Quantum computing program, quantum computing device, and quantum computing method

Abstract

To transform maximum likelihood detection performed in multi-antenna wireless communication into a binary optimization problem.SOLUTION: A quantum calculation program causes a computer to perform a transformation function that transforms maximum likelihood detection into a binary optimization problem by projecting a data symbol transmitted from at least two transmitting antennas, passing through a transmission path, and received by at least two receiving antennas from a binary variable to a complex number, and using the complex number to separate and expand the Frobenius norm of maximum likelihood detection that decodes the data symbol, into a real part and an imaginary part.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to a quantum computing program, a quantum computing device, and a quantum computing method. [Background technology]

[0002] Currently, research into quantum computers is progressing due to the physical limits of semiconductor miniaturization approaching and the expectation that quantum computers will be able to solve problems that are difficult to solve with classical computers. Wireless communication is one technical field in which quantum computers can be applied. Examples of literature relating to the application of quantum computers to wireless communication include Non-Patent Document 1, Non-Patent Document 2, and Non-Patent Document 3. [Prior art documents] [Non-patent literature]

[0003] [Non-Patent Document 1] P. Botsinis, D. Alanis, Z. Babar, SX Ng, and L. Hanzo “Iterative quantum-assisted multi-user detection for multi-carrier interleave division multiple access systems,” IEEE Transactions on Communications, vol. 63, no. 10, pp. 3713-3727, Oct. 2015. [Non-patent document 2] P. Botsinis, SX Ng, and L. Hanzo “Fixed-complexity quantum-assisted multi-user detection for CDMA and SDMA,” IEEE Transactions on Communications, vol. 62, no. 3, pp. 990-1000, Mar. 2014. [Non-patent document 3] S. Mondal, MR Laskar, and AK Dutta, “ML criterion based signal detection of a MIMO-OFDM system using quantum and semi-quantum assisted modified DHA / BBHT search algorithm,” IEEE Transactions on Vehicular Technology, vol. 70, no. 2, pp. 1688-1698, Feb. 2021.

[0004] Non-Patent Document 1 discloses a technique for applying the Dürr-Heuyer algorithm, which is a type of quantum search algorithm, to multi-user detection performed in wireless communications. Non-Patent Document 1 discloses the behavior of query complexity and bit error rate (BER) under the assumption that division, Frobenius norm, exponential function, and normal distribution can be calculated by quantum circuits.

[0005] Non-Patent Document 2 discloses a technology that accelerates the convergence of DHA to an optimal solution by determining the threshold value used in DHA based on a suboptimal solution calculated by a classical computer, under the same assumptions as Non-Patent Document 1.

[0006] Non-Patent Document 3 discloses a technology that uses a quasi-quantum algorithm to speed up maximum likelihood detection performed in multi-antenna wireless communication. Under the assumption that the objective function can be processed by a quantum circuit, this technology succeeds in reducing the amount of query calculations by treating the number of times the Grover operator is applied as a random variable that follows a gamma distribution.

[0007] However, the technologies disclosed in Non-Patent Document 1, Non-Patent Document 2, and Non-Patent Document 3 all merely evaluate the characteristics of wireless communication systems under the assumption that any objective function can be processed by a quantum circuit. However, it is not clear whether such an assumption actually holds, and there are technical hurdles to processing objective functions containing real coefficients with a quantum circuit. For this reason, in order to apply quantum computers to wireless communication, there is a need to develop a technology that converts problems into ones that can be easily processed by a quantum circuit. Summary of the Invention [Problem to be solved by the invention]

[0008] The present invention has been made in consideration of the above-mentioned circumstances, and aims to provide a quantum computing program, a quantum computing device, and a quantum computing method that can convert maximum likelihood detection performed in multi-antenna wireless communication into a binary optimization problem. [Means for solving the problem]

[0009] One aspect of the present invention is a quantum computing program that causes a computer to execute a transformation function that converts maximum likelihood detection into a binary optimization problem by projecting data symbols transmitted from at least two transmitting antennas, passing through a transmission path, and received by at least two receiving antennas from binary variables into complex numbers, and then using the complex numbers to decode the data symbols by expanding the Frobenius norm of the maximum likelihood detection into a real part and an imaginary part.

[0010] In one aspect of the present invention, the quantum computing program described above includes a quantum computing function that applies Grover adaptive search to the binary optimization problem and executes a quantum computing process to derive variables that have the potential to make the value of an objective function of the binary optimization problem smaller than a predetermined threshold; a classical computing function that executes a classical computing process to calculate the value of the objective function when the variable is given using a classical computer; and a program that updates the predetermined threshold to the value of the objective function when the variable is given if the value of the objective function when the variable is given is smaller than the predetermined threshold. New and a repetitive processing function that repeats the process of having the quantum computing function execute the quantum computing process, having the classical computing function execute the classical computing process, and having the update function execute the update process until a predetermined condition is satisfied.

[0011] In one aspect of the present invention, in the quantum computing program described above, the quantum computing function determines an initial variable, which is an initial value of the variable, using minimum mean square error estimation, and sets the smaller value of the value of the objective function when the initial variable is given and a value based on a distribution of minimum values ​​of the objective function as the initial value of the predetermined threshold, and uses the initial value of the predetermined threshold as the predetermined threshold in the quantum computing process to be executed for the first time.

[0012] One aspect of the present invention is a quantum computing device that includes a conversion unit that projects data symbols transmitted from at least two transmitting antennas, passed through a transmission path, and received by at least two receiving antennas from binary variables to complex numbers, and converts the maximum likelihood detection into a binary optimization problem by expanding the Frobenius norm of the maximum likelihood detection that decodes the data symbols using the complex numbers, by separating the Frobenius norm into a real part and an imaginary part.

[0013] In one aspect of the present invention, data symbols transmitted from at least two transmitting antennas, transmitted via a transmission path, and received by at least two receiving antennas are projected from binary variables to complex numbers, and the Frobenius norm of maximum likelihood detection for decoding the data symbols using the complex numbers is expanded by dividing the Frobenius norm into real and imaginary parts, thereby converting the maximum likelihood detection into a binary optimization problem. The computer runs It is a quantum computing method. [Effects of the Invention]

[0014] According to the present invention, it is possible to provide a quantum computing program, a quantum computing device, and a quantum computing method that can convert maximum likelihood detection performed in multi-antenna wireless communication into a binary optimization problem. [Brief explanation of the drawings]

[0015] [Figure 1] FIG. 2 is a diagram illustrating an example of a software configuration of a quantum computing device according to an embodiment of the present invention. [Figure 2] 1 is a diagram illustrating an example of a multi-antenna wireless communication system according to an embodiment of the present invention. [Figure 3] FIG. 1 is a diagram illustrating an example of a binary phase shift keying signal point arrangement according to an embodiment of the present invention. [Figure 4] FIG. 1 is a diagram illustrating an example of a signal point arrangement for quadrature phase shift keying according to an embodiment of the present invention. [Figure 5] FIG. 1 is a diagram illustrating an example of a signal point arrangement for quadrature amplitude modulation according to an embodiment of the present invention. [Figure 6] FIG. 10 is a diagram illustrating an example of a cumulative distribution function of the probability that a global optimum solution is greater than a threshold when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the number of transmitting antennas and the number of receiving antennas are two. [Figure 7] FIG. 10 is a diagram illustrating an example of the relationship between the communication quality of a transmission path and the bit error rate when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the number of transmitting antennas and the number of receiving antennas is two. [Figure 8] FIG. 10 is a diagram showing an example of the relationship between the number of queries in the classical domain and the value of the objective function when maximum likelihood detection in a comparative example is performed, and the relationship between the number of queries in the classical domain and the value of the objective function when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the number of transmitting antennas and the number of receiving antennas is two. [Figure 9] FIG. 10 is a diagram showing an example of the relationship between the number of queries to the quantum region and the value of the objective function when maximum likelihood detection in a comparative example is performed, and the relationship between the number of queries to the quantum region and the value of the objective function when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the number of transmitting antennas and the number of receiving antennas is two. [Figure 10] FIG. 10 is a diagram showing an example of the relationship between the sum of the number of queries in the classical domain and the number of queries in the quantum domain and the value of the objective function when maximum likelihood detection in a comparative example is performed, and the relationship between the sum of the number of queries in the classical domain and the number of queries in the quantum domain and the value of the objective function when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the number of transmitting antennas and the number of receiving antennas is two. [Figure 11] FIG. 10 is a diagram illustrating an example of the relationship between the communication quality of a transmission path and the bit error rate when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature amplitude modulation and the number of transmitting antennas and the number of receiving antennas is two. [Figure 12] FIG. 10 is a diagram showing an example of the relationship between the number of queries in the classical domain and the value of the objective function when maximum likelihood detection in a comparative example is performed, and the relationship between the number of queries in the classical domain and the value of the objective function when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature amplitude modulation and the number of transmitting antennas and the number of receiving antennas is two. [Figure 13] FIG. 10 illustrates an example of the relationship between transmission rate and query complexity for exhaustive search and Grover adaptive search. [Figure 14] 10A and 10B are diagrams illustrating an example of how to select a transmitting antenna and how to correspond to a bit string according to an embodiment of the present invention. [Figure 15] FIG. 10 is a diagram illustrating an example of the relationship between communication quality of a transmission path and a bit error rate when a multi-antenna wireless communication system according to an embodiment of the present invention employs space shift keying, three of six transmitting antennas are used, and the number of receiving antennas is six. [Figure 16] FIG. 10 is a diagram showing an example of the relationship between the number of queries in the classical domain and the value of the objective function when maximum likelihood detection or full search is performed in a comparative example, and the relationship between the number of queries in the classical domain and the value of the objective function when a multi-antenna wireless communication system according to an embodiment of the present invention employs space shift keying, three of six transmitting antennas are used, and the number of receiving antennas is six. [Figure 17] FIG. 1 is a diagram illustrating an example of a quantum circuit used in a simulation of a Grover adaptive search according to an embodiment of the present invention. [Figure 18] FIG. 10 is a diagram illustrating an example of the relationship between the number of queries in the classical domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 5 dB. [Figure 19] FIG. 10 is a diagram illustrating an example of the relationship between the number of queries to the quantum region and the bit error rate when the multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 5 dB. [Figure 20] FIG. 10 is a diagram illustrating an example of the relationship between the sum of the number of queries in the classical domain and the number of queries in the quantum domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 5 dB. [Figure 21] FIG. 10 is a diagram illustrating an example of the relationship between the number of queries in the classical domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 5 dB. [Figure 22]FIG. 10 is a diagram illustrating an example of the relationship between the number of queries to the quantum region and the bit error rate when the multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 5 dB. [Figure 23] FIG. 10 is a diagram illustrating an example of the relationship between the sum of the number of queries in the classical domain and the number of queries in the quantum domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 5 dB. [Figure 24] FIG. 1 is a diagram illustrating an example of a quantum circuit according to an embodiment of the present invention. [Figure 25] FIG. 10 is a diagram showing an example of the probability of observing each state when a Grover operator is applied 0 times to a quantum bit indicating a value of an objective function according to an embodiment of the present invention. [Figure 26] FIG. 10 is a diagram showing an example of the probability of observing each state when a Grover operator is applied once to a quantum bit indicating a value of an objective function according to an embodiment of the present invention. [Figure 27] FIG. 10 is a diagram showing an example of the probability of observing each state when a Grover operator is applied twice to a quantum bit indicating a value of an objective function according to an embodiment of the present invention. [Figure 28] FIG. 10 is a diagram showing an example of the probability of observing each state when a Grover operator is applied three times to a quantum bit indicating a value of an objective function according to an embodiment of the present invention. [Figure 29] FIG. 10 is a diagram showing an example of the probability of observing each state when a Grover operator is applied four times to a quantum bit indicating a value of an objective function according to an embodiment of the present invention. [Figure 30] FIG. 10 is a diagram showing an example of the relationship between the number of trials and the value of the objective function when a full search according to a comparative example is performed, and an example of the relationship between the number of trials and the value of the objective function when a process according to an embodiment of the present invention is performed. [Figure 31] 4 is a flowchart illustrating an example of processing executed by a quantum computing device according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0016] [Embodiment] A quantum computing program, a quantum computing device, and a quantum computing method according to the embodiments will be described with reference to FIGS.

[0017] Fig. 1 is a diagram showing an example of the software configuration of a quantum computing device according to an embodiment of the present invention. As shown in Fig. 1, the quantum computing device 1 includes a transformation function 11, a quantum computing function 12, a classical computing function 13, an update function 14, and an iterative processing function 15. The transformation function 11, the quantum computing function 12, the classical computing function 13, the update function 14, and the iterative processing function 15 are all realized by executing a quantum computing program 10.

[0018] 2 is a diagram illustrating an example of a multi-antenna wireless communication system according to an embodiment of the present invention. The quantum computing device 1 converts, for example, maximum likelihood detection executed in the multi-antenna wireless communication system illustrated in FIG. 2 into a binary optimization problem, and searches for a global optimal solution to the binary optimization problem. As illustrated in FIG. 2, the multi-antenna wireless communication system according to the embodiment includes N t a transmitter having N transmitting antennas; r The receiver has two receiving antennas. 11 , …, channel coefficient h 1Nt , …, channel coefficient h Nr1 , … and channel coefficients h NrNt These channel coefficients and the noise v i follows a complex normal distribution with mean 0 and variance 1. N denotes the transmitted symbol. t N-dimensional vector s, representing the received symbols r dimensional vector r and N showing fading r Row N t The relationship of the channel coefficient matrix H is expressed by the following equations (1) and (2): Equation (2) represents the signal-to-noise ratio of the transmission path, and the standard deviation of the signal-to-noise ratio of the transmission path σ v Contains:

[0019]

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[0020]

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[0021] Furthermore, the maximum likelihood detection executed in the multi-antenna wireless communication system shown in FIG. 2 is expressed by the following equation (3). In this case, the amount of query calculation is 2 R In addition, the quantum computing device 1 performs a Grover adaptive search by expanding equation (3).

[0022]

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[0023] However, since the Grover adaptive search targets a binary optimization problem, it cannot be directly applied to Equation (3). Therefore, transformation function 11 projects data symbols transmitted from at least two transmit antennas, transmitted via a transmission path, and received by at least two receive antennas from binary variables to complex numbers. Specifically, to treat maximum likelihood detection as a binary optimization problem, transformation function 11 uses the following Equation (4), Equation (5), or Equation (6), which is a relationship between bit b(i) and transmission symbol s(i) disclosed in the 5G NR specification, "3GPP, "TS 138 211 - V15.2.0 - 5G; NR; Physical channels and modulation (3GPPTS 38.211 version 15.2.0 Release 15)," 2018. Equation (4), Equation (5), and Equation (6) include the imaginary unit j.

[0024]

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[0027] Equation (4) is an equation used when a multi-antenna wireless communication system employs binary phase shift keying (BPSK). Fig. 3 is a diagram showing an example of a constellation of binary phase shift keying signals according to an embodiment of the present invention. Fig. 3 shows a constellation of signal points determined by equation (4) when bit b(i) is 0 and a constellation of signal points determined by equation (4) when bit b(i) is 1.

[0028] Equation (5) is an equation for the case where a multi-antenna wireless communication system employs quadrature phase shift keying (QPSK). Fig. 4 is a diagram showing an example of a signal point arrangement for quadrature phase shift keying according to an embodiment of the present invention. Fig. 4 shows the signal point arrangement determined by equation (5) when bit b(i) is 00, 01, 10, and 11, respectively.

[0029] Equation (6) is an equation when a multi-antenna wireless communication system employs quadrature amplitude modulation (QAM), specifically 16-QAM. Fig. 5 is a diagram showing an example of a signal point arrangement for quadrature amplitude modulation according to an embodiment of the present invention. Fig. 5 shows a signal point arrangement determined by equation (6) when bit b(i) takes each combination.

[0030] Next, the transformation function 11 converts the maximum likelihood detection into a binary optimization problem by dividing the Frobenius norm of the maximum likelihood detection that decodes the data symbols using the above-mentioned complex numbers into a real part and an imaginary part and expanding it. For example, the transformation function 11 substitutes, for example, equation (4), equation (5), or equation (6) into equation (3) and expands it, depending on the modulation scheme adopted in the multi-antenna wireless communication system, to obtain a variable vector b corresponding to the bit string.

[0031] As shown in Equation (3), maximum likelihood detection is expressed by the square of the Frobenius norm. Therefore, when the modulation scheme employed in the multi-antenna wireless communication system is binary phase-shift keying or quadrature phase-shift keying, maximum likelihood detection becomes a quadratic unconstrained binary optimization (QUBO) problem. Furthermore, when the modulation scheme employed in the multi-antenna wireless communication system is 16-QAM, maximum likelihood detection becomes a fourth-order binary optimization problem because the transmitted symbols are quadratic expressions of bit strings. However, the quantum computing device 1 can handle not only QUBO problems but also higher-order binary optimization problems by Grover adaptive search, and can therefore also handle fourth-order binary optimization problems. Furthermore, when an increase in the query computational complexity of the Grover adaptive search is tolerable, the quantum computing device 1 can also transform a higher-order binary optimization problem into a QUBO problem by adding variables and constraints.

[0032] Next, we will explain the application of Grover adaptive search to maximum likelihood detection in generalized spatial modulation. When generalized spatial modulation is adopted, a multi-antenna wireless communication system has N transmit antennas. t Information is assigned to K points selected from the set of points. In particular, in generalized spatial modulation where the number of modulation points L=1, the transmission rate R is expressed by the following equation (7). The parentheses in equation (7) represent binomial coefficients. Also, the square brackets with missing tops in equation (7) represent floor functions.

[0033]

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[0034] The transformation function 11 is expressed as follows: tIn order to adjust the number of combinations of allocating information to K symbols selected from the set of symbols to a power of 2, some of the transmission symbols are not used. Furthermore, since the number of modulation points L=1, the data symbols become binary variables, and therefore the conversion function 11 can treat maximum likelihood detection as a binary optimization problem. The conversion function 11 uses the term obtained by expanding the Frobenius norm included in equation (3) as the objective function E1(s), and adds constraints expressed by the second and third terms on the right-hand side of equation (8) to limit the transmission symbols to those that use the solution of the binary optimization problem.

[0035]

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[0036] Equation (3) is the constraint parameter λ C , a vector j having an index of the non-zero components, and a set χ of vector j of unused patterns among the K non-zero components. The second term on the right side of equation (8) represents an equality constraint, and t The value increases when K antennas are not selected from the set of N antennas. t The value increases if an unused pattern is selected when selecting K patterns from the set of K. Furthermore, although the maximum degree of the third term on the right-hand side of equation (8) depends on K, if the increased complexity of the quantum circuit due to the increase in control bits is acceptable, it can be handled by constructing a higher-order quantum circuit.

[0037] Furthermore, the search range for the global optimum is determined by the data symbols s i Therefore, 2 Nt However, the computational complexity of finding the global optimum has increased from O(2R) to O(2 Nt / 2 ) and therefore the transmitting antenna N tIt decreases when K is such that the transmission rate R is large within the range that can be expressed in a binary representation, that is, when K is such that equation (7) is large. Also, if it is possible to obtain a unique bit string by obtaining binary values ​​from estimated data symbols by applying Grover adaptive search to the binary optimization problem, the computational complexity of converting data symbols into bit strings is O(1).

[0038] The Grover adaptive search is performed by dividing the current minimum by the threshold y i The objective function is set to the threshold y i The quantum computing function 12 searches for a global optimum solution to a binary optimization problem by amplifying the probability of observing a state where the threshold is smaller than . Therefore, it is preferable that the quantum computing function 12 appropriately determines the initial value of the threshold to reduce the amount of query calculation. The quantum computing function 12 determines the initial value of the threshold by using the probability distribution of the minimum value based on the signal-to-noise ratio of the transmission path.

[0039] If there is no error, the minimum value of the objective function is considered to be the sum of squares of the norm as shown in the following equation (9) from equation (3).

[0040]

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[0041] The magnitude of the norm included in equation (9) follows a Rayleigh distribution because it is assumed that the noise in the transmission path follows a complex normal distribution. Since the square of the norm included in equation (9) can be calculated to follow an exponential distribution, equation (9) follows an Erlang distribution and is expressed as the probability density functions shown in the following equations (10) and (11).

[0042]

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[0044] The cumulative distribution function is expressed by the following equation (12).

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[0046] Even if an error is included in the data symbol, the quantum computing function 12 can know in advance that the global optimum solution of the binary optimization problem is below a certain value with a very high probability because the value of equation (12) is considered to be an upper bound. r When =2, it is expressed by the following equation (13).

[0047]

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[0048] 6 is a diagram showing an example of a cumulative distribution function of the probability that a global optimum solution is greater than a threshold when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the number of transmitting antennas and receiving antennas is two. The cumulative distribution function shown in FIG. 6 is obtained when the multi-antenna wireless communication system employs quadrature phase shift keying and N t = 2, and N r The formula is derived by adding a constant term obtained by expanding formula (3) to formula (1) when ∑ = 2, and using the minimum value obtained by minimizing the result and formula (13).

[0049] In FIG. 6, the thick solid line shows the simulation results when the signal-to-noise ratio of the transmission channel is 5 dB. Furthermore, in FIG. 6, the thick dotted line shows equation (13) when the signal-to-noise ratio of the transmission channel is 5 dB. Meanwhile, in FIG. 6, the thin solid line shows the simulation results when the signal-to-noise ratio of the transmission channel is 15 dB. Furthermore, in FIG. 6, the thin dotted line shows equation (13) when the signal-to-noise ratio of the transmission channel is 15 dB. Referring to FIG. 6, the lower the signal-to-noise ratio of the transmission channel, the larger the minimum value of the objective function, and the larger the difference from the value of equation (13). From the above results, the probability P that the threshold of the Grover adaptive search is smaller than the minimum value of the objective function is expressed by the following equation (14). Equation (14) is related to the signal-to-noise ratio of the transmission channel, σ v 2 The Grover adaptive search threshold y is calculated by the probability P that the Grover adaptive search threshold is smaller than the minimum value of the objective function. m This shows that the value is determined.

[0050]

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[0051] Grover adaptive search threshold y m Substituting into equation (14), y = xe x Using the Lambert W function, which is the inverse function of the W function, we transform it into the following equation (15). Since the W function is a two-valued function when x<0, we can see that W≦-1 is a branch. -1 Contains (x).

[0052]

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[0053] In the following explanation, the following two methods for determining the initial value of the threshold are shown and compared. The first method is a method in which the initial variable, which is the initial value of the variable, is determined using minimum mean squared error estimation (MMSE), and the smaller of the value of the objective function when the initial variable is given and a value based on the distribution of the minimum value of the objective function is set as the initial value of the threshold. The second method is a method in which the initial variable, which is the initial value of the variable, is determined randomly, and the value of the objective function when the initial variable is given is set as the initial value of the threshold.

[0054] Zero forcing (ZF) is a technique for canceling interference by applying a pseudo-inverse matrix expressed by the following equation (16) to received symbols r, thereby performing independent detection. Minimum mean square error estimation is a technique for canceling interference by applying a pseudo-inverse matrix expressed by the following equation (17) to received symbols r, thereby performing independent detection. Equation (18) expresses the pseudo-inverse matrix W ZF or the pseudoinverse matrix W MMSE When the pseudo-inverse matrix W is defined as the received symbol r, the estimated value s (hat) of the transmitted symbol is expressed in closed form by applying the pseudo-inverse matrix W to the received symbol r.

[0055]

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[0058] The above method uses Equation (14) which represents the probability distribution, so the initial value of the threshold may be smaller than the global optimum. The sum of squares of the norm is determined by the signal-to-noise ratio of the transmission path. mIf ρ is greater than ρ, the quantum computing function 12 performs a random search because there is nothing to amplify the observed probability for. If the upper bound of the probability that the initial value of the threshold will be smaller than the global optimum is the probability expressed by equation (14), this probability becomes the upper bound of the probability of performing a random search. The quantum computing function 12 controls the probability of performing a random search using the probability expressed by equation (14), and by setting an appropriate probability, it is possible to achieve average improvement, including the random search.

[0059] In maximum likelihood detection, the quantum computing function 12 must handle objective functions whose coefficients contain real numbers due to transmitted symbols, channel coefficients, noise, etc. Two methods for handling objective functions whose coefficients contain real numbers are approximation by integers and inputting real numbers as they are. When the integer approximation method is used, attention must be paid to the trade-off between the accuracy of the approximation and the number of quantum bits. On the other hand, by inputting real numbers as they are, the quantum computing function 12 can reduce the number of quantum bits m, which is advantageous when operating both a quantum computer and a classical computer.

[0060] However, if real coefficients are input directly to a quantum circuit, not only the binary variable portion but also the m quantum bits representing the objective function will be superposed, resulting in different probabilities. In such cases, the quantum circuit may not be able to output the correct objective function value. Therefore, as described below, the quantum computing device 1 applies Grover adaptive search to the binary optimization problem to derive variables that have the potential to reduce the value of the objective function of the binary optimization problem below a predetermined threshold, and then reevaluates the value of the objective function given the variables using a classical computer. This enables the quantum computing device 1 to cause the quantum circuit to output the correct value of the objective function.

[0061] Next, we obtain the objective function by quadrature phase shift keying and demonstrate the effectiveness of the objective function based on the relationship between the signal-to-noise ratio and bit error rate of the transmission path. t = 2, and Nr = 2. In this case, the objective function is to divide equation (5) by 1 / 2. 1 / 2 When the total power is set to 1, the following equations (19), (20) and (21) are obtained from equation (3).

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[0065] FIG. 7 is a diagram showing an example of the relationship between the communication quality of the transmission channel and the bit error rate when the multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the number of transmitting antennas and the number of receiving antennas are two. When the coefficients include real numbers, the relationship between the communication quality of the transmission channel and the bit error rate is equivalent to that of maximum likelihood detection, and it can be seen that the objective function is effective. Furthermore, referring to FIG. 7, it can be seen that performance varies significantly depending on the accuracy of approximation to integers. Therefore, even in Grover adaptive search, it is necessary to increase the number m of quantum bits expressing the objective function in order to approximate with large integers.

[0066] Next, we performed a simulation using Grover adaptive search to show the relationship between the number of queries and the value of the objective function. In this case, integer approximation was performed by tripling the real coefficients. In addition, the channel coefficients in this case were expressed as in Equation (22) below, with all minimum values ​​set to 0.

[0067]

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[0068] FIG. 8 is a diagram showing an example of the relationship between the number of queries in the classical domain and the value of the objective function when maximum likelihood detection according to a comparative example is performed, and the relationship between the number of queries in the classical domain and the value of the objective function when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the number of transmitting antennas and the number of receiving antennas is two.

[0069] 9 is a diagram showing an example of the relationship between the number of queries of the quantum region and the value of the objective function when maximum likelihood detection according to a comparative example is performed, and the relationship between the number of queries of the quantum region and the value of the objective function when the multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase shift keying and the number of transmitting antennas and the number of receiving antennas is 2. Here, the number of queries of the quantum region is the same objective function value among the number of operations of the Grover operator obtained from uniform random numbers, and the minimum value before measurement is shown as the result.

[0070] 10 is a diagram showing an example of a relationship between the sum of the number of queries in the classical domain and the number of queries in the quantum domain and the value of the objective function when maximum likelihood detection according to a comparative example is performed, and a relationship between the sum of the number of queries in the classical domain and the number of queries in the quantum domain and the value of the objective function when the multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature phase-shift keying and the number of transmitting antennas and the number of receiving antennas is 2. Here, the number of queries in the quantum domain is the same objective function value among the number of operations of the Grover operator obtained from uniform random numbers, and the minimum value before measurement is shown as the result.

[0071] As shown in Figure 8, Grover adaptive search converges the objective function value to 0 with fewer queries in the classical domain compared to maximum likelihood detection. Also, as shown in Figure 9, Grover adaptive search converges the objective function value to 0 with fewer queries in the quantum domain compared to maximum likelihood detection. On the other hand, as shown in Figure 10, Grover adaptive search converges the objective function value to 0 with the sum of the number of queries in the classical domain and the number of queries in the quantum domain, which is more than maximum likelihood detection. This is because the size of the binary optimization problem is 2 4This is thought to be because the value is relatively small at 16.

[0072] Fig. 11 is a diagram showing an example of the relationship between communication quality of a transmission path and a bit error rate when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature amplitude modulation and the number of transmitting antennas and the number of receiving antennas is 2. Referring to Fig. 11, it can be seen that the objective function is effective.

[0073] 12 is a diagram showing an example of the relationship between the number of queries in the classical domain and the value of the objective function when maximum likelihood detection according to a comparative example is performed, and the relationship between the number of queries in the classical domain and the value of the objective function when a multi-antenna wireless communication system according to an embodiment of the present invention employs quadrature amplitude modulation and has two transmitting antennas and two receiving antennas. Fig. 12 shows a comparison of average values ​​of the objective function in the classical domain when, as in the case of quadrature phase-shift keying, the channel is fixed, the signal-to-noise ratio of the transmission path is very high, the transmission bit sequence is 00110101, and real coefficients are multiplied by seven for integer approximation. Referring to Fig. 12, it can be seen that Grover adaptive search converges the value of the objective function to 0 with fewer queries in the classical domain than maximum likelihood detection.

[0074] Next, we obtain the objective function by using generalized space-shift keying (GSSK), and demonstrate the effectiveness of the objective function based on the relationship between the signal-to-noise ratio and bit error rate of the transmission path. In a multi-antenna wireless communication system, GSSK is adopted, and in order to maximize the transmission rate, the total number of transmitting antennas, N t Of which K=N t Consider the case where two transmit antennas are used. Fig. 13 shows an example of the relationship between the transmission rate and the query computational complexity for the full search and the Grover adaptive search. In this case, referring to Fig. 13, it can be seen that the Grover adaptive search has a smaller query computational complexity than the full search, and the difference becomes larger as the transmission rate increases.

[0075] Here, the total number of transmit antennas is N t= 6, K = 3 transmit antennas are used, the transmission rate is R = 4, and the total number of receive antennas is N r = 6. Here, for simplicity, the channel matrix has only diagonal elements. However, the case where the channel matrix H is dense can also be considered. FIG. 14 is a diagram showing an example of a method for selecting a transmitting antenna and its correspondence to a bit string according to an embodiment of the present invention. As an index selection pattern that can be converted into a QUBO problem, as shown in FIG. 14, a selection pattern that does not use a pattern in which s5 and s6 are simultaneously 1 is used. In this case, the third term can be simplified as in the following equation (24) in the case of the following equation (23), and the objective function is expressed by the following equations (25) and (26).

[0076]

number

[0077]

number

[0078]

number

[0079]

number

[0080] FIG. 15 is a diagram showing an example of the relationship between communication quality of a transmission path and bit error rate when a multi-antenna wireless communication system according to an embodiment of the present invention employs space shift keying, three of six transmitting antennas are used, and the number of receiving antennas is six. In this case, each data symbol is set to 1 (3 1 / 2 ) is set as . Figure 13 shows an example of the relationship between the communication quality of the transmission path and the bit error rate by minimization based on equation (26). The constraint parameter λ included in equation (26) Cis set to the smallest integer 2 at which no degradation in the bit error rate is observed. Referring to FIG. 15, it can be seen that the objective function is effective.

[0081] Fig. 16 is a diagram showing an example of the relationship between the number of queries in the classical domain and the value of the objective function when maximum likelihood detection or full search according to a comparative example is performed, and the relationship between the number of queries in the classical domain and the value of the objective function when the multi-antenna wireless communication system according to an embodiment of the present invention employs space shift keying, three of six transmitting antennas are used, and the number of receiving antennas is six. In this case, the channel coefficients are fixed to values ​​expressed by the following equation (27). Fig. 16 shows an example of the relationship between the number of queries in the classical domain and the value of the objective function when a Grover adaptive search is performed in addition to a 2-way search equivalent to maximum likelihood detection. R = 16 patterns searched by full search, Grover adaptive search 2 Nt =64 patterns and comparison.

[0082]

number

[0083] For visualization purposes, a constant term is added so that all minimum values ​​are 0, the initial values ​​are randomly selected from the usage patterns, and integer approximations are performed by multiplying the real coefficients by 3. Referring to Figure 16, the number of queries in the classical domain is comparable to that of maximum likelihood detection, and the effectiveness of the parameters used cannot be confirmed.

[0084] Next, we will show an example of how the number of queries is improved by taking spatial multiplexing when quadrature phase shift keying is adopted in a multi-antenna wireless communication system. A constant term c obtained by expanding equation (3) is added to equation (19). The constant term c is expressed by the following equation (28).

[0085]

number

[0086] Here, it is assumed that the number of quantum bits is sufficient and that real coefficients can be handled by approximating them with very large integers, and errors due to integer approximation are ignored. An example of the relationship between the number of queries and the bit error rate when real coefficients are handled by using a classical computer rather than a quantum computer to mark values ​​smaller than a threshold in a quantum circuit used to simulate a Grover adaptive search is shown. Figure 17 is a diagram showing an example of a quantum circuit used to simulate a Grover adaptive search according to an embodiment of the present invention. Here, as shown in Figure 17, m quantum bits representing the objective function are omitted in order to increase the number of queries and calculate the bit error rate. Furthermore, the following equation (29) is used as the Grover operator. Furthermore, the oracle operator O included in equation (29) is replaced with a classical operation. Since the above remains the case, namely a Grover adaptive search with 16 elements, the results are considered to be the same.

[0087]

number

[0088] The channel coefficients are taken from a complex normal distribution in various ways, and the signal-to-noise ratio of the transmission path is taken as 5 dB and 15 dB. Examples of the relationship between the number of queries and the bit error rate when each initial value is given are shown in Figures 18 to 23.

[0089] Fig. 18 is a diagram illustrating an example of the relationship between the number of queries in the classical domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 5 dB. Fig. 19 is a diagram illustrating an example of the relationship between the number of queries in the quantum domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 5 dB. Fig. 20 is a diagram illustrating an example of the relationship between the sum of the number of queries in the classical domain and the number of queries in the quantum domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 5 dB.

[0090] Fig. 21 is a diagram illustrating an example of the relationship between the number of queries in the classical domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 15 dB. Fig. 22 is a diagram illustrating an example of the relationship between the number of queries in the quantum domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 15 dB. Fig. 23 is a diagram illustrating an example of the relationship between the sum of the number of queries in the classical domain and the number of queries in the quantum domain and the bit error rate when the multi-antenna wireless communication system according to the embodiment of the present invention employs quadrature phase shift keying and the signal-to-noise ratio of the transmission path is 15 dB.

[0091] When only the minimum value distribution method was used, the bit error rate dropped rapidly for random initial values, clearly confirming the effectiveness of the objective function. Furthermore, although the minimum value distribution method was inferior to the minimum mean square error estimation method, when the signal-to-noise ratio of the transmission path was relatively high at 15 dB, the convergence was even faster when combined with the minimum mean square error estimation method.

[0092] The results for the method using minimum mean squared error estimation are due to the fact that the pseudoinverse matrix is ​​calculated and detected first. Here, because the size of the binary optimization problem being considered is small, the minimum mean squared error estimation process cannot be ignored, and a fair comparison with random initial values ​​cannot be made. However, the same conditions are used here to compare the effectiveness of the proposed method for each initial value. As for probability P, when the value is large, the converged bit error rate is high. This is thought to be because the probability of a random search occurring is high, and the search terminates before converging to the bit error rate of maximum likelihood detection due to the algorithm's termination condition. Therefore, it is necessary to determine an appropriate value for probability P.

[0093] Furthermore, the improvement achieved by this initial value is thought to depend on the number of bit sequences that have values ​​smaller than the initial threshold value y0. The minimum value depends only on noise when there are no errors. On the other hand, values ​​other than the minimum value depend on other factors such as channel coefficients. The higher the signal-to-noise ratio of the transmission path, the smaller the minimum value tends to be, so it is thought that improvement is more likely to be achieved. With the conventional random initial value, it can be seen that when the signal-to-noise ratio of the transmission path is as high as 15 dB, the bit error rate obtained by maximum likelihood detection does not fully converge. This is thought to be because, while the bit error rate of maximum likelihood detection improves when the signal-to-noise ratio of the transmission path is high, the Grover adaptive search is an algorithm that depends on the measurement probability of the states.

[0094] In other words, by obtaining the initial value from the probability distribution of the minimum value due to the signal-to-noise ratio of the transmission line, the number of queries in the classical domain and the quantum domain can be reduced on average. Also, unlike the case where minimum mean square error estimation is performed first, simply obtaining the initial value y m Convergence can be accelerated by pre-calculating and storing

[0095] The quantum computing function 12 expresses, as a quantum circuit, an objective function for a binary optimization problem whose coefficients include real numbers and which includes cubic or higher-order terms. Such an objective function is, for example, an objective function E(x) expressed by the following equation (30) that includes binary variables x1, ..., x8, each of which takes the value of 0 or 1, and real coefficients.

[0096]

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[0097] The quantum computing function 12 expresses the objective function by a quantum circuit that uses n=8 quantum bits that represent n=8 binary variables included in the objective function and m=6 quantum bits that represent the value of the objective function. FIG. 24 is a diagram showing an example of a quantum circuit according to an embodiment of the present invention. For example, the quantum computing function 12 expresses the objective function E(x) expressed by equation (30) by the quantum circuit shown in FIG. 24. The quantum circuit shown in FIG. 24 uses eight quantum bits that represent eight binary variables and a six-bit quantum bit that represents the value of the objective function E(x).

[0098] |x1> shown in Figure 24 is a quantum bit that represents the binary variable x1. Similarly, |x2>, |x3>, ..., and |x8> shown in Figure 2 are quantum bits that represent the binary variable x2, the binary variable x3, ..., and the binary variable x8, respectively. |z>6 shown in Figure 24 is a quantum bit that represents the value of the objective function E(x) expressed by equation (30).

[0099] The quantum circuit is a state preparation operator A that constructs a function E(x)-y obtained by subtracting a threshold y from the objective function E(x). y is applied to the initial state of the objective function E(x). This allows the quantum circuit to generate 2 pairs of n binary variables and the objective function values. n Generate a superposition of

[0100] Quantum circuits are defined by the unitary operator U GWe use the quantum gate represented by (θ) and the inverse quantum Fourier transform (IQFT). G The quantum gate represented by (θ) is the state preparation operator A y The objective function is calculated by using the quantum circuit. When realizing an integer k, the quantum circuit performs an inverse quantum Fourier transform, leaving only the state that represents one integer k. In other words, the quantum circuit performs integer addition and subtraction by rotating the phase. The quantum circuit then represents the results of the integer addition and subtraction using m=6 bits. The objective function uses the quantum bits that represent each of the n=8 variables as control bits, and the unitary operator U is calculated only when all the control bits are 1. G Therefore, the quantum computing function 12 can search for solutions to high-order unconstrained binary optimization problems by increasing the number of control bits.

[0101] To amplify the probability of outputting a negative state among the values ​​of the function E(x)-y calculated in parallel for the states represented by all combinations of n=8 variables, the quantum circuit uses an oracle operator O to invert and mark the phase θ at which the value of the function E(x)-y becomes negative. The phase θ at which the value of the function E(x)-y becomes negative is the phase θ at which the objective function E(x) becomes smaller than the threshold y. Furthermore, since the value of the objective function is expressed using two's complement, it can be constructed by applying a Z gate to only the first m=6 bits. The quantum circuit then uses a Grover diffusion operator D to amplify the probability of outputting a state represented by the marked phase θ. The Grover diffusion operator D inverts each state with respect to the average of the states represented by all combinations of n=8 variables. The quantum circuit then generates a state G by applying the Grover operator G L times. L A y |0> n+m Output.

[0102] The quantum computing function 12 determines an initial variable x0, which is the initial value of a variable, using minimum mean square error estimation, and sets the smaller value of the value of the objective function E(x0) when the initial variable x0 is given or a value based on the distribution of the minimum value of the objective function as the initial value of the predetermined threshold, and uses the initial value of the predetermined threshold as the predetermined threshold in the quantum computing process to be executed first.

[0103] In this case, the quantum circuit shown in Figure 24 is included in the objective function E(x), and operates a Hadamard gate H on the quantum bits representing eight initialized binary variables to generate a superposition of (x1, x2, ..., x8) with equal probability. The black circles shown in Figure 24 represent control bits. When all the control bits are 1, the unitary operator U is applied to the quantum bits representing the binary variables. G is applied.

[0104] The first unitary operator U shown in Figure 24 G The quantum gate expressed by (2.44π / 32) corresponds to the first term of the objective function shown in equation (30). The second unitary operator U G The quantum gate expressed by (1.22π / 32) corresponds to the second term of the objective function shown in Equation (30). The third unitary operator U G The quantum gate expressed by (0.27π / 32) corresponds to the third term of the objective function shown in equation (30). The fourth and subsequent terms of equation (30) are omitted in Figure 24. The quantum circuit shown in Figure 24 calculates the final phase by applying the inverse quantum Fourier transform after operating the quantum gate. Then, the quantum circuit shown in Figure 24 calculates the final phase by applying the Grover operator G to L i Apply the mixture twice.

[0105] The classical calculation function 13 executes classical calculation processing. Specifically, the classical calculation function 13 calculates a value y of an objective function E(x) when a variable x is given, using a classical computer.

[0106] The update function 14 executes the update process. Specifically, the update function 14 executes the update process when the value y of the objective function E(x) given a variable x is smaller than a predetermined threshold y i If it is less than a given threshold y i The value y of the objective function E(x) when the variable x is given i+1 On the other hand, the updating function 14 updates the value y of the objective function E(x) when the variable x is given to a predetermined threshold y i If it is equal to or greater than this, the predetermined natural number k is updated to a larger natural number. For example, the update function 14 updates the predetermined natural number k to a natural number expressed by the following equation (31). The value of λ included in equation (31) is, for example, 8 / 7.

[0107]

number

[0108] The iteration function 15 repeats the process of making the quantum computation function 12 perform quantum computation, the classical computation function 13 perform classical computation, and the update function 14 perform update until a predetermined condition is satisfied. The predetermined condition here is, for example, the number of times L that the Grover operator G is applied. i The predetermined condition is that the sum of the objective function E(x) is equal to or greater than a predetermined number of times. i+1 = y, the query computational complexity of the classical domain that is not updated to y is equal to or greater than a predetermined threshold. Alternatively, the predetermined condition here is that at least one of these two conditions is satisfied.

[0109] 25 to 29 are diagrams showing the probability of observing each state of eight binary variables when the Grover operator G acts on the quantum bit that indicates the value of the objective function E(x). Figures 25 to 29 show only the top 16 states with the highest observation probability.

[0110] 25 is a diagram showing an example of the probability of observing each state when the Grover operator is applied 0 times to a quantum bit indicating the value of an objective function according to an embodiment of the present invention. Fig. 25 shows the probability of observing each state of eight binary variables when the Grover operator G is not applied, indicating that each state is observed with equal probability.

[0111] Fig. 26 is a diagram showing an example of the probability of observing each state when a Grover operator is applied once to a quantum bit indicating the value of an objective function according to an embodiment of the present invention. Fig. 26 shows that by applying the Grover operator G once, the probability of observing the state (x1, x2, x3, x4, x5, x6, x7, x8) = (0, 0, 1, 1, 0, 1, 0, 1), in which the objective function E(x) is minimum, is amplified to approximately 0.01. Fig. 26 also shows that the state in which the value of the objective function E(x) is not negative is also slightly amplified.

[0112] 27 is a diagram showing an example of the probability of observing each state when a Grover operator is applied twice to a quantum bit indicating the value of an objective function according to an embodiment of the present invention. Fig. 27 shows that by applying the Grover operator G twice, the probability of observing the state (x1, x2, x3, x4, x5, x6, x7, x8) = (0, 0, 1, 1, 0, 1, 0, 1) in which the objective function E(x) is minimum is amplified to approximately 0.03. Fig. 27 also shows that the probability of observing states in which the value of the objective function E(x) is not negative is slightly amplified, although it is lower than in the case shown in Fig. 26.

[0113] Fig. 28 is a diagram showing an example of the probability of observing each state when a Grover operator is applied three times to a quantum bit indicating the value of an objective function according to an embodiment of the present invention. Fig. 28 shows that by applying the Grover operator G three times, the probability of observing the state (x1, x2, x3, x4, x5, x6, x7, x8) = (0, 0, 1, 1, 0, 1, 0, 1), in which the objective function E(x) is minimum, is amplified to approximately 0.05. Fig. 28 also shows that the state in which the value of the objective function E(x) is not negative is also slightly amplified.

[0114] 29 is a diagram showing an example of the probability of observing each state when a Grover operator is applied four times to a quantum bit indicating the value of an objective function according to an embodiment of the present invention. Fig. 29 shows that, by applying the Grover operator G four times, the probability of observing the state (x1, x2, x3, x4, x5, x6, x7, x8) = (0, 0, 1, 1, 0, 1, 0, 1), in which the objective function E(x) is minimized, is amplified to approximately 0.06. Fig. 29 also shows that the state in which the value of the objective function E(x) is not negative is also amplified slightly.

[0115] 25 to 29, it can be seen that there is a certain probability that a state in which the value of the objective function E(x) is not negative is observed. In order to avoid observing a state in which the value of the objective function E(x) is not negative, the classical calculation function 13 executes the classical calculation process described above.

[0116] FIG. 30 is a diagram showing an example of the relationship between the number of trials and the value of the objective function when an exhaustive search according to a comparative example is performed, and the relationship between the number of trials and the value of the objective function when a process according to an embodiment of the present invention is performed. The horizontal axis of FIG. 30 represents the number of trials. The vertical axis of FIG. 30 represents the value of the objective function E(x). The dashed line shown in FIG. 30 represents the relationship between the number of trials and the value of the objective function when an exhaustive search according to a comparative example is performed. The solid line shown in FIG. 30 represents the relationship between the number of trials and the value of the objective function when a process according to an embodiment of the present invention is performed.

[0117] 30, it can be seen that the process executed by the quantum computing device 1 can obtain a state that minimizes the objective function E(x) with fewer attempts than the exhaustive search according to the comparative example. The exhaustive search according to the comparative example obtains a state that minimizes the objective function E(x) with a query computational complexity of about O(N) in the classical domain, where N is the number of attempts. On the other hand, the process executed by the quantum computing device 1 obtains a state that minimizes the objective function E(x) with a query computational complexity of about O(N) in the quantum domain. 1 / 2) to obtain a state that minimizes the objective function E(x). Note that the ability to solve a problem with a smaller query computational effort by using quantum computing is called quadratic acceleration or quantum acceleration.

[0118] Next, an example of processing executed by the quantum computing device 1 will be described with reference to Fig. 31. Fig. 31 is a flowchart showing an example of processing executed by the quantum computing device according to an embodiment of the present invention.

[0119] In step S11, the transformation function 11 projects data symbols transmitted from at least two transmitting antennas, transmitted via a transmission path, and received by at least two receiving antennas from binary variables into complex numbers.

[0120] In step S12, the transformation function 11 transforms the maximum likelihood detection into a binary optimization problem by expanding the Frobenius norm of the maximum likelihood detection for decoding data symbols using complex numbers into a real part and an imaginary part.

[0121] In step S13, the quantum computing function 12 applies Grover adaptive search to the binary optimization problem and samples states that have the potential to make the value of the objective function of the binary optimization problem smaller than a predetermined threshold.

[0122] In step S14, the classical calculation function 13 executes a classical calculation process to calculate the value of the objective function when the variables are given, using a classical computer.

[0123] In step S15, the update function 14 determines whether the value of the objective function when the variables are obtained is smaller than a predetermined threshold. If the update function 14 determines that the value of the objective function when the variables are given is smaller than the predetermined threshold (step S15: YES), the process proceeds to step S16. On the other hand, if the update function 14 determines that the value of the objective function when the variables are given is equal to or greater than the predetermined threshold (step S15: NO), the process proceeds to step S17.

[0124] In step S16, the update function 14 updates the predetermined threshold to the value of the objective function when the variables are given, and the process proceeds to step S18.

[0125] In step S17, the update function 14 updates the predetermined natural number to a larger natural number, and the process proceeds to step S18.

[0126] In step S18, the repeat processing function 15 determines whether or not a predetermined condition is satisfied. If the repeat processing function 15 determines that the predetermined condition is satisfied (step S18: YES), it ends the process. On the other hand, if the repeat processing function 15 determines that the predetermined condition is not satisfied (step S18: NO), it returns the process to step S11.

[0127] The quantum computing device 1 according to the embodiment has been described above. The quantum computing device 1 includes a conversion function 11, a quantum computing function 12, a classical computing function 13, an update function 14, and an iterative processing function 15.

[0128] The transformation function 11 projects data symbols transmitted from at least two transmitting antennas, transmitted via a transmission path, and received by at least two receiving antennas from binary variables into complex numbers. The transformation function 11 then converts the maximum likelihood detection, which decodes the data symbols using complex numbers, into a binary optimization problem by expanding the Frobenius norm of the maximum likelihood detection into real and imaginary parts.

[0129] This allows the quantum computing device 1 to convert maximum likelihood detection performed in multi-antenna wireless communication into a binary optimization problem.

[0130] The quantum computing function 12 determines an initial variable x0, which is the initial value of a variable, using minimum mean square error estimation, and sets the smaller of the value of the objective function E(x0) when the initial variable x0 is given and a value based on the distribution of the minimum value of the objective function as the initial value of a predetermined threshold. The quantum computing function 12 then applies Grover adaptive search to the binary optimization problem to derive variables that have the potential to make the value of the objective function of the binary optimization problem smaller than the predetermined threshold. The classical computing function 13 uses a classical computer to calculate the value y of the objective function E(x) when a variable x is given.

[0131] The update function 14 is configured to update the value y of the objective function E(x) when a variable x is given, based on a predetermined threshold y i If it is less than a given threshold y i The value y of the objective function E(x) when the variables are given i+1 On the other hand, the updating function 14 updates the value y of the objective function E(x) when the variable x is given to a predetermined threshold y i If it is equal to or greater than this, the predetermined natural number k is updated to a larger natural number. The iteration processing function 15 repeats the process of having the quantum computing function 12 execute quantum computing processing, the classical computing function 13 execute classical computing processing, and the update function 14 execute update processing until a predetermined condition is satisfied.

[0132] This allows the quantum computing device 1 to obtain a solution even if the maximum likelihood detection performed in multi-antenna wireless communication converted into a binary optimization problem contains real numbers as coefficients.

[0133] Although the embodiments of the present invention have been described above with reference to the drawings, the quantum computing program, quantum computing device, and quantum computing method are not limited to the above-described embodiments, and various modifications, substitutions, combinations, and / or design changes can be made without departing from the spirit and scope of the present invention.

[0134] Furthermore, the effects of the above-described embodiments of the present invention are described as examples. Therefore, the embodiments of the present invention may also achieve other effects that a person skilled in the art can recognize from the description of the above-described embodiments in addition to the above-described effects. [Explanation of symbols]

[0135] 1... quantum computing device, 10... quantum computing program, 11... conversion function, 12... quantum computing function, 13... classical computing function, 14... update function, 15... iteration processing function

Claims

1. A transformation function that projects data symbols transmitted from at least two transmitting antennas, passed through a transmission path, and received by at least two receiving antennas from binary variables to complex numbers, and decodes the data symbols using the complex numbers, thereby transforming the maximum likelihood detection into a binary optimization problem by expanding the Frobenius norm of the maximum likelihood detection into a real part and an imaginary part. A quantum computing program to be executed by a computer.

2. a quantum computing function that applies a Grover adaptive search to the binary optimization problem and executes a quantum computing process to derive variables that have the potential to make the value of an objective function of the binary optimization problem smaller than a predetermined threshold; a classical computing function that executes a classical computing process to calculate the value of the objective function when the variables are given using a classical computer; an update function that, when a value of the objective function when the variable is given is smaller than the predetermined threshold, executes an update process to update the predetermined threshold to the value of the objective function when the variable is given; a repeating processing function that causes the quantum computing function to execute the quantum computing process, the classical computing function to execute the classical computing process, and the update function to execute the update process, repeating the process until a predetermined condition is satisfied; The quantum computing program according to claim 1, further comprising:

3. the quantum computing function determines an initial variable, which is an initial value of the variable, using minimum mean square error estimation, sets the smaller value of the value of the objective function when the initial variable is given and a value based on a distribution of minimum values ​​of the objective function as the initial value of the predetermined threshold, and uses the initial value of the predetermined threshold as the predetermined threshold in the quantum computing process to be executed for the first time; The quantum computing program according to claim 2 .

4. a conversion unit that converts the maximum likelihood detection into a binary optimization problem by projecting data symbols transmitted from at least two transmitting antennas, passing through a transmission path, and received by at least two receiving antennas from binary variables into complex numbers, and decoding the data symbols using the complex numbers, and by expanding the Frobenius norm of the maximum likelihood detection into a real part and an imaginary part; Quantum computing device.

5. a method for converting the maximum likelihood detection into a binary optimization problem by projecting data symbols transmitted from at least two transmitting antennas, passing through a transmission path, and received by at least two receiving antennas from binary variables into complex numbers, and decoding the data symbols using the complex numbers, and expanding the Frobenius norm of the maximum likelihood detection into a real part and an imaginary part; A computer-implemented quantum computing method.

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