Qudit quantum state tomography method

Abstract

The present invention relates to a method for quanta quantum state tomography that estimates a density matrix required for quanta quantum state reconstruction and optimizes the estimated density matrix using the Lindblad master equation, comprising: a density matrix estimation step for estimating a density matrix representing the quanta quantum state of a quanta system; a shot number calculation step for calculating the shot number for each measured IC-POVM basis by substituting the estimated density matrix into the Lindblad master equation; a cost function construction step for constructing a cost function as the sum of the differences between the shot number calculated for each basis and the actual shot number measured; and an update step for updating the estimated density matrix by finding the value of each term of the density matrix at which the value of the cost function becomes the minimum value.

Description

Technology Field

[0001] The present invention relates to a method for quantum state tomography of a cudd, and more specifically, to a method for quantum state tomography of a cudd that estimates a density matrix required for the reconstruction of a cudd quantum state and optimizes the estimated density matrix using the Lindblad master equation. Background Technology

[0002] A quantum bit, or qubit, is the basic unit of quantum information, operates in a two-dimensional state space, and is a quantum system with two states.

[0003] The quantum state of a single qubit is represented by a combination of the basis 0 and 1 (e.g., a quantum superposition state), and various quantum states of the qubit can be generated using quantum gates.

[0004] To perform quantum information processing, a process of measuring the quantum state of each qubit must be involved, and the measurement of the qubit's quantum state is performed by selectively measuring the electrons or photons emitted from each qubit.

[0005] Quantum state tomography is a method for determining the distribution of quantum states for all qubits. Conventional quantum state tomography is performed by measuring the quantum state for each qubit individually, as shown in Fig. 1.

[0006] Meanwhile, a qudit is a concept that extends the qubit to d-dimensionality, in which the quantum state is represented by multiple energy states (|0>, |1>, …, |d-1>), and high-dimensional quantum information can be processed by utilizing the d-dimensional state (for example, a qutrit when d=3, a ququart when d=4).

[0007] Qudits can store more information in a single quantum state than qubits, making them advantageous for data compression and complex quantum computations.

[0008] Furthermore, by reducing the number of quantum systems required for the same amount of information processing, it becomes possible to improve computational speed in specific algorithms.

[0009] As such, to measure and reconstruct the quantum state of a cudd having a d-dimensional state, cudd quantum state tomography is required.

[0010] Since QDIT-based systems utilize multiple energy levels, differences in the amount of light emitted at each energy level occur. These differences make the measurement more complex, and because the difference in the amount of light emitted at each level must be measured precisely, the measurement process and the amount of computation increase exponentially.

[0011] And if the measurement data is incomplete or unbalanced, it becomes difficult to accurately reconstruct the quantum state. Prior art literature

[0012] Registered Patent Publication No. 10-2624011 (Publication Date: January 12, 2024) The problem to be solved

[0013] The present invention was devised to solve the conventional problems described above, and aims to provide a method for quanta quantum state tomography that estimates a density matrix required for quanta quantum state reconstruction and optimizes the estimated density matrix using the Lindblad master equation. means of solving the problem

[0014] A method for qudit quantum state tomography according to the present invention for achieving the aforementioned purpose comprises: a density matrix estimation step for estimating a density matrix representing the qudit quantum state of a qudit system; a shot number calculation step for calculating the shot number for each measured IC-POVM basis by substituting the estimated density matrix into the Lindblad master equation; a cost function construction step for constructing a cost function as the sum of the differences between the shot number calculated for each basis and the actual shot number measured; and an update step for updating the estimated density matrix by finding the value of each term of the density matrix at which the value of the cost function becomes the minimum value.

[0015] In addition, the method for quantum state tomography according to the present invention is characterized by further including a parameter optimization step for optimizing the parameter values ​​of the Lindblad master equation before performing the density matrix estimation step.

[0016] In addition, in the method for quantum state tomography of a cudit according to the present invention, the parameter optimization step comprises: preparing the initial state of the cudit system as a specific ground state; measuring the number of photons emitted hourly from the specific ground state during a set time; estimating parameters in the Lindblad master equation based on the number of emitted photons measured hourly; calculating the number of emitted photons that change over time by substituting a density matrix (ρ) corresponding to the specific ground state into the Lindblad master equation with the estimated parameter values; constructing a cost function as the sum of the differences between the measured number of emitted photons and the calculated number of emitted photons; and finding the parameter value of the Lindblad master equation where the value of the cost function becomes a minimum value and updating the Lindblad master equation.

[0017] In addition, in the cudit quantum state tomography method according to the present invention, the step of updating the Lindblad master equation is characterized by finding a parameter value of the Lindblad master equation where the value of the cost function becomes a minimum value based on an optimization algorithm and updating the Lindblad master equation.

[0018] In addition, in the method for tomography of a cudit quantum state according to the present invention, the density matrix estimation step is characterized by estimating the value of each term of the density matrix representing the cudit quantum state using Maximum Likelihood Estimation (MLE).

[0019] In addition, in the qudit quantum state tomography method according to the present invention, the update step is characterized by finding the value of each term of the density matrix where the value of the cost function becomes the minimum value based on an optimization algorithm and updating the estimated density matrix.

[0020] Specific details of other embodiments are included in "Specific details for implementing the invention" and the attached "drawings".

[0021] The advantages and / or features of the present invention and the methods for achieving them will become clear by referring to the various embodiments described below in detail together with the accompanying drawings.

[0022] However, it should be understood that the present invention is not limited to the configurations of each embodiment disclosed below, but may be implemented in various different forms, and that each embodiment disclosed in this specification is provided merely to make the disclosure of the present invention complete and to fully inform those skilled in the art of the scope of the present invention, and that the present invention is defined only by the scope of each claim of the claims. Effects of the invention

[0024] According to the present invention, by estimating the density matrix required for the reconstruction of a qudit quantum state and optimizing the estimated density matrix using the Lindblad master equation, the accuracy of the quantum state reconstruction can be increased. Brief explanation of the drawing

[0025] Figure 1 is a diagram illustrating the quantum state tomography of a qubit. FIG. 2 is a flowchart illustrating a method for quantum state tomography of a cudd according to an embodiment of the present invention. Figure 3 is a diagram exemplifying the energy levels of a cutret composed of diamond NV centers. Figure 4 is a diagram exemplifying the optimized transition rate between energy levels of a cutret composed of the diamond NV center of Figure 3. FIGS. 5 to 8 are drawings illustrating a method for optimizing parameter values ​​of the Lindblad master equation applied to the present invention. FIG. 9 is a diagram illustrating a method for quantum state tomography of a cudd according to the present invention. FIG. 10 is a schematic diagram of a pulse sequence that creates a measurement ground state satisfying IC-POVM for cudd quantum state tomography. FIG. 11 is a diagram exemplarily showing density matrix result values ​​reconstructed by a quantum state tomography method according to the prior art. FIG. 12 is a diagram exemplarily showing density matrix result values ​​reconstructed by the cudd quantum state tomography method according to the present invention. Specific details for implementing the invention

[0026] Before describing the present invention in detail, it should be understood that the terms and words used in this specification should not be interpreted as being limited to their ordinary or dictionary meanings, and that the inventor of the present invention may appropriately define and use the concepts of various terms to best describe their invention, and furthermore, that these terms and words should be interpreted in a meaning and concept consistent with the technical spirit of the present invention.

[0027] In other words, it should be understood that the terms used in this specification are used merely to describe preferred embodiments of the present invention and are not intended to specifically limit the content of the present invention, and that these terms are defined in consideration of various possibilities of the present invention.

[0028] In addition, it should be noted that in this specification, singular expressions may include plural expressions unless the context clearly indicates a different meaning, and that even if they are expressed in a similarly plural form, they may include a singular meaning.

[0029] Throughout this specification, where it is stated that a component "includes" another component, unless specifically stated otherwise, this may mean that it does not exclude any other component but may include any other component.

[0030] Furthermore, it should be noted that in cases where it is stated that a component "exists inside or is installed in connection with" another component, this component may be installed in direct connection or contact with the other component, or it may be installed at a certain distance apart, and in the case where it is installed at a certain distance apart, there may be a third component or means for fixing or connecting the component to the other component, and a description of this third component or means may be omitted.

[0031] On the other hand, if it is stated that one component is "directly connected" or "directly connected" to another component, it should be understood that there is no third component or means.

[0032] Likewise, other expressions describing the relationship between each component, such as “between” and “right between”, or “adjacent to” and “directly adjacent to”, should be interpreted as having the same intent.

[0033] In addition, it should be understood that in this specification, terms such as “one side,” “other side,” “one side,” “other side,” “first,” “second,” etc., are used to clearly distinguish one component from another component, and that the meaning of the component is not restricted by such terms.

[0034] In addition, position-related terms such as "up," "down," "left," and "right" used in this specification should be understood as indicating the relative position of the corresponding component in the drawing, and unless an absolute position is specified, these position-related terms should not be understood as referring to an absolute position.

[0035] Furthermore, it should be understood that in the specification of the present invention, terms such as “…part,” “…unit,” “module,” and “device,” when used, refer to a unit capable of handling one or more functions or operations, and that this may be implemented in hardware or software, or a combination of hardware and software.

[0036] Furthermore, in specifying the reference numerals for each component of each drawing in this specification, the same component has the same reference numeral even if it is shown in different drawings; that is, the same reference numeral throughout the specification indicates the same component.

[0037] In the drawings attached to this specification, the size, position, connection relationships, etc., of each component constituting the present invention may be described in a partially exaggerated, reduced, or omitted manner for the convenience of explanation or to sufficiently clearly convey the concept of the present invention, and therefore, the proportions or scale may not be strictly accurate.

[0038] In addition, in describing the present invention below, detailed descriptions of components that are deemed to unnecessarily obscure the essence of the invention, such as known technologies including prior art, may be omitted.

[0040] Hereinafter, a method for quantum state tomography according to a preferred embodiment of the present invention will be described in detail with reference to the attached drawings.

[0041] Before describing the cudit quantum state tomography method according to the present invention, we will first describe the density matrix (ρ) and the Lindblad Master Equation applied to the present invention.

[0042] The density matrix (ρ) is a mathematical expression representing the probabilistic distribution of quantum states and can represent both pure states and mixed states.

[0043] The density matrix of a d-dimensional cudith is represented as a d×d complex matrix.

[0044] The Lindblad Master Equation is an equation that describes the change in quantum states (density matrices) over time in an open quantum system (a system interacting with an external environment), and it quantitatively explains the dynamics of quantum states, including the 'irreversible process' and 'non-dynamics' that appear when an open quantum system interacts with an external environment.

[0045] In other words, open quantum systems interact with the external environment and undergo irreversible processes such as decoherence, decay, and degradation; consequently, the system no longer remains in a pure quantum state but transforms into a mixed state, and changes in time cannot be explained by unitary operations.

[0046] The state change of such an open quantum system is described by the Lindblad master equation.

[0047] The Lindblad master equation can be expressed as follows.

[0048]

[0049] Here, ρ is the density matrix, H is the Hamiltonian, the first term is the unitary change of the quantum state due to the Hamiltonian (H), the second term is the non-unitary change due to environment interaction, L i is a Lindblad operator (an operator representing interaction with the environment).

[0050] FIG. 2 is a flowchart illustrating a method for quantum state tomography of a cudd according to an embodiment of the present invention.

[0051] The method for quantum state tomography of a qudit according to the present invention can be performed on a computer device having a memory that stores at least one program executed by at least one processor.

[0052] To reconstruct the quantum state for a physical cudit, in step S10, a density matrix (ρ) representing the cudit quantum state can be estimated.

[0053] Specifically, in the above-mentioned step S10, the value of each term of the density matrix (ρ) representing the cudit quantum state can be estimated using Maximum Likelihood Estimation (MLE).

[0054] In step S20, the density matrix (ρ) estimated through step S10 above can be substituted into the Lindblad master equation to calculate the shot number, which is the number of emitted photons that change over time for each measurement IC-POVM basis.

[0055] The IC-POVM (Informationally Complete Positive Operator-Valued Measure) is a set of measurement bases designed to completely reconstruct quantum states. By using IC-POVM bases, the probability distribution corresponding to each basis can be measured, and the density matrix (ρ) can be reconstructed based on this.

[0056] In qudit quantum tomography, the number of measurement grounds can be determined by the dimension of the qudit. For example, 9 measurement grounds are required for a 3-dimensional qudit (d=3), 16 measurement grounds are required for a 4-dimensional qudit (d=4), and N for an N-dimensional qudit (d=N). 2 A measurement basis is required.

[0057] In step S30, a cost function can be constructed as the sum of the differences between the number of photons calculated for each measurement IC-POVM basis through the above-mentioned step S20 and the number of photons actually measured for each measurement IC-POVM basis.

[0058] And in step S40, the density matrix can be updated by finding the value of each term of the density matrix where the value of the cost function constructed through the above-mentioned step S30 becomes the minimum value.

[0059] Specifically, in the above-mentioned step S40, the value of each term of the density matrix at which the cost function value is minimized is found based on a classical optimization algorithm, and the value of the term of the density matrix estimated through the above-mentioned step S10 can be updated.

[0060] The classical optimization algorithm applied in the above-mentioned step S40 can be implemented as a machine learning optimization algorithm, gradient descent, simulated annealing, genetic algorithm, Nelder-Mead (NM) algorithm, etc., but is not limited thereto.

[0061] In order for the QDIT quantum state tomography method of the present invention as described above to be implemented, each parameter value of the Lindblad master equation applied to the present invention must be optimized in advance.

[0062] As shown in Fig. 3, using a diamond NV (Nitrogen-Vacancy) center, three energy states (|m s =0>, |m s =+1>, |m s When constructing a qutrit having =-1>), transitions between each energy state proceed at a unique rate, which can vary over time.

[0063] The transition rate between each of these energy states is used as a rate parameter in the Lindblad master equation.

[0064] Therefore, if the rate of transition between each energy level used as a rate parameter in the Lindblad master equation is known, the Lindblad master equation can be constructed, and if the Lindblad master equation can be constructed, the quantum state can be reconstructed by substituting the density matrix into the Lindblad master equation.

[0065] Figure 4 is a diagram illustrating the optimized transition rates between energy levels of a cutret composed of diamond NV centers of Figure 3. As previously explained, the transition rates between each energy level may differ, and the transition rates between each energy level used as rate parameters in the Lindblad master equation must be optimized in advance.

[0066] The following describes how to optimize the rate parameter values ​​required for constructing the Lindblad master equation.

[0067] FIGS. 5 to 8 are drawings illustrating a method for optimizing parameter values ​​of the Lindblad master equation applied to the present invention.

[0068] To optimize the parameter values ​​of the Lindblad master equation, in step S110, as shown in FIG. 6, the initial state of the cutlet system is prepared as a specific ground state (quantum state) (e.g., any one of |0>, |+1>, |-1>).

[0069] In step S120, the number of photons emitted from a specific ground state is measured for a fixed time of, for example, 50 ns. At this time, the measurement is repeated several times while gradually shifting the measurement time in increments of, for example, 10 ns. Through this, the number of emitted photons (shot number) that changes over time can be collected.

[0070] In step S130, based on the number of emitted photons per hour collected through the above-mentioned step S120, the rate of transition between each energy level, which is used as a rate parameter in the Lindblad master equation, can be estimated.

[0071] In step S140, the density matrix (ρ) is substituted into the Lindblad master equation, for which parameter values ​​were estimated through the above-mentioned step S130, to calculate the number of emitted photons (shot number) that changes over time.

[0072] Since the cutlet system has been initialized to the desired quantum state through the above-described step S110, the quantum state is known. Thus, once the quantum state is known, the density matrix (ρ) can be calculated.

[0073] Accordingly, in step S140, the density matrix (ρ) can be substituted into the Lindblad master equation, for which parameter values ​​were estimated through the above-mentioned step S130, to calculate the number of emitted photons that change over time.

[0074] In step S150, the cost function can be constructed as the sum of the differences between the number of emitted photons measured through step S120 and the number of emitted photons calculated through step S140.

[0075] And in step S160, the Lindblad master equation can be updated by finding the parameter value of the Lindblad master equation where the value of the cost function configured through the above-mentioned step S150 becomes the minimum value.

[0076] Specifically, in the above-mentioned step S160, a parameter value of the Lindblad master equation where the value of the cost function is minimized is found based on a classical optimization algorithm, and the rate between each energy level used as a parameter (rate parameter) in the Lindblad master equation estimated through the above-mentioned step S130 can be updated.

[0077] FIG. 7 is a diagram illustrating the results of normalizing the number of photons measured by preparing a Q-Tret system in different ground states (e.g., |+0>, |-0>, |+1>, |-1>) and measuring the number of photons emitted according to the change in quantum state for a fixed time, for example, 50 ns, while gradually shifting the measurement time point in increments of 10 ns.

[0078] Figure 8 is a diagram showing the number of photons measured when the ground state in Figure 7 is |+1> or |-1>, and the dotted line in Figure 8 is a diagram showing the number of photons obtained by substituting the density matrix (ρ) into the Lindblad master equation with optimized parameter values.

[0079] Examples

[0080] When reconstructing the quantum state of a Qutrit system, the rate parameter values ​​required for constructing the Lindblad master equation are optimized and derived in advance, as shown in Fig. 9.

[0081] Subsequently, the value of each term in the density matrix (ρ) representing the cudit quantum state is estimated.

[0082] Then, the estimated density matrix (ρ) is substituted into the Lindblad master equation, in which the parameter values ​​were optimized earlier, to calculate the shot number, which is the number of emitted photons that change over time for each measurement basis.

[0083] Here, the simulation result (calculated value) for each basis obtained by substituting the estimated density matrix into the Lindbrand master equation is C s It is written as .

[0084] Subsequently, through experiments, the number of shots, which is the number of emitted photons that change over time for each measurement basis, is measured.

[0085] Here, the number of photons measured for each measurement basis through the experiment is C e It is written as .

[0086] And the number of photons calculated for each measurement basis (C s1 , C s2 , … , C s8 , C s9 ) and the actual number of photons (C) for each measurement basis e1 , C e2 , … , C e8 , C e9 Construct a cost function as the sum of the differences between ), and update the density matrix by finding the value of each term of the density matrix that becomes the minimum value at which the cost function converges to 0.

[0087] FIG. 10 is a schematic diagram of a pulse sequence that creates measurement ground states satisfying IC-POVM for cudd quantum state tomography, and is a schematic diagram of an example pulse sequence that creates nine measurement grounds necessary to completely reconstruct the quantum state of a cudd.

[0088] FIG. 11 is a diagram showing an exemplary result of a density matrix reconstructed by a quantum state tomography method according to the prior art, and is the result of performing quantum state tomography according to the prior art under the assumption that the number of photons emitted for each energy level is the same.

[0089] In Fig. 11, the density matrix (ρ) located at the top target ) is the quantum state to be inferred, which is initialized to 0, and the density matrix (ρ located below)est ) is the actual calculated value, and it can be seen that a value with lower accuracy (the similarity of the density matrix is ​​81.5%) is derived rather than being reconstructed with the state initialized to 0.

[0090] FIG. 12 is a diagram showing, exemplarily, the density matrix result value reconstructed by the qdit quantum state tomography method according to the present invention, and is the result of performing quantum state tomography according to the present invention under the assumption that the number of photons emitted for each energy level is the same.

[0091] Through Fig. 12, it can be confirmed that the state initialized to 0 was reconstructed with high accuracy (density matrix similarity is 99.8%) through correction according to the present invention.

[0092] As such, according to the present invention, the density matrix required for the reconstruction of a qudit quantum state is estimated, and by optimizing the estimated density matrix using the Lindblad master equation, the accuracy of the quantum state reconstruction can be increased.

[0093] Although various preferred embodiments of the present invention have been described above with some examples, the descriptions of various embodiments described in the "Specific details for carrying out the invention" section are merely illustrative, and those skilled in the art to which the present invention pertains will understand that the present invention can be modified in various ways or equivalent embodiments can be carried out based on the above description.

[0094] In addition, since the present invention can be implemented in various other forms, the present invention is not limited by the description above. The above description is provided merely to make the disclosure of the present invention complete and to fully inform those skilled in the art of the scope of the present invention, and it should be understood that the present invention is defined only by each claim of the claims.

Claims

Claim 1 A method for qudit quantum state tomography comprising: a density matrix estimation step for estimating a density matrix representing the qudit quantum state of a qudit system; a photon number calculation step for calculating the number of photons for each measured IC-POVM basis by substituting the estimated density matrix into the Lindblad master equation; a cost function construction step for constructing a cost function as the sum of the differences between the number of photons calculated for each basis and the number of photons actually measured; and an update step for updating the estimated density matrix by finding the value of each term of the density matrix at which the value of the cost function becomes a minimum value. Claim 2 A method for qudit quantum state tomography according to claim 1, further comprising a parameter optimization step of optimizing parameter values ​​of the Lindblad master equation before performing the density matrix estimation step. Claim 3 A method for quantum state tomography of a qudit according to claim 2, wherein the parameter optimization step comprises: preparing the initial state of the qudit system as a specific ground state; measuring the number of photons emitted hourly from the specific ground state during a set time; estimating parameters in the Lindblad master equation based on the number of emitted photons measured hourly; calculating the number of emitted photons that change over time by substituting a density matrix (ρ) corresponding to the specific ground state into the Lindblad master equation with the estimated parameter values; constructing a cost function as the sum of the differences between the measured number of emitted photons and the calculated number of emitted photons; and finding the parameter value of the Lindblad master equation where the value of the cost function becomes a minimum value and updating the Lindblad master equation. Claim 4 A method for Qudit quantum state tomography according to claim 3, wherein the step of updating the Lindblad master equation is characterized by finding a parameter value of the Lindblad master equation where the value of the cost function becomes a minimum value based on an optimization algorithm and updating the Lindblad master equation. Claim 5 A method for tomography of a cudd quantum state, wherein, in claim 1, the density matrix estimation step is characterized by estimating the value of each term of a density matrix representing a cudd quantum state using Maximum Likelihood Estimation (MLE). Claim 6 A method for quantum state tomography in which, in claim 1, the update step is characterized by finding the value of each term of the density matrix where the value of the cost function becomes the minimum value based on an optimization algorithm and updating the estimated density matrix.

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