## 3 Approximation by Removing the Number of Paths with Smaller Coefficients
### Approximation of Hamiltonians  
In quantum mechanics, the Hamiltonian of a system is a key quantity that determines the time evolution of a quantum state. The eigenvalue spectrum of the Hamiltonian provides important information about the system's behavior, such as its stability, spectral gaps, symmetries of system, and the transitions between energy levels that are possible. There are various techniques that try to characterize quantum systems by understanding their full Hamiltonian through various tomographic techniques [40; 41; 42; 43; 44; 45; 46; 47; 48; 49]. However, as the  
Figure 8: Histogram of the coefficients for VQC with different depth values: Here, \(n\) represents the number of qubits required for \(vec(VQC)\). The error grows with the depth of the circuit.  
Figure 7: A variational quantum circuit (VQC) with a depth of four. _rand_ indicates that each gate is generated with a random angle.  
size of the quantum systems increases, calculating the full eigenvalue spectrum can be computationally expensive, and a limited number of eigenvalues may be sufficient. For example, in quantum algorithms such as quantum Monte Carlo, it may only be necessary to know the lowest eigenvalue of the Hamiltonian and not the full spectrum [50]. In some optimization problems, such as the calculation of the ground state energy of a system, determining the lowest eigenvalue may be only sufficient and not the entire spectrum. Therefore, approximating the Hamiltonian through successive Schmidt decomposition can prove to be a useful tool for addressing such problems and also for tackling large quantum systems.  
To test our proposed approach for Hamiltonian approximation, we considered the Transverse Field Ising Model (TFIM) Hamiltonian, defined by \(H=h\sum_{i}^{n}\sigma_{x}^{i}+J\sum_{i,j=i+1}^{c}\sigma_{z}^{i}\sigma_{z}^{j}\) where \(h\) and \(J\) are the transverse field and coupling parameters, respectively, \(n\) is the total number of qubits, and \(c\) is the number of qubits that are coupled. For computing the results we choose \(h=0.1\) and \(J=0.5\) for a 10-qubit quantum system. Figure 9 shows the histogram of the coefficients with a cut-off probability of 0.04, for which the norm of the difference is 0.589 for the system where all spins are coupled and 0.252 for the system where only four spins are coupled. The interesting thing about such an approximation can be seen in the eigenvalue distribution of the true and the reconstructed Hamiltonian in Figure 10. Approximating the Hamiltonian using only a few Schmidt coefficients that are large, we are able to generate the eigenvalue spectrum that accurately reproduces the eigenvalues that are not close to 0. This is because if we perform the SVD of a Hamiltonian matrix, the singular values correspond to the magnitude of the eigenvalues, and since we ignore the Schmidt coefficients that are small, the reconstructed Hamiltonian has information on the eigenvalues that are away from 0, as is seen in Figure 10.  
Figure 9: Histogram of the Schmidt coefficients corresponding to TFIM Hamiltonian of a 10-qubit quantum system with \(c\)-qubits that are coupled. With fewer coupling terms, we see that the norm of difference reduces for the same cut-off probability.