[Abstract] For the terahertz massive MIMO system, a precoding scheme based on quantum machine learning is proposed. This scheme aims at maximizing the system reachability and rate by modeling the optimization problem in the communication system as the optimization problem in the quantum system. Hamiltonian transforms the design problem of optimal precoding into the ground state problem of obtaining the Hamiltonian of the quantum system. Then, a variational quantum eigensolver is used, combined with the optimization algorithm of classical machine learning, to obtain the optimal parameterized quantum circuit through training, and finally the ground state is extracted from the quantum circuit, which corresponds to the optimal precoding matrix. After verification and analysis by Google's tensorflow-quantum quantum machine learning platform, the proposed scheme can achieve exponential acceleration, and its performance is close to the classic SVD-based precoding scheme.
[Keywords] Terahertz massive MIMO; variational quantum eigenvalue solver; quantum variational algorithm; precoding
0 Preface
Terahertz and massive MIMO technologies are key technologies for current 5G and 6G communications. Terahertz communication has the characteristics of large bandwidth and high transmission rate. Communication technology for the terahertz frequency band is expected to solve the problems of spectrum scarcity and capacity limitation of current wireless communication systems. Massive MIMO technology is an extension of MIMO technology in the fourth generation wireless communication technology. It spatially increases system capacity and improves the performance of communication systems. However, as the number of antennas in massive MIMO increases, the complexity of the algorithm and the difficulty of implementation will also increase significantly. Studying how to reduce the complexity of the algorithm and how to reduce the overhead is the focus of current research. On the other hand, the combination of quantum computing and machine learning has become a new cross-research field. Due to its high parallelism, quantum computing can use n qubits to calculate 2n-dimensional matrices in traditional communications, which can greatly reduce the complexity of traditional machine learning algorithms. On current medium-scale noisy quantum computers, it can approach the performance of traditional algorithms. . Therefore, wireless communication technology combined with quantum machine learning will gradually become a possible focus of 6G wireless communication research based on artificial intelligence. This article innovatively proposes the application of quantum machine learning in the precoding design of terahertz massive MIMO systems, and explores new ways to combine artificial intelligence with specific technical aspects of 6G communications.
In MIMO systems, in order to suppress the impact of signal interference, precoding technology becomes the key to solving the problem. Precoding schemes are usually divided into two types: all-digital precoding and hybrid precoding. Full digital precoding can control the phase and amplitude of the transmitted signal at the same time, and can be used in both single-user systems and multi-user systems. Depending on whether the processing method is linear or not, all-digital precoding can be divided into linear precoding and nonlinear precoding. Linear precoding includes matched filtering (MF), singular value decomposition (SVD), zero-forcing precoding (ZF), minimum mean square error precoding (MMSE), etc. Nonlinear precoding mainly includes dirty paper coding (DPC) [1] and Tomlinson-Harashima precoding (THP) [2]. In massive MIMO systems, hybrid precoding has received widespread attention. In single-user systems, literature [3] proposes a spatially sparse hybrid precoder, and literature [4] proposes a hybrid precoder based on continuous interference cancellation. For multi-user systems, literature [5] proposed using the Orthogonal Matching Pursuit (OMP, Orthogonal Matching Pursuit) algorithm in a design based on quantized codebooks, and designed a hybrid minimum mean square error (MMSE, Minimum Mean Square Error) precoder has certain performance advantages for application scenarios with many users. Literature [6] proposes an efficient and simple hybrid precoding algorithm, which directly selects the phase of the ideal digital precoding as the analog precoding, and the digital precoding uses the least square method (LS, Least Square). In addition to the above methods that require perfect CSI, there are also some precoding methods that do not require perfect CSI. For example, a limited feedback method [8] uses the characteristics of the millimeter wave channel model to match the relatively best codeword in a limited set of candidate codewords. Codewords serve as analog precoders.
Current quantum computers only support a limited number of physical qubits and limited gate fidelity, and general quantum algorithms are difficult to implement on quantum computers. Therefore, an important direction for the development of quantum computing is to find an algorithm that can run effectively on a noisy intermediate-scale quantum computer (NISQ, Noisy Intermediate-Scale Quantum). Variational Quantum Algorithm (VQA) can be implemented in shallow quantum circuits that rely on external parameters. They are also called parameterized quantum circuits or quantum neural networks (QNN, Quantum Neural Network). Quantum network parameters can be optimized on classical computers using gradient algorithms in machine learning. In recent years, many quantum variational algorithms based on quantum neural networks have been proposed. Literature [9] proposed a variational quantum eigensolver (VQE, Variational Quantum Eigensolver), which is used to obtain the approximate ground state of a given Hamiltonian. Variational algorithm, after numerical simulation, compared with the traditional algorithm for solving matrix eigenvalues, VQE is more efficient in solving the eigenvalues of large-dimensional matrices, and can run on NISQ equipment. Literature [10] proposed a novel optimizer to solve the problem of excessive quantum state preparation and measurement times when solving complex problems using the variational hybrid quantum classical algorithm (VHQCA) on NISQ equipment. Reference [11] applies quantum variational algorithm to solve linear equations and matrix-vector multiplication, and conducts numerical tests. Literature [12] proposes to introduce a quantum singular value estimation algorithm to solve a system of linear equations, use the variation principle of singular values to design a new loss function, and train two quantum neural networks to learn singular vectors and output corresponding singular values. Literature [13] proposed a variational quantum circuit to generate singular value decomposition of bipartite pure states. Literature [14] proposed a quantum circuit composed of standard quantum gates to implement the gradient optimization algorithm, and demonstrated the solution to the homogeneous polynomial optimization problem in a four-qubit system, proving that the algorithm can effectively solve high-dimensional optimization question. In addition, unlike fault-tolerant quantum computers that have high requirements for error correction, noise can be effectively suppressed on quantum computers implemented with low-depth quantum circuits. It also shows that quantum computing has certain advantages in running on noisy medium-scale quantum computers. feasibility.
With the continuous development of quantum technology, major business giants, academic institutions, and government strategies are turning their attention to quantum computing as the main development direction in the next few decades. At the same time, with the rapid development of wireless communication technology, the number of wireless users is increasing, requiring higher data processing and data analysis capabilities. The high parallelism of quantum computing can reduce the cost of traditional machine learning algorithms. The complexity can be greatly reduced. Therefore, applying quantum computing to the field of wireless communications is considered to be a promising direction to assist the development of wireless communications technology in the future 6G and post-6G eras. Literature [15] designed an improved quantum algorithm to implement massive MIMO uplink detection based on MMSE. In this problem, a quantum algorithm based on quantum singular value estimation was first designed to obtain the emission signal in the form of a quantum state. Then, an improved quantum state information extraction algorithm is proposed to obtain the amplitude and phase of the quantum state emission signal so that the information can be used in conventional devices. Literature [16] proposed a complete set of quantum algorithms to implement multi-signal classification problems in wireless communication systems. Including quantum signal covariance matrix reconstruction based on quantum singular value decomposition and density matrix eigendecomposition based on variational quantum algorithms. This quantum algorithm can provide polynomial or even exponential acceleration compared to the traditional MUSIC algorithm. Literature [17] proposed a quantum ELM algorithm based on the quantum singular value estimation algorithm, introduced the quantum singular value estimation algorithm into wireless channel prediction, combined the classical ELM algorithm with the quantum singular value estimation algorithm, and used quantum singularity in the pseudo-inverse calculation part. Value estimation reduces the time complexity of the classic ELM algorithm. However, in the precoding technology of terahertz mobile communication system, there is currently no solution combined with quantum machine learning. This paper innovatively proposes to use quantum variation algorithm to design the precoding solution in terahertz massive MIMO system. , and conducted experimental simulations on the Google quantum machine learning platform, using the high parallelism of quantum computing to reduce the complexity of the algorithm and improve the performance of the entire communication system.
3 Numerical simulation
The simulation environment used in this article is a terahertz massive MIMO system model, and the training parameters are set as follows: the depth D of the quantum neural network is set to 2, and the total number of iterations is set to 80. In order to verify the effectiveness of the algorithm, it was verified on the Google Quantum Platform, using the quantum programming framework cirq, to compare the system reachability and rate with the following algorithms under different signal-to-noise ratios and different numbers of transmitting antennas: 1) SVD precoding; 2) MMSE precoding. Since simulating quantum computers in classical computers is currently a difficult problem, and the operational behavior of quantum computing systems requires an exponential number of operations in classical computers to simulate, the simulation experiments in this article were conducted with a small number of receiving antennas. The simulation results are shown in the figure .
Figures 5 and 6 respectively show the relationship between the reachable sum rate and signal-to-noise ratio of each algorithm system when the system is Nt=4, Nr=4 and Nt=16, Nr=16. It can be seen from Figures 5 and 6 It is found that the system attainable sum rate increases with the increase of signal-to-noise ratio. The system attainable sum rate performance of the algorithm proposed in this article is similar to the traditional SVD precoding performance and is better than the MMSE precoding performance.
Figure 7 shows the relationship between the reachability and rate of each algorithm system as the number of transmitting antennas changes from 16 to 196 when Nr=4. It can be seen from Figure 7 that the reachability and rate of each algorithm system increases with the increase of the number of transmitting antennas. In addition, the performance of the algorithm proposed in this article is similar to that of the SVD precoding algorithm and is better than the MMSE precoding algorithm.
4 Conclusion
The precoding scheme proposed in this article combined with the variational quantum eigenvalue solver aims at maximizing the system reachability and speed. Using the variational quantum eigenvalue solver, the problem is transformed into the expectation of solving the system Hamiltonian and its corresponding For the quantum state, prepare a parameterized trial wave function on the quantum machine learning platform, and then combine it with the optimization algorithm of classical machine learning to train to obtain a set of optimal parameters to update the parameterized quantum circuit, thereby obtaining the optimal precoding matrix. Verification analysis shows that this solution can achieve exponential acceleration. Compared with the classic algorithm, the performance is similar, but it has the advantage of computational complexity. The solution proposed in this article is an attempt to combine quantum machine learning with wireless communication technology, providing more possibilities for the development of wireless communication technology in the future.