Method, device and storage medium for measuring maximum entanglement fidelity of two ions

Abstract

This invention discloses a method, apparatus, and storage medium for measuring the fidelity of the maximum entangled state of two ions. Utilizing the exchange symmetry of the coefficient matrix in the maximum entangled state of two ions, without performing independent addressing operations on individual ions, the same global single-bit unitary operation is applied synchronously to both ions. This transforms the maximum entangled state of the two ions into a quantum state physically equivalent to the target Bell state. The transformation process relies solely on global operations, avoiding the errors and complexity caused by single-ion addressing. Furthermore, it eliminates the need for a global rotation operation around an axis; only two standard global rotation operations are required to convert the maximum entangled state into the target Bell state, reducing the operational depth and further improving the accuracy of fidelity measurement.

Description

Technical Field

[0001] This invention belongs to the field of quantum information technology, specifically relating to a method for measuring the maximum entanglement fidelity of two ions, and also to computer equipment and storage media. Background Technology

[0002] With the rapid development of quantum information science, trapped ion systems have been widely used in research directions such as quantum computing, quantum simulation and quantum precision measurement due to their advantages such as long coherence time, high manipulation precision and high quantum state readout efficiency. In ion trap quantum information experiments, the maximum entangled state of two ions (such as Bell state) is an important basic resource, and its preparation quality is usually quantitatively characterized by the fidelity of entangled state.

[0003] In existing technologies, the measurement of the fidelity of two-ion entangled states mainly relies on quantum state tomography. This method reconstructs the density matrix of the system by repeatedly measuring the system under multiple different measurement bases, and calculates the fidelity of the target entangled state accordingly. Although quantum state tomography has advantages such as wide applicability and complete information, the number of measurement settings increases rapidly with the system dimension, requiring a large number of repeated experiments in actual experiments, resulting in low measurement efficiency. In addition, existing methods usually rely on precise addressing operations for individual ions, but single-ion addressing is prone to introducing additional phase errors and crosstalk effects in experimental implementation, thereby reducing the accuracy and stability of the measurement results.

[0004] For standard Bell states, previous studies have proposed using methods such as parity oscillation to estimate the fidelity of entangled states by measuring some relevant operators, thus avoiding complete quantum state tomography. However, such methods usually assume that the target state is already in a specific Bell state form, or require local compensation transformation of the target state through single-ion addressing operations. When the maximally entangled state of two ions prepared in the experiment is equivalent to the standard Bell state only in the sense of local unitary transformation, it is still necessary to introduce single-ion addressing or revert to quantum state tomography, which limits the measurement efficiency and introduces additional errors.

[0005] In practical ion trap experiments, the following situation is common: two ions start from the same initial state and are prepared into a maximally entangled state through identical global quantum operations. Such entangled states usually have a certain symmetric structure, but existing measurement schemes fail to make full use of the structural features to complete the fidelity measurement without introducing single-ion addressing. Therefore, there is an urgent need for a new measurement method that can achieve a high-fidelity measurement with simple measurement steps, convenient experimental implementation, and high measurement accuracy for the maximally entangled state of two ions with this symmetric structure without the need for single-ion addressing operations. Summary of the Invention

[0006] The purpose of this invention is to address the aforementioned problems in the prior art by providing a method for measuring the maximum entanglement fidelity of two ions, and also providing a computer device and storage medium. When two ions are prepared into a maximum entangled state from the same initial state through global quantum operations, this invention can complete the measurement of the maximum entanglement fidelity without performing independent addressing operations on a single ion, thereby reducing the systematic errors introduced by addressing operations and improving measurement efficiency and the stability of measurement results.

[0007] The above-mentioned objectives of the present invention are achieved by the following technical means:

[0008] A method for measuring the fidelity of the maximum entangled state of two ions includes the following steps:

[0009] Step 1: Construct a coefficient matrix using the amplitude coefficients of the maximally entangled states of the two ions. ;

[0010] When the maximally entangled states of the two ions have exchange symmetry, the Takagi decomposition is used to pair the coefficient matrix. Singular value decomposition is performed to obtain the single-bit unitary matrix acting on the first ion. and the single-bit unitary matrix acting on the second ion ;

[0011] Step 2: Apply the same global single-bit unitary operation to both ions simultaneously. This transforms the maximally entangled state of the two ions into a quantum state equivalent to the target Bell state;

[0012] in, for The transpose of the matrix, for The inverse matrix, It is the tensor product;

[0013] Step 3: Perform joint measurements on the two ions and calculate the fidelity of the maximum entangled state of the two ions.

[0014] When the two ions are in their maximum entangled state When it has commutative symmetry, coefficient matrix ;

[0015] in, , , as well as All of them are computational bases for state space. It is the maximally entangled state In computational basis Amplitude on It is the maximally entangled state In computational basis and Amplitude on These are the maximally entangled states. In computational basis Amplitude on;

[0016] Directly on the coefficient matrix Perform Takagi decomposition to obtain the coefficient matrix. ;

[0017] Then the coefficient matrix is ​​obtained. .

[0018] As described above, global single-bit unitary operations Obtained through the following methods:

[0019] First, for a single-bit unitary matrix Do Decompose to obtain ,Will Substitute into the coefficient matrix Solving for the given information yields the following results. , as well as ;

[0020] in, and On Bloch's ball shaft and axis, The imaginary unit, For global phase factor, The Bloch vector of the quantum state of an ion revolves around the Bloch sphere. The first angle of rotation of the shaft, The Bloch vector of the quantum state of an ion revolves around the Bloch sphere. The angle of rotation of the shaft, To orbit the Bloch vector of the quantum state of an ion around the Bloch sphere The second angle of rotation of the shaft, The matrix representation is ; The matrix representation is ; The matrix representation is ;

[0021] Then, , as well as Substitution get and further obtained ;

[0022] in, The matrix representation is as follows , The matrix representation is as follows .

[0023] The fidelity of the maximally entangled state of the two ions, as described above, is obtained through the following steps:

[0024] First, the fluorescence signals obtained from fluorescence detection of the two ions are measured and calculated separately. With computational basis Population under and population The diagonal elements of the density matrix of the target Bell state are obtained. and diagonal elements diagonal element Both ions are The probability of a state, diagonal element Both ions are The probability of the state;

[0025] Secondly, applying different global rotation phases Oscillation curve of parity operator measured after pulse The oscillation amplitude is obtained by fitting. The coherence terms of the density matrix of the target Bell state are obtained, where, To be applied to Global rotation phase on the pulse;

[0026] Finally, according to The fidelity of the maximum entangled state of the two target ions was calculated. .

[0027] A computer device includes a memory and a processor, the memory storing a computer program, the processor executing the computer program to implement the steps of the method described above for measuring the maximum entangled state fidelity of two ions.

[0028] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method described above for measuring the maximum entangled state fidelity of two ions.

[0029] A computer program product includes a computer program that, when executed by a processor, implements the steps of the method described above for measuring the maximum entangled state fidelity of two ions.

[0030] Compared with the prior art, the present invention has the following advantages:

[0031] (1) This invention simplifies the local unitary transformation of a single ion to the same global single-bit unitary operation of two ions simultaneously. Without sacrificing fidelity, it completely avoids single-ion addressing, which has extremely high hardware requirements and is prone to errors, and reduces the system uncertainty introduced by addressing errors and crosstalk.

[0032] (2) Reduced operating depth: The method of the present invention also eliminates one winding path. Rotation angle of the shaft This also means one less detour The global rotation operation of the axis only requires two standard global rotation operations to convert the maximally entangled state into the target Bell state, reducing the depth of operation and further improving the accuracy of fidelity measurement.

[0033] (3) Simplified measurement process: This invention eliminates the need for complete quantum state tomography, significantly reducing the number of measurement setups and experimental repetitions, and improving measurement efficiency.

[0034] (4) High measurement accuracy and robustness: It makes full use of the structural features of the maximum entanglement state of the two ions, relies only on globally realizable quantum operations, and has low experimental implementation difficulty and strong robustness;

[0035] (5) Applicable to a variety of experimental platforms: The method of the present invention does not depend on a specific entangled state preparation mechanism and is applicable to a variety of two-bit physical systems with global single-bit manipulation capabilities, and has good versatility and promotion value. Attached Figure Description

[0036] Figure 1 This is a flowchart of the method of the present invention;

[0037] Figure 2 This is a graph showing the results of measuring the oscillation amplitude of the parity operator in Embodiment 2 of the present invention. Detailed Implementation

[0038] To facilitate understanding and implementation of the present invention by those skilled in the art, the present invention will be further described in detail below with reference to embodiments. The embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0039] Example 1:

[0040] like Figure 1 As shown, the method for measuring the fidelity of the maximum entangled state of two ions includes the following steps:

[0041] The two ions in this embodiment are of the same type (i.e., the same element and the same isotope).

[0042] Step 1: Prepare the maximally entangled state of the two ions, and construct the coefficient matrix corresponding to the maximally entangled state using the amplitude coefficient. And use singular value decomposition to transform the coefficient matrix Decompose into a set of target Bell states (the target Bell state is...) The equivalent unitary transformation relation yields the single-bit unitary matrix acting on the first ion. and the single-bit unitary matrix acting on the second ion The unitary transform can be equivalently implemented by applying a single-bit operation to two ions simultaneously, specifically including the following steps:

[0043] Step 1.1: Prepare the maximally entangled state of the two ions. Since the maximally entangled state must be a pure state, the maximally entangled state can be written as: The coefficient matrix constructed using the amplitude coefficient :

[0044] (1);

[0045] in, , , as well as It is a computational basis for state space, specifically, This indicates that both ions are state, The first ion and the second ion are respectively state and state, The first ion and the second ion are respectively state and state, Both ions are state; It is the maximally entangled state prepared. , , as well as These are the maximally entangled states. In computational basis , , as well as The amplitude on the above satisfies .

[0046] Step 1.2: Adjust the coefficient matrix. Perform singular value decomposition:

[0047] (2);

[0048] in, For the single-bit unitary matrix acting on the first ion, For the single-bit unitary matrix acting on the second ion, and Both are coefficient matrices singular values, and All are non-negative real numbers. In this embodiment, the maximally entangled state of the two ions is prepared. .

[0049] When the maximally entangled state of the two ions does not have exchange symmetry, the localization rotation of the first ion After rotation The corresponding coefficient matrix becomes Localization rotation of the second ion , The corresponding coefficient matrix is ; It is a tensor product. It is an identity matrix.

[0050] Therefore, it is possible to handle any maximally entangled state Perform a localized operation It is then transformed into the target Bell state; among which, It is a unitary matrix The conjugate matrix, To simultaneously perform local unitary transformation on the first ion and perform local unitary transformation on the second ion .

[0051] Step 2: When the maximally entangled states of the two ions have exchange symmetry, the coefficient matrix... It exhibits a complex symmetric structure, at which point we have Without performing independent addressing operations on individual ions, the same global single-bit unitary operation is applied synchronously to two ions. This method can transform the maximally entangled state of two ions into a quantum state that is physically equivalent to the target Bell state. The transformation process relies solely on global operations, avoiding the errors and complexity associated with single-ion addressing. Specifically, it includes the following steps:

[0052] When the maximally entangled state of two ions has exchange symmetry, the coefficient matrix It exhibits a complex symmetric structure, which can further simplify the global single-bit operation parameters (including...). , as well as This simplifies the determination process, making the measurement workflow more concise and efficient.

[0053] Step 2.1: If the maximally entangled state has This form is common; for example, the maximally entangled states prepared from two ions with the same initial state through identical operations all have this form.

[0054] The coefficient matrix is ​​simplified to After simplification, it is a complex symmetric matrix, where, The maximally entangled state of two ions has exchange symmetry. In computational basis and The amplitude on the coefficient matrix can be used to determine the amplitude of the amplitude. Perform Takagi decomposition (singular value decomposition of complex symmetric matrices):

[0055] (3);

[0056] In this embodiment, the maximally entangled state of the two ions is prepared, therefore .

[0057] Step 2.2: Without performing independent addressing operations on individual ions, apply the same global single-bit unitary operation synchronously to two ions. This transforms the maximally entangled state of the two ions into a quantum state that is physically equivalent to the target Bell state.

[0058] Step 2.3: For a single-bit unitary matrix Do Decompose and determine the single-bit unitary matrix. The operation parameters are used to determine the global single-bit unitary operation. The operating parameters specifically include the following steps:

[0059] Step 2.3.1: For a single-bit unitary matrix Do break down:

[0060] (4);

[0061] In the formula, and On Bloch's ball shaft and axis, The global phase factor (if two quantum states differ by only one complex factor) Therefore, these two states are physically indistinguishable, and the global phase will not affect any physical measurement results. (is a real number) The Bloch vector of the quantum state of an ion revolves around the Bloch sphere. The first angle of rotation of the shaft (that is, at) Before the action (angle of shaft rotation) The Bloch vector of the quantum state of an ion revolves around the Bloch sphere. The angle of rotation of the shaft, The Bloch vector of the quantum state of an ion revolves around the Bloch sphere. The second angle of rotation of the axis (that is, at) After the action (angle of shaft rotation) The Bloch vector, representing the quantum state of an ion, revolves around the Bloch sphere. Axis rotation , The corresponding matrix representation is ; The Bloch vector, representing the quantum state of an ion, revolves around the Bloch sphere. Axis rotation , The corresponding matrix representation is ; The Bloch vector, representing the quantum state of an ion, revolves around the Bloch sphere. Axis rotation , The corresponding matrix representation is .

[0062] A single-qubit system consisting of two selected energy levels of an ion (such as...) 40 Ca + Ionic Sublevels after Zeeman splitting and The sublevels following Zeeman splitting can serve as qubits. state and Both the pure state and the mixed state of an ion can be represented by a vector on the Bloch sphere, namely the Bloch vector. The Bloch vector of the quantum state of an ion is a vector pointing from the center of the Bloch sphere to a point on the surface of the Bloch sphere.

[0063] Step 2.3.2, will Substitution ,get:

[0064] (5);

[0065] In the formula, The Bloch vector, representing the quantum state of an ion, revolves around the Bloch sphere. Axis rotation , The matrix representation is ; The Bloch vector, representing the quantum state of an ion, revolves around the Bloch sphere. Axis rotation ; Calculate according to formula (5) , as well as .

[0066] Step 2.3.3, will , as well as Substitution ,get and further obtained .

[0067] in, The Bloch vector of the quantum state of an ion revolves around the Bloch sphere. Axis rotation , The matrix representation is as follows .

[0068] In this embodiment, a global single-bit unitary operation can be used. Transform the maximally entangled state of the two ions into the target Bell state. Since the global phase does not affect the measurement properties, this is in 40 Ca + In an ion trap system, it is equivalent to two 40 Ca + Maximum entangled states of ions Two 729nm pulses are applied to convert the target Bell state; compared to quantum state tomography, this greatly simplifies the measurement of the maximum entanglement fidelity in the symmetric case.

[0069] This invention utilizes the exchange symmetry generated by a specific preparation process. By simplifying the application of local unitary transformations to individual ions into the application of the same global single-bit unitary operation to two ions simultaneously, it completely avoids single-ion addressing, which is extremely demanding on hardware and prone to errors, without sacrificing fidelity.

[0070] Furthermore, the method of the present invention lacks a winding... Rotation angle of the shaft This also means one less detour The global rotation operation of the axis only requires two standard global rotation operations to convert the maximally entangled state into the target Bell state, reducing the depth of operation and further improving the accuracy of fidelity measurement.

[0071] Step 3: After converting the maximally entangled state of the two ions into the target Bell state, the two ions are jointly measured using the parity oscillation method to calculate the fidelity of the maximally entangled state. This specifically includes the following steps:

[0072] Step 3.1: Perform fluorescence detection on the two ions, and measure... Computational basis and Population under computational basis and population The diagonal elements of the density matrix of the target Bell state are obtained. and diagonal elements diagonal element Both ions are The probability of a state, diagonal element Both ions are The probability of the state.

[0073] Step 3.2: Apply different global rotation phases Oscillation curve of parity operator measured after pulse The oscillation amplitude is obtained by fitting. According to the relation Obtain the coherence terms of the density matrix of the target Bell state, where, To be applied to Global rotation phase on the pulse.

[0074] Step 3.3: Finally, the fidelity of the maximum entangled state of the two target ions is calculated based on the diagonal and coherent terms. .

[0075] Example 2:

[0076] In this embodiment, the method for measuring the maximum entanglement fidelity of two ions as described in Example 1 is used. 40 Ca + Two ion trap systems 40 Ca + The transformation of the ion's maximally entangled state into the target Bell state involves the following process:

[0077] This embodiment 40 Ca + The ion trap system employs a linear Paul ion trap to trap two electrons. 40 Ca + Ions, selection 40 Ca + Ionic Sublevels after Zeeman splitting and The sub-levels after Zeeman splitting serve as qubits. state and The system utilizes a narrow-linewidth laser with a wavelength of 729 nm to achieve coherent manipulation of the internal states of ions.

[0078] Perform step 1 of Example 1 to prepare two 40 Ca + The maximally entangled state of an ion is specifically:

[0079] First, the two are cooled by laser and pumped by optical means. 40 Ca + Ions are initialized to the same ground state .

[0080] Subsequently, four-color lasers were simultaneously applied to the two ions, at both... 40 Ca + Ionic Energy levels and The subspace preparation of energy levels yields two symmetric forms. 40 Ca + The density matrix of the maximally entangled ions, after normalization, is as follows: .

[0081] By performing step 2 of Example 1, we obtain and They are respectively and Then we get This further yields global single-bit unitary operations. .

[0082] two 40 Ca + A global 729nm laser pulse is applied to the maximally entangled state of the ions. By adjusting the phase, duration, and amplitude of the global 729nm laser pulse, the entanglement of the two ions can be controlled. 40 Ca + Synchronous global single-bit unitary operation of ions , will two 40 Ca + The maximally entangled state of an ion is transformed into a quantum state equivalent to the target Bell state. Since the global single-bit unitary operation is identical for both ions, single-ion addressing is not required.

[0083] Perform step 3 of Example 1 to detect the fluorescence of the two ions:

[0084] First, under the statistical calculation basis and The population probability.

[0085] Secondly, the oscillation amplitudes of the parity operators are measured under different global rotation phases to obtain the coherent terms of the target Bell state; such as Figure 2 As shown, the measured oscillation curve of the parity operator exhibits a clear sinusoidal oscillation (the red solid line represents the theoretical fitted curve). The oscillation amplitude is then extracted by fitting the experimental data points (marked in blue). .

[0086] Finally, through the formula Achieve fidelity.

[0087] In one embodiment, a computer device is also provided, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps in the above method embodiments.

[0088] In one embodiment, a computer-readable storage medium is provided having a computer program stored thereon that, when executed by a processor, implements the steps in the above method embodiments.

[0089] In one embodiment, a computer program product is provided, including a computer program that, when executed by a processor, implements the steps in the above method embodiments.

[0090] It should be noted that the embodiments described in this invention are merely illustrative of the spirit of the invention. Those skilled in the art to which this invention pertains can make various modifications or additions to the described embodiments or use similar methods to substitute them, without departing from the spirit of the invention or exceeding the scope defined by the appended claims.

Claims

1. A method of measuring the maximum entanglement fidelity of two ions, characterized in that, Includes the following steps: Step 1, Constructing the coefficient matrix by using the amplitude coefficients of the maximum entangled state of two ions ; When the maximally entangled states of the two ions have exchange symmetry, the Takagi decomposition is used to pair the coefficient matrix. Singular value decomposition is performed to obtain the single-bit unitary matrix acting on the first ion. and the single-bit unitary matrix acting on the second ion ; Step 2: Apply the same global single-bit unitary operation to both ions simultaneously. This transforms the maximally entangled state of the two ions into a quantum state equivalent to the target Bell state; in, for The transpose of the matrix, for The inverse matrix, It is the tensor product; Step 3: Perform joint measurements on the two ions and calculate the fidelity of the maximum entangled state of the two ions.

2. The method for measuring the maximum entanglement fidelity of two ions according to claim 1, characterized in that, When the two ions are in their maximum entangled state When it has commutative symmetry, coefficient matrix ; in, , , as well as All of them are computational bases for state space. It is the maximally entangled state In computational basis Amplitude on It is the maximally entangled state In computational basis and Amplitude on These are the maximally entangled states. In computational basis Amplitude on; Directly on the coefficient matrix Perform Takagi decomposition to obtain the coefficient matrix. ; Then we can further obtain the coefficient matrix. .

3. The method for measuring the maximum entanglement fidelity of two ions according to claim 2, characterized in that, The global single-bit unitary operation Obtained through the following methods: First, for a single-bit unitary matrix Do Decompose to obtain ,Will Substitute into the coefficient matrix Solving for the given information yields the following results. , as well as ; in, and On Bloch's ball shaft and axis, The imaginary unit, For global phase factor, The Bloch vector of the quantum state of an ion revolves around the Bloch sphere. The first angle of rotation of the shaft, The Bloch vector of the quantum state of an ion revolves around the Bloch sphere. The angle of rotation of the shaft, To orbit the Bloch vector of the quantum state of an ion around the Bloch sphere The second angle of rotation of the shaft, The matrix representation is ; The matrix representation is ; The matrix representation is ; Then, , as well as Substitution get and further obtained ; in, The matrix representation is as follows , The matrix representation is as follows .

4. The method for measuring the maximum entanglement fidelity of two ions according to claim 3, characterized in that, The fidelity of the maximum entangled state of the two ions is obtained through the following steps: First, the fluorescence signals obtained from fluorescence detection of the two ions are measured and calculated separately. With computational basis Population under and population The diagonal elements of the density matrix of the target Bell state are obtained. and diagonal elements diagonal element Both ions are The probability of a state, diagonal element Both ions are The probability of the state; Secondly, applying different global rotation phases Oscillation curve of parity operator measured after pulse The oscillation amplitude is obtained by fitting. The coherence terms of the density matrix of the target Bell state are obtained, where, To be applied to Global rotation phase on the pulse; Finally, according to The fidelity of the maximum entangled state of the two target ions was calculated. .

5. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method for measuring the maximum entangled state fidelity of two ions as described in any one of claims 1 to 4.

6. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of the method for measuring the maximum entangled state fidelity of two ions as described in any one of claims 1 to 4.

7. A computer program product, comprising a computer program, characterized in that, When executed by a processor, the computer program implements the steps of the method for measuring the maximum entangled state fidelity of two ions as described in any one of claims 1 to 4.

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