System Types

Up until this point, we have only considered quantum mechanical systems composed of "spinless fermions". HQS Quantum Solver, however, provides further types of quantum systems. Each of these systems is implemented in its own module, and all these modules are structured in a similar way, meaning that they all provide a VectorSpace, an Operator, and a set of functions providing operator terms.

Spins

The spins module implements systems consisting of $M$ spin-$\frac{1}{2}$ particles. The VectorSpace is spanned by vectors of the form $$\underset{j=0}{\overset{M-1}{⨂}}|{\sigma }_j⟩\text{where}{\sigma }_j\in \{\uparrow ,\downarrow \}.$$ As a shorthand, we usually write, e.g., $$|\uparrow ⟩\otimes |\downarrow ⟩\otimes |\downarrow ⟩\otimes |\uparrow ⟩=|\uparrow \downarrow \downarrow \uparrow ⟩.$$

With this module, e.g., the quantum Heisenberg model, $$H=-J\sum _{(j,k)\in \mathcal{N}}{\vec{S}}_j\cdot {\vec{S}}_k,$$ can be simulated. See the spin_waves.ipynb example for details.

Spinless Fermions

The spinless_fermions module implements systems of fermions having a site index but no (dedicated) spin index. The VectorSpace is spanned by vectors of the form $$\prod _{j=0}^{M-1}({\hat{c}}_j^{†}{)}^{n_j}|0⟩\text{where}{n}_j\in \{0,1\},$$ and $c_j^{†}$ is the creation operator for site $j$.

With this module, e.g., the tight-binding model, $$H=-t\sum _{(j,k)\in \mathcal{N}}{\hat{c}}_j^{†}{\hat{c}}_k,$$ can be simulated, as was shown in the "Getting Started" section.

Spinful Fermions

The spinful_fermions module implements system of fermions having a site index and a spin index. The VectorSpace is spanned by vectors of the form $$\prod _{\sigma \in \{\uparrow ,\downarrow \}}\prod _{j=0}^{M-1}({\hat{c}}_{j\sigma }^{†}{)}^{n_{j\sigma }}|0⟩\text{where}{n}_{j\sigma }\in \{0,1\},$$ and $c_{j\sigma }^{†}$ is the creation operator for site $j$ and spin polarization $\sigma$.

With this module, e.g., the Hubbard model, $$H=-t\sum _{\sigma \in \{\uparrow ,\downarrow \}}\sum _{(j,k)\in \mathcal{N}}{\hat{c}}_{j\sigma }^{†}{\hat{c}}_{k\sigma }+U\sum _j{\hat{n}}_{j\uparrow }{\hat{n}}_{j\downarrow },$$ can be simulated. See the spinful_fermions_groundstate.ipynb example for details.

Bosons

The bosons module implements systems of bosons. The VectorSpace is spanned by vectors of the form $$\prod _{j=0}^{M-1}\sqrt{n_j!}({\hat{b}}_j^{†}{)}^{n_j}|0⟩\text{where}{n}_j\in \{0,1,\dots ,N_j\},$$ and $b_j^{†}$ is the creation operator for site $j$. The variable $N_j$ is the maximal occupation of site $j$.

With this module, e.g., the Bose-Hubbard model, $$H=-t\sum _{(i,j)\in \mathcal{N}}\left({\hat{b}}_j^{†}{\hat{b}}_k+{\hat{b}}_k^{†}{\hat{b}}_j\right)+\frac{U}{2}\sum _j{\hat{n}}_j({\hat{n}}_j-1),$$ can be simulated. See the bose_hubbard_groundstate.ipynb example for details.