Using Machine Learning to Calculate Quantum Wavefunctions

Liam Bernheimer

Classical mechanics

Hamilton's equations
$\frac{d\mathbf{q}_{i}}{dt}=\frac{\partial H}{\partial\mathbf{p}_{i}},\frac{d\mathbf{p}_{i}}{dt}=-\frac{\partial H}{\partial\mathbf{q}_{i}}$
Classical mechanics

Hamilton's equations
$\frac{d\mathbf{q}_{i}}{dt}=\frac{\partial H}{\partial\mathbf{p}_{i}},\frac{d\mathbf{p}_{i}}{dt}=-\frac{\partial H}{\partial\mathbf{q}_{i}}$
Classical mechanics

Hamilton's equations
$\frac{d\mathbf{q}_{i}}{dt}=\frac{\partial H}{\partial\mathbf{p}_{i}},\frac{d\mathbf{p}_{i}}{dt}=-\frac{\partial H}{\partial\mathbf{q}_{i}}$
Quantum mechanics

The Schrödinger equation
$i\hbar\frac{d}{dt}\left|\psi\left(t\right)\right\rangle =\hat{H}\left|\psi\left(t\right)\right\rangle$

Spin chain

Classical mechanics

$N$-dimensional phase space

  • Linear scaling with system size

Quantum mechanics

\( 2^N \)-dimensional Hilbert space

  • Exponential scaling with system size

Spin chain

$N=1$

Classical mechanics

Quantum mechanics

Spin chain

$N=2$

Classical mechanics

Quantum mechanics

Spin chain

$N=3$

Classical mechanics

Quantum mechanics

Spin chain

Classical mechanics

$N$-dimensional phase space

  • Linear scaling with system size

Quantum mechanics

\( 2^N \)-dimensional Hilbert space

  • Exponential scaling with system size

Spin chain

Classical mechanics

$N$-dimensional phase space

  • Linear scaling with system size

Quantum mechanics

\( 2^N \)-dimensional Hilbert space

  • Exponential scaling with system size
  • Big data - AI!

Neural networks can learn high dimensional space from samples

  • $256^{3}=16,777,216$ colors
  • $256^{2}=65,536$ pixels

$16,777,216^{65,536}=$ astronomical number

Neural networks can learn high dimensional space from samples

Works for wavefunctions too!

Ground state wavefunctions


At room temperature, most molecules are found in their ground state wavefunction, which has the lowest energy. This makes the ground state wavefunction a key focus in quantum chemistry.



$\frac{3}{2}k_{B}T\overset{\underbrace{T=298K}}{=}0.04\text{eV}\ll10.2\text{eV}$
$\left(\text{Excitation energy of H atom}\right)$

Finding the ground state wavefunction

Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.

Finding the ground state wavefunction

Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.

Finding the ground state wavefunction

Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.

Finding the ground state wavefunction

Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.

Finding the ground state wavefunction

Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.

Finding the ground state wavefunction

Fermions

Fermions exhibit antisymmetry under exchange, meaning their wavefunction changes sign when swapped.


$\psi\left(x_{1},x_{2}\right)=-\psi\left(x_{2},x_{1}\right)$

$\frac{1}{\sqrt{2!}}\left|\begin{array}{cc} \chi_{1}\left(x_{1}\right) & \chi_{2}\left(x_{1}\right)\\ \chi_{1}\left(x_{2}\right) & \chi_{2}\left(x_{2}\right) \end{array}\right|$
Complexity of $\mathscr{O}\left(N^{3}\right)$

Fermions

Symmetrization by sorting!

Store only the wavefunction where
$x_{1}\leq x_{2}$
Complexity of $\mathscr{O}\left(N\log N\right)$

Fermions

Symmetrization by sorting!

Store only the wavefunction where
$x_{1}\leq x_{2}$
Complexity of $\mathscr{O}\left(N\log N\right)$

Wigner crystallization

Thank You

- Bernheimer, L., Atanasova, H. & Cohen, G. Reports on Progress in Physics vol. 87 118001 (2024).
- Atanasova, H., Bernheimer, L. & Cohen, G. Nature Communications vol. 14 (2023).