Noise learning in dynamic quantum circuits

Abstract

Systems and techniques are provided that facilitate noise learning in dynamic quantum circuits. In various embodiments, the system may learn noise associated with in-circuit non-unitary operation of a dynamic quantum circuit by modifying in-circuit non-unitary operation with a probabilistic Pauli-Z gate and a rotating Pauli operator.

Description

[0001] Government License Rights

[0002] This invention was made with government support under W911NF-21-1-0002 awarded by the Army Research Office. The government has certain rights in the invention. BACKGROUND

[0003] The present disclosure relates to quantum circuits, and more specifically to noise learning in dynamic quantum circuits.

[0004] A dynamic quantum circuit can be a quantum circuit that contains one or more mid-circuit non-unitary operations, such as a mid-circuit quantum bit measurement that feeds forward to a classical controlled quantum gate. Dynamic quantum circuits can be considered more powerful and more general purpose than conventional quantum circuits. Unfortunately, however, there is a dearth of knowledge about noise in dynamic quantum circuits, which has resulted in a lack of effective error mitigation or suppression techniques for dynamic quantum circuits. Indeed, existing error mitigation or suppression techniques are only effectively compatible with conventional quantum circuits, where all non-unitary operations are pushed to the end of the circuit. Such existing error mitigation or suppression techniques are not compatible with mid-circuit non-unitary operations.

[0005] Accordingly, a system or technique that can address one or more of these technical problems can be desirable. SUMMARY

[0006] The following presents a summary to provide a basic understanding of one or more embodiments of the application. This summary is not intended to identify key or important elements, nor is it intended to delineate any scope of any of the claims. Its sole purpose is to present concepts in a simplified form as a prelude to the more detailed description that is presented later. In one or more embodiments described herein, an apparatus, system, method, or device that can facilitate noise learning in dynamic quantum circuits is described.

[0007] According to one or more embodiments, a system is provided. In various aspects, the system can include a processor that can execute computer executable components stored in a non-transitory computer-readable storage medium. In various instances, the computer executable components can include a learning component that can learn noise associated with a mid-circuit non-unitary operation of a circuit of a dynamic quantum circuit by modifying the mid-circuit non-unitary operation in the circuit with a probabilistic Pauli-Z gate and a twirled Pauli operator.

[0008] According to one or more embodiments, a computer-implemented method is provided. In various aspects, the computer-implemented method can include learning, by a device operably coupled to a processor, noise associated with non-identity operations in a circuit of a dynamic quantum circuit by modifying the non-identity operations in the circuit with a probabilistic Pauli-Z gate and a rotating Pauli operator.

[0009] According to one or more embodiments, a computer program product for facilitating noise learning in a dynamic quantum circuit is provided. In various aspects, the computer program product can include a non-transitory computer-readable memory having program instructions embodied thereon. In various instances, the program instructions can be executable by a processor to cause the processor to learn noise associated with non-identity operations in a circuit of a dynamic quantum circuit by modifying the non-identity operations in the circuit with a probabilistic Pauli-Z gate and a rotating Pauli operator.

[0010] Various other details of various embodiments described herein are provided in the following clauses:

[0011] Clause 1 : A system comprising: a processor that executes computer-executable components stored in a non-transitory computer-readable memory, wherein the computer-executable components comprise: a learning component that learns noise associated with non-identity operations in a circuit of a dynamic quantum circuit by modifying the non-identity operations in the circuit with a probabilistic Pauli-Z gate and a rotating Pauli operator.

[0012] Clause 2: The system of any of the preceding clauses, wherein the non-identity operations in the circuit comprise quantum bit measurements in the circuit that feed forward to at least one quantum gate that is classically controlled.

[0013] Clause 3: The system of any of the preceding clauses, wherein the at least one quantum gate that is classically controlled is located between the rotating Pauli operators.

[0014] Clause 4: The system of any of the preceding clauses, wherein the at least one quantum gate that is classically controlled is not located between the rotating Pauli operators.

[0015] Clause 5: The system of any of the preceding clauses, wherein the learning component learns the noise by repeatedly executing the non-identity operations in the circuit modified with the probabilistic Pauli-Z gate and the rotating Pauli operator over a set of Pauli bases and over a set of repetition depths and extracting a set of basis fidelities respectively corresponding to the set of Pauli bases based on such repeated executions.

[0016] Clause 6: The system of any preceding clause, wherein the learning component learns the noise associated with the non-unitary operation in the circuit by inverting a set of basis fidelities via commutation relations defined between a set of Pauli bases and a set of Pauli generators associated with the noise.

[0017] Clause 7: The system of any preceding clause, wherein the computer executable components further comprise: a mitigation component that mitigates the noise by inserting an inverse of the noise into the dynamic quantum circuit.

[0018] In various aspects, any combination or multiple combinations of Clauses 1-7 can be implemented.

[0019] Clause 8: A computer-implemented method comprising: learning, by a device operatively coupled to a processor, a noise associated with a non-unitary operation in a circuit of a dynamic quantum circuit by modifying the non-unitary operation in the circuit with a probabilistic Pauli-Z gate and a rotating Pauli operator.

[0020] Clause 9: The computer-implemented method of any preceding clause, wherein the non-unitary operation in the circuit comprises a quantum bit measurement in the circuit that feeds forward to at least one quantum gate of a classical control.

[0021] Clause 10: The computer-implemented method of any preceding clause, wherein the at least one quantum gate of a classical control is located between the rotating Pauli operators.

[0022] Clause 11: The computer-implemented method of any preceding clause, wherein the at least one quantum gate of a classical control is not located between the rotating Pauli operators.

[0023] Clause 12: The computer-implemented method of any preceding clause, wherein learning the noise comprises: performing, by the device and repeatedly over a set of repetition depths and over a set of Pauli bases, the non-unitary operation in the circuit modified with the probabilistic Pauli-Z gate and the rotating Pauli operator; and extracting, by the device and based on such repeated performance, a set of basis fidelities respectively corresponding to the set of Pauli bases.

[0024] Clause 13: The computer-implemented method of any preceding clause, wherein learning the noise comprises: inverting, by the device, a set of basis fidelities via commutation relations defined between a set of Pauli bases and a set of Pauli generators associated with the noise.

[0025] Clause 14: The computer-implemented method of any preceding clause, further comprising: mitigating, by the device, the noise by inserting an inverse of the noise into the dynamic quantum circuit.

[0026] In various aspects, any combination or multiple combinations of any of Clauses 8-14 can be implemented.

[0027] Clause 15: A computer program product for facilitating noise learning in a dynamic quantum circuit, the computer program product comprising a non-transitory computer-readable memory having program instructions embodied thereon, the program instructions executable by a processor to cause the processor to: learn noise associated with a non-identity operation in a circuit of a dynamic quantum circuit by modifying the non-identity operation in the circuit with a probabilistic Pauli-Z gate and a rotating Pauli operator.

[0028] Clause 16: The computer program product of any preceding clause, wherein the non-identity operation in the circuit comprises a quantum bit measurement of a quantum gate fed forward to at least one classical control.

[0029] Clause 17: The computer program product of any preceding clause, wherein the at least one classical controlled quantum gate is located between the rotating Pauli operators.

[0030] Clause 18: The computer program product of any preceding clause, wherein the at least one classical controlled quantum gate is not located between the rotating Pauli operators.

[0031] Clause 19: The computer program product of any preceding clause, wherein the program instructions are further executable to cause the processor to: repeatedly perform the non-identity operation in the circuit modified with the probabilistic Pauli-Z gate and the rotating Pauli operator over a set of Pauli bases and over a set of repetition depths; and extract a set of basis fidelities respectively corresponding to the set of Pauli bases based on such repeated performance.

[0032] Clause 20: The computer program product of any preceding clause, wherein the program instructions are further executable to cause the processor to: invert the set of basis fidelities via a commutation relation defined between the set of Pauli bases and a set of Pauli generators associated with the noise.

[0033] In various aspects, any combination or multiple combinations of any of Clauses 15-20 can be implemented. BRIEF DESCRIPTION OF DRAWINGS

[0034] Figure 1 A block diagram illustrating an example, non-limiting system that facilitates noise learning in a dynamic quantum circuit in accordance with one or more embodiments described herein is shown.

[0035] Figure 2An exemplary, non-limiting diagram of a dynamic quantum circuit is shown in accordance with one or more embodiments described herein.

[0036] Figure 3 A block diagram of an exemplary, non-limiting system that facilitates learning of noise in a dynamic quantum circuit is shown in accordance with one or more embodiments described herein, the system including a probabilistic Pauli-Z gate, a rotated Pauli operator, and a learned noise model coefficient set.

[0037] Figures 4-5 An exemplary, non-limiting block diagram of a non-unitary operation in a circuit modified with a probabilistic Pauli-Z gate and a rotated Pauli operator is shown in accordance with one or more embodiments described herein.

[0038] Figure 6 An exemplary, non-limiting block diagram of how noise associated with a non-unitary operation in a circuit can be composed of a noise model Pauli generator and a noise model coefficient set is shown in accordance with one or more embodiments described herein.

[0039] Figure 7 An exemplary, non-limiting block diagram of how an observable expectation value can be obtained with respect to a non-unitary operation in a circuit is shown in accordance with one or more embodiments described herein.

[0040] Figure 8 An exemplary, non-limiting block diagram of how a plurality of sets of observable expectation values can be obtained is shown in accordance with one or more embodiments described herein.

[0041] Figure 9 An exemplary, non-limiting block diagram of how a basis fidelity can be obtained from a plurality of sets of observable expectation values is shown in accordance with one or more embodiments described herein.

[0042] Figure 10 An exemplary, non-limiting block diagram of how learned noise model coefficients can be obtained from a basis fidelity is shown in accordance with one or more embodiments described herein.

[0043] Figure 11 An exemplary, non-limiting block diagram of an exemplary, non-limiting experimental result is shown in accordance with one or more embodiments described herein.

[0044] Figure 12 A block diagram of an exemplary, non-limiting system that includes an inverted noise operation that facilitates learning of noise in a dynamic quantum circuit is shown in accordance with one or more embodiments described herein.

[0045] Figures 13-14 An exemplary, non-limiting block diagram of a dynamic quantum circuit with an inverted noise operation inserted is shown in accordance with one or more embodiments described herein.

[0046] Figures 15-16 Example, non-limiting experimental results are shown that facilitate noise learning in dynamic quantum circuits in accordance with one or more embodiments described herein.

[0047] Figure 17 A flowchart of an example, non-limiting computer-implemented method that facilitates noise learning in dynamic quantum circuits in accordance with one or more embodiments described herein is shown.

[0048] Figure 18 A block diagram of an example, non-limiting operating environment in which one or more embodiments described herein can be facilitated is shown. DETAILED DESCRIPTION

[0049] The following detailed description is merely illustrative and is not intended to limit or restrict the embodiments or the application or uses of such embodiments to the embodiments described in the specification, drawings, and claims. Furthermore, there is no intention to be bound by any expressed or implied information presented in the preceding Background or Summary sections, or in the Detailed Description section.

[0050] One or more embodiments will now be described with reference to the attached drawings, wherein like reference numerals are used to refer to like elements throughout. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a more thorough understanding of one or more embodiments. It is evident, however, that one or more embodiments can be practiced without these specific details.

[0051] A quantum computer can be any suitable device that utilizes a lattice of qubits (e.g., a plurality of superconducting qubits fabricated on one or more quantum substrates and exhibiting any suitable connectivity topology) for information processing. A quantum circuit can be any suitable sequence of any suitable number of parallel or serial 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 otherwise affect the state of a qubit. As some non-limiting 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).

[0052] A conventional quantum circuit can be a quantum circuit whose only non-identity operation is a qubit measurement located at the end of the quantum circuit (e.g., in the final layer). That is, a conventional quantum circuit must not contain an in-circuit non-identity operation. In contrast, a dynamic quantum circuit can be a quantum circuit that contains one or more in-circuit non-identity operations. As a non-limiting example, an in-circuit non-identity operation can include an in-circuit qubit measurement. In various aspects, an in-circuit qubit measurement can be a qubit measurement operation that is not pushed to the end of the circuit. In other words, an in-circuit qubit measurement can 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 instances, an in-circuit qubit measurement can be fed forward to one or more classically controlled quantum gates. In various aspects, a classically controlled quantum gate can be any suitable quantum gate whose performance depends on or is otherwise controlled by a qubit measurement operation (e.g., if the qubit measurement operation produces a measurement value of 0, the classically controlled quantum gate is not executed; conversely, if the qubit measurement operation produces a measurement value of 1, the classically controlled quantum gate can be executed). In various instances, 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 to a classical bit that has only two possible values (0 or 1).

[0053] A dynamic quantum circuit can be considered more computationally powerful and more general than a conventional quantum circuit. Indeed, if a dynamic quantum circuit contains one or more in-circuit qubit measurements, in some cases, the one or more in-circuit qubit measurements can influence, in real-time, which quantum gates are applied or not applied during the remainder of the dynamic quantum circuit. For example, assume a dynamic quantum circuit includes an in-circuit qubit measurement on a first qubit, where such in-circuit qubit measurement is fed forward to a particular quantum gate on a second qubit. In various instances, if the in-circuit qubit measurement produces a value of 1, the particular quantum gate can be applied to the second qubit. However, if the in-circuit qubit measurement produces a value of 0, the particular quantum gate cannot be applied to the second qubit. Such flexibility of a dynamic quantum circuit can help reduce circuit depth or otherwise alleviate qubit connectivity limitations.

[0054] However, implementation of dynamic quantum circuits can be hindered by a lack of effective error mitigation or suppression techniques. Whether a quantum circuit is regular or dynamic, it experiences noise (e.g., although single-qubit quantum gates can be noiseless, entangling quantum gates and qubit measurements are not noiseless). This noise degrades circuit performance, which is undesirable. Error mitigation or suppression techniques can be thought of as quantum computing protocols that can help reduce this noise. Unfortunately, existing error mitigation or suppression techniques (e.g., probabilistic error cancellation (PEC)) are only effectively applicable to regular quantum circuits, not dynamic quantum circuits. In particular, existing error mitigation techniques assume that all non-unitary operations are pushed to the end of the circuit, so they cannot properly learn or mitigate noise when non-unitary operations in the circuit are involved. Furthermore, although existing error suppression techniques (e.g., in-circuit pauses or segmentation to allow noise to dissipate) can technically be performed on dynamic quantum circuits, it has been empirically found that such existing error suppression techniques produce significantly worse than expected noise suppression for dynamic quantum circuits than for regular quantum circuits. In other words, existing error mitigation or suppression techniques are either incompatible with non-unitary operations in the circuit or are not effectively compatible with non-unitary operations in the circuit, such as qubit measurements in circuits that feed forward to classically controlled quantum gates.

[0055] Accordingly, it is desirable for a technical system that can address one or more of these technical problems.

[0056] The various embodiments described herein can solve one or more of these technical problems. In particular, the various embodiments described herein can facilitate noise learning in dynamic quantum circuits. That is, the inventors of the various embodiments described herein devised various techniques for identifying, determining, measuring, or otherwise learning noise caused by non-unitary operations in a circuit (e.g., qubit measurements in a circuit that feed forward to quantum gates that are controlled classically). In particular, such various techniques can involve modifying non-unitary operations in a circuit with a probabilistic Pauli-Z gate and a rotated Pauli operator. In various aspects, implementation of the rotated Pauli operator can be considered to reshape the noise caused by the non-unitary operations in the circuit such that the noise can be considered to be a function of a plurality of Pauli generators corresponding to a plurality of noise coefficients. In various instances, such various techniques can also include repeatedly performing the non-unitary operations in the circuit modified with the probabilistic Pauli-Z gate and the rotated Pauli operator over a plurality of Pauli bases and over a plurality of repetition depths. In various cases, measurements taken during such repeated performance can yield a plurality of basis fidelities associated with the plurality of Pauli bases. In various aspects, the plurality of basis fidelities can be inverted via an anti-commutation relation between the plurality of Pauli bases and the plurality of Pauli generators, thereby determining the plurality of noise coefficients. When the plurality of noise coefficients is determined, the noise caused by the non-unitary operations in the circuit can be considered to have been learned. In various instances, such learned noise can be inverted and such noise inversion can be performed prior to the non-unitary operations in the circuit, thereby mitigating or otherwise reducing the noise caused by the non-unitary operations in the circuit. However, this is merely one non-limiting example. In various other instances, such learned noise can be used for any suitable purpose, such as benchmarking techniques, designing noise tailored quantum algorithms or quantum error correction protocols, facilitating real-time device adjustments, or facilitating dynamic decoupling.

[0057] The various embodiments described herein can be considered to be computerized tools (e.g., any suitable combination of computer-executable hardware or computer-executable software) that can facilitate noise learning in dynamic quantum circuits. In various aspects, such computerized tools can include an accessing component, a learning component, or a mitigating component.

[0058] In various embodiments, there can be a quantum computer. In various aspects, the quantum computer can include any suitable number of qubits. In various instances, such qubits can exhibit any suitable structure, construction, or architecture (e.g., can be superconducting qubits, spin qubits, or quantum dots).

[0059] In various instances, there can be a dynamic quantum circuit. In various aspects, the dynamic quantum circuit can be any suitable quantum circuit that can be executed or otherwise performed on a quantum computer, and it can include in-circuit non-Clifford operations. As a non-limiting example, the in-circuit non-Clifford operation can be an in-circuit qubit measurement operation that can be fed forward to subsequent classical controls of quantum gates of the dynamic quantum circuit.

[0060] In various instances, it can be desirable to learn or mitigate any noise caused by the in-circuit non-Clifford operation. In various cases, the computerized tool as described herein can learn or mitigate such noise.

[0061] In various aspects, the accessing component of the computerized tool can electronically access the quantum computer via any suitable wired or wireless electronic connection. In various instances, the accessing component can also access or otherwise receive, retrieve, or import the dynamic quantum circuit from any suitable data source. For example, the accessing component can obtain the dynamic quantum circuit from any suitable centralized or decentralized data structure (e.g., graph data structure, relational data structure, hybrid data structure), whether remote from or local to the accessing component. In any case, the accessing component can access the quantum computer or the dynamic quantum circuit such that other components of the computerized tool can electronically interact with the quantum computer (e.g., start, stop, initialize, control) or can electronically interact with the dynamic quantum circuit (e.g., read, write, edit, copy, manipulate, execute).

[0062] In various aspects, the learning component of the computerized tool can electronically learn the noise caused by the in-circuit non-Clifford operation based on a probabilistic Pauli-Z gate and based on a rotated Pauli operator. More specifically, the probabilistic Pauli-Z gate can be a Pauli-Z gate that is executed with a 50% probability instead of a 100% probability. Moreover, the rotated Pauli operator can be a random tensor product of Pauli gates. In various instances, the learning component can apply the probabilistic Pauli-Z gate to any qubit on which the in-circuit non-Clifford operation is performed such that the probabilistic Pauli-Z gate follows (e.g., occurs after) the in-circuit non-Clifford operation. In various cases, the learning component can deploy the rotated Pauli operator on both sides of the in-circuit non-Clifford operation and the probabilistic Pauli-Z gate.

[0063] In various aspects, the probabilistic Pauli-Z gate can be considered to remove, cancel, or otherwise reduce any post-execution phase error that can be caused by the non- unitary operation in the circuit. In various instances, the rotation Pauli operator can be considered to reshape noise caused by the non-unitary operation in the circuit into a Pauli structure (e.g., into a diagonal structure). In various cases, based on this reshaping, the noise caused by the non-unitary operation in the circuit can be considered to be a function of a set of Pauli generators weighted by a set of coefficients, respectively. In various aspects, the values or magnitudes of such coefficients can initially be unknown.

[0064] In various instances, the learning component can determine such coefficients, and thus can determine the noise caused by the non-unitary operation in the circuit. In particular, the learning component can repeatedly execute the non-unitary operation in the circuit modified with the probabilistic Pauli-Z gate and the rotation Pauli operator over a set of Pauli bases and over a set of progressively increasing repetition depths. In various cases, the learning component can measure the observable expectation values for each base-depth pair, resulting in a set of observable expectation values for each Pauli basis. In various aspects, for each particular Pauli basis, the learning component can fit an exponential decay curve to the set of observable expectation values corresponding to that particular Pauli basis, and such fitted exponential decay curve can yield a basis fidelity corresponding to that particular Pauli basis. In other words, the learning component can obtain a set of basis fidelities corresponding to the set of Pauli bases, respectively. In various instances, the learning component can determine, identify, or otherwise estimate the values for the unknown coefficients corresponding to the set of Pauli generators by inverting the set of basis fidelities via the commutation relations defined between the set of Pauli bases and the set of Pauli generators. In various cases, the commutation relations can be obtained by applying the symplectic inner product to the respective base-generator pairs. In this way, the learning component can determine the set of coefficients corresponding to the set of Pauli generators, and thus can be considered to have determined the noise caused by the non-unitary operation in the circuit.

[0065] In various aspects, the mitigation component of the computerized tool can mitigate the noise caused by the non-unitary operation in the circuit. More specifically, because the noise can be considered to be a function of the set of Pauli generators and the set of coefficients, the noise can be considered to be an effective quantum gate or quantum operation that can be executed on a quantum computer. Thus, the mitigation component can compute an inverse operation of this noise, where the inverse operation is likewise an effective quantum gate or quantum operation that can be executed on a quantum computer. Accordingly, the mitigation component can insert this inverse operation into the dynamic quantum circuit prior to the non-unitary operation in the circuit, and such inverse operation can cause the noise of the non-unitary operation in the circuit to be canceled out or otherwise reduced.

[0066] Accordingly, various embodiments described herein can be considered computerized tools that are capable of learning noise caused by or otherwise associated with non- unitary operations in the circuit of a dynamic quantum circuit. Once learned, such noise can be mitigated by inverting. However, this is merely a non-limiting example. In other cases, once learned, such noise can be used for any other suitable purpose (e.g., such noise can be used to create a noise tailored quantum protocol).

[0067] Various embodiments described herein can be used to solve problems that are highly technical in nature (e.g., facilitating noise learning in dynamic quantum circuits) using hardware or software, that are not abstract and cannot be performed as a set of mental acts by a human. Moreover, some of the processes performed can be performed by a special purpose computer (e.g., a quantum computer that can execute or implement a dynamic quantum circuit including real qubits). In various aspects, some of the defined tasks associated with various embodiments described herein can include, by a device operatively coupled to a processor, learning noise associated with non-unitary operations in the circuit of a dynamic quantum circuit by modifying the non-unitary operations in the circuit with a probabilistic Pauli-Z gate and a rotated Pauli operator.

[0068] Neither a human mind nor a human with paper and pencil can electronically access a dynamic quantum circuit that includes non-unitary operations in the circuit (e.g., qubit measurements in the circuit with feedforward) and electronically learn noise caused by the non-unitary operations in the circuit by applying a probabilistic Pauli-Z gate and a rotated Pauli operator to the non-unitary operations in the circuit. After all, a quantum computer is a special purpose computing hardware that leverages physical qubits (e.g., superconducting qubits such as transmons) to process information. Physical qubits cannot be implemented by a human mind or a human with paper and pencil. Moreover, a quantum circuit can be a sequence of quantum gates that can be executed on a quantum computer. Neither a human mind nor a human with paper and pencil can execute quantum gates (e.g., probabilistic Pauli-Z gates, rotated Pauli operators) on physical qubits. Thus, a computerized tool that learns noise caused by non-unitary operations in the circuit by implementing probabilistic Pauli-Z gates and rotated Pauli operators is computerized in nature and cannot be implemented without a computer in any practical, useful, or realistic way.

[0069] In various examples, one or more embodiments described herein can integrate the teachings described herein into practical applications. As noted above, existing error mitigation or suppression techniques (e.g., PEC) are only effectively compatible with conventional quantum circuits (e.g., quantum circuits in which all non-gauge operations, such as qubit measurements, are pushed to the end of the circuit). In fact, these existing error mitigation or suppression techniques are specifically designed to learn noise caused by or otherwise associated with gauge operations (such as Clifford layers), but not noise caused by or otherwise associated with non-gauge operations (such as qubit measurements). In fact, while gauge operations can be noisy, they can still be considered reversible, and thus capable of fully preserving quantum information. In sharp contrast, non-gauge operations can be considered irreversible, and thus incapable of fully preserving quantum information. In other words, even if they are noiseless, non-gauge operations can be considered to irreversibly discard at least some quantum information. Existing error mitigation or suppression techniques are unable to learn noise in this discarded quantum information. Additionally, noise caused by gauge operations can only affect those qubits on which the gauge operations are performed. That is, noise from gauge operations can at most propagate through directly coupled neighboring qubits. In sharp contrast, noise caused by non-gauge operations can affect any or all qubits of a quantum computer. In other words, noise from non-gauge operations can not only propagate through directly coupled neighboring qubits, but also through qubit readout lines or otherwise between non-adjacent qubits. This can make noise from non-gauge operations highly complex, device-dependent, and difficult to handle. For at least these reasons, existing error mitigation or suppression techniques are not effectively compatible with non-gauge operations in a circuit.

[0070] The various embodiments described herein can address one or more of the technical problems of existing error mitigation or suppression techniques. In other words, the various embodiments described herein can enable noise caused by a non-Clifford operation in a circuit to be learned. Indeed, as described herein, the inventors have realized that noise associated with a non-Clifford operation in a circuit can be learned by modifying the non-Clifford operation in the circuit with a probabilistic Pauli-Z gate and a rotation Pauli operator. In particular, such modified non-Clifford operation in the circuit can be iteratively performed over multiple Pauli bases and over multiple iteration depths. In various aspects, basis fidelities can be extracted from measurements made during such iterative performance, and noise associated with the non-Clifford operation in the circuit can be identified by inverting these basis fidelities via commutation relations corresponding to the multiple Pauli bases. In this way, noise caused by a non-Clifford operation in a circuit can be learned, identified, or otherwise determined. In various instances, such learned noise can be used for any suitable purpose. As one non-limiting example, an inverse operation of such learned noise can be computed, and such inverse operation can be inserted or placed prior to the non-Clifford operation in the circuit, thereby canceling, mitigating, or otherwise reducing noise of the non-Clifford operation in the circuit. In other words, the various embodiments described herein can learn or mitigate noise caused by a non-Clifford operation in a circuit, which is in stark contrast to existing error mitigation or suppression techniques that are not able to effectively learn or mitigate such noise.

[0071] Learning or mitigating noise caused by a non-Clifford operation in a circuit can be considered to provide quantifiable performance improvements in the field of quantum circuits. For example, learning or mitigating such noise can enable a quantum computer to achieve meaningful quantum computing results using fewer runs (e.g., by fewer operations than otherwise required). As another example, learning or mitigating such noise can make it possible to construct and execute quantum circuits that are otherwise more complex or more delicate. As yet another example, learning or mitigating such noise can enable a quantum computer to run with less pre-processing or post-processing. In these ways, learning or mitigating noise associated with a non-Clifford operation in a circuit can be considered to help reduce or more efficiently use quantum computing resources. This is a specific and tangible technical improvement in the field of quantum circuits. For at least these reasons, the various embodiments described herein no doubt constitute useful and practical applications of computers.

[0072] It should be understood that the drawings and the descriptions herein disclose non-limiting examples of various embodiments. It should also be understood that the drawings are not necessarily to scale.

[0073] Figure 1A block diagram of an exemplary, non-limited system 100 that can facilitate noise learning in dynamic quantum circuits in accordance with one or more embodiments described herein is shown. As shown, a 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.

[0074] In various embodiments, the quantum computer 104 can be any suitable quantum computing device or quantum computing hardware. In various aspects, the quantum computer 104 can include a set of qubits 106. In various instances, the set of qubits 106 can include n qubits: qubit 106(1) through qubit 106(n) for any suitable positive integer n. In various cases, any one of the set of qubits 106 can exhibit any suitable structure or architecture. As a non-limiting example, a qubit from the set of qubits 106 can exhibit a superconducting qubit architecture (e.g., such a qubit can be made of any suitable number of Josephson structures shunted with any suitable number of planar capacitor electrode plates). As another non-limiting example, a qubit from the set of qubits 106 can exhibit a quantum dot architecture. As yet another non-limiting example, a qubit from the set of qubits 106 can exhibit a spin qubit architecture. In various aspects, different qubits of the set of qubits 106 can exhibit the same or different structures or architectures from one another. In various instances, the set of qubits 106 can exhibit any suitable inter-qubit coupling topology (e.g., a linear connection topology, a heavy hexagonal connection topology). Although not explicitly shown in FIG. 1, the quantum computer 104 can include or otherwise be associated with any suitable hardware or software (e.g., a real-time controller implemented in a field programmable gate array of the quantum computer 104) that can be used to initialize any one of the set of qubits 106 or perform any suitable quantum operation on the set of qubits 106 (e.g., a quantum gate, a qubit measurement, a qubit idling). Figure 1 Although not explicitly shown in FIG. 1, the quantum computer 104 can include or otherwise be associated with any suitable hardware or software (e.g., a real-time controller implemented in a field programmable gate array of the quantum computer 104) that can be used to initialize any one of the set of qubits 106 or perform any suitable quantum operation on the set of qubits 106 (e.g., a quantum gate, a qubit measurement, a qubit idling).

[0075] In various aspects, the dynamic quantum circuit 108 can be any suitable sequence of quantum gates or quantum operations that can be executed on or otherwise performed on the quantum computer 104. Thus, in various instances, the dynamic quantum circuit 108 can be considered an n-qubit quantum circuit (e.g., can be considered a quantum circuit that can operate on n qubits). In various cases, the dynamic quantum circuit 108 can include an in-circuit non-unitary operation 110, hence the term “dynamic.” In various aspects, the in-circuit non-unitary operation 110 can not be located at the end of the dynamic quantum circuit 108, hence the term “in-circuit.” In other words, the in-circuit non-unitary operation 110 can 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 instances, the in-circuit non-unitary operation 110 can be any suitable type of non-unitary (e.g., irreversible) quantum computing operation. As a non-limiting example, the in-circuit non-unitary operation 110 can be an in-circuit qubit measurement operation. In various cases, such an in-circuit qubit measurement operation can feed forward to any suitable quantum gate of the dynamic quantum circuit 108 placed or located after the in-circuit non-unitary operation 110.

[0076] Figure 2 An exemplary, non-limiting diagram 200 of a dynamic quantum circuit 108 is shown in accordance with one or more embodiments described herein. That is, Figure 2 A non-limiting example embodiment of a dynamic quantum circuit 108 and an in-circuit non-unitary operation 110 is depicted.

[0077] As shown, the dynamic quantum circuit 108 can operate on a set of qubits 106, and thus can be considered an n-qubit quantum circuit. In various aspects, the dynamic quantum circuit 108 can include an in-circuit non-unitary layer 202, at least one layer 204, and at least one layer 206. In various instances, the in-circuit non-unitary layer 202 can be any layer of the dynamic quantum circuit 108 that contains the in-circuit non-unitary operation 110.

[0078] In various cases, the at least one layer 204 can include any layer of the dynamic quantum circuit 108 that precedes the in-circuit non-unitary layer 202. Thus, the at least one layer 204 can include any suitable number of any suitable type of quantum gate or quantum operation, and such quantum gate or quantum operation can be considered to precede (e.g., be placed or located before) the in-circuit non-unitary operation 110. As a non-limiting example, the at least one layer 204 can include a Hadamard gate (represented by “H”) applied to qubit 106(1) (represented by “Q1”), a CNOT gate (represented by “CNOT”) applied to qubit 106(2) (represented by “Q2”) and qubit 106(1), and a CNOT gate (represented by “CNOT”) applied to qubit 106(3) (represented by “Q3”) and qubit 106(2).n "Y" and a controlled Pauli-X gate (denoted by "CX") having qubit 106(1) as the source and qubit 106(n) as the target. Note that these quantum gates are merely non-limiting examples that are convenient for illustration and explanation. Also, although Figure 2 quantum gates applied in the at least one layer 204 to any of the qubits in the set of qubits 106 other than qubit 106(1) and qubit 106(n), this is also merely a non-limiting example that is convenient for illustration and explanation. In various aspects, the at least one layer 204 can include any suitable number of any suitable types of quantum gates arranged in any suitable order of execution, and such quantum gates can be applied to any of the qubits in the set of qubits 106.

[0079] In various aspects, the at least one layer 206 can include any layer of the dynamic quantum circuit 108 that is subsequent to the non-identity layer 202 in the circuit. Thus, the at least one layer 206 can include any suitable number of any suitable types of quantum gates or quantum operations, and such quantum gates or quantum operations can be considered to be subsequent to (e.g., placed or positioned after) the non-identity operation 110 in the circuit. As one non-limiting example, the at least one layer 206 can include a rotation about the z-axis (denoted by "R z "Y" and a controlled Pauli-X gate (denoted by "CX") having qubit 106(1) as the source and qubit 106(n) as the target. Note that these quantum gates are merely non-limiting examples that are convenient for illustration and explanation. Also, although Figure 2 quantum gates applied in the at least one layer 206 to any of the qubits in the set of qubits 106 other than qubit 106(1) and qubit 106(n), this is also merely a non-limiting example that is convenient for illustration and explanation. In various aspects, the at least one layer 206 can include any suitable number of any suitable types of quantum gates arranged in any suitable order of execution, and such quantum gates can be applied to any of the qubits in the set of qubits 106.

[0080] In Figure 2In the non-limiting example of FIG. 1, the in-circuit non-unitary operation 110 can be an in-circuit qubit measurement applied to the qubit 106(n). However, this is merely a non-limiting example for ease of illustration and explanation. In various aspects, the in-circuit non-unitary operation 110 can be an in-circuit qubit measurement applied to any other qubit in the set of qubits 106. In various instances, when the in-circuit non-unitary operation 110 is an in-circuit qubit measurement, such in-circuit qubit measurement can be fed forward to the classically controlled quantum gate 208. In other words, whether the classically controlled quantum gate 208 is actually executed can depend on the outcome of the in-circuit qubit measurement. For example, if the in-circuit qubit measurement produces a measurement state of 1, then the classically controlled quantum gate 208 can be executed. However, if the in-circuit qubit measurement instead produces a measurement state of 0, then the classically controlled quantum gate 208 can not be executed. In Figure 2 In the non-limiting example of FIG. 1, the classically controlled quantum gate 208 is a Pauli-X gate (denoted by “X”) applied to the qubit 106(1). However, this is merely a non-limiting example for ease of illustration and explanation. In various instances, the classically controlled quantum gate 208 can be any other suitable quantum gate applied to any one of the qubits in the set of qubits 106.

[0081] Although Figure 2 only one instance of an in-circuit non-unitary operation (e.g., 110) is shown as being located within the in-circuit non-unitary layer 202, this is merely a non-limiting example for ease of illustration and explanation. In various instances, the in-circuit non-unitary layer 202 can include any suitable number of in-circuit non-unitary operations, where any of the in-circuit non-unitary operations can be applied to any one of the qubits in the set of qubits 106.

[0082] In any case, the in-circuit non-unitary operation 110 can be considered to cause, generate, or otherwise be associated with noise 210 (denoted by “Λ”). As shown, the noise 210 can be considered to be a quantum operation that can affect, corrupt, contaminate, or otherwise propagate to any or all of the set of qubits 106. In other words, the noise 210 of the in-circuit non-unitary operation 110 can not only be contained on the particular qubit on which the in-circuit non-unitary operation 110 is executed (e.g., 106(n) in the non-limiting example of FIG. 1). In various instances, the noise 210 can be considered to be a Completely Positive Trace Preserving map (CPTP). Figure 2 In the non-limiting example of FIG. 1, the classically controlled quantum gate 208 is a Pauli-X gate (denoted by “X”) applied to the qubit 106(1). However, this is merely a non-limiting example for ease of illustration and explanation. In various instances, the classically controlled quantum gate 208 can be any other suitable quantum gate applied to any one of the qubits in the set of qubits 106.

[0083] Returning to Figure 1 , it is desirable to learn and subsequently mitigate the noise 210 caused or otherwise associated with the in-circuit non-unitary operation 110. As described herein, the dynamic circuit noise learning and mitigation system 102 can facilitate such learning and mitigation.

[0084] In various embodiments, the dynamic circuit noise learning and mitigation system 102 can include a processor 112 (e.g., a computer processing unit, a microprocessor) and a non-transitory computer-readable memory 114 operably connected or coupled to the processor 112. The memory 114 can store computer-executable instructions that, when executed by the processor 112, can cause the processor 112 or other components of the dynamic circuit noise learning and mitigation system 102 (e.g., the accessing component 116, the learning component 118, the mitigation component 120) to perform one or more actions. In various embodiments, the memory 114 can store computer-executable components (e.g., the accessing component 116, the learning component 118, the mitigation component 120), and the processor 112 can execute the computer-executable components.

[0085] In various embodiments, the dynamic circuit noise learning and mitigation system 102 can include an accessing component 116. In various aspects, the accessing component 116 can electronically access the quantum computer 104 in any suitable manner such that the dynamic circuit noise learning and mitigation system 102 can initialize, electronically activate (e.g., start up), electronically deactivate (e.g., shut down), or otherwise electronically control the quantum computer 104. Moreover, in various instances, the accessing component 116 can electronically receive, retrieve, obtain, import, or otherwise access the dynamic quantum circuit 108 from any suitable data structure or from any suitable computing device. In any case, the accessing component 116 can electronically access (e.g., send data or program instructions to or receive data or program instructions from) the quantum computer 104 or the dynamic quantum circuit 108 such that other components of the dynamic circuit noise learning and mitigation system 102 can electronically interact with the quantum computer 104 or with the dynamic quantum circuit 108.

[0086] In various embodiments, the dynamic circuit noise learning and mitigation system 102 can include a learning component 118. In various aspects, the learning component 118 can learn the noise 210 of the non-unitary operation 110 in the circuit based on a probabilistic Pauli-Z gate and based on a rotated Pauli operator, as described herein.

[0087] In various embodiments, the dynamic circuit noise learning and mitigation system 102 can include a mitigation component 120. In various instances, the mitigation component 120 can mitigate the noise 210 of the non-unitary operation 110 in the circuit by inserting an inverse operation of the noise 210 into the dynamic quantum circuit 108, as described herein.

[0088] Figure 3A block diagram of an exemplary, non-limiting system 300 including a probabilistic Pauli-Z gate, a pair of rotated Pauli operators, and a learned noise model coefficient set, which can facilitate noise learning in a dynamic quantum circuit, is shown in accordance with one or more embodiments described herein. As shown, system 300 can include, in some cases, the same components as system 100, and can further include a probabilistic Pauli-Z gate 302, a pair of rotated Pauli operators 304, and a learned noise model coefficient set 306.

[0089] In various aspects, learned noise model coefficient set 306 can be considered a set of scalars that characterizes or defines noise 210. In various instances, learning component 118 can electronically determine, identify, or estimate learned noise model coefficient set 306 by using probabilistic Pauli-Z gate 302 and by using a pair of rotated Pauli operators 304. More specifically, learning component 118 can electronically modify non- unitary operation 110 in a circuit with probabilistic Pauli-Z gate 302 and rotated Pauli operators 304. In various cases, learning component 118 can repeatedly or iteratively execute non-unitary operation 110 in a circuit modified with probabilistic Pauli-Z gate 302 and a pair of rotated Pauli operators 304 on quantum computer 104 and over a set of Pauli basis sets and a set of repetition depths. Based on such repeated or iterative execution, learning component 118 can measure or extract a set of basis fidelities. In various aspects, learning component 118 can determine or identify learned noise model coefficient set 306 based on the set of basis fidelities. With respect to Figures 4-10 Various non-limiting aspects are further described.

[0090] Figures 4-5 An exemplary, non-limiting block diagram of non-unitary operation 110 in a circuit modified with probabilistic Pauli-Z gate 302 and a pair of rotated Pauli operators 304 is shown in accordance with one or more embodiments described herein.

[0091] First, consider Figure 4 In various embodiments, as shown, learning component 118 can generate or create quantum circuit segment 400 that includes non-unitary operation 110 (and its associated noise 210) in a circuit and excludes the rest of dynamic quantum circuit 108.

[0092] In various aspects, learning component 118 can use probabilistic Pauli-Z gate 302 (denoted by “Z 0.5modifying the in-circuit non-identity operation 110. More specifically, as shown, the learning component 118 can place a probabilistic Pauli-Z gate 302 in the circuit after the in-circuit non-identity operation 110 and on any one of the set of qubits 106 corresponding to the in-circuit non-identity operation 110. In Figure 4 In the non-limiting example of FIG. 4, the in-circuit non-identity operation 110 is applied to qubit 106(n). Accordingly, the probabilistic Pauli-Z gate 302 can be applied to qubit 106(n) after (e.g., downstream of) the in-circuit non-identity operation 110. In any case, the probabilistic Pauli-Z gate 302 can be a single-qubit Pauli-Z gate that is applied with a 50% probability or 50% likelihood, and thus labeled “0.5.” In other words, during execution on a quantum computer (e.g., 104), the probabilistic Pauli-Z gate 302 will have a 50% chance of performing a Pauli-Z operation and a 50% chance of performing no operation. In any case, the in-circuit non-identity operation 110 can be considered to have a residual phase error, and the probabilistic Pauli-Z gate 302 can be considered to remove, zero-out, or otherwise reduce such residual phase error.

[0093] In various aspects, the learning component 118 can modify the in-circuit non-identity operation 110 within the quantum circuit segment 400 with a pair of rotated Pauli operators 304. More specifically, the pair of rotated Pauli operators 304 can be the same pair of n-qubit Paulis (each denoted by “P n ”), where one of the pair is applied before (e.g., upstream of) the in-circuit non-identity operation 110 and the other of the pair is applied after (e.g., downstream of) the probabilistic Pauli-Z gate 302. In various instances, the pair of rotated Pauli operators 304 can be randomly selected from a set of all possible n-qubit Paulis. In particular, may be randomly selected, where I can be a single-qubit identity matrix, where X can be a single-qubit Pauli-X gate, where Y can be a single-qubit Pauli-Y gate, and where Z can be a single-qubit Pauli-Z gate. When P n are positioned on both sides of the in-circuit non-identity operation 110 and the probabilistic Pauli-Z gate 302, the in-circuit non-identity operation 110 and the probabilistic Pauli-Z gate 302 can be considered to be rotated by P n and thus the term “rotated Pauli operator.”

[0094] As Figure 4As shown, when the in-circuit non-unitary operation 110 is a quantum bit measurement in the circuit of the feedforwarded classical controlled quantum gate 208, the classical controlled quantum gate 208 can be omitted from the quantum circuit segment 400. That is, in some embodiments, the classical controlled quantum gate 208 can not be placed, located, or applied between the pair of rotated Pauli operators 304. However, this is merely one non-limiting example. In other embodiments, the classical controlled quantum gate 208 can be placed, located, or applied between the pair of rotated Pauli operators 304. Indeed, this is shown in a non-limiting manner in Figure 5 Particularly, Figure 5 A quantum circuit segment 500 that can contain or include the same components as the quantum circuit segment 400 is shown, and this quantum circuit segment 500 can further include the classical controlled quantum gate 208. For ease of explanation and illustration, the remaining figures depict various non-limiting embodiments with respect to the quantum circuit segment 400. However, it should be understood and appreciated that such embodiments can be similarly applied to the quantum circuit segment 500.

[0095] Moreover, it is noted that, as described above, the figures show one in-circuit non-unitary operation (e.g., a single instance of 110) located within the in-circuit non-unitary layer 202 of the dynamic quantum circuit 108. For this reason, this one in-circuit non-unitary operation is shown within the quantum circuit segment 400 (and the quantum circuit segment 500). However, this is merely a non-limiting example for ease of illustration and explanation. In various embodiments, as described above, the in-circuit non-unitary layer 202 can include multiple in-circuit non-unitary operations that can be applied to multiple quantum bits in the set of quantum bits 106. In such embodiments, the learning component 118 can include, within the quantum circuit segment 400 (or 500), all of the in-circuit non-unitary operations within the in-circuit non-unitary layer 202. Moreover, in such embodiments, the learning component 118 can modify each of such in-circuit non-unitary operations with a respective probabilistic Pauli-Z gate (e.g., a respective instance of 302), and all such in-circuit non-unitary operations and their respective probabilistic Pauli-Z gates can be P n(e.g., by the pair of rotating Pauli operators 304) collectively enclose on both sides. In such embodiments, the noise 210 can be considered to be a collective or total noise created or generated by all such multiple in-circuit non-unitary operations. For example, assume that for any suitable positive integer a < n, the in-circuit non-unitary layer 202 includes a number of a in-circuit non-unitary operations (e.g., a number of instances of 110, each applied to a number of a qubits of the set of qubits 106). In this case, the learning component 118 can insert all such a in-circuit non-unitary operations on their respective qubits into the quantum circuit segment 400 (or 500), and can apply a different probabilistic Pauli-Z gate to each of such a in-circuit non-unitary operations. This would result in a total of a probabilistic Pauli-Z gates (e.g., a number of instances of 302) within the quantum circuit segment 400 (or 500). Moreover, in this case, the pair of rotating Pauli operators 304 can be considered to collectively rotate or enclose on both sides all a such in-circuit non-unitary operations and all a such probabilistic Pauli-Z gates. In these cases, the noise 210 can be considered to be a collective or total noise generated by all a in-circuit non-unitary operations.

[0096] In any case, enclosing the in-circuit non-unitary operation 110 on both sides with the pair of rotating Pauli operators 304 can be considered to reshape or reformat the noise 210 into a Pauli or diagonal structure. More specifically, such rotations can make the noise 210 expressible or definable with a set of Pauli generators and a set of coefficients respectively corresponding to those Pauli generators. With respect to Figure 6 Various non-limiting aspects are further described.

[0097] Figure 6 An exemplary, non-limiting block diagram 600 is shown in accordance with one or more embodiments described herein, showing how the noise 210 can be considered to be composed of a noise model Pauli generator and a noise model coefficient array after being reshaped by the pair of rotating Pauli operators 304.

[0098] For ease of explanation, the noise 210, after being reshaped or reformatted by the pair of rotated Pauli operators 304, may be referred to as reshaped noise 606. In various embodiments, the reshaped noise 606 may be considered to be defined by or otherwise be a function of the set of noise model Pauli generators 602 and the 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: noise model Pauli generator 602(1) to noise model Pauli generator 602(j). In various instances, each of the set of noise model Pauli generators 602 may be a unique or different n-qubit Pauli operator that is not an n-qubit identity matrix. For example, the noise model Pauli generator 602(1) may be A member of , and the noise model Pauli generator 602(n) can be Because the set of all possible n-qubit Pauli operators minus the n-qubit identity matrix can have 4 n -1 cardinality, so the case can be j≤4 n -1. Additionally, in some cases, sparsity can be inferred, applied, or otherwise enforced, in which case j < 4 n In fact, because probabilistic Pauli-Z gate 302 can be considered to zero out residual phase errors associated with non-unitary operation 110 in the circuit, set of noise model Pauli generators 602 can omit or otherwise exclude any n-qubit Pauli operators having Pauli-Y noise or Pauli-Z noise applied to the qubit corresponding to non-unitary operation 110 in the circuit (e.g., applied to qubit 106(n) in the non-limiting example shown in the figures).

[0099] As a non-limiting example, assume n=2, and assume that in the circuit non-unitary operation 110 is applied to qubit 2 instead of qubit 1. In various aspects, the total set of possible 2-qubit Pauli operators can be a 16-element set given by:

[0100]

[0101] In various aspects, the 2-qubit identity operator can be omitted, which can yield a 15-element set given by:

[0102] In some cases, the set of noise model Pauli generators 602 may be equivalent to such a 15-element set.

[0103] Now, since the probabilistic Pauli-Z gate 302 is implemented, any 2-qubit Pauli operator associated with Pauli-Y noise or Pauli-Z noise on qubit 2 can be omitted from such a 15-element set, which can result in a 7-element set given by:

[0104]

[0105] Accordingly, in such embodiments, the set of noise model Pauli generators 602 can be equivalent to such a 7-element set. The reduction from 15 elements to 7 elements can be considered a significant improvement in sparsity, which can help reduce computational complexity in learning the noise 210.

[0106] As another non-limiting example, assume n > 2. In this case, as the number of qubits increases, only low-weight noise model Pauli generators can be retained to reduce complexity, where the weight of an n-qubit Pauli is the number of non-identity single-qubit Pauli operations in the n-qubit Pauli. In particular, assume n = 5, where the non-unitary operation 110 in the circuit is a qubit measurement applied to qubit 5, but not qubits 1-4. Note that the set of all possible 5-qubit Pauli operators has a cardinality of 1024. By excluding the 5-qubit identity operator, this cardinality can be reduced to 1023. Additionally, since the probabilistic Pauli-Z gate 302 is implemented, this cardinality can be further reduced from 1023 to 511, since any noise model Pauli generator associated with Pauli-Y noise or Pauli-Z noise on qubit 5 can be omitted. Additionally, retaining only Pauli operators with a weight of at most 2 on unmeasured qubits (e.g., qubits 1-4 in this example) can allow terms of the form to be omitted, and terms of the form or to be retained. This can further reduce the cardinality from 511 to 133. In various cases, the set of noise model Pauli generators 602 can be equivalent to such a 133-element set.

[0107] In various aspects, the set of unknown noise model coefficients 604 can correspond to the set of noise model Pauli generators 602, respectively. Accordingly, since the set of noise model Pauli generators 602 can include j generators, the set of unknown noise model coefficients 604 can include j coefficients: unknown noise model coefficient 604(1) through unknown noise model coefficient 604(j). In various instances, each coefficient in the set of unknown noise model coefficients 604 can be a real-valued, non-negative scalar having a magnitude that is presently unknown and corresponding to a respective one of the set of noise model Pauli generators 602. As one non-limiting example, unknown noise model coefficient 604(1) can be a real-valued, non-negative scalar having a magnitude that is unknown and corresponding to noise model Pauli generator 602(1). Similarly, unknown noise model coefficient 604(j) can be a real-valued, non-negative scalar having a magnitude that is unknown and corresponding to noise model Pauli generator 602(j).

[0108] In various aspects, as noted above, the reshaped noise 606 (e.g., the noise 210 after being reformatted or reshaped by the pair of rotation Pauli operators 304) can be expressed or defined with the set of noise model Pauli generators 602 and the set of unknown noise model coefficients 604. In particular, the implementation of the Lindblad master equation can yield the following:

[0109]

[0110] where where Λ can be the reshaped noise 606 (e.g., can be the noise 210 after being reshaped via Pauli rotations), where p can be an input quantum state (or density matrix), where · can represent a combination of individual noise channels applied to p, where G can be an index set corresponding to the set of noise model Pauli generators 602 and the set of unknown noise model coefficients 604, where P g may be a member of the set of noise model Pauli generators 602, and where l g may be any member of the set of unknown noise model coefficients 604 corresponding to P g Accordingly, the reshaped noise 606 (e.g., the noise 210 after being reshaped by the pair of rotation Pauli operators 304) can be determined, identified, or otherwise learned by determining, identifying, or otherwise learning the set of unknown noise model coefficients 604. In various instances, the learned values or learned magnitudes of the set of unknown noise model coefficients 604 can be considered the set of learned noise model coefficients 306.

[0111] In various embodiments, the learning component 118 can determine or otherwise identify the learned noise model coefficient set 306 by repeatedly executing the quantum circuit segment 400 (or 500) on the quantum computer 104 over the set of Pauli bases and over the set of repetition depths. Based on this repeated execution, the learning component 118 can measure or extract a set of basis fidelities corresponding to the set of Pauli bases, respectively. In various instances, the learning component 118 can generate the learned noise model coefficient set 306 by inverting the easy relation on the set of basis fidelities. With respect to Figures 7-10 Various non-limiting aspects are further described.

[0112] Figure 7 An exemplary, non-limiting block diagram 700 showing how to obtain observable expectation values with respect to non-unitary operations in a circuit is shown in accordance with one or more embodiments described herein. In other words, Figure 7 An exemplary, non-limiting depiction of how the learning component 118 repeatedly executes the quantum circuit segment 400 (or 500) is shown.

[0113] In various embodiments, the learning component 118 can initialize the set of qubits 106 to any suitable initial quantum state. As a non-limiting example, the learning component 118 can initialize the set of qubits 106 to a zero state (e.g., the learning component 118 can cause each qubit in the set of qubits 106 to enter the |0> state).

[0114] In various aspects, the learning component 118 can run or otherwise execute a Pauli basis 702 (denoted with “B”) on the quantum computer 104. In various instances, the Pauli basis 702 can be any suitable n-qubit Pauli operator that is not an n-qubit identity matrix. That is,

[0115] After running or executing the Pauli basis 702, the learning component 118 can repeat running or executing the quantum circuit segment 400 (or 500) k times for any suitable positive integer k as shown. In other words, the learning component 118 can run or execute the quantum circuit segment 400 (or 500) to a repetition depth of k. After such a repeated running or executing the quantum circuit segment 400 (or 500), the learning component 118 can run or execute the conjugate transpose of the Pauli basis 702 as indicated by the numeral 704. After running or executing the conjugate transpose of the Pauli basis 702, the learning component 118 can run or execute one of the pair of rotation Pauli operators 304 as indicated by the numeral 706. In various aspects, the learning component 118 can subsequently measure the resulting quantum state of the set of quantum bits 106 as indicated by the numeral 708. In various instances, such a measurement can collectively be considered to yield an observable expectation value 710. In various cases, the observable expectation value 710 can be a scalar that can be considered a function of the Pauli basis 702 and the repetition depth k.

[0116] Figure 8 An exemplary, non-limiting block diagram 800 is shown that illustrates how a plurality of sets of observable expectation values can be obtained according to one or more embodiments described herein. As described above, the observable expectation value 710 can be considered a function of the Pauli basis 702 and the repetition depth k. Thus, the learning component 118 can execute the protocol shown in Figure 7 over a plurality of Pauli bases and over a plurality of repetition depths, thereby yielding a plurality of observable expectation values.

[0117] In particular, there can be a set of Pauli bases 802. In various aspects, the set of Pauli bases 802 can have the same cardinality as the set of noise model Pauli generators 602. Thus, since the set of noise model Pauli generators 602 can include j generators, the set of Pauli bases 802 can include j bases: Pauli base 802(1) through Pauli base 802(j). In various aspects, each Pauli base in the set of Pauli bases 802 can be a unique or different member of the set of n-qubit Pauli operators minus the n-qubit identity matrix. As one non-limiting example, Pauli base 802(1) can be a member of while Pauli base 802(j) can be a different member of .

[0118] In some cases (e.g., when sparsity is not inferred, implemented, or otherwise enforced), both the set of Pauli bases 802 and the set of noise model Pauli generators 602 can be identical to However, this is just one non-limiting example. In other cases (e.g., when sparsity is inferred, implemented, or enforced), the set of Pauli basis 802 can be a strict subset of and the set of noise model Pauli generators 602 can be a different (albeit possibly overlapping) strict subset of In such cases, the set of Pauli basis 802 can be considered not equivalent to the set of noise model Pauli generators 602.

[0119] As a non-limiting example, again assume n = 2. Now, assume that sparsity is not enforced. In such cases, both the set of Pauli basis 802 and the set of noise model Pauli generators 602 can be equivalent to the 15-element set given by:

[0120]

[0121] However, assume that sparsity is instead enforced (e.g., due to the probabilistic Pauli-Z gate 302 canceling residual phase errors). In such cases, the set of noise model Pauli generators 602 can be equivalent to the 7-element set given by:

[0122]

[0123] and the set of Pauli basis 802 can instead be equivalent to the 7-element set given by:

[0124]

[0125] In such cases, the set of Pauli basis 802 and the set of noise model Pauli generators 602 can be considered to have the same cardinality as each other, but they can still be considered not equivalent to each other.

[0126] Further, in various aspects, there can be a set of repetition depths 804. In various instances, the set of repetition depths 804 can include d depths for any suitable positive integer d: depth 804(1) through depth 804(d). In various cases, each repetition depth in the set of repetition depths 804 can be considered a positive integer value that can be assigned as k. In various aspects, the set of repetition depths 804 can increase gradually. In other words, the value or size of each repetition depth in the set of repetition depths 804 can be greater than the value or size of the repetition depth that precedes it (e.g., depth 804(2) (not shown) can be greater than depth 804(1), depth 804(3) (not shown) can be greater than depth 804(2),..., depth 804(d) can be greater than depth 804(d-1) (not shown)).

[0127] In various aspects, the learning component 118 can perform the protocol illustrated in Figure 7 Figure 7 In various instances, this can result in a plurality of sets of observable expectation values 806.

[0128] As one non-limiting example, the learning component 118 can select Pauli basis 802(1) as B, and can select depth 804(1) as k. With this selection, the learning component 118 can perform the protocol illustrated in Figure 7 Figure 7 As another non-limiting example, the learning component 118 can select Pauli basis 802(1) as B, and can select depth 804(d) as k. With this selection, the learning component 118 can perform the protocol illustrated in

[0129] As yet another non-limiting example, the learning component 118 can select Pauli basis 802(j) as B, and can select depth 804(1) as k. With this selection, the learning component 118 can perform the protocol illustrated in Figure 7 Figure 7 As still another non-limiting example, the learning component 118 can select Pauli basis 802(j) as B, and can select depth 804(d) as k. With this selection, the learning component 118 can perform the protocol illustrated in

[0130] In various aspects, the sets of observable expectation values 806(1) through 806(j) can be considered collectively to form a plurality of sets of observable expectation values 806.

[0131] Figure 9 An exemplary, non-limiting block diagram 900 is shown illustrating how a basis fidelity can be obtained from a plurality of sets of observable expectation values, in accordance with one or more embodiments described herein. ​​​

[0132] In various embodiments, the learning component 118 can determine, identify, or otherwise extract a set of basis fidelities 902 based on the plurality of sets of observable expectation values 806. In various aspects, the learning component 118 can do so by fitting an exponential decay curve to the plurality of sets of observable expectation values 806.

[0133] As one non-limiting example, the learning component 118 can fit an exponential decay curve to the set of observable expectation values 806(1). In various instances, such an exponential decay curve can be of the form Af B k , where k can be from the set of repetition depths 804, where B can be the Pauli basis 802(1), where A can be considered to be state preparation and measurement errors, and where f B may be a fidelity (e.g., a real-valued scalar ranging in value from 0 to 1) corresponding to the Pauli basis 802(1). In other words, k can be considered to be an independent variable of such an exponential decay curve, and A and f B may be considered to be constants of such an exponential decay curve. By fitting (e.g., by least squares or any other suitable fitting technique) an exponential decay curve to the set of observable expectation values 806(1), f B may be estimated or approximated. Such an estimate or approximation can be considered or otherwise referred to as the basis fidelity 902(1). Note that the basis fidelity 902(1) can be considered to correspond to the Pauli basis 802(1).

[0134] As another non-limiting example, the learning component 118 can fit an exponential decay curve to the set of observable expectation values 806(j). As above, in various instances, such an exponential decay curve can be of the form Af B k , where k can be from the set of repetition depths 804, where B can be the Pauli basis 802(j), where A can be considered to be state preparation and measurement errors, and where f B may be a fidelity (e.g., a real-valued scalar ranging in value from 0 to 1) corresponding to the Pauli basis 802(j). In other words, k can be considered to be an independent variable of such an exponential decay curve, and A and f B may be considered to be constants of such an exponential decay curve. By fitting an exponential decay curve to the set of observable expectation values 806(j), f B may be estimated or approximated. Such an estimate or approximation can be considered or otherwise referred to as the basis fidelity 902(j). Note that the basis fidelity 902(j) can be considered to correspond to the Pauli basis 802(j).

[0135] In various aspects, the base fidelities 902(1) through 902(j) can be collectively considered a set of base fidelities 902.

[0136] Figure 10 An exemplary, non-limiting block diagram 1000 showing how to obtain learned noise coefficients from base fidelities is shown in accordance with one or more embodiments described herein.

[0137] In various embodiments, the learning component 118 can compute, determine, estimate, or otherwise learn a set of learned noise model coefficients 306 based on the set of base fidelities 902. Specifically, the learning component 118 can invert the set of base fidelities 902 via the anti-commutation relations defined between the set of Pauli bases 802 and the set of noise model Pauli generators 602. More specifically, the learning component 118 can utilize the following equation:

[0138]

[0139] where may be a column vector representing the set of base fidelities 902 (which can be indexed according to the set of Pauli bases 802), where may be a column vector representing the set of unknown noise model coefficients 604 (which can be indexed according to the set of noise model Pauli generators 602), and where M can be a square matrix summarizing the anti-commutation relations defined between the set of Pauli bases 802 and the set of noise model Pauli generators 602. In various aspects, the rows of M can be indexed according to the set of Pauli bases 802, and the columns of M can be indexed according to the set of noise model Pauli generators 602. In some cases, M can be considered an application of the symplectic inner product between the set of Pauli bases 802 and the set of noise model Pauli generators 602, as shown by the following equation:

[0140] M r，c = <B r , P c >

[0141] where M r,c may be an element of M located in the rth row and cth column of M for any suitable positive integer r < j and c < j, where B r may be the rth Pauli base in the set of Pauli bases 802, where P c may be the cth noise model Pauli generator in the set of noise model Pauli generators 602, and where <B r , P c > can be the Pauli operator B r and P cIn each case, if {B r , P c}=0, then r , P c >=1, where {B r , P c} is B r and P c In all respects, if [B r , P c ]=0, then r , P c >=0, where [B r , P c ] is B r and P c The exchange child.

[0142] In various aspects, the learning component 118 can solve for M by inverting M and applying such inverse to the left side of each side of the above formula via matrix multiplication. . In particular, this inversion of M can enable the learning component 118 to obtain, determine, or otherwise identify a total of j coefficients: learned noise model coefficients 306(1) through learned noise model coefficients 306(j). In various cases, the learned noise model coefficients 306(1) through learned noise model coefficients 306(j) can be collectively considered to be the learned noise model coefficient set 306. In various aspects, the learned noise model coefficients 306(1) can be considered to be the obtained, determined, or identified values ​​or magnitudes of the unknown noise model coefficients 604(1). Accordingly, the learned noise model coefficients 306(1) can be considered to correspond to the noise model Pauli generator 602(1). Similarly, the learned noise model coefficients 306(j) can be considered to be the obtained, determined, or identified values ​​or magnitudes of the unknown noise model coefficients 604(j). Therefore, the learned noise model coefficients 306(j) can be considered to correspond to the noise model Pauli generator 602(j).

[0143] Thus far, various embodiments have been described in which the Pauli basis set 802 has the same cardinality (e.g., j) as the noise model Pauli generator set 602. However, this is merely a non-limiting example for ease of illustration and explanation. In various other embodiments, the Pauli basis set 802 may have a different cardinality (e.g., may have a different number of elements) than the noise model Pauli generator set 602. In fact, whenever M has full column rank (e.g., whenever M has at least as many rows as it has columns), it is possible to ​​solving the above equation. Thus, in various embodiments, the set of Pauli generators 802 can have a larger cardinality than the set of noise model Pauli generators 602 (e.g., the set of Pauli bases 802 can have more than j elements), and still obtain a unique In other words, various embodiments can involve a set of noise model Pauli generators 602 having a cardinality j for any suitable positive integer l > j and a set of Pauli bases 802 having a cardinality l.

[0144] In any case, the learning component 118 can determine the set of learned noise model coefficients 306, and thus can be said to have learned the reshaped noise 606 (e.g., to have learned a reshaped or rotated version of the noise 210).

[0145] Figure 11 Exemplary, non-limiting experimental results in accordance with one or more embodiments described herein are shown. In particular, various experiments were conducted by the inventors in which various embodiments described herein were put into practice. During such experiments, the following parameters were implemented: n = 2; the non-identity operation 110 in the circuit was a qubit measurement in the circuit on the first qubit; the classically controlled quantum gate 208 was performed on a second qubit that was not directly coupled to the first qubit; the set of noise model Pauli generators 602 was a 7-element set given by the following:

[0146]

[0147] and the set of Pauli bases 802 was a 7-element set given by the following:

[0148]

[0149] As shown, Figure 11 includes graph 1102 and graph 1104. Graph 1102 shows exponential decay curves fitted to the observable expectation values over the repetition depth and the above seven Pauli bases. In particular, the horizontal axis of graph 1102 can be considered to vary over the repetition depth, and the different dashed lines in graph 1102 can be considered to represent different Pauli bases. Based on such fitting exponential decay curves, a different basis fidelity was obtained for each of the seven Pauli bases used: a first basis fidelity for , a second basis fidelity for , a third basis fidelity for , a fourth basis fidelity for , a fifth basis fidelity for , a sixth basis fidelity for , and a seventh basis fidelity for Furthermore, based on these seven basis fidelities, different learned noise model coefficients are obtained for each of the seven noise model Pauli generators. Such different learned noise model coefficients are shown in graph 1104.

[0150] Note that, as shown in graph 1104, The noise model Pauli generator acquires non-zero learned noise model coefficients. This can be considered as correlated noise between the first qubit and the second qubit. However, it is important to note that during this experiment, the first qubit and the second qubit were not directly coupled to each other. In the absence of non-unitary operations in the circuit, such correlated noise between uncoupled qubits would not occur. After all, the noise from unitary operations would only affect the qubits to which such unitary operations were applied. However, as described above, noise from non-unitary operations can propagate throughout the quantum computer, thereby affecting qubits that are not directly coupled. Unlike the various embodiments described herein, existing error mitigation or suppression techniques are unable to detect such correlated noise between uncoupled qubits.

[0151] Note that in some cases, seven Pauli basis can be measured directly. However, in other cases, fewer than all seven such Pauli basis can be measured directly, and the remaining Pauli basis can be obtained via post-processing. As a non-limiting example, one can directly measure basis, or can be measured directly Base and can be used Post-processing to effectively measure As another non-limiting example, one can directly measure basis, or can be measured directly Base and can be used Post-processing to effectively measure This post-processing can be performed to help reduce the number of Pauli bases in which direct measurements are performed.

[0152] Figure 12 A block diagram of an exemplary, non-limiting system 1200 including an inverse noise operation that can facilitate noise learning in dynamic quantum circuits according to one or more embodiments described herein is shown. As shown, system 1200 can, in some cases, include the same components as system 300 and can also include an inverse noise operation 1202.

[0153] In various embodiments, as noted above, upon obtaining the set of learned noise model coefficients 306, it can be considered that the learning component 118 has learned, determined, or otherwise identified a reshaped or rotated version of the noise 210 (e.g., has learned the reshaped noise 606). In various aspects, the mitigation component 120 can electronically mitigate the reshaped or rotated version of the noise 210 by inserting an inverse of such noise into the dynamic quantum circuit 108. More specifically, upon obtaining the set of learned noise model coefficients 306, it can be considered that the reshaped noise 606 (e.g., the noise 210 surrounded by a pair of rotated Pauli operators 304 from both sides) has been learned. In various instances, the mitigation component 120 can electronically invert the reshaped noise 606 (e.g., via any suitable matrix inversion technique), thereby producing an inverse noise operation 1202. In other words, the inverse noise operation 1202 can be considered to be any n-qubit quantum operation that is the inverse of the reshaped noise 606 (e.g., the noise 210 surrounded by a pair of rotated Pauli operators 304 from both sides). In various cases, the mitigation component 120 can mitigate, cancel, or otherwise reduce the reshaped noise 606 (e.g., the noise 210 surrounded by a pair of rotated Pauli operators 304 from both sides) by inserting the inverse noise operation 1202 into the dynamic quantum circuit 108. With respect to Figures 13-14 Various non-limiting aspects are illustrated.

[0154] Figures 13-14 Exemplary, non-limiting block diagrams 1300 and 1400 of the dynamic quantum circuit 108 after inserting the inverse noise operation 1202 are illustrated in accordance with one or more embodiments described herein.

[0155] First, consider Figure 13 As illustrated, at least one layer 204 and at least one layer 206 of the dynamic quantum circuit 108 can remain unchanged. However, as also illustrated, the mitigation component 120 can change the in-circuit non-identity layer 202 to an in-circuit non-identity layer 1302 in the circuit.

[0156] In various aspects, the in-circuit non-identity layer 1302 can include the in-circuit non-identity operation 110, the classically controlled quantum gate 208, and the noise 210, just like the in-circuit non-identity layer 202. In various instances, the mitigation component 120 can insert the probabilistic Pauli-Z gate 302 and the pair of rotated Pauli operators 304 into the in-circuit non-identity layer 1302. However, this is merely one non-limiting example. In some cases, the mitigation component 120 can omit the probabilistic Pauli-Z gate 302. Moreover, in various aspects, the mitigation component 120 can insert the inverse noise operation 1202 (surrounded by the pair of rotated Pauli operators 304 from both sides) into the in-circuit non-identity layer 1302. -1inserted into the non-Clifford layer 1302 in the circuit. Again, the inverse noise operation 1202 can be considered an n-qubit quantum operation that performs an inverse of a reshaped or rotated version of the noise 210 (e.g., performs an inverse of the reshaped noise 606). Accordingly, the mitigation component 120 can place or position the inverse noise operation 1202 before (e.g., upstream of) the reshaped or rotated version of the noise 210 (e.g., place or position the inverse noise operation 1202 before the first or most upstream one of the pair of rotated Pauli operators 304). Thus, when the dynamic quantum circuit 108 is executed, the inverse noise operation 1202 can cancel out or otherwise reduce the effects of the noise 210.

[0157] As Figure 13 shown, in some embodiments, the classically controlled quantum gate 208 can be located or placed between the pair of rotated Pauli operators 304. Note that this can be the case regardless of whether the learning component 118 obtains the learned noise model coefficient set 306 by utilizing the quantum circuit segment 400 or the quantum circuit segment 500. However, this is merely one non-limiting example. In some cases, the classically controlled quantum gate 208 can not be located or positioned between the pair of rotated Pauli operators 304. In particular, the classically controlled quantum gate 208 can be located within the non-Clifford layer 1302 in the circuit, but can be located after the second or most downstream one of the pair of rotated Pauli operators 304. In other cases, as shown with respect to Figure 14 the classically controlled quantum gate 208 can be omitted entirely. Indeed, as Figure 14 shown, in various embodiments, the mitigation component 120 can cause the dynamic quantum circuit 108 to include a non-Clifford layer 1402 in the circuit instead of the non-Clifford layer 1302. In various aspects, the non-Clifford layer 1402 can be identical to the non-Clifford layer 1302 except that the classically controlled quantum gate 208 can be omitted or otherwise removed. After all, in various embodiments, it can still be useful to implement the in-circuit qubit measurement even in the absence of feedforward (e.g., even if such in-circuit qubit measurement is not used to classically control the quantum gate).

[0158] In any case, the mitigation component 120 can mitigate the effects of the noise 210 by inserting the inverse noise operation 1202 into the dynamic quantum circuit 108.

[0159] Figures 15-16 Exemplary, non-limiting experimental results in accordance with one or more embodiments described herein are shown.

[0160] First, consider Figure 15As noted above, the inventors conducted various experiments in which they put various embodiments described herein into practice. During these experiments, the following parameters were implemented: n = 2; the non-unitary-in-circuit operation 110 was a qubit measurement in the circuit on the first qubit; the classically controlled quantum gate 208 was performed on a second qubit that was not directly coupled to the first qubit; the noise model Pauli generator set 602 was a 7-element set given by the following:

[0161]

[0162] and the Pauli basis set 802 was a 7-element set given by the following:

[0163]

[0164] As noted above, the obtained learned noise model coefficients are shown in graph 1104. These learned noise model coefficients were used to compute an inverse noise operation (e.g., 1202). To verify this inverse noise operation, a new quantum circuit segment was created, where this new quantum circuit segment was identical to Figures 4-5 and where this new quantum circuit segment also had the inverse noise operation placed or positioned before the non-unitary-in-circuit operation 110 (e.g., before the first or most upstream one of the pair of rotation Pauli operators 304). In accordance with Figures 7-10 , a plurality of sets of observable expectation values were obtained using the new quantum circuit segment, an exponential decay curve was fit accordingly, as shown in graph 1502, and subsequently the basis fidelities were extracted for the seven Pauli bases from the fitted exponential decay curve. As shown in graph 1504, the basis fidelities obtained after implementing the inverse noise operation (labeled “Mitigated”) were significantly closer to 1 on all seven Pauli bases than the basis fidelities obtained before implementing the inverse noise operation (labeled “Learned”). These experimental results demonstrate that various embodiments described herein successfully mitigated the noise associated with the non-unitary-in-circuit operation (e.g., a qubit measurement in the circuit with a feedforward to a classically controlled quantum gate).

[0165] Now consider Figure 16 To further verify various embodiments described herein, the inventors conducted various embodiments (according to the two-qubit system mentioned in the above experiments) in which the noise model coefficients were obtained for a qubit measurement in the circuit, and in which the amplitude of this qubit measurement in the circuit was varied between weak and strong. Figure 16 includes graph 1602, which shows the learned noise model coefficients for each of the seven noise model Pauli generators used. As shown, the learned noise model coefficients tend to increase with stronger measurement amplitudes (especially for and generator). Moreover, Figure 16 includes a graph 1604 illustrating how the total noise scalar (labeled “y”) varies with the measured amplitude in the circuit, where y = exp(2∑ g∈G λ g ), where G can be a set of indices corresponding to the noise model Pauli generators, and where λ g may be a learned noise model coefficient corresponding to the gth noise model Pauli generator. As shown, the total noise scalar increases significantly as the measured amplitude in the circuit increases. This experimental result can be considered as further validation of the various embodiments described herein.

[0166] Figure 17 A flow diagram illustrating an example, non-limiting computer-implemented method 1700 that facilitates noise learning in dynamic quantum circuits in accordance with one or more embodiments described herein is shown. In various instances, the computer-implemented method 1700 can be facilitated by the dynamic circuit noise learning and mitigation system 102.

[0167] In various embodiments, the act 1702 can include accessing, by a device operatively coupled with a processor (e.g., 112) (e.g., via 116), a dynamic quantum circuit (e.g., 104).

[0168] In various aspects, the act 1704 can include learning, by the device (e.g., via 118), a noise (e.g., 210) associated with a circuit-in-non-identity operation (e.g., 110) of the dynamic quantum circuit by modifying the circuit-in-non-identity operation with a probabilistic Pauli-Z gate (e.g., 302) and a rotation Pauli operator (e.g., 304).

[0169] Although not explicitly shown in Figure 17 , the circuit-in-non-identity operation can include a circuit-in-qubit measurement that feeds forward to at least one classically controlled quantum gate (e.g., 208). In some instances, the at least one classically controlled quantum gate can be between rotation Pauli operators (e.g., as shown in Figure 5 ). In other instances, the at least one classically controlled quantum gate can not be between rotation Pauli operators (e.g., as shown in Figure 4 ).

[0170] Although not explicitly shown in Figure 17While not explicitly shown in Figures 7-9

[0171] While not explicitly shown in Figure 17 , learning the noise can include: performing, by the device (e.g., via 118) and on a set of Pauli bases (e.g., 802) and a set of repetition depths (e.g., 804), a non-unitary operation in a circuit modified with a probabilistic Pauli-Z gate and a rotation Pauli operator; and extracting, by the device (e.g., via 118) and based on such repeated performance, a set of basis fidelities (e.g., 902) corresponding to the set of Pauli bases, respectively (e.g., as described with respect to

[0172] While not explicitly shown in Figure 17 , the computer-implemented method 1700 can include mitigating, by the device (e.g., via 120), the noise by inserting an inverse of the noise (e.g., 1202) into the dynamic quantum circuit.

[0173] Note that when the in-circuit non-unitary operation 110 is a quantum bit measurement in the circuit, the various embodiments described herein can be considered or otherwise referred to as measurement-based probabilistic error correction (e.g., mPEC).

[0174] Further, note that when the in-circuit non-unitary operation 110 is a quantum bit measurement in the circuit, if the pair of rotation Pauli operators 304 apply a Pauli-X gate or a Pauli-Y gate to any quantum bit to which the in-circuit non-unitary operation is applied (e.g., quantum bit 106(n) in the non-limiting example shown in the figure), the dynamic circuit noise learning and mitigation system 102 can flip, in various aspects, any classical bit measured by the in-circuit non-unitary operation 110. However, if the pair of rotation Pauli operators 304 apply a Pauli-Z gate or an identity gate to any quantum bit to which the in-circuit non-unitary operation is applied (e.g., quantum bit 106(n) in the non-limiting example shown in the figure), such flipping of the measured classical bit can be omitted.

[0175] The various embodiments described herein can be considered a computerized tool for learning or mitigating noise caused by an in-circuit non-unitary operation. Such embodiments can be applied regardless of quantum bit connectivity and can be extended to quantum bit lattices of any suitable size. Such embodiments undoubtedly constitute a specific and substantial improvement in the field of dynamic quantum circuits.

[0176] Figure 18 ​The following discussion and the drawings intend to provide a brief, general description of a suitable computing environment 1800 in which one or more embodiments described herein can be implemented. For example, various aspects of the disclosure are described in terms of a sequence of actions, flowcharts, block diagrams, or block diagrams of machine logic, including computational representations such as flowchart diagrams, a flowchart, a flow diagram, a structure diagram, a program structure diagram, or a block diagram of a computer program product (CPP) embodiment. With respect to any flowchart, flow diagram, flowchart diagram, structure diagram, program structure diagram, or block diagram, the order in which the operations are described or depicted in the flowchart, flow diagram, flowchart diagram, structure diagram, program structure diagram, or block diagram is not intended to be limiting. For example, the operations can be performed in a different order than that depicted in any flowchart, flow diagram, flowchart diagram, structure diagram, program structure diagram, or block diagram. Further, the depicted operations can be performed concurrently, in an order different than that depicted in any flowchart, flow diagram, flowchart diagram, structure diagram, program structure diagram, or block diagram. For example, two operations depicted in consecutive flowchart, flow diagram, flowchart diagram, structure diagram, program structure diagram, or block diagram frames can in fact be executed concurrently or be executed in the reverse order indicated in the flowchart, flow diagram, flowchart diagram, structure diagram, program structure diagram, or block diagram. Similarly, the flowcharts, flow diagrams, flowchart diagrams, structure diagrams, program structure diagrams, or block diagrams can describe or depict one or more steps having an order or sequence of steps, or combinations of steps. The steps can be executed in different orders than those described or depicted in any flowcharts, flow diagrams, flowchart diagrams, structure diagrams, program structure diagrams, or block diagrams. Further, the steps can be executed concurrently or be executed in reverse order depending on the implementation. Furthermore, steps can be omitted or combined, and additional steps can be added, in different embodiments.

[0177] A computer program product embodiment ("CPP embodiment" or "CPP") is a term of art used in this disclosure to describe any collection of one or more storage media (also referred to as "media") collectively including machine readable code corresponding to instructions or data used in performing the computer operations specified in a given CPP claim, collectively included in a set of one or more storage devices. A "storage device" is any tangible device that can retain and store instructions for use by a computer processor. Without limitation, computer readable storage media can be electronic storage media, magnetic storage media, optical storage media, electromagnetic storage media, semiconductor storage media, mechanical storage media, or any suitable combination of the foregoing. Some known types of storage devices comprising these media include a diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), a static random access memory (SRAM), a compact disk read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanical encoding device such as punch cards or holes formed on the main surface of a disk, or any suitable combination of the foregoing. The term "computer readable storage media" as used herein is not to be construed as being a transitory signal per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission media, pulses of light propagating through a fiber-optic cable, electrical signals propagating through an electrical wire, or other transitory medium. As those skilled in the art will appreciate, data is typically moved at some occasional point in time during normal operation of a storage device, such as during access, defragmentation, or garbage collection, but this does not make the storage device transitory as the data is not transitory when it is stored.

[0178] Computing environment 1800 includes an example of at least some environment for executing computer code related to performing the inventive method, such as dynamic quantum circuit noise learning code 1880. In addition to block 1880, computing environment 1800 also includes, for example, computer 1801, wide area network (WAN) 1802, end user device (EUD) 1803, remote server 1804, public cloud 1805, and private cloud 1806. In this embodiment, computer 1801 includes processor set 1810 (including processing circuitry 1820 and cache 1821), communication fabric 1811, volatile memory 1812, persistent storage 1813 (including operating system 1822 and block 1880, as described above), peripheral set 1814 (including user interface (UI), device set 1823, storage 1824, and Internet of Things (IoT) sensor set 1825), and network module 1815. Remote server 1804 includes remote database 1830. Public cloud 1805 includes gateway 1840, cloud orchestration module 1841, host physical machine set 1842, virtual machine set 1843, and container set 1844.

[0179] Computer 1801 can take the form of a desktop computer, laptop computer, tablet computer, smart phone, smart watch or other wearable computer, mainframe computer, quantum computer, or any form of computer or mobile device now known or hereafter developed that is capable of running a program, accessing a network, or querying a database, such as remote database 1830. As is well understood in the computer arts, and depending on the technology, execution of a computer-implemented method can be distributed among multiple computers or among multiple locations. On the other hand, in this introduction to computing environment 1800, the detailed discussion is focused on a single computer, specifically computer 1801, to make the introduction as simple as possible. Computer 1801 can be located in the cloud, even though it is not shown in the cloud in this figure. On the other hand, unless it is affirmatively indicated, computer 1801 need not be located in the cloud. Figure 18

[0180] ​The processor set 1810 includes one or more of any type of computer processors now known or developed in the future. The processing circuitry 1820 can be distributed across multiple packages, for example, multiple coordinated integrated circuit chips. The processing circuitry 1820 can implement multiple processor threads or multiple processor cores. The cache 1821 is memory located in the processor chip package and is typically used for data or code that should be quickly accessible to threads or cores that should be running on the processor set 1810. Cache memory is typically organized into multiple levels according to relative proximity to the processing circuitry. Alternatively, some or all of the cache of the processor set can be located "off-chip." In some computing environments, the processor set 1810 can be designed to work with qubits and perform quantum computations.

[0181] Computer readable program instructions generally be loaded onto the computer 1801 to cause the processor set 1810 of the computer 1801 to perform a series of operational steps to implement the computer-implemented method so that the instructions thus executed will instantiate the method specified in the flow diagram or narrative description of the computer-implemented method included in this document (collectively, the "inventive method"). These computer readable program instructions are stored in various types of computer readable storage media such as the cache 1821 and other storage media discussed below. The processor set 1810 accesses program instructions and associated data, to control and direct the execution of the inventive method. In the computing environment 1800, at least some of the instructions for performing the inventive method can be stored in block 1880 in the persistent storage 1813.

[0182] The communication fabric 1811 is the signal-conducting pathway that allows the various components of the computer 1801 to communicate with each other. Typically, this fabric is made of switches and electrical conductors, such as those that make up a bus, a bridge, a physical input / output port, and the like. Other types of signal communication pathways can be used, such as fiber-optic communication pathways or wireless communication pathways.

[0183] The volatile memory 1812 is any type of volatile memory now known or developed in the future. Examples include dynamic random access memory (RAM) or static RAM. Typically, volatile memory is characterized by random access, but this is not required unless explicitly stated. In the computer 1801, the volatile memory 1812 is located in a single package and is internal to the computer 1801, but alternatively or additionally, the volatile memory can be distributed across multiple packages or located externally with respect to the computer 1801.

[0184] Persistent storage 1813 is any form of non-volatile storage for a computer now known or developed in the future. The non-volatile nature of such storage means that the stored data is retained regardless of whether power is supplied to the computer 1801 or directly to the persistent storage 1813. The persistent storage 1813 can be read-only memory (ROM), but typically at least a portion of the persistent storage allows data to be written, deleted, and overwritten. Some familiar forms of persistent storage include magnetic disks and solid-state storage devices. The operating system 1822 can take a variety of forms, such as various known proprietary operating systems or an open-source portable operating system interface type of operating system employing a kernel. The code included in the block 1880 typically includes at least some of the computer code related to performing the innovative method.

[0185] The set of peripheral devices 1814 includes a collection of peripheral devices of the computer 1801. The data communication connections between the peripheral devices and other components of the computer 1801 can be implemented in various ways, such as a Bluetooth connection, a near-field communication (NFC) connection, a connection made by a cable such as a universal serial bus (USB) type cable, a plug-in connection (e.g., a secure digital (SD) card), a connection made through a local area communication network, and even a connection made through a wide area network such as the Internet. In various embodiments, the set of UI devices 1823 can include components such as a display, a speaker, a microphone, a wearable device such as eyewear and a smartwatch, a keyboard, a mouse, a printer, a touchpad, a game controller, and a haptic device. The storage 1824 is external storage, such as an external hard drive, or a pluggable storage such as an SD card. The storage 1824 can be persistent or volatile. In some embodiments, the storage 1824 can take the form of a quantum computing storage device for storing data in the form of qubits. In embodiments where the computer 1801 is required to have a large amount of storage (e.g., where the computer 1801 stores and manages a large database locally), then that storage can be provided by a peripheral storage device designed to store large amounts of data, such as a storage area network (SAN) shared by multiple geographically distributed computers. The set of IoT sensors 1825 consists of sensors that can be used in Internet of Things applications. For example, one sensor can be a thermometer, another sensor can be a motion detector.

[0186] The network module 1815 is a collection of computer software, hardware, and firmware that allows the computer 1801 to communicate with other computers via the WAN 1802. The network module 1815 can include hardware such as a modem or Wi-Fi signal transceiver, software for packetizing or depacketizing data for transmission over a communications network, or web browser software for communicating data via the Internet. In some embodiments, the network control functions and network forwarding functions of the network module 1815 are performed on the same physical hardware device. In other embodiments, such as embodiments utilizing software defined networking (SDN), the control functions and forwarding functions of the network module 1815 are performed on physically separate devices, such that the control functions manage multiple different network hardware devices. Computer readable program instructions for performing the methods of the present application can generally be downloaded to the computer 1801 from an external computer or external storage device through the network adapter card or network interface included in the network module 1815.

[0187] The WAN 1802 is any wide area network (e.g., the Internet) that can transmit computer data over non-local distances through any technology now known or future developed for communicating computer data. In some embodiments, the WAN can be replaced or supplemented with a local area network (LAN) (e.g., a Wi-Fi network) designed to communicate data between devices located in a local area. WANs or LANs typically include computer hardware such as copper transmission cables, optical transmission fibers, wireless transmissions, routers, firewalls, switches, gateway computers, and edge servers.

[0188] The end user device (EUD) 1803 is any computer system that can take any form discussed above with respect to the computer 1801 that is used and controlled by an end user (e.g., a business customer operating the computer 1801). The EUD 1803 typically receives helpful and useful data from the operation of the computer 1801. For example, in the hypothetical case where the computer 1801 is designed to provide recommendations to an end user, then the recommendation is typically sent from the network module 1815 of the computer 1801 to the EUD 1803 via the WAN 1802. In this way, the EUD 1803 can display or otherwise present the recommendation to the end user. In some embodiments, the EUD 1803 can be a client device, such as a thin client, a thick client, a mainframe computer, or a desktop computer.

[0189] Remote server 1804 is any computer system that serves at least some data or functionality of computer 1801. Remote server 1804 can be controlled and used by the same entity that operates computer 1801. Remote server 1804 represents a machine that collects and stores helpful and useful data for use by other computers, such as computer 1801. For example, if computer 1801 is designed and programmed to provide recommended hypothetical situations based on historical data, then that historical data can be provided to computer 1801 from a remote database 1830 of remote server 1804.

[0190] Public cloud 1805 is any computer system that can be used by multiple entities that provides on-demand availability of computer system resources or other computing power, especially data storage (cloud storage) and computing power, without direct active management by the scale of the resources. Direct and active management of the computing resources of public cloud 1805 is performed by computer hardware or software of cloud orchestration module 1841. The computing resources provided by public cloud 1805 are typically implemented by virtual computing environments (VCEs) from a set of virtual machines 1843 or containers from a set of containers 1844 running on individual computers that make up a set of host physical machines 1842, which is the universe of physical computers in or available to public cloud 1805. The VCEs are typically in the form of virtual machines or containers. It should be understood that these VCEs can be stored as images and can be transferred as images between various physical machine hosts or after instantiation of the VCE. Cloud orchestration module 1841 manages the transfer and storage of the images, deploys new VCE instantiations and manages active VCE deployment instantiations. Gateway 1840 is a collection of computer software, hardware, and firmware that allows public cloud 1805 to communicate over WAN 1802.

[0191] Some further explanation of virtualized computing environments (VCEs) will now be provided. VCEs can be stored as “images.” New active instances of VCEs can be instantiated from the images. Two familiar types of VCEs are virtual machines and containers. Containers are VCEs that use operating system-level virtualization. This refers to an operating system feature that allows the existence of multiple isolated user space instances, called containers, that have their own process and network memory spaces. From the perspective of the programs running in them, these isolated user space instances often behave like real computers. Computer programs running on an ordinary operating system can utilize all of the resources of that computer, such as 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 devices allocated to the container, a feature that is known as containerization.

[0192] The private cloud 1806 is similar to the public cloud 1805 except that the computing resources are available only to a single enterprise. Although the private cloud 1806 is depicted as being in communication with the WAN 1802, in other embodiments the private cloud can be completely disconnected from the Internet and accessible only through a local / private network. A hybrid cloud is a combination of different types (e.g., private, community, or public cloud types) of clouds, typically implemented by different vendors respectively. Each of the multiple clouds remains as a separate and distinct entity, but the larger hybrid cloud architecture is bound together by standardized or proprietary technologies that enable orchestration, management, or data / application portability between the multiple constituent clouds. In this embodiment, the public cloud 1805 and the private cloud 1806 are both part of a larger hybrid cloud.

[0193] Non-limiting examples of various embodiments of the present application are described herein. For ease of description or explanation, various portions of the present disclosure use the term “each” when discussing various embodiments of the present application. The use of this term “each” is a non-limiting example. In other words, when the present disclosure provides a description of “each” that applies to a certain particular object or component, it should be understood that this is a non-limiting example of various embodiments of the present application, and it should be further understood that in other various embodiments of the present application it can be the case that less than “each” of that particular object or component is subject to such a description.

[0194] The embodiments described herein can be directed to one or more of the following: systems, methods, apparatus, or computer program products at any possible technical detail level of integration. A computer program product can include a computer readable storage medium (or media) having computer readable program instructions thereon for causing a processor to carry out aspects of one or more embodiments described herein. The computer readable storage medium can be a tangible device that can retain and store instructions for use by an instruction execution device. The computer readable storage medium can be, for example, 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 suitable combination of the foregoing. A non-exhaustive list of more specific computer readable storage mediums can also include the following: a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), a static random access memory (SRAM), a portable compact disc read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanically encoded device such as punch-cards or raised structures in a groove having instructions recorded thereon, and any suitable combination of the foregoing. A computer readable storage medium, as used herein, is not to be construed as being transitory signals per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission media (e.g., light pulses passing through a fiber-optic cable), or electrical signals transmitted through a wire.

[0195] The computer readable program instructions described herein can be downloaded to respective computing / processing devices from a computer readable storage medium or to external computers or external storage devices via a network, for example, the Internet, a local area network, a wide area network or a wireless network. The network can comprise copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers or edge servers. A network adapter card or network interface in each computing / processing device receives computer readable program instructions from the network and forwards the computer readable program instructions for storage in a computer readable storage medium within the respective computing / processing device. Computer readable program instructions for carrying out operations of one or more embodiments described herein can be in assembly

[0196] The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational acts to be performed on the computer, other programmable apparatus or other device to produce a computer implemented process, such that the instructions which execute on the computer, other programmable apparatus, or other device implement the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0197] The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational acts to be performed on the computer, other programmable apparatus or other device to produce a computer implemented process, such that the instructions which execute on the computer, other programmable apparatus, or other device implement the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0198] Although subject matter has been described in the general context of computer-executable instructions of a computer program product running on a computer or multiple computers, it will be appreciated by those skilled in the art that one or more embodiments described herein may also be implemented at least in part in parallel with one or more other program modules. Typically, a program module includes a routine, program, component, or data structure that performs a specific task or implements a specific abstract data type. In addition, the above-mentioned computer-implemented method can be practiced with other computer system configurations, including single-processor or multi-processor computer systems, small computing devices, mainframe computers, and computers, handheld computing devices (e.g., PDAs, phones), or microprocessor-based or programmable consumer or industrial electronic devices. The illustrated aspects may also be practiced in a distributed computing environment, in which tasks are performed by remote processing devices linked by a communication network. However, if not all, one or more aspects of one or more embodiments described herein may be practiced on a stand-alone computer. In a distributed computing environment, program modules may be located in local and remote memory storage devices.

[0199] As used herein, the terms "component," "system," "platform," or "interface" may refer to or include computer-related entities or entities associated with an operating machine having one or more specific functions. The entities described herein may be hardware, a combination of hardware and software, software, or software being executed. For example, a component may be, but is not limited to, a process, a processor, an object, an executable file, an execution thread, a program, or a computer running on a processor. By way of illustration, both an application running on a server and a server may be components. One or more components may reside within a process or execution thread, and a component may be located on a single computer or distributed between two or more computers. In another example, each component may be executed from various computer-readable media having various data structures stored thereon. Components may communicate via local or remote processes, for example, based on signals having one or more data packets (e.g., data from one component interacts via signals with other components in a local system, a distributed system, or across a network with other systems such as the Internet). As another example, a component may be a device that provides specific functionality by means of mechanical parts operated by electrical or electronic circuits operated by software or firmware applications executed by a processor. In this case, the processor may be a processor internal or external to the device and may execute at least a portion of the software or firmware application. As yet another example, a component can be a device that provides specific functionality through electronic components without mechanical parts, where the electronic components can include a processor or other means to execute software or firmware that at least partially imparts the functionality of the electronic components. In one aspect, the component can emulate the electronic components through a virtual machine (e.g., within a cloud computing system).

[0200] Additionally, the term "or" is intended to mean an inclusive "or" rather than an exclusive "or." That is, unless specified otherwise, or clear from context, "X employs A or B" is intended to mean any of the natural inclusive permutations. That is, if X employs A; X employs B; or X employs both A and B, then "X employs A or B" is satisfied under any of the foregoing instances. As used herein, the term "and / or" is intended to mean "and" or "or," with the same meaning as "or." Furthermore, the articles "a" and "an" as used in the subject specification and annexed drawings should generally be construed to mean "one or more" unless specified otherwise or clear from context to be directed to a singular form. As used herein, the term "example" or "exemplary" is utilized to mean serving as an example, instance, or illustration. For the avoidance of doubt, the subject matter described herein is not limited by such examples. In addition, any aspect or design described herein as "example" or "exemplary" is not necessarily to be construed as preferred or advantageous over other aspects or designs, nor is it meant to preclude equivalent example structures and techniques known to those of ordinary skill in the art.

[0201] As used in the subject specification, the term "processor" can refer to virtually any computing processing unit or device, including but not limited to a single-core processor; a single processor with software multithread execution capability; a multi-core processor; a multi-core processor with software multithread execution capability; a multi-core processor with hardware multithread technology; a parallel platform; or a parallel platform with distributed shared memory. Additionally, a processor can refer to an integrated circuit, an application-specific integrated circuit (ASIC), a digital signal processor (DSP), a field- programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. Further, a processor can utilize nanoscale architectures such as, but not limited to, molecular and quantum-dot based transistors, switches or gates, in order to optimize space usage or enhance performance of related devices. A processor can be implemented as a combination of computing processing units.

[0202] Herein, terms such as "warehouse," "memory," "data warehouse," "data storage device," "database," and virtually any other information storage component associated with the operation and functionality of components are utilized to refer to an entity in "memory components," "memory," or components comprising memory. The memory or memory components described herein can be volatile memory or nonvolatile memory, or can include both volatile and nonvolatile memory. By way of illustration, and not limitation, nonvolatile memory can include read only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable ROM (EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). Volatile memory can include RAM, which can act as external cache memory, for example. By way of illustration, and not limitation, RAM can be provided in a variety of forms, such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), or Rambus dynamic RAM (RDRAM). Additionally, the memory components of the systems or computer-implemented methods described herein are intended to include, without being limited to, these or any other suitable types of memory.

[0203] The above-described examples include only systems and computer-implemented methods. Of course, not all possible combinations of components or computer-implemented methods can be described for purposes of describing one or more embodiments, but one of ordinary skill in the art can recognize that many further combinations and permutations of the described components or computer-implemented methods are possible. Moreover, where the use of terms such as "including," "having," "with," "containing," or "featuring," etc., are used in the detailed description, claims, appendices or drawings, such terms are always meant to be interpreted as encompassing their equivalents.

[0204] The description of the various embodiments is presented for purposes of illustration and description, but is not intended to be exhaustive or to limit the embodiments to the precise forms described. Many modifications and variations will be apparent to practitioners skilled in the art. The terminology used herein was chosen for the best reasons described in the specification and for the most effective and clear description of the principles and practical applications and technical improvements of the embodiments described herein, or to make others skilled in the art understand the embodiments described herein.

Claims

1. A system comprising: a processor that executes computer-executable components stored in a non-transitory computer-readable storage medium, wherein the computer-executable components comprise: a learning component that learns a noise associated with a non-gate-preserving operation in a circuit of a dynamic quantum circuit by modifying the non-gate-preserving operation in the circuit with a probabilistic Pauli-Z gate and a rotating Pauli operator.

2. The system of the preceding claim, wherein the non-gate-preserving operation in the circuit comprises a circuit-in quantum bit measurement that feeds forward to at least one classically controlled quantum gate.

3. The system of the preceding claim, wherein the at least one classically controlled quantum gate is located between the rotating Pauli operators.

4. The system of claim 2, wherein the at least one classically controlled quantum gate is not located between the rotating Pauli operators.

5. The system of any preceding claim, wherein the learning component learns the noise by repeatedly executing the non-gate-preserving operation in the circuit modified with the probabilistic Pauli-Z gate and the rotating Pauli operator over a set of Pauli basis sets and over a set of repetition depths, and extracting a set of basis fidelities corresponding to the set of Pauli basis sets, respectively, based on such repeated executions.

6. The system of the preceding claim, wherein the learning component learns the noise associated with the non-gate-preserving operation in the circuit by inverting the set of basis fidelities via an algebraic relation defined between the set of Pauli basis sets and a set of Pauli generators associated with the noise.

7. The system of any preceding claim, wherein the computer-executable components further comprise: a mitigation component that mitigates the noise by inserting an inverse of the noise into the dynamic quantum circuit.

8. A computer-implemented method comprising: by a device operatively coupled with a processor, learning a noise associated with a non-gate-preserving operation in a circuit of a dynamic quantum circuit by modifying the non-gate-preserving operation in the circuit with a probabilistic Pauli-Z gate and a rotating Pauli operator.

9. The computer-implemented method of the preceding claim, wherein the non-gate-preserving operation in the circuit comprises a circuit-in quantum bit measurement that feeds forward to at least one classically controlled quantum gate.

10. The computer-implemented method of the preceding claim, wherein the at least one classically controlled quantum gate is located between the rotating Pauli operators.

11. The computer-implemented method of claim 9, wherein the at least one classically controlled quantum gate is not located between the rotating Pauli operators.

12. The computer-implemented method of any of the preceding four claims, wherein learning the noise comprises: by the device, repeatedly executing the non-gate-preserving operation in the circuit modified with the probabilistic Pauli-Z gate and the rotating Pauli operator over both a set of Pauli basis sets and a set of repetition depths; and by the device, extracting a set of basis fidelities corresponding to the set of Pauli basis sets, respectively, based on such repeated executions.

13. The computer-implemented method of the preceding claim, wherein learning the noise comprises: inverting, by the device, the set of basis fidelities via commutation relations defined between the set of Pauli bases and a set of Pauli generators associated with the noise.

14. The computer-implemented method of any one of the preceding six claims, further comprising: mitigating, by the device, the noise by inserting an inverse of the noise into the dynamic quantum circuit.

15. A computer program product for facilitating learning of noise in a dynamic quantum circuit, the computer program product comprising a non-transitory computer-readable memory having program instructions embodied thereon, the program instructions executable by a processor to cause the processor to: learn a noise associated with a non-unitary operation in a circuit of a dynamic quantum circuit by modifying the non-unitary operation in the circuit with a probabilistic Pauli-Z gate and a rotating Pauli operator.

16. The computer program product of the preceding claim, wherein the non-unitary operation in the circuit comprises a circuit-in quantum bit measurement fed forward to at least one classically controlled quantum gate.

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

18. The computer program product of claim 16, wherein the at least one classically controlled quantum gate is not located between the rotating Pauli operators.

19. The computer program product of any one of the preceding four claims, wherein the program instructions are further executable to cause the processor to: perform the non-unitary operation in the circuit modified with the probabilistic Pauli-Z gate and the rotating Pauli operator repeatedly over a set of Pauli bases and over a set of repetition depths; and extract, based on such repeated performance, a set of basis fidelities respectively corresponding to the set of Pauli bases.

20. The computer program product of the preceding claim, wherein the program instructions are further executable to cause the processor to: invert the set of basis fidelities via commutation relations defined between the set of Pauli bases and a set of Pauli generators associated with the noise.

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