MEASURING THE RECORD FAITHFULNESS OF A QUANTUM GATTER WITH RESPECT TO A UNITARITY

Abstract

Systems and methods for quantum computing devices are provided. For example, a method might involve preparing one or more qubits in a selected initial state from a set of initial states.The method may involve implementing a first quantum circuit for n repetitions on the one or more qubits, wherein the first quantum circuit comprises one or more quantum gates; the method may involve implementing a second quantum circuit to map a state of the one or more qubits to a target state, wherein the second quantum circuit is based on a unitarity associated with the first quantum circuit; the method may involve performing a measurement of the one or more qubits; the method may involve determining a fidelity between the first quantum circuit and the unitarity, based at least partially on the measurement of the one or more qubits.

Description

PRIORITY CLAIM

[0001] The present application claims priority over the preliminary US application with serial number 63 / 436,221 entitled “Measuring Quantum Gate Fidelity Relative to a Unitary”, filed on December 20, 2022, which is incorporated herein by reference. AREA

[0002] The present disclosure relates in general to quantum computing systems and in particular to the characterization and reduction of errors in operations (e.g. quantum gates) implemented by quantum computing systems. GENERAL STATE OF THE ART

[0003] Quantum computing is a method of computation that uses quantum effects such as the superposition of basis states and entanglement to perform certain calculations more efficiently than a classical digital computer. Unlike a digital computer, which stores and processes information in the form of bits, such as a "1" or "0", quantum computing systems can process information using quantum bits ("qubits"). A qubit can refer to a quantum device that allows the superposition of multiple states, such as data in both the "0" and "1" states, and / or to the superposition of data itself across those multiple states. The superposition of a "0" and a "1" state in a quantum system can be represented, using conventional terminology, as, for example, |0) + b |1). The "0" and "1" states of a digital computer are analogous to the basis states |0) and |01) of a qubit, respectively. SUMMARY

[0004] Aspects and advantages of embodiments of the present disclosure are partly set forth in the following description, or can be learned from the description, or can be learned through practical implementation of the embodiments.

[0005] An exemplary aspect of the present disclosure relates to a method. The method may involve preparing one or more qubits in a selected initial state of a set of initial states by one or more quantum computing devices. The method may involve implementing a first quantum circuit for n repetitions on the one or more qubits by the one or more quantum computing devices, wherein the first quantum circuit comprises one or more quantum gates. The method may involve implementing a second quantum circuit by the one or more quantum computing devices to map a state of the one or more qubits to a target state, wherein the second quantum circuit is based on a unitarity associated with the first quantum circuit.The method may involve performing a measurement of one or more qubits by one or more quantum computing devices. The method may also involve determining a fidelity between the first quantum circuit and the unitarity by one or more quantum computing devices, wherein the fidelity is based at least partially on the measurement of one or more qubits.

[0006] Other aspects of the present disclosure relate to various systems, methods, devices, non-transitory computer-readable media, computer-readable instructions, and computing devices.

[0007] These and other features, aspects, and advantages of various embodiments of the present disclosure will be better understood with reference to the following description and the accompanying claims. The accompanying drawings, which are incorporated into and form part of this patent specification, illustrate exemplary embodiments of the present disclosure and, together with the description, explain the relevant principles. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] A detailed discussion of embodiments, intended for a person skilled in the art, is set out in the description relating to the accompanying figures, in which the following applies: Fig. Figure 1 shows an exemplary quantum computing system according to exemplary embodiments of the present disclosure; Fig. Figure 2 shows a flowchart of an exemplary process according to exemplary embodiments of the present disclosure; Fig. Figure 3 shows an exemplary 2-elaboration according to exemplary embodiments of the present disclosure; Fig. Figure 4 shows an exemplary circuit for determining fidelity according to exemplary embodiments of the present disclosure; Fig. Figure 5 shows an exemplary circuit for assigning qubit(s) to a target state according to exemplary embodiments of the present disclosure; Fig. Figure 6 shows a flowchart of an exemplary process according to exemplary embodiments of the present disclosure; Fig. Figure 7 shows exemplary fidelity data according to exemplary embodiments of the present disclosure; and Fig. Figure 8 shows exemplary fidelity data according to exemplary embodiments of the present disclosure. DETAILED DESCRIPTION

[0009] Exemplary aspects of this disclosure are aimed at characterizing and reducing errors in the implementation of operations (e.g., quantum gates) in a quantum computing system. A reliable understanding of the structure of noise in quantum processors can be advantageous for the further development of quantum computing devices. A variety of characterization methods exist to quantify and / or validate the performance of quantum operations, such as qubit state preparation, single-qubit quantum gates, composite quantum gates (e.g., quantum gates acting on two or more qubits), measurement, and reset.For example, characterization methods can include randomized benchmarking (RB), cross-entropy benchmarking (XEB), unitary tomography (UT), quantum process tomography (QPT), gate set tomography (GST), and direct fidelity estimation (DFE).

[0010] As quantum computer systems become more complex, interactions between components emerge, such as control crosstalk or gate bleeding, which can make the performance of a particular quantum operation highly context-dependent. Therefore, there is a growing need for methods that account for such non-local effects when characterizing quantum operations and system dynamics.

[0011] Aspects of this disclosure relate to systems and methods for characterizing quantum operations, providing the determination of fidelity between an experimentally implemented quantum operation and a unitarity. The systems and methods according to aspects of this disclosure also provide the extraction of coherent and incoherent error information from the fidelity data, while remaining fast, reliable, and robust against state preparation and measurement (SPAM) errors. Coherent errors are errors that apply a reversible (but potentially unknown) transformation. Incoherent errors are errors that cause a decoherence of the quantum state and are irreversible.

[0012] In some examples, the systems and methods of the present disclosure provide the determination of an average quantum gate reproduction fidelity for n repetitions of a quantum operation of interest using an appropriate number of experiments, while adjusting the quantum circuit context intended for the quantum operation (e.g., both spatial and temporal context). The spatial context may, for example, include operations applied to qubits in spatial proximity to the qubit or qubits on which the quantum operation is performed, or other environmental properties associated with a spatial environment of the operation of interest. The temporal context may, for example, include operations applied close in time to the operation of interest (e.g.,before and / or after the operation of interest, such as quantum gates or operations immediately before and / or immediately after the operation of interest) or other temporal effects or temporal correlations (e.g. residual pulse tails, etc.).

[0013] In one example, the systems and procedures might involve preparing one or more qubits in a selected initial state from a set of initial states. The set of initial states might approximate a Haar random state. The set of initial states might be associated with a 2-design, such as a symmetric, information-complete, positive operator-valued measure-2 design.

[0014] The systems and procedures may involve implementing a first quantum circuit for n repetitions on one or more qubits. The first quantum circuit may include one or more quantum gates of interest (e.g., the quantum gates / operations to be characterized). In some examples, implementing the first quantum circuit may involve implementing one or more context-dependent quantum gates on one or more context-dependent qubits in spatial proximity to the first quantum circuit. In some examples, implementing the first quantum circuit may involve implementing one or more context-dependent quantum gates in temporal proximity to the first quantum circuit.

[0015] After the n repetitions of the first quantum circuit, the systems and procedures can implement a second quantum circuit to map a state of one or more qubits to a target state (e.g., |00〉 in an example where the one or more qubits are two qubits). The second quantum circuit can be implemented based on a unitarity used to characterize the first quantum circuit. This unitarity can be called a reference unitarity. In some examples, the reference unitarity can be an ideal unitarity associated with the first quantum circuit. In other examples, the reference unitarity can be a characterizing unitarity (e.g., a derived unitarity) associated with the first quantum circuit. The second quantum circuit can be configured to map the one or more qubits to the target state and can perform operations (e.g.,These include quantum gates, which are determined based on reference unitarity. In some examples, the systems and methods can implement the second quantum circuit to map the state of one or more qubits to a target state in a single operation.

[0016] After the implementation of the second circuit, the systems and procedures may involve performing a measurement of one or more qubits. A fidelity between the first quantum circuit and the reference unitarity can be determined based on the measurement. For example, the fidelity between n repetitions of the first quantum circuit and the nth power of the reference unitarity can be determined by averaging the probability of measuring the target state over the set of all initial states (e.g., over the states of the 2-configuration). The above operations can be repeated for different values ​​of n to determine fidelity data that provide fidelity according to aspects of this disclosure for each value of n. The fidelity data can be analyzed to evaluate the reference unitarity used to characterize the first quantum circuit.In some cases, a multitude of reference units can be compared (e.g., in a multitude of successive tests) to evaluate a variety of methods for characterizing unitarity. In some embodiments, the systems and methods can extract coherent and incoherent error information from the fidelity data. For example, the coherent error information can be extracted from linear contributions or the linear behavior of the fidelity data with respect to n. The incoherent information can be extracted, for example, from quadratic contributions or the quadratic behavior of the fidelity data with respect to n.

[0017] In some cases, a context can be selected or modified based on one or more test objectives. For example, the tests described above can be repeated using a variety of contexts to quantify the respective effects of a variety of environmental variables or error types (e.g., crosstalk, measurement-induced phase shift, etc.) on the operation in question. In some cases, additional context (e.g., spatially close operations such as microwave operations, measurements on surrounding qubits, etc.) can be added to increase the error rate of a particular type (e.g., crosstalk, etc.). In some cases, a variety of error-inducing contexts can be tested in combination with a variety of reference units to evaluate the robustness of a variety of methods for characterizing unitarity against a variety of error types.In some cases, additional context can be added to reduce the error rate of one or more types. For example, a near-time dynamic decoupling circuit can be added between repetitions of a quantum operation of interest. Such dynamic decoupling can, for instance, couple out single-qubit phase errors and attenuate low-frequency noise. Advantageously, the effect of such an error-reducing modification may, in some cases, be negligible compared to one or more error channels isolated by the additional context.

[0018] Aspects of this disclosure provide a number of technical effects and advantages. For example, some implementations can be used to directly evaluate the performance of quantum circuits while accurately capturing the effects of scattering interactions, gate bleeding, and other error models that prevent isolated gate characterizations from predicting experimental performance. The systems and methods according to the example aspects of this disclosure can be contrasted with general randomized benchmarking techniques that use Haar random circuits to transform coherent errors into incoherent errors, effectively randomizing the context of the operation.The systems and methods described in the exemplary aspects of this disclosure enable the determination of the fidelity of quantum operations in a manner consistent with a target experiment in both spatial and temporal contexts. Fidelity can be determined to measure it against various unitarity characterizations, making it a useful tool for comparing the effectiveness of competing characterization methods. In some examples, aspects of this disclosure can enable the separation of contributions to error behavior from coherent and incoherent sources. In some cases, aspects of this disclosure can provide a more accurate fidelity estimation than previous methods (e.g., nested randomized benchmarking) while requiring significantly fewer experimental resources.

[0019] As used here, the use of the expression "about" or "approximately" in conjunction with a given numerical value refers to a value within 10% of the given numerical value. As used here, "close to the maximum" refers to a value within 10% of a maximum. As used here, "close to the minimum" refers to a value within 10% of a minimum.

[0020] With reference to the FIG., exemplary embodiments of the present disclosure are further explained in detail.

[0021] Fig. Figure 1 shows an exemplary quantum computing system 100. System 100 is an example of a system consisting of one or more classical computers and / or quantum computing devices at one or more locations where the systems, components, and techniques described below can be implemented. Those skilled in the art will understand, using the disclosures provided herein, that other quantum computing devices or systems can be used without deviating from the scope of this disclosure.

[0022] System 100 includes quantum hardware 102 in data communication with one or more classical processors 104. The classical processors 104 can be configured to execute computer-readable instructions stored in one or more memory devices to perform operations, such as any of the operations described here. The quantum hardware 102 includes components for performing quantum computations. For example, the quantum hardware 102 includes a quantum system 110, control device(s) 112, and readout device(s) 114 (e.g., readout resonator(s)). The quantum system 110 can include one or more multi-level quantum subsystems, such as a register of qubits (e.g., qubits 120). In some implementations, the multi-level quantum subsystems can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, etc.

[0023] The type of multi-level quantum subsystems used by System 100 can vary. For example, in some cases it may be advantageous to include one or more readout devices 114 attached to one or more superconducting qubits, such as transmon, fluxmon, gmon, xmon, or other qubits. In other cases, ion traps, photonic devices, or superconducting cavities (e.g., for preparing qubit-free states) may be used. Other examples of realizations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots, or phosphorus impurity qubits.

[0024] Quantum circuits can be constructed and applied to the register of qubits contained in the quantum system 110 via multiple control lines coupled to one or more control devices 112. Exemplary control devices 112 operating on the register of qubits can be used to implement quantum gates or quantum circuits with a variety of quantum gates, such as Pauli gates, Hadamard gates, controlled-NOT gates (CNOT gates), controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc. The one or more control devices 112 can be configured to act on the quantum system 110 via one or more respective control parameters (e.g., one or more physical control parameters).For example, in some implementations the multi-level quantum subsystems can be superconducting qubits, and the control devices 112 can be configured to provide control pulses to control lines to generate magnetic fields to adjust the frequency of the qubits.

[0025] The quantum hardware 102 may further include readout devices 114 (e.g., readout resonators). Measurement results 108 obtained via measurement devices can be provided to the classical processors 104 for processing and analysis. In some implementations, the quantum hardware 102 may include a quantum circuit, and the control device(s) 112 and readout device(s) 114 may implement one or more quantum logic gates that operate on the quantum system 102 by physical control parameters (e.g., microwave pulses) sent through wires contained in the quantum hardware 102. Other examples of control devices include arbitrary waveform generators, with a DAC (digital-to-analog converter) generating the signal.

[0026] The readout device(s) 114 can be configured to perform quantum measurements on the quantum system 110 and send measurement results 108 to the classical processors 104. Furthermore, the quantum hardware 102 can be configured to receive data specifying physical control qubit parameter values ​​106 from the classical processors 104. The quantum hardware 102 can use the received physical control qubit parameter values ​​106 to update the action of the control device(s) 112 and the readout device(s) 114 on the quantum system 110. For example, the quantum hardware 102 can receive data specifying new values ​​representing the voltage levels of one or more DACs contained in the control devices 112 and update the action of the DACs on the quantum system 110 accordingly.The classical processors 104 can be configured to initialize the quantum system 110 into an initial quantum state, e.g. by sending data to the quantum hardware 102 that specifies an initial set of parameters 106.

[0027] In some implementations, the readout device(s) 114 can exploit a difference in impedance for the states |0〉 and |1〉 of an element of the quantum system, such as a qubit, to measure the state of the element (e.g., the qubit). For example, the resonant frequency of a readout resonator can take on different values ​​when a qubit is in state |0〉 or state |1〉 due to the nonlinearity of the qubit. Therefore, a microwave pulse reflected by the readout device 114 carries an amplitude and a phase shift that depend on the state of the qubit. In some implementations, a Purcell filter can be used in conjunction with the readout device(s) 114 to prevent microwave propagation at the qubit frequency.

[0028] In some embodiments, the quantum system 110 can include a plurality of qubits 120, which are arranged, for example, in a two-dimensional lattice 122. For clarification, the following is included in Fig. Figure 1 shows a two-dimensional lattice 122 with 4x4 qubits; however, in some implementations, the system 110 can include a smaller or larger number of qubits. In some embodiments, the multiple qubits 120 can interact with each other via multiple qubit couplers, e.g., the qubit coupler 124. The qubit couplers can define the interactions between the nearest neighbors of the multiple qubits 120. In some implementations, the strengths of the multiple qubit couplers are tunable parameters. In some cases, the multiple qubit couplers included in the quantum computing system 100 can be couplers with a fixed coupling strength.

[0029] In some implementations, the multiple qubits can include 120 data qubits, such as qubit 126, and 128 measurement qubits. A data qubit is a qubit that participates in a computation performed by the system. A measurement qubit is a qubit that can be used to determine the result of a computation performed by the data qubit. That is, during a computation, an unknown state of the data qubit is transferred to the measurement qubit using a suitable physical operation and measured via a suitable measurement operation performed on the measurement qubit.

[0030] In some implementations, each qubit in the multiple qubits 120 can be operated using specific operating frequencies, such as an idle frequency and / or an interaction frequency (or frequencies) and / or a read frequency and / or a reset frequency. The operating frequencies can vary from qubit to qubit. For example, each qubit can operate at a different idle frequency. The operating frequencies for the qubits 120 can be selected before performing a computation.

[0031] Fig. Figure 1 shows an exemplary quantum computing system that can be used to implement the procedures and operations according to exemplary aspects of this disclosure. Other quantum computing systems can be used without deviating from the scope of this disclosure.

[0032] Fig. Figure 2 shows a flowchart of an exemplary method 200 according to exemplary embodiments of the present disclosure. The method 200 can be implemented using any suitable classical and / or quantum computing system, such as the quantum computing system 100 from Fig. 1, will be implemented. Fig. Section 2 shows operations that are carried out in a specific order for the purposes of illustration and discussion. Those skilled in the art will understand from the disclosures provided herein that the operations of all the procedures described herein can be rearranged, adapted, extended, include steps not illustrated, and / or modified in various ways without departing from the scope of this disclosure.

[0033] In 202, the method can involve preparing one or more qubits in a selected initial state of a set of initial states. The set of initial states can, in some embodiments, be a 2-configuration to approximate a Haar random state. The 2-configuration can be a symmetric, information-complete, positive operator-valued measure (SIC-POVM). This can be an ensemble of d 2 pure states |ψ〉 that satisfy the following condition: |〈ψi|ψj〉|={1i=j1d+1i≠j

[0034] An example of a 2-design 250 is shown in Fig. Figure 3 illustrates the 2-elaboration 250. This can be constructed using the states 250.1, 250.2, 250.3, 250.4, ... at the vertices of a tetrahedron 252 inscribed in a Bloch sphere 254 representation of qubits. Other exemplary sets of initial states may be used without departing from the scope of this disclosure.

[0035] In example 204, the procedure 200 may involve implementing a first quantum circuit for n repetitions on one or more qubits. The first quantum circuit may implement one or more quantum operations (e.g., quantum gates) that are to be characterized. In some examples, implementing the first quantum circuit may involve implementing one or more context-dependent quantum gates on one or more context-dependent qubits in spatial proximity to the first quantum circuit. In some examples, implementing the first quantum circuit may involve implementing one or more context-dependent quantum gates in temporal proximity to the first quantum circuit.

[0036] In 206, the procedure 200 can involve implementing a second quantum circuit to map a state of one or more qubits to a target state, such as the state |00〉 in the example of a quantum circuit acting on two qubits. The mapping to the target state can be based on the reference unitarity associated with the first quantum circuit. For example, the reference unitarity can be configured to model the first quantum circuit (e.g., including context-dependent gates and context-dependent qubits). In some examples, the reference unitarity can be an ideal unitarity. In other examples, the reference unitarity can be a characterized unitarity derived from one or more fault models and / or characterization procedures.Details of an example of a quantum circuit used to assign one or more qubits to the target state are given with reference to . Fig. 5 discussed.

[0037] With reference to Fig. In reference 208, procedure 200 may involve performing a measurement of one or more qubits. The measurement may cause the one or more qubits to collapse into a measured state. In reference 210, procedure 200 may involve determining a fidelity (e.g., context-aware fidelity, CAFE) between the first quantum circuit and the unitarity, based at least partially on the measurement of one or more qubits. For example, the measurement of one or more qubits may be used to determine a probability that the measured state is the target state (e.g., |00〉 in the example of a quantum circuit acting on two qubits). The probability over all initial states of the set of initial states may be the average fidelity for the first quantum circuit.

[0038] In 212, the procedure 200 can involve modifying one or more control signals used to implement the first quantum circuit in the quantum computing system, based at least partially on fidelity. For example, the one or more control signals can be modified based on fidelity to reduce the probability of an error during the implementation of the first quantum circuit (e.g., one or more quantum gates in the first quantum circuit).

[0039] A specific example of procedure 200 is given with reference to the Fig. 3-5 below. The specific example is discussed using the implementation of a first quantum circuit that includes one or more CZ gates for the purposes of illustration and discussion. Those skilled in the art will understand, using the disclosures provided here, that the first quantum circuit may include other types of quantum gates.

[0040] Fig. Figure 4 shows an exemplary circuit diagram 300 for determining a fidelity (e.g., CAFE) according to exemplary embodiments of the present disclosure. The circuit diagram 300 includes a circuit 310. At 312, the circuit 310 prepares a state |ψ〉 from an m-qubit-2 configuration (as explained above). The circuit 310 implements the first quantum circuit 314 (e.g., illustrated as a cycle circuit) on the m qubits for n repetitions. This implementation includes the implementation of operations on neighboring contextual qubits using circuits 320 and 330. For example, the n repetitions of the first quantum circuit 314 can involve performing n repetitions of contextual quantum gates 322 on contextual qubits in circuit 320 and n repetitions of contextual quantum gates 332 on contextual qubits in circuit 330.

[0041] The first quantum circuit 314 can include several operations, such as one or more single-qubit quantum gates, one or more composite quantum gates, or other operations. In some embodiments, the first quantum circuit 314 can include one or more dynamic decoupling (DD) gates. For example, in some cases, a quantum operation of interest might involve a CZ gate. In such cases, dynamic decoupling might involve adding an X gate to both qubits in the first quantum circuit 314. Such a DD gate can, in some cases, couple out single-qubit phase errors and attenuate low-frequency noise. DD gates can be robust to certain coherent error parameters. Including DD gates in the first quantum circuit 314 can allow a focus on error characterization that affects the performance of the quantum computing system.The insertion of DD gates into the first quantum circuit 314 is an exemplary method to facilitate the separation of coherent and incoherent contributions to quantum gate errors.

[0042] In some cases, the first quantum circuit 314 can consist solely of a quantum circuit of interest operating on one or more qubits of interest, and the first quantum circuit 314 can be repeated n times without any transiently adjacent quantum operations being performed on the qubits of interest between each of the n repetitions. Such a lack of interleaved operations can, for example, reduce unwanted errors compared to interleaved randomized benchmarking (IRB), where single-qubit gates used by IRB to generate random Clifford gates can introduce unwanted sources of error through decoherence and systematic errors.

[0043] Circuit 310 can implement a second circuit 316 (e.g., a measurement circuit). The second circuit 316 attempts to reassign the state of the qubits to a target state using the reference unitarity (e.g., |00〉 in the example of a quantum circuit acting on two qubits). The average gate fidelity of the first quantum circuit 314 can be found by averaging over the experiments for all initial 2-configuration states {ψi}, as outlined below: F=〈ℙ0…0〉{ψi}

[0044] More precisely, a state {ψi}, drawn from an m-qubit-2 configuration, is first provided in 312. Then, the cycle circuit to be characterized is applied the desired number n in 314. Finally, the state is mapped back to |00) using the reference unitarity and then measured in 316. The fidelity between n repetitions of the applied operation and the nth power of the reference unitarity can be found by averaging the probability of obtaining |00〉 over all initial states.

[0045] In some cases, one or more contexts (e.g., contextual quantum gates 332, 322) can be selected or modified based on one or more test targets. For example, a first set of context variables (e.g., quantum gates 332, 322) can be selected. Circuit 310 can prepare a state |ψ〉 from an m-qubit-2 configuration (as explained above). Circuit 310 can implement the first quantum circuit 314 (e.g., illustrated as a cycle circuit) on the m qubits for n repetitions, depending on the first set of context variables. In some cases, for example, circuit 300 can implement a first set of contextual quantum gates 322, 332 for n repetitions. The m qubits can be mapped to a target state (e.g., based on a reference unitarity). One or more first fidelity values ​​(e.g.,Fidelity relative to a characterized unitarity or ideal unitarity; incoherent error rate; coherent error rate, etc.) can be determined based on one or more measurements at 316. A second set of context variables can be selected; a state |ψ〉 can be prepared; the first quantum circuit 314 can be repeated for n repetitions depending on the second set of context variables; and one or more second fidelity values ​​can be determined. In some cases, the first and second fidelity values ​​can be compared to determine the robustness of a quantum circuit 314 or the robustness of a characterized unitarity against one or more error types.

[0046] Fig. Figure 5 shows an exemplary circuit 350 for assigning a plurality of qubits to a target state (e.g., |00〉) according to exemplary embodiments of the present disclosure. More precisely, the circuit 350 assigns |00〉 to an arbitrary two-qubit state |ψ〉 using only one CZ operation. The overlap of any two-qubit state with the state |ψ〉 can be obtained by performing the inverse operation of this circuit 350 and subsequently specifying the probability of measuring |00〉.

[0047] More precisely, a general two-qubit target state can be characterized as follows: |ψ〉=A|00〉+B|01〉+C|10〉+D|11〉,|A|2+|B|2+|C|2+|D|2=1.

[0048] A matrix containing the amplitudes of this state is provided below: Mψ=[ABCD]

[0049] The singular value decomposition (SVD) of this matrix yields: Mψ=UψSψVψ

[0050] Taking into account the initial singular value, Sψ00, The degree of entanglement in the target state can be quantified.

[0051] The first operation to create the target state is to prepare a state with a suitable entanglement signature. In the example using a CZ gate, one qubit can be set to |+〉. The other qubit can be subjected to a Y-rotation with an angle of α=2arccos(Sψ00) be applied. This condition, which is in Fig. The matrix labelled |CZ (5) corresponds to the final state, except for single-qubit rotations. To find these rotations, the SVD of the 2x2 matrix is ​​calculated according to an intermediate state shown below: UCSSCSVCZ†

[0052] The single-qubit unitarity for the first qubit is given by: U1=UψUCZ−1

[0053] The single-qubit unitarity for the second qubit is given by: U2=VψVCZ−1

[0054] According to exemplary aspects of this disclosure, the fidelity of a quantum operation can be estimated and expressed as a single number as a performance metric. This can be useful for validating and estimating the performance of various quantum algorithms, quantum circuits, and / or other operations. However, aspects of this disclosure also provide for the extraction of the coherent and incoherent contributions to the error.

[0055] Fig. Figure 6 shows a flowchart of an exemplary method 400 according to exemplary embodiments of the present disclosure. The method 400 can be implemented using any suitable classical and / or quantum computing system, such as the quantum computing system 100 from Fig. 1, will be implemented. Fig. Figure 6 shows operations that are carried out in a specific order for the purposes of illustration and discussion. Those skilled in the art will understand from the disclosures provided here that the operations of all the procedures described herein can be rearranged, adapted, extended, include steps not illustrated, and / or modified in various ways without departing from the scope of this disclosure.

[0056] In case 402, procedure 400 can perform procedure 200. Fig. 2. In 404, procedure 400 may involve storing the fidelity as part of fidelity data. For example, the average fidelity for a given value of n may be stored as part of fidelity data. In 406, it may be determined whether to repeat procedure 400 for a new value of n. If so, procedure 400 continues to 408, where n is modified (e.g., incremented, changed to a different value, decremented, etc.). Procedure 400 may return to 402, where it continues procedure 200 from Fig. 2 is repeated for a different value of n. This can be repeated for several different values ​​of n to generate the fidelity data.

[0057] Fig. Figure 7 shows an exemplary graphical representation of fidelity data 500 according to exemplary embodiments of the present disclosure. Fig. Figure 7 represents the fidelity along the vertical axis and n (number of repetitions) along the horizontal axis. The fidelity data 500 includes fidelity 510.1, which is assigned to n=0, fidelity 510.2, which is assigned to n=2, fidelity 510.3, which is assigned to n=4, fidelity 510.4, which is assigned to n=6, fidelity 510.5, which is assigned to n=8, and so on. In some examples, a fidelity decay curve 520 can be fitted to the fidelity data using a model. Details of an example model that can be used to extract and separate coherent and incoherent error contributions are detailed below.

[0058] In the case of 410, the procedure 400 may involve analyzing the fidelity data 410. For example, the procedure 400 may involve analyzing the fidelity data 410 to extract coherent error information. The procedure 400 may also involve analyzing the fidelity data 410 to extract incoherent error information (e.g., using a model or by exploiting DD pulses).

[0059] For example, in some embodiments, a fidelity falloff curve can be fitted to the fidelity data. The coherent error information can be determined from a linear contribution associated with the fidelity falloff curve. The incoherent error information can be determined from a quadratic contribution associated with the fidelity falloff curve.

[0060] In some cases, extracting coherent and incoherent error information may involve fitting the fidelity data 410 to a model. An example model for fitting a fidelity decay curve for a quantum circuit that is a CZ gate is described below. The unitarity for the two-qubit CZ gate can be provided as follows: U:

[0061] A noisy quantum channel can be used to implement the first quantum circuit, as described by a two-qubit depolarization channel: ε(ρ)=(1−pdepol)U˜ρU˜†+pdepolId / d

[0062] The two-qubit depolarization channel yields a completely mixed state with probability p. depolIt selects the first quantum circuit, represented by cycle unitarity, and otherwise applies the first quantum circuit. In some cases, a state preparation and measurement (SPAM) error can be modeled as an offset in the fidelity curve that is constant with respect to n, which in some cases allows the extraction of coherent and incoherent error data that are robust to SPAM errors. With such a depolarizing channel and a SPAM offset, a model that correlates the average gate fidelity for n repetitions can be specified as follows: Fn=14−εspam−120(1−pdepol)n⋅(1−|1+2e−inΔγcos(nΔθ)+e−in(2Δγ+Δϕ)|2) where ε spam represents the state preparation and measurement (SPAM) error.

[0063] Using this exemplary model, the fidelity data can be adjusted to account for the incoherent error contributions ε. incoh and the coherent error contributions εcoh To maintain fidelity, the following can be used to obtain these parameters: 1−F=1−F1(1−εspam), εincoh=1−F1(pdepol=0)(1−εspam), εcoh=1−F1(Δθ=Δγ=Δϕ=0)(1−εspam),

[0064] In some embodiments, aspects of the present disclosure may involve modifying the first quantum circuit (e.g., the quantum circuit to be analyzed) to isolate specific defects. For example, dynamic decoupling gates may be inserted between repetitions of the n repetitions of the first quantum circuit. This approach may be useful for characterizing coherent defects present in the first quantum circuit without requiring full unitarity tomography. In the example of a first quantum circuit that is a CZ gate, adding an X gate to both qubits in the first quantum circuit may couple out all single-qubit phase errors, in addition to attenuating low-frequency noise.

[0065] With reference to Fig. In Section 6 of 412, Method 400 can involve modifying a quantum computing system based on the fidelity data. For example, the fidelity data can be used to evaluate the performance of a unitarity used to characterize the first quantum circuit. More specifically, in some embodiments, aspects of the present disclosure can involve implementing a second quantum circuit to map the state of one or more qubits to the target state in a single operation. By performing the entire inversion in a single step, similar to randomized benchmarking but using the reference unitarity for the inversion, different unitarity characterizations can be validated, respecting the non-Clifford nature of most coherent fault models.As such, the determination of fidelity according to exemplary embodiments of the present disclosure can be used to benchmark unitarity characterizations by simply changing the last measurement step and observing which predictions most accurately map the final state back to the target state (e.g. |00〉).

[0066] For example, it shows Fig. 8 a diagram of exemplary fidelity data assigned to three different unitarity characterizations of a first quantum circuit according to exemplary embodiments of the present disclosure. Fig. Figure 8 represents the fidelity along the vertical axis and n (number of repetitions) along the horizontal axis. Curve 530 can be associated with a first unitarity characterization. Curve 540 can be associated with a second unitarity characterization. Curve 550 can be associated with a third unitarity characterization. The first unitarity characterization can provide the isolation and measurement of different unitarity parameters, for example, in an FSIM gate. The second unitarity characterization can be a unitarity characterization extracted from XEB. The third unitarity characterization can be an ideal unitarity.

[0067] As through Fig. As demonstrated in Figure 8, certain unitarity characterizations are better suited to removing coherent errors. Furthermore, by increasing the amount of context around the implementation of the first quantum circuit when determining fidelity (e.g., by including contextual gates and / or contextual qubits), unitarity characterizations that break down in the face of context information can be used to determine the dominant effects on the quantum processor / quantum hardware that impact algorithm performance.

[0068] In some cases, a variety of contexts (e.g., error-amplifying contextual elements) can be tested in combination with a variety of reference unitarities to evaluate the robustness of a variety of unitarity characterization procedures against a variety of error types. For example, a first system according to Fig. 4. Implemented using a first set of context variables (e.g., contextual quantum gates 322, 332) and a first mapping circuit (e.g., 350) configured to map m qubits to a target state based on a first reference unitarity (e.g., characterized unitarity). A second system according to Fig. System 4 can be implemented using a second set of context variables and the first mapping circuit based on the first reference unitarity. A third and fourth system according to Fig. Four systems can be implemented using the first or second set of context variables with a second mapping circuit based on a second reference unitarity. One or more fidelity levels can be derived from each of the first, second, third, and fourth systems according to... Fig. 4 are determined and compared.

[0069] Based on the characterized unitarity(ies), the error performance of a quantum computing system can be evaluated. Design parameters of the quantum computing system (e.g., heat, qubit structure, control signals, hardware layout, qubit arrangement, etc.) can be modified based on the characterized unitarity(ies) to improve the performance of a quantum computing system.

[0070] Implementations of the digital and / or quantum-related subject matter and the digital functional and quantum operations described in this specification may be implemented in digital electronic circuit arrangements, suitable quantum circuit arrangements, or, more generally, in quantum computing systems, in specifically implemented digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of these. The term "quantum computing systems" may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0071] Implementations of the digital and / or quantum-related object described in this description can be implemented as one or more digital and / or quantum computer programs, i.e., as one or more modules of digital and / or quantum-related computer program instructions encoded on a tangible, non-transitory storage medium for execution by a data processing device or for controlling the operation of a data processing device. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random-access or serial-access storage device, one or more qubits / qubit structures, or a combination of one or more of these. Alternatively or additionally, the program instructions can be encoded on an artificially generated propagated signal capable of transmitting digital and / or quantum information (e.g.,to encode a machine-generated electrical, optical or electromagnetic signal) that is generated to encode digital and / or quantum information for transmission to a suitable receiving device for execution by a data processing device.

[0072] The terms "quantum information" and "quantum data" refer to information or data transmitted, captured, or stored by quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as two-level systems in the relevant context. Such quantum systems may include multi-level systems, e.g., with two or more levels. For example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational basis states are identified with the ground state and the first excited state; however, it is clear that other configurations are possible where the computational states are identified with excited states of higher levels (e.g., qudits).

[0073] The term "data processing device" refers to digital and / or quantum data processing hardware and includes all types of equipment, devices, and machines for processing digital and / or quantum data, including, for example, a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors or computers, and combinations thereof. The device may also be, or further include, a specific logic circuit arrangement, such as an FPGA (Field Programmable Gate Array), an ASIC (Application Specific Integrated Circuit), or a quantum simulator, i.e., a quantum data processing device designed to simulate or generate information about a particular quantum system.In particular, a quantum simulator is a special type of quantum computer that is not capable of performing general-purpose quantum calculations. In addition to the hardware, the device may optionally include code that creates an execution environment for digital and / or quantum computer programs, such as code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of these.

[0074] A digital computer program, which may also be called or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages ​​or declarative or procedural languages, and it may be used in any form, including as a standalone program or as a module, component, subroutine or other unit suitable for use in a digital computing environment.A quantum computer program, which may also be referred to or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages ​​or declarative or procedural languages, and translated into a suitable quantum programming language, or it may be written in a quantum programming language, e.g. QCL, Quipper, Cirq, etc.

[0075] A digital and / or quantum computer program can, but does not have to, correspond to a file in a file system. A program can be stored in a portion of a file containing other programs or data, such as one or more scripts stored in a markup language document, in a single file dedicated to the program, or in multiple coordinated files, such as files containing one or more modules, subroutines, or portions of code. A digital and / or quantum computer program can be used to run on a digital or quantum computer, or on multiple digital and / or quantum computers located in one location or distributed across multiple locations and connected by a digital and / or quantum data communication network. A quantum data communication network is a network capable of transmitting quantum data using quantum systems, such as qubits.In general, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0076] The processes and logic sequences described in this document can be performed by one or more programmable digital and / or quantum computers operating with one or more digital and / or quantum processors, which may optionally execute one or more digital and / or quantum computer programs to perform functions by processing input digital and quantum data and generating outputs. The processes and logic sequences can also be performed by dedicated logic circuit arrangements, such as an FPGA, an ASIC, or a quantum simulator, or devices can be implemented as such, or by a combination of dedicated logic circuit arrangements or quantum simulators and one or more programmed digital and / or quantum computers.

[0077] For a system consisting of one or more digital and / or quantum computers or processors to be "configured" or "operable" to perform certain operations or actions, the system must have software, firmware, hardware, or a combination thereof that, when operating, causes the system to perform those operations or actions. For one or more digital and / or quantum computer programs to be configured to perform certain operations or actions, this means that the one or more programs contain instructions which, when executed by a digital and / or quantum computing device, cause the device to perform the operations or actions. A quantum computer can receive instructions from a digital computer which, when executed by the quantum computing device, cause the device to perform the operations or actions.

[0078] Digital and / or quantum computers capable of executing a digital and / or quantum computer program may be based on general-purpose or specialized digital and / or quantum microprocessors, or both, or any other type of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit receives instructions and digital and / or quantum data from a read-only memory, a random-access memory, or quantum systems capable of transmitting quantum data, such as photons, or combinations thereof.

[0079] Some exemplary elements of a digital and / or quantum computer are a central processing unit for carrying out or executing instructions and one or more storage devices for storing instructions and digital and / or quantum data. The central processing unit and storage may be augmented by or integrated with special logic circuit arrangements or quantum simulators. Generally, a digital and / or quantum computer will also include one or more mass storage devices for storing digital and / or quantum data, such as magnetic, magneto-optical, or optical disks, or quantum systems suitable for storing quantum information, or be operationally coupled to receive digital and / or quantum data from them, or to transmit digital and / or quantum data to them, or both.However, a digital and / or quantum computer does not necessarily need to have such devices.

[0080] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer-programmable instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memories, media, and storage devices, including, for example, semiconductor memory devices such as EPROM, EEPROM, and flash memory devices; magnetic disk storage devices such as internal hard disks or removable disks; magneto-optical disks; and CD-ROM and DVD-ROM disks; and quantum systems such as trapped atoms or electrons. Quantum memories are understood to be devices capable of storing quantum data for extended periods with high fidelity and efficiency, such as light-matter interfaces where light is used for transmission and matter for storage and preservation of the quantum properties of quantum data, such as superposition or quantum coherence.

[0081] The control of the various systems or parts thereof described in this description can be implemented in a digital and / or quantum computer program product that includes instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or parts thereof described in this description can each be implemented as a device, method, or electronic system that may include one or more digital and / or quantum processing devices and a memory for storing executable instructions for performing the operations described in this description.

[0082] While this description contains many specific implementation details, these should not be interpreted as limitations on the scope of what can be claimed, but rather as descriptions of features that may be specific to certain implementations. Certain features described in this description in connection with separate implementations may also be implemented in combination within a single implementation. Conversely, various features described in connection with a single implementation may also be implemented separately in multiple implementations or in any suitable subcombination.Furthermore, although features described above may be effective in certain combinations and may even have been originally claimed as such, in some cases one or more features from a claimed combination may be removed from the combination, and the claimed combination may be directed to a subcombination or a variation of a subcombination.

[0083] Similarly, the operations depicted in the drawings in a particular order should not be interpreted as requiring them to be performed in the shown sequence or in sequential order, nor as requiring all illustrated operations to be performed to achieve desired results. Multitasking and parallel processing may be advantageous under certain circumstances. Furthermore, the separation of different system modules and components in the implementations described above should not be interpreted as requiring such separation in all implementations. It should be understood that the described program components and systems can generally be integrated into a single software product or bundled into multiple software products.

[0084] Certain implementations of the subject matter have been described. Other implementations fall within the scope of the following claims. For example, the actions listed in the claims can be performed in a different order and still achieve desirable results. As an example, the processes depicted in the accompanying figures do not necessarily require the specific or sequential order shown to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. QUOTES INCLUDED IN THE DESCRIPTION

[0000] This list of documents cited by the applicant was automatically generated and is included solely for the reader's convenience. The list is not part of the German patent or utility model application. The DPMA accepts no liability for any errors or omissions. Cited patent literature

[0000] US 63 / 436,221

[0001]

Claims

[1] Procedure, encompassing: Preparing one or more qubits in a selected initial state of a set of initial states by one or more quantum computing devices; Implementing a first quantum circuit for n repetitions on the one or more qubits by the one or more quantum computing devices, wherein the first quantum circuit comprises one or more quantum gates; Implementing a second quantum circuit by the one or more quantum computing devices to map a state of the one or more qubits to a target state, wherein the second quantum circuit is based on a unitarity associated with the first quantum circuit; Performing a measurement of one or more qubits by one or more quantum computing devices; Determining a fidelity between the first quantum circuit and the unitarity by the one or more quantum computing devices, based at least partially on the measurement of the one or more qubits. [2] Method according to claim 1, wherein the implementation of the first quantum circuit comprises implementing one or more context-dependent quantum gates on one or more context-dependent qubits in spatial proximity to the one or more qubits. [3] Method according to claim 1, wherein the implementation of the first quantum circuit comprises the implementation of one or more context-dependent quantum gates in close temporal proximity to the first quantum circuit. [4] Method according to claim 1, wherein the set of initial states approximates a Haar random state. [5] Method according to claim 4, wherein the set of initial states is assigned to a 2-configuration. [6] Method according to claim 5, wherein the 2-elaboration is a symmetrical, information-complete, positive operator-valued measure. [7] Method according to claim 1, wherein determining the fidelity comprises averaging a probability of measuring the target state over the set of initial states. [8] Method according to claim 1, wherein the one or more qubits comprise two qubits and the target state is |00〉. [9] Method according to claim 1, wherein implementing a second quantum circuit by the one or more quantum computing devices for assigning a state of one or more qubits to a target state comprises implementing the second quantum circuit by the one or more quantum computing devices for assigning the state of one or more qubits to the target state in a single operation. [10] Method according to claim 1, wherein the method comprises repeating the method according to claim 1 for a plurality of different values ​​of n in order to generate fidelity data over the different values ​​of n. [11] Method according to claim 10, wherein the method comprises extracting coherent error information from the fidelity data. [12] Method according to claim 10, wherein the method comprises extracting incoherent error information from the fidelity data. [13] Method according to claim 1, wherein the one or more quantum gates of the first quantum circuit comprise a composite quantum gate. [14] Method according to claim 1, wherein the method comprises modifying one or more control signals of the quantum computing system, based at least partially on fidelity. [15] Quantum computing system, comprising: a large number of qubits; one or more control devices that are operable to implement one or more quantum gates on the plurality of qubits; one or more classical or quantum processors capable of implementing computer-readable instructions stored in one or more memory devices to cause the one or more classical or quantum processors to perform operations, wherein the operations include: Preparing one or more qubits in a selected initial state of a set of initial states; Implementing a first quantum circuit for n repetitions on the one or more qubits, wherein the first quantum circuit comprises one or more quantum gates; Implementing a second quantum circuit to map a state of one or more qubits to a target state, wherein the second quantum circuit is based on a unitarity associated with the first quantum circuit; Performing a measurement of one or more qubits; and Determining a fidelity between the first quantum circuit and unitarity, based at least partially on the measurement of one or more qubits. [16] Quantum computing system according to claim 15, wherein the one or more control devices are configured to provide control signals to implement the one or more quantum gates based on fidelity. [17] Quantum computing system according to claim 15, wherein the operation of implementing the first quantum circuit comprises implementing one or more context-dependent quantum gates on one or more context-dependent qubits in spatial proximity to the one or more qubits. [18] Quantum computing system according to claim 15, wherein the operation of implementing the first quantum circuit comprises implementing one or more context-dependent quantum gates in close temporal proximity to the first quantum circuit. [19] Tangible, non-transitory, computer-readable medium that stores computer-readable instructions which, when executed by one or more classical or quantum processors, cause the one or more classical or quantum processors to perform operations, the operations comprising: Implementing a first quantum circuit for n repetitions on the one or more qubits, wherein the first quantum circuit comprises one or more quantum gates; Implementing a second quantum circuit to map a state of one or more qubits to a target state, wherein the second quantum circuit is based on a unitarity associated with the first quantum circuit; Performing a measurement of one or more qubits; Determining a fidelity between the first quantum circuit and unitarity, based at least partially on the measurement of one or more qubits. [20] Tangible, non-transitory, computer-readable medium according to claim 19, wherein the operations further comprise modifying the unitarity based at least partially on the fidelity.

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