A method for constructing a double quantum bit controlled gate based on an ensemble of rare earth ions

Abstract

The application discloses a kind of based on ensemble rare earth ions double quantum bit controlled gate construction method, it is related to quantum control technical field, it includes to double quantum bit frequency site selection, and respectively as control bit, target bit;Utilize the blocking effect generated by the dipole-dipole interaction between double quantum bits to construct controlled gate, including: to control bit apply light pulse driving control bit is from ground state transition to excited state;To target bit apply single quantum gate to drive its evolution, and apply compensation pulse;To control bit apply light pulse driving control bit from excited state de-excitation to ground state.The application utilizes the interaction of the light pulse and quantum system medium that can be optimized to produce arbitrary geometric quantum logic gate on target quantum bit, and simultaneously by using the blocking effect caused by dipole-dipole interaction, the evolution of target bit quantum bit can be constrained by control bit.

Description

Technical Field

[0001] This invention relates to a method for constructing a controlled gate for two qubits based on ensemble rare-earth ions, belonging to the field of quantum manipulation technology. Background Technology

[0002] Quantum computing primarily follows the laws of quantum mechanics and quantum dynamics. Compared to classical computing, it boasts significantly faster computation speeds and can solve many problems that are difficult to address in classical computing. Therefore, it has wide applications in quantum neural network simulation, artificial intelligence, large prime factorization, and unordered database retrieval. In recent years, the development of high-fidelity single-qubit logic gates has progressed rapidly, and how to expand the number of qubits to achieve fast, high-fidelity two-qubit gates has also become a research hotspot.

[0003] In the field of two-qubit systems, the Rydberg atom system, due to its Rydberg blocking effect, is considered the ideal choice for constructing two-qubit and multi-qubit gates. Rare-earth ions, when doped into solid crystals, can also generate strong dipole-dipole interactions. This effect is very similar to Rydberg blocking, thus allowing the construction of controlled two-qubit gates in ensemble rare-earth ion systems. Currently, the methods for constructing two-qubit gates using the blocking effect are broadly classified into two categories based on the blocking strength: weak blocking gates and strong blocking gates. Weak blocking gates typically refer to cases where the dipole-dipole interaction is less than the Rabi frequency of the optical pulse. In this case, qubit addressing is relatively convenient, but the resulting gate fidelity is generally low. Strong blocking, on the other hand, involves dipole-dipole interactions much greater than the Rabi frequency of the optical pulse. The gate fidelity is generally high, but qubit addressing is more difficult. Strong blocking gates can be further divided into asynchronous driving gates and synchronous driving gates based on their evolution. Synchronous driving gates based on strong blocking mainly rely on non-resonant driving and collective excitation effects, requiring precise pulse driving, which is experimentally challenging and prone to introducing additional errors. The asynchronous drive gate based on the strong blocking effect relies on fast first-order dynamics. The two qubits that need to be addressed independently are driven step by step by the laser field, and the two qubit gates can be implemented relatively easily in experiments.

[0004] Furthermore, since the selected system is an imperfect system, when driving two qubits, a single quantum manipulation must not only be robust to the frequency detuning caused by non-uniform broadening in the ensemble qubit, but also have a strong suppression effect on other ions near the addressing frequency of the qubit ion, especially the non-resonant excitation of the other qubit being addressed.

[0005] Using a rare earth ion system Eu with a doping concentration of 0.05% 3+Taking Y₂SiO₅ as an example, the qubits are composed of a set of ensemble Eu ions. The center frequency of the optical transitions between these ensemble ions is 500 THz (579.88 nm), and the full width at half maximum (FWHM) of the optical absorption peak is about 170 kHz. The energy levels of the two qubits are... and The coupling between the two qubits is achieved through optical transitions between each qubit and an excited state. In such a system, to create a two-qubit gate quickly and with high fidelity, several requirements must be met: (1) the frequency difference between the two independently addressable qubits must be sufficiently large; (2) the non-resonant excitation of other ions beyond approximately 8.9 MHz of the ensemble qubit ion by the pulse must be sufficiently small; (3) the Rabi frequency of the driving optical pulse must be much smaller than the dipole-dipole interaction between the two qubits, satisfying the blocking condition; and (4) the optical pulse must be able to [interact with] the qubits within a short time. Ensemble qubits within the 170 kHz frequency detuning range are manipulated equally, meaning that the manipulation fidelity in this range is close to 1.

[0006] Currently, high-fidelity quantum logic gate manipulation has been achieved in ensemble rare-earth ion qubit systems by designing and optimizing optical pulses with multiple degrees of freedom. However, this work has not been extended to the two-qubit dimension. Therefore, achieving high-fidelity two-qubit manipulation in ensemble rare-earth ion qubit systems is a pressing issue. Summary of the Invention

[0007] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for constructing controlled gates of two qubits based on ensemble rare earth ions. This method utilizes optimizable two-color light pulses and the interaction between the quantum system medium to generate arbitrary geometric quantum logic gates on the target qubit. At the same time, by using the blocking effect caused by dipole-dipole interactions, the evolution of the target qubit can be constrained by the control bit.

[0008] To achieve the above objectives, the present invention is implemented using the following technical solution:

[0009] On one hand, this invention provides a method for constructing a controlled two-qubit gate based on an ensemble of rare-earth ions, comprising:

[0010] Frequency addressing is performed on a two-qubit system, where one qubit is used as a control bit and the other qubit is used as a target bit.

[0011] Controlled gates are constructed using the blocking effect generated by the dipole-dipole interaction between two qubits, including:

[0012] A light pulse is applied to the control bit, driving it to transition from the ground state to the excited state.

[0013] A single quantum gate is applied to the target bit to drive its evolution, and then a compensation pulse is applied to the target bit.

[0014] A light pulse is applied to the control bit to drive it to de-excite from the excited state to the ground state.

[0015] Furthermore, the frequency addressing satisfies the following conditions:

[0016] The frequency separation of two qubits satisfies the condition that driving one qubit does not directly affect the other qubit.

[0017] The dipole-dipole interaction between two qubits exceeds the Rabi frequency of the optical pulse applied to the control bit, the optical pulse involved in a single quantum gate, and the compensation pulse by at least four times.

[0018] Furthermore, the light pulse applied to the control bit is a monochromatic pulse, while the light pulse applied to the target bit is a dual-color Rabi pulse;

[0019] The monochromatic pulse and the dual-color Rabi pulse are both obtained by converting the amplitude of the corresponding radio signal. The method for obtaining the amplitude of the corresponding radio signal includes:

[0020] Set the Rabi frequency of the monochromatic pulse and the equivalent Rabi frequency of the dual-color Rabi pulse;

[0021] The Rabi frequency of the dual-color Rabi pulse is calculated based on the equivalent Rabi frequency of the dual-color Rabi pulse.

[0022] The amplitude of the radio signal corresponding to the dual-color Rabi pulse is calculated based on the Rabi frequency of the dual-color Rabi pulse.

[0023] The amplitude of the radio signal corresponding to the monochromatic pulse is calculated based on the Rabi frequency of the monochromatic pulse.

[0024] Furthermore, the Rabi frequency of the monochromatic pulse and the equivalent Rabi frequency of the dual-color Rabi pulse are set according to the following expression:

[0025] ;

[0026] in, This represents the Rabi frequency of a monochromatic pulse. This represents the equivalent Rabi frequency of the two-color Rabi pulse. Indicates time, Indicates the duration of a single pulse segment. Indicates the phase of a monochromatic pulse. , Both represent the phase of the dual-pulse. express and The relative phase difference between them This represents the geometric phase that needs to be accumulated to construct a two-quantum gate;

[0027] The expression for calculating the pull ratio frequency of the dual-pulse based on the equivalent pull ratio frequency of the dual-pulse is as follows:

[0028] ;

[0029] ;

[0030] ;

[0031] in, This represents the equivalent Rabi frequency of the two-color Rabi pulse. , Both represent the pull ratio frequency of the two-color pull ratio pulse. This represents the normalized amplitude coefficient of the dual-Seraphim pulse. Represents the imaginary unit. , Both represent the phase of the dual-Syracbean pulse applied to the target position;

[0032] The expression for calculating the radio signal amplitude corresponding to the dual-wavelength Rabi pulse based on the Rabi frequency of the dual-wavelength Rabi pulse is as follows:

[0033]

[0034]

[0035] in, , This represents the radio signal amplitude of an arbitrary generator that applies a dual-Serabi pulse to the target position. Denotes the reduced Planck constant. The conversion factor representing the Rabi frequency of a two-color Rabi pulse to the amplitude of a radio signal. Indicates the target position is from the ground state The optical transition dipole moment to the excited state Indicates the target position is from the ground state The optical transition dipole moment that transitions to the excited state;

[0036] The expression for calculating the radio signal amplitude corresponding to the monochromatic pulse based on the Rabi frequency of the monochromatic pulse is as follows:

[0037] ;

[0038] in, The radio signal amplitude representing an arbitrary generator of monochromatic pulses. This represents the optical transition dipole moment of the control state as it transitions from the ground state to the excited state.

[0039] Furthermore, both the monochromatic pulse and the dual-color Rabi pulse involved in the single quantum gate are obtained by superimposing a series of Fourier cosine components, and their calculation expression is as follows:

[0040] ;

[0041] ;

[0042] in, This represents the Rabi frequency of a monochromatic pulse. This represents the equivalent Rabi frequency of the two-color Rabi pulse. Indicates the duration of a single pulse segment. Indicates the first Each pulse degree of freedom Indicates time.

[0043] Furthermore, the aforementioned The even-numbered and odd-numbered terms satisfy the following expression:

[0044] ;

[0045] ;

[0046] in, Indicates an index variable.

[0047] Furthermore, the aforementioned The value of is determined by a multi-objective optimization genetic algorithm, the objective function of which is:

[0048] ;

[0049] ;

[0050] in, Denotes the first objective function. This represents the second objective function. Indicates the gate fidelity at the final moment. Represents all non-initial states The sum of the layout numbers, This indicates other energy levels that can be reached by population leakage.

[0051] Furthermore, when the two quantum gates are CNOT gates, At that time, the light pulse applied to the control bit The values ​​are respectively , , , The light pulse applied to the target position The values ​​are respectively , , , .

[0052] Furthermore, the compensation pulse is reset , , We obtained, among which, This represents the normalized amplitude coefficient of the dual-Seraphim pulse. This represents the accumulated geometric phase required to construct a two-quantum gate. This indicates the phase difference between the two-color pulses. , ;

[0053] The expression for calculating the equivalent Rabi frequency after superimposing a compensation pulse onto a dual-color Rabi pulse is as follows:

[0054] ;

[0055] in, This represents the equivalent Rabi frequency of the two-color Rabi pulse. Indicates the duration of a single pulse segment. Indicates the first Each pulse degree of freedom Indicates time.

[0056] On the other hand, the present invention also provides a high-fidelity two-qubit controlled gate based on ensemble rare-earth ions, which is obtained by the two-qubit controlled gate construction method based on ensemble rare-earth ions as described in any of the above claims.

[0057] Compared with the prior art, the beneficial effects achieved by the present invention are as follows:

[0058] This invention creates arbitrary two-qubit geometric logic gates in an ensemble rare-earth ion qubit system using optimizable two-color light pulses. The controlled two-qubit logic gate is characterized by the evolution of the target bit being influenced by the control bit qubit; when the control bit qubit is located... When the target bit is in the dipole-dipole state, due to the dipole-dipole interaction, the excited state energy level of the target bit will be raised, causing the target bit qubit to be locked and not to evolve. When the control bit qubit is in the... In the initial state of the target bit, it will be normally subjected to an arbitrary quantum logic gate. An arbitrary quantum logic gate is defined for any initial state of the target bit. Light pulses can be made to orbit around an arbitrary n-axis (by angle). and To define), rotate at any angle ( ), which becomes the final target state. The target bit logic gate is constructed in The computational subspace contains a unitary operator consisting of these three parameters. To express, and ,in, For parameters The function expression; The duration of the pulse.

[0059] Ensemble rare-earth ion quantum bit system For example, under certain conditions, the method according to the present invention can have the following characteristics: the pulse Rabi frequency is small, not exceeding 3 MHz; the fidelity of the generated logic gate is not less than 99%; it is robust to frequency detuning in the quantum system within a range of at least ±170 kHz; and it does not cause excitation to other ions located outside the center frequency of 8.9 MHz of the quantum bit ion.

[0060] This invention utilizes the interaction between optimizable two-color light pulses and the quantum system medium to generate arbitrary geometric quantum logic gates on target qubits. Simultaneously, by employing the blocking effect caused by dipole-dipole interactions, the evolution of the target qubit can be constrained by control bits. This enables the expansion of single quantum gates to dual quantum gates in ensemble rare-earth ion systems, shortens the timing adiabatic pulse action time, and consequently reduces the evolution time of quantum states in dual-quantum and multi-quantum systems, avoids decoherence, improves robustness to frequency detuning during pulse action, and enhances fidelity. Attached Figure Description

[0061] Figure 1 This is a flowchart illustrating a method for constructing a controlled dual-qubit gate based on ensemble rare-earth ions in one embodiment of the present invention.

[0062] Figure 2 This is a schematic diagram illustrating the evolution path of the control bit and the target bit on the Bloch sphere in a dual-qubit controlled gate construction method based on ensemble rare-earth ions in one embodiment of the present invention, wherein (a) is the evolution path of the control bit on the Bloch sphere. and (a) Schematic diagram of the evolution path; (b) shows the target bit in and A schematic diagram of the evolutionary path;

[0063] Figure 3 In one embodiment of the present invention, a method for constructing a controlled dual-qubit gate based on ensemble rare-earth ions is used, in which ions are doped with... In crystals Schematic diagram of the relevant energy level structure;

[0064] Figure 4 This is a simplified schematic diagram of the two-qubit blocking effect in a two-qubit controlled gate construction method based on ensemble rare earth ions in one embodiment of the present invention;

[0065] Figure 5 This is a schematic diagram showing the evolution of the Rabi frequency of the two-color light pulse applied to the target position in the CNOT gate over time in Embodiment 1 of the present invention;

[0066] Figure 6 In Embodiment 2 of the present invention, when the control bit is A schematic diagram illustrating the evolution of the layout number of each energy level in the system over time when the CNOT gate is applied, where (a) is... In the initial state, the control bits are in , and A schematic diagram illustrating the evolution of the number of layouts on energy levels over time; (b) is... In the initial state, the target bit is , and A schematic diagram illustrating the evolution of the number of layouts on energy levels over time;

[0067] Figure 7 In Embodiment 2 of the present invention, when the control bit is A schematic diagram illustrating the evolution of the layout number of each energy level in the system over time when the CNOT gate is applied, where (a) is... In the initial state, the control bits are in , and A schematic diagram illustrating the evolution of the number of layouts on energy levels over time; (b) is... In the initial state, the target bit is , and A schematic diagram illustrating the evolution of the number of layouts on energy levels over time;

[0068] Figure 8 This is a schematic diagram showing the truth table of the CNOT gate and the fidelity of the quantum gate formed by each initial state in Embodiment 2 of the present invention.

[0069] Figure 9 This is a schematic diagram showing the fidelity of the quantum gate formed by the initial states of the CZ gate truth table in Embodiment 3 of the present invention.

[0070] Figure 10 The optimized monochromatic pulse applied to the control bit in the CNOT gate in Embodiment 4 of the present invention. A schematic diagram illustrating the evolution of the Rabi frequency over time with respect to the dual-color Rabi pulse applied to the target position;

[0071] Figure 11This is a schematic diagram illustrating the evolution of the layout number of each energy level over time when the CNOT gate with optimized pulses is applied to the system in Embodiment 4 of the present invention. In this diagram, (a) shows the layout number of each energy level as a function of the control bit. The population evolution diagram of the control bit in the current state, (b) is the population evolution diagram when the control bit is in the current state. The population evolution diagram of the target bit in the state, (c) is when the control bit is The population evolution over time for the control bit, (d) is the population evolution of the control bit when it is in a certain state. Population evolution over time at the target site;

[0072] Figure 12 This is a schematic diagram illustrating the dependence of fidelity on frequency detuning when the CNOT gate with optimized pulses is applied to the system in Embodiment 4 of the present invention. In this diagram, (a) shows the initial state as follows: At that time, the fidelity changes with frequency detuning, (b) is the initial state. At that time, the fidelity changes with frequency detuning;

[0073] Figure 13 This is a schematic diagram illustrating the dependence of the non-resonant excitation population on the frequency detuning when the CNOT gate with the optimized pulse is applied to the system in Embodiment 4 of the present invention. In this diagram, (a) represents the initial state as... At that time, the non-resonant excitation changes with frequency detuning, (b) is the initial state. At that time, the non-resonant excitation changes with frequency detuning. Detailed Implementation

[0074] The present invention will be further described below with reference to the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and should not be used to limit the scope of protection of the present invention.

[0075] like Figure 1 As shown, this embodiment of the invention provides a method for constructing a controlled two-qubit gate based on an ensemble of rare-earth ions, comprising the following steps:

[0076] S1: Frequency addressing is performed on a two-qubit system, where one qubit is used as the control bit and the other as the target bit. Specifically:

[0077] Frequency addressing of two qubits is performed by laser focusing, and two qubits with permanent dipole-dipole interactions much greater than the Rabi frequency of the control optical pulse (the permanent dipole-dipole interaction is at least 4 times the Rabi frequency of the control optical pulse) are identified as the control bit and the target bit.

[0078] In this embodiment, the two qubits obtained by addressing should be far apart in frequency, that is, driving one qubit will not directly affect the other qubit.

[0079] S2: Constructing a controlled gate using the blocking effect generated by the dipole-dipole interaction between two qubits, specifically including four steps:

[0080] The first step is to apply a light pulse to the control bit, driving it to transition from the ground state to the excited state, i.e., in The total area of ​​pulses applied to the control bit at each moment is Monochromatic pulse Driving qubits can be achieved from Transition to excited state ;

[0081] The second step is to apply a single quantum gate to the target bit to drive its evolution, that is, in The total area of ​​the pulse applied to the target position at each moment is Dual-color pulse This makes a complete single quantum gate Acts on the target bit;

[0082] The third step is to apply a compensation pulse to the target position, that is, in By changing the phase of the two-color light pulse to apply a compensation pulse to the target position, the dark state of the target position accumulates the same additional phase as the bright state. Only in this way can the robustness of the logic gate control fidelity to frequency detuning be achieved after the introduction of frequency detuning.

[0083] Step 4: Apply a light pulse to the control bit to drive it to de-excite from the excited state to the ground state, that is, in The total area of ​​the pulse applied to the control bit is Monochromatic pulse And add an extra one Phase, driving the qubit from the excited state Deactivation to state.

[0084] In this embodiment, The duration of a single pulse segment is indicated, preferably 0.5~1. This is because if Excitations smaller than this preferred range will result in higher levels of non-resonant excitation, while the driving field will fail to meet the conditions for pulses much smaller than those for dipole-dipole interactions. Conversely, exceeding this preferred range will increase the pulse duration, leading to decoherence. Suppression of non-resonant excitations can be achieved by appropriately increasing the pulse duration and by optimizing the optical pulse containing degrees of freedom.

[0085] Example 1:

[0086] Frequency addressing is performed on a two-qubit system where the frequency distance between the two qubits is such that driving one qubit does not directly affect the other. Frequency addressing also satisfies that the dipole-dipole interaction between the two qubits exceeds at least four times the Rabi frequency of the optical pulse applied to the control bit, the optical pulse involved in a single quantum gate, and the compensation pulse. One qubit serves as the control bit, and the other as the target bit.

[0087] Controlled gates are constructed using the blocking effect generated by the dipole-dipole interaction between two qubits, including:

[0088] This embodiment designs two optical pulses with degrees of freedom in a two-qubit three-level system. The optical pulse applied to the control bit is a monochromatic pulse, and the optical pulses involved in the single quantum gate and the compensation pulse are both bichromatic pulses, specifically including:

[0089] Select the parameters of the dual quantum gate to be constructed, and input their corresponding amplitudes and phases into an arbitrary wave generator to generate a radio signal with the same amplitude and phase as the corresponding optical pulse. The laser needs to output the control pulse and the target pulse asynchronously in the order specified in the instruction manual. The pulse resonating with the target position needs to be deflected by an acousto-optic modulator to obtain +1 or -1 order deflected output light, generating a set of dual-Chrabi pulses and acting on the target position.

[0090] The driving frequency of the acousto-optic modulator for the control bit is f. c-aom The phase of the radio signal for the control bit is represented as follows: In a continuous laser optical path, the laser frequency is f. laser The control bit consists of two energy levels and To characterize, the frequency difference between them is f c-0—1 Electrons from energy levels to energy level The optical transition frequency is ν c The driving acousto-optic modulator generates an effect on... The frequency of the radio signal of the transitioning light pulse is f c， Satisfy f c =f c-aom + f c-0—1 with f laser + f c = ν c .

[0091] The target bit's driving frequency is f t-aom The target bit level consists of two levels. and To characterize, the frequency difference between them is f t-0—1 Electrons from energy levels to energy level The optical transition frequency is ν p From energy level to energy level The optical transition frequency is ν s The frequency of the radio signal driving the light pulses generated by the acousto-optic modulator also includes f. s with f p These satisfy f s =f t-aom + f t-0—1 f p = f t-aom ;f laser + f s = ν s f laser+ f p = ν p The phase of the two radio signals at the target location is represented as follows: and .

[0092] The radio signal amplitudes corresponding to monochromatic pulses and dual-color Rabi pulses are expressed as follows: , and These all change with time and are calculated based on the Rabi frequency of the corresponding pulse. Their expressions are as follows:

[0093] ;

[0094]

[0095] ;

[0096] in, , The radio signal amplitude representing an arbitrary generator of a dual-Serabi pulse. Denotes the reduced Planck constant. The conversion factor (determined by the experimental system) represents the Rabi frequency of the two-color Rabi pulse to the amplitude of the radio signal. Indicates the target position is from the ground state Transition to excited state The optical transition dipole moment, Indicates the target position is from the ground state Transition to excited state The optical transition dipole moment, , This indicates the Rabi frequency of the two-color Rabi pulse. , All are light pulses determined by radio signals generated by arbitrary waveform generators, and satisfy the following conditions: ,in, This represents the equivalent Rabi frequency of the two-color Rabi pulse. The radio signal amplitude representing an arbitrary generator of monochromatic pulses. Indicates the control bit is from the ground state Transition to excited state The optical transition dipole moment, This represents the Rabi frequency of the monochromatic pulse.

[0097] The pull ratio frequency of the two-color pull ratio pulse is calculated based on the equivalent pull ratio frequency of the two-color pull ratio pulse, and its expression is:

[0098] ;

[0099] ;

[0100] ;

[0101] in, This represents the equivalent Rabi frequency of the two-color Rabi pulse. , Both represent the pull ratio frequency of the two-color pull ratio pulse. This represents the normalized amplitude coefficient of the dual-Seraphim pulse. Represents the imaginary unit. , Both represent the phase of the dual-color pulse.

[0102] In this scheme, the amplitude of the output control pulse is set to be equal to that of the target pulse, that is... The pull ratio frequency of a monochromatic pulse and the equivalent pull ratio frequency of a dual-color pull ratio pulse are set according to the following expression:

[0103] ;

[0104] in, This represents the Rabi frequency of a monochromatic pulse. This represents the equivalent Rabi frequency of the two-color Rabi pulse. Indicates time, Indicates the duration of a single pulse segment. Indicates the phase of a monochromatic pulse. , Both represent the phase of the dual-pulse. express and The relative phase difference between them This represents the accumulated geometric phase required to construct a two-quantum gate. Both the monochromatic pulse and the two-chroma pulse involved in a single quantum gate are obtained by superimposing a series of Fourier cosine components, and their calculation expression is:

[0105] ;

[0106] ;

[0107] in, This represents the Rabi frequency of a monochromatic pulse. This represents the equivalent Rabi frequency of the two-color Rabi pulse. In this embodiment, the duration of a single pulse segment is indicated. , Indicates the first Each pulse degree of freedom Indicates time.

[0108] In this embodiment, the frequency of the cosine wave may be limited by the time accuracy of the instruments and equipment used, so this is considered here. The maximum value is 4. Since the control bit pulse and the target bit pulse have different functions (the control bit pulse is responsible for rapid population transition, while the target bit pulse is responsible for quantum gate construction), each requires a set of optimized pulses. , respectively denoted as and .

[0109] exist At any moment, the target qubit is manipulated using this dual-Serabi pulse. , Under the basis vectors, the time evolution operator can be written as ,in , These represent the bright and dark states of the target bit, respectively: It can be seen that the two-color light pulse pairs... A logic gate operation was performed, and the logic gate is mapped to The computational subspace contains any of the following logic gates:

[0110] ;

[0111] in, Within the range [0, π], , Within the range [0, 2π]. The overall logic gate is composed of a control qubit and a target qubit, where the control qubit itself only performs... The evolution of logic gates, therefore the overall logic gate form When written in matrix form, it is ,in This represents a 2x2 zero matrix.

[0112] Figure 2 The evolution paths of the control position and the target position on the Bloch sphere are given, where, This represents the polar axis (Z) of the Bloch sphere. and Indicates and Two orthogonal directions in a perpendicular plane This indicates the initial state of the control bits, i.e. State (Bloch's North Pole). This indicates the initial state of the target bit. The basis vectors representing the control bits. The basis vector representing the target position.

[0113] Figure 2 (a) is the control bit in and The two evolutionary paths overlap and are in completely opposite directions, without accumulating any geometric phase; Figure 2 (b) is the target location in and The evolutionary path consists of two segments that evolve in different directions along the meridian of the Bloch sphere, forming a geometric phase angle. In subsequent embodiments, CNOT gates (Controlled-NOT Gates) are used. ,Right now And CZ gate (Controlled-Z Gate). Let's take an example to illustrate the shape and performance of a light pulse. Substituting the aforementioned light pulses sequentially into the coupled differential equations of a two-qubit system, the software simulation of the construction of the two-qubit gate and the pulse's performance, along with the fidelity F of generating the target quantum state, is defined as follows:

[0114] ;

[0115] in, For the two-qubit gate that needs to be implemented; For the initial quantum state, This is the evolved system density matrix.

[0116] It is worth noting that in the pulse and target position During the interaction between states, a dynamic phase factor that depends on frequency detuning accumulates. In order to extract this phase factor as a global phase, Each state needs to accumulate the same phase factor, and therefore needs to undergo a similar evolution. Therefore... At that moment, a phase compensation pulse needs to be applied to the target bit. Interaction, will the initial The initial state changes to a new dark state, while the initial state is changed to a new dark state. The state changes to a new bright state. This compensation pulse is achieved by resetting... , , We obtained, among which, This represents the normalized amplitude coefficient of the dual-Syracbean pulse applied to the target position. This represents the accumulated geometric phase required to construct a two-quantum gate. This indicates the phase difference between the two-color pulses. And satisfy the following expression:

[0117] ;

[0118] in, This represents the equivalent Rabi frequency of the two-color Rabi pulse. Indicates the duration of a single pulse segment. Indicates the first Each pulse degree of freedom Indicates time.

[0119] To achieve a unit transformation effect from the phase compensation pulse without creating any logic gates, Moment = 0, only phase compensation is performed. To prevent degree-of-freedom overflow from causing insufficient experimental accuracy, the compensation pulse is specified to be 0. AND gates construct pulses They are the same set of parameters.

[0120] Figure 3 Eu is doped in Y2SiO5 crystal 3+ The diagram shows the relevant energy level structure, which is used as an example to illustrate the system to which this invention is applicable. The ground state and excited state in the diagram each contain three hyperfine levels. It is an excited state. and These are two energy levels characterizing a quantum state. For the control bit, the coupling between the energy levels occurs through optical transitions. To achieve this, for the target bit, the coupling between energy levels is achieved through optical transitions. as well as To achieve this. For control bits and target bits, using... The Hamiltonians for basis vectors are as follows:

[0121]

[0122] in, The system Hamiltonian representing the control bit. The system Hamiltonian representing the target bit. Denotes the reduced Planck constant. , This indicates the Rabi frequency of the two-color Rabi pulse.

[0123] To establish a connection between two qubits, a dipole-dipole interaction needs to be introduced. The total Hamiltonian of the entire two-bit system is ;in This is a 3x3 identity matrix used to expand the qubit dimension. The final total Hamiltonian for the two qubits is... The basis vectors are as follows:

[0124]

[0125] in, This represents the total Hamiltonian of a two-bit system. This represents dipole-dipole interaction.

[0126] The total Hamiltonian is a 9x9 matrix, where... This represents a 3x3 zero matrix.

[0127] Figure 4 This is a simplified diagram of the two-qubit blocking effect in rare-earth ion systems. To control the Rabi frequency applied to the bits, , The Rabi frequency applied to the target bit represents one beam in a Raman dual transition. This represents the shift in the excited state energy level of the target qubit after dipole-dipole interaction. This represents the position of the excited state energy level of the target qubit after the shift. As can be seen from the figure, when the control qubit is located... In this state, the light pulse acting on the control qubit will drive State reaches excited state Because of the blocking effect of rare-earth ion systems, a qubit reaching an excited state will prevent other qubits from reaching an excited state. By applying dipole-dipole interactions, the energy level of the excited state of the target qubit is raised, causing the optical pulse driving the target qubit to be unable to reach the target qubit. The number of state layouts is excited to In this state, the target bit qubit will not evolve at all. However, when the control bit qubit is in the... In the state of control, the light pulse acting on the control qubit and State is uncoupled and will not Excitation of the target qubit. The excited state energy level of the target qubit remains unchanged, and the light pulse driving the target qubit will normally construct an arbitrary quantum logic gate.

[0128] Example 2:

[0129] Based on Example 1, the amplitude of the light pulse is zero at the initial moment, that is... Rabi frequency of the monochromatic pulse applied to the control bit and the equivalent Rabi frequency of the dual-color Rabi pulse applied to the target position In the expression, the first Each pulse degree of freedom The even-numbered and odd-numbered terms satisfy the following expression:

[0130] ;

[0131] ;

[0132] in, Indicates an index variable.

[0133] In this embodiment, take , , Construct a CNOT gate, and first consider a quantum system without frequency detuning and non-resonant excitation, and apply an optical pulse to the control bit. The values ​​are respectively , , , The light pulse applied to the target position The values ​​are respectively , , , .

[0134] Figure 5 This refers to the Rabi frequency of the optical pulses acting on the control and target bits in the CNOT gate generated in this embodiment. , To control the Rabi frequency applied to the bits, , The Rabi frequency applied to the target bit is one of the beams driven by the Raman dual transition. It can be seen that the advantage of this pulse is that the absolute value of the instantaneous Rabi frequency is less than 2 MHz, and the change at the start and end times is relatively slow, which reduces the response time requirement of the acousto-optic modulator.

[0135] Figure 6 This describes the evolution of the quantum state over time by the CNOT gate generated in this embodiment in a quantum system without detuning but with spontaneous emission loss, where (a) is... In the initial state, the control bits are in , and A schematic diagram illustrating the evolution of the number of layouts on energy levels over time; (b) is... In the initial state, the target bit is , and A schematic diagram illustrating the evolution of the number of energy levels over time. The diagram shows that the initial state of the system is... The terminated state remains after passing through the CNOT gate. .

[0136] Figure 7 This describes the evolution of the quantum state over time by the CNOT gate generated in this embodiment in a quantum system without detuning but with spontaneous emission loss, where (a) is... In the initial state, the control bits are in , and A schematic diagram illustrating the evolution of the number of layouts on energy levels over time; (b) is... In the initial state, the target bit is , and A schematic diagram illustrating the evolution of the number of energy levels over time. The diagram shows that the initial state of the system is... The final state after passing through the CNOT gate is .

[0137] The truth table for the CNOT gate is shown in Table 1:

[0138] Table 1: Truth Table for CNOT Gates

[0139]

[0140] Table 1 can be transformed into a truth matrix. The corresponding position indicates the fidelity of the two quantum gate in each initial state. Figure 8 The truth matrix generated in this embodiment, considering the spontaneous emission loss and decoherence effect of the system, is the CNOT gate obtained from simulation. =10 MHz, which is a dipole interaction size that is very easy to implement in this system. It can be seen that when the control bit is... At this time, the fidelity of the CNOT gate can reach over 99.9%. When the control bit is... When the dual quantum gates exhibit a blocking effect, the target position is affected by the blocking effect, and the fidelity will decrease slightly, but it will still be above 99.7%. In addition, in this example, the optical pulse does not take into account the frequency detuning and non-resonant excitation of the system, so the fidelity will fluctuate when frequency detuning occurs.

[0141] Example 3:

[0142] Based on Example 1, take , , The pulse degrees of freedom are consistent with those in Example 2, and a CZ gate is constructed. If the input state is... Then, the target qubit will have an additional phase after passing through this gate. The truth matrix of the CZ gate is .

[0143] Figure 9 This is the truth table obtained from the simulation of the CZ gate generated in this embodiment, taking into account the system's spontaneous emission loss and decoherence effect. It can be seen that it is very similar to that of Embodiment 2, but when the control bit is... At this time, the fidelity of the CZ gate can reach over 99.9%, when the control bit is The time fidelity is over 99.7%. This case also does not consider frequency detuning and non-resonant excitation present in the system.

[0144] Example 4:

[0145] Based on Example 2, through optimization The value of is used to detect the performance of the quantum gate at the termination moment of the interaction between the light pulse and the quantum system. For ensemble rare-earth ion systems, there are two main constraints: first, for ions within the selected frequency window, it is expected that the quantum gates constructed by these ions will maintain high fidelity (for...). The average fidelity is greater than 99% within the range of [-170, 170] kHz; secondly, it has a good suppression effect on the non-resonant excitation of other ions far from the driving frequency window (for...). To be honest Non-resonant excitation outside 8.9MHz does not exceed 1%.

[0146] The value of is determined by scanning using a multi-objective optimization genetic algorithm. The objective function of the multi-objective optimization genetic algorithm is:

[0147] ;

[0148] ;

[0149] in, Denotes the first objective function. This represents the second objective function. Indicates the gate fidelity at the final moment. Represents all non-initial states The sum of the layout numbers, such as hour, , This indicates other energy levels that can be reached by population leakage.

[0150] The optical pulses acting on the control and target bits are optimized by ensuring that both loss functions are as close to zero as possible simultaneously. It's important to note that, unlike the optimization process for single-quantum gates, the evolution path of dual-quantum controlled gates differs depending on the control bit. Initial state The value needs to be considered simultaneously with the control bit. Harmony The situation.

[0151] Typically, there is a competing trade-off between the two constraints of high robustness and low non-resonant excitation: improving operational fidelity within the frequency window often requires stronger drive or wider spectral coverage, but this may lead to the failure of the blockade between the two quanta, while simultaneously enhancing the non-resonant excitation of ions outside the window; conversely, strengthening the suppression of non-resonant excitation may weaken the gate operation efficiency within the window. Therefore, theoretically, there is no optimal solution that simultaneously maximizes both objectives.

[0152] For this type of multi-objective optimization problem, this paper adopts an optimization strategy based on a multi-objective genetic algorithm. This algorithm can systematically search the parameter space and obtain a set of Pareto optimal solutions, where improvement in any objective inevitably leads to degradation of another. By analyzing this Pareto front, we can minimize the proportion of non-resonant excitations while maintaining high robustness (i.e., achieving the required fidelity within the frequency window), thus achieving the optimal balance between physical realizability and operational accuracy. The optimized pulse parameters for creating the CNOT gate in the ensemble rare-earth ion system are:

[0153] The light pulse applied to the control bit The values ​​are respectively , , , The light pulse applied to the target position The values ​​are respectively , , , .

[0154] Figure 10 It is the Rabi frequency of the dual-color light pulse acting on the target position in the CNOT gate generated in this embodiment. With attachment Figure 4 In contrast, this optical pulse can create fast and high-fidelity CNOT gates in ensemble rare-earth ion systems with frequency detuning and non-resonant excitation.

[0155] Figure 11 This is the CNOT gate constructed in this embodiment, where (a) is the gate when the control bit is The population evolution diagram of the control bit in the current state, (b) is the population evolution diagram when the control bit is in the current state. The population evolution diagram of the target bit in the state, (c) is when the control bit is The population evolution over time for the control bit, (d) is the population evolution of the control bit when it is in a certain state. The population evolution over time for the target bit is shown in the graph. As can be seen from the graph, when the control bit is... State, target position is When the target bit is locked, it will not evolve; however, when both the control bit and the target bit are locked... At that time, the target position evolves normally and flips. state and The population of states. That is, the initial state of the entire two-qubit system is... The final state is .

[0156] Figure 12 and Figure 13 The fidelity (non-resonant excitation population) of the CNOT gate created in this embodiment after optimizing the interaction between the optical pulse and the quantum system with frequency detuning varies with frequency detuning. The relationship diagram of the changes, in which It is the difference between the center frequency of the light pulse and the corresponding transition frequency of the quantum state ion or background ion, i.e., the non-resonant frequency detuning. (a) are all initial states. In all cases (b), In the case of the solid line, the optimized pulse of Example 4 is represented, and the initial pulse of Example 2 is represented.

[0157] To demonstrate the optimization effect, this embodiment is compared with Embodiment 2. The solid line represents the optimized pulse result, and the dashed line represents the unoptimized pulse result of Embodiment 2. The results show that within ±200kHz of the center frequency, the CNOT gate fidelity remains above 99%, exhibiting strong robustness. For Eu 3+ For the Y₂SiO₅ system, the full width at half maximum (FWHM) of the quantum state absorption peak is 170 kHz. The pulse generated in this embodiment is sufficient to meet the requirements of this system. The fidelity behavior between ±170 kHz and ±8.9 MHz is irrelevant because the quantum state ions are located within a zero-absorption frequency window, with the center frequency of the qubit ions approximately 8.9 MHz from the boundary of the window. For background ions with detuning between 8.9 and 10 MHz, non-resonant excitation can be observed by calculating the population within this range. The closer the population is to 0 within this range, the lower the probability of non-resonant excitation of the background ions. In this embodiment, the population in the 8.9–10 MHz range does not exceed 0.2%, indicating that non-resonant excitation of the background ions is almost nonexistent.

[0158] The advantage of this embodiment is that it is suitable for Eu. 3+The Y2SiO5 system exhibits up to 99% fidelity for the logic gates created by optical pulses within a frequency detuning range of ±200 kHz, while exhibiting almost no non-resonant excitation outside the frequency range of ±8.9 MHz detuning.

[0159] The aforementioned technical solution addresses the challenge of expanding a single qubit into a two-qubit system when constructing a two-qubit gate based on an ensemble rare-earth ion system. It also significantly shortens the pulse duration for gate construction, resolving the issue of decoherence. Furthermore, it enhances the robustness of quantum manipulation within a specific frequency detuning range for the ensemble rare-earth ion quantum state system, suppressing non-resonant excitations outside this range, and providing a technical reference for multi-quantum manipulation in other imperfect systems.

[0160] The above technical solution can construct arbitrary two-bit logic gates based on ensemble rare earth ion systems, which can be used to prepare quantum chips or quantum memories and applied to frequency-addressable quantum computing systems.

[0161] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the technical principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for constructing a controlled two-qubit gate based on ensemble rare-earth ions, characterized in that, include: Frequency addressing is performed on a two-qubit system, where one qubit is used as a control bit and the other qubit is used as a target bit. Controlled gates are constructed using the blocking effect generated by the dipole-dipole interaction between two qubits, including: A light pulse is applied to the control bit, driving it to transition from the ground state to the excited state. A single quantum gate is applied to the target bit to drive its evolution, and then a compensation pulse is applied to the target bit. A light pulse is applied to the control bit to drive it to de-excite from the excited state to the ground state; The frequency addressing satisfies the following conditions: The frequency separation of two qubits satisfies the condition that driving one qubit does not directly affect the other qubit. The dipole-dipole interaction between two qubits exceeds the Rabi frequency of the optical pulse applied to the control bit, the optical pulse involved in a single quantum gate, and the compensation pulse by at least four times. The light pulse applied to the control bit is a monochromatic pulse, and the light pulses and compensation pulses involved in the single quantum gate are both bichromatic pulses; The monochromatic pulse and the dual-color Rabi pulse are both obtained by converting the amplitude of the corresponding radio signal. The method for obtaining the amplitude of the corresponding radio signal includes: The pull ratio frequency of the monochromatic pulse and the equivalent pull ratio frequency of the dual-color pull ratio pulse are set according to the following expression: ； in, This represents the Rabi frequency of a monochromatic pulse. This represents the equivalent Rabi frequency of the two-color Rabi pulse. Indicates time, Indicates the duration of a single pulse segment. Indicates the phase of a monochromatic pulse. , Both represent the phase of the dual-pulse. express and The relative phase difference between them This represents the geometric phase that needs to be accumulated to construct a two-quantum gate; The pull ratio frequency of the two-color pull ratio pulse is calculated based on the equivalent pull ratio frequency of the two-color pull ratio pulse. The expression for calculating the pull ratio frequency of the two-color pull ratio pulse based on the equivalent pull ratio frequency is as follows: ； ； ； in, This represents the equivalent Rabi frequency of the two-color Rabi pulse. , Both represent the pull ratio frequency of the two-color pull ratio pulse. This represents the normalized amplitude coefficient of the dual-Seraphim pulse. Represents the imaginary unit. , Both represent the phase of the dual-pulse; The amplitude of the radio signal corresponding to the two-color Rabi pulse is calculated based on the Rabi frequency of the two-color Rabi pulse. The expression for calculating the amplitude of the radio signal corresponding to the two-color Rabi pulse based on the Rabi frequency is as follows: ； ； in, , The radio signal amplitude representing an arbitrary generator of a dual-Serabi pulse. Denotes the reduced Planck constant. The conversion factor representing the Rabi frequency of a two-color Rabi pulse to the amplitude of a radio signal. Indicates the target position is from the ground state The optical transition dipole moment to the excited state Indicates the target position is from the ground state The optical transition dipole moment that transitions to the excited state; The amplitude of the radio signal corresponding to the monochromatic pulse is calculated based on the Rabi frequency of the monochromatic pulse. The expression for calculating the amplitude of the radio signal corresponding to the monochromatic pulse based on the Rabi frequency of the monochromatic pulse is as follows: ； in, The radio signal amplitude representing an arbitrary generator of monochromatic pulses. This represents the optical transition dipole moment of the control state as it transitions from the ground state to the excited state.

2. The method for constructing a controlled dual-qubit gate based on ensemble rare-earth ions according to claim 1, characterized in that, Both the monochromatic pulse and the dual-color Rabi pulse involved in the single quantum gate are obtained by superimposing a series of Fourier cosine components, and their calculation expressions are as follows: ； ； in, This represents the Rabi frequency of a monochromatic pulse. This represents the equivalent Rabi frequency of the two-color Rabi pulse. Indicates the duration of a single pulse segment. Indicates the first Each pulse degree of freedom Indicates time.

3. The method for constructing a controlled dual-qubit gate based on ensemble rare-earth ions according to claim 2, characterized in that, The The even-numbered and odd-numbered terms satisfy the following expression: ； ； in, Indicates an index variable.

4. The method for constructing a controlled dual-qubit gate based on ensemble rare-earth ions according to claim 3, characterized in that, The The value of is determined by a multi-objective optimization genetic algorithm, the objective function of which is: ； ； in, Denotes the first objective function. This represents the second objective function. Indicates the gate fidelity at the final moment. Represents all non-initial states The sum of the layout numbers, This indicates other energy levels that can be reached by population leakage.

5. The method for constructing a controlled dual-qubit gate based on ensemble rare-earth ions according to claim 4, characterized in that, When the two quantum gates are CNOT gates At that time, the light pulse applied to the control bit The values ​​are respectively , , , The light pulse applied to the target position The values ​​are respectively ， ， ， 。 6. The method for constructing a controlled dual-qubit gate based on ensemble rare-earth ions according to claim 1, characterized in that, The compensation pulse is reset , , We obtained, among which, This represents the normalized amplitude coefficient of the dual-Seraphim pulse. This represents the accumulated geometric phase required to construct a two-quantum gate. This indicates the phase difference between the two-color pulses. ; The expression for calculating the equivalent Rabi frequency after superimposing the compensation pulse onto the dual Rabi pulse involved in the single quantum gate is as follows: ； in, This represents the equivalent Rabi frequency of the two-color Rabi pulse. Indicates the duration of a single pulse segment. Indicates the first Each pulse degree of freedom Indicates time.

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