Quantum fidelity estimation method based on resonance algorithm

Abstract

The invention discloses a quantum fidelity estimation method based on a resonance algorithm, and relates to the technical field of quantum computing, and the method comprises the steps: providing a target quantum system with the fidelity to be estimated and a detection quantum bit, initializing the detection quantum bit into a preset energy eigenstate, and enabling the target quantum system to be a quantum state prepared by an experiment; coupling the two to form a coupling system; controlling the coupling system to evolve for a preset time under a preset resonance Hamiltonian; measuring the probability that the detection quantum bit is in a preset excited state; frequency sweeping parameters in the resonance Hamiltonian are adjusted, evolution and measurement are repeated, when the frequency sweeping parameters are matched with the target energy level corresponding to the target quantum system, a peak value appears in the excited state probability, and the peak value is the fidelity of the target quantum system and the quantum state corresponding to the target energy level. According to the method, a large amount of multi-bit measurement is not needed, quantum fidelity estimation can be realized only through single-bit measurement, the measurement times are reduced, and the measurement realization difficulty is reduced.

Description

Technical Field

[0001] This invention relates to the field of quantum computing technology, and in particular to a quantum fidelity estimation method based on the resonance algorithm. Background Technology

[0002] Quantum Fidelity Estimation is a core technology in the field of quantum computing. It is used to evaluate the differences between intermediate states and final states generated during the operation of quantum devices and theoretical states under ideal working conditions. Essentially, it quantifies the degree of overlap between experimentally prepared quantum states and theoretical quantum states. It is widely used in various fields of quantum computing to evaluate the operational performance indicators of quantum devices and the accuracy of quantum operations.

[0003] The Variational Quantum Eigensolver (VQE), a quantum-classical hybrid algorithm, is adapted to noisy intermediate-scale quantum (NISQ) devices and aims to efficiently calculate the ground-state energy of quantum systems such as molecules. Specifically, it generates trial quantum states through parameterized quantum circuits, measures the expected energy, and then uses a classical optimizer to adjust the parameters to minimize the expected energy, thereby approximating the ground-state energy. It is one of the key quantum algorithms in the current NISQ era.

[0004] However, existing fidelity estimation schemes have significant limitations for test quantum states prepared by VQE circuits. Direct fidelity estimation methods require a large number of repeated multi-qubit measurements to obtain overlap information with the target ground state. In high-qubit scenarios, they also face exponential resource consumption due to tomography of high-dimensional quantum systems, which seriously contradicts the limited qubits and measurement resources of NISQ devices. This results in low estimation efficiency and high implementation difficulty, restricting the practical application of algorithms such as VQE.

[0005] Therefore, there is an urgent need for a method that can reduce the number of measurements and adapt to the resource constraints of NISQ devices to achieve quantum fidelity estimation, in order to fill the gap in existing technologies. Summary of the Invention

[0006] The technical problem this invention aims to solve is that, in the field of quantum computing, existing fidelity estimation methods for quantum states prepared by variable quantum eigenvalue solver circuits require a large number of repeated multi-qubit measurements. In high-qubit scenarios, these methods also face exponential resource consumption due to the tomography of high-dimensional quantum systems, making them difficult to adapt to the resource constraints of medium-scale quantum devices with noise. This results in low estimation efficiency and high implementation difficulty. Therefore, an effective solution is urgently needed to address these technical problems.

[0007] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows: In a first aspect, the present invention provides a quantum fidelity estimation method based on a resonance algorithm, comprising: Provide a target quantum system whose fidelity is to be estimated and a probe qubit, initialize the probe qubit to a preset energy eigenstate, and the target quantum system is an experimentally prepared quantum state; The target quantum system is coupled with the probe qubit to form a coupled system; The coupling system is controlled to evolve for a preset time under a preset resonant Hamiltonian; Measure the probability that the probe qubit is in a preset excited state; The frequency sweep parameter in the resonant Hamiltonian is adjusted and the evolution and measurement are repeated. When the frequency sweep parameter matches the target energy level corresponding to the target quantum system, the probability of the excited state reaches a peak value. The value of the peak value is the fidelity of the quantum state corresponding to the target quantum system and the target energy level.

[0008] In one implementation, the provision of a target quantum system and a probe qubit for estimating fidelity includes initializing the probe qubit to a preset energy eigenstate, and the target quantum system being an experimentally prepared quantum state, comprising: To obtain the target quantum system whose fidelity needs to be estimated, wherein the target quantum system is fabricated as a variational quantum eigenvalue solver circuit. Bit experiment final state ; An auxiliary probe qubit is provided, which contains two energy levels: a ground state and an excited state. The probe qubit is initialized to a preset energy eigenstate, which is the ground state of the probe qubit. .

[0009] In one implementation, the resonant Hamiltonian includes a Zeeman splitting term, a resonance condition term, and a driving field term. The Zeeman splitting term is used to determine the energy level structure of the probe qubit, and the resonance condition term is used to correlate the target Hamiltonian with the excited state of the probe qubit. The resonance is controlled by coupling and frequency sweeping parameters, and the driving field term is used to control the probe qubit to perform Rabi oscillation.

[0010] In one implementation, controlling the coupling system to evolve for a preset time under a preset resonant Hamiltonian includes: Set the evolution time to ,in, and For the parameters of the driving field, , It is a local minimum; The coupling system is controlled to evolve over the preset time under the influence of the preset resonant Hamiltonian. .

[0011] In one implementation, measuring the probability that the probe qubit is in a preset excited state includes: Excite the probe qubit A single-bit measurement was performed to obtain the transition of the probe qubit to an excited state. probability Among them, evolution time Fix the oscillation phase to be , making the excited state The probability amplitude has reached its maximum value.

[0012] In one implementation, the frequency sweep parameters are matched with the target energy level corresponding to the target quantum system, including: The frequency sweep parameters match the eigenlevels of the target Hamiltonian, satisfying the resonance condition in the resonance condition term. The probe bits The state is only the eigenstate corresponding to the eigenlevel of the target Hamiltonian. They become entangled and continue to resonate during their evolution.

[0013] In one implementation, the probability of the excited state exhibits a peak value, the value of which represents the fidelity between the target quantum system and the quantum state corresponding to the target energy level, including: The probability obtained from all measurements The plotting yielded several discrete peaks, and the corresponding probabilities were obtained. The value is prepared by the variable quantum eigenvalue solver circuit. Bit experiment final state eigenstates of the target Hamiltonian fidelity between ; The fidelity of the ground state of the target quantum system is obtained by selecting the first peak among the discrete peaks.

[0014] Beneficial Effects: A quantum fidelity estimation method based on a resonance algorithm is disclosed, relating to the field of quantum computing technology. The method first provides a target quantum system and a probe qubit whose fidelity is to be estimated. The probe qubit is initialized to a preset energy eigenstate, and the target quantum system is an experimentally prepared quantum state. Then, the target quantum system and the probe qubit are coupled to form a coupled system. Next, the coupled system is controlled to evolve for a preset time under a preset resonant Hamiltonian, and the probability of the probe qubit being in a preset excited state is measured. Finally, the frequency sweep parameter in the resonant Hamiltonian is adjusted, and the evolution and measurement are repeated. When the frequency sweep parameter matches the target energy level corresponding to the target quantum system, a peak value appears in the probability of the excited state. The value of the peak value is the fidelity of the target quantum system and the quantum state corresponding to the target energy level. This invention eliminates the need for numerous repetitive multi-qubit measurements, achieving quantum fidelity estimation solely through single-qubit measurements. This significantly reduces the complexity of measurement implementation and avoids the exponential resource consumption of tomography in high-dimensional quantum systems, requiring only one additional auxiliary probe qubit. This substantially saves quantum device resources and is adaptable to constraints in noisy, medium-scale quantum devices. Furthermore, the method can be fine-tuned to extend to fidelity estimation of arbitrary target pure states, improving its applicability and effectively supporting the practical application of quantum algorithms such as variable quantum eigenvalue solvers. Attached Figure Description

[0015] Figure 1 A flowchart illustrating a specific implementation of the quantum fidelity estimation method based on the resonance algorithm provided in this embodiment of the invention.

[0016] Figure 2 This is a schematic diagram of the circuit principle for estimating the fidelity of a 2-bit VQE experimental state in the quantum fidelity estimation method based on the resonance algorithm provided in this embodiment of the invention.

[0017] Figure 3 This is a schematic diagram of the simulation results of estimating the ground state fidelity of the 2-bit Ising transverse field Hamiltonian using the resonance algorithm in the quantum fidelity estimation method based on the resonance algorithm provided in this embodiment of the invention. Detailed Implementation

[0018] To make the objectives, technical solutions, and effects of this invention clearer and more explicit, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0019] The flowchart shown in the attached diagram is for illustrative purposes only and does not necessarily include all content, operations, or steps, nor does it require execution in the described order. For example, some operations or steps can be broken down, combined, or partially merged, so the actual execution order may change depending on the actual situation.

[0020] It should be understood that the terminology used in this specification is for the purpose of describing particular embodiments only and is not intended to limit the invention. As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” are intended to include the plural forms unless the context clearly indicates otherwise. It should be understood that, in order to clearly describe the technical solutions of the embodiments of the present invention, the terms "first" and "second" are used in the embodiments of the present invention to distinguish identical or similar items with essentially the same function and effect. For example, "first control information" and "second control information" are only used to distinguish different control information and do not limit their order. Those skilled in the art will understand that the words "first" and "second" do not limit the quantity or the order of execution, and that the words "first" and "second" do not necessarily imply that they are different. It should also be understood that the term "and / or" as used in this specification and the appended claims refers to any combination of one or more of the associated listed items and all possible combinations, and includes such combinations.

[0021] Quantum fidelity estimation is a technique used to assess the difference between the intermediate and final states of a quantum device during operation and the theoretical state under ideal operating conditions. Quantum fidelity can be viewed as the degree of overlap between experimentally prepared quantum states and theoretical quantum states, which can be expressed in linear algebra as the square of the inner product between two state vectors, i.e. ,in, For theoretical quantum state The conjugate left vector, which is related to the experimentally prepared quantum state The inner product between them is expressed as This inner product can also be viewed as... The measurement results are as follows The probability amplitude. Quantum fidelity estimation techniques are widely used in various fields of quantum computing, primarily to evaluate the operational performance indicators of quantum devices and the accuracy of quantum operations.

[0022] This invention proposes a quantum fidelity estimation scheme based on the Quantum Resonance Transition algorithm (QRT). The task scenario is to estimate the fidelity of the ground state of a target Hamiltonian prepared by a Variable Quantum Eigenvalue Solver (VQE) circuit. VQE is a quantum-classical hybrid algorithm designed to efficiently compute the ground state energy of quantum systems (such as molecules). It combines the parallelism of quantum computing with the flexibility of classical optimization, making it particularly suitable for current noisy medium-scale quantum (NISQ) devices. VQE generates a trial quantum state through parameterized quantum circuits and measures the expected energy. Based on the measurement results, a classical optimizer is used to adjust the parameters to minimize the expected energy, thereby approximating the ground state energy. For the trial quantum state prepared by the VQE circuit, the direct fidelity estimation method requires a large number of repeated multi-qubit measurements to obtain the fidelity between it and the target ground state. Therefore, the QRT-based fidelity estimation scheme can achieve fidelity estimation with only single-qubit measurements, and the number of measurements is greatly reduced. Furthermore, without altering the core principles, fine-tuning the technical solution of this invention can also achieve fidelity estimation of any target pure state.

[0023] The steps based on the quantum resonance transition algorithm specifically include: coupling the working quantum system to a probe qubit with a specific frequency; setting the probe qubit to one of its energy eigenstates (e.g., an excited state); and placing the entire coupled system under a specifically constructed resonant Hamiltonian environment. After the entire coupled system has evolved for a period of time, a measurement is performed on the probe qubit. When the frequency of the probe qubit matches the transition frequency between the two energy levels of the quantum system, the probe qubit achieves resonance with the target quantum system, and resonance occurs. The decay rate of the probe qubit can be observed to reach its peak. Therefore, by changing the frequency of the probe qubit and measuring it, the transition frequency of the quantum system can be located. In this invention, the quantum resonance transition algorithm is used to characterize the fidelity of the final state prepared by the VQE task.

[0024] The proposed common application scenario in this embodiment is that, in quantum chemical simulation tasks on multi-qubit quantum computing devices, when researchers need to solve for the eigenstate of the simulated Hamiltonian's ground state, they often need the help of a variable quantum eigenvalue solver. This involves using a quantum circuit with a fixed structure but adjustable parameters (often the rotation angle of a single-qubit quantum gate) to simulate and prepare the ground state of the Hamiltonian. To obtain the fidelity between the prepared circuit state and the actual theoretical ground state, complex quantum measurements are often required. In this case, the resonance algorithm for fidelity estimation can achieve fidelity estimation using fewer measurement resources.

[0025] This embodiment provides a quantum fidelity estimation method based on the resonance algorithm, such as... Figure 1 As shown, the specific steps include the following: Step S100: Provide a target quantum system and a probe qubit whose fidelity is to be estimated, initialize the probe qubit to a preset energy eigenstate, and the target quantum system is an experimentally prepared quantum state.

[0026] In this embodiment, the target quantum system is the quantum system whose degree of overlap with the theoretical ideal state needs to be evaluated. It carries the quantum state information actually generated in the quantum computing task, specifically a multi-qubit experimental final state prepared by a variable quantum eigenvalue solver circuit, such as those used to approximate the molecular ground state in quantum chemical simulations. The fidelity of qubit quantum states determines whether the precision of quantum operations meets the task requirements. A probe qubit is an auxiliary qubit used to couple with the target quantum system to detect resonance signals. It indirectly reflects the energy level characteristics of the target system through its own state changes. Specifically, it can be a single qubit with two energy levels: a ground state and an excited state. Compared to directly measuring the target system, the single-qubit structure reduces measurement complexity. The preset energy eigenstate is the initial energy stationary state of the probe qubit. Its energy does not change over time, providing a stable starting point for subsequent coupling evolution. Specifically, it can be the ground state of the probe qubit. Choosing the ground state as the initial state avoids additional interference from the initial excited state, ensuring the accuracy of subsequent resonance detection. Experimentally prepared quantum states are quantum states actually generated on a quantum device through quantum circuits, distinct from theoretically derived ideal states. Specifically, it can be a trial quantum state output by a VQE after adjusting parameters such as rotation angle through parameterized circuits. This type of quantum state is the object of practical application in quantum computing tasks, and its difference from the theoretical state is an evaluation criterion for fidelity estimation.

[0027] The probe qubit is initialized to a preset energy eigenstate. A stable initial state can reduce interference factors in the evolution process and ensure that the subsequent state changes of the probe qubit are dominated only by the coupling and resonant Hamiltonian with the target system, so that the resonant signal can be accurately captured.

[0028] In one implementation, the provision of a target quantum system and a probe qubit with the desired fidelity, and the initialization of the probe qubit to a preset energy eigenstate, wherein the target quantum system is an experimentally prepared quantum state, specifically includes the following steps: Step S110: Obtain the target quantum system whose fidelity needs to be estimated, wherein the target quantum system is fabricated from a variable quantum eigenvalue solver circuit. Bit experiment final state ; Step S120: Provide an auxiliary probe qubit, the probe qubit comprising two energy levels: a ground state and an excited state; Step S130: Initialize the probe qubit to a preset energy eigenstate, wherein the preset energy eigenstate is the ground state of the probe qubit. .

[0029] In this embodiment, the specific implementation of step S100 is described in detail, aiming to clarify the specific source of the target quantum system, the structural characteristics of the probe qubit, and the specific selection basis for the initialization operation.

[0030] First, regarding the target quantum system for obtaining the fidelity to be estimated, considering the application scenarios of current noisy medium-scale quantum (NISQ) devices, the target quantum system selected in this embodiment is specifically a variable quantum eigenfunction solver (VQE) circuit. Bit experiment final state As a quantum-classical hybrid algorithm, VQE outputs trial quantum states, which are core objects in tasks such as quantum chemical simulations and solving quantum many-body problems. Accurate estimation of the fidelity of these quantum states is crucial to the reliability of results in related quantum computing tasks, thus having clear practical application requirements. Furthermore, VQE circuits generate trial states through parameterized quantum gates, and their output... Bit experiment final state It belongs to the quantum state prepared experimentally.

[0031] Secondly, the auxiliary probe qubit must possess two energy levels: a ground state and an excited state. From the perspective of quantum mechanics, a two-level structure can stably generate Rabi oscillations. In the subsequent evolution of the coupled system, the driving field term needs to control the probe qubit to repeatedly transition between the two energy levels to generate an observable oscillating signal. If the probe qubit has more energy levels, it will introduce additional transition channels, causing the oscillation signal to be mixed with irrelevant noise, interfering with the determination of the resonance condition and the accurate measurement of the excited state probability. Therefore, choosing a two-level probe qubit containing only the ground state and the excited state can simplify the detection process to the greatest extent, reduce the difficulty of signal analysis, and ensure the purity of the subsequent resonance signal.

[0032] Finally, the probe qubit is initialized to its ground state. The purpose of using the preset energy eigenstate is to avoid additional interference from the initial excited state. The ground state of the probe qubit is the lowest energy stable state, which will not spontaneously exchange energy or become quantumly entangled with the target quantum system at the initial moment. This ensures that the initial state of the coupled system is strictly the ground state of the probe qubit. With the experimental state of the target system The product state. If initialized as an excited state, the probe qubit may spontaneously emit before coupling, or generate non-resonant interactions with the target system, causing the initial state to deviate from the preset conditions, affecting the accuracy of the resonance signal in the subsequent evolution process, and failing to accurately reflect the degree of overlap between the target system and the theoretical state.

[0033] Step S200: Couple the target quantum system with the probe qubit to form a coupled system.

[0034] In this embodiment, coupling is the operation that enables the target quantum system and the probe qubit to interact, allowing their quantum states to evolve co-evolve and information to be exchanged. This association is achieved through quantum mechanical interaction terms, specifically by constructing a Hamiltonian that incorporates the interaction between the two. This interaction allows a direct correlation between the oscillation state of the probe qubit and the energy level information of the target system, making the probe qubit a probe for sensing the characteristics of the target system. The coupled system is the overall quantum system formed by the coupling of the target quantum system and the probe qubit. Its quantum state is the composite state of the initial product states of the two subsystems after their interaction. Specifically, it can be a combination of... A composite system consisting of a 1-bit target quantum system and a 1-bit probe quantum bit has a unified evolution law. Under the control of a preset resonant Hamiltonian, the energy level information of the target system can be gradually encoded into the state of the probe quantum bit.

[0035] Through coupling operations, two originally independent systems form a co-evolving whole, enabling indirect detection. The construction of the coupled system allows the probe qubit to act as an intermediary, transforming the high-dimensional information of the target system into its own single-qubit state changes, thus solving the problem of high resource consumption for multi-qubit measurements in traditional direct measurement schemes.

[0036] Step S300: Control the coupling system to evolve for a preset time under a preset resonant Hamiltonian.

[0037] In this embodiment, the preset resonance Hamiltonian is the physical quantity that dominates the time-varying quantum state of the coupled system. It contains the main action terms required to achieve resonance detection and precisely controls the evolution direction of the coupled system. Specifically, it can be a Hamiltonian composed of a Zeeman splitting term, a resonance condition term, and a driving field term. The Zeeman splitting term determines the two-level structure of the probe qubit, providing a basis for subsequent oscillations; the resonance condition term correlates the target Hamiltonian with the excited state of the probe qubit, determining whether resonance occurs; and the driving field term controls the probe qubit to perform Rabi oscillations between energy levels, generating observable state change signals. The three terms work together to construct the necessary evolutionary environment for resonance detection. Evolution is the process by which the quantum state of the coupled system gradually changes over time under the influence of the preset resonance Hamiltonian. Specifically, it can be the process by which the system gradually transitions from the initial product state of the probe qubit's ground state and the experimental state of the target system to a superposition state or entangled state containing information about each energy level of the target system. During this process, the oscillation state of the probe qubit changes regularly according to the energy level characteristics of the target system. The preset time is the evolution duration of the coupling system set to ensure detection effectiveness. It is calculated based on the driving field parameters and can specifically be the time required to ensure the oscillation phase of the detection qubit reaches a certain value. At this time, the probability amplitude of the detected qubit being in an excited state reaches its maximum value, minimizing signal errors during measurement and improving the accuracy of probability measurement.

[0038] The coordinated action of multiple terms of the preset resonant Hamiltonian ensures stable oscillation of the probe qubit, and the resonance condition term enables the screening of specific energy levels of the target system, avoiding interference from irrelevant energy level information. The preset time setting ensures that the measurement timing falls at the peak of the probability amplitude, solving the problem of insufficient measurement accuracy caused by weak signals.

[0039] In one implementation, the resonant Hamiltonian includes a Zeeman splitting term, a resonance condition term, and a driving field term. The Zeeman splitting term is used to determine the energy level structure of the probe qubit, and the resonance condition term is used to correlate the target Hamiltonian with the excited state of the probe qubit. The resonance is controlled by coupling and frequency sweeping parameters, and the driving field term is used to control the probe qubit to perform Rabi oscillation.

[0040] In this embodiment, the coupled system is first allowed to evolve under the resonant Hamiltonian described by the QRT principle, denoted as the system Hamiltonian. It consists of a Zeeman splitting term, a resonance condition term, and a driving field term. The Zeeman splitting term determines the energy level structure of the probe qubit, and the resonance condition term determines the target Hamiltonian of the VQE mission. With probe bits The state (i.e., the self-excited state) is coupled and connected through parameters. The choice of [the term] determines whether the system resonates and the object of resonance. Due to the existence of the driving field term, the probe bit will continuously undergo Rabi oscillations as the system evolves over time, and its transition to [the desired state]... The probability amplitude of the state will oscillate continuously over time.

[0041] In a typical In a fermion system with 1 / 2 spin of a qubit, a resonant Hamiltonian is designed to couple the target quantum system with a probe qubit, as shown in Equations 1 and 2.

[0042] Formula 1:

[0043] Formula 2:

[0044] in, The symbol for matrix direct product. and Represents the Pauli matrix. for The identity matrix is ​​expressed as shown in Equation 3.

[0045] Formula 3:

[0046] for The target Hamiltonian on a qubit-oriented quantum system. and Represent The eigenlevels and eigenstates are themselves composed of the ground state. and It is composed of the linear superposition of the density matrices of the excited states. and The parameters of the driving field are generally taken as follows: for , This is a local minimum value, representing the coupling strength between the probe qubit and the target quantum system. These represent the adjustable parameters of the algorithm. The expression indicates that the entire system consists of a probe bit and It is formed by coupling the working bit system.

[0047] In the Hamiltonian described by Equation 1, the first term is... The controlled Zeeman term dominates the energy level of the probe qubit, producing (Detecting the ground state of a bit) and (Detecting the excited state of the qubit) Two-level splitting. The last term is... The dominant driving field term controls the spin direction of the probe bit, causing it to repeatedly oscillate between the split energy levels using Rabi oscillations, thereby generating an oscillating signal, which manifests as... The probability amplitude of the state oscillates over time. The second and third terms act on the probe bit. The state subspace together constitutes the resonance condition term, and the second term serves as the modulation parameter condition. This determines whether a resonant transition occurs; the third term targets the Hamiltonian. Encoded to State subspace. The second and third terms combine to act on the probe bit. The state subspace ensures that only states satisfying the resonance condition (and) The intrinsic energy matching, i.e. Only by specifying the parameters can the probe bits be ensured. State and target quantum systems The states are uniquely entangled and resonate. In one implementation, controlling the coupling system to evolve for a preset time under a preset resonant Hamiltonian specifically includes the following steps: Step S310: Set the evolution time to... ,in, and For the parameters of the driving field, , It is a local minimum; Step S320: Control the coupling system to evolve the preset time under the action of the preset resonant Hamiltonian. .

[0048] In this embodiment, the specific implementation of step S300 is explained, with the core being to clarify the specific value of the evolution time, the meaning of the parameters, and the basis for their selection.

[0049] First, in this embodiment, the specific value of the preset evolution time is set as follows: ,in and These are all parameters of the driving field. From the parameter definitions, The value is fixed as This is directly related to the role of the driving field term, which functions to control the probe qubit to generate stable Rabi oscillations between the ground and excited states. This ensures that the rotational amplitude of the driving field matches the transition characteristics of the two-level system; otherwise, it will lead to periodic disruption of the Rabi oscillation, making it impossible to form a regular signal waveform, and subsequently making it difficult to determine the position of the probability peak through phase. Secondly, The value of is set to a minimum, which physically represents the coupling strength between the probe qubit and the target quantum system. The purpose of choosing a minimum value is to achieve weak coupling. In a weakly coupled state, the quantum state of the target quantum system will not be destroyed by its interaction with the probe qubit, and its overlap with the theoretical state can be fully preserved. Otherwise, strong entanglement would occur, and the state of the target quantum system would be disturbed by the probe qubit, failing to accurately reflect the fidelity between the target system and the theoretical state. Furthermore, the choice... The purpose of using evolution time is to ensure that the Rabi oscillation phase of the probe qubit reaches exactly the desired value. From the perspective of the oscillation laws in quantum mechanics, the probability amplitude of detecting a qubit in an excited state varies sinusoidally with time, when the phase is... When the sine function reaches its maximum value of 1, the probability amplitude of the excited state also reaches its peak. Measuring at this moment can maximize the observed value of the excited state probability and reduce measurement errors caused by weak signals.

[0050] Furthermore, the aforementioned timing settings are adapted to the characteristics of medium-scale quantum devices with noise, as NISQ devices have shorter quantum state decoherence times. When it is a minimum value, The product of is also correspondingly smaller, making The value is extremely large. Choose the smallest one. The aim is to reduce the full width at half maximum (FWHM) of the resonance peak corresponding to the target energy level and reduce the crosstalk effects caused by the resonance peaks of other energy levels.

[0051] Step S400: Measure the probability that the probe qubit is in a preset excited state.

[0052] In this embodiment, the measurement involves observing the quantum state of the probe qubit to obtain its state percentage. This operation is performed on a single qubit only, specifically through projection measurement of the probe qubit in the excited-state subspace. The measurement method does not require multi-qubit manipulation of the target system; it is achieved using a simple single-qubit observation device, reducing the difficulty and resource consumption of the experimental operation. The preset excited state is an energy eigenstate of the probe qubit with an energy higher than the ground state. It represents the state to which the probe qubit may transition under the influence of the driving field. Specifically, it can be the excited state of the probe qubit... The occupancy of a state is related to the resonance degree between the probe qubit and the target system. When the two resonate, the probability of the probe qubit transitioning to that state increases significantly. Probability is the likelihood that the probe qubit will be in a predetermined excited state after evolution. It is a quantitative indicator obtained through statistical measurement results. Specifically, it can be achieved by repeatedly measuring the state of the probe qubit and counting the number of times it is in the predetermined excited state out of the total number of measurements. This proportion has a definite correlation with the degree of overlap between the target quantum system and the corresponding theoretical state, and serves as intermediate data for obtaining fidelity.

[0053] Traditional fidelity estimation schemes require numerous multi-qubit measurements of the target system, which can lead to exponential resource consumption, especially in high-qubit scenarios. This embodiment, however, performs measurements only on the probe qubit, thus solving the resource consumption problem. Furthermore, single-qubit measurements are simple to perform, have controllable errors, and can quickly acquire stable probability data, avoiding the error accumulation caused by inter-qubit crosstalk in multi-qubit measurements.

[0054] In one implementation, measuring the probability that the probe qubit is in a preset excited state specifically includes the following steps: Step S410: Excite the probe qubit. A single-bit measurement was performed to obtain the transition of the probe qubit to an excited state. probability Among them, evolution time Fix the oscillation phase to be , making the excited state The probability amplitude has reached its maximum value.

[0055] In this embodiment, a specific evolution time is selected. The probe bits of the coupling system are State measurement, time Fix the oscillation phase to be ,at this time The state probability amplitude reaches its maximum value. After measuring the probe bit, its transition to... probability of state .

[0056] Specifically, the initial quantum state of the algorithm's input can be decomposed into the experimental state of the working bits. and the probe bit are in The entire system is currently in the product state of the quantum states of the probe bit and the working bit, i.e., written as... Starting from this initial state, the system Hamiltonian in Equation 1... Evolution time under controlled evolution Its final state evolution expression is shown in Equation 4.

[0057] Formula 4:

[0058] The first item is the probe bit in Components of the state subspace This represents the probability amplitude of each component. and They respectively represent the descriptions in Equation 2 The Each eigenlevel and eigenstate. The remaining components are represented by... To represent this. According to the first-order approximation, the probe bit measurement is in an excited state. The probability expression is shown in Equation 5.

[0059] Formula 5:

[0060] As can be seen from Equation 5, the probe bit state simultaneously with The eigenstates corresponding to all energy levels are entangled. for The probability amplitude corresponding to the Rabi oscillation of each eigenlevel is The superposition of state subspaces, whose magnitudes are related to their respective... Related. If by The parameters represented are modulated to the same level as... a certain eigenlevel Matching (i.e.) Its Rabi oscillation amplitude can reach the same time. Maximum value below For other items where the parameters do not match (i.e.) ),because It is a minimum value. and The extremely small difference will cause this term to approach 0 infinitely, and therefore can be ignored; only the smallest difference is retained. One item.

[0061] Step S500: Adjust the frequency sweep parameter in the resonant Hamiltonian and repeat the evolution and measurement. When the frequency sweep parameter matches the target energy level corresponding to the target quantum system, the probability of the excited state reaches a peak value. The value of the peak value is the fidelity of the quantum state corresponding to the target quantum system and the target energy level.

[0062] In this embodiment, the frequency sweep parameter is an adjustable parameter in the resonant Hamiltonian used to match different energy levels of the target quantum system. By changing its value, it scans the energy level range of the target system. Specifically, it can be a parameter in the resonance condition term used to modulate whether resonance occurs. Its value determines whether the probe qubit can resonate with a certain energy level of the target system, and it is an adjustable variable used to achieve multi-level fidelity estimation. The target energy level is the theoretical energy level corresponding to the target quantum system, and it is an eigenvalue of the target Hamiltonian. Specifically, it can be the ground state energy level or excited state energy level of the target Hamiltonian. Each target energy level corresponds to a specific theoretical quantum state. The quantum state of the target system is a superposition of these corresponding states. Probing different target energy levels can achieve fidelity estimation of multiple theoretical states. The peak value is the maximum probability of the probe qubit being in a preset excited state during the adjustment of the frequency sweep parameters. It is a direct indicator of resonance. Specifically, it can be the obvious bulge in the probability curve when the frequency sweep parameters are perfectly matched with a target energy level. The appearance of this bulge indicates that the probe qubit and the corresponding energy level of the target system have resonated, and the probability value at this point has specific physical meaning. Fidelity is a quantitative indicator of the degree of overlap between the experimentally prepared target quantum system and the theoretical quantum state, reflecting the closeness between the experimental state and the ideal state. Specifically, it can be the square of the inner product of two quantum state vectors. When the frequency sweep parameters are matched with the target energy level, the peak value of the probability of the probe qubit being in an excited state is equal to the fidelity between the target system and the corresponding quantum state of that energy level.

[0063] On the one hand, the adjustability of the sweep parameters allows the method provided by this invention to adapt to multiple energy levels of the target system, estimating not only the ground state fidelity but also the excited state fidelity, thus expanding the method's applicability. On the other hand, the correspondence between peak values ​​and fidelity means that obtaining fidelity does not require prior knowledge of the target system's eigenstates or complex quantum state tomography calculations; it can be achieved simply by observing the peak value of the probability curve. This design solves the technical problems of traditional schemes requiring prior knowledge of the target eigenstates and complex calculations.

[0064] In one implementation, the frequency sweep parameters are matched with the target energy level corresponding to the target quantum system, specifically including the following steps: Step S510: The frequency sweep parameters match the eigenlevels of the target Hamiltonian, satisfying the resonance condition in the resonance condition term; Step S520, the probe bit The state is only the eigenstate corresponding to the eigenlevel of the target Hamiltonian. They become entangled and continue to resonate during their evolution.

[0065] In this embodiment, the ground state energy level matching case (i.e.) Taking evolution time as an example, for At that time, its probe bit measurement is in an excited state. The expression for the probability is shown in Equation 6.

[0066] Formula 6:

[0067] This is the experimental state. ground state of the target Hamiltonian The fidelity between them.

[0068] In one implementation, the probability of the excited state exhibits a peak value, the value of which represents the fidelity between the target quantum system and the quantum state corresponding to the target energy level. This specifically includes the following steps: Step S530: Obtain all measured probabilities The plotting yielded several discrete peaks, and the corresponding probabilities were obtained. The value is prepared by the variable quantum eigenvalue solver circuit. Bit experiment final state eigenstates of the target Hamiltonian fidelity between ; Step S540: Select the first peak among the plurality of discrete peaks to obtain the fidelity of the ground state of the target quantum system.

[0069] In this embodiment, adjustment Parameters of the probe bits and repeat the probability Measurement. When The choice of the target Hamiltonian for the VQE task eigenlevels When a match is found (i.e., the resonance condition in the resonance condition term is satisfied), the probe bit's The state will only be with Corresponding eigenstates Once entangled, the two will continue to resonate in subsequent evolution. This will change... Repeated measurements The values ​​are plotted on a chart, which will show multiple discrete peaks, all located in... Each point, corresponding The value is and eigenstates fidelity between Meanwhile, the first peak on the graph represents the fidelity of the ground state. .

[0070] when When matching different eigenlevels, the probe bits can be measured in the same way. The probability is obtained The fidelity to the corresponding eigenstate. Additionally, if it is necessary to estimate the experimental state... With a certain target pure state To maintain the fidelity between them, we only need to adjust the formula in equation 5. Replace with density matrix Based on the above principles, the evolution time can be obtained. Post-probe bit measurement is excited state The expression for the probability is shown in Equation 7.

[0071]

[0072] make Evolutionary time for Then for The fidelity between the two can be obtained by measurement. Therefore, it can be achieved by measuring a single bit of the probe bit. and Fidelity estimation between them.

[0073] For example, using a two-qubit Ising transverse-field Hamiltonian system as an example, this demonstrates how to use the resonance algorithm to estimate the fidelity between it and a quantum state generated by a random quantum circuit. Target Hamiltonian The Hamiltonian is chosen as the transverse field, and its expression is shown in Equation 8.

[0074] Formula 8

[0075] in, and Representing the coupling strength and transverse field strength, respectively, the ground state energy can be obtained by directly diagonalizing the Hamiltonian. and the ground state is Next, a VQE prepared using a random circuit is used to test the quantum state. By estimating its relationship with the ground state This demonstrates the operation of the QRT algorithm in estimating quantum fidelity. The algorithm's quantum circuitry operates in a coupled system consisting of one probe qubit and two working qubits, with the circuit structure as follows: Figure 2 As shown. Figure 2 The circuit for estimating the fidelity of a 2-bit VQE experimental state is shown, in which, This represents a vector containing multiple parameters. The evolution of a parameterized random quantum circuit, where a two-bit working qubit system undergoes this evolution to obtain the experimental state. Then let it be in the position A working bit system in a certain state and a state in which The probe bits are coupled in a state, allowing them to be in the resonance algorithm circuit. The following evolution occurs, in which This is the Hamiltonian of the system described by Formula 5.

[0076] When parameter and target Hamiltonian ground state energy When they are equal, the Rabi oscillation amplitude of the probe bit over time reaches its maximum value, i.e., the experimental state. With ground state The probability amplitude between .let Evolutionary time Set as At this point, according to the function in Equation 6, the probability that the probe bit is in an excited state is... Reaching the maximum value is Therefore, the probability can be obtained through a simple single-bit measurement. This enables the control of experimental states. With the target ground state The fidelity of the readings.

[0077] Figure 3 Simulation results demonstrating the ground-state fidelity of the 2-bit Ising transverse Hamiltonian estimated using the resonance algorithm are presented. Let Given an arbitrary VQE circuit state, its relationship with the target ground state is... True Fidelity Value The markings are at the red solid line. The blue solid line represents the coupled system. As it evolved, the probe bit measurement became probability The red dashed line representing the change over time is shown in Equation 6. The function fitting curve approximation over time. For example... Figure 3 As shown, the vertical axis represents the probability that the measurement result of the corresponding probe bit is an excited state. The horizontal axis represents the parameter. The selection of parameters. and The corresponding Hamiltonian ground state energy level is At the same time, let those in The evolution time of the system state under the Hamiltonian of the resonant system ,make Reaching the maximum. Using a trial quantum state with a random circuit. To estimate its relationship with the target Hamiltonian ground state The fidelity between them, according to theoretical calculations, is as follows: This refers to the position of the vertical axis marked by the red solid line. By selecting different... and measure available Figure 3 The blue curve represents the first-order approximation curve, while the red dashed line represents the fitted curve. It can be seen that when... and During matching, the following will occur A peak value, and the peak point of both the actual measured curve (blue solid line) and the first-order approximate fitted line (red dashed line) is located at... And corresponding The value is equal to the actual fidelity (the peak point falls on the red solid line), indicating that the resonance algorithm can effectively estimate the fidelity of the ground state.

[0078] In summary, this invention provides a quantum fidelity estimation method based on the resonance algorithm. By utilizing the principle of the resonance transition algorithm, it is possible to estimate the fidelity between the eigenstates and the experimental state of a Hamiltonian without preparing the quantum eigenstates or even without knowledge of them. This effectively saves measurement resources for quantum devices and, in the case of high qubit counts, avoids the exponential resource consumption of high-dimensional quantum system tomography, requiring only the additional quantum resources of one auxiliary probe qubit. Furthermore, to extract the fidelity information, only the probe qubit needs to be used in... The measurement can be performed by projecting the state subspace, which also has a significant advantage in terms of the difficulty of implementation.

[0079] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0080] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. A quantum fidelity estimation method based on the resonance algorithm, characterized in that, The method includes: Provide a target quantum system whose fidelity is to be estimated and a probe qubit, initialize the probe qubit to a preset energy eigenstate, and the target quantum system is an experimentally prepared quantum state; The target quantum system is coupled with the probe qubit to form a coupled system; The coupling system is controlled to evolve for a preset time under a preset resonant Hamiltonian; Measure the probability that the probe qubit is in a preset excited state; The frequency sweep parameter in the resonant Hamiltonian is adjusted and the evolution and measurement are repeated. When the frequency sweep parameter matches the target energy level corresponding to the target quantum system, the probability of the excited state reaches a peak value. The value of the peak value is the fidelity of the quantum state corresponding to the target quantum system and the target energy level.

2. The quantum fidelity estimation method based on the resonance algorithm according to claim 1, characterized in that, The target quantum system and probe qubit, which provide the fidelity to be estimated, are initialized to a preset energy eigenstate. The target quantum system is an experimentally prepared quantum state, including: To obtain the target quantum system whose fidelity needs to be estimated, wherein the target quantum system is fabricated as a variational quantum eigenvalue solver circuit. Bit experiment final state ; An auxiliary probe qubit is provided, which comprises two energy levels: a ground state and an excited state. The probe qubit is initialized to a preset energy eigenstate, which is the ground state of the probe qubit. .

3. The quantum fidelity estimation method based on the resonance algorithm according to claim 2, characterized in that, The resonant Hamiltonian includes a Zeeman splitting term, a resonance condition term, and a driving field term. The Zeeman splitting term is used to determine the energy level structure of the probe qubit, and the resonance condition term is used to correlate the target Hamiltonian with the excited state of the probe qubit. The resonance is controlled by coupling and frequency sweeping parameters, and the driving field term is used to control the probe qubit to perform Rabi oscillation.

4. The quantum fidelity estimation method based on the resonance algorithm according to claim 3, characterized in that, The control of the coupling system to evolve for a preset time under a preset resonant Hamiltonian includes: Set the evolution time to ,in, and For the parameters of the driving field, , It is a local minimum; The coupling system is controlled to evolve over the preset time under the influence of the preset resonant Hamiltonian. .

5. The quantum fidelity estimation method based on the resonance algorithm according to claim 4, characterized in that, Measuring the probability that the probe qubit is in a preset excited state includes: Excite the probe qubit A single-bit measurement was performed to obtain the transition of the probe qubit to an excited state. probability Among them, evolution time Fix the oscillation phase to be , making the excited state The probability amplitude has reached its maximum value.

6. The quantum fidelity estimation method based on the resonance algorithm according to claim 5, characterized in that, The frequency sweep parameters are matched with the target energy level corresponding to the target quantum system, including: The frequency sweep parameters match the eigenlevels of the target Hamiltonian, satisfying the resonance condition in the resonance condition term. The probe bits The state is only the eigenstate corresponding to the eigenlevel of the target Hamiltonian. They become entangled and continue to resonate during their evolution.

7. The quantum fidelity estimation method based on the resonance algorithm according to claim 6, characterized in that, The probability of the excited state reaching a peak value, the value of which is the fidelity between the target quantum system and the quantum state corresponding to the target energy level, includes: The probability obtained from all measurements The plotting yielded several discrete peaks, and the corresponding probabilities were obtained. The value is prepared by the variable quantum eigenvalue solver circuit. Bit experiment final state eigenstates of the target Hamiltonian fidelity between ; The fidelity of the ground state of the target quantum system is obtained by selecting the first peak among the discrete peaks.

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