Information processing device and information processing method

Abstract

The present invention addresses the problem of visualizing states of two quantum bits, including quantum entanglement, in a form that can be intuitively understood with good visibility.　One aspect of the present invention is an information processing device characterized by comprising: an input interface that receives at least one of information pertaining to the states of a plurality of quantum bits and information pertaining to operations related to the plurality of quantum bits; a visualization processing unit that generates a parameter for visualizing at least one of the states of arbitrary two quantum bits among the plurality of quantum bits and operations related to the arbitrary two quantum bits among the operations related to the plurality of quantum bits; and an output interface that outputs output information based on the parameter, wherein a diagram which represents a Bloch sphere and quantum entanglement is displayed on an output device on the basis of the output information.

Description

[Technical Field]

[0001] This invention relates to quantum computers, and more particularly to a technique for visualizing quantum states in quantum computers. [Background technology]

[0002] Quantum computers are a new type of computer that uses the quantum states of a two-level system as the smallest unit of information, and are expected to surpass the computational power of classical computers. The potential of quantum computers lies in the unique properties of quantum systems (quantum nature), including the superposition and entanglement of quantum states, and understanding quantum nature is essential to understanding how quantum computers work.

[0003] Conventional techniques exist for visualizing quantum states, and Patent Document 1 discloses a technique for visualizing quantum states using Bloch vectors and density matrices. [Prior art documents] [Patent Documents]

[0004] [Patent Document 1] US2020 / 0104739 A1 [Overview of the project] [Problems that the invention aims to solve]

[0005] Currently, there is no known method for visualizing quantum entanglement in a clear and understandable way. For example, Bloch vectors, a typical representation method, cannot represent quantum entanglement. Furthermore, while density matrices contain information including quantum entanglement, their representation methods are complex, such as using two 4x4 bar graphs, making it difficult to immediately determine the degree of quantum entanglement. Therefore, in order to further support the development of quantum developers, it is necessary to develop a representation method that allows for an intuitive understanding of quantum entanglement.

[0006] Therefore, the challenge is to visualize a 2-qubit state, including quantum entanglement, in a way that is clear, intuitive, and easy to understand. [Means for solving the problem]

[0007] One aspect of the present invention is an information processing device comprising: an input interface that receives at least one of information on the states of a plurality of qubits and information on operations relating to a plurality of qubits; a visualization processing unit that generates parameters for visualizing at least one of the states of any two qubits among the plurality of qubits and operations relating to any two qubits among the operations relating to the plurality of qubits; and an output interface that outputs output information based on the parameters, wherein the information processing device displays a diagram representing a Bloch sphere and quantum entanglement on an output device based on the output information.

[0008] Another aspect of the present invention is an information processing method performed by an information processing apparatus comprising an input interface, an output interface, a storage device, and an arithmetic unit, and having a visualization processing unit, wherein the visualization processing unit performs an input step of receiving at least one of information on the states of a plurality of qubits and information on operations relating to the plurality of qubits; a visualization step of generating a set of parameters for visualizing at least one of the states of any two qubits among the plurality of qubits and operations relating to any two qubits among the operations relating to the plurality of qubits; and an output step of outputting output information based on the set of parameters in order to display a diagram representing a Bloch sphere and quantum entanglement on an output device.

[0009] In a specific and preferred aspect of the present invention, the input step receives information about the states of the plurality of qubits; the visualization step performs a first step of obtaining a density matrix of two qubit states from the information about the states of the plurality of qubits; a second step of calculating a qubit correlation matrix and a density matrix for each individual qubit from the density matrix; and the output step outputs a first parameter for displaying a 3x3 matrix representing quantum entanglement based on the density matrix of two qubit states, and a second parameter for displaying Bloch vectors to be displayed on the Bloch sphere based on the density matrix for each individual qubit.

[0010] In another specific and preferred aspect of the present invention, the input step receives information about operations relating to the plurality of qubits; the visualization step performs a third step of obtaining a unitary matrix representing a 2-qubit operation from the information about operations relating to the plurality of qubits; a fourth step of decomposing the unitary matrix to obtain a first operation acting on one qubit alone and a second operation acting on two qubits; the output step outputs a third parameter for displaying Bloch vectors to be displayed on the Bloch sphere based on the first operation; and a fourth parameter for displaying points inside or outside a tetrahedron superimposed on a cube with side length π and a tetrahedron inside the cube based on the second operation. [Effects of the Invention]

[0011] It is possible to visualize a 2-qubit state, including quantum entanglement, in a way that is easy to understand intuitively and clearly. [Brief explanation of the drawing]

[0012] [Figure 1] This is a quantum circuit diagram explaining quantum circuits. [Figure 2] This is a block diagram illustrating the overall system of Example 1. [Figure 3] This is a flowchart illustrating the process for representing a quantum state in Example 1. [Figure 4] This is an explanatory diagram illustrating an example of a 2-qubit state (unentangled state) in relation to Example 1. [Figure 5] This is an explanatory diagram illustrating an example of a 2-qubit state (moderately entangled state) in relation to Example 1. [Figure 6] This is an explanatory diagram illustrating an example of a 2-qubit state (maximum entanglement state) in relation to Example 1. [Figure 7] This is an explanatory diagram illustrating another representation of a 2-qubit state (moderately entangled state) in relation to Example 1. [Figure 8] This is a flowchart illustrating the process for displaying quantum computations in Example 2. [Figure 9] This flowchart illustrates the method for selecting a 2-qubit operation in Example 2. [Figure 10] This is an explanatory diagram illustrating the Cartan decomposition of a two-qubit operation, relating to Example 2. [Figure 11A] This is an explanatory diagram illustrating the Bloch vector (K11, K21) for a two-qubit operation, relating to Example 2. [Figure 11B] This is an explanatory diagram illustrating the Bloch vector (K12, K22) for a two-qubit operation, relating to Example 2. [Figure 12] This is a perspective view illustrating a tetrahedron representing the entanglement generation structure of a two-qubit operation, relating to Example 2. [Figure 13] This is an explanatory diagram illustrating Bloch vectors in a two-qubit operation, relating to Example 2. [Figure 14] This is an explanatory diagram illustrating Bloch vectors in a two-qubit operation, relating to Example 2. [Figure 15] This is an explanatory diagram illustrating the quantum entanglement strength in Example 2. [Figure 16] This flowchart illustrates a method for obtaining tetrahedral coordinates representing the entanglement generation structure of a two-qubit operation, relating to Example 2. [Figure 17] This is an explanatory diagram illustrating the local equivalence operation 1 necessary for obtaining coordinates on a tetrahedron, relating to Example 2. [Figure 18] This is an explanatory diagram illustrating the local equivalence operation 2 necessary for obtaining coordinates on a tetrahedron, relating to Example 2. [Figure 19] This is a diagram illustrating the local equivalence operations necessary for obtaining coordinates on a tetrahedron, relating to Example 2. [Figure 20] This is an explanatory diagram illustrating an example of a 2-qubit operation (quantum entanglement-free operation) relating to Example 2. [Figure 21] This is an explanatory diagram illustrating an example of a 2-qubit operation (moderate entanglement generation operation) in relation to Example 2. [Figure 22] This is an explanatory diagram illustrating an example of a 2-qubit operation (maximum entanglement generation operation) in relation to Example 2. [Figure 23] This is a table-like diagram illustrating an example of the display of the moderate entanglement generation operation in Example 2. [Figure 24] This is a block diagram illustrating the general outline of a system that receives input information from an emulator in a classical computer, relating to Example 3. [Figure 25] This block diagram illustrates the general structure of a system in which input information is received from an emulator in an externally connected classical computer, relating to Example 3. [Figure 26] This is a block diagram illustrating the general outline of a system that receives input information from a quantum computer, relating to Example 3. [Modes for carrying out the invention]

[0013] Embodiments will be described in detail with reference to the drawings. However, the present invention is not to be construed as being limited to the embodiments described below. It will be readily apparent to those skilled in the art that the specific configuration can be modified without departing from the spirit or intent of the present invention.

[0014] In the configurations of the embodiments described below, the same reference numerals are used in common across different drawings for identical parts or parts having similar functions, and redundant explanations may be omitted.

[0015] When there are multiple elements with the same or similar function, they may be described using the same symbol but with different subscripts. However, if there is no need to distinguish between multiple elements, the subscript may be omitted in the description.

[0016] In this specification, notations such as "Part 1," "Part 2," and "Part 3" are used to identify components and do not necessarily limit their number, order, or content. Furthermore, the numbers used to identify components are used on a context-by-context basis, and a number used in one context does not necessarily indicate the same component in another context. Moreover, this does not prevent a component identified by one number from also performing the function of a component identified by another number.

[0017] The positions, sizes, shapes, and ranges of each component shown in drawings, etc., may not represent their actual positions, sizes, shapes, and ranges in order to facilitate understanding of the invention. Therefore, the present invention is not necessarily limited to the positions, sizes, shapes, and ranges disclosed in drawings, etc.

[0018] The publications, patents, and patent applications cited herein constitute part of the description herein.

[0019] In this specification, elements expressed in the singular form shall include the plural form unless otherwise clearly indicated in the context.

[0020] In one example of the embodiment described below, a two-qubit state is displayed using a Bloch sphere and a diagram representing quantum entanglement. The diagram representing quantum entanglement utilizes a 3x3 bar graph or an ellipse. Furthermore, this invention can display not only a two-qubit state but also a two-qubit operation, in which case the quantum entanglement is represented by a single point within a cube or a single point within a tetrahedron.

[0021] According to the technology of the embodiment, two-qubit states and two-qubit operations, including quantum entanglement, can be clearly displayed. As a result, quantum entanglement and the quantum entanglement generation structure can be intuitively understood, and the development of quantum computers and quantum algorithms can be made more efficient.

[0022] This embodiment is based on quantum mechanics. Therefore, first, the general properties of quantum mechanics will be described.

[0023] In quantum mechanics, an observable quantity (physical quantity) a is represented by the eigenvalue of a Hermitian matrix A. The value obtained when measuring a is one of the eigenvalues {a1, a2,...} of A. The quantum state |ψ> is described by a linear combination of the eigenvectors (eigenstates) {|a1>, |a2>,...}. That is, it can be written as |ψ> = c1|a1> + c2|a2> +... using complex numbers {c1, c2,...}. The time evolution of a quantum system is described by a unitary operator acting on the quantum state. In this quantum system, when measuring a, the quantum state of the system irreversibly transitions to one of {|a 2 , , 2 ,

[0025] , , T >}. For example, if a1 is observed, the state of the system transitions from |ψ> to |a1>. This phenomenon is also called "collapse of the wave packet". Physical quantity a k is measured, that is, the probability that the quantum system transitions to |a k > is given by |c k | 2 . From the probability conservation law, |c1| 2 + |c2| 2 ... = 1 is required.

[0024] A quantum computer is a computer that artificially constructs a quantum system (two-level system) composed of two eigenstates {|0>, |1>}, and uses the quantum state of this system as the minimum unit of information. This minimum information unit is called a qubit and can be expressed as |ψ> = c1|0> + c2|1> using complex numbers c1, c2 that satisfy |c1| 2 + |c2| 2 = 1.

[0025] A qubit can be represented by a vector with complex numbers as components. For example, |0> = (1 0) T, |1>=(0 1) T Using a vector representation where (superscript T indicates transpose), a 1-qubit state is |ψ1>=c1|0>+c2|1>=(c1,c2) T This can be expressed as follows. Operations on a single qubit in a quantum computer are described by a 2x2 unitary matrix U1 acting on this vector |ψ1>. U1 can generally be written as a linear combination of 2x2 Hermitian matrices defined by (Equation 1).

[0026]

number

[0027] A multi-qubit state is described as a linear combination of the direct product (tensor product) of the eigenstates {|0>,|1>}. For example, in the case of a 2-qubit state, |0>*|0>=|00>=(1 0 0 0) T , |0>*|1>=|01>=(0 1 0 0) T , |1>*|0>=|10>=(0 0 1 0) T , |1>*|1>=|11>=(0 0 0 1) T It can be written as a linear combination of , (where * represents the vector cross product). That is, |c1| 2 +|c2| 2 +|c3| 2 +|c4| 2 Using complex numbers c1, c2, c3, and c4 that satisfy =1, |ψ2>=c1|00>+ c2|01>+ c3|10>+ c4|11> can be written. Operations on two qubits are described by a 4x4 unitary matrix U2 acting on this vector |ψ2>. In general, U2 can be written as a linear combination of the direct products of two Hermitian matrices (Equation 1). Below, σ a and σ b The Cartesian product of (a,b=0,1,2,3) is σ a σ b This will be expressed as follows: n>2 qubit state and operation U n It is defined similarly (where n is a natural number).

[0028] A diagram that shows the operations performed on a qubit in chronological order is called a quantum circuit diagram.

[0029] Figure 1 shows an example of a quantum circuit diagram for n>2 qubits. In a quantum circuit diagram, qubits are arranged vertically, and the operations for each qubit are written in chronological order from left to right. In this diagram, U1, U2, U n These represent 1-qubit operation, 2-qubit operation, and n-qubit operation, respectively.

[0030] A one-qubit pure state |ψ1> and operation U1 can be represented using vectors on the surface of a three-dimensional sphere of radius 1. These vectors are called "Bloch vectors," and the sphere is called a "Bloch sphere." The relationship between a one-qubit state and the Bloch vector (x,y,z) is given by x=Tr[X.ρ1], y=Tr[Y.ρ1], and z=Tr[Z.ρ1], using the density matrix ρ1=|ψ1><ψ1| (where <ψ1| represents the complex conjugate of |ψ1>) defined from |ψ1>. Here, . represents matrix product, and Tr[A] represents the trace of matrix A. Bloch sphere x 2 +y 2 +z 2 =1 is Tr[(ρ1) 2 This is determined by the condition ]=Tr[ρ1]=1. A single-qubit operation U1 is represented by the difference between a Bloch vector representing the input state and a Bloch vector representing the output state. In a multi-qubit system, if the qubit states can be divided into a direct product of single-qubit states, the state of the system can be represented by multiple Bloch vectors.

[0031] However, the Bloch vector alone cannot represent quantum entanglement. Quantum entanglement is a phenomenon in which measuring one qubit state determines the states of other qubits, and it is a quantum property unique to multi-qubit systems. For example, the 2-qubit state |Bell>=(1 / 2) 1 / 2 (|00>+|11>) is an example of a quantum entangled state. This quantum state |Bell> cannot be divided into a direct product of 1-qubit states and therefore cannot be described by a Bloch vector. Consequently, quantum operations that generate quantum entanglement also cannot be described solely by changes in Bloch vectors.

[0032] This embodiment provides a novel quantum system representation method that explicitly represents quantum entanglement for a two-qubit system. Quantum entanglement is an important indicator in designing quantum computers, and there is a high demand for quantitatively representing quantum entanglement. The embodiment will be described below with reference to the drawings. [Examples]

[0033] This embodiment is an information processing device that takes multiple qubit states or multiple qubit operations as input and outputs their properties in a way that reveals quantum entanglement. This embodiment describes the outline of this information processing device.

[0034] Figure 2 shows an example of the configuration of the information processing device in this embodiment. This information processing device (classical computer 1) performs calculations by exchanging data between the main memory (memory) 2, which is the storage unit, and the classical arithmetic unit (processor) 3, which is the arithmetic unit, just like a normal classical computer. These are controlled by the control device 4. The processing performed by the classical arithmetic unit 3 is stored in the main memory 2. If the storage capacity of the main memory 2 is insufficient, the auxiliary storage device 5 is used. Data and programs are input using the input device 6, and the input information there is sent to the classical computer 1 through the input interface 7. The output results are displayed on the output device 9 through the output interface 8.

[0035] In addition to the above, this information processing device has a visualization processing unit 10 in the main memory 2. The visualization processing unit 10 is, for example, software executed in the classical operation unit 3. The visualization processing unit 10 generates a set of parameters for visualizing the qubit state and operations. The information of the quantum circuit diagram 11, which is the input information for this information processing device, is stored in the main memory 2 and, if necessary, in the auxiliary storage device 5, for example, through the input interface 7 from the input device 6.

[0036] In the information processing device of Example 1, information to be visualized is generated based on the quantum circuit diagram 11 within the information processing device. Two qubits to be visualized can be selected by displaying the quantum circuit diagram 11 on the output device 9 via the output interface 8 and specifying any two qubits from the input interface 7. The qubits to be visualized can be the result of an operation or the state of a qubit in the middle of an operation. The state of qubits in the result of an operation or in the middle of an operation can be determined, for example, by performing a simulation based on the quantum circuit diagram 11.

[0037] Therefore, the visualization processing unit 10 receives the information from the quantum circuit diagram 11 and generates a set of parameters for visualizing the qubit state and operations. The generated set of parameters is displayed on the output device 9 via the output interface 8. The output result here is a diagram representing the Bloch sphere, Bloch vector, and quantum entanglement based on the set of parameters.

[0038] Next, we will discuss the representation of a two-qubit state. The input is the density matrix of the two-qubit quantum state, and the output is two Bloch vectors and a diagram representing quantum entanglement.

[0039] Figure 3 illustrates the computational processing performed by the visualization processing unit 10 to output the state display. First, the visualization processing unit 10 reads the quantum circuit diagram 11 and obtains the density matrix ρ of the input 2-qubit state, which is obtained from calculations according to the quantum circuit diagram 11 (S301). If the number of qubits in the input state is three or more, only two qubits are selected from the multiple qubits, and the behavior of the other two qubits is ignored. The density matrix ρ of the input 2-qubit state is given as a 4×4 Hermitian matrix and can be decomposed as shown in (Equation 2).

[0040]

number

[0041] Each expansion coefficient is given as shown in (Equation 3).

[0042]

number

[0043] Next, we decompose the density matrix ρ as shown in (Equation 4) (S302).

[0044]

number

[0045] Here e ij This is a matrix representing the correlation between two qubits. In this specification, e ij This is called the "qubit correlation matrix." The qubit correlation matrix is ​​generally a 3x3 real matrix and represents quantum entanglement between two qubits. When the qubit correlation matrix is ​​zero, it corresponds to a state without quantum entanglement.

[0046] Next, the density matrix ρ of each individual qubit obtained by the decomposition a and ρ b Express ρ in terms of Bloch vectors (S303). a and ρ b If vectors a and b represent the Bloch vectors, then each component is calculated by (Equation 5).

[0047]

number

[0048] These Bloch vectors are used as part of the output of the state representation.

[0049] The norm of these Bloch vectors is generally less than 1. This constraint is Tr[ρ 2 This comes from the condition ] <= Tr[ρ] = 1. In particular, when ρ is in a pure state, Tr[ρ 2 ]=Tr[ρ]=1, and as a result, the norm of the Bloch vector is given by (Equation 6).

[0050]

number

[0051] Here, E is a quantity that has a non-zero value when the qubit correlation matrix is ​​a non-zero matrix. In other words, it can be seen that in general these two Bloch vectors are vectors on the Bloch sphere with a radius less than 1. The degree of radius contraction is proportional to the degree of quantum entanglement. That is, the length of the Bloch vector represents the degree of quantum entanglement. Note that in the maximum entanglement state, the Bloch sphere contracts completely, and the Bloch vector becomes the zero vector.

[0052] The two Bloch vectors obtained above are plotted on two Bloch spheres (S304). This becomes part of the output of the two-qubit state.

[0053] Next, a diagram representing quantum entanglement is introduced. The diagram representing quantum entanglement displays at least one of the following: a diagram showing the numerical values ​​of the qubit correlation matrix, or the "quantum entanglement shape diagram" introduced below. The user can choose to display only the quantum entanglement shape diagram, only the qubit correlation matrix, or both the quantum entanglement shape diagram and the qubit correlation matrix as the diagram representing quantum entanglement, and can freely select which to display according to the user's choice (S305).

[0054] The numerical values ​​of the qubit correlation matrix representing a 2-qubit operation are represented by a 3x3 graph. This representation can be either a 3D graph or a 2D table.

[0055] The quantum entanglement shape diagram is obtained using the non-negative singular values ​​(e1, e2, e3) of the qubit correlation matrix, x 2 / e1+y 2 / e2+z 2We define it as a 3-dimensional (x,y,z) real space satisfying / e3 <= 1. In the case of maximum entanglement, the quantum entanglement shape diagram is a 3-dimensional sphere of radius 1, and in the case of no entanglement, it is just the origin. A general entangled state is a space bounded by an ellipse inside a 3-dimensional sphere of radius 1. The quantum entanglement shape diagram is a diagram that intuitively represents the properties of quantum entanglement, and users can know the degree of quantum entanglement by seeing how much the quantum entanglement shape diagram has collapsed from a sphere. The quantum entanglement shape diagram is generated in this way (S306).

[0056] The generated quantum entanglement shape diagram is plotted (S307), and it is selected whether the diagram representing quantum entanglement should consist only of the quantum entanglement shape diagram or include the qubit correlation matrix (S308). If the qubit correlation matrix is ​​required, the qubit correlation matrix is ​​also plotted (S309).

[0057] The two Bloch vectors obtained through the above processing and the diagram representing quantum entanglement are drawn as the final output on the display, which is the output device 9, via the output interface 8. Here, "diagram representing quantum entanglement" refers to at least one of the quantum entanglement shape diagram or the quantum entanglement correlation matrix. Quantum entanglement can be represented by either the quantum entanglement shape diagram or the quantum entanglement correlation matrix. The quantum entanglement shape diagram allows for an intuitive understanding of quantum entanglement, while the quantum entanglement correlation matrix can include more detailed information. These can be arbitrarily selected by the user.

[0058] We have discussed methods for representing a two-qubit state. Below, we will show specific examples of how to represent a two-qubit state using Figures 4 to 6.

[0059] Here, we consider |a>=(H1)|00>, |b>=(rootSWAP)(HI)|00>, and |c>=(CNOT)(HI)|00> as examples of a quantum entanglement-free state |a>, a moderately entangled state |b>, and a maximum entangled state |c>, and show examples of how each of these two-qubit states is represented.

[0060] Figure 4 shows an example of the representation of the unentangled quantum state |a>. The left column of the table shows the Bloch vectors, the middle column shows the qubit correlation matrix and its graphical representation, and the right column shows the quantum entanglement shape diagram (the same applies to Figures 5 to 7). The qubit correlation matrix for the unentangled quantum state is the zero matrix. Therefore, the quantum entanglement shape diagram consists only of the origin. In this example, the quantum entanglement shape diagram is displayed based on a cube with side length 1, and its origin (0,0,0) is the centroid of the cube. The Bloch vectors are vector a = (0,0,1) and vector b = (1,0,0).

[0061] Figure 5 shows an example of a representation of a moderately entangled state |b>. The qubit correlation matrix of a moderately entangled state is a non-zero matrix. In this example, it is {{-0.25,0.25,0.25},{-0.25,0.25,-0.25},{0.25,0.25,-0.25}}. The graphical representation in the middle column of Figure 5 is a 2D or 3D representation of these matrix components. In this example, the matrix components are taken on a 2D plane, and the component values ​​are shown as prisms with positive and negative directions. Calculating the singular values ​​of this matrix, we get e1=1 / 2, e2=1 / 2, e3=1 / 4, and the quantum entanglement shape diagram is 2x, as shown in the example in the right column of Figure 5. 2 +2y 2 +4z 2 This results in a space that satisfies <= 1. The Bloch vectors are vector a = (0.5, -0.5, 0.5) and vector b = (0.5, 0.5, 0.5). The length of the Bloch vectors is less than 1, indicating that quantum entanglement exists.

[0062] Figure 6 shows an example of the maximum entanglement state |c>. The correlation matrix of the maximum entanglement state in this state is a diagonal matrix and takes the value of -1 or 1. In the case of state |c> in this embodiment, it is {{1,0,0},{0,-1,0},{0,0,1}}. The graphical representation of these matrix components in two or three dimensions is shown in the middle column of Figure 6. When the singular values ​​of this matrix are calculated, e1=e2=e3=1, and the quantum entanglement shape diagram is a sphere of radius 1 as shown in the right column of Figure 6. The Bloch vectors are (0,0,0) for both vector a and vector b. The length of the Bloch vector is 0, indicating that there is maximum quantum entanglement.

[0063] The two Bloch vectors (vector a and vector b) representing a two-qubit state may be represented together in a single Bloch sphere.

[0064] Figure 7 shows an example of plotting the Bloch vectors of the moderate quantum entanglement state |b> on a Bloch sphere. The solid and dotted arrows in the left column of Figure 7 represent vector a = (0.5, -0.5, 0.5) and vector b = (0.5, 0.5, 0.5), respectively. The display of the qubit correlation matrix and quantum entanglement shape diagram is the same as described above.

[0065] As mentioned earlier, in this embodiment, the Bloch spheres depicted in Figures 4 to 7 show that the radius of the Bloch sphere varies, reflecting the degree of quantum entanglement. While the size of the Bloch sphere alone can indicate the magnitude of quantum entanglement, the quantum entanglement shape diagram and quantum entanglement correlation matrix allow for the acquisition of more detailed information, such as the direction of entanglement. [Examples]

[0066] Up to this point, we have discussed the representation of a two-qubit state. Below, we will discuss the representation of changes in a two-qubit state, that is, a two-qubit operation. This embodiment enables a representation that allows us to consider whether the operation generates quantum entanglement.

[0067] While the operation can also be represented as a change in the two-qubit state, this does not express the properties inherent to the operation. Therefore, in this embodiment, we describe a method for obtaining a diagram that explicitly represents the quantum entanglement generation structure of the operation.

[0068] Figure 8 shows the processing performed by the visualization processing unit 10 to obtain a representation of a two-qubit operation. The input is a unitary matrix representing the two-qubit operation, and the output is a diagram showing the changes in two Bloch vectors and quantum entanglement. First, the subsystem to be analyzed is selected from the quantum circuit diagram 11 (S801).

[0069] Figure 9 shows the method for selecting a subsystem. First, two qubits are selected from multiple qubits, and an appropriate start and end stage are chosen. Below, we focus on operations that act only on the two selected qubits within the range from the specified start to end stage. In Figure 9, qubits 2 and 3 are selected from n>3 qubits. Below, qubit 2 will be referred to as qubit a, and qubit 3 as qubit b. The operations included in the part labeled "selected portion" in Figure 9 will be the subject of analysis.

[0070] The quantum circuit diagram shown in Figure 9 is displayed on the output device 9 as a visually recognizable display by the visualization processing unit 10, based on the quantum circuit diagram 11 recorded in the main memory 2 and auxiliary memory 5. The user can perform necessary processing, such as selecting the range of operations for which they want to visualize the quantum entanglement generation status, via the input device 6.

[0071] The selected two-qubit operation can be represented using a 4x4 unitary matrix V. Matrix V is represented by 16 real parameters. One of these corresponds to an operation that globally transforms the phase of the quantum state, and such an operation has no special significance in quantum computing. Therefore, below we consider a special unitary matrix U, excluding the one parameter corresponding to the global transformation. U is expressed using V as U=(detV) -1 / 4 V can be defined as such, where detV represents the determinant of V. In this way, the visualization processing unit 10 obtains a special unitary matrix U to display the calculation unit (S802).

[0072] Next, we decompose the matrix U as shown in (Equation 7) (S803).

[0073]

number

[0074] Here the coefficient is ``r'' ij , l ij , c i The variables are real numbers between -π and +π (inclusive). This type of decomposition is called "Cartan decomposition".

[0075] Figure 10 shows a quantum circuit diagram representing the Cartan decomposition. The method for obtaining (K1, K2, A) and (c1, c2, c3) from U will be described later. In this example, A and K obtained from the Cartan decomposition are shown. 11 ,K 12 ,K 21 ,K 22 All of this will be illustrated and visualized. Not only A, which involves two qubits and where the origin of quantum entanglement is concentrated, but also K, which involves one qubit. 11 ,K 12 ,K 21 ,K 22 By visualizing this, it is possible to comprehensively visualize quantum computation. In the following example, K 11 ,K 12 ,K 21 ,K 22 Represent A as a Bloch sphere, and A as a cube or tetrahedron.

[0076] The aforementioned K1 and K2 can be expressed using two Bloch vectors. This is because, as shown in Figure 10, K 11 ,K 12 ,K 21 ,K 22 These are all operations that act on a single qubit and do not generate quantum entanglement. For example, K 11 and K 21 Since this operation acts only on qubit a, it can be expressed as a change in the Bloch vector a. That is, vec{a}, K 21 vec{a}, K11 K 21 By displaying the three Bloch vectors of $\vec{a}$ (K 11 ,K 21 ), it can be expressed. Here, $\vec{a}$ means the a vector. Similarly, (K 12 ,K 22 ) can also be expressed as the change of the Bloch vector $\vec{b}$. That is, $\vec{b}, K 22 \vec{b}, K 12 K 22 By displaying the three Bloch vectors of $\vec{b}$ (K 12 ,K 22 ), it can be expressed (S804). Here, $\vec{b}$ means the b vector.

[0077] Examples of the Bloch vector representations of (K 11 ,K 12, K 21 ,K 22 ) are shown in FIGS. 11A and 11B. FIG. 11A is a diagram representing (K 11 ,K 21 ), and the dotted line, dashed line, and solid line respectively represent $\vec{a}, K 21 \vec{a}, K 11 K 21 \vec{a}$. Similarly, FIG. 11B is a diagram representing (K 12 ,K 22 ), and the dotted line, dashed line, and solid line respectively represent $\vec{b}, K 22 \vec{b}, K 12 K 22 \vec{b}$. The changes of these Bloch vectors (two Bloch spheres) are made part of the output of the arithmetic display. This display method requires appropriately specifying the reference state ($\vec{a}, \vec{b}$). The user specifies the reference Bloch vector using the input device 6 before drawing.

[0078] Here, (K 11 ,K 12, K 21 ,K 22While we have described a method using two Bloch spheres as a Bloch vector representation method, there are also other representation methods using one or four Bloch spheres. Details of these methods are described below. Users can select their preferred representation method using input device 6.

[0079] On the other hand, operation A in Figure 10 cannot be represented by a Bloch vector. This is because, as shown in Figure 10, A is an operation spanning two qubits and is the part that generates quantum entanglement. The quantum entanglement generation operation A can be represented as a point on a cube with side length π. This is because, as shown in (Equation 7), A is represented by a set of real numbers (c1, c2, c3) between -π and +π. By drawing this point (c1, c2, c3) on the cube, A can be visualized.

[0080] Figure 12 is a perspective view (an illustrative diagram on the display screen) illustrating the tetrahedron 1200, which represents the entanglement generation structure of a two-qubit operation. A relevant publicly available document is "Geometric theory of nonlocal two-qubit operations" by Jun Zhang, Jiri Vala, Shankar Sastry, and K. Birgitta Whaley, Phys. Rev. A 67, 042313 - Published 18 April 2003.

[0081] Note that, in general, the solution (K1, K2, A) to the Cartan decomposition shown in (Equation 7) or Figure 10 is not uniquely determined. However, the decomposition such that (c1, c2, c3) is a point on the tetrahedron 1200 shown in Figure 12 is uniquely determined. This tetrahedron is defined as (c1, c2, c3) satisfying c1 <= π / 2 when 0 <= c3 <= c2 <= c1 < π and c1 + c2 <= π and c3 = 0. Therefore, as a diagram that uniquely represents the properties of the quantum entanglement generation operation, we draw not only the coordinates (c1, c2, c3) inside the cube but also the coordinates on the tetrahedron 1200. This visualizes the properties of A. The method for obtaining the coordinates on the tetrahedron that uniquely represent the properties of A (S805) will be described later.

[0082] The acquired coordinates of A are drawn within the cube (shown in Figure 12) (S806). In a preferred embodiment, it is determined whether the coordinates of A are within the tetrahedron 1200 (S807). If the coordinates of A are not within the tetrahedron 1200, the coordinates are acquired within the tetrahedron 1200 by a local equivalence operation (S808) shown later in Figure 16.

[0083] The resulting diagram representing the Bloch sphere and quantum entanglement is plotted on the output device 9 as the final output (S809).

[0084] Up to six Bloch vectors are plotted on each Bloch sphere, representing K1 and K2 as described in (Equation 7). If the Bloch vectors are not altered by a single-qubit operation, the displayed Bloch vectors may be grouped together.

[0085] The "diagram representing quantum entanglement" refers to a single point within a cube or a single point within a tetrahedron that represents the quantum entanglement generator A. Since the combination of two Bloch spheres and a single point within a cube is not unique, multiple combinations can generally be displayed. Users can select any combination and plot its properties.

[0086] Figure 13 shows another example of display. The Bloch vectors may be displayed together on a single Bloch sphere. In this display method, six Bloch vectors vec{a}, K 21 vec{a}, K 11 K 21 vec{a}, vec{b}, K 22 vec{b}, K 12 K 22 The vector vec{b} is displayed. Figure 13 shows an example of displaying the above six Bloch vectors together on a single Bloch sphere.

[0087] Figure 14 shows other display examples. Also, using four Bloch spheres (K 11 ,K 12, K 21 ,K 22) can also be represented. Each Bloch sphere is plotted with two Bloch vectors, vec{a} and K 21 vec{a}, vec{a} and K 11 vec{a}, vec{b} and K 22 vec{b}, vec{b} and K 12 The vector {vec{b} is displayed. Figure 14 shows an example of displaying using the four Bloch spheres described above.

[0088] The quantum entanglement generation structure of a two-qubit operation can be uniquely represented by a single point within the tetrahedron 1200 shown in Figure 12. Therefore, by displaying the degree of quantum entanglement generation of a two-qubit operation on the tetrahedron, it is possible to visualize how much quantum entanglement a particular operation generates. In this embodiment, the quantum entanglement strength ep(U) is introduced as an indicator, and its relationship with the coordinates within the tetrahedron is described.

[0089] The quantum entanglement intensity ep(U) of a 4x4 special unitary matrix U representing a 2-qubit operation is defined as follows (Equation 8), using (c1, c2, c3) defined in (Equation 7).

[0090]

number

[0091] The range of ep(U) is between 0 and 2 / 9. The point where ep(U) = 2 / 9 corresponds to the operation that gives the maximum quantum entanglement strength.

[0092] Figure 15 shows the relationship between each point in the tetrahedron and the quantum entanglement intensity. Solid lines indicate contour lines where the ratio of quantum entanglement intensity to the maximum quantum entanglement intensity is 0.2 or 0.4. These contour lines represent the quantum entanglement intensity distribution. The user can freely select the values ​​of the contour lines to display. In this embodiment, a method for obtaining (K1, K2, A) and (c1, c2, c3) from the special unitary matrix U is described.

[0093] First U B =Q +We calculate UQ. Here, Q refers to the matrix defined in (Equation 9), and Q + This represents the complex conjugate matrix of Q.

[0094]

number

[0095] Next, m=U B T U B Diagonalize the matrix m. This matrix m can be diagonalized using the SO(4) matrix O2, so m = O2 T This can be written as F2O2, where F2 is a diagonal matrix.

[0096] Next, the U B Using F2 and O2, we find K1, K2, and A. K2 is K2 = QO2Q + A is a diagonal matrix F satisfying FF=F2, where A=QFQ + K1 is the inverse matrix F of F -1 The matrix O1 = U is defined using B O2 T F -1 Using K1=QO1Q + It can be obtained from this.

[0097] Finally, we will show the procedure for obtaining (c1, c2, c3) from matrix A. First, we calculate (g1, g2, g3) defined by (Equation 10).

[0098]

number

[0099] Since it is known that (g1, g2, g3) are related to (c1, c2, c3) in equation (11), we solve equation (11) to find (c1, c2, c3).

[0100]

number

[0101] Generally, multiple solutions exist, but we select (c1, c2, c3) from among them that satisfy the definition of A as shown in (Equation 7).

[0102] Next, we will describe the local equivalence operation (S808) that maps the coordinates (c1, c2, c3) representing the quantum entanglement generation operation A into the tetrahedron shown in Figure 12. The local equivalence operation means a redefinition of (K1, K2).

[0103] Figure 16 shows the flow of the local equivalence operation. First, it is checked whether (c1, c2, c3) are all between 0 and π (S1601). If there are components that are not between 0 and π, local equivalence operation 1 (S1602) described below is performed.

[0104] Next, check if c1>=c2>=c3 is satisfied (S1603), and if this condition is not met, perform local equivalence operation 2 (S1604) described below.

[0105] Next, check whether the condition c1 + c2 <= π is satisfied (S1605), and if it is not satisfied, perform the local equivalence operation 3 (S1606) described below.

[0106] Finally, check the values ​​of c1 and c3 (S1607). If c3=0 and c1>π / 2, perform local equivalence operation 4 (S1608) described below.

[0107] The (c1, c2, c3) obtained through the above local equivalence operations become a point within the tetrahedron (S1609). The coordinates obtained within the tetrahedron as a result of redefining (K1, K2) through the local equivalence operations are equivalent to the original coordinates in the sense that they represent quantum entanglement. By being displayed within the tetrahedron, the relationship with the quantum entanglement intensity is more clearly shown.

[0108] The results of each local equivalence operation can be represented using Bloch vectors and tetrahedral coordinates by the method described above. The user can select any stage of the calculation from the input device 6 and plot the Bloch vectors and tetrahedral coordinates for the calculation results at that time. Next, local equivalence operation 1 will be described.

[0109] Figure 17 shows the details of local equivalence operation 1 (S1602). If c1 is greater than or equal to -π and less than 0, and both c2 and c3 are greater than or equal to 0 and less than or equal to π, then perform the local equivalence operation 1A described below. If c2 is greater than or equal to -π and less than 0, and both c3 and c1 are greater than or equal to 0 and less than or equal to π, then perform the local equivalence operation 1B described below. If c3 is greater than or equal to -π and less than 0, and both c1 and c2 are greater than or equal to 0 and less than or equal to π, then the local equivalence operation 1C described below is performed. If c1 is between 0 and π (inclusive), and both c2 and c3 are greater than or equal to -π and less than 0, then perform the local equivalence operation 1D described below. If c2 is between 0 and π, and both c3 and c1 are greater than or equal to -π and less than 0, then perform the local equivalence operation 1E described below. If c3 is between 0 and π (inclusive), and both c1 and c2 are greater than or equal to -π and less than 0, then the local equivalence operation 1F described below is performed. If c1, c2, and c3 are all greater than or equal to -π and less than 0, then perform the local equivalence operation 1G described below.

[0110] Next, we will discuss local equivalence operation 2.

[0111] Figure 18 shows the details of local equivalence operation 2 (S1604). If c1 >= c3 >= c2, then perform local equivalence operation 2A as described below. If c2 >= c1 >= c3, perform the local equivalence operation 2B described below. If c2 >= c3 >= c1, then perform the local equivalence operation 2C described below. If c3 >= c2 >= c1, then perform the local equivalence operation 2D described below. If c3 >= c1 >= c2, then perform the local equivalence operation 2E described below.

[0112] The local equivalence operations 1A-1G, 2A-2E, 3, and 4 will be discussed below. Each of these local equivalence operations refers to the substitution of (K1, K2, c1, c2, c3).

[0113] Figure 19 shows the definitions of each operation in the local equivalence operation. Substitute (K1,K2,c1,c2,c3) with (K'1,K'2,c'1,c'2,c'3) as follows. Local equivalence operation 1A refers to the operation of changing K1 to K1exp(-i(π / 2)XX) and (c1,c2,c3) to (c1+π,c2,c3). Local equivalence operation 1B refers to the operation of changing K1 to K1exp(-i(π / 2)YY) and (c1,c2,c3) to (c1,c2+π,c3). Local equivalence operation 1C refers to the operation where K1 is changed to K1exp(-i(π / 2)ZZ) and (c1,c2,c3) is changed to (c1,c2,c3+π). Local equivalence operation 1D refers to the operation where K1 is changed to K1exp(-i(π / 2)(ZZ+XX)) and (c1,c2,c3) is changed to (c1+π,c2,c3+π). Local equivalence operation 1E refers to the operation where K1 is changed to K1exp(-i(π / 2)(XX+YY)) and (c1,c2,c3) is changed to (c1,c2+π,c3+π). The local equivalence operation 1F refers to the operation where K1 is K1exp(-i(π / 2)(YY+ZZ)) and (c1,c2,c3) is (c1+π,c2+π,c3). Local equivalence operation 1G refers to the operation where K1 is K1exp(-i(π / 2)(XX+YY+ZZ)) and (c1,c2,c3) is (c1+π,c2+π,c3+π). Local equivalence operation 2A refers to the operation where K1 is set to K1exp(i(π / 4)(XI+IX)), K2 is set to exp(-i(π / 4)(XI+IX))K2, and (c1,c2,c3) is changed to (c1,c3,c2). Local equivalence operation 2B refers to the operation where K1 is set to K1exp(i(π / 4)(ZI+IZ)) and K2 is set to exp(-i(π / 4)(ZI+IZ))K2, and (c1,c2,c3) is changed to (c2,c1,c3). The local equivalence operation 2C refers to the operation where K1 is set to K1exp(i(π / 4)(ZI+IZ)) exp(i(π / 4)(XI+IX)) and K2 is set to exp(-i(π / 4)(XI+IX)) exp(-i(π / 4)(ZI+IZ))K2, and (c1,c2,c3) is changed to (c2,c3,c1). The local equivalence operation 2D refers to the operation where K1 is K1exp(i(π / 4)(YI+IY)) and K2 is exp(-i(π / 4)(YI+IY))K2, and (c1,c2,c3) is changed to (c3,c2,c1). Local equivalence operation 2E refers to the operation where K1 is set to K1exp(i(π / 4)(YI+IY)) exp(i(π / 4)(XI+IX)) and K2 is set to 2exp(-i(π / 4)(XI+IX)) exp(-i(π / 4)(YI+IY))K2, and (c1,c2,c3) is changed to (c3,c1,c2). Local equivalence operation 3 refers to the operation where K1 is set to K1exp(i(π / 4)(ZI-IZ)) exp(-i(π / 2)XX) exp(-i(π / 2)YY), K2 is set to exp(-i(π / 4)(ZI-IZ))K2, and (c1,c2,c3) is set to (π-c2,π-c1,c3). Local equivalence operation 4 refers to the operation where K1 is set to K1exp(i(π / 4)(YI-IY)) exp(-i(π / 2)XX) and K2 is set to exp(-i(π / 4)(YI-IY))K2, and (c1,c2,c3) is set to (π-c1,π-c2,0).

[0114] We have discussed how to represent two-qubit operations up to this point. Below, we will show specific examples of how to represent two-qubit operations. Here, we will discuss quantum entanglement-free operations U. a , intermediate quantum entanglement generation operation U b , Maximum entanglement generation operation U c As an example, U a =H1, U b =(rootSWAP)(HI), U c Considering =(CNOT)(HI), examples of how to represent each of the 2-qubit operations are shown in Figures 20-22.

[0115] Figure 20 shows quantum entanglement-free operation U. aAn example of the display is shown. In the table in Figure 20, the left column shows Bloch vectors, the middle column shows a cube diagram representing quantum entanglement, and the right column shows a tetrahedron diagram representing the quantum entanglement generation structure. The cube has a side length of π and its centroid is taken at the origin. When the region satisfying -π ≤ c1 ≤ π, -π ≤ c2 ≤ π, -π ≤ c1 ≤ π, 0 ≤ c3 < c2 < c1 < π, and c1 + c2 ≤ π is plotted, a tetrahedron is illustrated (the same applies to Figures 21 and 22).

[0116] Quantum entanglement-free operation U a Since this does not involve operations that generate quantum entanglement, in the decomposition of (Equation 7), A = II (4 × 4 identity matrix). Therefore, c1 = c2 = c3 = 0, and A corresponds to the origin in the cube and tetrahedron. K1 and K2 are K1 = II and K2 = HI, respectively. Therefore, the Bloch vector of qubit a changes by the amount of the H operation, and the Bloch vector of qubit b does not change. If we take the Bloch vector before the operation as a reference, vec{a} = vec{b} = (0,0,1), then K 21 vec{a}=(1,0,0), K 11 K 21 vec{a}=(1,0,0), K 22 vec{b}=(0,0,1), K 12 K 22 vec{b}=(0,0,1).

[0117] Figure 21 shows the intermediate quantum entanglement generation operation U. b An example of the display is shown. Moderate entanglement generation operation U b This can be decomposed as K1=II, K2=HI, and A=exp(-iπ / 8(XX+YY+ZZ)). The quantum entanglement generation operation is nontrivial, and c1=c2=c3=π / 4. This coordinate system is located within a tetrahedron. K1 and K2 are quantum entanglement-free operations U a Since it is equal to the same, the Bloch vector can be expressed in the same way.

[0118] Figure 22 shows the maximum quantum entanglement generation operation U. c An example of the display is shown. Maximum entanglement generation operation U cThe equation can be decomposed as K1=IH, K2=(RzRz)*HH, and A=exp(-iπ / 4(ZZ)). Here, Rz is defined as Rz=exp(iπ / 4(Z)). The quantum entanglement generation operation is A=exp(-iπ / 4(ZZ)), so the corresponding coordinates are c1=c2=0 and c3=-π / 2. Thus, these are points outside the tetrahedron. The coordinates inside the tetrahedron can be obtained by the local equivalence operation described above, and the result is c1=π / 2 and c2=c3=0. Next, we consider the change in the Bloch vectors of qubits a and b from K1 and K2. For qubit a, the Rz operation acts first, and then H acts. On the other hand, for qubit b, H acts first, and then the Rz operation and H operation act in order. As a reference, if we set the Bloch vector before the operation to vec{a}=vec{b}=(0,0,1), then K 21 vec{a}=(0,-1,0), K 11 K 21 vec{a}=(0,-1,0), K 22 vec{b}=(1,0,0), K 12 K 22 vec{b}=(0,1,0).

[0119] As shown in Figures 20 to 22, in this embodiment, the state of quantum entanglement generated by computation can be represented by the position of a point relative to a cube or tetrahedron.

[0120] As mentioned above, the solution (K1, K2, A) to the Cartan decomposition shown in (Equation 7) or Figure 10 is not uniquely determined. Therefore, there are generally multiple Bloch vectors and cubic coordinates that represent the same operation U. Each decomposition is related to the others by local equivalence operations. Users can output any Bloch vector and cubic coordinates related by local equivalence operations.

[0121] Figure 23 shows the moderate entanglement generation operation U. b This shows various representations of =(rootSWAP)(HI). Here, three types of (K1, K2, A) and their corresponding Bloch spheres and cubes are drawn.

[0122] The upper panel of Figure 23 shows vec{a}=vec{b}=(0,0,1), K21 vec{a}=(0,1,0), K 11 K 21 vec{a}=(-2 1 / 2 ,-2 1 / 2 ,0),K 22 vec{b}=(0,-1,0), K 12 K 22 vec{b}=(2 1 / 2 , 2 1 / 2 ,0), (c1,c2,c3)=(-3π / 4, 3π / 4, 3π / 4,), Figure 23 middle section shows vec{a}=vec{b}=(0,0,1), K 21 vec{a}=(0,-1,0), K 11 K 21 vec{a}=(-2 1 / 2 ,-2 1 / 2 ,0),K 22 vec{b}=(0,1,0), K 12 K 22 vec{b}=(2 1 / 2 , 2 1 / 2 ,0), (c1,c2,c3)=(π / 4, 3π / 4, 3π / 4,), Figure 23 lower panel shows vec{a}=vec{b}=(0,0,1), K 21 vec{a}=(2 1 / 2 ,0,-2 1 / 2 ), K 11 K 21 vec{a}=(2 1 / 2 ,0,2 1 / 2 ), K 22 vec{b}=(2 1 / 2 ,0,-2 1 / 2 ), K 12 K 22 vec{b}=(-2 1 / 2 ,0,-2 1 / 2 ), (c1,c2,c3)=(3π / 4, 3π / 4, π / 4,). In both cases, U can be calculated from (Equation 7) to U b This is equivalent to (rootSWAP)(HI). Performing the local equivalence operation described above on any decomposition and finding the coordinates within the tetrahedron yields c1=c2=c3=π / 4.

[0123] The quantum operation to be input can also be specifically specified as an operation implemented in a quantum computer. In this case, the quantum operation U is defined as shown in (Equation 12) using the Hamiltonian operator H which represents the physical system of the quantum computer.

[0124]

number

[0125] Here, t represents time, and hbar represents the reduced Planck constant. The T on the right-hand side signifies the time-ordered product. If the Hamiltonian is time-independent, then U = exp(-iHt / hbar). In this case, the Bloch vector and coordinates within the cube or tetrahedron described in Example 2 become functions of the Hamiltonian's parameters. By examining how much of the tetrahedron can be occupied when the Hamiltonian's parameters are changed, it is possible to analyze how broad a class of quantum entanglement generation operations can be implemented on a quantum computer. Furthermore, by comparing the Bloch vector and cubic coordinates of the operation constructed from the Hamiltonian with those of the desired operation and observing the difference, it is possible to determine how much the parameters of the quantum computer (Hamiltonian) need to be adjusted to realize the desired operation. Thus, the visualization technique of this embodiment is also useful in the design of quantum computers. [Examples]

[0126] This embodiment takes a two-qubit state or a two-qubit operation as input and outputs the properties of the specified two-qubit state and operation in a form that reveals quantum entanglement. This embodiment specifically shows an example of the input in Embodiment 1 and illustrates the use case of this embodiment. When visualizing a two-qubit operation, the quantum circuit diagram becomes the input information, and when visualizing a two-qubit state, the operation result becomes the input information.

[0127] Figure 24 shows an example of a use case. Here, we describe a case where the calculation is performed by an emulator 12 within a classical computer 1. An emulator is a device that simulates a quantum computer using the main memory 2, classical arithmetic unit 3, control unit 4, and auxiliary memory 5 within the classical computer 1. The emulator 12 has the function of simulating quantum computation by performing matrix operations corresponding to quantum circuit diagrams on the classical computer.

[0128] The visualization processing unit 10 receives the calculation results performed by the emulator 12 based on the quantum circuit diagram 11 as input information and executes the process described in Example 1. The output result (a diagram representing the Bloch sphere and quantum entanglement) is displayed on an output device 9 such as a display via the output interface 8 (in the case of state display). When displaying calculations, the quantum circuit diagram 11 becomes the input information, and the output result is displayed on the output device 9 after the process described in Example 2.

[0129] Figure 25 shows another example of a use case. Here, we describe a case where the quantum operation, which serves as input information, is performed by an emulator 14 in a classical computer 13 located outside of classical computer 1. Classical computer 13 has the same properties as classical computer 1, except for the visualization processing unit. The calculation result performed by the emulator 14 in the external classical computer 13 is transmitted to classical computer 1 via the input / output interface 15, and this calculation result is stored in the main memory 2 or auxiliary memory 5 of classical computer 1. The connection between classical computer 1 and classical computer 13 can be physical or over a network. The visualization processing unit 10 in classical computer 1 receives this calculation result as input information and executes the processing of Example 1. The output result (a diagram representing Bloch spheres and quantum entanglement) is displayed on an output device 9 such as a display via the output interface 8. Similarly, when displaying the calculation, the processing described in Example 2 is performed based on the information of the quantum circuit diagram 11 stored in classical computer 1.

[0130] Figure 26 shows another example of a use case. Here, we describe the case where the computation result is received from a quantum computer 16 outside of the classical computer 1. The quantum computer 16 has multiple qubits, and operations on these qubits are executed according to computation instructions from the classical computer 1. These computation instructions are transmitted via the input / output interface 15. Information on the qubits and operations in the quantum computer 16 is transmitted to the classical computer via the input / output interface 15, and this information is stored as a quantum circuit diagram 11 in the main memory 2 or auxiliary memory 5 of the classical computer 1. The connection between the classical computer 1 and the quantum computer 16 can be physical or over a network. The visualization processing unit 10 in the classical computer 1 receives this quantum circuit diagram 11 as input information and executes the process described in Example 1. The output result (a diagram representing a Bloch sphere and quantum entanglement) is displayed on an output device 9 such as a display via the output interface 8. When displaying the computation, the process described in Example 2 is performed based on the information of the quantum circuit diagram 11 stored in the classical computer 1, as in Example 18.

[0131] Example 3 envisions not only the visualization of the state but also the visualization of the operations. In either case, the input information is obtained from either an emulator or the results of a quantum computer. Operations in a quantum computer can be defined from the operation instructions sent from the classical computer shown in Figure 26. Furthermore, operations in a quantum computer can also be defined from the changes in the quantum state within the quantum computer.

[0132] Thus, the information for visualization can be obtained from quantum circuit diagrams, emulators within an information processing device having a visualization processing unit, emulators in a computer outside the information processing device, and quantum computers located inside or outside the information processing device.

[0133] As described in detail above, the two-qubit state is displayed using a Bloch sphere and a diagram illustrating quantum entanglement. The diagram illustrating quantum entanglement uses, for example, a 3x3 bar graph or an ellipse. In addition to displaying the two-qubit state, the embodiment can also display two-qubit operations, in which case the quantum entanglement is represented by a single point within a cube or a single point within a tetrahedron. As a result, quantum entanglement and the quantum entanglement generation structure become intuitively understandable, and the development of quantum computers and quantum algorithms becomes more efficient.

[0134] According to the above embodiment, efficient operation of quantum computers becomes possible, resulting in lower energy consumption, reduced carbon emissions, prevention of global warming, and contribution to the realization of a sustainable society. [Explanation of Symbols]

[0135] 1. Classical computer with a visualization processing unit 2 Main memory 3 Classical calculation section 4. Control device 5 Auxiliary storage 6 Input devices 7 Input Interfaces 8 output interfaces 9 Output device 10 Visualization Processing Unit Quantum circuit diagram with 11 inputs 12. Internal emulator of Classical Computer 1 13 Classical computer outside of classical computer 1 14. Internal emulator of the classical computer 13 15 Input / Output Interfaces Connected to External Computers 16 Quantum computer

Claims

1. An input interface that receives at least one of information about the states of a plurality of qubits and information about operations performed on a plurality of qubits, A visualization processing unit that generates parameters for visualizing at least one of the states of any two qubits among the plurality of qubits, and any two operations on any two qubits among the operations on the plurality of qubits, The system includes an output interface that outputs output information based on the aforementioned parameters, An information processing device that displays a diagram representing a Bloch sphere and quantum entanglement on an output device based on the output information, The visualization processing unit calculates the parameters based on the density matrix of the states of the two qubits. The diagram representing the quantum entanglement is a diagram that shows the qubit correlation matrix obtained based on the density matrix in at least one form: numerical and graphical representation. Information processing device.

2. An input interface that receives at least one of information about the states of a plurality of qubits and information about operations performed on a plurality of qubits, A visualization processing unit that generates parameters for visualizing at least one of the states of any two qubits among the plurality of qubits, and any two operations on any two qubits among the operations on the plurality of qubits, The system includes an output interface that outputs output information based on the aforementioned parameters, An information processing device that displays a diagram representing a Bloch sphere and quantum entanglement on an output device based on the output information, The visualization processing unit calculates the parameters based on the density matrix of the states of the two qubits. The diagram representing the quantum entanglement is a diagram showing a space bounded by an ellipse inside a three-dimensional sphere of radius 1, based on the non-negative singular values ​​of the qubit correlation matrix obtained based on the density matrix. Information processing device.

3. An input interface that receives at least one of information about the states of a plurality of qubits and information about operations performed on a plurality of qubits, A visualization processing unit that generates parameters for visualizing at least one of the states of any two qubits among the plurality of qubits, and any two operations on any two qubits among the operations on the plurality of qubits, The system includes an output interface that outputs output information based on the aforementioned parameters, An information processing device that displays a diagram representing a Bloch sphere and quantum entanglement on an output device based on the output information, The visualization processing unit calculates the parameters based on the density matrix of the states of the two qubits. The Bloch vector shown in the Bloch sphere is generated based on elements obtained by decomposing the density matrix, and the length of the Bloch vector represents the degree of quantum entanglement. Information processing device.

4. Either there are two Bloch spheres, each displaying a Bloch vector, or there is one Bloch sphere, and two Bloch vectors are displayed corresponding to the states of the two qubits. The information processing apparatus according to claim 3.

5. An input interface that receives at least one of information about the states of a plurality of qubits and information about operations performed on a plurality of qubits, A visualization processing unit that generates parameters for visualizing at least one of the states of any two qubits among the plurality of qubits, and any two operations on any two qubits among the operations on the plurality of qubits, The system includes an output interface that outputs output information based on the aforementioned parameters, An information processing device that displays a diagram representing a Bloch sphere and quantum entanglement on an output device based on the output information, The visualization processing unit calculates the parameters based on a unitary matrix that shows the operations on the two qubits. Information processing device.

6. The Bloch vector shown in the Bloch sphere is generated based on an operation that acts on only one qubit obtained by decomposing the unitary matrix. The information processing apparatus according to claim 5.

7. The Bloch vectors shown in the Bloch sphere are six vectors, including a vector generated based on four operations that act on only one qubit obtained by cartan decomposition of the unitary matrix, and the six vectors are represented collectively or dispersed across one or more Bloch spheres. The information processing apparatus according to claim 6.

8. The figure representing the quantum entanglement is a cube with a side length of π and its center of gravity at the origin, and 0 <= c 3 <= c 2 <= c 1 < π and c 1 + c 2 <= π and c 3 = 0 when c 1 <= π / 2 is satisfied (c 1 , c 2 , c 3 ), which is obtained by superimposing points represented by a set of three real numbers generated based on a two-qubit operation obtained by performing a Cartan decomposition on the unitary matrix on a tetrahedron composed of the set. The information processing apparatus according to claim 5.

9. When the point represented by the set of three real numbers is located outside the tetrahedron, a local equivalence operation is performed to displace the point represented by the set of three real numbers inside the tetrahedron, and the degree of quantum entanglement is displayed on the tetrahedron as an intensity distribution. The information processing apparatus according to claim 8.

10. An input interface that receives at least one of information about the states of a plurality of qubits and information about operations performed on a plurality of qubits, A visualization processing unit that generates parameters for visualizing at least one of the states of any two qubits among the plurality of qubits, and any two operations on any two qubits among the operations on the plurality of qubits, The system includes an output interface that outputs output information based on the aforementioned parameters, An information processing device that displays a diagram representing a Bloch sphere and quantum entanglement on an output device based on the output information, At least one of the information regarding the states of the plurality of qubits and the information regarding operations on the plurality of qubits is obtained from a quantum circuit diagram, an emulator in the information processing device which is a classical computer having the visualization processing unit, an emulator in another classical computer outside the information processing device, and a quantum computer located inside or outside the information processing device. Information processing device.

11. An information processing method performed on an information processing device comprising an input interface, an output interface, a storage device, and an arithmetic unit, and having a visualization processing unit, The visualization processing unit is An input step that receives at least one of the following: information about the states of multiple qubits, and information about operations performed on multiple qubits. A visualization step of generating a set of parameters for visualizing at least one of the states of any two qubits among the plurality of qubits, and any two operations on any two qubits among the operations on the plurality of qubits, Based on the aforementioned set of parameters, an output step is performed to output output information based on the set of parameters in order to display a diagram representing the Bloch sphere and quantum entanglement on the output device, Execute, In the input step, information about the state of the plurality of qubits is received, The visualization step includes a first step of obtaining a density matrix of two-qubit states from the state information of the plurality of qubits, and a second step of calculating a qubit correlation matrix and a density matrix for each individual qubit from the density matrix. In the output step, a first parameter is output for displaying a 3x3 matrix representing quantum entanglement based on the density matrix of the two qubit states, and a second parameter is output for displaying Bloch vectors to be displayed on the Bloch sphere based on the density matrix of each individual qubit. Information processing methods.

12. An information processing method performed on an information processing device comprising an input interface, an output interface, a storage device, and an arithmetic unit, and having a visualization processing unit, The visualization processing unit is An input step that receives at least one of the following: information about the states of multiple qubits, and information about operations performed on multiple qubits. A visualization step of generating a set of parameters for visualizing at least one of the states of any two qubits among the plurality of qubits, and any two operations on any two qubits among the operations on the plurality of qubits, Based on the aforementioned set of parameters, an output step is performed to output output information based on the set of parameters in order to display a diagram representing the Bloch sphere and quantum entanglement on the output device, Execute, In the input step, information about the operations on the plurality of qubits is received, The visualization step includes a third step of obtaining a unitary matrix representing a two-qubit operation from the information of operations on the plurality of qubits, and a fourth step of decomposing the unitary matrix to obtain a first operation that acts on one qubit alone and a second operation that acts on two qubits. In the output step, a third parameter is output for displaying Bloch vectors to be shown on the Bloch sphere based on the first calculation, and a fourth parameter is output for displaying points inside or outside a tetrahedron, superimposed on a cube with side length π and a tetrahedron inside the cube, based on the second calculation. Information processing methods.

Images