Noise learning in dynamic quantum circuits

Abstract

This invention provides a system and techniques for facilitating noise learning in dynamic quantum circuits. In various embodiments, the system can learn noise associated with non-unit operations in the circuit of a dynamic quantum circuit by correcting non-unit operations in the circuit using stochastic Pauli-Z gates and orbiting Pauli operators. (Figure 1)

Description

[Background technology]

[0001] [Government licensing rights] This invention was made with government support granted by the Army Research Bureau, W911NF-21-1-0002. The government has certain rights to this invention.

[0002] This disclosure relates to quantum circuits, and more specifically, to noise learning in dynamic quantum circuits.

[0003] A dynamic quantum circuit can be a quantum circuit that includes one or more unit-apart operations along the circuit, for example, a qubit measurement along the circuit being fedforward to a classically controlled quantum gate. Dynamic quantum circuits can be considered to have higher computational power and greater versatility than ordinary quantum circuits. However, unfortunately, noise in dynamic quantum circuits is poorly understood, and therefore there are few effective error mitigation or suppression techniques for dynamic quantum circuits. In fact, existing error mitigation or suppression techniques are only effectively or efficiently suited to ordinary quantum circuits where all unit-apart operations are pushed to the end of the circuit. Such existing error mitigation or suppression techniques are incompatible with unit-apart operations along the circuit.

[0004] Therefore, a system or technology that can address one or more of these technical problems would be desirable. [Overview of the project]

[0005] The following is an overview to provide a basic understanding of one or more embodiments of the invention. This overview is not intended to identify any important or essential elements or to define any scope of any particular embodiment or any scope of any claim. Its sole purpose is to present the concepts in a simplified form as a prelude to a more detailed description to be presented later. One or more embodiments described herein describe a device, system, method, or apparatus that can facilitate noise learning in dynamic quantum circuits.

[0006] According to one or more embodiments, a system is provided. In various embodiments, the system may comprise a processor capable of executing computer-executable components stored in non-temporary computer-readable memory. In various cases, the computer-executable components may comprise learning a learning component capable of learning noise associated with unity operations in the circuit of a dynamic quantum circuit by correcting unity operations in the circuit with stochastic Pauli-Z gates and orbiting Pauli operators.

[0007] According to one or more embodiments, a computer implementation method is provided. In various embodiments, the computer implementation method may include a step of learning, by a device operablely coupled to a processor, the noise associated with non-unit operations in the middle of a dynamic quantum circuit by correcting non-unit operations in the middle of the circuit with a stochastic Pauli-Z gate and a swirling Pauli operator.

[0008] According to one or more embodiments, a computer program product for facilitating noise learning in dynamic quantum circuits is provided. In various embodiments, the computer program product may comprise a non-temporary computer-readable memory having program instructions embodied therein. In various cases, the program instructions may be executable to cause a processor to learn noise associated with non-unit operations in the circuit of a dynamic quantum circuit by modifying non-unit operations in the circuit with stochastic Pauli-Z gates and orbiting Pauli operators.

[0009] Various other details of the various embodiments described herein are provided in the following sections:

[0010] Item 1: A system comprising a processor that executes computer-executable components stored in a non-transitory computer-readable memory, the computer-executable components having a learning component that learns noise associated with non-unitary operations in the middle of a dynamic quantum circuit by correcting non-unitary operations in the middle of the circuit with probabilistic Pauli-Z gates and rotation Pauli operators.

[0011] Item 2: The system according to any one of the preceding clauses, wherein the non-unitary operations in the middle of the circuit include qubit measurements in the middle of the circuit that feed forward to at least one classically controlled quantum gate.

[0012] Item 3: The system according to any one of the preceding clauses, wherein the at least one classically controlled quantum gate is between rotation Pauli operators.

[0013] Item 4: The system according to any one of the preceding clauses, wherein the at least one classically controlled quantum gate is not between rotation Pauli operators.

[0014] Item 5: The system according to any one of the preceding clauses, wherein the learning component repeatedly executes non-unitary operations in the middle of the circuit corrected by the probabilistic Pauli-Z gates and the rotation Pauli operators over a set of Pauli bases and over a set of repetition depths, and extracts a set of basis fidelities corresponding to the set of Pauli bases respectively based on such repeated executions to learn the noise.

[0015] Item 6: The system according to any one of the preceding clauses, wherein the learning component learns the noise associated with the non-unitary operations in the middle of the circuit by inverting the set of basis fidelities through an exchange relation defined between the set of Pauli bases and the set of Pauli generators in relation to the noise.

[0016] Item 7: The system according to any one of the preceding clauses, wherein the computer-executable component further includes a mitigation component that mitigates the noise by inserting the reciprocal of the noise into the dynamic quantum circuit.

[0017] In various aspects, any single or plural combination of Items 1 to 7 can be implemented.

[0018] Item 8: A computer-implemented method comprising the step of learning noise associated with non-unitary operations in the middle of a dynamic quantum circuit by a device operably coupled to a processor and correcting the non-unitary operations in the middle of the circuit with probabilistic Pauli-Z gates and rotation Pauli operators.

[0019] Item 9: The computer-implemented method according to any one of the preceding clauses, wherein the non-unitary operation in the middle of the circuit includes quantum bit measurement in the middle of the circuit that feeds forward to at least one classically controlled quantum gate.

[0020] Item 10: The computer-implemented method according to any one of the preceding clauses, wherein the at least one classically controlled quantum gate is between the rotation Pauli operators.

[0021] Item 11: The computer-implemented method according to any one of the preceding clauses, wherein the at least one classically controlled quantum gate is not between the rotation Pauli operators.

[0022] Item 12: The step of learning the noise includes the step of repeatedly executing, by the device, non-unitary operations in the middle of the circuit corrected with the probabilistic Pauli-Z gates and the rotation Pauli operators over both a set of Pauli bases and a set of repetition depths; and the step of extracting, by the device, a set of base fidelities corresponding to the set of Pauli bases based on such repeated executions. The computer-implemented method according to any one of the preceding clauses.

[0023] Item 13: The computer implementation method according to any one of the preceding clauses, wherein the step of learning the noise comprises the step of the device inverting the set of base fidelity via a defined exchange relationship between the set of Pauli bases and the set of Pauli generators associated with the noise.

[0024] Item 14: A computer implementation method according to any one of the preceding clauses, further comprising the step of reducing the noise by inserting the reciprocal of the noise into the dynamic quantum circuit using the device.

[0025] In various forms, any single or multiple combination of items 8-14 can be implemented.

[0026] Item 15: A computer program product for facilitating noise learning in a dynamic quantum circuit, the computer program product comprising a non-temporary computer-readable memory having program instructions embodied therein, wherein the program instructions, which are executable by a processor, cause the processor to learn noise associated with a non-unit operation in the circuit of a dynamic quantum circuit by modifying the non-unit operation in the circuit with a stochastic Pauli-Z gate and a swirling Pauli operator.

[0027] Item 16: A computer program product as described in any of the preceding clauses, wherein the non-unit operations in the circuit include a qubit measurement in the circuit that feeds forward to at least one classically controlled quantum gate.

[0028] Item 17: The computer program product described in any one of the preceding clauses, wherein the at least one classically controlled quantum gate lies between the orbital Pauli operators.

[0029] Item 18: The computer program product described in any one of the preceding clauses, wherein the at least one classically controlled quantum gate is not between the orbital Pauli operators.

[0030] Item 19: The computer program product described in any of the preceding clauses, wherein the program instruction is executable to cause the processor to repeatedly perform the non-unit operation in the middle of the circuit, modified by the stochastic Pauli-Z gate and the orbital Pauli operator, over a set of Pauli bases and over a set of iteration depths; and further to extract a set of base fidelity corresponding to each set of Pauli bases, based on such repeated executions.

[0031] Item 20: The computer program product described in any of the preceding clauses, wherein the program instructions are executable to cause the processor to further perform, by the device, invert the set of base fidelity via exchange relations defined between the set of Pauli bases and the set of Pauli generators relating to the noise.

[0032] In various embodiments, any single or multiple combination of items 15-20 can be implemented. [Brief explanation of the drawing]

[0033] [Figure 1] This figure shows a block diagram of an exemplary non-restrictive system that facilitates noise learning in a dynamic quantum circuit according to one or more embodiments described herein.

[0034] [Figure 2] This figure shows an exemplary, non-limiting diagram of a dynamic quantum circuit according to one or more embodiments described herein.

[0035] [Figure 3]This figure shows a block diagram of an exemplary non-restrictive system that facilitates noise learning in a dynamic quantum circuit according to one or more embodiments described herein, including a stochastic Pauli-Z gate, a swirling Pauli operator, and a set of learned noise model coefficients.

[0036] [Figure 4] This figure shows an exemplary, non-restrictive block diagram of a non-unit operation in the middle of a circuit modified with a stochastic Pauli-Z gate and a swirling Pauli operator, according to one or more embodiments described herein. [Figure 5] This figure shows an exemplary, non-restrictive block diagram of a non-unit operation in the middle of a circuit modified with a stochastic Pauli-Z gate and a swirling Pauli operator, according to one or more embodiments described herein.

[0037] [Figure 6] This figure shows an exemplary, non-limiting block diagram illustrating how noise associated with unitless operations in a circuit can be constructed using a noise model Pauli generator and noise model coefficients according to one or more embodiments described herein.

[0038] [Figure 7] This figure shows an exemplary, non-restrictive block diagram illustrating how observable expectations can be obtained for unitless operations in a circuit according to one or more embodiments described herein.

[0039] [Figure 8] This figure shows an exemplary, non-restrictive block diagram illustrating how multiple sets of observable expectation values ​​can be obtained by one or more embodiments described herein.

[0040] [Figure 9] This figure shows an exemplary, non-limiting block diagram illustrating how base fidelity can be obtained from multiple sets of observable expectation values ​​by one or more embodiments described herein.

[0041] [Figure 10] This figure shows an exemplary, non-restrictive block diagram illustrating how noise model coefficients learned from basis fidelity can be obtained by one or more embodiments described herein.

[0042] [Figure 11] The results of exemplary, non-limiting experiments by one or more embodiments described herein are shown below.

[0043] [Figure 12] This figure shows a block diagram of an exemplary, non-limiting system that includes an inverse noise operation to facilitate noise learning in a dynamic quantum circuit according to one or more embodiments described herein.

[0044] [Figure 13] This figure shows an exemplary, non-restrictive block diagram of a dynamic quantum circuit into which an inverse noise operation is inserted, according to one or more embodiments described herein. [Figure 14] This figure shows an exemplary, non-restrictive block diagram of a dynamic quantum circuit into which an inverse noise operation is inserted, according to one or more embodiments described herein.

[0045] [Figure 15] This figure shows the results of exemplary, non-limiting experiments according to one or more embodiments described herein. [Figure 16] This figure shows the results of exemplary, non-limiting experiments according to one or more embodiments described herein.

[0046] [Figure 17] This figure shows a flowchart illustrating an exemplary non-restrictive computer implementation method for facilitating noise learning in dynamic quantum circuits according to one or more embodiments described herein.

[0047] [Figure 18]A block diagram of an example non-limiting operating environment in which one or more embodiments described herein may be facilitated is shown. [Modes for carrying out the invention]

[0048] The following detailed description is illustrative only and is not intended to limit the embodiments or the application or use of any embodiments. Furthermore, it is not intended to be bound by any express or implied information presented in the preceding background art or summary section of the invention, or the section on embodiments for carrying out the invention.

[0049] Herein, one or more embodiments are described with reference to the drawings, and throughout, similar reference numerals are used to refer to similar elements. In the following description, for illustrative purposes, numerous specific details are provided to give a more complete understanding of one or more embodiments. However, it is clear that in various cases one or more embodiments may be carried out without these specific details.

[0050] A quantum computer can be any suitable device that utilizes a qubit lattice (e.g., multiple superconducting qubits fabricated on one or more quantum substrates and exhibiting any suitable connection topology) for information processing. A quantum circuit can be any suitable sequence of any number of parallel or series quantum gates that can be executed on a quantum computer. A quantum gate can be a basic component of a quantum circuit that can change, modify, or influence the state of a qubit. In some non-restrictive examples, a quantum gate can be any suitable single-qubit gate (e.g., Pauli X gate, Pauli Y gate, Pauli Z gate, phase gate, rotation gate, Hadamard gate) or any suitable entangling or two-qubit gate (e.g., controlled NOT gate, controlled phase gate).

[0051] A typical quantum circuit can be a quantum circuit in which its only non-unit operation is a qubit measurement located at the end of the quantum circuit (e.g., in the final layer). In other words, a typical quantum circuit may not include any non-unit operations in the middle of the circuit. In contrast, a dynamic quantum circuit can be a quantum circuit that includes one or more non-unit operations in the middle of the circuit. As an unrestricted example, a non-unit operation in the middle of the circuit may include a qubit measurement in the middle of the circuit. In various embodiments, a qubit measurement in the middle of the circuit may be a qubit measurement operation that is not pushed to the end of the circuit. In other words, a qubit measurement in the middle of the circuit may be a qubit measurement operation that is performed after at least one first quantum gate of the circuit and before at least one second quantum gate of the circuit. In various cases, a qubit measurement in the middle of the circuit may be feedforward to one or more classically controlled quantum gates. In various embodiments, a classically controlled quantum gate can be any suitable quantum gate, and its performance depends on, or is controlled by, a qubit measurement operation (for example, if the qubit measurement operation yields a measurement of 0, the classically controlled quantum gate may not be executed; instead, if the qubit measurement operation yields a measurement of 1, the classically controlled quantum gate may be executed). In various cases, the term "classically controlled" can be used to refer to such a quantum gate because the qubit measurement operation collapses the superposition quantum state of the qubit into a classical bit that has only two possible values, 0 or 1.

[0052] Dynamic quantum circuits can be considered to have greater computational power and greater versatility compared to conventional quantum circuits. In fact, if a dynamic quantum circuit includes one or more intermediate qubit measurements, one or more of these intermediate qubit measurements can, in some cases, influence in real time which quantum gates are applied or not applied for the remainder of the dynamic quantum circuit. For example, consider a dynamic quantum circuit that includes an intermediate qubit measurement for a first qubit, and such an intermediate qubit measurement feedforward to a particular quantum gate for a second qubit. In various cases, if the intermediate qubit measurement obtains a value of 1, the particular quantum gate can be applied to the second qubit. However, if the intermediate qubit measurement instead obtains a value of 0, the particular quantum gate may not be applied to the second qubit. This flexibility of dynamic quantum circuits can help reduce circuit depth or improve qubit connectivity constraints.

[0053] However, the implementation of dynamic quantum circuits can be hindered by the lack of effective error mitigation or suppression techniques. Quantum circuits, whether conventional or dynamic, can be noisy (for example, single-qubit quantum gates may be noiseless, but entangling quantum gates and qubit measurements are not). Such noise can degrade circuit performance and is undesirable. Error mitigation or suppression techniques can be considered quantum computing protocols that can help reduce such noise. Unfortunately, existing error mitigation or suppression techniques, such as probabilistic error cancellation (PEC), are only effectively or efficiently applicable to conventional quantum circuits and not to dynamic quantum circuits. In particular, existing error mitigation techniques assume that all non-unit operations are pushed to the end of the circuit, and therefore cannot adequately learn or mitigate noise when non-unit operations are included in the middle of the circuit. Furthermore, existing error suppression techniques (e.g., circuit pauses or segmentation that allow noise to escape) are technically feasible for dynamic quantum circuits, and it has been empirically proven that such existing error suppression techniques provide far worse noise suppression in dynamic quantum circuits than in conventional quantum circuits. In other words, existing error mitigation or suppression techniques are incompatible with, or not effectively suited to, intermediate non-unit operations in the circuit, such as intermediate qubit measurements that feedforward to classically controlled quantum gates.

[0054] Therefore, a system or technology that can address one or more of these technical problems would be desirable.

[0055] The various embodiments described herein can address one or more of these technical problems. Specifically, the various embodiments described herein can facilitate noise learning in dynamic quantum circuits. That is, the inventors of the various embodiments described herein have devised various techniques for identifying, determining, measuring, or learning noise generated by intermediate non-unit operations, such as intermediate qubit measurements that feedforward to classically controlled quantum gates. In particular, such various techniques may include modifying intermediate non-unit operations with stochastic Pauli-Z gates and orbital Pauli operators. In various embodiments, the implementation of orbital Pauli operators may be considered as shaping the noise generated by intermediate non-unit operations, thereby the noise may be considered as a function of multiple Pauli generators corresponding to multiple noise coefficients. In various cases, such various techniques may further include repeatedly executing intermediate non-unit operations modified with stochastic Pauli-Z gates and orbital Pauli operators over multiple Pauli bases and over multiple iteration depths. In various cases, measurements taken during such repeated executions can yield multiple basis fidelity values ​​associated with multiple Pauli bases. In various embodiments, multiple noise coefficients can be determined by inverting multiple basis fidelity values ​​via exchange relations between multiple Pauli bases and multiple Pauli generators. When multiple noise coefficients are determined, the noise generated by non-unit operations in the circuit can be considered learned. In various cases, such learned noise can be inverted, and such inverted noise can be executed before non-unit operations in the circuit, thereby mitigating or reducing the noise generated by non-unit operations in the circuit. However, this is merely an unrestricted example. In various other cases, such learned noise can be used for any appropriate purpose, such as benchmarking techniques, designing noise-matched quantum algorithms or quantum error correction protocols, facilitating real-time device tuning, or facilitating dynamic decoupling.

[0056] Various embodiments described herein may be considered computerized tools (e.g., any preferred combination of computer-executable hardware or computer-executable software) capable of facilitating noise learning in dynamic quantum circuits. In various embodiments, such computerized tools may comprise access components, learning components, or mitigation components.

[0057] In various embodiments, a quantum computer may exist. In various embodiments, a quantum computer may have any suitable number of qubits. In various cases, such qubits may exhibit any suitable structure, configuration, or architecture (which may be, for example, superconducting qubits, spin qubits, or quantum dots).

[0058] In various cases, there can be dynamic quantum circuits. In various embodiments, a dynamic quantum circuit can be any suitable quantum circuit that can be implemented or executed on a quantum computer and may include non-unit operations along the circuit. As a non-restrictive example, a non-unit operation along the circuit may be a qubit measurement operation along the circuit that can be fedforward to a subsequent classically controlled quantum gate of the dynamic quantum circuit.

[0059] In various cases, it may be desirable to learn or reduce any noise generated by non-unit operations in the circuit. In various cases, the computerized tools described herein can learn or reduce such noise.

[0060] In various embodiments, the access component of a computerized tool may electronically access a quantum computer via any suitable wired or wireless electronic connection. In various cases, the access component may further access or receive, acquire, or import dynamic quantum circuits from any suitable source. For example, the access component may acquire dynamic quantum circuits from any suitable centralized or distributed data structure (e.g., graph data structure, relational data structure, hybrid data structure), whether remote or local from the access component. In any case, the access component may access a quantum computer or dynamic quantum circuits, thereby allowing other components of the computerized tool to electronically interact with the quantum computer (e.g., power up, power down, initialize, control) or with dynamic quantum circuits (e.g., read, write, edit, copy, manipulate, perform).

[0061] In various embodiments, the learning component of a computerized tool can electronically learn noise generated by non-unit operations in the circuit based on stochastic Pauli-Z gates and orbital Pauli operators. More specifically, the stochastic Pauli-Z gate may be a Pauli-Z gate that is performed with a 50% probability rather than a 100% probability. Furthermore, the orbital Pauli operator may be a random tensor product of Pauli gates. In various cases, the learning component may apply the stochastic Pauli-Z gate to any qubit, thereby performing non-unit operations in the circuit such that the stochastic Pauli-Z gate follows (e.g., occurs after) the non-unit operations in the circuit. In various cases, the learning component may sandwich both the non-unit operations in the circuit and the stochastic Pauli-Z gate with orbital Pauli operators.

[0062] In various embodiments, a stochastic Pauli-Z gate can be thought of as eliminating, excluding, or reducing any post-implementation errors that may arise from non-unit operations in the circuit. In various cases, a revolving Pauli operator can be thought of as shaping noise generated by non-unit operations in the circuit into a Pauli structure (e.g., a diagonal structure). In various cases, based on such shaping, the noise generated by non-unit operations in the circuit can be thought of as a function obtained by weighting a set of Pauli generators, each with a set of coefficients. In various embodiments, the values ​​or magnitudes of such coefficients may be initially unknown.

[0063] In various cases, the learning component can determine such coefficients and thus determine the noise generated by non-unit operations in the circuit. In particular, the learning component can repeatedly perform non-unit operations in the circuit, modified by stochastic Pauli-Z gates and orbiting Pauli operators, over a set of Pauli bases and over a set of gradually increasing iteration depths. In various cases, the learning component can measure the observable expectation value for each base depth pair, thereby obtaining a set of observable expectation values ​​for each Pauli base. In various embodiments, for each particular Pauli base, the learning component can fit an exponential decay curve to the set of observable expectation values ​​corresponding to that particular Pauli base, and such fitted exponential decay curve can obtain the base fidelity corresponding to that particular Pauli base. In other words, the learning component can obtain a set of base fidelity corresponding to each set of Pauli bases. In various cases, the learning component can determine, identify, or estimate values ​​for unknown coefficients corresponding to a set of Pauli generators by inverting the set of basis fidelity via a defined commutation relation between the set of Pauli basis and the set of Pauli generators. In various cases, the commutation relation can be obtained by applying the symplectic inner product to each basis-generator pair. Thus, the learning component can determine the set of coefficients corresponding to the set of Pauli generators, and therefore can be considered to have determined the noise generated by non-unit operations in the circuit.

[0064] In various embodiments, mitigation components of computerized tools can mitigate noise generated by non-unit operations in the circuit. More specifically, since noise can be considered a function of a set of Pauli generators and a set of coefficients, noise can be considered a quantum operation that can be performed on an active quantum gate or quantum computer. Accordingly, the mitigation component can compute the inverse operation of the noise, where the inverse operation is similarly a quantum operation that can be performed on an active quantum gate or quantum computer. Therefore, the mitigation component can insert the inverse operation into the dynamic quantum circuit before the non-unit operation in the circuit, and such an inverse operation may generate noise from the non-unit operation in the circuit that is canceled or reduced.

[0065] Accordingly, the various embodiments described herein can be considered as computerized tools capable of learning noise generated by or associated with non-unit operations in the circuit of a dynamic quantum circuit. Once learned, such noise can be mitigated by inversion. However, this is merely an unrestricted example. In other cases, once learned, such noise can be utilized for any other suitable purpose (for example, such noise can be utilized to create noise-adapted quantum protocols).

[0066] The various embodiments described herein may be used, utilizing hardware or software, to solve problems that are inherently highly technical, non-abstract, and cannot be performed by humans as a set of mental activities (for example, to facilitate noise learning in dynamic quantum circuits). Furthermore, some of the processes to be performed may be carried out by a dedicated computer (for example, a quantum computer containing tangible qubits capable of implementing or integrating dynamic quantum circuits). In various embodiments, some defined tasks related to the various embodiments described herein may include a step of learning, by a device operably coupled to a processor, noise associated with non-unit operations in the circuit of a dynamic quantum circuit by correcting non-unit operations in the circuit with stochastic Pauli-Z gates and orbiting Pauli operators.

[0067] Neither the human mind nor a person with a pen and paper can electronically access dynamic quantum circuits, including non-unit operations in the circuit (e.g., feedforward measurement of qubits in the circuit), nor can they electronically learn the noise generated by non-unit operations in the circuit by applying stochastic Pauli-Z gates and orbital Pauli operators to those non-unit operations. Ultimately, a quantum computer is a dedicated computing hardware that processes information using physical qubits (e.g., superconducting qubits such as transmons). Physical qubits cannot be implemented by the human mind or by a person with a pen and paper. Furthermore, a quantum circuit can be a sequence of quantum gates that can be implemented on a quantum computer. Neither the human mind nor a person with a pen and paper can implement quantum gates (e.g., stochastic Pauli-Z gates, orbital Pauli operators) on physical qubits. Therefore, a computerized tool capable of learning the noise generated by non-unit operations in a circuit through the implementation of stochastic Pauli-Z gates and orbiting Pauli operators is inherently computerized and cannot be implemented in any clever, practical, or rational way without a computer.

[0068] In various cases, one or more embodiments described herein may integrate the teachings described herein into practical applications. As stated above, existing error mitigation or suppression techniques (e.g., PEC) are effectively suited only to ordinary quantum circuits (e.g., quantum circuits where all non-unit operations, such as qubit measurements, are pushed to the circuit termination). In fact, such existing error mitigation or suppression techniques are specifically designed to learn noise generated by or associated with unit operations, such as Clifford layers, rather than noise generated by or associated with non-unit operations, such as qubit measurements. Indeed, unit operations may contain noise, but nevertheless they can be considered reversible operations and thus can fully preserve quantum information. Obviously, in contrast, non-unit operations are irreversible and can therefore be considered not to fully preserve quantum information. In other words, even if noiseless, non-unit operations can be considered to irreversibly discard at least some quantum information. Existing error mitigation or suppression techniques cannot learn noise in such discarded quantum information. Additionally, noise generated by unit operations can only affect the qubits on which the unit operation is performed. That is, noise from unit operations can propagate at most through directly coupled adjacent qubits. Clearly in contrast, noise generated by non-unit operations can affect any or all qubits in a quantum computer. In other words, noise from non-unit operations can propagate not only through directly coupled adjacent qubits, but also through qubit readout lines or non-adjacent qubits. This makes noise from non-unit operations very complex, device-dependent, and difficult to handle. For these reasons at least, existing error mitigation or suppression techniques are not effectively suited to non-unit operations in the middle of a circuit.

[0069] The various embodiments described herein can address one or more of these technical problems of existing error mitigation or suppression techniques. In other words, the various embodiments described herein can enable noise generated by non-unit operations in the circuit that are learned. Indeed, as described herein, the inventors have recognized that noise associated with non-unit operations in the circuit can be learned by modifying those non-unit operations in the circuit with stochastic Pauli-Z gates and orbital Pauli operators. In particular, such modified non-unit operations in the circuit can be performed iteratively across multiple Pauli bases and across multiple iteration depths. In various embodiments, base fidelity can be extracted from measurements taken during such iterative executions, and noise associated with non-unit operations in the circuit can be identified by inverting their base fidelity via commutation relations corresponding to multiple Pauli bases. Thus, noise generated by non-unit operations in the circuit can be learned, identified, or determined. In various cases, such learned noise can be utilized for any appropriate purpose. As a non-limiting example, the inverse operation of such learned noise may be computed, and such inverse operation may be inserted or positioned before non-unit operations in the circuit, thereby eliminating, mitigating, or reducing the noise of non-unit operations in the circuit. In other words, the various embodiments described herein can learn or mitigate noise generated by non-unit operations in the circuit, in stark contrast to existing error mitigation or suppression techniques that cannot effectively learn or mitigate such noise.

[0070] Learning or mitigating noise generated by non-unit operations in a circuit can be seen as providing a quantifiable performance improvement in the field of quantum circuits. By learning or mitigating such noise, a quantum computer may be able to achieve meaningful quantum computation results using fewer shots (e.g., with less operation than normally required). As another example, learning or mitigating such noise may enable the construction and implementation of more complex or advanced quantum circuits that would otherwise be impossible. As yet another example, learning or mitigating such noise may allow a quantum computer to operate with less pre-processing or post-processing. Thus, learning or mitigating noise associated with non-unit operations in a circuit can be seen as helping to reduce or optimize the use of quantum computing resources. This is a concrete and visible technical improvement in the field of quantum circuits. For at least these reasons, the various embodiments described herein certainly constitute useful and practical applications of computers.

[0071] It should be understood that the drawings and disclosures herein illustrate non-limiting examples of various embodiments. Furthermore, it should be noted that the drawings are not necessarily drawn to scale.

[0072] Figure 1 shows a block diagram of an example of a non-restrictive system 100 that can facilitate noise learning in a dynamic quantum circuit according to one or more embodiments described herein. As shown, the dynamic circuit noise learning and mitigation system 102 can be electronically integrated with a quantum computer 104 and a dynamic quantum circuit 108 via any suitable wired or wireless electronic connection.

[0073] In various embodiments, the quantum computer 104 may be any suitable quantum computing device or quantum computing hardware. In various embodiments, the quantum computer 104 may include a set of qubits 106. In various embodiments, the set of qubits 106 may include n qubits for any suitable positive integer from qubit 106(1) to qubit 106(n). In various embodiments, any of the qubits in the set of qubits 106 may represent any suitable structure or architecture. As an unrestricted example, qubits from the set of qubits 106 may represent a superconducting qubit architecture (for example, such qubits can be constructed from any number of suitable Josephson junctions shunted by any number of suitable planar capacitor pads). As another unrestricted example, qubits from the set of qubits 106 may represent a quantum dot architecture. As yet another unrestricted example, qubits from the set of qubits 106 may represent a spin qubit architecture. In various embodiments, different qubits of the set of qubits 106 may represent the same or different structures or architectures as one another. In various cases, the set of qubits 106 can represent any suitable interqubit coupling topology (e.g., linear connection topology, heavy hex connection topology). Although not explicitly shown in Figure 1, the quantum computer 104 may include, or be associated with, any suitable hardware or software (e.g., a real-time controller implemented on the field-programmable gate array of the quantum computer 104) that can be used for initializing any set of qubits 106, or for performing any suitable quantum operations (e.g., quantum gates, qubit measurements, qubit idling) on ​​the set of qubits 106.

[0074] In various embodiments, the dynamic quantum circuit 108 may be any suitable sequence of quantum gates, or a quantum operation that can be performed or executed on the quantum computer 104. Accordingly, in various cases, the dynamic quantum circuit 108 may be considered an n-qubit quantum circuit (for example, a quantum circuit that can operate on n qubits). In various cases, the dynamic quantum circuit 108 may include an intermediate non-unit operation 110, hence the term "dynamic." In various embodiments, the intermediate non-unit operation 110 may be located at a position not at the end of the dynamic quantum circuit 108, hence the term "intermediate." In other words, the intermediate non-unit operation 110 may be located within the dynamic quantum circuit 108 after at least one quantum gate of the dynamic quantum circuit 108 and before at least one other quantum gate of the dynamic quantum circuit 108. In various cases, the intermediate non-unit operation 110 may be any suitable type of quantum that computes an operation that is non-unit (e.g., irreversible). As a non-restrictive example, the intermediate unit operation 110 in the circuit may be an intermediate qubit measurement operation. In various cases, such an intermediate qubit measurement operation may be feedforward to any suitable quantum gate of the dynamic quantum circuit 108 which is placed or positioned after the intermediate unit operation 110.

[0075] Figure 2 shows an example of a non-limiting diagram 200 of a dynamic quantum circuit 108 according to one or more embodiments described herein. That is, Figure 2 shows a non-limiting, exemplary embodiment of the dynamic quantum circuit 108 and the non-unit operations 110 in the circuit.

[0076] As shown, the dynamic quantum circuit 108 can operate on a set of qubits 106 and can therefore be considered an n-qubit quantum circuit. In various embodiments, the dynamic quantum circuit 108 may include an intermediate non-unit layer 202, at least one layer 204, and at least one layer 206. In various cases, the intermediate non-unit layer 202 may be any layer of the dynamic quantum circuit 108 that includes an intermediate non-unit operation 110.

[0077] In various cases, at least one layer 204 may include any layer of the dynamic quantum circuit 108 that precedes the non-unit layer 202 in the middle of the circuit. Accordingly, at least one layer 204 may include any preferred number of quantum gates or quantum operations of any preferred type, and such quantum gates or quantum operations may be considered to precede (e.g., placed before) the non-unit operation 110 in the middle of the circuit. As a non-restrictive example, at least one layer 204 may include an Hadamard gate (indicated as "H") applied to a qubit 106(1) (indicated as "Q1"), (indicated as "Q n This may include a Pauli Y gate (indicated as "Y") applied to qubit 106(n) (indicated as "), and a controlled Pauli X gate (indicated as "CX") that takes qubit 106(1) as the source and targets qubit 106(n). Note that these quantum gates are merely non-restrictive examples for ease of illustration and explanation. Furthermore, Figure 2 does not show quantum gates in at least one layer 204 that are applied to any set of qubits 106 other than qubit 106(1) and qubit 106(n), but these are also merely non-restrictive examples for ease of illustration and explanation. In various embodiments, at least one layer 204 may contain any preferred number of any preferred type of quantum gates placed at any appropriate order of implementation, and such quantum gates may be applied to any set of qubits 106.

[0078] In various embodiments, at least one layer 206 may include any layer of the dynamic quantum circuit 108 that follows the non-unit layer 202 in the middle of the circuit. Accordingly, at least one layer 206 may include any preferred number of quantum gates or quantum operations of any preferred type, and such quantum gates or quantum operations may be considered to follow (e.g., be placed or positioned after) the non-unit operation 110 in the middle of the circuit. As a non-limiting example, at least one layer 206 may be applied to a qubit 106(1) ("R zThis may include a rotation about the z-axis (indicated as "), a phase gate (indicated as "S") applied to qubit 106(1), and a Hadamard gate applied to qubit 106(n). Note that, as just above, these quantum gates are merely non-restrictive examples for ease of illustration and explanation. Additionally, Figure 2 does not show quantum gates in at least one layer 206 that would be applied to any set of qubits 106 other than qubit 106(1) and qubit 106(n), but this is also merely a non-restrictive example for ease of illustration and explanation. In various embodiments, at least one layer 206 may contain any preferred number of any preferred type of quantum gates placed in any suitable order of implementation, and such quantum gates may be applied to any set of qubits 106.

[0079] In the non-restrictive example of Figure 2, the intermediate unit operation 110 could be an intermediate qubit measurement applied to qubit 106(n). However, this is merely a non-restrictive example for the sake of illustration and explanation. In various embodiments, the intermediate unit operation 110 could be an intermediate qubit measurement applied to any other set of qubits 106. In various cases, if the intermediate unit operation 110 is an intermediate qubit measurement, such an intermediate qubit measurement can be fedforward to a classically controlled quantum gate 208. In other words, whether or not the classically controlled quantum gate 208 is actually executed may depend on the result of the intermediate qubit measurement. For example, if the intermediate qubit measurement yields a measured state of 1, the classically controlled quantum gate 208 may be executed. However, if the intermediate qubit measurement instead yields a measured state of 0, the classically controlled quantum gate 208 may not be executed. In the non-restrictive example in Figure 2, the classically controlled quantum gate 208 is a Pauli X gate (indicated as "X") applied to qubit 106(1). However, it should be noted that this is merely a non-restrictive example for the sake of ease of illustration and explanation. In various cases, the classically controlled quantum gate 208 could be any other suitable quantum gate applied to any set of qubits 106.

[0080] Figure 2 shows only a single instance of an intermediate non-unit operation (e.g., 110) within the intermediate non-unit layer 202 of the circuit, but this is merely a non-restrictive example for ease of illustration and explanation. In various cases, the intermediate non-unit layer 202 of the circuit may contain any suitable number of intermediate non-unit operations, any of which can be applied to any set of qubits 106.

[0081] In any case, a non-unit operation 110 in the circuit may be thought to generate, produce, or be associated with noise 210 (denoted as "Λ"). As shown, noise 210 may be thought of as a quantum operation that potentially affects, damages, contaminates, or otherwise propagates any or all sets of qubits 106. In other words, the noise 210 of a non-unit operation 110 in the circuit may not be contained only in the specific qubit on which the non-unit operation 110 is performed (e.g., 106(n) in the non-restrictive example of Figure 2). In various cases, noise 210 may be thought of as a completely positive trace preserving map (CPTP).

[0082] Referring back to Figure 1, it may be desirable to learn and then mitigate noise 210 generated in or associated with non-unit operations 110 along the circuit. As described herein, a dynamic circuit noise learning and mitigation system 102 can facilitate such learning and mitigation.

[0083] In various embodiments, the dynamic circuit noise learning and mitigation system 102 may include a processor 112 (e.g., a computer processing unit, a microprocessor) and a non-temporary computer-readable memory 114 operably connected to or coupled to the processor 112. The memory 114 may store computer-executable instructions, which, when executed by the processor 112, cause the processor 112 or other components of the dynamic circuit noise learning and mitigation system 102 (e.g., an access component 116, a learning component 118, a mitigation component 120) to perform one or more activities. In various embodiments, the memory 114 may store computer-executable components (e.g., an access component 116, a learning component 118, a mitigation component 120), and the processor 112 may execute the computer-executable components.

[0084] In various embodiments, the dynamic circuit noise learning and mitigation system 102 may include an access component 116. In various embodiments, the access component 116 may electronically access the quantum computer 104 in any suitable manner, thereby enabling the dynamic circuit noise learning and mitigation system 102 to initialize, electronically activate (e.g., power up), electronically deactivate (e.g., power down), or electronically control the quantum computer 104. Furthermore, in various cases, the access component 116 may electronically receive, retrieve, acquire, import, or access the dynamic quantum circuit 108 from any suitable data structure or from any suitable computing device. In any case, the access component 116 may electronically access the quantum computer 104 or the dynamic quantum circuit 108 (for example, by transmitting or receiving data or program instructions to or from the quantum computer 104 or the dynamic quantum circuit 108), thereby allowing other components of the dynamic circuit noise learning and mitigation system 102 to electronically interact with the quantum computer 104 or the dynamic quantum circuit 108.

[0085] In various embodiments, the dynamic circuit noise learning and mitigation system 102 may include a learning component 118. In various embodiments, as described herein, the learning component 118 may learn noise 210 of non-unit operations 110 along the circuit based on a stochastic Pauli-Z gate and a swirling Pauli operator.

[0086] In various embodiments, the dynamic circuit noise learning and mitigation system 102 may include a mitigation component 120. In various cases, as described herein, the mitigation component 120 can mitigate noise 210 of non-unit operations 110 in the circuit by inserting the inverse operation of the noise 210 into the dynamic quantum circuit 108.

[0087] Figure 3 shows a block diagram of an exemplary non-restrictive system 300, which includes a stochastic Pauli-Z gate, a swirling Pauli operator, and a set of learned noise model coefficients, which can facilitate noise learning in a dynamic quantum circuit according to one or more embodiments described herein. As shown, in some cases system 300 may include the same components as system 100, and may further include a stochastic Pauli-Z gate 302, a pair of swirling Pauli operators 304, and a set of learned noise model coefficients 306.

[0088] In various embodiments, the learned set of noise model coefficients 306 can be considered a set of scalars that characterize or define the noise 210. In various cases, the learning component 118 can electronically determine, identify, or estimate the learned set of noise model coefficients 306 by using a stochastic Pauli-Z gate 302 and a pair of orbital Pauli operators 304. More specifically, the learning component 118 can electronically modify the non-unit operation 110 in the circuit with the stochastic Pauli-Z gate 302 and the orbital Pauli operator 304. In various cases, the learning component 118 can repeatedly or iteratively execute the modified non-unit operation 110 in the circuit with the stochastic Pauli-Z gate 302 and the pair of orbital Pauli operators 304 over both a set of Pauli bases and a set of iteration depths on the quantum computer 104. Based on such repeated or iterative executions, the learning component 118 can measure or extract a set of base fidelity. In various embodiments, the learning component 118 may determine or identify a set of learned noise model coefficients 306 based on a set of base fidelity. Various non-limiting embodiments are further described with respect to Figures 4 to 10.

[0089] Figures 4-5 show exemplary non-limiting block diagrams of a non-unit operation 110 in the middle of the circuit, which is modified by a stochastic Pauli-Z gate 302 and a pair of orbital Pauli operators 304 according to one or more embodiments described herein.

[0090] Firstly, consider Figure 4. In various embodiments, as shown, the learning component 118 may generate or create a quantum circuit segment 400 that includes the non-unit operations 110 (and their associated noise 210) in the middle of the circuit and excludes the remaining dynamic quantum circuit 108.

[0091] In various embodiments, the learning component 118 is located within the quantum circuit segment 400, ("Z 0.5 The non-unit operation 110 in the circuit can be modified with a stochastic Pauli-Z gate 302 (as shown in "). More specifically, as shown, the learning component 118 can place the stochastic Pauli-Z gate 302 after the non-unit operation 110 in the circuit and on any set of qubits 106 corresponding to the non-unit operation 110 in the circuit. In the non-restrictive example of Figure 4, the non-unit operation 110 in the circuit is applied to qubit 106(n). Accordingly, the stochastic Pauli-Z gate 302 can be applied to qubit 106(n) after the non-unit operation 110 in the circuit (e.g., downstream thereof). In any case, the stochastic Pauli-Z gate 302 can be a single-qubit Pauli-Z gate applied with a 50% probability or 50% likelihood, and is therefore denoted as "0.5". In other words, during implementation on a quantum computer (e.g., 10⁴), the stochastic Pauli-Z gate 302 performs a Pauli-Z operation with a 50% probability and does not perform the operation with a 50% probability. In any case, the non-unit operation 110 in the circuit can be considered to have residual phase errors, and the stochastic Pauli-Z gate 302 can be considered to remove, zero out, or otherwise reduce such residual phase errors.

[0092] In various embodiments, the learning component 118 may modify the non-unit operation 110 in the middle of the circuit with a pair of orbital Pauli operators 304 within the quantum circuit segment 400. More specifically, the pair of orbital Pauli operators 304 are (each "P nThe pairs of n-qubit Pauli operators shown as "(...)" can be identical pairs, where one such pair is applied before (e.g., upstream of) the non-unit operation 110 in the circuit, and the other such pair is applied after (e.g., downstream of) the stochastic Pauli-Z gate 302. In various cases, the pair of orbiting Pauli operators 304 can be randomly selected from all possible sets of n-qubit Pauli operators. In particular,

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[0093] As shown in Figure 4, if the non-unit operation 110 in the circuit is a qubit measurement in the circuit that feeds forward to a classically controlled quantum gate 208, the classically controlled quantum gate 208 may be omitted from the quantum circuit segment 400. That is, in some embodiments, the classically controlled quantum gate 208 may not be placed, positioned, or applied between pairs of orbital Pauli operators 304. However, this is merely an unrestrictive example. In other embodiments, the classically controlled quantum gate 208 may be placed, positioned, or applied between pairs of orbital Pauli operators 304. In fact, this is shown in an unrestrictive manner in Figure 5. In particular, Figure 5 shows a quantum circuit segment 500 which may include or comprise the same components as the quantum circuit segment 400 and may further comprise the classically controlled quantum gate 208. For ease of explanation and illustration, many of the remaining figures show various unrestrictive embodiments relating to the quantum circuit segment 400. However, it is understood and recognized that such embodiments may also apply to the quantum circuit segment 500.

[0094] Furthermore, as mentioned above, it should be noted that the diagram shows one intermediate non-unit operation (e.g., a single instance 110) within the intermediate non-unit layer 202 of the dynamic quantum circuit 108. For this reason, one intermediate non-unit operation is shown within the quantum circuit segment 400 (and quantum circuit segment 500). However, this is merely a non-limiting example for ease of illustration and explanation. In various embodiments, as mentioned above, the intermediate non-unit layer 202 may contain multiple intermediate non-unit operations that can be applied to multiple sets of qubits 106. In such embodiments, the learning component 118 may include all intermediate non-unit operations within the intermediate non-unit layer 202 within the quantum circuit segment 400 (or 500). Furthermore, in such embodiments, the learning component 118 can modify each of the non-unit operations in such a circuit with each stochastic Pauli-Z gate (e.g., each of the 302 cases), and all such non-unit operations in such a circuit and their respective stochastic Pauli-Z gates can be collectively processed by P nIt can be sandwiched by (for example, by a pair of revolving Pauli operators 304). In such embodiments, noise 210 can be considered as a collective or overall noise created or generated by all such multiple intermediate non-unit operations in the circuit. For example, suppose an intermediate non-unit layer 202 contains intermediate non-unit operations in the circuit for any suitable positive integer a ≤ n (for example, an example of 110 a applied to a set of qubits 106 each). In such a case, the learning component 118 may insert all of the a of such intermediate non-unit operations into the quantum circuit segment 400 (or 500) for each of those qubits, and may apply a separate stochastic Pauli-Z gate to each of such intermediate non-unit operations. This results in a sum of stochastic Pauli-Z gates (for example, an example of 302 a) in the quantum circuit segment 400 (or 500). Furthermore, in such cases, the pair of revolving Pauli operators 304 can be considered to twirl or sandwich all the 'a' of the non-unit operations in the circuit and all the 'a' of the stochastic Pauli-Z gates in a single unit. In such cases, the noise 210 can be considered to be a collective or overall noise generated by all the 'a' of the non-unit operations in the circuit.

[0095] In any case, inserting a non-unit operation 110 in the circuit with a pair of swirling Pauli operators 304 can be considered as shaping or reformatting the noise 210 into a Pauli or diagonal structure. More specifically, such twiling can make the noise 210 expressible or defined in terms of a set of Pauli generators and a set of coefficients corresponding to each of those Pauli generators. Various non-limiting embodiments are further described with respect to Figure 6.

[0096] Figure 6 shows an exemplary, non-limiting block diagram 600 illustrating how noise 210, after being shaped by a pair of revolving Pauli operators 304, may be thought to consist of noise model Pauli generators and noise model coefficients according to one or more embodiments described herein.

[0097] For ease of explanation, after being shaped or reformatted by a pair of rotation Pauli operators 304, the noise 210 may be referred to as shaped noise 606. In various embodiments, the shaped noise 606 may be defined by or considered to be a function of a set of noise model Pauli generators 602 and a set of unknown noise model coefficients 604. In various aspects, the set of noise model Pauli generators 602 may include j generators for any suitable positive integer j, where there are noise model Pauli generators 602(1) through noise model Pauli generators 602(j). In various cases, each of the set of noise model Pauli generators 602 may be a unique or distinct n - qubit Pauli operator that is not the n - qubit identity matrix. For example, noise model Pauli generator 602(1) may be

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[0098] As a non-restrictive example, we assume n=2 and that the non-unit operation 110 in the circuit is applied to qubit 2 instead of qubit 1. In various embodiments, the total set of possible 2-qubit Pauli operators is:

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[0099] In various ways,

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[0100] Here, due to the implementation of the stochastic Pauli-Z gate 302, any 2-qubit Pauli operator associated with Pauli Y noise or Pauli Z noise on qubit 2 is,

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[0101] As another non-restrictive example, assume n > 2. In such a case, to reduce complexity as the number of qubits increases, we can maintain only low-weight noise model Pauli generators, where the weight of an n-qubit Pauli is the number of non-identity single-qubit Pauli operations in that n-qubit Pauli. In particular, assume n = 5, where the non-identity operation 110 in the circuit is a qubit measurement applied to qubit 5 rather than qubits 1-4. Note that the entire set of possible 5-qubit Pauli operators has a cardinality of 1024. This cardinality can be reduced to 1023 by excluding 5-qubit identity operators. Additionally, due to the implementation of the stochastic Pauli-Z gate 302, any noise model Pauli generator associated with Pauli Y noise or Pauli Z noise on qubit 5 can be omitted, so this cardinality can be further reduced from 1023 to 511. Additionally, by maintaining at most a weighted -2 ​​Pauli operator on unmeasured qubits (e.g., qubits 1-4 in this example), the form

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[0102] In various embodiments, a set of unknown noise model coefficients 604 can correspond to a set of noise model Pauli generators 602. Accordingly, since a set of noise model Pauli generators 602 can contain j generators, a set of unknown noise model coefficients 604 can contain j coefficients: from unknown noise model coefficient 604(1) to unknown noise model coefficient 604(j). In various cases, each of the unknown noise model coefficients 604 can be a real-valued non-negative scalar with a currently unknown magnitude, corresponding to one of each of the sets of noise model Pauli generators 602. As an unrestricted example, an unknown noise model coefficient 604(1) can be a real-valued non-negative scalar corresponding to a noise model Pauli generator 602(1) whose magnitude is not yet known. Similarly, an unknown noise model coefficient 604(j) can be a real-valued non-negative scalar corresponding to a noise model Pauli generator 602(j) whose magnitude is not yet known.

[0103] In various embodiments, as described above, the shaped noise 606 (e.g., the noise 210 after being reformatted by a pair of revolving Pauli operators 304) can be represented or defined in terms of a set of noise model Pauli generators 602 and a set of unknown noise model coefficients 604. In particular, an implementation of the Lindbladmaster equation can be obtained as follows:

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[0104] In various embodiments, the learning component 118 may determine or identify a set of noise model coefficients 306 learned by repeatedly executing the quantum circuit segment 400 (or 500) on the quantum computer 104 over a set of Pauli bases and over a set of iteration depths. Based on such iterative executions, the learning component 118 may measure or extract a set of base fidelity corresponding to each set of Pauli bases. In various cases, the learning component 118 may invert the set of base fidelity via a commutation relation to obtain the learned set of noise model coefficients 306. Various non-limiting embodiments are further described with reference to Figures 7 to 10.

[0105] Figure 7 shows an exemplary non-restrictive block diagram 700 illustrating how an observable expectation can be obtained with respect to unitless operations along the circuit according to one or more embodiments described herein. In other words, Figure 7 shows an exemplary non-restrictive illustration of how the learning component 118 can iteratively execute the quantum circuit segment 400 (or 500).

[0106] In various embodiments, the learning component 118 can initialize the set of qubits 106 to any suitable first quantum state. As a non-restrictive example, the learning component 118 can initialize the set of qubits 106 to a zero state (for example, each of the set of qubits 106 can be initialized by the learning component 118).

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[0107] In various embodiments, the learning component 118 may execute or implement the Pauli basis 702 (denoted as "B") on the quantum computer 104. In various cases, the Pauli basis 702 may be any suitable n-qubit Pauli operator other than the n-qubit identity matrix. That is,

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[0108] After executing or performing the Pauli basis 702, the learning component 118 may repeatedly execute or perform the quantum circuit segment 400 (or 500) k times for any preferred positive integer k, as shown. In other words, the learning component 118 may execute or perform the quantum circuit segment 400 (or 500) w times to an iteration depth of k. Following such repeated execution or performance of the quantum circuit segment 400 (or 500), the learning component 118 may execute or perform the conjugate transpose of the Pauli basis 702, as shown by code 704. After executing or performing the conjugate transpose of the Pauli basis 702, the learning component 118 may execute or perform one of the pairs of revolving Pauli operators 304, as shown by code 706. In various embodiments, the learning component 118 may then measure the quantum state which is the result of a set of qubits 106, as shown by code 708. In various cases, such measurements can be collectively considered as yielding an observable expectation value 710. In various cases, the observable expectation value 710 may be a scalar that can be considered a function of the Pauli basis 702 and the iteration depth k.

[0109] Figure 8 is a diagram of an exemplary non-restrictive block diagram 800 showing how multiple sets of observable expectations can be obtained by one or more embodiments described herein. As stated above, the observable expectations 710 can be considered a function of the Pauli basis 702 and the iteration depth k. Accordingly, the learning component 118 may execute the protocol shown in Figure 7 over multiple Pauli basis and multiple iteration depths, thereby obtaining multiple observable expectations.

[0110] In particular, there can be a set of Pauli basis 802. In various embodiments, the set of Pauli basis 802 may have the same cardinality as the set of noise model Pauli generators 602. Therefore, since the set of noise model Pauli generators 602 may contain j generators, the set of Pauli basis 802 may contain j basis vectors: from Pauli basis 802(1) to Pauli basis 802(j). In various embodiments, each of the set of Pauli basis 802 may be the result of subtracting an n-qubit identity matrix from a unique or distinct member of a set of n-qubit Pauli operators. As a non-restrictive example, Pauli basis 802(1) is,

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[0111] In some cases (for example, when sparsity is not assumed, implemented, or enforced), both the set of Pauli basis 802 and the set of noise model Pauli generator 602 are

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[0112] As a non-restrictive example, let us again assume n=2. Here, we assume that sparsity is not enforced. In such a case, both the set of Pauli basis 802 and the set of noise model Pauli generator 602 are:

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[0113] Furthermore, in various embodiments, there can be sets of iteration depths 804. In various cases, a set of iteration depths 804 may include depths for any suitable positive integer d: from depth 804(1) to depth 804(d). In various cases, each of the sets of iteration depths 804 may be considered a positive integer value that can be assigned to k. In various embodiments, a set of iteration depths 804 may increase gradually. In other words, each iteration depth in a set of iteration depths 804 may be greater in value or magnitude than the iteration that came before it (for example, depth 804(2) (not shown) may be greater than depth 804(1), depth 804(3) (not shown) may be greater than depth 804(2), ..., depth 804(d) may be greater than depth 804(1), ..., depth 804(2) may be greater than depth 804(d-1) (not shown).

[0114] In various embodiments, the learning component 118 may execute the protocol shown in Figure 7 over a set of Pauli basis 802 and over a set of iteration depths 804. In other words, the learning component 118 may execute the protocol shown in Figure 7 for each unique basis depth pair. In various cases, this can yield multiple sets of observable expectations 806.

[0115] As an unrestricted example, the learning component 118 may select Pauli basis 802(1) as B and depth 804(1) as k. With such selections, the learning component 118 may execute the protocol shown in Figure 7, which yields the observable expectation 806(1)(1). As another unrestricted example, the learning component 118 may select Pauli basis 802(1) as B and depth 804(d) as k. With such selections, the learning component 118 may execute the protocol shown in Figure 7, which yields the observable expectation 806(1)(d). In various cases, the observable expectation 806(1)(d) from the observable expectation 806(1)(1) can be considered collectively as forming a set of observable expectation 806(1) corresponding to Pauli basis 802(1).

[0116] As yet another non-restrictive example, the learning component 118 may select Pauli basis 802(j) as B and depth 804(1) as k. With such selection, the learning component 118 may execute the protocol shown in Figure 7, which yields the observable expectation 806(j)(1). As yet another non-restrictive example, the learning component 118 may select Pauli basis 802(j) as B and depth 804(d) as k. With such selection, the learning component 118 may execute the protocol shown in Figure 7, which yields the observable expectation 806(j)(d). In various cases, the observable expectation 806(j)(d) from the observable expectation 806(j)(1) can be considered collectively as forming a set of observable expectation 806(j) corresponding to Pauli basis 802(j).

[0117] In various ways, the set of observable expectation values ​​806(j) derived from the set of observable expectation values ​​806(1) can be considered to collectively form multiple sets of observable expectation values ​​806.

[0118] Figure 9 shows an exemplary, non-restrictive block diagram 900 illustrating how, in one or more embodiments described herein, basis fidelity may be obtained from multiple sets of observable expectation values.

[0119] In various embodiments, the learning component 118 may determine, identify, or extract a set of base fidelity 902 based on a set of observable expectation values ​​806. In various embodiments, the learning component 118 may complete this by fitting an exponential decay curve to a set of observable expectation values ​​806.

[0120] As a non-restrictive example, the learning component 118 may fit an exponential decay curve to a set of observable expected values ​​806(1). In various cases, such an exponential decay curve can be formalized as Af B k This can be the case, where k can be from a set of repeating depths 804, B can be the Pauli base 802(1), A can be considered the state preparation and measurement error, and f B k can be the fidelity corresponding to the Pauli basis 802(1) (for example, a real-valued scalar in the range of 0 to 1). In other words, k can be considered the independent variable of such an exponential decay curve, and A and f B f can be considered a constant of such an exponential decay curve. By fitting the exponential decay curve to a set of observable expectation values ​​806(1) (e.g., via the least squares sum or any other suitable fitting technique), f B This can be estimated or approximated. Such estimated or approximated values ​​may be considered or referred to as the basis fidelity 902(1). Note that the basis fidelity 902(1) may be considered to correspond to the Pauli basis 802(1).

[0121] As another non-restrictive example, the learning component 118 may fit an exponential decay curve to a set of observable expected values ​​806(j). Just as described above, in various cases, such an exponential decay curve is formalized by the form Af B kThis can be the case, where k can be from a set of repeating depths of 804, B can be the Pauli base 802(j), A can be considered to be the state preparation and measurement error, and f B k can be the fidelity corresponding to the Pauli basis 802(j) (for example, a real-valued scalar in the range of 0 to 1). In other words, k can be considered the independent variable of such an exponential decay curve, and A and f B This can be considered a constant of such an exponential decay curve. By fitting the exponential decay curve to a set of observable expectation values ​​806(j), f B This can be estimated or approximated. Such estimated or approximated values ​​may be considered or referred to as the basis fidelity 902(j). Note that the basis fidelity 902(j) may be considered to correspond to the Pauli basis 802(j).

[0122] In various ways, base fidelity 902(1) through base fidelity 902(j) can be considered collectively as a set of base fidelity 902.

[0123] Figure 10 shows an exemplary, non-limiting block diagram 1000 illustrating how noise coefficients learned from basis fidelity can be obtained by one or more embodiments described herein.

[0124] In various embodiments, the learning component 118 may compute, determine, estimate, or learn a set of learned noise model coefficients 306 based on a set of base fidelity 902. In particular, the learning component 118 may invert the set of base fidelity 902 via exchange relations that may be defined between a set of Pauli bases 802 and a set of noise model Pauli generators 602. More specifically, the learning component 118 may utilize subsequent formulations:

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[0125] In various embodiments, the learning component 118 inverts M and applies such inverse matrix to the left side of each edge of the above formulation via matrix multiplication,

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[0126] Up to this point, various embodiments have been described in which the set of Pauli bases 802 has the same concentration (e.g., j) as the set of noise model Pauli generators 602. However, this is merely a non-limiting example for ease of illustration and explanation. In various other embodiments, the set of Pauli bases 802 may have a different concentration (e.g., a different number of elements) than the set of noise model Pauli generators 602. In fact, the above formulations are used in all cases where M is a complete column rank (e.g., all cases where M has at least as many rows as columns),

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[0127] In any case, the learning component 118 may determine a set of learned noise model coefficients 306, and thus it can be considered that a shaped noise 606 has been learned (for example, a shaped or twirl version of the noise 210 has been learned).

[0128] Figure 11 shows the results of exemplary, non-limiting experiments using one or more embodiments described herein. In particular, the inventors performed a variety of experiments in which the various embodiments described herein were put into practice. During such experiments, the following parameters were implemented: n=2; the intermediate non-unit operation 110 is an intermediate qubit measurement on the first qubit; the classically controlled quantum gate 208 is performed on the second qubit which is not directly coupled to the first qubit; and the set of noise model Pauli generators 602 is

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[0129] As shown, Figure 11 includes graphs 1102 and 1104. Graph 1102 shows exponential decay curves fitted to observable expectations over the iteration depth and across the seven Pauli bases described above. In particular, the horizontal axis of graph 1102 can be considered to represent the range over the iteration depth, and the different dotted lines in graph 1102 can be considered to represent different Pauli bases. Based on such fitted exponential decay curves, distinct base fidelity was obtained for each of the seven Pauli bases used:

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[0130] As shown in Graph 1104, the non-zero learned noise model coefficients are:

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[0131] Note that in some cases, the seven Pauli bases can be measured directly. However, in other cases, not all seven of these Pauli bases can be measured directly, and the remaining Pauli bases can be obtained through post-processing. As a non-limiting example,

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[0132] Figure 12 shows a block diagram of an exemplary, non-limiting system 1200 that includes an inverse noise operation that can facilitate noise learning in a dynamic quantum circuit according to one or more embodiments described herein. As shown, system 1200 may in some cases have the same components as system 300 and may further include an inverse noise operation 1202.

[0133] In various embodiments, as described above, the learning component 118 may be considered to have learned, determined, or identified a shaped or twirl version of noise 210 (e.g., learned shaped noise 606) upon acquiring a set of learned noise model coefficients 306. In various embodiments, the mitigation component 120 may electronically mitigate the shaped or twirl version of noise 210 by inserting the reciprocal of such noise into the dynamic quantum circuit 108. More specifically, once a set of learned noise model coefficients 306 is acquired, it may be considered that noise 210 (e.g., noise 210 sandwiched between pairs of shaped noise 606) (e.g., swirling Pauli operators 304) has been learned. In various cases, the mitigation component 120 may electronically invert the shaped noise 606 (e.g., via any suitable matrix inversion technique) to obtain the inverse noise operation 1202. In other words, the inverse noise operation 1202 can be thought of as the reciprocal of the shaped noise 606 (of which any n-qubit quantum operation would be sandwiched between pairs of (e.g., circling Pauli operators 304)). In various cases, the mitigation component 120 can mitigate, eliminate, or reduce the shaped noise 606 (e.g., reduce the noise 210 sandwiched between pairs of (circling Pauli operators 304)) by inserting the inverse noise operation 1202 into the dynamic quantum circuit 108. Various non-limiting embodiments are shown with respect to Figures 13 and 14.

[0134] Figures 13 and 14 show exemplary non-limiting block diagrams 1300 and 1400 of the dynamic quantum circuit 108 after the inverse noise operation 1202 is inserted according to one or more embodiments described herein.

[0135] Firstly, consider Figure 13. As shown, at least one layer 204 and at least one layer 206 of the dynamic quantum circuit 108 may remain unchanged. However, as also shown, the mitigation component 120 may change the non-unit layer 202 in the circuit to the non-unit layer 1302 in the circuit.

[0136] In various embodiments, the intermediate non-unit layer 1302 may include intermediate non-unit operations 110, classically controlled quantum gates 208, and noise 210, just like the intermediate non-unit layer 202. In various cases, the mitigation component 120 may insert a pair of stochastic Pauli-Z gates 302 and orbital Pauli operators 304 into the intermediate non-unit layer 1302. However, this is merely an unrestricted example. In some cases, the mitigation component 120 may omit the stochastic Pauli-Z gates 302. Also, in various embodiments, the mitigation component 120 may include (

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[0137] As shown in Figure 13, the classically controlled quantum gate 208 may, in some embodiments, be located or positioned between pairs of orbital Pauli operators 304. It should be noted that this may be independent of whether the learning component 118 obtains a set of learned noise model coefficients 306 by utilizing quantum circuit segment 400 or quantum circuit segment 500. However, this is merely an unrestricted example. In some cases, the classically controlled quantum gate 208 may not be located between pairs of orbital Pauli operators 304. In particular, the classically controlled quantum gate 208 may be located in an ununit layer 1302 midway through the circuit, and may be after the second or furthest downstream pair of orbital Pauli operators 304. In yet another instance, the classically controlled quantum gate 208 may be omitted entirely, as shown with respect to Figure 14. In fact, as shown in Figure 14, the mitigation component 120 allows the dynamic quantum circuit 108 to include an intermediate non-unit layer 1402 instead of the intermediate non-unit layer 1302 in various embodiments. In various embodiments, the intermediate non-unit layer 1402 may be identical to the intermediate non-unit layer 1302, except that the classically controlled quantum gate 208 may be omitted or removed. Ultimately, in various embodiments, the implementation of intermediate qubit measurement may be useful even without further feedforward (for example, such intermediate qubit measurement is also not used to classically control the quantum gate).

[0138] In any case, the mitigation component 120 can improve the effect of the noise 210 by inserting an inverse noise operation 1202 into the dynamic quantum circuit 108.

[0139] Figures 15-16 show the results of exemplary, non-limiting experiments according to one or more embodiments described herein.

[0140] Firstly, consider Figure 15. As described above, the inventors have performed various experiments to implement the various embodiments described herein. As described above, during such experiments, the following parameters were implemented: n=2; the intermediate non-unit operation 110 is an intermediate qubit measurement on the first qubit; the classically controlled quantum gate 208 is performed on the second qubit which is not directly coupled to the first qubit; and the set of noise model Pauli generators 602 is

number

number

[0141] Now consider Figure 16. To further verify the various embodiments described herein, the inventors performed various embodiments (using the 2-qubit system referred to in the above experiment). Here, noise model coefficients were acquired for qubit measurements in the middle of the circuit, and the amplitude of the qubit measurements in the middle of the circuit was varied between weak and strong. Figure 16 includes Graph 1602 showing the learned noise model coefficients for each of the seven noise model Pauli generators used. As shown, the learned noise model coefficients are (in particular,

number

number

number

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[0142] Figure 17 shows a flowchart of an exemplary, non-limiting computer implementation method 1700 that can facilitate noise learning in a dynamic quantum circuit according to one or more embodiments described herein. In various cases, the computer implementation method 1700 can be facilitated by a dynamic circuit noise learning and mitigation system 102.

[0143] In various embodiments, activity 1702 may include accessing a dynamic quantum circuit (e.g., 104) by a device (e.g., via 116) operably coupled to a processor (e.g., 112).

[0144] In various embodiments, activity 1704 may involve a device (e.g., via 118) learning noise (e.g., 210) associated with non-unit operations (e.g., 110) in the middle of a dynamic quantum circuit by correcting non-unit operations in the middle of the circuit with a stochastic Pauli-Z gate (e.g., 302) and a revolving Pauli operator (e.g., 304).

[0145] Although not explicitly shown in Figure 17, non-unit operations in the circuit may include intermediate qubit measurements that feedforward to at least one classically controlled quantum gate (e.g., 208). In some cases, at least one classically controlled quantum gate may be between orbital Pauli operators (e.g., as shown in Figure 5). In other cases, at least one classically controlled quantum gate may not be between orbital Pauli operators (e.g., as shown in Figure 4).

[0146] Although not explicitly shown in Figure 17, learning noise may involve a device (e.g., via 118) repeatedly performing non-unit operations in the circuit modified by stochastic Pauli-Z gates and orbital Pauli operators over both a set of Pauli bases (e.g., 802) and a set of iteration depths (e.g., 804), and the device (e.g., via 118) extracting a set of base fidelity (e.g., 902) corresponding to each set of Pauli bases (e.g., as shown in Figures 7-9) based on such repeated executions.

[0147] Although not explicitly shown in Figure 17, learning noise may involve the device (e.g., via 118) inverting the set of base fidelity via a defined exchange relation (e.g., M) between the set of Pauli bases and the set of Pauli generators (e.g., 602) associated with the noise.

[0148] Although not explicitly shown in Figure 17, the computer implementation method 1700 may include reducing the noise by a device (e.g., via 120) by inserting the reciprocal of the noise (e.g., 1202) into the dynamic quantum circuit.

[0149] It should be noted that when the non-unit operation 110 in the circuit is a qubit measurement in the circuit, the various embodiments described herein may be considered or referred to as measurement-based probabilistic error elimination (e.g., mPEC).

[0150] Furthermore, it should be noted that if the intermediate non-unit operation 110 is an intermediate qubit measurement, the dynamic circuit noise learning and mitigation system 102 may, in various embodiments, electronically flip any classical bit measured by the intermediate non-unit operation 110 if the pair of circling Pauli operators 304 applies a Pauli X gate or a Pauli Y gate to any qubit to which the intermediate non-unit operation is applied (e.g., qubit 106(n) in the non-restrictive example shown in the figure). However, if such a pair of circling Pauli operators 304 of the measured classical bit applies a Pauli Z gate or the same gate instead to any qubit to which the intermediate non-unit operation is applied (e.g., qubit 106(n) in the non-restrictive example shown in the figure), the flip may be omitted.

[0151] The various embodiments described herein can be considered as computerized tools for learning or mitigating noise generated by non-unit operations in a circuit. Such embodiments can be applied regardless of qubit connectivity and can be scaled to qubit lattices of any suitable size. Such embodiments certainly constitute concrete and visible improvements in the field of dynamic quantum circuits.

[0152] Figure 18 and the following discussion are intended to provide a brief, general description of a preferred computing environment 1800 in which one or more embodiments described herein may be implemented. For example, various aspects of this disclosure are described by narrative text, flowcharts, block diagrams of computer systems, or block diagrams of machine logic included in embodiments of computer program products (CPPs). With respect to any flowchart, depending on the technology involved, operations may be performed in a different order than those shown in a given flowchart. For example, again depending on the technology involved, two operations shown in consecutive blocks of a flowchart may be performed in reverse order, as a single integrated stage, simultaneously, or with at least partial time overlap.

[0153] An embodiment of a computer program product ("CPP embodiment" or "CPP") is a term used in this disclosure to describe any set of one or more storage media (also called "mediums") that are collectively contained in one or more storage media that collectively contain machine-readable code corresponding to instructions or data for performing a computer operation specified in a given CPP claim. A "storage media" is any tangible device capable of holding and storing instructions for use by a computer processor. Computer-readable storage media may be, but are not limited to, electronic storage media, magnetic storage media, optical storage media, electromagnetic storage media, semiconductor storage media, mechanical storage media, or any preferred combination thereof. Some known types of storage devices, including these media, include diskettes, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random-access memory (SRAM), compact disc read-only memory (CD-ROM), digital versatile disk (DVD), memory stick, floppy disk, mechanically encoded devices (such as pits / lands formed on the main surface of a punch card or disk), or any preferred combination of the above. The computer-readable storage media as used in this disclosure are not to be interpreted as storage in the form of temporary signals themselves, such as freely propagating electromagnetic waves like radio waves, electromagnetic waves propagating through waveguides, optical pulses passing through fiber optic cables, electrical signals communicated over wires, or other transmission media.As those skilled in the art will understand, data is typically moved at several intermittent points during the normal operation of the storage device, such as during access, defragmentation, or garbage collection; however, data is not temporary while it is stored, and therefore the storage device is not considered temporary.

[0154] The computing environment 1800 includes an example of an environment for implementing at least a portion of the computer code involved in performing the method of the present invention, such as the dynamic quantum circuit noise learning code 1880. In addition to block 1880, the computing environment 1800 includes, for example, a computer 1801, a wide area network (WAN) 1802, an end user device (EUD) 1803, a remote server 1804, a public cloud 1805, and a private cloud 1806. In this embodiment, the computer 1801 includes a processor set 1810 (including processing circuits 1820 and a cache 1821), a communication fabric 1811, volatile memory 1812, persistent storage 1813 (including the operating system 1822 and block 1880 as identified above), a peripheral device set 1814 (including a user interface (UI) device set 1823, storage 1824, and an Internet of Things (IoT) sensor set 1825), and a network module 1815. The remote server 1804 includes the remote database 1830. The public cloud 1805 includes the gateway 1840, the cloud orchestration module 1841, the host physical machine set 1842, the virtual machine set 1843, and the container set 1844.

[0155] Computer 1801 may take the form of a desktop computer, laptop computer, tablet computer, smartphone, smartwatch, or other wearable computer, mainframe computer, quantum computer, or any other form of computer or mobile device, currently known or to be developed in the future, that can run programs, access networks, or query databases such as remote database 1830. As is well understood in the field of computer technology, and depending on the technology, the execution of a computer implementation can be distributed among multiple computers or locations. On the other hand, in this presentation concerning the computing environment 1800, in order to keep the presentation as concise as possible, the detailed discussion focuses on a single computer, specifically computer 1801. Although computer 1801 is not shown in the cloud in Figure 18, it may be located in the cloud. On the other hand, computer 1801 is not required to be located in the cloud, except to any extent that may be explicitly shown.

[0156] The processor set 1810 includes one or more computer processors of any type currently known or to be developed in the future. The processing circuitry 1820 may be distributed across multiple packages, for example, multiple coordinated integrated circuit chips. The processing circuitry 1820 may implement multiple processor threads or multiple processor cores. The cache 1821 is memory located within the processor chip package and is typically used for data or code that should be available for high-speed access by threads or cores running on the processor set 1810. The cache memory is typically organized into multiple levels, depending on its relative proximity to the processing circuitry. Alternatively, some or all of the cache for the processor set may be located "off-chip". In some computing environments, the processor set 1810 may operate with qubits and be designed to perform quantum computations.

[0157] Computer-readable program instructions are typically loaded into computer 1801 to cause the processor set 1810 of computer 1801 to execute a series of operational steps, thereby realizing a computer implementation method. The instructions thus executed instantiate a method specified in a flowchart or descriptive description of a computer implementation method included herein (collectively referred to as the "Method of the Invention"). These computer-readable program instructions are stored in various types of computer-readable storage media, such as a cache 1821 and other storage media discussed below. The program instructions and associated data are accessed by the processor set 1810 to control and direct the execution of the Method of the Invention. In the computing environment 1800, at least some of the instructions for executing the Method of the Invention may be stored in block 1880 in persistent storage 1813.

[0158] The communication fabric 1811 is a signal conduction path that enables various components of the computer 1801 to communicate with one another. Typically, this fabric is made up of switches and conductive paths, such as buses, bridges, physical input / output ports, and similar components. Other types of signal communication paths, such as fiber optic communication paths or wireless communication paths, may also be used.

[0159] Volatile memory 1812 is any type of volatile memory currently known or to be developed in the future. Examples include dynamic random-access memory (RAM) or static RAM. Volatile memory typically features random access, but this is not required unless explicitly stated. In computer 1801, volatile memory 1812 is located in a single package and is internal to computer 1801, but alternatively or additionally, volatile memory can be distributed across multiple packages or located externally to computer 1801.

[0160] Persistent storage 1813 is any form of non-volatile storage for a computer, currently known or to be developed in the future. The non-volatility of this storage means that the stored data is maintained regardless of whether power is supplied to computer 1801 or directly to persistent storage 1813. Persistent storage 1813 may be read-only memory (ROM), but typically at least a portion of the persistent storage allows for data writing, data erasure, and data rewriting. Some well-known forms of persistent storage include magnetic disks and solid-state storage devices. Operating system 1822 can take multiple forms, including various known proprietary operating systems or open-source portable operating system interface (CSI) types employing a kernel. The code contained in block 1880 typically includes at least a portion of computer code involved in performing the method of the present invention.

[0161] The peripheral device set 1814 includes a set of peripheral devices for the computer 1801. Data communication connections between the computer 1801's peripheral devices and other components can be implemented in various ways, including Bluetooth connections, near-field communication (NFC) connections, connections made by cables (such as universal serial bus (USB) type cables), insert-type connections (e.g., secure digital (SD) cards), connections made through local area communication networks, and even connections made through wide area networks such as the internet. In various embodiments, the UI device set 1823 may include components such as a display screen, speaker, microphone, wearable devices (such as goggles and smartwatches), keyboard, mouse, printer, touchpad, game controller, and haptic devices. Storage 1824 is external storage such as an external hard drive, or insertable storage such as an SD card. Storage 1824 may be persistent or volatile. In some embodiments, storage 1824 may take the form of a quantum computing memory device for storing data in the form of qubits. In embodiments where computer 1801 is required to have a large amount of storage (for example, when computer 1801 locally stores and manages a large database), this storage may be provided by peripheral memory devices designed to store large amounts of data, such as a storage area network (SAN) shared by multiple geographically distributed computers. The IoT sensor set 1825 consists of sensors that can be used in Internet of Things applications. For example, one sensor may be a thermometer and another may be a motion detector.

[0162] The network module 1815 is a collection of computer software, hardware, and firmware that enables computer 1801 to communicate with other computers via the WAN 1802. The network module 1815 may include hardware such as a modem or Wi-Fi signal transceiver, software for packetizing or depackaging data for communication network transmission, or web browser software for communicating data over the Internet. In some embodiments, the network control and network forwarding functions of the network module 1815 are performed on the same physical hardware device. In other embodiments (e.g., embodiments utilizing software-defined networking (SDN)), the control and forwarding functions of the network module 1815 are performed on physically separate devices, such that the control function manages several different network hardware devices. Computer-readable program instructions for performing the methods of the present invention can typically be downloaded to computer 1801 from an external computer or external storage device through a network adapter card or network interface included in the network module 1815.

[0163] WAN1802 is any wide area network (e.g., the Internet) that can communicate computer data over non-local distances using any currently known or future-developed technology for communicating computer data. In some embodiments, a WAN may be replaced or complemented by a local area network (LAN), such as a Wi-Fi network, which is designed to communicate data between devices located in a local area. A WAN or LAN typically includes computer hardware such as copper transmission cables, optical transmission fibers, wireless transmissions, routers, firewalls, switches, gateway computers, or edge servers, or a combination thereof.

[0164] The end-user device (EUD) 1803 is any computer system used and controlled by an end-user (e.g., a customer of the company operating computer 1801) and can take any of the forms discussed above in relation to computer 1801. Typically, EUD 1803 receives useful and valuable data from the operation of computer 1801. For example, in a hypothetical case where computer 1801 is designed to provide recommendations to the end-user, these recommendations would typically be communicated from the network module 1815 of computer 1801 to EUD 1803 via WAN 1802. In this way, EUD 1803 can display or otherwise present the recommendations to the end-user. In some embodiments, EUD 1803 may be a client device such as a thin client, heavy client, mainframe computer, or desktop computer.

[0165] The remote server 1804 is any computer system that provides at least some data or functionality to computer 1801. The remote server 1804 may be controlled and used by the same entity that operates computer 1801. The remote server 1804 represents a machine that collects and stores useful and valuable data for use by other computers, such as computer 1801. For example, in a hypothetical case where computer 1801 is designed and programmed to provide recommendations based on historical data, this historical data may be provided to computer 1801 from the remote database 1830 of the remote server 1804.

[0166] Public Cloud 1805 is any computer system available to multiple entities that provides computer system resources or other computer functions, particularly data storage (cloud storage) and computing power, on demand without direct, active management at scale. Direct and active management of the computing resources of Public Cloud 1805 is performed by the computer hardware or software of the Cloud Orchestration Module 1841. The computing resources provided by Public Cloud 1805 are typically implemented by virtual computing environments that run within Public Cloud 1805 or on various computers that make up the host physical machine set 1842, which is the universe of physical computers available in Public Cloud 1805. Virtual computing environments (VCEs) typically take the form of virtual machines in the virtual machine set 1843 or containers in the container set 1844. These VCEs may be stored as images and may be transferred either as images or after instantiation of the VCEs between and between hosts of various physical machines. The cloud orchestration module 1841 manages image transfer and storage, deploys new VCE instances, and manages active instanceizations of VCE deployments. The gateway 1840 is a collection of computer software, hardware, and firmware that enables the public cloud 1805 to communicate over the WAN 1802.

[0167] Here, some further explanation of virtualized computing environments (VCEs) is provided. A VCE can be stored as an "image." From this image, a new active instance of the VCE can be instantiated. Two well-known types of VCEs are virtual machines and containers. A container is a VCE that uses operating system-level virtualization. This refers to an operating system feature where the kernel allows for the existence of multiple isolated user-space instances called containers. These isolated user-space instances typically behave like actual computers in terms of the programs running within them. Computer programs running on a normal operating system can utilize all of that computer's resources, including connected devices, files and folders, network shares, CPU power, and quantifiable hardware capabilities. However, programs running inside a container can only use the contents of the container and the devices allocated to the container, which is a known feature of containerization.

[0168] Private Cloud 1806 is similar to Public Cloud 1805, except that its computing resources are available for use by a single enterprise only. While Private Cloud 1806 is shown as being in communication with WAN 1802, in other embodiments, the private cloud may be completely isolated from the internet and accessible only through a local / private network. A hybrid cloud is a combination of multiple clouds of different types (e.g., private, community, or public cloud types), often implemented by different vendors. While each cloud is an independent discrete entity, larger hybrid cloud architectures are coupled together by standardized or proprietary technologies that enable orchestration, management, or data / application portability across multiple configuration clouds. In this embodiment, both Public Cloud 1805 and Private Cloud 1806 are part of a larger hybrid cloud.

[0169] The disclosures herein describe non-limiting examples of various embodiments of the subject innovation. For ease of description or explanation, various parts of the disclosure herein use the term “each” when discussing various embodiments of the subject innovation. Such use of the term “each” is non-limiting. In other words, where the disclosure herein provides a description that applies to “each” of any particular object or component, it should be understood that this is a non-limiting example of various embodiments of the subject innovation, and it should be further understood that in various other embodiments of the subject innovation, such a description may apply to fewer than “each” of the particular object or component.

[0170] The embodiments described herein may refer to one or more systems, methods, apparatus, or computer program products or combinations thereof at any possible level of technical detail of integration. A computer program product may include a computer-readable storage medium (or more media) having computer-readable program instructions for causing a processor to execute an aspect of one or more embodiments described herein. A computer-readable storage medium can be a tangible device capable of holding and storing instructions for use by an instruction-executing device. A computer-readable storage medium can be, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any preferred combination thereof. A non-exclusive list of more specific examples of computer-readable storage media may also include portable computer diskettes, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disk read-only memory (CD-ROM), digital multipurpose disks (DVDs), memory sticks, floppy disks, mechanically encoded devices such as punched cards or grooved structures on which instructions are recorded, or any preferred combination thereof. When used herein, computer-readable storage media should not be construed as transient signals themselves, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through waveguides or other transmission media (e.g., light pulses passing through optical fiber cables), or electrical signals transmitted through wires.

[0171] The computer-readable program instructions described herein may be downloaded from a computer-readable storage medium to each computing / processing device, or downloaded to an external computer or 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. The network may include copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers, or edge servers, or a combination thereof. A network adapter card or network interface within each computing / processing device receives computer-readable program instructions from the network and transfers the computer-readable program instructions for storage in a computer-readable storage medium within each computing / processing device. Computer-readable program instructions for performing operations of one or more embodiments described herein may be assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state setting data, 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®, C++, or similar, or conventional procedural programming languages ​​such as the C programming language or similar programming languages. Computer-readable program instructions may run entirely on a computer, partially on a computer, run as a standalone software package, partially on a computer, partially on a remote computer, or entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or wide area network (WAN), or the connection may be made to an external computer (for example, via the Internet using an Internet service provider).In one or more embodiments, to perform an aspect of one or more embodiments described herein, an electronic circuit including, for example, a programmable logic circuit, a field-programmable gate array (FPGA), or a programmable logic array (PLA) may execute a computer-readable program instruction by personalizing the electronic circuit using state information of the computer-readable program instruction.

[0172] Aspects of one or more embodiments described herein will be described with reference to flowcharts or block diagrams of methods, apparatus (systems), and computer program products according to one or more embodiments described herein. It will be understood that each block of a flowchart or block diagram, or both, and any combination of blocks of a flowchart or block diagram, or both, can be implemented by computer-readable program instructions. These computer-readable program instructions may be provided to a processor of a general-purpose computer, a dedicated computer, or other programmable data processing device for creating a machine. Thus, instructions executed via the processor of a computer or other programmable data processing device may form means for implementing the functions / actions specified in the blocks or combinations of blocks of a flowchart or block diagram. These computer-readable program instructions may also be stored in a computer-readable storage medium that can instruct a computer, a programmable data processing device, or other device or combination thereof to function in a particular manner. Thus, a computer-readable storage medium storing instructions includes a product containing instructions that can implement the modes of functions / actions specified in the blocks or combinations of blocks of a flowchart or block diagram. Computer-readable program instructions can also be loaded onto a computer, other programmable data processing device, or other device to cause a series of actions to be executed on the computer, other programmable device, or other device, creating a computer implementation process in which the instructions executed on the computer, other programmable device, or other device implement the functions / actions specified in a block or multiple blocks of a flowchart or block diagram or a combination thereof.

[0173] The flowcharts and block diagrams in the figures illustrate the architecture, functionality, or operations of a system, computer-implementable method, or possible implementation of a computer program product according to one or more embodiments described herein. In this regard, each block in a flowchart or block diagram may represent a module, segment, or part of an instruction containing one or more executable instructions for implementing a specified logical function. In one or more implementations, the functions described in a block may occur in an order other than that shown in the figure. For example, two consecutively shown blocks may be executed substantially simultaneously, depending on the functionality involved, or the blocks may, in some cases, be executed in reverse order. In addition, it will be noted that each block in a block diagram or flowchart diagram, or both, and any combination of blocks in a block diagram or flowchart diagram, or both, may be implemented by a dedicated hardware-based system that can perform a specified function or operation, or by executing one or more combinations of dedicated hardware and computer instructions.

[0174] While the subject matter has been described above in the general context of computer executable instructions for computers or computer program products running on computers, those skilled in the art will recognize that one or more embodiments described herein can also be implemented at least partially in parallel with one or more other program modules. Generally, program modules include routines, programs, components, or data structures that perform a particular task or implement a particular abstract data type. Furthermore, the computer implementation methods described above can be implemented in single-processor or multi-processor computer systems, mini-computer devices, mainframe computers, and other computer system configurations, including computers, handheld computing devices (e.g., PDAs, telephones), or microprocessor-based or programmable consumer or industrial electronic devices. The embodiments shown can also be implemented in distributed computing environments where tasks are performed by remote processing devices connected via a communication network. However, embodiments of one or more embodiments described herein can be implemented on standalone computers, though not all of them. In a distributed computing environment, program modules can reside in both local and remote memory storage devices.

[0175] Where used herein, terms such as “component,” “system,” “platform,” or “interface” may refer to and / or include computer-related entities or entities relating to operating machines having one or more inherent functionalities. Entities described herein may be hardware, a combination of hardware and software, software, or software in operation. For example, a component may 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, an application running on a server and the server itself may both be components. One or more components may reside in an execution process or thread or both, and components may be localized in one computer, distributed across two or more computers, or both. In another example, each component may be executed from various computer-readable media containing various data structures. Components may communicate via local, remote, or both processes, according to signals, etc., which may contain one or more data packets (for example, data from one component interacting with another component in a network such as the Internet, with a local system, a distributed system, or other systems or a combination thereof via signals). As another example, a component may be a device having inherent functionality provided by mechanical parts operated by electrical or electronic circuits operated by software or firmware applications run by a processor. In such a case, the processor may be inside or outside the device and may execute at least part of the software or firmware application.As yet another example, a component can be a device that provides unique functionality through electronic components without using mechanical parts, and the electronic components can include a processor or other means for executing software or firmware that at least partially provides the functionality of the electronic components. In some embodiments, a component can emulate an electronic component, for example, via a virtual machine within a cloud computing system.

[0176] In addition, 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 in any of the cases where X uses A, X uses B, or X uses both A and B. As used herein, the term "and / or" is intended to have the same meaning as "or". Moreover, the articles "a" and "an" used in the specification and drawings of the present subject matter should generally be construed to mean "one or more" unless otherwise specified or clear from the context that they refer to the singular form. As used herein, the terms "example" and / or "exemplary" are utilized to mean serving as an example, instance, or illustration. To avoid misunderstanding, the subject matter described herein is not limited to such examples. In addition, any aspect or design described herein as an "example" and / or "exemplary" should not necessarily be construed as more preferable or advantageous than other aspects or designs, nor is it intended to exclude equivalent exemplary structures and techniques known to those of ordinary skill in the art.

[0177] As used herein, the term “processor” may refer to substantially any computing unit or device, including, but not limited to, single-core processors, single-processors with software multithreading capabilities, multi-core processors, multi-core processors with software multithreading capabilities, multi-core processors with hardware multithreading technology, parallel platforms, and parallel platforms with distributed shared memory. Furthermore, a processor may refer to integrated circuits, application-specific integrated circuits (ASICs), digital signal processors (DSPs), field-programmable gate arrays (FPGAs), programmable logic controllers (PLCs), complex-programmable logic devices (CPLDs), discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. Furthermore, a processor may utilize nanoscale architectures such as molecular and quantum dot-based transistors, switches, or gates, but not limited to, to optimize space utilization or enhance the performance of associated equipment. A processor may be implemented as a combination of computing units.

[0178] As used herein, terms such as "storage", "storage", "data storage", "data storage", "database", and substantially any other information storage component related to the operation and functionality of a component are used to refer to a "memory component" entity embodied in a "memory" or a component including a memory. The memory or memory component described herein can be either volatile memory or non-volatile memory, or can include both volatile and non-volatile memory. By way of example and not limitation, non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable ROM (EEPROM), flash memory, or non-volatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). Volatile memory can include, for example, RAM that can operate as an external cache memory. By way of example and not limitation, RAM is available in many forms such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), extended SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), or Rambus dynamic RAM (RDRAM). Also, the memory components of the systems or computer implemented methods described herein are intended to include these or any other suitable type of memory without limitation thereto.

[0179] The above descriptions include only examples of systems and computer implementations. Of course, it is impossible to describe all possible combinations of components or computer implementations for the purpose of illustrating one or more embodiments, but those skilled in the art will recognize that many further combinations or permutations of one or more embodiments are possible. Furthermore, to the extent that the terms “includes,” “has,” “possesses,” and similar terms are used in the detailed description, claims, appendices, or drawings, such terms are intended to be comprehensive in the same manner as the term “comprising,” as is interpreted when “comprising” is used as a transitional term in the claims.

[0180] While descriptions of various embodiments have been presented for illustrative purposes, they are not intended to be exhaustive or to limit the embodiments described herein. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the embodiments described. The terminology used herein has been selected to best describe the principles, practical applications, or technical improvements to the technologies available on the market, or to enable other those skilled in the art to understand the embodiments described herein.

Claims

1. The system includes a processor that executes computer executable components stored in non-temporary computer-readable memory, and the computer executable components are: A learning component that learns noise associated with non-unit operations in the circuit of a dynamic quantum circuit by correcting non-unit operations in the circuit using a stochastic Pauli-Z gate and a swirling Pauli operator. A system that has

2. The system according to the preceding claim, wherein the non-unit operation in the circuit includes a qubit measurement in the circuit that feeds forward to at least one classically controlled quantum gate.

3. The system according to the preceding claim, wherein the at least one classically controlled quantum gate lies between orbital Pauli operators.

4. The system according to claim 2, wherein the at least one classically controlled quantum gate is not between the orbital Pauli operators.

5. The system according to any one of the preceding claims, wherein the learning component learns the noise by repeatedly performing the non-unit operations in the circuit, modified by the stochastic Pauli-Z gate and the orbiting Pauli operator, over a set of Pauli bases and over a set of iteration depths, and by extracting a set of base fidelity corresponding to each of the sets of Pauli bases based on such repeated executions.

6. The system according to the preceding claim, wherein the learning component learns the noise associated with non-unit operations in the circuit by inverting the set of base fidelity via a commutation relationship defined between the set of Pauli bases and the set of Pauli generators in relation to the noise.

7. The aforementioned computer executable component is A noise reduction component that reduces the noise by inserting the reciprocal of the noise into the dynamic quantum circuit. The system according to any one of the preceding claims, further comprising:

8. The step of learning noise associated with unity operations in the circuit of a dynamic quantum circuit by correcting unity operations in the circuit using a stochastic Pauli-Z gate and a swirling Pauli operator with a device operablely coupled to the processor. A computer implementation method comprising the following:

9. The computer implementation method according to the preceding claim, wherein the non-unit operation in the circuit includes a qubit measurement in the circuit that feeds forward to at least one classically controlled quantum gate.

10. The computer implementation method according to the preceding claim, wherein the at least one classically controlled quantum gate is located between the orbital Pauli operators.

11. The computer implementation method according to claim 9, wherein the at least one classically controlled quantum gate is not between the orbital Pauli operators.

12. The step of learning the aforementioned noise is, The device repeatedly performs non-unit operations in the circuit modified by the stochastic Pauli-Z gate and the orbital Pauli operator over both the set of Pauli bases and the set of iteration depths; and The device then extracts a set of base fidelity corresponding to each set of Pauli bases based on such repeated executions. A computer implementation method according to any one of the four preceding claims, having the following:

13. The step of learning the aforementioned noise is, The device inverts the set of base fidelity via a defined exchange relationship between the set of Pauli bases and the set of Pauli generators related to the noise. A computer implementation method according to a prior claim, having the following:

14. The device reduces the noise by inserting the reciprocal of the noise into the dynamic quantum circuit. A computer implementation method according to any one of the preceding six claims, further comprising:

15. A computer program product for facilitating noise learning in dynamic quantum circuits, wherein the computer program product comprises a non-temporary computer-readable memory having program instructions embodied therein, and the program instructions, which are executable by a processor, are provided to the processor. By correcting non-unit operations in the circuit using a stochastic Pauli-Z gate and a swirling Pauli operator, the noise associated with non-unit operations in the circuit of a dynamic quantum circuit is learned. A computer program product that executes a command.

16. The computer program product according to the preceding claim, wherein the non-unit operation in the circuit includes a qubit measurement in the circuit that feeds forward to at least one classically controlled quantum gate.

17. The computer program product according to the preceding claim, wherein the at least one classically controlled quantum gate is located between the orbital Pauli operators.

18. The computer program product according to claim 16, wherein the at least one classically controlled quantum gate is not between the orbital Pauli operators.

19. The aforementioned program instruction is given to the processor, Repeatedly performing the non-unit operations in the circuit, modified by the stochastic Pauli-Z gate and the orbiting Pauli operator, over a set of Pauli bases and over a set of iteration depths; and Extracting a set of base fidelity corresponding to each of the aforementioned sets of Pauli bases based on such repeated executions. A computer program product according to any one of the four preceding claims, which is executable for further execution of the above.

20. The aforementioned program instruction is given to the processor, Inverting the set of base fidelity via a defined exchange relationship between the set of Pauli bases and the set of Pauli generators related to the noise. A computer program product according to a prior claim, which is executable for further execution.

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