Two-dimensional and three-dimensional quantum bit forecast type quantum gate method based on cavity QED system

Abstract

The invention discloses a two-dimensional and three-dimensional quantum bit forecast type quantum gate method based on a cavity QED system, in the method, an optical cavity constrains an auxiliary atom and two target atoms, state evolution is carried out under the action of an external weak driving field, the auxiliary atom can be utilized to realize error forecast elimination, and the accuracy of state evolution is improved. The auxiliary atoms serve as an integrated error detector in the optical cavity and also play roles of a single-photon source and a single-photon detector; by means of alternating-current stark displacement generated under the effect of a non-resonance light field, quantum state phase regulation and control can be achieved, and therefore the three-dimensional two-qutrit prediction type condition phase gate is achieved. The method is of great significance in realizing more complex quantum algorithms and quantum computing tasks and promoting quantum communication and quantum network development.

Description

Technical Field

[0001] The present invention belongs to the field of quantum computing technology, and in particular relates to a two- and three-dimensional quantum bit predictive quantum gate method based on a cavity QED system. Background Art

[0002] Quantum gates based on optical cavities are often experimentally challenging due to atomic spontaneous emission and dissipation in the optical cavity. To effectively avoid these detectability errors, Borregaard et al. proposed and analyzed the concept of a quantum predictor gate in an optical cavity in 2015. The core concept of this predictor operation is to convert the fidelity of the resulting deterministic gate into a failure probability predicted by an auxiliary atom. Such a predicted quantum gate has only an arbitrarily small error if it succeeds. Subsequently, Qin et al. applied this scheme to a circuit QED system constructed with two remotely distributed resonators in 2017, using this setup to realize a phase-controlled two-qubit controlled Z gate and a multi-qubit Toffoli gate. Furthermore, in 2023, Su et al. extended Borrgaard et al.'s optical cavity scheme to a three-cavity system, realizing distributed quantum computing with qubit atoms using a fiber-cavity coupling system. Furthermore, they utilized decoherence-free photon space (DFS) coding to further enhance the reliability of this predicted quantum gate scheme. However, the carriers used in these works are all qubits, and there is still a gap in high-dimensional quantum high-fidelity manipulation. For quantum circuits and quantum networks that require a lot of resources, the use of high-dimensional quantum gates is undoubtedly an effective solution. Summary of the Invention

[0003] The purpose of the present invention is to provide a two-dimensional qubit predictive quantum gate method based on a cavity QED system, which solves the problem of high-fidelity control of high-dimensional quantum gates by integrating error detection and phase control.

[0004] The technical solutions for achieving the purpose of the present invention are:

[0005] A two-dimensional quantum bit predictive quantum gate method based on a cavity QED system, comprising:

[0006] Step 1: Construct a two-dimensional quantum bit prediction quantum gate physical model;

[0007] Step 2: Perform dissipative dynamics analysis on the Hamiltonian of the physical model through the Lindblad form master equation and perform analytical calculations to describe the dissipative dynamics evolution process of the physical model;

[0008] Step 3: Based on the dissipative dynamics evolution process of the physical model, the decay factors corresponding to different coupled states are made the same, and the phase caused by decay is removed as the global phase;

[0009] Step 4: Based on the analytical calculation of the physical model, set the pulse time, and after the pulse ends, combine the single three-dimensional quantum bit operation to realize the predicted two-dimensional quantum bit conditional phase gate.

[0010] Furthermore, the two-dimensional quantum bit prediction quantum gate physical model includes an optical microcavity, in which an auxiliary atom and two target atoms are bound. The auxiliary atom and the target atom both have four-level structures, and the energy levels of the auxiliary atom include a ground state |g>, an excited state |E>, and two states |f> coupled to the cavity field. α >、|f β >, the energy levels of the target atom include the ground state |0>, |1>, |2> and the excited state |e>, among which |0> and |1> are coupled with the cavity field; the external weak laser drives the auxiliary atom to make the |g>→|E> transition; the coupling strengths between the optical cavity and the auxiliary atom and the target atom are g1 and g2 respectively, and they meet the strong coupling condition; the decay of the excited state |E> in the auxiliary atom to the three ground states is respectively determined by γ α , γ β and γ g The decay of the |e> state in the target atom to the three ground states is described by γ0, γ1 and γ2, respectively, and the cavity leakage rate is κ.

[0011] Furthermore, the effective Hamiltonian is obtained by analyzing the Lindblad form master equation:

[0012]

[0013] Among them, Δ n is the AC Stark shift, P n represents the projection of the system state on the subspace, and n represents the number of target atoms in the cavity coupling state |1> state or |2> state.

[0014] Further analytical calculations yield:

[0015]

[0016] Among them, r n,g 、r n,f 、r n,c 、r n,k is the corresponding effective decay rate, and f represents f α or f β state, d is any of the two states 1 and 2, k is any of the three states 0, 1, and 2, r n,g is the effective decay rate corresponding to the auxiliary atom decaying to the |g> state, are the effective Lindblad operators corresponding to the spontaneous radiation of the auxiliary atom to the |g> state and |f> state, are the effective Lindblad operators corresponding to cavity leakage and various types of spontaneous radiation of target atoms, respectively.

[0017] Furthermore, the dissipative dynamics evolution process of the physical model is described as follows:

[0018]

[0019] in, is the density matrix, L k and are the Lindblad operator and the effective Lindbald operator effective quantity, respectively. is the effective Hamiltonian, and i is the imaginary number symbol.

[0020] Furthermore, step three specifically includes:

[0021] The decay factors corresponding to different coupling states are calculated using the effective Hamiltonian and the effective Linblad algorithm.

[0022] Γ n =|r n,g | 2 +|r n,f | 2 +|r n,c | 2 +n|r n,k | 2 ;

[0023] The three decay factors are made to satisfy Γ0=Γ1=Γ2 to ensure that the same phase shift is applied to different states, which is treated as a global phase shift to be removed. The detuning amount is adjusted by derivation and calculation to satisfy the following relationship:

[0024]

[0025] are the auxiliary atomic excited state, the target atomic excited state and the effective detuning of the cavity field, respectively, C f is the cavity coherence; under such a detuning setting, the decay factor caused by The phase shift is eliminated, i.e. the auxiliary atom is in the |g> state under measurement, suppressing the influence of the undetectable error part on the system.

[0026] Furthermore, the single three-dimensional qubit operation in step 4 is:

[0027]

[0028] Where U is the unitary transformation, and Δ0, Δ1, and Δ2 are the AC Stark shifts corresponding to different numbers of coupled atoms.

[0029] Furthermore, the pulse length in step 4 is:

[0030]

[0031] where φ is the phase.

[0032] Compared with the prior art, the present invention has the following beneficial effects: the optical cavity of the present invention confines an auxiliary atom and two target atoms, and their states evolve under the action of an external weak driving field; the present invention can utilize the auxiliary atoms to achieve error prediction elimination, and the auxiliary atoms not only serve as integrated error detectors in the optical cavity, but also play the role of single-photon source and single-photon detector; the present invention can realize quantum state phase control by utilizing the AC Stark shift generated under the action of a non-resonant light field, thereby realizing a three-dimensional two-qutrit predictive conditional phase gate; this method is of great significance for realizing more complex quantum algorithms and quantum computing tasks, and promoting the development of quantum communications and quantum networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] In order to more clearly illustrate the objectives, technical implementation details and significant advantages of the present invention, the present invention is hereby attached with the following figures as a supplement to the detailed description:

[0034] Figure 1 Schematic diagram of the experimental setup of a predicted two-qubit three-dimensional quantum gate based on an optical cavity.

[0035] Figure 2 Schematic diagram of the atomic energy level structure in the optical cavity. Figure 2 (a) Schematic diagram of the auxiliary atomic energy level structure, (b) Schematic diagram of the target atomic energy level structure.

[0036] Figure 3 The relationship between the fidelity and error rate of the conditional phase gate and the cavity cooperativity (Δ c / κ=1).

[0037] Figure 4 Figure 3 is the relationship between the success probability and error rate of the conditional phase gate and the cavity mode detuning amount (C=300). DETAILED DESCRIPTION

[0038] The present application will be further described in detail below in conjunction with the accompanying drawings and examples. It will be understood that the specific embodiments described herein are only intended to explain the relevant invention and are not intended to limit the invention. After reading the present invention, modifications to various equivalent forms of the present invention made by those skilled in the art all fall within the scope defined by the claims appended hereto.

[0039] The present invention provides a two- and three-dimensional qubit predictive quantum gate method based on a cavity QED system, specifically comprising:

[0040] Step 1: Construct a two-qutrit (three-dimensional quantum bit) prediction quantum gate physical model, specifically:

[0041] like Figure 1 The device diagram shown utilizes an optical microcavity, a four-level auxiliary atom, two four-level target atoms and an external weak laser driving field. The auxiliary atom and the target atom realize dynamic phase control through light field coupling, and eliminate detectable errors through state detection to realize a high-fidelity quantum gate. The core concept of this prediction operation is to convert the infidelity of the deterministic gate in the obtained gate into the failure probability predicted by the auxiliary atom. Such a predicted quantum gate will only have an arbitrarily small error as long as it is successful. The auxiliary atom not only acts as an error predictor for the predicted quantum gate, but also acts as a photon source and detector in the cavity, solving the problems of photon loss, detector inefficiency and non-ideal single photon source. The energy level structure of the auxiliary atom and the target atom is shown in the figure. Figure 2 As shown in (a), the auxiliary atom's |f α >、|f β The |E> and |g> states are both ground states, and |E> is an excited state. The decays of |E> state to the three ground states are respectively determined by γ α , γ β and γ g Description. The |g> state is adiabatically excited to the |E> state by an external weak laser field with a driving strength of Ω. |f α >、|f β The transition between the |0> and |E> states is considered a closed transition and is coupled to the light field mode in the cavity with the same coupling strength as g1. Meanwhile, the arrangement of the target atom is similar to that of the auxiliary atom, except that the external weak drive between the |0> and |e> states is missing, e.g. Figure 2 (b) Except for the correlated states between the cavity and the auxiliary and target atoms, the other unrelated states are completely uncoupled. This ensures that the system does not change any unrelated states in the bare cavity mode. However, when any one or more atoms in the cavity are in the cavity-coupled state, an AC Stark shift and a controllable phase are introduced into the overall state of the system.

[0042] Step 2: Based on the physical model, the dissipative dynamics analysis of the Hamiltonian of the physical model is performed through the Lindblad form master equation, and analytical calculation is performed:

[0043] Step 2.1, select a suitable rotating coordinate system according to the model Hamiltonian. The present invention can write the total Hamiltonian as In the form of H e Represents the main part of the system and provides the basic physical image of the system. This step will distinguish the external driving term V of the system and conduct subsequent main equation derivation to observe the impact of perturbation on the system. e , V can be expressed as:

[0044]

[0045] Here Δ E , Δ e , Δ c where |E> and |e> are the excitation energy levels, respectively, and the detuning between the cavity mode frequency and the laser field frequency. a is the annihilation operator for the cavity mode, and Hc represents the conjugate portion. Furthermore, because the system's evolutionary timescale is much larger than the period of the external weak laser field, the present invention can assume that the system's state changes are relatively slow compared to the oscillations of the driving field. Therefore, the present invention removes the rapid oscillation term of the driving field from the system.

[0046] Step 2.2, based on the background of the physical model, the correlation time between the system and the environment is much smaller than the evolution time scale of the system itself. The present invention can consider the evolution of the system to be independent of the historical environmental time, thereby ignoring the memory effect of the environment. This approximation is called the Markov approximation. Combined with the assumption that the system and the environment are in a weakly coupled state, it is further considered to be the Born-Markov approximation. Through such an approximation, an integral-differential equation containing a time memory term can be simplified to a differential equation that only focuses on the local area. Through such an approximation, the Markov master equation of the system is obtained. Combined with the rotating wave approximation made in the previous article, the present invention can write the master equation in Lindblad form:

[0047]

[0048] Step 2.3, further use the effective Hamiltonian to describe the system evolution process, as follows:

[0049]

[0050] in, is the density matrix, L k and are the Lindblad operator and its effective quantity, is the effective Hamiltonian. The effective Hamiltonian and the effective Lindbald operator are given by the following formulas:

[0051]

[0052] The present invention can obtain the effective amount of Hamiltonian and Lindblad operator by using the above formula, thereby greatly reducing the computational complexity, where H NH is the transition-free operator, which represents the evolution of the system without quantum jumps. Note that the quantum system does not undergo transitions, but still exhibits non-unit evolution due to coupling with the environment, which is described by the Lindblad operator.

[0053] Step 2.4, based on the non-transition operator obtained in the previous step, expand it in the single excitation subspace, so as to calculate the effective Hamiltonian and effective Lindblad operator with a lower dimensional matrix. Through derivation and organization, the effective Hamiltonian can be written as:

[0054] where Δ n is the AC Stark shift, P n Represents the projection of the system state on the subspace, and n represents the number of target atoms in the cavity coupling state |1> or |2> state. All atomic spontaneous emission parameters are given a value, that is, γ α =γ β =γ g =γ0=γ1=γ2=γ, and the effective Lindblad operator is calculated, thus obtaining the following relationship:

[0055]

[0056]

[0057] where r n,g 、r n,f 、r n,c 、r n,k is the corresponding effective decay rate, and f represents f α or f β state, d is any state between 1 and 2, and k is any state between 0, 1, and 2. n,g is the effective decay rate corresponding to the decay of the auxiliary atom to the |g> state.

[0058] Step 3: The effective Hamiltonian and effective Linblad algorithm obtained in step 2 can be used to calculate the decay factors Γ corresponding to different coupling states. n =|r n,g | 2 +|r n,f | 2 +|r n,c | 2 +n|r n,k | 2 By making the three decay factors satisfy Γ0=Γ1=Γ2, we ensure that they apply the same phase shift to different states, so that the present invention can treat it as a global phase shift to remove it. The detuning amount is adjusted by derivation and calculation to satisfy the following relationship:

[0059]

[0060] in, are the auxiliary atomic excited state, the target atomic excited state and the effective detuning of the cavity field, respectively. Cf is the cavity coherence. Under such a detuning setting, the decay factor caused by The phase shift can be eliminated, that is, the auxiliary atom is in the |g> state during measurement, which can effectively suppress the influence of the undetectable error part on the system.

[0061] Step 4: Based on the parameter calculation in the previous step, since Δ0≠Δ1≠Δ2, the present invention needs to introduce an appropriate unitary transformation, that is, the following single qutrit operation:

[0062]

[0063] Therefore, the present invention can ensure that the corresponding quantum states when the number of atoms in the coupled state is 1 and 2 will not produce any phase shift. At the same time, the present invention adjusts the pulse length to:

[0064]

[0065] In this way, the quantum gate will only impose an arbitrary phase φ on the quantum state |00> corresponding to the time when the number of atoms in the coupled state is 0, without changing any other states, thus realizing a two-qutrit predictive conditional phase gate.

[0066] Step 5: Use the qutip library in Python to solve the main equations of the physical model to simulate the actual evolution process of the system to verify the accuracy of the derived calculations, and obtain the fidelity and success probability of the quantum gate constructed by the scheme to discuss its reliability and advantages and disadvantages.

[0067] Step 5.1: During the simulation, first construct the Hilbert space of the multi-subsystem (cavity mode, auxiliary atom, two target atoms) described by the scheme, and define their transition operators.

[0068] In step 5.2, based on this simulation system, by constructing the total Hamiltonian and the corresponding Lindblad master equation, the time evolution of the entire system under a given initial state is solved with appropriate parameters. The initial state of the system is assumed to be that the optical cavity is in the cavity vacuum state |vac>, the auxiliary atom is in the |g> state, and the two target atoms are in a uniform superposition state. This can be expressed as:

[0069]

[0070] By defining the original total Hamiltonian to perform quantum system evolution, the results, when the parameters calculated by this method are substituted, accurately reflect the actual system performance and verify the correctness of the parameters. The original Hamiltonian does not undergo master equation transformation and adiabatic elimination, as this simulation is unaffected by various approximations and simplifications.

[0071] In step 5.3, the fidelity of the final state is calculated by extracting the density matrix of the system at the end of the evolution, thereby verifying the implementation effect of the conditional phase gate operation. After simulating the system evolution through the mesolve function, the success probability and fidelity of the quantum gate are calculated using the following formula:

[0072]

[0073] where ρ T is the total density operator, ρ qutrit is the conditional density operator after measuring the auxiliary atom, U represents a single qutrit operation, |Ψ f > is the ideal state of the target atom after time evolution, and ρ qutrit It is derived from the following formula:

[0074]

[0075] Here Tr cav is the trace operation performed on the cavity.

[0076] In order to verify the effectiveness of this scheme, qutip in Python is selected to simulate the evolution of the quantum system. The specific process is as follows:

[0077] First, the Hilbert space of the multi-subsystem (cavity mode, auxiliary atom, two target atoms) described by the scheme is constructed, and their transition operators are defined. By constructing the total Hamiltonian and the corresponding Lindblad master equation, the time evolution process of the entire system under a given initial state is solved under appropriate parameter settings. The initial state of the system is that the optical cavity is in the cavity vacuum state |vac>, the auxiliary atom is in the |g> state, and the two target atoms are in a uniform superposition state. By extracting the density matrix of the system at the end of the evolution to calculate the fidelity of the final state, the effect of the conditional phase gate operation is verified. The success probability and fidelity of the quantum gate are obtained by the following formula:

[0078]

[0079] where ρ T is the total density operator, ρ qutrit is the conditional density operator after measuring the auxiliary atom, U represents a single qutrit operation, |Ψ f > is the ideal state of the target atom after time evolution, and ρ qutrit It is derived from the following formula:

[0080]

[0081] The results are as follows:

[0082] like Figure 3As shown, the overall decay rate γ of the system is assumed to be 0.1, and the decay of |E>→|g> is assumed to be one tenth of the overall decay rate, and the cavity detuning Δ c / κ is defined as 1. At this time, observing the graph, we can see that the error rate 1-F of the quantum gate system decreases as the cavity cooperativity C increases from 100 to 1000, but even when C is equal to 100, the quantum gate error rate is still less than 4×10 -3 As C increases from 100 to 500, the decrease in error rate is more pronounced than subsequent changes; as C increases from 500 to 1000, the magnitude of the change in error rate decreases. Theoretically, after eliminating detectable errors such as cavity leakage and atomic spontaneous emission, the error rate of this predicted quantum gate can be made arbitrarily small.

[0083] like Figure 4 As shown, when the cavity detuning is small (Δ c / κ<1), the success rate of quantum gate is almost unaffected by detuning. c When / κ>10, the success probability will drop sharply. After analysis, it is believed that this is due to the detuning amount Δ c Too large will result in the cavity mode being unable to effectively couple with the atom. Therefore, the auxiliary atom in the cavity will transition to the excited state |E> under the drive of the external weak laser field, and no energy exchange will occur with the cavity. c When it is at a smaller level, it will not affect the success probability and error rate.

[0084] The above-described embodiments merely represent several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present application. It should be noted that a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, and these modifications and improvements fall within the scope of protection of the present application. Therefore, the scope of protection of the present application shall be determined by the appended claims.

Claims

1. A two-dimensional qubit predictive quantum gate method based on a cavity QED system, characterized in that: include: Step 1: Construct a two-dimensional quantum bit prediction quantum gate physical model; Step 2: Perform dissipative dynamics analysis on the Hamiltonian of the physical model through the Lindblad form master equation and perform analytical calculations to describe the dissipative dynamics evolution process of the physical model; Step 3: Based on the dissipative dynamics evolution process of the physical model, the decay factors corresponding to different coupled states are made the same, and the phase caused by decay is removed as the global phase; Step 4: Based on the analytical calculation of the physical model, set the pulse time, and after the pulse ends, combine the single three-dimensional quantum bit operation to realize the predicted two-dimensional quantum bit conditional phase gate.

2. A two-dimensional qubit predictive quantum gate method based on a cavity QED system according to claim 1, characterized in that: The two-dimensional quantum bit prediction quantum gate physical model includes an optical microcavity, in which an auxiliary atom and two target atoms are bound. The auxiliary atom and the target atoms both have a four-level structure. The energy levels of the auxiliary atom include a ground state |g>, an excited state |E>, and two states |f> coupled to the cavity field. α >、|f β >, the energy levels of the target atom include the ground state |0>, |1>, |2> and the excited state |e>, among which |0> and |1> are coupled with the cavity field; the external weak laser drives the auxiliary atom to make the |g>→|E> transition; the coupling strengths between the optical cavity and the auxiliary atom and the target atom are g1 and g2 respectively, and they meet the strong coupling condition; the decay of the excited state |E> in the auxiliary atom to the three ground states is respectively determined by γ α , γ β and γ g The decay of the |e> state in the target atom to the three ground states is described by γ0, γ1 and γ2, respectively, and the cavity leakage rate is κ.

3. The two-dimensional qubit predictive quantum gate method based on a cavity QED system according to claim 2, characterized in that: The effective Hamiltonian obtained by analyzing the Lindblad form master equation is: Among them, Δ n is the AC Stark shift, P n represents the projection of the system state on the subspace, and n represents the number of target atoms in the cavity coupling state |1> state or |2> state.

4. The two-dimensional qubit predictive quantum gate method based on a cavity QED system according to claim 3, characterized in that: Analytical calculation yields: Among them, r n,g 、r n,f 、r n,c 、r n,k is the corresponding effective decay rate, and f represents f α or f β state, d is any of the two states 1 and 2, k is any of the three states 0, 1, and 2, r n,g is the effective decay rate corresponding to the auxiliary atom decaying to the |g> state, are the effective Lindblad operators corresponding to the spontaneous radiation of the auxiliary atom to the |g> state and |f> state, are the effective Lindblad operators corresponding to cavity leakage and various types of spontaneous radiation of target atoms, respectively.

5. The two-dimensional qubit predictive quantum gate method based on a cavity QED system according to claim 4, characterized in that: The dissipative dynamics evolution process describing the physical model is: in, is the density matrix, L k and are the Lindblad operator and the effective Lindbald operator effective quantity, respectively. is the effective Hamiltonian, and i is the imaginary number symbol.

6. The two-dimensional qubit predictive quantum gate method based on a cavity QED system according to claim 5, characterized in that: Step three specifically includes: The decay factors corresponding to different coupling states are calculated using the effective Hamiltonian and the effective Linblad algorithm. The three decay factors are made to satisfy Γ0=Γ1=Γ2 to ensure that the same phase shift is applied to different states. This is treated as a global phase shift to be removed. The detuning amount is adjusted by derivation and calculation to satisfy the following relationship: are the auxiliary atomic excited state, the target atomic excited state and the effective detuning of the cavity field, respectively, C f is the cavity coherence; under such a detuning setting, the decay factor caused by The phase shift is eliminated, i.e. the auxiliary atom is in the |g> state under measurement, suppressing the influence of the undetectable error part on the system.

7. The two-dimensional qubit predictive quantum gate method based on a cavity QED system according to claim 6, characterized in that: The single 3D qubit operation in step 4 is: Where U is the unitary transformation, and Δ0, Δ1, and Δ2 are the AC Stark shifts corresponding to different numbers of coupled atoms.

8. The two-dimensional qubit predictive quantum gate method based on a cavity QED system according to claim 7, characterized in that: The pulse length in step 4 is: where φ is the phase.

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