is as follows.

[0102] Software as a Service (SaaS): This function provided to consumers is the function of using the provider's applications running on cloud infrastructure. These applications can be accessed from various client devices through a thin client interface such as a web browser (e.g., web-based email). Except for limited settings of user-specific application configurations where possible, consumers do not manage or control the underlying cloud infrastructure, including the network, servers, operating systems, storage, or individual application functions.

[0103] Platform as a Service (PaaS): This function provided to consumers is the function of deploying consumer-created or -acquired applications created using the programming languages and tools supported by the provider on cloud infrastructure. Consumers do not manage or control the underlying cloud infrastructure, including the network, servers, operating systems, or storage, but can control the deployed applications and, in some cases, the application hosting environment configuration.

[0104] Infrastructure as a Service (IaaS): This function provided to consumers is the function of supplying processing, storage, network, and other basic computing resources, and consumers can deploy and run any software, which can include operating systems and applications. Consumers do not manage or control the underlying cloud infrastructure, but can control the operating systems, storage, and deployed applications, and in some cases, can limitedly control selected network components (e.g., host firewalls).

[0105] The deployment model is as follows.

[0106] Private cloud: This cloud infrastructure is operated solely for an organization. The infrastructure can be managed by the organization or a third party and can exist on-premises or off-premises.

[0107] Community cloud: This cloud infrastructure is shared by several organizations and supports a specific community that shares interests (e.g., mission, security requirements, policies, and compliance issues). The infrastructure can be managed by the organization or a third party and can exist on-premises or off-premises.

[0108] Public cloud: This cloud infrastructure is available to the general public or large industrial groups and is owned by an organization that sells cloud services.

[0109] Hybrid cloud: This cloud infrastructure is a composite of two or more clouds (private, community, or public) that maintain distinct entities but are joined together by standardized or proprietary technologies (e.g., cloud bursting for load balancing between clouds) that enable data and application portability.

[0110] The cloud computing environment is a service-oriented environment that emphasizes statelessness, loose coupling, modularity, and semantic interoperability. At the center of cloud computing is an infrastructure that includes a network of interconnected nodes.

[0111] For simplicity of explanation, these computer-implemented methodologies are shown and described as a series of operations. The innovations of the subject matter are not limited by the operations shown, or by the order of the operations, or by both. For example, the operations can be performed in various orders or simultaneously, or both, and can be performed with other operations not shown herein and other operations not described herein. It should be understood and recognized that not all of the operations shown are necessary to implement the computer-implemented methodologies in accordance with the disclosed subject matter. Further, as an alternative, one of ordinary skill in the art will understand and recognize that the computer-implemented methodologies can be represented as a series of states related to each other by a state diagram or events. Further, it should be recognized that the computer-implemented methodologies disclosed below and throughout this specification can be stored on a product to facilitate transferring and conveying such computer-implemented methodologies to a computer. As used herein, the term product is intended to encompass a computer program accessible from a computer-readable device or storage medium.

[0112] To provide background for various aspects of the disclosed subject matter, FIG. 10 and the following discussion are intended to provide a general description of a suitable environment in which various aspects of the disclosed subject matter can be implemented. FIG. 10 shows a block diagram of an exemplary, non-limiting operating environment that can facilitate one or more embodiments described herein. For simplicity, repeated descriptions of the same elements used in other embodiments described herein are omitted.

[0113] Referring to FIG. 10, a suitable operating environment 1000 for implementing various aspects of the present disclosure can further include a computer 1012. The computer 1012 can further include a processing unit 1014, a system memory 1016, and a system bus 1018. The system bus 1018 couples system components including, but not limited to, the system memory 1016 to the processing unit 1014. The processing unit 1014 can be any of a variety of available processors. Dual microprocessors and other multiprocessor architectures can also be used as the processing unit 1014. The system bus 1018 can use any of a variety of available bus architectures, including, but not limited to, Industry Standard Architecture (ISA), Micro Channel Architecture (MCA), Enhanced ISA (EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB), Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus (USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and Small Computer Systems Interface (SCSI), to couple memory bus or memory controller, peripheral bus or external bus, or local bus, or combinations thereof, to any of several types of bus structures including any of these bus structures.

[0114] System memory 1016 can further include volatile memory 1020 and non-volatile memory 1022. The basic input / output system (BIOS) is stored in non-volatile memory 1022, and the BIOS includes basic routines for transferring information among elements within computer 1012, such as during startup. Computer 1012 can further include removable / non-removable volatile / non-volatile computer storage media. FIG. 10 shows, for example, disk storage 1024. Disk storage 1024 can further include devices such as, but not limited to, magnetic disk drives, floppy disk drives, tape drives, Jaz drives, Zip drives, LS-100 drives, flash memory cards, or memory sticks. Disk storage 1024 can further include storage media that are separate from, or combined with, other storage media. To facilitate connection of disk storage 1024 to system bus 1018, a removable or non-removable interface, such as interface 1026, is typically used. FIG. 10 further shows software that functions as an intermediary between the basic computer resources and the user within a suitable operating environment 1000. Such software can further include, for example, operating system 1028. Operating system 1028, which can be stored in disk storage 1024, functions to execute control and allocation of the resources of computer 1012.

[0115] System application 1030 utilizes the management of resources by operating system 1028 via, for example, program modules 1032 and program data 1034 stored in system memory 1016 or disk storage 1024. It should be recognized that the present disclosure can be implemented using various operating systems or combinations of operating systems. A user inputs commands or information into computer 1012 via input device 1036. Input device 1036 includes, but is not limited to, pointing devices such as a mouse, trackball, stylus, touch pad, keyboard, microphone, joystick, game pad, satellite antenna, scanner, TV tuner card, digital camera, digital video camera, web camera, etc. These input devices and other input devices are connected to processing unit 1014 through system bus 1018 via interface port 1038. Interface port 1038 includes, for example, serial port, parallel port, game port, and universal serial bus (USB). Output device 1040 uses some of the same type of ports as input device 1036. Thus, for example, a USB port can be used to provide input to computer 1012 and output information from computer 1012 to output device 1040. An output adapter 1042 is provided to show that there are some output devices 1040 that require a dedicated adapter, such as a monitor, speaker, and printer. By way of example, output adapter 1042 includes, but is not limited to, video cards and sound cards that provide connection means between output device 1040 and system bus 1018. It should be noted that other devices or device systems or both, such as remote computer 1044, provide both input and output functions.

[0116] Computer 1012 can operate using a logical connection to one or more remote computers, such as remote computer 1044, within a networked environment. Remote computer 1044 can be a computer, server, router, network PC, workstation, microprocessor-based device, peer device, or other common network node, and typically can further include many or all of the elements described with respect to computer 1012. For simplicity, only memory storage device 1046 is shown for remote computer 1044. Remote computer 1044 is logically connected to computer 1012 via network interface 1048 and then physically connected via communication connection 1050. Network interface 1048 includes wired and / or wireless communication networks such as local area networks (LANs), wide area networks (WANs), cellular networks, and / or other wired and / or wireless communication networks. LAN technologies include, but are not limited to, Fiber Distributed Data Interface (FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ring, etc. WAN technologies include, but are not limited to, point-to-point links, circuit-switched networks such as Integrated Services Digital Network (ISDN) and its variants, packet-switched networks, and Digital Subscriber Line (DSL). Communication connection 1050 refers to the hardware / software used to connect network interface 1048 to system bus 1018. For clarity of illustration, communication connection 1050 is shown inside computer 1012, but communication connection 1050 can also be placed outside computer 1012. By way of example only, the hardware / software for connecting to network interface 1048 can further include internal and external technologies such as conventional telephone-grade modems, cable modems, and DSL modems, ISDN adapters, and Ethernet cards.

[0117] Next, referring to FIG. 11, an exemplary cloud computing environment 1150 is shown. As shown, cloud computing environment 1150 includes one or more cloud computing nodes 1110, and local computing devices used by cloud consumers, such as personal digital assistants (PDAs) or mobile phones 1154A, desktop computers 1154B, laptop computers 1154C, or automotive computer systems 1154N, or combinations thereof, can communicate with these nodes. Although not shown in FIG. 11, cloud computing node 1110 can further include a quantum platform (such as a quantum computer, quantum hardware, quantum software, etc.) with which local computing devices used by cloud consumers can communicate. Nodes 1110 can communicate with each other. Those nodes may be physically or virtually grouped (not shown) into one or more networks, such as the private, community, public, or hybrid clouds described above, or combinations thereof. Thereby, cloud computing environment 1150 can provide infrastructure, platform, software, or combinations thereof as services, so that cloud consumers do not need to maintain resources on local computing devices. It is intended that computing devices 1154A - N of the type shown in FIG. 11 are merely examples, and that cloud computing node 1110 and cloud computing environment 1150 can communicate with any type of computerized device on any type of network or addressable network connection, or both (e.g., using a web browser).

[0118] Next, referring to FIG. 12, a set of functional abstraction layers provided by the cloud computing environment 1150 (FIG. 11) is shown. It should be understood in advance that the components, layers, and functions shown in FIG. 12 are merely examples and that embodiments of the present invention are not limited thereto. As shown, the following layers and corresponding functions are provided.

[0119] The hardware and software layer 1260 includes hardware components and software components. Examples of hardware components include mainframe 1261, RISC (Reduced Instruction Set Computer) architecture-based server 1262, server 1263, blade server 1264, storage device 1265, and network and networking components 1226. In some embodiments, software components include network application server software 1267, database software 1268, quantum platform routing software (not shown in FIG. 12), or quantum software (not shown in FIG. 12), or combinations thereof.

[0120] The virtualization layer 1270 provides an abstraction layer that can provide the following examples of virtual entities: virtual server 1271, virtual storage 1272, virtual network 1273 including a virtual private network, virtual applications and operating systems 1274, and virtual clients 1275.

[0121] In one example, the management layer 1280 can provide the following functions. Resource provisioning 1281 provides for the dynamic procurement of computing resources and other resources used for the purpose of executing tasks within a cloud computing environment. Metering and pricing 1282 provides for cost tracking when resources are utilized within a cloud computing environment and for charging or billing for the consumption of these resources. In one example, these resources may include application software licenses. Security provides for the identification and authentication of cloud consumers and tasks and for the protection of data and other resources. The user portal 1283 provides access to the cloud computing environment for consumers and system administrators. Service level management 1284 provides for the allocation and management of cloud computing resources such that the required service levels are achieved. Service Level Agreement (SLA) planning and fulfillment 1285 provides for the pre - adjustment and procurement of cloud computing resources expected to be required in the future in accordance with the SLA.

[0122] The workload layer 1290 provides examples of functions that can utilize a cloud computing environment. Non - limiting examples of workloads and functions that can be provided from this layer include mapping and navigation 1291, software development and life cycle management 1292, virtual classroom education delivery 1293, data analysis processing 1294, transaction processing 1295, and quantum gate operation software 1296.

[0123] The present invention can be a system, a method, an apparatus, or a computer program product, or a combination thereof, at any technical detail level capable of integration. The computer program product can include a computer-readable storage medium having thereon computer-readable program instructions for causing a processor to execute aspects of the present invention. This computer-readable storage medium can be a tangible device capable of holding and storing instructions for use by an instruction execution device. This computer-readable storage medium can be, for example, but not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or a suitable combination thereof. A non-exhaustive list of more specific examples of computer-readable storage media can further include a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), a static random access memory (SRAM), a portable compact disk read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanically encoded device such as a punch card or a raised structure in a groove having instructions recorded thereon, and suitable combinations thereof. As used herein, a computer-readable storage medium should not be construed to be a transitory signal per se, such as a radio wave or other freely propagating electromagnetic wave, an electromagnetic wave propagating through a waveguide or other transmission medium (e.g., an optical pulse through an optical fiber cable), or an electrical signal transmitted through a wire.

[0124] The computer-readable program instructions described herein can be downloaded from a computer-readable storage medium to respective corresponding computing / processing devices, or can be downloaded from an external computer or an external storage device via a network, such as the Internet, a local area network, a wide area network, or a wireless network, or a combination thereof. This network can include copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers or edge servers, or a combination thereof. The network adapter card or network interface within each computing / processing device receives the computer-readable program instructions from the network and transfers those computer-readable program instructions to be stored on a computer-readable storage medium within the respective corresponding computing / processing device. The computer-readable program instructions for carrying out operations of the present invention may be assembly instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state-setting data, or configuration data for integrated circuits, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages such as Smalltalk(R), C++, and procedural programming languages such as the "C" programming language or similar programming languages. These computer-readable program instructions may be executed entirely on the user's computer, partly on the user's computer, executed as a stand-alone software package, partly on the user's computer and partly on a remote computer, or can be executed entirely on a remote computer or a remote server.In the last scenario above, the remote computer can be connected to the user's computer via any type of network including a local area network (LAN) or a wide area network (WAN), or this connection can be implemented to an external computer (e.g., via the Internet using an Internet service provider). In some embodiments, to execute aspects of the present invention, an electronic circuit including, for example, a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA) can utilize the state information of the computer-readable program instructions to personalize the electronic circuit, thereby executing the computer-readable program instructions.

[0125] In this specification, aspects of the present invention are described with reference to flowcharts and / or block diagrams of methods, apparatuses (systems), and computer program products according to embodiments of the present invention. It is understood that each block of these flowcharts and / or block diagrams and combinations of blocks therein can be implemented by computer-readable program instructions. These computer-readable program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing apparatus to generate means for implementing the functions / operations specified in the blocks of these flowcharts and / or block diagrams and / or combinations thereof by the processor of these computers or other programmable data processing apparatuses. These computer-readable program instructions can further be stored in a computer-readable storage medium that can direct a computer, a programmable data processing apparatus, or other devices or combinations thereof to function in a particular manner, such that the computer-readable storage medium storing instructions therein includes a product that includes instructions for implementing the functions / operations specified in the blocks of these flowcharts and / or block diagrams and / or combinations thereof. These computer-readable program instructions can further be loaded onto a computer, other programmable apparatus, or other device to cause a series of operational steps to be executed on the computer, other programmable apparatus, or other device to generate a process implemented by the computer, such that the instructions executed on the computer, other programmable apparatus, or other device implement the functions / operations specified in the blocks of these flowcharts and / or block diagrams and / or combinations thereof.

[0126] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible embodiments of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in these flowcharts or block diagrams can represent a module, segment, or portion of instructions that include one or more executable instructions for implementing the specified logical function. In some alternative embodiments, the functions represented by these blocks can be performed in an order different from that shown in the figures. For example, two blocks shown in succession can in fact be executed substantially simultaneously, or, depending on the functions involved, these blocks can also be executed in the reverse order. It should also be noted that each block of these block diagrams or flowcharts or both, as well as combinations of blocks of these block diagrams or flowcharts or both, can be implemented by a dedicated hardware-based system that performs the specified function or operation or implements a combination of dedicated hardware and computer instructions.

[0127] While the foregoing has described the subject matter in the general context of computer-executable instructions of a computer program product that executes on one or more computers or both, those skilled in the art will recognize that the present disclosure may also be implemented in combination with other program modules. In general, a program module includes routines, programs, components, data structures, or other program modules, or combinations thereof, that perform particular tasks or implement particular abstract data types or both. Further, those skilled in the art will recognize that the computer-implemented methods of the present invention may also be implemented using other computer system configurations, including single-processor or multi-processor computer systems, minicomputing devices, mainframe computers, computers, handheld computing devices (e.g., PDAs, telephones), microprocessor-based or programmable consumer or industrial electronic devices, and the like. The illustrated embodiments may also be implemented in a distributed computing environment where tasks are performed by remote processing devices linked through a communications network. However, at least some aspects of the present disclosure may also be implemented on stand-alone computers. In a distributed computing environment, program modules may be located in both local and remote memory storage devices. For example, in one or more embodiments, computer-executable components may be executed from a memory that can include one or more distributed memory units or from a memory that can consist of one or more distributed memory units. As used herein, the terms "memory" and "memory unit" are interchangeable. Further, one or more embodiments described herein may be executed by a number of processors that are combined or cooperate to function in a distributed fashion to execute the code of the computer-executable components, for example, by executing code from one or more distributed memory units.As used herein, the term "memory" can include a single memory or memory unit at one location, or multiple memories or memory units at one or more locations.

[0128] As used in this application, the terms "component", "system", "platform", "interface", etc. can refer to an entity having one or more specific functions, an entity related to a computer or an entity related to an operational machine, or can include such an entity, or can be both. Entities disclosed herein can be hardware, a combination of hardware and software, software, or software in execution. For example, a component can be, but is not limited to, a process running on a processor, a processor, an object, an executable file, an execution thread, a program or a computer, or a combination thereof. As an example, both an application running on a server and the server can be components. One or more components can exist within a process or an execution thread or both, and a component can be limited to one computer or can be distributed among two or more computers or both. In other examples, corresponding respective components can be executed from various computer-readable media on which various data structures are stored. A component can communicate according to a signal having, for example, one or more data packets (e.g., data from one component interacting with another component via a signal within a local system, within a distributed system, or across a network such as the Internet or a combination thereof, together with other systems). As another example, a component can be a device having a specific function provided by a mechanical part operated by an electrical or electronic circuit, and this electrical or electronic circuit is operated by software or a firmware application executed by a processor.In such cases, the processor can be placed inside or outside the device and can execute at least a part of a software application or a firmware application. As another example, a component can be a device that provides a specific function via electronic components that do not include mechanical parts, and those electronic components can include a processor or other means for executing software or firmware that at least partially provides the functions of the electronic components. In one aspect, a component can emulate an electronic component via a virtual machine, for example, within a cloud computing system.

[0129] Furthermore, the term "or" is intended to mean an inclusive "or" rather than an exclusive "or". That is, unless otherwise specified or clear from the context, "X uses A or B" is intended to mean any of the natural inclusive permutations. That is, "X uses A or B" is satisfied under any of the following cases: X uses A, X uses B, or X uses both A and B. Further, unless otherwise specified or clear from the context that a singular form is indicated, the articles "a" and "an" used in this specification and the accompanying drawings should generally be construed to mean "one or more". As used herein, the terms "example" or "exemplary" or both are utilized to mean something that serves as an example, instance, or illustration. To avoid misunderstanding, the subject matter disclosed herein is not limited by such examples. Further, it is not necessary to interpret any aspect or design described herein as "example" or "exemplary" or both as being more preferred or advantageous than other aspects or designs, nor does it mean excluding equivalent exemplary structures and techniques known to those skilled in the art.

[0130] As used herein, the term "processor" can refer to substantially any computing processing unit or device, including but not limited to a single-core processor, a single-core processor with software multithreading capabilities, a multi-core processor, a multi-core processor with software multithreading capabilities, a multi-core processor with hardware multithreading technology, a parallel platform, and a parallel platform with distributed shared memory. Further, a processor can refer to an integrated circuit designed to execute the functions described herein, an application-specific integrated circuit (ASIC), a digital signal processing processor (DSP), a field-programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), discrete gate or transistor logic, discrete hardware components, or any combination thereof. Further, a processor can utilize nanoscale architectures such as, but not limited to, molecule-based and quantum dot-based transistors, switches, and gates to optimize space usage or enhance the performance of a user device. A processor can also be implemented as a combination of computing processing units. In the present disclosure, terms such as "store", "storage", "data store", "data storage", "database", and substantially any other information storage component related to the operation and function of a component are utilized to refer to an entity implemented as a "memory component" that is a "memory" or a component including a memory. It should be recognized that the memory or memory component or both described herein can be volatile memory or non-volatile memory, or can include both volatile memory and non-volatile memory.By way of example, non-volatile memory can include, but is not limited to, read-only memory (ROM), programmable ROM (PROM), erasable programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), flash memory, or non-volatile random access memory (RAM) (such as ferroelectric RAM (FeRAM)). Volatile memory can include RAM, which can function, for example, as an external cache memory. By way of example, and without limitation, many forms of RAM can be used, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM (RDRAM). Further, the disclosed memory components of the systems or computer-implemented methods herein are intended to include, but are not limited to, these types of memory and other suitable types of memory.

[0131] The foregoing are merely examples of systems and computer-implemented methods. Of course, it is impossible to describe every possible combination of components or computer-implemented methods for purposes of illustrating the present disclosure, but those skilled in the art can understand that many other combinations and permutations of the present disclosure are possible. Further, to the extent that terms such as "includes", "has", "possesses", etc. are used in the detailed description, claims, appendices, and drawings, such terms are intended to be inclusive in the same manner as the term "comprising" is construed when used as a transitional word in a claim.

[0132] The above description of various embodiments has been presented for purposes of illustration, and it is not intended that the above description be exhaustive or that it be limited to the disclosed embodiments only. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terms used herein have been selected to best explain the principles of the embodiments, a practical application, or a technical improvement over technologies found in the marketplace, or to enable those skilled in the art to understand the embodiments disclosed herein.

Claims

1. A system comprising a processor that executes computer-executable components stored in a memory wherein the computer-executable components include a gauge fixing component that applies a gauge fixing operation to a subsystem code of encoded qubits to generate a switched subsystem code, and a transversal component that applies a transversal Hadamard operation to the switched subsystem code to generate a rotated subsystem code A system.

2. The computer-executable components further include a rotation component that generates code for executing a fault-tolerant logical Hadamard gate on a quantum code having a property of reducing at least one of frequency collision or crosstalk error related to a quantum device by rotating the rotated subsystem code The system according to claim 1.

3. The system according to claim 2, wherein the rotation component executes two pairwise swap gate operations between qubits that are symmetric with respect to the vertical axis and the diagonal axis of the rotated subsystem code.

4. The system according to any one of claims 1 to 3, wherein the gauge fixing component applies the gauge fixing operation to the subsystem code to exchange two-body gauge operators and four-body gauge operators in a subset of the subsystem code.

5. The system according to any one of claims 1 to 4, wherein the transversal component applies the transversal Hadamard operation to the switched subsystem code to apply a Hadamard gate on data qubits and to exchange X-type gauge operators and Z-type gauge operators, X-type stabilizers and Z-type stabilizers, and X-type logical operators and Z-type logical operators.

6. A computer-implemented method comprising applying, by a system operably coupled to a processor, a gauge fixing operation to a subsystem code of encoded qubits to generate a switched subsystem code, and applying, by the system, a transversal Hadamard operation to the switched subsystem code to generate a rotated subsystem code A computer-implemented method including

7. Generating, by the system, code for performing a fault-tolerant logical Hadamard gate on a quantum code having a property of reducing at least one of frequency collision or crosstalk errors associated with a quantum device by rotating the rotated subsystem code The computer-implemented method according to claim 6, further comprising

8. Exchanging, by the system, two-body gauge operators and four-body gauge operators in a subset of the subsystem code by applying the gauge fixing operation to the subsystem code The computer-implemented method according to any one of claims 6 and 7, further comprising

9. Applying, by the system, the transversal Hadamard operation to the switched subsystem code to apply a Hadamard gate on data qubits and to exchange X-type gauge operators and Z-type gauge operators, X-type stabilizers and Z-type stabilizers, and X-type logical operators and Z-type logical operators The computer-implemented method according to any one of claims 6 to 8, further comprising

10. Performing, by the system, two pairwise swap gate operations between qubits that are symmetric with respect to the vertical and diagonal axes of the rotated subsystem code The computer-implemented method according to any one of claims 6 to 9, further comprising

11. A computer program, comprising Causing a computer to Apply a gauge fixing operation to a subsystem code of encoded qubits to generate a switched subsystem code, and Apply a transversal Hadamard operation to the switched subsystem code to generate a rotated subsystem code The computer program for causing the execution

12. Further causing the computer to Generate code for performing a fault-tolerant logical Hadamard gate on a quantum code having a property of reducing at least one of frequency collision or crosstalk errors associated with a quantum device by rotating the rotated subsystem code The computer program according to claim 11, for causing the execution

13. Further causing the computer to Apply the gauge fixing operation to the subsystem code to swap the two-body gauge operator and the four-body gauge operator in a subset of the subsystem code The computer program according to any one of claims 11 and 12, which causes the above to be executed

14. Further, to the computer Apply the transversal Hadamard operation to the switched subsystem code to apply a Hadamard gate on the data qubits and swap the X-type gauge operator and the Z-type gauge operator, the X-type stabilizer and the Z-type stabilizer, and the X-type logical operator and the Z-type logical operator The computer program according to any one of claims 11 to 13, which causes the above to be executed

15. Further, to the computer Execute two pairwise swap gate operations between qubits that are symmetric with respect to the vertical axis and the diagonal axis of the rotated subsystem code The computer program according to any one of claims 11 to 14, which causes the above to be executed

16. A system comprising A processor that executes computer-executable components stored in a memory And the computer-executable component includes A gauge fixing component that applies a gauge fixing operation to the subsystem code of the encoded qubits to generate a switched subsystem code, and A lattice shift component that shifts the lattice of the switched subsystem code to generate a shifted and switched subsystem code A system

17. The computer-executable component further includes A transversal component that applies a transversal Hadamard operation to the shifted and switched subsystem code to generate a rotated subsystem code, and A rotation component that rotates the rotated subsystem code to generate a code that executes a fault-tolerant logical Hadamard gate having a property of reducing at least one of frequency collision or crosstalk error related to a quantum device The system according to claim 16, which includes

18. The system of claim 17, wherein the transverse component applies the transversal Hadamard operation to the shifted and switched subsystem code to apply a Hadamard gate on the data qubits and to swap X-type gauge operators with Z-type gauge operators, X-type stabilizers with Z-type stabilizers, and X-type logical operators with Z-type logical operators.

19. The system according to any one of claims 17 and 18, wherein the rotation component performs two pairwise swap gate operations between qubits that are symmetric with respect to the vertical and diagonal axes of the rotated subsystem code.

20. The system according to any one of claims 16 to 19, wherein the gauge fixing component applies the gauge fixing operation to the subsystem code to swap two-body gauge operators and four-body gauge operators in a subset of the subsystem code.

21. A computer-implemented method comprising: applying, by a system operably coupled to a processor, a gauge fixing operation to a subsystem code of encoded qubits to generate a switched subsystem code; and shifting, by the system, a lattice of the switched subsystem code to generate a shifted and switched subsystem code A computer-implemented method.

22. generating, by the system, a rotated subsystem code by applying a transversal Hadamard operation to the shifted and switched subsystem code; and generating, by the system, code for performing a fault-tolerant logical Hadamard gate on a quantum code having a property of reducing at least one of frequency collisions or crosstalk errors associated with a quantum device by rotating the rotated subsystem code The computer-implemented method of claim 21, further comprising.

23. The system applies the transversal Hadamard operation to the shifted and switched subsystem code to apply a Hadamard gate on the data qubits and to swap X-type gauge operators with Z-type gauge operators, X-type stabilizers with Z-type stabilizers, and X-type logical operators with Z-type logical operators The computer-implemented method according to claim 22, further comprising the above

24. The system executes two pairwise swap gate operations between qubits that are symmetric with respect to the vertical and diagonal axes of the rotated subsystem code The computer-implemented method according to any one of claims 22 and 23, further comprising the above

25. The system applies the gauge fixing operation to the subsystem code to swap two-body gauge operators and four-body gauge operators in a subset of the subsystem code The computer-implemented method according to any one of claims 21 to 24, further comprising the above

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