A method for preparing high-efficiency concatenated GHZ state based on adiabatic rapid passage and application thereof

Abstract

This invention discloses a highly efficient cascaded GHZ state preparation method and its application based on an adiabatic fast channel, comprising: arranging atoms in a laser field according to four scales of GHZ states; for the initial state with a single-atom arrangement, directly preparing the GHZ state; for the initial state with a 2k atom array arrangement, inserting a π space in the ground state space between two RAP pulses. g The pulse sequence structure is used, and a ground state space π is applied after the evolution is complete, consisting of non-adjacent bits. g The laser pulse enables the preparation of cascaded GHZ states. By optimizing the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level, the spatial modulation characteristics of the Rydberg interaction are combined with the robustness of adiabatic manipulation. This unifies the preparation process of GHZ states of different scales, significantly reduces the operational complexity, and thus achieves high-fidelity and robust preparation of multi-scale cascaded GHZ states.

Description

Technical Field

[0001] This invention belongs to the field of quantum information and quantum precision measurement technology, specifically, it relates to a method for preparing efficient cascaded GHZ states based on adiabatic fast channels and its application. Background Technology

[0002] The core challenge of quantum precision measurement lies in breaking the Standard Quantum Limit (SQL). For measurements using N independent quantum sensors, the uncertainty follows... The SQL scale, a limit inherently stemming from the statistical properties of quantum projection noise, limits the accuracy of phase measurement when qubits are in an incoherent superposition state. Specifically, when qubits are in an incoherent superposition state, their phase measurement accuracy is limited by the random superposition of independent sampling by each particle, preventing the measurement sensitivity from exceeding the classical statistical limit dominated by shot noise. However, applying quantum entangled states can improve accuracy to the 1 / N scale of the Heisenberg Limit (HL). In 1997, the pioneering work of Huelga et al. first proposed a theoretical framework for breaking the SQL limit using spin-squeezed states. Building upon this, the Greenberger-Horne-Zeilinger (GHZ) state, due to the shared quantum coherence among all particles, amplifies the overall phase accumulation rate of laser light in Ramsey interference by a factor of N, making it an ideal candidate for realizing the HL. Specifically, the N-particle GHZ state... After Ramsey evolution, the phase sensitivity can reach Δφ = 1 / N, but it is limited by a dynamic range of 2π / N, which means that a multi-scale measurement strategy is needed to extend it in practical applications.

[0003] The cascaded GHZ state scheme combines GHZ states of different scales to measure k GHZ states with varying particle numbers: large-scale GHZ states provide high-precision but narrow-range phase estimation, while small-scale GHZ states are used to extend the unpacking range. A recursive phase deblurring algorithm ultimately restores the dynamic range to the 2π order of magnitude while maintaining HL-level measurement accuracy. In 2024, Adam M. Kaufman's team realized a four-cascaded GHZ state in a strontium atomic optical lattice and verified its potential in next-generation atomic clocks by measuring phase accuracy.

[0004] However, existing preparation methods have significant bottlenecks: different scales of GHZ states require independently designed preparation sequences; as the number of atoms increases, the depth of the quantum circuit increases exponentially, leading to an exacerbation of decoherence effects; and existing preparation methods are very sensitive to fluctuations in laser parameters, with laser intensity or phase jitter causing a rapid decrease in the fidelity of the prepared entangled states; and the need for multi-scale synchronous control, as different scales of GHZ require different preparation times, necessitates controlling the timing sequence to achieve the generation of cascaded GHZs, increasing the operational difficulty. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention provides a method and application for the preparation of highly efficient cascaded GHZ states based on adiabatic rapid passages. By optimizing the pulse sequence of the combination of adiabatic rapid passage (RAP) and ground state manipulation techniques, high-fidelity and robust preparation of multi-scale cascaded GHZ states can be achieved.

[0006] To achieve the above technical objectives, the present invention adopts the following technical solution: a method for preparing efficient cascaded GHZ states based on adiabatic fast channels, specifically including the following steps:

[0007] Step S1: Arrange the atoms in the laser field according to four scales of the GHZ state, including: single atom arrangement and 2k atom array arrangement, where k = 1, 2, 4;

[0008] Step S2: For the initial state with a single-atom arrangement, directly prepare the GHZ state;

[0009] Step S3: For the initial state using a 2k atom array, the ground state space π is inserted between two RAP pulses. g The pulse sequence structure is used, and a ground state space π is applied after the evolution is complete, consisting of non-adjacent bits. g Pulse, to achieve the preparation of cascaded GHZ states.

[0010] Furthermore, step S3 includes the following sub-steps:

[0011] Step S3.1: Based on the maximum pull ratio frequency Ω0 of the RAP pulse, set the dimensionless time τ = Ω0t, where t represents the actual time;

[0012] Step S3.2, in τ∈[0,τ] D During this period, non-adjacent or diagonal atoms in the GHZ states of each 2k atom array arrangement are adiabatically transferred from the ground state |1> to the Rydberg state |r> via the first PAP pulse, and the multi-excited states are suppressed by the interaction strength between atoms, where, where, τ D =τ tot / 2,τ tot Represents the dimensionless total time;

[0013] Step S3.3, at τ = τ D Applying ground state space π g Pulse, realizing the ground states of all atoms in the GHZ state for each 2k atom array arrangement. The population flip;

[0014] Step S3.4, in τ∈[τ] D ,τtot During this period, a reverse adiabatic evolution is performed by applying a second RAP pulse to all atoms in each GHZ state of a 2k atom array arrangement to form the GHZ state.

[0015] Step S3.5: After the evolution is complete, apply the ground state space π of non-adjacent bits to the GHZ state |ψ>. g Pulse, to obtain standard GHZ state form

[0016] Furthermore, before implementing step S3, it is necessary to optimize the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level until the fidelity of different scale GHZ states reaches its maximum.

[0017] Furthermore, the specific process for optimizing the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level is as follows:

[0018] i. Set the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level;

[0019] ii. Calculate the Hamiltonian of the global laser field coupled ground state |1> and Rydberg state |r> in the rotating coordinate system for each GHz state scale based on the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level;

[0020] iii. Calculate the final state of the initial density matrix after the evolution process by combining the Hamiltonian, and determine the fidelity based on the density matrix of the standard GHZ state;

[0021] iv. By updating the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level, repeat steps ii-iii until the atomic fidelity at the corresponding GHz state scale reaches its maximum, thus completing the optimization of the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level.

[0022] Furthermore, the pully frequency Ω(τ) of the RAP pulse is set as follows:

[0023]

[0024] The frequency difference Δ(τ) between the laser pulse and the atomic transition energy level of the RAP pulse is set as follows:

[0025]

[0026] Where, τ k =(2s-1)τ tot / 4, s∈{1,2,...} represents the s-th RAP pulse; τ R =0.175τtot ; a represents the correction coefficient for Ω(τ), a = exp[-(τ)] tot / (4τ R )) 4 ];Δ max =1.30Ω0.

[0027] Furthermore, the Hamiltonian of the global laser field coupled ground state |1> and Rydberg state |r> in the rotating coordinate system at each GHZ state scale is... The calculation process is as follows:

[0028]

[0029] Where n represents the number of atoms at each GHZ state scale, i represents the index of n, and |r> i <r| i Represents the Rydberg state operator for the i-th atom, |r> i <1| i Hc represents the transition of the i-th atom between the ground states |1> and |r>, where Hc represents |r>. i <1| i Hermitian conjugate term, Represents the Rydberg interaction term. j represents the index of n, j ≠ i, V ij V represents the interaction strength between the i-th atom and the j-th atom. ij =C6 / |r i -r j | 6 C6 represents the coefficient of the van der Waals interaction, r i r j These represent the position coordinates of the i-th atom and the j-th atom, respectively. This indicates the direct product operation.

[0030] Furthermore, the differentiation process for the density matrix of the standard GHZ state is as follows:

[0031]

[0032] Where ρ represents the density matrix of the standard GHZ state, ρ=|ψ final ><ψ final |; g represents the index of the number of decoherence effects. The operator representing the g-th decoherence effect, express Hermitian conjugate operator, Represent ρ and commutative relations express The anti-commutative relation of ρ,

[0033] Furthermore, the present invention also provides a method for preparing cascaded GHZ states based on the efficient cascaded GHZ state preparation method of adiabatic fast channel, which is applied to an optical lattice atomic clock system.

[0034] Compared with existing technologies, this invention has the following advantages: The efficient cascaded GHZ state preparation method based on adiabatic fast channels optimizes the Rabi frequency of the RAP pulse and the frequency difference between the laser and atomic transition energy levels during the cascading of GHZ states using a 2k atom array. This combines the spatial modulation characteristics of the Rydberg interaction with the robustness of adiabatic manipulation, unifying the preparation process for GHZ states of different scales and significantly reducing operational complexity. This achieves high-fidelity and robust preparation of multi-scale cascaded GHZ states, solving key technical challenges in traditional cascaded GHZ state preparation, such as the exponential increase in circuit depth with increasing atom number, high noise sensitivity, and limited dynamic range. Applying the efficient cascaded GHZ states prepared by this method to an optical lattice atomic clock system can significantly improve quantum projection noise suppression capabilities and expand the dynamic range while maintaining the measurement accuracy of the Heisenberg limiting scale. Attached Figure Description

[0035] Figure 1 This is a schematic diagram of the atomic arrangement in the laser field of the present invention;

[0036] Figure 2 This is a schematic diagram of the mean square error of the phase estimation of the effective cascaded GHZ state prepared by the present invention. Detailed Implementation

[0037] The technical solution of the present invention will be further explained and described below with reference to the accompanying drawings.

[0038] This invention provides a highly efficient cascaded GHZ state preparation method based on adiabatic fast channels, specifically including the following steps:

[0039] Step S1: Arrange the atoms in the laser field according to four scales of GHZ state, and there are multiple copies of each scale of GHZ state. The four scales include: single atom arrangement and arrangement using a 2k atom array, where k = 1, 2, 4.

[0040] Step S2: For the initial state with a single-atom arrangement, directly prepare the GHZ state;

[0041] Step S3: For the initial state using a 2k atom array, the ground state space π is inserted between two RAP pulses. g The pulse sequence structure is used, and a ground state space π is applied after the evolution is complete, consisting of non-adjacent bits. gPulses are generated to rapidly and efficiently produce four cascaded GHZ states of different scales under a global laser field; this includes the following sub-steps:

[0042] Step S3.1: Based on the maximum pull ratio frequency Ω0 of the RAP pulse, set the dimensionless time τ = Ω0t, where t represents the actual time;

[0043] Step S3.2, in τ∈[0,τ] D During this period, non-adjacent or diagonal atoms in the GHZ states of each 2k atom array arrangement are adiabatically transferred from the ground state |1> to the Rydberg state |r> via the first PAP pulse, and the multi-excited states are suppressed by the interaction strength between atoms, where, where, τ D =τ tot / 2,τ tot Represents the dimensionless total time;

[0044] Step S3.3, at τ = τ D Applying ground state space π g Pulse, realizing the ground states of all atoms in the GHZ state for each 2k atom array arrangement. The population flip is negligible in terms of pulse duration;

[0045] Step S3.4, in τ∈[τ] D ,τ tot During this period, a reverse adiabatic evolution is performed by applying a second RAP pulse to all atoms in each GHZ state of a 2k atom array arrangement to form the GHZ state.

[0046] Step S3.5: After the evolution is complete, apply the ground state space π of non-adjacent bits to the GHZ state |ψ>. g Pulse, to obtain standard GHZ state form

[0047] like Figure 1 The atoms are arranged in a 2k geometric configuration. The orange and yellow solid lines and the white dashed lines between the atoms represent Rydberg interactions, and their size gradually decreases. The interaction between the atoms separated by the white dashed lines is negligible. Figure 1 On the right side, atoms The states are coupled via RAP pulses, with |1> and |0> forming the ground state space, and all atoms are coupled to the global laser field Ω(t).

[0048] Before implementing step S3, it is necessary to optimize the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level until the fidelity of different scale GHZ states is maximized. This combines the spatial modulation characteristics of the Rydberg interaction with the robustness of adiabatic control, unifies the preparation process of different scale GHZ states, significantly reduces operational complexity, and thus achieves high-fidelity and robust preparation of multi-scale cascaded GHZ states. This solves key technical problems in traditional cascaded GHZ state preparation, such as the circuit depth increasing exponentially with the number of atoms, high noise sensitivity, and limited dynamic range. Specifically, i. Set the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level;

[0049] In this invention, the Rabi frequency Ω(τ) of the RAP pulse is related to the laser intensity and determines the transition rate, and is set as follows:

[0050]

[0051] Wherein, Ω0 = 2π × 4MHz;

[0052] The frequency difference Δ(τ) between the laser pulse and the atomic transition energy level in the RAP pulse is set as follows:

[0053]

[0054] Where, τ k =(2s-1)τ tot / 4, s∈{1,2,...} represents the s-th RAP pulse; τ R =0.175τ tot ; a represents the correction coefficient for Ω(τ), a = exp[-(τ)] tot / (4τ R )) 4 ];Δ max =1.30Ω0.

[0055] ii. Calculate the Hamiltonian of the global laser field coupled ground state |1> and Rydberg state |r> in the rotating coordinate system for each GHz state scale based on the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level.

[0056]

[0057] The first two terms of the Hamiltonian are derived from the coupling between the laser and the atomic energy levels. The first term is the detuning term, where n represents the number of atoms at each GHz scale, and i represents the index of n. |r> i <r| i The second term represents the Rydberg state operator for the i-th atom; the third term is a transition term that causes the atom to transition between |1> and |r>, where |r>i <1| i Hc represents the transition of the i-th atom between the ground states |1> and |r>, where Hc represents |r>. i <1| i Hermitian conjugate term, Represents the Rydberg interaction term. j represents the index of n, j ≠ i, V ij V represents the interaction strength between the i-th atom and the j-th atom. ij =C6 / |r i -r j | 6 C6 represents the coefficient of the van der Waals interaction, C6 = 2π × 10.4 GHz·μm 6 r i r j These represent the position coordinates of the i-th atom and the j-th atom, respectively. The magnitude of the interaction between atoms depends on the distance between them. This represents a direct product operation. Because the required Rydberg interaction strength between atoms is less than 2π × 400 MHz, and the interatomic distances are relatively large, the aforementioned V... ij The formula can be stably satisfied. Furthermore, in the qubit space, the ground states |0> and |1> can be bridged by stimulated Raman transitions with a fast π-wave transition. g Pulse, to achieve effective coherent population reversal.

[0058] iii. In order to take into account the possible decoherence effect, the final state of the initial density matrix after the evolution process is calculated by combining the Hamiltonian, and the fidelity is determined according to the density matrix of the standard GHZ state.

[0059] The differentiation process of the density matrix of the standard GHZ state in this invention is as follows:

[0060]

[0061] Where ρ represents the density matrix of the standard GHZ state, ρ=|ψ final ><ψ final |; g represents the index of the number of decoherence effects, and the second term is the impact of the decoherence effect. Different decoherence effects are simulated using different decoherence operators. The operator representing the g-th decoherence effect, express Hermitian conjugate operator, Represent ρ and commutative relations express The anti-commutative relation of ρ,

[0062] iv. By updating the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level, repeat steps ii-iii until the atomic fidelity at the corresponding GHz state scale reaches its maximum, thus completing the optimization of the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level.

[0063] Through the optimization process of the RAP pulse described above, the fidelity of GHZ states of different scales can be improved. The adiabatic robustness of RAP can be used to unify the preparation process of GHZ states of different scales, significantly reducing the operational complexity, thereby achieving high-fidelity and robust preparation of multi-scale cascaded GHZ states.

[0064] By incorporating Rydberg state dissipation and employing the aforementioned RAP pulse optimization method, efficient cascaded GHZ states were prepared. The fidelity of the evolved GHZ states at various scales was calculated. For Strontium-88 atoms, the Rydberg state lifetime τ... ryd =51μs, with a total preparation time of 13μs: the fidelity for two bits is 0.987, and the fidelity for four bits is 0.968. For the fidelity calculation of eight bits, due to the large Hilbert space and the huge computational resources required, a fitting method is used: in many experimental cases, the fidelity loss is linearly related to the number of atoms, but for the sake of conservatism, an exponential fitting y = ce is used. bx Where c and b are fitting parameters, the fidelity of the eight bits is 0.927. This is significantly better than the experimental data in the 2024 paper by Adam M. Kaufman's team. Furthermore, the fidelity of a single bit is set to 1 in this invention because a single bit only requires a final π / 2 pulse, avoiding Rydberg states and the effects of dissipation.

[0065] In one technical solution of this invention, taking a four-atom square array as a reference, by optimizing the Rabi frequency of the RAP pulse, the frequency difference between the laser and the atomic transition energy levels, and the strength of the interaction between atoms, the final state under the Hamiltonian evolution achieves the highest fidelity to the ideal GHZ state, thus obtaining Δ max =1.30Ω0 and the interaction strength between adjacent atoms V0 =9.55Ω0. The optimized results can be directly applied to the GHZ state scale of 2 atoms and 8 atoms. This is because the interaction between four atoms includes both adjacent atoms and diagonal interactions. Ignoring the diagonal interactions degenerates to the two-atom case. Extending to the eight-atom case only adds some weaker interactions.

[0066] Applying the cascaded GHZ states prepared by the efficient cascaded GHZ state preparation method based on adiabatic fast channels of this invention to an optical lattice atomic clock system can significantly improve measurement accuracy. Specifically, a Ramsey sequence is applied to the final state of the cascaded GHZ states. From the perspective of the Bloch sphere, this means accumulating a certain phase around the z-axis and then rotating the y-axis by an angle of π / 2. The parity operator is then calculated. The expected value and curve C v sin[(φ-φ v ) / n v ]-y v Phase fitting yields C v ,φ v and y v The value of , where v represents the index of the GHZ state scale, v = 1, 2, 3, 4, corresponding to GHZ states with scales of 1, 2, 4, 8 atoms respectively. This represents the Pauli operator for the i-th atom, where φ is the accumulated phase. v n represents the phase shift at the v-th GHz state scale. v y represents the number of atoms at the v-th GHZ state scale. v Representing the vertical offset at the v-th GHz state scale, we obtain two-bit C2 = 0.993 and four-bit C3 = 0.981. Similarly, through exponential fitting, we obtain eight-bit C4 = 0.957, and find that φ k =y k =0. Using the above data, and assuming the number of atoms in each cluster is M. v = 2 + 8(4 - v), so for the four scales of atoms 1, 2, 4, and 8, a total of 118 atoms are estimated. The mean square error (MSE) under different accumulated phases is obtained through the phase estimation scheme, as shown in the appendix. Figure 2 As shown, it can be seen that the cascaded GHZ state scheme generated by RAP using the present invention achieves the same accuracy as the standard GHZ state. However, compared with the data in the paper Nature 634,315 (2024) by Adam M. Kaufman's team in 2024, it can be seen that the cascaded GHZ state generated by RAP using the present invention has better accuracy.

[0067] The above are merely preferred embodiments of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A method for preparing efficient cascaded GHZ states based on adiabatic fast channels, characterized in that, Specifically, the steps include the following: Step S1: Arrange the atoms in the laser field according to four scales of the GHZ state, including: single-atom arrangement and using 2 k Atom array arrangement, in which, k =1,2,4; Step S2: For the initial state with a single-atom arrangement, simply add one... Preparation of GHZ state by pulse; Step S3, for using 2 k The initial state of the atomic array arrangement is achieved by inserting the ground state space between two RAP pulses. The pulse sequence structure is defined, and a ground state space of non-adjacent bits is applied after the evolution is complete. The pulse is used to prepare cascaded GHZ states; this includes the following sub-steps: Step S3.1: Based on the maximum pull ratio frequency of the RAP pulse. Set dimensionless time , where t represents the actual time; Step S3.2, in During this period, each type of 2 k In a GHZ state of an atomic array, non-adjacent or diagonal atoms are activated by the first RAP pulse from the ground state. Adiabatic transfer to the Ridgburg state And by utilizing the strength of interatomic interactions to suppress multiple excited states, where, where, , Represents the dimensionless total time; Step S3.3, in Apply ground state space Pulse, to achieve each of 2 k All atomic ground states in the GHZ state of the atomic array The population flip; Step S3.4, in During this period, through each of 2 k In a GHZ state with an atomic array, all atoms undergo a reverse adiabatic evolution by applying a second RAP pulse, thus forming the GHZ state. ; Step S3.5: After the evolution is complete, in the GHZ state Apply non-adjacent bits to the ground state space Pulse, to obtain standard GHZ state form .

2. The method for preparing efficient cascaded GHZ states based on adiabatic fast channels according to claim 1, characterized in that, Before implementing step S3, it is necessary to optimize the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level until the fidelity of different scale GHZ states reaches the maximum.

3. The method for preparing a highly efficient cascaded GHZ state based on an adiabatic fast channel according to claim 2, characterized in that, The specific process for optimizing the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level is as follows: i. Set the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level; ii. Calculate the global laser field coupled ground state at each GHz scale based on the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level. With Ridburg Hamiltonian in a rotating coordinate system; iii. Calculate the final state of the initial density matrix after the evolution process by combining the Hamiltonian, and determine the fidelity based on the density matrix of the standard GHZ state; iv. By updating the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level, repeat steps ii-iii until the atomic fidelity at the corresponding GHz state scale reaches its maximum, thus completing the optimization of the Rabi frequency of the RAP pulse and the frequency difference between the laser and the atomic transition energy level.

4. The method for preparing a high-efficiency cascaded GHZ state based on an adiabatic fast channel according to claim 3, characterized in that, The Rabi frequency of the RAP pulse Set to: The frequency difference between the laser of the RAP pulse and the atomic transition energy level Set to: in, (2s-1) s∈{1,2,...} represents the s-th RAP pulse; ; a represents Correction factor, ; .

5. The method for preparing a highly efficient cascaded GHZ state based on an adiabatic fast channel according to claim 4, characterized in that, The global laser field coupling ground state at each GHz state scale With Ridburg Hamiltonian in a rotating coordinate system The calculation process is as follows: Where n represents the number of atoms at each GHZ state scale, and i represents the index of n. Represents the Rydberg state operator of the i-th atom. This indicates that the i-th atom is in the ground state. and Leap between, express Hermitian conjugate term, Represents the Rydberg interaction term. j represents the index of n, j ≠ i. This represents the interaction strength between the i-th atom and the j-th atom. , The coefficients representing the van der Waals interactions, , These represent the position coordinates of the i-th atom and the j-th atom, respectively. This indicates the direct product operation.

6. The method for preparing a high-efficiency cascaded GHZ state based on an adiabatic fast channel according to claim 5, characterized in that, The differentiation process for the density matrix of the standard GHZ state is as follows: in, The density matrix representing the standard GHZ state. ; An index representing the number of decoherence effects. Indicates the first Operators for decoherence effects, express Hermitian conjugate operator, express and commutative relations - ; express and The opposition to the relationship, .

7. The cascaded GHZ state prepared by the efficient cascaded GHZ state preparation method based on adiabatic fast channel as described in any one of claims 1-6 is applied to an optical lattice atomic clock system.

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