Title: ETH_Sem5 / AdvQC / 16_DMRG

Q: When is Full Configuration Interaction really needed?
A: - Full-CI needed when the HF reference is inaccurate \(\rightarrow\) nearly-degenerate orbitals (and, therefore, Slater determinants), i.e., strongly-correlated molecules.
- Active space idea: full-CI ``within a subspace'' (recovers static correlation).
- Missing correlation effects (dynamical correlation) recovered a posteriori. But still a major challenge!
- FCI solves a quantum many-body problem in terms of one-particle entities, the orbitals, which may be calculated with HF.
- FCI offers a route towards ``exact'' quantum-chemical simulations, provided that a sufficiently accurate one-particle basis is available.
- Second quantization defines the ideal framework for designing FCI-based algorithms.
- Computational realizations of FCI are limited by the curse of the dimensionality.
- Alternative strategies (CC, CI, PT2 , \dots) have been designed, but are tailored for weakly-correlated molecules.

Q: Explain SVD!
A: Let \(M\) be a \(n \times m\) complex-valued matrix. The singular value decomposition (SVD) of \(M\) is defined as:
\[\bm{M}=\bm{U} \bm{\Sigma} \bm{V}^{\dagger}\]
- \(\bm{U}\) is a \(n \times \min (n, m)\) matrix such that \(\bm{U}^{\dagger} \bm{U}=\bm{1}\) (left singular vectors).
- \(\bm{V}\) is a \(m \times \min (n, m)\) matrix such that \(\bm{V}^{\dagger} \bm{V}=\bm{1}\) (right singular vectors).
- \(\bm{\Sigma}\) is a \(\min (n, m) \times \min (n, m)\) diagonal matrix, with non-negative entries.
- Equivalent to the diagonalization for square matrices.
- Applications: solution of least-squares fitting problems, principal component analysis, calculation of the rank/null-space of a matrix \dots
\[\begin{aligned} \bm{M} & =\left(\begin{array}{ccccc} M_{11} & & \vdots & & M_{51} \\ & \cdots & M_{i j} & \cdots & \\ M_{13} & & \vdots & & M_{53} \end{array}\right) \end{aligned}\]

Q: How can one quantify the quality of approximating a matrix by trashing the singular values which are below a given threshold?
A: SVD can be used to define a low-rank approximation of a matrix.
Definition: Frobenius norm of a matrix \(A\):
\[\|\boldsymbol{A}\|_F:=\sqrt{\sum_{i j} A_{i j}^2}\]
Theorem: The Frobenius norm of a matrix is the square root of the sum of the squared values of its singular values.
Proof:
\[\begin{aligned} \|\boldsymbol{A}\|_F:=\sqrt{\sum_{i j} A_{i j}^2} & =\sqrt{\operatorname{Tr}\left(\boldsymbol{A}^{\dagger} \boldsymbol{A}\right)} \\ &=\sqrt{\operatorname{Tr}\left(\boldsymbol{V} \boldsymbol{\Sigma} \boldsymbol{U}^{\dagger} \boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{\dagger}\right)} \\ &=\sqrt{\operatorname{Tr}\left(\boldsymbol{V} \boldsymbol{\Sigma} \boldsymbol{\Sigma} \boldsymbol{V}^{\dagger}\right)} \\ & =\sqrt{\operatorname{Tr}\left(\boldsymbol{V}^{\dagger} \boldsymbol{V} \boldsymbol{\Sigma} \boldsymbol{\Sigma}\right)}\\ &=\sqrt{\operatorname{Tr}(\boldsymbol{\Sigma} \boldsymbol{\Sigma})}=\sqrt{\sum_i\left(\sigma_i^2\right)} \end{aligned}\]
Truncated SVD: singular values below a given threshold \(\eta\) neglected \((\widetilde{M})\).
Quality of the approximation can be quantified based on the Frobenius norms.
\[\|M-\tilde{\boldsymbol{M}}\|_F=\sqrt{\operatorname{Tr}(\boldsymbol{\Sigma}-\tilde{\boldsymbol{\Sigma}})^2}=\sqrt{\sum_{k \in \text {discarded}} \sigma_k^2}\]
If many of the singular values are \(\approx 0\) (low-rank matrices), \(M\) can be largely compressed.

Q: What is the Schmidt decomposition
A: The SVD can be applied to compress many-body states.
Partition an \(L\)-body full \(\mathrm{CI}\) wave function in two parts (left and right).
Calculate the SVD to the matrix obtained by reshaping the \(C\) tensor.
Number of ``small'' singular values determines the possibility of compressing the state.
1. Reshape the tensor to a matrix of dimension \(4^{\dim{\bm{\sigma}_L}}\times4^{\dim{\bm{\sigma}_R}}\).
2. Apply SVD
3. Interpret the result as change of basis, i.e. tensor product of two bases.
4. \(U_{\boldsymbol{\sigma}_L, m}\) and \(V_{\boldsymbol{\sigma}_R, m}\) represent rotations of the left and right block, respectively.
\[\begin{aligned} |\Psi\rangle & =\sum_{\sigma_1 \in\{-, \uparrow, \downarrow, \uparrow \downarrow\}} \cdots \sum_{\sigma_N \in\{-, \uparrow, \downarrow, \uparrow \downarrow\}} C_{\sigma_1, \ldots, \sigma_N}\left|\sigma_1 \cdots \sigma_N\right\rangle \\ &=\sum_{\boldsymbol{\sigma}_L, \boldsymbol{\sigma}_R} C_{\boldsymbol{\sigma}_L, \boldsymbol{\sigma}_R}\left|\boldsymbol{\sigma}_L \boldsymbol{\sigma}_R\right\rangle \\ & =\sum_{\boldsymbol{\sigma}_L, \boldsymbol{\sigma}_R} \sum_{m=1}^{N_{\max }} U_{\left(\boldsymbol{\sigma}_L, m\right)} \Sigma_m V_{\left(\boldsymbol{\sigma}_R, m\right)}\left|\boldsymbol{\sigma}_L \boldsymbol{\sigma}_R\right\rangle \\ &=\sum_{m=1}^{N_{\max }} \Sigma_m\left(\sum_{\boldsymbol{\sigma}_L} U_{\boldsymbol{\sigma}_L, m}\left|\boldsymbol{\sigma}_L\right\rangle\right) \otimes\left(\sum_{\boldsymbol{\sigma}_R} V_{\boldsymbol{\sigma}_R, m}\left|\boldsymbol{\sigma}_R\right\rangle\right) \\ &=\sum_{m=1}^{N_{\max }} \Sigma_m\left|a_m^{(L)}\right\rangle \otimes\left|a_m^{(R)}\right\rangle \end{aligned}\]

Q: Explain schematically how the the FCI wavefunction is rewritten into a MPS!
A: Any pair of orbitals can be taken as a reference site for the SVD.
Starting from the first orbital, we can isolate one orbital at the time and apply the SVD to the resulting system-bath partitioning.
Rank-\(L\) tensor converted into \(L\) rank-3 tensor.

Q: What is the area law?
A: The Area Law:
The entanglement entropy of
1. the ground-state wave functions of
2. a short-ranged Hamiltonian that
3. has a finite gap between the ground and the first excited state, e.g., no semicondunctors, \dots
\(\ldots\) is independent of the system size, e.g., spin chain: adding one spin to the end does not influence spins at front
- The entanglement entropy is related to the magnitude of the singular values in the Schmidt decomposition.
- The number of singular values different from 0 is independent on \(L\),
- The size of the tensors is independent on \(L\), the complexity of the wave functions grows linearly with \(L\) and not exponentially as in the FCI matrix product state \(\dim{m_i} = \min{(m_{i-1},4^{L-i})}\), i.e. the tensors do not grow.
- ... but the hypotheses are not met by quantum chemical Hamiltonian

Q: What is the Matrix Product State parametrization? What tunes the accuracy?
A: \[ \left|\Phi_{\mathrm{MPS}}\right\rangle=\sum_{\boldsymbol{\sigma}} \sum_{m_1}^{m_{\max }} \cdots \sum_{m_L}^{m_{\max }} M_{1, m_1}^{\sigma_1} M_{m_1, m_2}^{\sigma_2} \cdots M_{m_{L-1}, 1}^{\sigma_L}|\boldsymbol{\sigma}\rangle\]
The tensor factorization is known as matrix product state (also tensor train in numerical analysis).
The mapping of the orbitals onto a one-dimensional sorting is known as lattice. For QC problems the sorting is crucial (as \(\hat{H}\) is not short ranged)! Orbitals with strongest correlation should be placed next to each other.
Tensors depend on the physical indices \((\sigma)\) and on the (auxiliary indices) \(m\).
Maximum dimension of the auxiliary indices (bond dimension) tunes the accuracy of the parameterization.
Original tensor obtained contracting over the auxiliary indices.
Conversion of the full-CI wave function into MPS only formal, but not useful in practice.
DMRG: optimize the wave function directly in the MPS format?

Q: How can one schmematically minimize the energy functional \[E[\Psi]=\left\langle\Psi\left|\mathcal{H}_{\mathrm{ele}}\right| \Psi\right\rangle-E\langle\Psi \mid \Psi\rangle\] with MPS and MPO?
A: \[\mathcal{O}=\sum_{\boldsymbol{\sigma}} \sum_{\boldsymbol{\sigma}^{\prime}} \sum_{b_1} \cdots \sum_{b_{L-1}} O_{1, b_1}^{\sigma_1, \sigma_1^{\prime}} O_{b_1, b_2}^{\sigma_2, \sigma_2^{\prime}} \cdots O_{b_{L-1}, 1}^{\sigma_L, \sigma_L^{\prime}}|\boldsymbol{\sigma}\rangle\left\langle\boldsymbol{\sigma}^{\prime}\right|\]
- Variational parameters: entries of the MPS \(M^{\sigma_i}_{m_{i-1},m_i}\).
- Derivative w.r.t. a given entry \(\rightarrow\) depends on all the other entries.

Q: What is the problem of optimizing the energy by taking the derivative w.r.t. a given entry directly? Which alternative can be used?
A: - Variational parameters strongly coupled, hard to optimize with gradient-based methods.
- Alternative: minimize with respect to the entries of only a single tensor by keeping the other constant.
- Setting the resulting gradient to 0 yields a conventional eigenvalue problem, analogously to full-CI.
- Procedure repeated for every tensor until convergence is reached (alternating least squares).

Q: Which parameters need to be set in a DMRG calculation?
A: - Number of orbitals (lattice size \(L\)).
- Number of \(\alpha\) and \(\beta\) electrons.
- Spin state (for spin-adapted DMRG calculations).
- Bond dimension \(m\) (\(m\) ranges between 500 and 2000 for 30-60 orbitals).
- Number of optimization sweeps (10/20 sweeps are usually sufficient).
- Definition of the second-quantization Hamiltonian.
- Parameters for the iterative Davidson diagonalization algorithm.

Q: What are open challenges of DMRG?
A: 1. Can target up to \(\approx 100\) particles/orbitals, memory bottleneck hit for larger systems.
2. Efficiently encode strong correlation effects, weak correlation can be added a posteriori CC, second-order perturbation theory, DFT but difficult to combine with DMRG by preserving its efficiency.
3. Inherently designed for ground-states, targeting excited states non trivial.
4. Optimization may get trapped in local minima for very large lattices \((L>50)\).
5. Molecular properties (nuclear energy gradients, response properties \dots) difficult to calculate.

Q: How can one determine which orbitals are strongly entangled, i.e., how should one build the lattice?
A: Entanglement entropy from the eigenvalues \(\omega_j\) of the reduced density matrix.
\[ s_i(1)=-\sum_j \omega_j \ln \left(\omega_j\right) \]
Single-orbital entanglement \(s_i(1)\) (system = orbitals, \(i\)).
\[ \rho_{\sigma_i, \sigma_i^{\prime}}^{[i]}=\sum_{\boldsymbol{\sigma}_e}\left\langle\boldsymbol{\sigma}_e \sigma_i \mid \Psi\right\rangle\left\langle\Psi \mid \boldsymbol{\sigma}_e \sigma_i^{\prime}\right\rangle \]
Two-orbital entanglement \(s_{i j}(2)\) (system = pair of two orbitals, \(i\) and \(j\)).
\[ \rho_{\sigma_i \sigma_j, \sigma_i^{\prime} \sigma_j^{\prime}}^{[i, j]}=\sum_{\boldsymbol{\sigma}_e}\left\langle\boldsymbol{\sigma}_e \sigma_i \sigma_j \mid \Psi\right\rangle\left\langle\Psi \mid \boldsymbol{\sigma}_e \sigma_i^{\prime} \sigma_j^{\prime}\right\rangle \]
Mutual information \(I_{i j}\), isolates the "pure" two-particle correlation.
\[ I_{i j}=\frac{1}{2}\left(s_i(1)+s_j(1)-s_{i j}(2)\right) \]
\(I_{ij}\) as a metric for identifying correlation effects.
Entanglement diagram: graphical representation of \(s_i(1)\) and \(I_{i j}\).
Can guide the selection of an active space.
Can be used to calculate the optimal DMRG sorting (\(\rightarrow\)~Fiedler ordering).
Qualitatively correct entanglement obtained with partially converged (i.e. small \(m\)) DMRG calculation.
``Quick-and-dirty'' DMRG calculation sufficient for generating reliable correlation metrics.