Self-learning Emulators and Eigenvector Continuation

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I Supplemental Material

i.1 Geometrical picture of eigenvector continuation error

We will present a geometrical picture of eigenvector continuation (EC) error as well as some additional insight into the error estimate that appears in Eq. (3) of the main text. We consider a Hamiltonian manifold $H (c)$ that depends on the control parameters $c$ . We write $| v (c) ⟩$ for the eigenvector of interest and $E (c)$ for the corresponding energy eigenvalue. Suppose we know the eigenvectors at $M$ different training points, ${c^{(1)}, \dots, c^{(M)}}$ . We label the set of $M$ training eigenvectors as $S_{M} = {| v (c^{(1)}) ⟩, \dots, | v (c^{(M)}) ⟩}$ . Let us define the norm matrix $N (S_{M})$ as

(S1)

and let $Ω^{2} (S_{M})$ be the determinant of $N (S_{M})$ . Then $Ω^{2} (S_{M})$ corresponds to the square of the volume of the $M$

-dimensional parallelopiped defined by the vectors in the set

S_{M}

. If all the eigenvectors are normalized, then the maximum possible volume is 1, which is attained when all the eigenvectors are orthogonal.

Let us now consider selecting the next training point, $c_{M + 1}$ . Let $P$ be the projection operator onto the linear span of $S_{M}$ , and let $Q$ be the orthogonal complement so that $Q = 1 - P$ . Suppose we now expand our training set $S_{M}$ by adding another training vector $| v (c) ⟩$ to form $S_{M + 1}$ . Let us define the perpendicular projection vector $| v_{⊥} (c) ⟩$ as

| v_{⊥} (c) ⟩ = Q | v (c) ⟩ .

(S2)

Since $Ω^{2} (S_{M})$ is the squared volume of the parallelopiped defined by the vectors in $S_{M}$ and $Ω^{2} (S_{M + 1})$ is the squared volume of the parallelopiped defined by the vectors in $S_{M + 1}$ , it follows that the ratio $Ω^{2} (S_{M + 1})$ to $Ω^{2} (S_{M})$ is given by the squared norm of $| v_{⊥} (c) ⟩$ ,

(S3)

Let us define the projections of $H$ onto $P$ and $Q$ subspaces as

H^{P} (c) = P H (c) P, H^{Q} (c) = Q H (c) Q .

(S4)

The EC approximation is nothing more than the approximation of $| v (c) ⟩$ by some eigenvector of $H^{P} (c)$ , which we denote as $| v^{P} (c) ⟩$ . Let the corresponding energy be labelled $E^{P} (c)$ so that

H^{P} (c) | v^{P} (c) ⟩ = E^{P} (c) | v^{P} (c) ⟩ .

(S5)

We also label the eigenvectors of $H^{Q} (c)$ contained in the orthogonal complement $Q$ as,

H^{Q} (c) | v_{j}^{Q} (c) ⟩ = E^{Q} (c) | v_{j}^{Q} (c) ⟩ .

(S6)

When the difference between the exact eigenvector and the eigenvector continuation approximation of the eigenvector is small, we can use first order perturbation theory to write

(S7)

To first order in perturbation theory, the residual vector is just $| v_{⊥} (c) ⟩ \approx | v (c) ⟩ - | v^{P} (c) ⟩$ . We therefore have

(S8)

If we now combine with Eq. (S3), we get

(S9)

We can now connect this result with the error or loss function in the main text. The second part of the equation gives an expression for the error term $∥ | v_{⊥} (c) ⟩ ∥$ using first-order perturbation theory, and the first part of the equation is a geometrical interpretation of the error term as the ratio of the squared volumes, $Ω^{2} (S_{M + 1})$ to $Ω^{2} (S_{M})$ . Taking the logarithm of the square root, we get

(S10)

The term in the numerator,

(S11)

will go to zero at each of the training points, causing large variations in the logarithm of the error as we add more and more training points. In contrast, the term in the denominator, $[E^{P} (c) - E_{j}^{Q} (c)]^{2}$ , will be smooth as a function of $c$ . Similarly, will also be a smooth function of $c$ . We can write

(S12)

where $A$ is a constant and $B (c)$ averages to zero over the entire domain of $c$ . While the function $B (c)$ is unknown, it will be dominated by the large variations in the logarithm of the error as more and more training points are added. We note that

	$\sum j$
				(S13)

We therefore arrive at the variance error estimate used in the main text,

(S14)

i.2 Model 2 Hamiltonian

Model 2 describes four-distinguishable particles with equal masses $m$ on a three-dimensional lattice with pairwise point interactions with coefficients $c_{i j}$ for each pair $i < j$ . We use lattice units where physical quantities are multiplied by powers of the spatial lattice spacing to make the combinations dimensionless. We take the common mass $m$ to equal $1$ in lattice units. We let $n$ denote the spatial lattice points on our three dimensional $L^{3}$ periodic lattice. Let the lattice annihilation and creation operators for particle $i$ be written as $a_{i} (n)$ and $a_{i}^{†} (n)$ respectively. The free non-relativistic lattice Hamiltonian has the form

H_{free} = \frac{3}{m} \sum i = 1, 2, 3, 4 \sum n a_{i}^{†} (n) a_{i} (n) -

\frac{1}{2 m} \sum i = 1, 2, 3, 4 \sum^l =^1,^2,^3 \sum n a_{i}^{†} (n) [a_{i} (n +^l) + a_{i} (n -^l)] .

(S15)

We add to the free Hamiltonian single-site contact interactions, and the resulting Hamiltonian then has the form

H = H_{free} + \sum i < j \sum n c_{i j} ρ_{i} (n) ρ_{j} (n),

(S16)

where $ρ_{i} (n)$ is the density operator for particle $i$ ,

ρ_{i} (n)

= a_{i}^{†} (n) a_{i} (n) .

(S17)

Self-learning Emulators and Eigenvector Continuation

Singular algebraic equations with empirical data

Secant acceleration of sequential residual methods for solving large-scale nonlinear systems of equations

Gaussian Process Assisted Active Learning of Physical Laws

Polynomial spline collocation method for solving weakly regular Volterra integral equations of the first kind

Exact splitting methods for kinetic and Schrödinger equations

The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

Learning from the Kernel and the Range Space

References

I Supplemental Material

i.1 Geometrical picture of eigenvector continuation error

i.2 Model 2 Hamiltonian

Self-learning Emulators and Eigenvector Continuation

Related Research

Singular algebraic equations with empirical data

Secant acceleration of sequential residual methods for solving large-scale nonlinear systems of equations

Gaussian Process Assisted Active Learning of Physical Laws

Polynomial spline collocation method for solving weakly regular Volterra integral equations of the first kind

Exact splitting methods for kinetic and Schrödinger equations

The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

Learning from the Kernel and the Range Space

References

I Supplemental Material

i.1 Geometrical picture of eigenvector continuation error

i.2 Model 2 Hamiltonian