Many exciting phenomena in quantum many-body systems are due to the interplay of quantum fluctuations and correlations. Celebrated examples are the superfluid Helium [1] [2] , the fractional quantum Hall effect [3] [4] , the Haldane phase in quantum spin chains [5] [6] , quantum spin liquids [7] , and high-temperature superconductivity [8] . The aim of theoretical physics is to understand the emergent properties for such challenging quantum many-body systems. The main difficulty in investigating quantum many-body problems is due to the fact that the Hilbert space spanned by the possible microstates grows exponentially with the system size. To unravel the physics of microscopic model systems and to study the robustness of quantum phases of matter, large scale numerical simulations are essential. The exact diagonization method is only possible for small many-body systems. For large systems, efficient quantum Monte Carlo (QMC) methods can be applied. In a large class of quantum many-body systems (e.g., fermionic degrees of freedom, geometric frustration), however, these QMC sampling techniques cannot be used effectively due to the sign problem [9] . In this case, the variational methods have been shown to be a powerful tool to efficiently simulate quantum many-body systems.

The first type of variational method is the density matrix renormalization group (DMRG) method which was originally developed to study ground state properties of one-dimensional systems. The success of the DMRG method is based on the area law of the quantum ground state [10] , and thus can be represented efficiently using matrix product states (MPS) [11] - [19] . This algorithm has been generalized to study two-dimensional systems [20] , Abelian and non-Abelian symmetries [21] - [25] , single-site optimization with density matrix perturbation [26] [27] , hybrid real-momentum space representation [28] [29] , and the development of real-space parallelization [30] , continuous matrix product state (cMPS) [31] [32] . The cMPS can also be used to study the Lieb-Liniger model [33] [34] , the Gaudin-Yang model [35] , periodic bc atomtronics [36] and 1 + 1 relative Bose theories [37] .

The machine learning has been extensively used in physical sciences [38] . The second type of variational method is based on machine learning, where neutral network variational ansatz efficiently represents highly correlated quantum states and their parameters are easily optimized by means of the variational Monte Carlo (VMC) method [39] , which has powerful expressibility for quantum state [40] - [43] . Alongside with restricted Boltzmann machine (RBM) [44] - [59] , other network structures such as a feed-forward [60] - [63] , recurrent neural networks [64] [65] , autoregressive neural TensorNet [66] , and convolutional neural networks [67] - [78] , have been adopted. Recurrent neural networks (RNNs) are tremendously powerful tools that have been used in language modeling [79] , speech recognition [80] , and image generation [81] [82] . RNNs process data streams by maintaining a hidden state, which is updated by applying an identical function to the previous hidden state and the input at the next time step.

At the same time, neural networks using self-attention layers, like the Transformer [83] , have had a profound impact on much of machine learning. They have led to breakthroughs in natural language processing [84] , language modeling [85] , image recognition [86] , and protein folding [87] . However, the application of Transformer in this field is still rather limited, with a few results concerning quantum lattice models [88] , open systems [89] , quantum state tomography [90] and quantum circuit simulation [91] . Therefore, the full potential of the transformer architecture has yet to be explored.

The third type of variational method is called the variational quantum eigensolver (VQE), which is extensively used in quantum calculation. This is a hybrid algorithm, where the quantum state is prepared by a quantum algorithm, which is implemented according to a quantum circuit with many quantum gates. Classical computer is used to optimize the parameters used in quantum circuit. Compared with the matrix product state and machine learning ansatz, the variational state in VQE is realized by quantum algorithm, which can be implemented in the quantum hardware devices, and thus exponential speedup over classical methods becomes possible. The previous quantum algorithms to find the ground state of a given Hamiltonian were based on adiabatic state preparation and quantum phase estimation subroutines [92] [93] , both of which have circuit depth requirements beyond those available in the NISQ era. Here we listed both the original VQE architecture [94] and some more advanced methods: orthogonality constrained VQE [95] , subspace expansion method [96] , subspace VQE [97] [98] , multistate contracted VQE [99] , adiabatically assisted VQE [100] [101] , and accelerated VQE [102] - [104] . To study the dynamics of many-body quantum systems, the conventional quantum Hamiltonian simulation algorithm such as the Trotter-Suzuki product formula [105] was used. However, the circuit depth of standard Trotterization methods can rapidly exceed the coherence time of noisy quantum computers. This has led to recent proposals for variational approaches to dynamical simulation, including iterative variational algorithms [106] [107] , imaginary time evolution [108] , general first order derivative equations with non-Hermitian Hamiltonians [109] , adaptive ansatz to reduce the circuit depth [110] [111] and variational fast forwarding [112] - [115] .

In this paper, these three variational methods are discussed in detail and are used to solve the ground state energy of the Fermi-Hubbard model and Schwinger model. The paper is organized as follows: The details of the three variational methods are given in Sections 2, 3, and 4. In Sections 5 and 6, the Fermi-Hubbard model and Schwinger model have been mapped to spin 1/2 systems using the Jordan-Wigner representation. In Section 7, the ground state energy of the Fermi-Hubbard model and Schwinger model are calculated using these three variational methods. The discussion is given in the final Section 8.

A general discrete Hilbert space of the many-body quantum systems with $N$ sites has the structure

$\mathscr{H}=\text{span}\left\{|s_0\rangle \otimes \cdots \otimes |s_{N-1}\rangle |s_i\in {ℒ}_i,i\in \left\{0,\cdots ,N-1\right\}\right\}$(1)

where ${ℒ}_i$ is the set of discrete quantum numbers at site $i$. ${\left\{|s_i\rangle \right\}}_{s_i\in {ℒ}_i}$ is the standard orthogonal basis for the local Hilbert space ${\mathscr{H}}_i$ corresponding to the $i$th site, $i=0,\cdots ,N-1$. The whole Hilbert space $\mathscr{H}$ is the tensor product of individual local Hilbert space: $\mathscr{H}={\otimes }_{i=0}^{N-1}{\mathscr{H}}_i$. The standard orthogonal basis $|s_0\cdots s_{N-1}\rangle \equiv |s_0\rangle \otimes \cdots \otimes |s_{N-1}\rangle$ in $\mathscr{H}$ is also called the computational basis. For the qubit systems, ${ℒ}_i=\left\{0,1\right\}$. For spin-1/2, ${ℒ}_i=\left\{-1,1\right\}$, which is the two eigenvalues of Pauli matrix $Z$. For general spin- $S$ with integral or half integral $S=1/2,1,3/2,2,\cdots$, ${ℒ}_i=\left\{-2S,-2S+2,\cdots ,2S-2,2S\right\}$ and ${\mathscr{H}}_i$ has the dimension $2S+1$. Sometimes, the dimension of Hilbert space is reduced under some kind of restriction. For example, the spin 1/2 system with the restriction ${\displaystyle {\sum }_{i=0}^{N-1}{Z}_i}=0$, the dimension 16 of $\mathscr{H}$ is reduced to 6 if $N=4$.

The state $|\psi \rangle \in \mathscr{H}$ can be represented by

$|\psi \rangle =\underset{s}{{\displaystyle \sum }}\psi \left(s\right)|s\rangle =\underset{s_0,\cdots ,s_{N-1}}{{\displaystyle \sum }}\psi \left(s_0,\cdots ,s_{N-1}\right)|s_0\cdots s_{N-1}\rangle$(2)

where the sum $s$ over all quantum numbers is assumed. Since the computational basis is standard orthogonal, the wave function $\psi \left(s\right)=\langle s|\psi \rangle$ is the inner product of $|\psi \rangle$ and $|s\rangle$.

The Hamiltonian $H$ of many-body quantum system is a Hermitian operator from $d^N$ dimensional Hilbert space $\mathscr{H}$ into itself. We assume that the dimension $d$ of ${\mathscr{H}}_i$ are same for $i=0,\cdots ,N-1$. The calculation of the spectra of $H$ is difficult due to the exponentially large dimension over $N$. For small $N$, exact diagonalization can be adopted since $H$ can be represented as the Hermitian matrix in the computation basis $|s\rangle$.

A general operator $\hat{O}$ in the basis $|s\rangle$ can be written as

$\hat{O}=\underset{s,s^{\prime }}{{\displaystyle \sum }}{O}_{{s^{\prime }}_0\cdots {s^{\prime }}_{N-1}}^{s_0\cdots s_{N-1}}|s_0\cdots s_{N-1}\rangle \langle {s^{\prime }}_{N-1}\cdots {s^{\prime }}_0|$(3)

Since the basis $|s\rangle$ is orthogonal to each other,

$\hat{O}|{s^{\prime }}_0\cdots {s^{\prime }}_{N-1}\rangle =\underset{s}{{\displaystyle \sum }}{O}_{{s^{\prime }}_0\cdots {s^{\prime }}_{N-1}}^{s_0\cdots s_{N-1}}|s_0\cdots s_{N-1}\rangle$(4)

i.e., $\hat{O}|s^{\prime }\rangle ={\displaystyle {\sum }_s{O}_{s^{\prime }}^s|s\rangle }$. Apply this operator $\hat{O}$ to a general state $|\psi \rangle$ in (2), one has

$\hat{O}|\psi \rangle =\underset{s,s^{\prime }}{{\displaystyle \sum }}{O}_{s^{\prime }}^s\psi \left(s^{\prime }\right)|s\rangle$(5)

The inner product between $\hat{O}|\psi \rangle$ and $|\psi \rangle$ is

$\langle \psi |\hat{O}|\psi \rangle =\underset{s,s^{\prime }}{{\displaystyle \sum }}{O}_{s^{\prime }}^s\psi \left(s^{\prime }\right)\overline{\psi \left(s\right)}$(6)

where $\overline{\psi \left(s\right)}$ denotes the complex conjugate of $\psi \left(s\right)$.

The wave function $\psi \left(s_0,\cdots ,s_{N-1}\right)$ can be regarded as the tensor ${\psi }_{s_0\cdots s_{N-1}}$ of order $N$, and thus the manipulation of the wave function is reduced to be that of tensor ${\psi }_{s_0\cdots s_{N-1}}$. Similarly, $O_{s^{\prime }}^s=O_{{s^{\prime }}_0\cdots {s^{\prime }}_{N-1}}^{s_0\cdots s_{N-1}}$ is the tensor of order $2N$. The computational cost of ${\displaystyle {\sum }_{s^{\prime }}{O}_{s^{\prime }}^s\psi \left(s^{\prime }\right)}={\displaystyle {\sum }_{{s^{\prime }}_0,\cdots ,{s^{\prime }}_{N-1}}{O}_{{s^{\prime }}_0\cdots {s^{\prime }}_{N-1}}^{s_0\cdots s_{N-1}}{\psi }_{{s^{\prime }}_0,\cdots ,{s^{\prime }}_{N-1}}}$ is $O\left(d^{2N}\right)$ which is not acceptable. The trick of reducing computational cost is the decomposition of tensor and then truncation. A very important operation for tensor is the decomposition as follows:

${\psi }_{s_0\cdots s_{N-1}}=M^{\left[0\right]s_0}{M}^{\left[1\right]s_1}\cdots M^{\left[N-1\right]s_{N-1}}$(7)

where $M^{\left[i\right]s_i}={\left(M^{\left[i\right]s_i}\right)}_{{\alpha }_i{\alpha }_{i+1}}$ is the ${\chi }_i\times {\chi }_{i+1}$ matrix for fixed (physical) indices $s_i$, $i=0,\cdots ,N-1$, i.e., $M^{\left[i\right]}$ is a tensor of order 3. The index ${\left\{{\alpha }_i\right\}}_{i=0}^N$ is contracted in the multiplication of all $M^{\left[i\right]s_i}$, and ${\alpha }_0={\chi }_0={\alpha }_N={\chi }_N=1$ since the contracted result ${\psi }_{s_0\cdots s_{N-1}}$ is a complex number. Since the index ${\left\{{\alpha }_i\right\}}_{i=0}^N$ is not physical, they are called virtual indices.

The decomposition (7) can be realized by $N-1$ steps of singular value decomposition (SVD) of matrices, which is given in detail in Appendix. Inserting (7) into (2), we get the matrix product state

$|\psi \rangle =\underset{s_0,\cdots ,s_{N-1}}{{\displaystyle \sum }}{M}^{\left[0\right]s_0}{M}^{\left[1\right]s_1}\cdots M^{\left[N-1\right]s_{N-1}}|s_0\cdots s_{N-1}\rangle$(8)

If $s_i=1,\cdots ,d$ for $i=0,\cdots ,N-1$, then ${\chi }_1=d$ and ${\chi }_{i+1}\le d{\chi }_i$, $i=1,\cdots ,N-2$, and thus ${\chi }_i\approx d^i$. The exact decomposition of (7) will cause exponential disaster. Fortunately, if the state of interest is not highly entangled, the truncation in SVD will not cause much error for the approximation of this state.

As shown in Appendix, if the tensor ${\psi }_{s_0\cdots s_{N-1}}$ is normalized

$\underset{s_0,\cdots ,s_{N-1}}{{\displaystyle \sum }}{\left|{\psi }_{s_0\cdots s_{N-1}}\right|}^2=1$(9)

the $N$ tensors ${\left\{M^{\left[i\right]}\right\}}_{i=0}^{N-1}$ of order 3 satisfy the left canonical condition

$\underset{s_i}{{\displaystyle \sum }}{\left(M^{\left[i\right]s_i}\right)}^{†}{M}^{\left[i\right]s_i}={\mathbb{I}}_{{\chi }_{i+1}},i=0,\cdots ,N-1$(10)

where ${\mathbb{I}}_{{\chi }_{i+1}}$ is the ${\chi }_{i+1}\times {\chi }_{i+1}$ identity matrix. For a general tenor ${\psi }_{s_0\cdots s_{N-1}}$ of order $N$, we have the left canonical form

${\psi }_{s_0\cdots s_{N-1}}=A^{\left[0\right]s_0}{A}^{\left[1\right]s_1}\cdots A^{\left[N-1\right]s_{N-1}}{\text{Λ}}^{\left[N\right]}$(11)

with the left canonical condition for the $N$ tensors ${\left\{A^{\left[i\right]}\right\}}_{i=0}^{N-1}$

$\underset{s_i}{{\displaystyle \sum }}{\left(A^{\left[i\right]s_i}\right)}^{†}{A}^{\left[i\right]s_i}={\mathbb{I}}_{{\chi }_{i+1}},i=0,\cdots ,N-1$(12)

Here we intentionally introduce a number ${\text{Λ}}^{\left[N\right]}$ which becomes 1 if ${\psi }_{s_0\cdots s_{N-1}}$ satisfies (9). In fact, this left canonical form can also be obtained from the general matrix product form (7) where $M^{\left[i\right]}$ does not satisfy (10). This can be realized by $N-1$ steps of SVD. First,

${\left(M^{\left[0\right]s_0}\right)}_{{\alpha }_0,{\alpha }_1}=\underset{k}{{\displaystyle \sum }}{U}_{\left({\alpha }_0{s}_0\right),k}{S}_k{\left(V^{†}\right)}_{k,{\alpha }_1}$(13)

where $U$ and $V$ are unitary matrices and $\left\{S_k\right\}$ are the singular values. Then ${\left(A^{\left[0\right]s_0}\right)}_{{\alpha }_0,k}=U_{\left({\alpha }_0{s}_0\right),k}$ satisfies the left canonical condition. The remaining part $S_k{\left(V^{†}\right)}_{k,{\alpha }_1}$ is combined with ${\left(M^{\left[1\right]s_1}\right)}_{{\alpha }_1,{\alpha }_2}$ to get ${\displaystyle {\sum }_{{\alpha }_1}{S}_k{\left(V^{†}\right)}_{k,{\alpha }_1}{\left(M^{\left[1\right]s_1}\right)}_{{\alpha }_1,{\alpha }_2}}$ and use SVD for this matrix again, we can find $A^{\left[1\right]s_1}$ which satisfies the left canonical condition. Continuing this process, we can get the left canonical form (11).

Similarly, if the SVD is implemented from the right-hand side, the general tensor has the right canonical form

${\psi }_{s_0\cdots s_{N-1}}={\text{Λ}}^{\left[0\right]}{B}^{\left[0\right]s_0}{B}^{\left[1\right]s_1}\cdots B^{\left[N-1\right]s_{N-1}}$(14)

with the right canonical condition for the tensor ${\left\{B^{\left[i\right]}\right\}}_{i=0}^{N-1}$ of order 3

$\underset{s_i}{{\displaystyle \sum }}{B}^{\left[i\right]s_i}{\left(B^{\left[i\right]s_i}\right)}^{†}={\mathbb{I}}_{{\chi }_i},i=0,\cdots ,N-1$(15)

If the SVD is implemented from both sides, they will meet at some bonds, e.g.,

${\psi }_{s_0\cdots s_{N-1}}={\text{Λ}}^{\left[0\right]}{A}^{\left[0\right]s_0}{A}^{\left[1\right]s_1}{\text{Λ}}^{\left[2\right]}{B}^{\left[2\right]s_2}\cdots B^{\left[N-1\right]s_{N-1}}{\text{Λ}}^{\left[N\right]}$(16)

We call this the mixed canonical form. Here ${\text{Λ}}^{\left[2\right]}$ is a ${\chi }_2\times {\chi }_2$ diagonal matrix.

A full decomposition is

${\psi }_{s_0\cdots s_{N-1}}={\text{Λ}}^{\left[0\right]}{\text{Γ}}^{\left[0\right]s_0}\cdots {\text{Λ}}^{\left[N-1\right]}{\text{Γ}}^{\left[N-1\right]s_{N-1}}{\text{Λ}}^{\left[N\right]}$(17)

where ${\text{Γ}}^{\left[i\right]s_i}$ is the ${\chi }_i\times {\chi }_{i+1}$ matrix and ${\text{Λ}}^{\left[i\right]}$ is the ${\chi }_i\times {\chi }_i$ diagonal matrix. Obviously, these three canonical forms are related to each other

$A^{\left[i\right]s_i}={\text{Λ}}^{\left[i\right]}{\text{Γ}}^{\left[i\right]s_i}={\text{Λ}}^{\left[i\right]}{B}^{\left[i\right]s_i}{\left({\text{Λ}}^{\left[i+1\right]}\right)}^{-1},B^{\left[i\right]s_i}={\text{Γ}}^{\left[i\right]s_i}{\text{Λ}}^{\left[i+1\right]}={\left({\text{Λ}}^{\left[i\right]}\right)}^{-1}{A}^{\left[i\right]s_i}{\text{Λ}}^{\left[i+1\right]}$(18)

for $i=0,\cdots ,N-1$.

Assume that $O_{s^{\prime }}^s=O_{{s^{\prime }}_0\cdots {s^{\prime }}_{N-1}}^{s_0\cdots s_{N-1}}$ has the decomposition

$O=v^L{W}^{\left[0\right]}\cdots W^{\left[N-1\right]}{v}^R$(19)

where $W^{\left[i\right]}={\left(W_{s_i{s^{\prime }}_i}^{\left[i\right]}\right)}_{{\alpha }_i{\alpha }_{i+1}}$ is the tensor of order 4 with $1\le s_i,{s^{\prime }}_i\le d$, $1\le {\alpha }_i\le {\chi }_i$ and $1\le {\alpha }_{i+1}\le {\chi }_{i+1}$, $v^L$ and $v^R$ are the $1\times {\chi }_1$ and ${\chi }_N\times 1$ matrices, respectively. For example, the Hamiltonian

$H=-\underset{i=0}{\overset{N-2}{{\displaystyle \sum }}}J{X}_i{X}_{i+1}-\underset{i=0}{\overset{N-1}{{\displaystyle \sum }}}g{Z}_i$(20)

can be written in the form (19) with

$v^L=\left(1,0,0\right),v^R={\left(0,0,1\right)}^{\text{T}},W^{\left[i\right]}=\left(\begin{array}{ccc}1& X& -gZ\\ 0& 0& -JX\\ 0& 0& 1\end{array}\right),i=0,\cdots ,N-1$

Here $X$, $Y$ and $Z$ are the Pauli $x$, $y$ and $z$ matrices, respectively.

From the decompositions in (7) and (19), one has

$\begin{array}{l}{O}_{{s^{\prime }}_0\cdots {s^{\prime }}_{N-1}}^{s_0\cdots s_{N-1}}{\psi }_{{s^{\prime }}_0,\cdots ,{s^{\prime }}_{N-1}}=v_{{\alpha }_0}^L{\left(W_{s_0{s^{\prime }}_0}^{\left[0\right]}\right)}_{{\alpha }_0{\alpha }_1}\cdots {\left(W_{s_{N-1}{s^{\prime }}_{N-1}}^{\left[N-1\right]}\right)}_{{\alpha }_{N-1}{\alpha }_N}{v}_{{\alpha }_N}^R\\ {\left(M^{\left[0\right]{s^{\prime }}_0}\right)}_{{\beta }_0{\beta }_1}\cdots {\left(M^{\left[N-1\right]{s^{\prime }}_{N-1}}\right)}_{{\beta }_{N-1}{\beta }_N}\end{array}$(21)

The inner product $O|\psi \rangle$ with $|\psi \rangle$ in (6) is a scalar

$\begin{array}{c}\langle \psi |\hat{O}|\psi \rangle =O_{{s^{\prime }}_0\cdots {s^{\prime }}_{N-1}}^{s_0\cdots s_{N-1}}{\psi }_{{s^{\prime }}_0,\cdots ,{s^{\prime }}_{N-1}}\overline{{\psi }_{s_0,\cdots ,s_{N-1}}}\\ =v_{{\alpha }_0}^L{\left(W_{s_0{s^{\prime }}_0}^{\left[0\right]}\right)}_{{\alpha }_0{\alpha }_1}\cdots {\left(W_{s_{N-1}{s^{\prime }}_{N-1}}^{\left[N-1\right]}\right)}_{{\alpha }_{N-1}{\alpha }_N}{v}_{{\alpha }_N}^R\\ {\left(M^{\left[0\right]{s^{\prime }}_0}\right)}_{{\beta }_0{\beta }_1}\cdots {\left(M^{\left[N-1\right]{s^{\prime }}_{N-1}}\right)}_{{\beta }_{N-1}{\beta }_N}\\ {\overline{\left(M^{\left[0\right]s_0}\right)}}_{{\gamma }_0{\gamma }_1}\cdots {\overline{\left(M^{\left[N-1\right]s_{N-1}}\right)}}_{{\gamma }_{N-1}{\gamma }_N}\end{array}$(22)

The summation over one virtual index ${\alpha }_0$ and two physical indices $s_0$ and ${s^{\prime }}_0$ of $v_{{\alpha }_0}^L{\left(W_{s_0{s^{\prime }}_0}^{\left[0\right]}\right)}_{{\alpha }_0{\alpha }_1}$, ${\left(M^{\left[0\right]{s^{\prime }}_0}\right)}_{{\beta }_0{\beta }_1}$ and ${\overline{\left(M^{\left[0\right]s_0}\right)}}_{{\gamma }_0{\gamma }_1}$ is a tensor of order 3 with virtual indices ${\alpha }_1$, ${\beta }_1$ and ${\gamma }_1$. Note that ${\beta }_0={\gamma }_0=1$.

If the operator $\hat{O}$ is local and ${\psi }_s$ is a canonical form, the expression in (22) can be simplified. For example, ${\psi }_s$ has the mixed canonical form (16), $\hat{O}$ is the operator acting on local site 2 and site 3. For example, ${\left(W_{s_i{s^{\prime }}_i}^{\left[i\right]}\right)}_{{\alpha }_i,{\alpha }_{i+1}}={\delta }_{s_i,{s^{\prime }}_i}{\delta }_{{\alpha }_i,{\alpha }_{i+1}}$ for $i\ne 2,3$,

(23)

The ground state of Hamiltonian $\hat{H}$ can be calculated by DMRG algorithm [11] , which is a variational method by minimizing $\langle \psi |\hat{H}|\psi \rangle$ over the tensor ${\psi }_s$ of order $N$. Here we always require that ${\displaystyle {\sum }_s{\left|{\psi }_s\right|}^2}=1$. Since ${\psi }_s$ has the structure of matrix product state, this global minimization can be realized by local minimization step by step. For example, $\hat{H}$ and ${\psi }_s$ have a form (19), and (16), respectively, then $\langle \psi |\hat{H}|\psi \rangle$ has a form (23) can be written as

$\langle \psi |\hat{H}|\psi \rangle =\langle \Theta |H_{\text{eff}}|\Theta \rangle$(24)

where $\text{Θ}={\text{Λ}}^{\left[2\right]}{B}^{\left[2\right]}{B}^{\left[3\right]}$ is a tensor of order 4. $H_{\text{eff}}$, which is the tensor of order 8, is obtained by the contraction of the left environment $L^{\left[2\right]}$, right environment $R^{\left[3\right]}$, $W^{\left[2\right]}$ and $W^{\left[3\right]}$. Here the left environment $L^{\left[i\right]}$ is obtained from the contraction of $L^{\left[i-1\right]}$, $W^{\left[i-1\right]}$, $A^{\left[i-1\right]}$ and $\overline{A^{\left[i-1\right]}}$, $L_{{\beta }_0{\gamma }_0{\alpha }_0}^{\left[0\right]}={\delta }_{{\beta }_0{\gamma }_0}{v}_{{\alpha }_0}^L$. Similarly, $R^{\left[i\right]}$ is obtained from the contraction of $R^{\left[i+1\right]}$, $W^{\left[i+1\right]}$ and $B^{\left[i+1\right]}$, $\overline{B^{\left[i+1\right]}}$, $R_{{\beta }_N{\gamma }_N{\alpha }_N}^{\left[N\right]}={\delta }_{{\beta }_N{\gamma }_N}{v}_{{\alpha }_N}^R$.

To keep the mixed canonical form of tensor of order $N$, THE SVD for the minimum ${\Theta }_{\min }$ is

${\Theta }_{\min }={\tilde{A}}^{\left[2\right]}{\tilde{\Lambda }}^{\left[3\right]}{\tilde{B}}^{\left[3\right]}\iff {\Theta }_{\left({\beta }_2{s^{\prime }}_2\right),\left({\beta }_4{s^{\prime }}_3\right)}=\underset{{\beta }_3=1}{\overset{\min \left(d{\chi }_2,d{\chi }_4\right)}{{\displaystyle \sum }}}{\tilde{A}}_{\left({\beta }_2{s^{\prime }}_2\right),{\beta }_3}^{\left[2\right]}{\tilde{\Lambda }}_{{\beta }_3{\beta }_3}^{\left[3\right]}{\tilde{B}}_{{\beta }_3,\left({\beta }_4{s^{\prime }}_3\right)}^{\left[3\right]}$(25)

Here the tensor ${\text{Θ}}_{{\beta }_2{\beta }_4}^{{s^{\prime }}_2{s^{\prime }}_3}$ of order 4 is rewritten as a matrix ${\text{Θ}}_{\left({\beta }_2{s^{\prime }}_2\right),\left({\beta }_4{s^{\prime }}_3\right)}$. The bond dimension becomes $d$ times ${\chi }_2$ or ${\chi }_4$ which is almost $d$ times of the bond dimension for the original ${\displaystyle {\sum }_{{\beta }_3=1}^{{\chi }_3}{\left(A^{\left[2\right]{s^{\prime }}_2}\right)}_{{\beta }_2,{\beta }_3}{\text{Λ}}_{{\beta }_3{\beta }_3}^{\left[3\right]}{\left(B^{\left[3\right]{s^{\prime }}_3}\right)}_{{\beta }_3,{\beta }_4}}$. Here we assume that ${\chi }_2~{\chi }_3~{\chi }_4$. To keep the bond dimension under control, the SVD in (25) is truncated for $k=1,\cdots ,{\chi }_{\text{max}}$. In practice, the largest ${\chi }_{\text{max}}$ singular values in magnitude ${\left\{{\text{Λ}}_{{\beta }_3{\beta }_3}^{\left[3\right]}\right\}}_{{\beta }_3=1}^{{\chi }_{\text{max}}}$ are kept and then scale these singular values such that ${\displaystyle {\sum }_{k=1}^{{\chi }_{\text{max}}}{\left|{\text{Λ}}_{{\beta }_3{\beta }_3}^{\left[3\right]}\right|}^2}={\displaystyle {\sum }_{k=1}^{\text{min}\left(d{\chi }_2,d{\chi }_4\right)}{\left|{}_{\widetilde{\text{Λ}}}^{{\beta }_3{\beta }_3}\right|}^2}$ to ensure ${\left|{\psi }_s\right|}^2=1$. For the ground state with not highly entangled, the singular values in magnitude decays vary fast. This truncation will not cause much error. Replacing $\text{Θ}={\text{Λ}}^{\left[2\right]}{B}^{\left[2\right]}{B}^{\left[3\right]}$ by ${\text{Θ}}_{\text{min}}={\tilde{A}}^{\left[2\right]}{}^{\widetilde{\text{Λ}}}{\tilde{B}}^{\left[3\right]}$, we get another better tensor with the less value of $\langle \psi |\hat{O}|\psi \rangle$. This better tensor is also in the mixed canonical form. This is called one step of two-site DMRG algorithm since the local tensors ${\text{Λ}}^{\left[2\right]}{B}^{\left[2\right]}$ at site 2 and $B^{\left[3\right]}$ at site 3 are updated. To update the local tensors ${\text{Λ}}^{\left[3\right]}{B}^{\left[3\right]}$ at site 3 and $B^{\left[4\right]}$ at site 4 in the next step of DMRG, we should prepare the left environment $L^{\left[3\right]}$, right environment $R^{\left[4\right]}$. $L^{\left[3\right]}$ is the contraction of $L^{\left[2\right]}$, $W^{\left[2\right]}$, ${\tilde{A}}^{\left[2\right]}$ and $\overline{{\stackrel{˜}{A}}^{\left[2\right]}}$. So we use $N-1$ steps of DMRG to update all $N$ sub-tensors from the left to the right-hand side. This is called one sweep of DMRG. The next sweep is implemented from the right to the left-hand side. This algorithm will stop if $\langle \psi |\hat{H}|\psi \rangle$ will not decrease.

The quantum expectation value of an operator $\hat{O}$ on a non-normalized state $|\psi \rangle$ can be written as a classical expectation value over the distribution $p\left(s\right)={\left|\psi \left(s\right)\right|}^2/\langle \psi |\psi \rangle$ using

$\langle \hat{O}\rangle \equiv \frac{\langle \psi |\hat{O}|\psi \rangle }{\langle \psi |\psi \rangle }=\mathbb{E}\left[\tilde{O}\right]\equiv \underset{s}{{\displaystyle \sum }}p\left(s\right)\tilde{O}\left(s\right)$(26)

where

$\tilde{O}\left(s\right)=\frac{\langle s|\hat{O}|\psi \rangle }{\langle s|\psi \rangle }=\underset{s^{\prime }}{{\displaystyle \sum }}\frac{\psi \left(s^{\prime }\right)}{\psi \left(s\right)}\langle s|\hat{O}|s^{\prime }\rangle$(27)

If the matrix presentation $O$ of the operator $\hat{O}$ under the basis $|s^{\prime }\rangle$ is sparse, there are only $O\left(n\right)$ basis $|s\rangle$ such that $\langle s|\hat{O}|s^{\prime }\rangle$ are nonzero. Equation (26) shows that $\langle \hat{O}\rangle$ can be calculated by Monte Carlo sampling of $\tilde{O}\left(s\right)$ with the probability $p\left(s\right)$.

We use neutral network to describe the wave function $\psi \left(s\right)$, which depends on real parameters $\theta ={\left\{{\theta }_k\right\}}_{k=1}^p$, and thus denoted by ${\psi }_{\theta }\left(s\right)$. Denote by $\delta {\theta }_k$ the change of the kth component ${\theta }_k$. ${\psi }_{\theta +\delta {\theta }_k}\left(s\right)$ has the expansion

${\psi }_{\theta +\delta {\theta }_k}\left(s\right)={\psi }_{\theta }\left(s\right)+\delta {\theta }_k\frac{\partial {\psi }_{\theta }\left(s\right)}{\partial {\theta }_k}+O\left(\delta {\theta }_k^2\right)$(28)

Define the local operator ${\hat{\Psi }}_k$ with matrix elements

$\langle s|{\hat{\Psi }}_k|s^{\prime }\rangle ={\delta }_{s,s^{\prime }}{\Psi }_k\left(s\right)$(29)

where we introduced the log-derivative of ${\psi }_{\theta }\left(s\right)$

${\Psi }_k\left(s\right)=\frac{\partial \ln {\psi }_{\theta }\left(s\right)}{\partial {\theta }_k}=\frac{1}{{\psi }_{\theta }\left(s\right)}\frac{\partial {\psi }_{\theta }\left(s\right)}{\partial {\theta }_k}$(30)

Here ${\text{Ψ}}_k\left(s\right)$ depends on $\theta$. We thus have

$\langle s|{\hat{\Psi }}_k|{\psi }_{\theta }\rangle ={\Psi }_k\left(s\right){\psi }_{\theta }\left(s\right)=\frac{\partial {\psi }_{\theta }\left(s\right)}{\partial {\theta }_k}$(31)

From (28), one has

$|{\psi }_{\theta +{\theta }_k}\rangle =\left(1+\delta {\theta }_k{}_{\widehat{\text{Ψ}}}\right)|{\psi }_{\theta }\rangle$(32)

where we omitted $O\left(\delta {\theta }_k^2\right)$.

$|{\psi }_{\theta }\rangle$ is normalized to be

$|v_{0,\theta }\rangle =\frac{|{\psi }_{\theta }\rangle }{\sqrt{\langle {\psi }_{\theta }|{\psi }_{\theta }\rangle }}$(33)

Define

$|v_{k,\theta }\rangle =\left({\hat{\Psi }}_k-\langle {\hat{\Psi }}_k\rangle \right)|v_{0,\theta }\rangle ,k=1,\cdots ,p$(34)

where

$\langle {\hat{\Psi }}_k\rangle =\langle v_{0,\theta }|{\hat{\Psi }}_k|v_{0,\theta }\rangle =\frac{\langle {\psi }_{\theta }|{\hat{\Psi }}_k|{\psi }_{\theta }\rangle }{\langle {\psi }_{\theta }|{\psi }_{\theta }\rangle }$(35)

Thus $|v_{k,\theta }\rangle$ and $|v_{0,\theta }\rangle$ are orthogonal, $k=1,\cdots ,p$. But $\langle v_{k,\theta }|v_{k^{\prime },\theta }\rangle \ne 0$ for $k,k^{\prime }=1,\cdots ,p$.

The norm squared of $|{\psi }_{\theta +\delta {\theta }_k}\rangle$ is

$\begin{array}{c}\langle {\psi }_{\theta +\delta {\theta }_k}|{\psi }_{\theta +\delta {\theta }_k}\rangle =\langle {\psi }_{\theta }|{\left(1+\delta {\theta }_k{\hat{\Psi }}_k\right)}^{*}\left(1+\delta {\theta }_k{\hat{\Psi }}_k\right)|{\psi }_{\theta }\rangle \\ =\langle {\psi }_{\theta }|{\psi }_{\theta }\rangle \left[1+2\mathrm{Re}\left(\delta {\theta }_k\langle {\hat{\Psi }}_k\rangle \right)+O\left(\delta {\theta }_k^2\right)\right]\end{array}$(36)

where $\text{Re}$ denotes the real part. One has from (32) (36)

$\begin{array}{c}|v_{0,\theta +\delta {\theta }_k}\rangle =\frac{|{\psi }_{\theta +\delta {\theta }_k}\rangle }{\sqrt{\langle {\psi }_{\theta +\delta {\theta }_k}|{\psi }_{\theta +\delta {\theta }_k}\rangle }}\\ =|v_{0,\theta }\rangle +\left[\delta {\theta }_k{\hat{\Psi }}_k-\mathrm{Re}\left(\delta {\theta }_k\langle {\hat{\Psi }}_k\rangle \right)\right]|v_{0,\theta }\rangle +O\left(\delta {\theta }_k^2\right)\\ =\left[1+i\mathrm{Im}\left(\delta {\theta }_k\langle {\hat{\Psi }}_k\rangle \right)\right]|v_{0,\theta }\rangle +\delta {\theta }_k|v_{k,\theta }\rangle +O\left(\delta {\theta }_k^2\right)\left(\text{by}\left(34\right)\right)\\ ={\text{e}}^{i\delta \varphi }\left[|v_{0,\theta }\rangle +\delta {\theta }_k|v_{k,\theta }\rangle \right]+O\left(\delta {\theta }_k^2\right)\end{array}$(37)

where $\delta \varphi =\mathrm{Im}\left(\delta {\theta }_k\langle {\hat{\Psi }}_k\rangle \right)$ is the imaginary part of $\delta {\theta }_k\langle {\hat{\Psi }}_k\rangle$.

Let $\hat{H}$ be the Hamiltonian operator and using (37), one has

$\begin{array}{l}\frac{\partial }{\partial {\theta }_k}\frac{\langle {\psi }_{\theta }|\hat{H}|{\psi }_{\theta }\rangle }{\langle {\psi }_{\theta }|{\psi }_{\theta }\rangle }=\frac{\partial }{\partial {\theta }_k}\langle v_{0,\theta }|\hat{H}|v_{0,\theta }\rangle \\ =\underset{\delta {\theta }_k\to 0}{\lim }\frac{\langle v_{0,\theta +\delta {\theta }_k}|\hat{H}|v_{0,\theta +\delta {\theta }_k}\rangle -\langle v_{0,\theta }|\hat{H}|v_{0,\theta }\rangle }{\delta {\theta }_k}\end{array}$

$\begin{array}{l}=\langle v_{k,\theta }|\hat{H}|v_{0,\theta }\rangle +\langle v_{0,\theta }|\hat{H}|v_{k,\theta }\rangle \\ =2\mathrm{Re}\left[\frac{\langle {\psi }_{\theta }|\hat{H}\left({\hat{\Psi }}_k-\langle {\hat{\Psi }}_k\rangle \right)|{\psi }_{\theta }\rangle }{\langle {\psi }_{\theta }|{\psi }_{\theta }\rangle }\right]\\ =2\mathrm{Re}\left[\frac{\langle {\psi }_{\theta }|\hat{H}{\hat{\Psi }}_k|{\psi }_{\theta }\rangle }{\langle {\psi }_{\theta }|{\psi }_{\theta }\rangle }-\frac{\langle {\psi }_{\theta }|\hat{H}|{\psi }_{\theta }\rangle }{\langle {\psi }_{\theta }|{\psi }_{\theta }\rangle }\frac{\langle {\psi }_{\theta }|{\hat{\Psi }}_k|{\psi }_{\theta }\rangle }{\langle {\psi }_{\theta }|{\psi }_{\theta }\rangle }\right]\end{array}$(38)

The $k$th parameter ${\theta }_k$ is updated according to the gradient descent method

${\theta }_k\leftarrow {\theta }_k-\eta f_k$(39)

where $f_k=\frac{\partial E_{\theta }}{\partial {\theta }_k}$, $E_{\theta }=\frac{\langle {\psi }_{\theta }|\hat{H}|{\psi }_{\theta }\rangle }{\langle {\psi }_{\theta }|{\psi }_{\theta }\rangle }$, $\eta >0$ is the learning rate.

If each component ${\theta }_k$ has a change $\delta {\theta }_k$, the wave function ${\psi }_{\theta }\left(s\right)$ has changed to be ${\psi }_{\theta +\delta \theta }\left(s\right)$. Let $|v_{0,\theta +\delta \theta }\rangle =|{\psi }_{\theta +\delta \theta }\rangle /\sqrt{\langle {\psi }_{\theta +\delta {\theta }_k}|{\psi }_{\theta +\delta {\theta }_k}\rangle }$. Equation (37) is generalized to be

$|v_{0,\theta +\delta \theta }\rangle ={\text{e}}^{i\delta \varphi }\left[|v_{0,\theta }\rangle +\underset{k=1}{\overset{p}{{\displaystyle \sum }}}\delta {\theta }_k|v_{k,\theta }\rangle \right]+O\left(\delta {\theta }^2\right)$(40)

where $\delta \varphi ={\displaystyle {\sum }_{k=1}^p\mathrm{Im}\left(\delta {\theta }_k\langle {\hat{\Psi }}_k\rangle \right)}$.

The distance between $|v_{0,\theta }\rangle$ and $|v_{0,\theta +\delta \theta }\rangle$ is

$\delta s^2=\underset{\delta \alpha }{\text{min}}{\Vert {\text{e}}^{-i\delta \alpha }|v_{0,\theta +\delta \theta }\rangle -|v_{0,\theta }\rangle \Vert }^2$(41)

When $\delta \alpha =\delta \varphi$, $\delta s^2$ is minimized to be

$\delta s^2=\underset{1\le k,k^{\prime }\le p}{{\displaystyle \sum }}\langle v_{k,\theta }|v_{k^{\prime },\theta }\rangle \delta {\theta }_k\delta {\theta }_{k^{\prime }}+o\left(\delta {\theta }^2\right)=\underset{1\le k,k^{\prime }\le p}{{\displaystyle \sum }}{S}_{k,k^{\prime }}\delta {\theta }_k\delta {\theta }_{k^{\prime }}+o\left(\delta {\theta }^2\right)$(42)

with

$\begin{array}{c}{S}_{k,k^{\prime }}=\frac{\langle v_{k,\theta }|v_{k^{\prime },\theta }\rangle +\langle v_{k^{\prime },\theta }|v_{k,\theta }\rangle }{2}=\mathrm{Re}\left(\langle v_{k,\theta }|v_{k^{\prime },\theta }\rangle \right)\\ =\mathrm{Re}\left(\langle {\hat{\Psi }}_k^{*}{\hat{\Psi }}_{k^{\prime }}\rangle -{\langle {\hat{\Psi }}_k\rangle }^{*}\langle {\Psi }_{k^{\prime }}\rangle \right)\end{array}$(43)

where we used

$\langle v_{k,\theta }|v_{k^{\prime },\theta }\rangle =\langle v_{0,\theta }|{\left({\hat{\Psi }}_k-\langle {\hat{\Psi }}_k\rangle \right)}^{*}\left({\hat{\Psi }}_{k^{\prime }}-\langle {\hat{\Psi }}_{k^{\prime }}\rangle \right)|v_{0,\theta }\rangle$(44)

Find $\delta \theta$ such that: 1) $\text{Δ}E=E_{\theta +\delta \theta }-E_{\theta }$ is minimized; 2) the distance $\delta s^2$ between $|v_{0,\theta }\rangle$ and $|v_{0,\theta +\delta \theta }\rangle$ is as small as possible. Minimize

$\text{Δ}E+\mu \delta s^2=\underset{k}{{\displaystyle \sum }}\frac{\partial E_{\theta }}{\partial {\theta }_k}\delta {\theta }_k+\mu \underset{k,k^{\prime }}{{\displaystyle \sum }}{S}_{k,k^{\prime }}\delta {\theta }_k\delta {\theta }_{k^{\prime }}+o\left(\delta {\theta }^2\right)$(45)

to find that $\delta \theta$ satisfies

$\underset{k^{\prime }}{{\displaystyle \sum }}{S}_{k,k^{\prime }}\delta {\theta }_{k^{\prime }}=-\frac{f_k}{2\mu }$(46)

which can be used to update the parameter $\theta$. This algorithm is called the stochastic reconfiguration method.

In the Boltzmann machine ansatz, the wave function is chosen to be

${\psi }_{\theta }\left(s\right)=\exp \left(\underset{j=0}{\overset{N-1}{{\displaystyle \sum }}}{a}_j{s}_j\right)\underset{i=0}{\overset{M-1}{{\displaystyle \prod }}}2\cosh \left(b_i+\underset{j=0}{\overset{N-1}{{\displaystyle \sum }}}{W}_{ij}{s}_j\right)$(47)

where $\theta =\left\{a_j,b_i,W_{ij}\right\}$ are the real parameters. Here $M$ is a given positive integer number. $\alpha =M/N$ is the ratio of $M$ and $N$. Compared to the other neutral network ansatz, ${\Psi }_k$ for the Boltzmann machine ansatz can be calculated analytically [39] .

The variational quantum eigensolver is aimed at finding the ground state energy of a Hamiltonian $\hat{H}$ by minimizing the cost function $L\left(\theta \right)=\langle {\psi }_{\theta }|\hat{H}|{\psi }_{\theta }\rangle$, where he parameter $\theta$ is updated by $\theta -\eta \nabla L\left(\theta \right)$ by iterations. Here $\eta >0$ is the learning rate. We choose the trial state $|{\psi }_{\theta }\rangle =U_{\theta }|{\psi }_0\rangle$ for some ansatz $U_{\theta }$ and initial state $|{\psi }_0\rangle$. The variational Hamiltonian ansatz also aims to prepare a trial ground state for a given Hamiltonian $\hat{H}={\displaystyle {\sum }_k{\hat{H}}_k}$ (where ${\hat{H}}_k$ are Hermitian operators, usual Pauli strings) by Trotterizing an adiabatic state preparation process [116] . Here, each Trotter step corresponds to a variational ansatz so that the unitary is given $U_{\theta }={\displaystyle {\prod }_l\left({\displaystyle {\prod }_k{\text{e}}^{-i{\theta }_{l,k}{\hat{H}}_k}}\right)}$.

Consider a general Hamiltonian

$\begin{array}{c}\hat{H}=\underset{\sigma =\uparrow ,\downarrow }{{\displaystyle \sum }}\underset{j=0}{\overset{N-2}{{\displaystyle \sum }}}\left(C_X^{\text{h}}{X}_{j,\sigma }{X}_{j+1,\sigma }+C_Y^{\text{h}}{Y}_{j,\sigma }{Y}_{j+1,\sigma }+C_Z^{\text{h}}{Z}_{j,\sigma }{Z}_{j+1,\sigma }\right)\\ +\underset{j=0}{\overset{N-1}{{\displaystyle \sum }}}\left(C_X^{\text{v}}{X}_{j,\uparrow }{X}_{j,\downarrow }+C_Y^{\text{v}}{Y}_{j,\uparrow }{Y}_{j,\downarrow }+C_Z^{\text{v}}{Z}_{j,\uparrow }{Z}_{j,\downarrow }\right)\end{array}$(48)

where $N$ is an even integer and the coefficients $C_{\alpha }^{\text{h}}>0$ and $C_{\alpha }^{\text{v}}$ do not depend on sites. $X_{j,\sigma }$, etc., denotes the Pauli $X$ matrix acting on site $j$ and spin $\sigma =\uparrow ,\downarrow$. Splitting the $N$ sites to be even sites and odd sites, this Hamiltonian can be written as

$\hat{H}=\underset{\alpha =X,Y,Z}{{\displaystyle \sum }}\left(H_{\alpha }^{\text{h},\text{e}}+H_{\alpha }^{\text{h},\text{o}}+H_{\alpha }^{\text{v},\text{e}}+H_{\alpha }^{\text{v},\text{o}}\right)=H^{\text{e}}+H^{\text{o}}$(49)

where

$H^{\text{e}}=\underset{\alpha =X,Y,Z}{{\displaystyle \sum }}\left(H_{\alpha }^{\text{h},\text{e}}+H_{\alpha }^{\text{v},\text{e}}\right),H^{\text{o}}=\underset{\alpha =X,Y,Z}{{\displaystyle \sum }}\left(H_{\alpha }^{\text{h},\text{o}}+H_{\alpha }^{\text{v},\text{o}}\right)$(50)

$H_{\alpha }^{\text{h},\text{e}}=\underset{\sigma =\uparrow ,\downarrow }{{\displaystyle \sum }}\underset{\text{even}j=0}{\overset{N-2}{{\displaystyle \sum }}}{C}_{\alpha }^{\text{h}}{\alpha }_{j,\sigma }{\alpha }_{j+1,\sigma },H_{\alpha }^{\text{v},\text{e}}=\underset{\text{even}j=0}{\overset{N-1}{{\displaystyle \sum }}}{C}_{\alpha }^{\text{v}}{\alpha }_{j,\sigma }{\alpha }_{j,\uparrow }{\alpha }_{j,\downarrow },\alpha =X,Y,Z$(51)

$H_{\alpha }^{\text{h},\text{o}}$ and $H_{\alpha }^{\text{v},\text{o}}$ are defined similarly by replacing summation over even site $j$ with odd site $j$.

Inspired by Ref. [117] [118] , the quantum circuit is $U_{\theta }={\displaystyle {\prod }_{k=1}^p{U}^{\text{h}}\left({\theta }_k^{\text{h}}\right)U^{\text{v}}\left({\theta }_k^{\text{v}}\right)}$, where

$\begin{array}{l}{U}^{\text{h}}\left({\theta }_k^{\text{h}}\right)=G\left({\theta }_{k,X}^{\text{h},\text{e}},H_X^{\text{h},\text{e}}\right)G\left({\theta }_{k,Y}^{\text{h},\text{e}},H_Y^{\text{h},\text{e}}\right)G\left({\theta }_{k,Z}^{\text{h},\text{e}},H_Z^{\text{h},\text{e}}\right)\\ G\left({\theta }_{k,Z}^{\text{h},\text{o}},H_Z^{\text{h},\text{o}}\right)G\left({\theta }_{k,Y}^{\text{h},\text{o}},H_Y^{\text{h},\text{o}}\right)G\left({\theta }_{k,X}^{\text{h},\text{o}},H_X^{\text{h},\text{o}}\right)\end{array}$(52)