CN111860550A - Method for obtaining threshold line for confirming quantum state of quantum bit - Google Patents
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Abstract
The invention discloses a method for acquiring a threshold line for confirming quantum state of a quantum bit, and relates to the field of quantum measurement and control; preparing the qubits into a first quantum state and a second quantum state, respectively carrying out repeated measurement on the qubits to obtain coordinate point data of a plurality of qubit reading signals on an orthogonal plane coordinate system, respectively recording the coordinate point data as a first set and a second set, respectively carrying out Gaussian fitting on the coordinate points of the first set and the second set to obtain a first statistical center coordinate point and a second statistical center coordinate point of Gaussian fitting graphs respectively corresponding to the first set and the second set, and respectively corresponding to a first standard deviation and a second standard deviation; determining a first probability density distribution function in the first set and a second probability density distribution function in the second set, respectively; determining a fidelity function, and determining a corresponding threshold line as an optimal threshold line when the fidelity function takes the maximum value; the threshold line established according to the method of the invention can provide more accurate reference for judging unknown quantum states.
Description
Technical Field
The invention belongs to the field of quantum measurement and control, and particularly relates to a method for acquiring a threshold line for confirming quantum states of quantum bits.
Background
The qubit information refers to the quantum state of the qubit, the basic quantum states are |0> state and |1> state, the quantum state of the qubit changes after the qubit is operated, and the qubit information is embodied as an execution result of the quantum chip, which is the quantum state of the qubit after the quantum chip is executed, and the execution result is carried by a qubit reading signal and is transmitted.
The fast resolution of the qubit quantum state by means of the qubit read signal is a key task to know the performance of the quantum chip execution, and a qubit quantum state determination method is provided in the previously applied patent, which comprises the following steps: acquiring distribution patterns of corresponding quantum bit reading signals in an orthogonal plane coordinate system when the quantum bits are in two different known quantum states; acquiring the central positions of the two distribution patterns in an orthogonal plane coordinate system, and determining a perpendicular bisector of a connecting line of the two central positions as a threshold dividing line; and taking the threshold dividing line as a basis for judging the quantum state of the quantum bit.
The problem in the prior art is that, in an ideal situation, the arrangement that the perpendicular bisector of the connecting line between the central points of the two distributed patterns is used as the threshold dividing line can meet the requirement in most of the time, but due to factors such as preparation errors of quantum states, the threshold line has errors which may affect the reading result of unknown quantum states.
Disclosure of Invention
It is an object of the present invention to provide a method for obtaining a threshold line for identifying qubit quantum states, which solves the deficiencies of the prior art and which provides a more accurate threshold split line for use in quantum state discrimination.
The technical scheme adopted by the invention is as follows:
an acquisition method for identifying a threshold line of qubit quantum states, comprising:
preparing the quantum bit into a first quantum state and repeatedly measuring the quantum bit to obtain coordinate point data of a plurality of quantum bit reading signals on an orthogonal plane coordinate system, and marking the coordinate point data as a first set R|0>(ii) a Preparing the quantum bit into a second quantum state and repeatedly measuring the quantum bit to obtain coordinate point data of a plurality of quantum bit reading signals on an orthogonal plane coordinate system, and marking the coordinate point data as a second set R|1>Wherein: the first and second quantum states are both known quantum states and are different from each other, wherein: the orthogonal plane coordinate system is set as an I-Q coordinate system;
Respectively comparing all coordinate points in the first set with all coordinate points in the second setPerforming Gaussian fitting to obtain a first statistical center coordinate point (I) of the Gaussian fitting graph corresponding to the first set and the second set respectively|0>,Q|0>) And a second statistical center coordinate point (I)|1>,Q|1>) Respectively corresponding first standard deviation σ1And a second standard deviation σ2(ii) a Wherein: for coordinating the first statistical center point in the I-Q coordinate system|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) The straight line divided in two spaces is marked as a threshold line, and the threshold line is perpendicular to the first statistical center point coordinate (I)|0>,Q|0>) And a second statistical center point coordinate (I)|1>,Q|1>) The connecting line of (1); the two spaces are respectively marked as a first space and a second space;
according to the first statistical centre point coordinates (I)|0>,Q|0>) And the first standard deviation σ1Determining a first probability density distribution function p (R) of all coordinate points in the first set in an I-Q coordinate system|0＞) According to said second statistical centre point coordinate (I)|1>,Q|1>) And the second standard deviation σ2Determining a second probability density distribution function p (R) of all coordinate points in the second set in an I-Q coordinate system|1)；
Determining a fidelity function as the first probability density distribution function p (R) |0＞) An integration function in the first space and the second probability density distribution function p (R)|1) A sum of integral functions in the second space;
and determining the corresponding threshold line as the optimal threshold line when the fidelity function takes the maximum value.
Further, the first probability density distribution function p (R)|0＞) And a second probability density distribution function p (R)|1) Respectively as follows:
The evaluation formula of the optimal threshold line is as follows:
further, the center point coordinates (I) are calculated according to the first statistic|0>,Q|0>) And the first standard deviation σ1Determining a first probability density distribution function p (R) of all coordinate points in the first set in an I-Q coordinate system|0>) According to said second statistical centre point coordinate (I)|1>,Q|1>) And the second standard deviation σ2Determining a second probability density distribution function p (R) of all coordinate points in the second set in an I-Q coordinate system|1) The method also comprises the following steps:
according to the first statistical centre point coordinates (I)|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) Determining a first included angle between the connecting line of the first and second coordinate axes and any coordinate axis of the I-Q coordinate system;
rotating and updating the first statistical center point coordinate (I) according to the first angle|0>,Q|0>) And said second statistical centre point coordinate (I) |1>,Q|1>)；
And rotating and updating all coordinate points in the first set and all coordinate points in the second set according to the first included angle.
Further, the first statistic center point coordinate (I) is rotated and updated according to the first included angle|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) The method specifically comprises the following steps:
determining the first included angle as the first statistical center point coordinate (I)|0>,Q|0>) And the second center point coordinate (I)|1>,Q|1>) The connecting line of the two forms an included angle with the I axis of the I-Q coordinate system;
clockwise rotating the first statistical center point coordinate (I) according to the first angle|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>)；
Updating the first statistical center point coordinates (I)|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) Are respectively (I'|0>,Q′|0>)、(I′|1>,Q′|1>) Wherein: q'|0>＝Q′|1>。
Further, the first statistical center point coordinate (I ') when updated'|0>,Q′|0>) And the updated second statistical center point coordinate (I'|1>,Q′|1>) When the vertical axes of the first and second threshold lines are equal, the threshold line is a vertical threshold line perpendicular to the axis I;
determining a threshold line corresponding to the fidelity function when the fidelity function takes the maximum value as an optimal threshold line, specifically comprising:
when the first space is the updated first statistical center point coordinate (I'|0>,Q′|0>) The space and the first space are located to the right of the vertical threshold line, and the second space is the updated second statistical center point coordinate (I' |1>,Q′|1>) If the space is located and the second space is located on the left side of the vertical threshold line, determining that the corresponding threshold line at the maximum value of the fidelity function is the optimal threshold line, and otherwise, determining that the corresponding threshold line at the minimum value of the fidelity function is the optimal threshold line; wherein the sum of the maximum value of the fidelity function and the minimum value of the fidelity function is 1.
Further, the sine value of the first included angle θ is:
further, the evaluation formula of the optimal threshold line can be converted into the following formula by simplification:
order:
wherein: gl′(I') is monotonically decreasing and intersects the I-axis, then solving equation (3) translates into solving the following equation:
the solution resulting in equation (4) is a real solution I '═ a, then the expression to obtain the optimal threshold line l' is: i ═ a.
Further, when the solution of the equation (5) is a real solution I '═ a, the expression of the optimal threshold line l' is obtained as follows: after I' ═ a, the method further comprises:
repeatedly acquiring a plurality of coordinate point data of the corresponding quantum bit reading signal on the orthogonal plane coordinate system when the quantum bit is in a quantum state, and recording the coordinate point data as a third set;
rotating and updating all coordinate point data in the third set according to the first angle;
Setting a termination condition by taking the optimal threshold line l' as an initial threshold line;
dividing a third set into two clusters by using the initial threshold line, wherein the two clusters are a first cluster and a second cluster respectively, and the counting number n is 1;
respectively carrying out weightless averaging on all coordinate point data in the first cluster and the second cluster to obtain corresponding expected coordinates which are respectively a first coordinate and a second coordinate, determining a second included angle between a connecting line of the first coordinate and the second coordinate and any coordinate axis of an I-Q coordinate system according to the first coordinate and the second coordinate, rotating by using the second included angle and updating all coordinate point data in the first cluster and the second cluster, the first coordinate and the second coordinate, wherein: the line connecting the updated first coordinate and the second coordinate is parallel to the axis I;
determining an updated threshold line with the initial threshold line and the second coordinates before and after updating, wherein: the expression of the updated threshold line is the sum of the updated I-axis coordinate of the second coordinate and the expression of the initial threshold line minus the I-axis coordinate of the second coordinate before updating;
and returning to the execution step by taking the updated threshold line as an initial threshold line: dividing the third set into two clusters by using the initial threshold line, wherein the two clusters are a first cluster and a second cluster respectively, and the counting number n is equal to n + 1;
And stopping executing until a termination condition is reached, and determining the updated threshold line as the optimal threshold line required to be obtained.
Further, the setting of the termination condition specifically includes:
setting a maximum execution number N, and stopping execution when N is equal to N, wherein: the maximum number of executions N is selected manually.
Further, the setting of the termination condition specifically includes:
setting a first threshold, wherein: the first threshold value is selected according to the actual required processing precision;
stopping execution when a maximum value of a distance between the first coordinates before and after the update and a distance between the second coordinates before and after the update is smaller than the first threshold.
Compared with the prior art, the method comprises the steps of preparing two different qubit quantum states, namely a first quantum state and a second quantum state, respectively, performing a large number of repeated reading operations to obtain two sets, performing Gaussian fitting respectively to obtain a first statistical center coordinate point and a second statistical center coordinate point, respectively corresponding to a first standard deviation and a second standard deviation, and respectively determining a corresponding first probability density distribution function and a corresponding second probability density distribution function, dividing a coordinate system into a first space and a second space by a threshold line, determining a fidelity function according to the sum of the integral of the first probability density distribution function in the first space and the integral of the second probability density distribution function in the second space, and when the fidelity function takes the maximum value, determining the corresponding threshold line to be an optimal threshold line; the invention firstly obtains the data of the quantum bit in the determined quantum state through a large number of repetitions, and performs Gaussian fitting on the data through a computer, determining each probability density distribution function according to the result of Gaussian fitting, taking the integral of each probability density distribution function distributed on the corresponding space as a fidelity function, when the fidelity function takes the maximum value, the threshold line used for dividing the space makes the reading fidelity effect of the quantum bit reading signal on both sides of the threshold line best, because the purpose of establishing the threshold line is to directly judge and obtain the quantum state of the quantum bit through the relation between the position of the measurement result in the coordinate system and the threshold line in any subsequent single measurement process, the threshold line established according to the judgment standard for enabling the fidelity function to take the maximum value can provide more accurate reference for judging the unknown quantum state.
Drawings
Fig. 1 is a flow chart of an acquisition method for identifying a threshold line of a qubit quantum state according to an embodiment of the invention.
Detailed Description
The embodiments described below with reference to the drawings are illustrative only and should not be construed as limiting the invention.
With reference to fig. 1, the present invention provides a method for obtaining a threshold line for confirming quantum states of qubits, comprising the following steps:
preparing the quantum bit into a first quantum state and repeatedly measuring the quantum bit to obtain coordinate point data of a plurality of quantum bit reading signals on an orthogonal plane coordinate system, and marking the coordinate point data as a first set R|0>(ii) a Preparing the quantum bit into a second quantum state and repeatedly measuring the quantum bit to obtain coordinate point data of a plurality of quantum bit reading signals on an orthogonal plane coordinate system, and marking the coordinate point data as a second set R|1>Wherein: the first and second quantum states are both known quantum states and are different from each other, wherein: the orthogonal plane coordinate system is providedSetting an I-Q coordinate system; specifically, the representation of an arbitrary quantum state in hilbert space is Φ ═ α |0 > + β |1 >, where |0>And |1>Which are two orthogonal basis vectors of the hilbert space, corresponding to the first quantum state and the second quantum state described in this embodiment. Specifically, when the first quantum state is quantum state |0 >In this state, the second quantum state is |1>(ii) a Or vice versa.
Respectively carrying out Gaussian fitting on all coordinate points in the first set and all coordinate points in the second set to obtain first statistical center coordinate points (I) of Gaussian fitting graphs corresponding to the first set and the second set respectively|0>,Q|0>) And a second statistical center coordinate point (I)|1>,Q|1>) Respectively corresponding first standard deviation σ1And a second standard deviation σ2(ii) a Wherein: for coordinating the first statistical center point in the I-Q coordinate system|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) The straight line divided in two spaces is marked as a threshold line, and the threshold line is perpendicular to the first statistical center point coordinate (I)|0>,Q|0>) And a second statistical center point coordinate (I)|1>,Q|1>) The connecting line of (1); the two spaces are respectively marked as a first space and a second space;
according to the first statistical centre point coordinates (I)|0>,Q|0>) And the first standard deviation σ1Determining a first probability density distribution function p (R) of all coordinate points in the first set in an I-Q coordinate system|0＞) According to said second statistical centre point coordinate (I)|1>,Q|1>) And the second standard deviation σ2Determining a second probability density distribution function p (R) of all coordinate points in the second set in an I-Q coordinate system |1)；
Determining a fidelity function as the first probability density distribution function p (R)|0＞) An integration function in the first space and the second probability density distribution function p (R)|1) A sum of integral functions in the second space;
and determining the corresponding threshold line as the optimal threshold line when the fidelity function takes the maximum value.
The method has the advantages that compared with the prior art, two different quantum states, namely a first quantum state and a second quantum state, are respectively prepared, a large number of repeated reading operations are carried out to obtain two sets, Gaussian fitting is respectively carried out to obtain a statistical center coordinate point and a second statistical center coordinate point, corresponding first standard deviation and second standard deviation are respectively obtained, corresponding first probability density distribution function and second probability density distribution function are respectively determined, the coordinate system is divided into a first space and a second space by the obtained threshold line, the fidelity function is determined according to the sum of the integral of the first probability density distribution function in the first space and the integral of the second probability density distribution function in the second space, and when the fidelity function takes the maximum value, the corresponding threshold line is the optimal threshold line; the invention firstly obtains the data of the quantum bit in the determined quantum state through a large number of repetitions, and performs Gaussian fitting on the data through a computer, determining each probability density distribution function according to the result of Gaussian fitting, taking the integral of each probability density distribution function distributed on the corresponding space as a fidelity function, when the fidelity function takes the maximum value, the threshold line used for dividing the space makes the reading fidelity effect of the quantum bit reading signal on both sides of the threshold line best, because the purpose of establishing the threshold line is to directly judge and obtain the quantum state of the quantum bit through the relation between the position of the measurement result in the coordinate system and the threshold line in any subsequent single measurement process, the threshold line established according to the judgment standard for enabling the fidelity function to take the maximum value can provide more accurate reference for judging the unknown quantum state.
Example 1
Specifically, with reference to fig. 1, an embodiment 1 of the method for obtaining a threshold line for confirming a quantum state of a qubit according to the present invention includes the following steps:
step 10, preparing the qubits into a first quantum state and repeatedly measuring the qubits to obtain coordinate point data of a plurality of qubit reading signals on an orthogonal plane coordinate system, and marking the coordinate point data as a first quantum stateSet R|0>(ii) a Preparing the quantum bit into a second quantum state and repeatedly measuring the quantum bit to obtain coordinate point data of a plurality of quantum bit reading signals on an orthogonal plane coordinate system, and marking the coordinate point data as a second set R|1>Wherein: the first and second quantum states are both known quantum states and are different from each other, wherein: the orthogonal plane coordinate system is set as an I-Q coordinate system;
specifically, the first quantum state may be an |0> state quantum state, the second quantum state may be an |1> state quantum state, and the orthogonal plane coordinate system is set as an I-Q coordinate system, where I is a horizontal axis and Q is a vertical axis.
Preparing the quantum bit into a first quantum state, repeatedly measuring the first quantum state to obtain coordinate point data of a plurality of quantum bit reading signals on an orthogonal plane coordinate system, and recording the coordinate point data as a first set R |0>The first set R is obtained by obtaining and analyzing through a quantum bit signal reading device|0>The data in (2) is stored in the computer, similarly to the second set R|1>The data of the computer is also stored in the computer;
step 20, performing gaussian fitting on all coordinate points in the first set and all coordinate points in the second set respectively to obtain first statistical center coordinate points (I) of gaussian fitting graphs corresponding to the first set and the second set respectively|0>,Q|0>) And a second statistical center coordinate point (I)|1>,Q|1>) Respectively corresponding first standard deviation σ1And a second standard deviation σ2(ii) a Wherein: for coordinating the first statistical center point in the I-Q coordinate system|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) The straight line divided in two spaces is marked as a threshold line, and the threshold line is perpendicular to the first statistical center point coordinate (I)|0>,Q|0>) And a second statistical center point coordinate (I)|1>,Q|1>) The connecting line of (1); the two spaces are respectively marked as a first space and a second space;
wherein the first set is obtained by comparing all coordinate points in the first set with the pointThe respective Gaussian fitting of all coordinate points in the second set is performed by a computer, and the computer program performs the fitting of all coordinate points in the first set|0>And a second set R |0>Performing two-dimensional Gaussian fitting on the data to obtain a two-dimensional Gaussian distribution graph, and obtaining a first statistical center coordinate point (I) of the Gaussian fitting graph corresponding to the first set and the second set respectively|0>,Q|0>) And a second statistical center coordinate point (I)|1>,Q|1>) Respectively corresponding first standard deviation σ1And a second standard deviation σ2；
Step 30, according to the first statistical center point coordinate (I)|0>,Q|0>) And the first standard deviation σ1Determining a first probability density distribution function p (R) of all coordinate points in the first set in an I-Q coordinate system|0>) According to said second statistical centre point coordinate (I)|1>,Q|1>) And the second standard deviation σ2Determining a second probability density distribution function p (R) of all coordinate points in the second set in an I-Q coordinate system|1)；
Wherein: the first probability density distribution function p (R)|0>) And a second probability density distribution function p (R)|1) Respectively as follows:
The above formula is a probability density distribution function corresponding to a gaussian distribution, and the formula can be directly obtained by fitting the first set and the second set with a computer, but the first probability density distribution function p (R) is the probability density distribution function corresponding to a gaussian distribution|0>) And a second probability density distribution function p (R) |1) The derivation is not limited to this method.
Step 40, determining the fidelity function as said first probability density distribution function p (R)|0＞) An integration function in the first space and the second probability density distribution function p (R)|1) A sum of integral functions in the second space;
and step 50, determining the corresponding threshold line as the optimal threshold line when the fidelity function takes the maximum value.
Wherein: the evaluation formula of the optimal threshold line is as follows:
by solving the above equation, an expression about the optimal threshold line can be obtained, and the sum of the fidelity on both sides of the threshold line can be the maximum value by satisfying the equation, so that the optimal threshold line required by the present invention can be obtained.
It should be noted that the fidelity function takes the maximum value or the minimum value, where the first statistical center point coordinate (I) is the maximum value or the minimum value|0>,Q|0>) Coordinates (I) located at the second statistical center point|1>,Q|1>) On the right, the fidelity function needs to take the maximum value when the first statistical center point coordinate (I)|0>，Q|0>) Coordinates (I) located at the second statistical center point|1>，Q|1>) On the left, the fidelity function needs to take the minimum value.
Example 2
It should be noted that, on the premise of satisfying the above two-dimensional double-gaussian distribution statistical model, the obtained current threshold line can be proved by mathematics that the current threshold line and the coordinate (I) of the first statistical center point |0>，Q|0>) And said second statistical centre point coordinate (I)|1>，Q|1>) And is vertical.
The procedure was demonstrated as follows:
it is known that:
the expression of the finally obtained optimal threshold line in the IQ coordinate system is not assumed to be:
aI + bQ + c is 0, wherein ab is not equal to 0, b is not less than 0, a2+b2＝1
Obtaining an included angle phi between the optimal threshold line and the axis I, and rotating all coordinate point data in the first set and the second set clockwise by the angle phi by taking the origin of coordinates as the center in an IQ coordinate system, wherein the updated coordinate of the center point of the first statistic (I)|0>，Q|0>) And said second statistical centre point coordinate (I)|1>，Q|1>) Is described as (I)|0>，new，Q|0>，new)、(I|1>,new,Q|1>,new) The expression of the rotated optimal threshold line becomes Q ≦ -c, and we assume that in space Q ≦ -c; wherein: φ can be obtained by solving the following equation:
at this time, the space divided by Q ═ c and the fidelity correspondence calculation formula are:
and since the optimal threshold line is a straight line that maximizes fidelity, namely:
namely:
wherein:
the maximum optimization method of the multivariate function g (a, b, c) is as follows: in this problem, the maximum value must exist, so we only need to solve all the stationing points and then find the maximum value point among the stationing points.
When the constraint condition ab is not equal to 0, b is not less than 0, a2+b2Under 1, the stagnation point can be solved using the lagrange multiplier method:
Where λ is an auxiliary parameter, that is:
the stagnation point can be collated from the above equation set and satisfies: (Q)|0>-Q|1>)a＝(I|0>-I|1>)b。
Due to the fact that
Is the first statistical center point coordinate (I)|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) Slope k of the lineo1，
Is the slope k of the optimal threshold linelI.e. ko1klAnd-1, so that an optimal threshold line is obtained and is necessarily perpendicular to a connecting line of the first statistical center point coordinate and the second center point coordinate, and the certification is finished.
Based on the above facts, the present invention provides another embodiment, and further, on the basis of embodiment 1, before the step 30, the method further includes:
step 22, according to the first statistical center point coordinate (I)|0>，Q|0>) And said second statistical centre point coordinate (I)|1>，Q|1>) Determining a first included angle between the connecting line of the first and second coordinate axes and any coordinate axis of the I-Q coordinate system;
step 24, rotating and updating the coordinate (I) of the center point of the first statistic according to the first included angle|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>)；
And 26, rotating and updating all coordinate points in the first set and all coordinate points in the second set according to the first included angle.
By adopting the technical scheme of the above steps, in this embodiment, the first included angle is obtained, and then the coordinate (I) of the first statistical center point is rotated and updated according to the first included angle |0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) And all the coordinate points in the first set and all the coordinate points in the second set, so as to reduce the degree of freedom of the fidelity function through rotation operation and facilitate the later calculation of the maximum value of the fidelity function.
Further, the first included angle θ is the coordinate (I) of the first statistical center point|0>,Q|0>) And the second center point coordinate (I)|1>,Q|1>) The included angle between the connecting line of the two and the I axis of the I-Q coordinate system is determined according to the first included angleTheta clockwise rotation of the first statistical center point coordinate (I)|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) And all coordinate points in the first set and all coordinate points in the second set, wherein: updated coordinates (I) of the first statistical center point|0>,Q|0>) And said second statistical centre point coordinate (I)|1>，Q|1>) Are respectively (I'|0>，Q′|0>)、(I′|1>，Q′|1>) And Q'|0>＝Q′|1>Obtaining the updated first statistical center point coordinate (I ') by a rotation operation'|0>，Q′|0>) And a second statistical center point coordinate (I'|1>，Q′|1>) So that the optimal threshold line sought will be parallel to the Q axis.
Preferably, the first included angle θ can be obtained by calculating the following formula:
it should be noted that the obtaining of the first included angle θ includes and is not limited to the above method.
Further, the evaluation formula of the optimal threshold line is:
the following transformations can be carried out:
wherein: taking into account the actual physical meaning P1-P3>0, then formula (2) is converted to:
order:
wherein: gl′If the image of the function (I') is monotonically decreasing and intersects the I-axis, then solving equation (3) can be converted to solving the following equation according to the integral property:
the solution of formula (5) is obtained as a real solution I ' ═ a, and then the first threshold line l ' is obtained as an expression I ' ═ a.
In particular, if σ1＝σ2The expression for the current threshold line can be found as:
through the embodiment, the originally obtained current threshold line is converted into a straight line perpendicular to the Q axis, namely the first threshold line, through rotation operation, so that the difficulty of solving the original equation (1) is greatly simplified in algorithm, and the efficiency of obtaining the threshold straight line is improved.
Example 3
It should be noted that repeatedly acquiring the qubit at |0>State or |1>The quantum bit in state reads the data after signal analysis, and the first statistical center point coordinate (I) of the two-dimensional double-Gaussian distribution graph is obtained due to different statistical data|0>，Q|0>) And a second statistical center point coordinate (I)|1>，Q|1>) Will vary in a floating manner, but the spacing of the two center coordinates, i.e. the
Keeping the original shape; secondly, the noise level of the system does not vary much, and can still be approximated by σ 1And σ2. Finally, the distribution of the quantum bit reading result on the i-q coordinate system still obeys two-dimensional double heightA statistical distribution of the si. On the premise of the three conditions, the same rotation transformation method as that in embodiment 2 can still be adopted to convert the theoretical threshold straight line into the one for solving a single variable, that is:
in the form of (1). Meanwhile, under the premise that the three above conditions are satisfied, I ' -I ' can be proved mathematically '|0>Is a constant, the numerical values only sum
σ1And σ2In this connection, the difference between the optimal threshold line l and the abscissa of the desired coordinate is a constant c.
The procedure was demonstrated as follows:
the threshold straight line is known to be perpendicular to the line connecting the coordinates of the center point. And after the rotating operation, satisfies:
then if the first statistical center point coordinate (I)|0>，Q|0>) And a second statistical center point coordinate (I)|1>，Q|1>) The floating change, equivalent to which one needs to solve:
due to the fact that
And therefore after the rotation transformation, the following still exist:
and because the purpose of the rotation transformation is to make the coordinate connecting line of the central point parallel to the I axisLine, i.e. Q'|0>-Q′|1>＝Q″|0>-Q″|1>0, thus:
i.e. | I'|0>-I′|1>|＝|I″|0>-I″|1>|＝m。
Comparing the first statistical center point coordinates (I) respectively|0＞，Q|0＞) And a second statistical center point coordinate (I) |1＞，Q|1＞) The calculation process of the formula (4) before and after the floating change:
as can be seen from formula (a) and formula (b), the two equations are other than I'|0>And I ″)|0>Apart from this, the rest is completely the same (note: considering that the actual limiting condition is that the threshold straight line is between two central point coordinates, and the sign does not affect the process and conclusion of the solution), so it should be understood that the solutions are in the same form: i '-I'|0>＝I″-I″|0>。
The certification is over.
Based on the facts, the invention further provides an obtaining method for obtaining the updated more reliable threshold straight line by combining the K-mean clustering method in the machine learning;
the basic idea of the k-means clustering method is as follows: initializing k different center points [ mu ](1)，…，μ(k)Then iteratively swapping two different steps until convergence. Step one, each training sample is distributed to the nearest central point mu(i)The represented cluster i. Step two, each central point mu(i)Updated as all training samples x in cluster i(j)Mean value of。
Then, on the basis of embodiment 2, after obtaining the expression of the optimal threshold line, the following steps are further included:
step 60, repeatedly obtaining a plurality of coordinate point data of the corresponding quantum bit reading signal on the orthogonal plane coordinate system when the quantum bit is in a certain quantum state, and recording the coordinate point data as a third set;
Wherein: the repeatedly acquiring a plurality of coordinate point data of the corresponding qubit reading signal on the orthogonal plane coordinate system when the qubit is in a certain quantum state, and marking as the third set, means that the qubit is prepared to a certain quantum state, and whether the qubit is unknown or not, the repeatedly acquiring a plurality of coordinate point data of the corresponding qubit reading signal of the qubit on the orthogonal plane coordinate system for a plurality of times, and marking as the third set
Step 70, rotating and updating all coordinate point data in the third set according to the first angle;
wherein all coordinate point data in the third set are rotated and updated according to the first angle θ in order to make the third set
Is consistent with the first set and the second set.
Step 80, setting a termination condition by taking the optimal threshold value line l' as an initial threshold value line;
step 90, dividing the third set into two clusters by using the initial threshold line, wherein the two clusters are respectively a first cluster and a second cluster, and the counting number n is 1;
in particular, using the initial threshold line/' will
Segmentation into two clusters
And
counting n is 1;
step 100, respectively carrying out weighted average on all coordinate point data in the first cluster and the second cluster to obtain corresponding expected coordinates which are respectively a first coordinate and a second coordinate, determining a second included angle between a connecting line of the first coordinate and the second coordinate and any coordinate axis of an I-Q coordinate system according to the first coordinate and the second coordinate, rotating by the second included angle and updating all coordinate point data in the first cluster and the second cluster, the first coordinate and the second coordinate, wherein: the line connecting the updated first coordinate and the second coordinate is parallel to the axis I;
In particular, the clusters are
And
all sample coordinates in the system are respectively subjected to weightless averaging to respectively obtain corresponding expected coordinates
And
obtaining a second angle θ ', wherein the sine of the second angle θ' is:
will cluster
And
all sample coordinates in (1), the first coordinate and the second coordinate are in I-Q coordinatesThe system rotates clockwise by an angle theta' with the origin of coordinates as the center, and the purpose is to perform subsequent calculation by using the important property that the difference value between the optimal threshold line and the horizontal coordinate of the expected coordinate is a constant;
step 110, determining an updated threshold line by the initial threshold line and the second coordinates before and after updating, wherein: the expression of the updated threshold line is the sum of the updated I-axis coordinate of the second coordinate and the expression of the initial threshold line minus the I-axis coordinate of the second coordinate before updating;
specifically, the method comprises the following steps: at this time, an updated threshold line l is obtained1Namely, it is
The following formula will be satisfied:
Step 110, adding1Namely, it is
And returning to the execution step as a new initial threshold line: with an initial threshold line will
Segmentation into two clusters
And
counting n as n + 1;
and step 120, stopping executing until a termination condition is reached, and determining that the updated threshold line is the threshold line required to be obtained.
Through the steps, the core idea of the K-means clustering algorithm is used, the threshold line is adopted to divide a large cluster into two small clusters, the rotation angle is determined according to the expected coordinates of the two small clusters, the large cluster is rotated clockwise, then the threshold line is determined again, the large cluster is divided into the two small clusters by the re-determined threshold line, the execution is sequentially executed, and the execution is stopped after the termination condition is met.
Further, the step 80 of setting the termination condition specifically includes:
setting a maximum execution number N, and stopping execution when N is equal to N, wherein: the maximum execution times N are selected manually, and the numerical value of the maximum execution times N can be determined according to the requirement of the actually required running time; thus, the execution time can be effectively controlled.
Further, the step 80 of setting the termination condition specifically includes:
Setting a first threshold, wherein: the first threshold value is selected according to the actual required processing precision;
stopping execution when a maximum value of a distance between the first coordinates before and after the update and a distance between the second coordinates before and after the update is smaller than the first threshold.
Specifically, a first threshold e is set, wherein: the first threshold value epsilon is selected according to the actual required processing precision; when in use
When it is time to stop execution.
The first threshold value epsilon is determined artificially, and the physical meaning of the first threshold value epsilon is that when the distance between the expected center coordinate of the small cluster obtained after the new threshold value line is re-segmented and the expected center coordinate of the small cluster obtained after the previous threshold value line is segmented is smaller than the first threshold value epsilon, the execution is stopped, namely the finally obtained threshold value line meets the precision requirement.
The construction, features and functions of the present invention are described in detail in the embodiments illustrated in the drawings, which are only preferred embodiments of the present invention, but the present invention is not limited by the drawings, and all equivalent embodiments modified or changed according to the idea of the present invention should fall within the protection scope of the present invention without departing from the spirit of the present invention covered by the description and the drawings.
Claims (10)
1. A method of obtaining a threshold line for validating a qubit quantum state, comprising:
preparing the quantum bit into a first quantum state and repeatedly measuring the quantum bit to obtain coordinate point data of a plurality of quantum bit reading signals on an orthogonal plane coordinate system, and marking the coordinate point data as a first set R|0>(ii) a Preparing the quantum bit into a second quantum state and repeatedly measuring the quantum bit to obtain coordinate point data of a plurality of quantum bit reading signals on an orthogonal plane coordinate system, and marking the coordinate point data as a second set R|1>Wherein: the first and second quantum states are both known quantum states and are different from each other, wherein: the orthogonal plane coordinate system is set as an I-Q coordinate system;
respectively carrying out Gaussian fitting on all coordinate points in the first set and all coordinate points in the second set to obtain first statistical center coordinate points (I) of Gaussian fitting graphs corresponding to the first set and the second set respectively|0>,Q|0>) And a second statistical center coordinate point (I)|1>,Q|1>) Respectively corresponding first standard deviation σ1And a second standard deviation σ2(ii) a Wherein: for coordinating the first statistical center point in the I-Q coordinate system|0>,Q|0>) And said second statistical centre point coordinate (I) |1>,Q|1>) The straight line divided in two spaces is marked as a threshold line, and the threshold line is perpendicular to the first statistical center point coordinate (I)|0>,Q|0>) And a second statistical centerPoint coordinates (I)|1>,Q|1>) The connecting line of (1); the two spaces are respectively marked as a first space and a second space;
according to the first statistical centre point coordinates (I)|0>,Q|0>) And the first standard deviation σ1Determining a first probability density distribution function p (R) of all coordinate points in the first set in an I-Q coordinate system|0＞) According to said second statistical centre point coordinate (I)|1>,Q|1>) And the second standard deviation σ2Determining a second probability density distribution function p (R) of all coordinate points in the second set in an I-Q coordinate system|1)；
Determining a fidelity function as the first probability density distribution function p (R)|0＞) An integration function in the first space and the second probability density distribution function p (R)|1) A sum of integral functions in the second space;
and determining the corresponding threshold line as the optimal threshold line when the fidelity function takes the maximum value.
2. The method of claim 1, wherein said first probability density distribution function p (R) is a function of the maximum likelihood density of the qubit quantum state|0>) And a second probability density distribution function p (R)|1) Respectively as follows:
The evaluation formula of the optimal threshold line is as follows:
3. the method of claim 2, wherein the first statistical center point coordinate (I) is obtained according to a first statistical center point coordinate (I)|0>,Q|0>) And the first standard deviation σ1Determining a first probability density distribution function p (R) of all coordinate points in the first set in an I-Q coordinate system|0＞) According to said second statistical centre point coordinate (I)|1>,Q|1>) And the second standard deviation σ2Determining a second probability density distribution function p (R) of all coordinate points in the second set in an I-Q coordinate system|1) The method also comprises the following steps:
according to the first statistical centre point coordinates (I)|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) Determining a first included angle between the connecting line of the first and second coordinate axes and any coordinate axis of the I-Q coordinate system;
rotating and updating the first statistical center point coordinate (I) according to the first angle|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>)；
And rotating and updating all coordinate points in the first set and all coordinate points in the second set according to the first included angle.
4. The method of claim 3, wherein said rotating and updating the first statistic center point coordinate (I) according to the first angle |0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) The method specifically comprises the following steps:
determining the first included angle as the first statistical center point coordinate (I)|0>,Q|0>) And the second center point coordinate (I)|1>,Q|1>) The connecting line of the two forms an included angle with the I axis of the I-Q coordinate system;
clockwise rotating the first statistical center point coordinate (I) according to the first angle|0>,Q|0>) And saidSecond statistical center point coordinate (I)|1>,Q|1>)；
Updating the first statistical center point coordinates (I)|0>,Q|0>) And said second statistical centre point coordinate (I)|1>,Q|1>) Are respectively (I'|0>,Q′|0>)、(I′|1>,Q′|1>) Wherein: q'|0>＝Q′|1>。
5. The method of claim 4, wherein the threshold line is a line of bits that is a quantum state of a quantum bit,
when the updated first statistical center point coordinate (I'|0>,Q′|0>) And the updated second statistical center point coordinate (I'|1>,Q′|1>) When the vertical axes of the first and second threshold lines are equal, the threshold line is a vertical threshold line perpendicular to the axis I;
determining a threshold line corresponding to the fidelity function when the fidelity function takes the maximum value as an optimal threshold line, specifically comprising:
when the first space is the updated first statistical center point coordinate (I'|0>,Q′|0>) The space and the first space are located to the right of the vertical threshold line, and the second space is the updated second statistical center point coordinate (I' |1>,Q′|1>) If the space is located and the second space is located on the left side of the vertical threshold line, determining that the corresponding threshold line at the maximum value of the fidelity function is the optimal threshold line, and otherwise, determining that the corresponding threshold line at the minimum value of the fidelity function is the optimal threshold line; wherein the sum of the maximum value of the fidelity function and the minimum value of the fidelity function is 1.
6. The method according to claim 5, wherein the sine of the first angle θ is:
7. the method of claim 6, wherein the evaluation formula of the optimal threshold line can be converted into the following formula by simplification:
order:
wherein: gl′(I') is monotonically decreasing and intersects the I-axis, then solving equation (3) translates into solving the following equation:
the solution resulting in equation (4) is a real solution I '═ a, then the expression to obtain the optimal threshold line l' is: i ═ a.
8. The method of claim 7, wherein when the solution of equation (5) is a real solution I '═ a, then the expression for obtaining an optimal threshold line l' is: after I' ═ a, the method further comprises:
Repeatedly acquiring a plurality of coordinate point data of the corresponding quantum bit reading signal on the orthogonal plane coordinate system when the quantum bit is in a quantum state, and recording the coordinate point data as a third set;
rotating and updating all coordinate point data in the third set according to the first angle;
setting a termination condition by taking the optimal threshold line l' as an initial threshold line;
dividing a third set into two clusters by using the initial threshold line, wherein the two clusters are a first cluster and a second cluster respectively, and the counting number n is 1;
respectively carrying out weightless averaging on all coordinate point data in the first cluster and the second cluster to obtain corresponding expected coordinates which are respectively a first coordinate and a second coordinate, determining a second included angle between a connecting line of the first coordinate and the second coordinate and any coordinate axis of an I-Q coordinate system according to the first coordinate and the second coordinate, rotating by using the second included angle and updating all coordinate point data in the first cluster and the second cluster, the first coordinate and the second coordinate, wherein: the line connecting the updated first coordinate and the second coordinate is parallel to the axis I;
determining an updated threshold line with the initial threshold line and the second coordinates before and after updating, wherein: the expression of the updated threshold line is the sum of the updated I-axis coordinate of the second coordinate and the expression of the initial threshold line minus the I-axis coordinate of the second coordinate before updating;
And returning to the execution step by taking the updated threshold line as an initial threshold line: dividing the third set into two clusters by using the initial threshold line, wherein the two clusters are a first cluster and a second cluster respectively, and the counting number n is equal to n + 1;
and stopping executing until a termination condition is reached, and determining the updated threshold line as the optimal threshold line required to be obtained.
9. The method according to claim 8, wherein the setting of the termination condition specifically includes:
setting a maximum execution number N, and stopping execution when N is equal to N, wherein: the maximum number of executions N is selected manually.
10. The method according to claim 8, wherein the setting of the termination condition specifically includes:
setting a first threshold, wherein: the first threshold value is selected according to the actual required processing precision;
stopping execution when a maximum value of a distance between the first coordinates before and after the update and a distance between the second coordinates before and after the update is smaller than the first threshold.
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