Cascaded variational quantum eigensolver algorithm

We present a cascaded variational quantum eigensolver algorithm that only requires the execution of a set of quantum circuits once rather than at every iteration during the parameter optimization process, thereby increasing the computational throughput. This algorithm uses a quantum processing unit to probe the needed probability mass functions and a classical processing unit perform the remaining calculations, including the energy minimization. The ansatz form does not restrict the Fock space and provides full control over the trial state, including the implementation of symmetry and other physically motivated constraints.

Figure 1

Schematic of an implementation of the cascaded variational quantum eigensolver algorithm. The QPU executes a set of quantum circuits, each generating a unique quantum state $\widehat{R}\widehat{U}|0\rangle$ that when measured yield a family of occupation numbers $(n_1,n_2,\cdots ,n_Q)$ recorded as $n_s$. Repeating the same measurements multiple times for different $\widehat{R}$ produces collections of families $\left(n_s^{\widehat{R}}\right)$ that are passed on as input to the CPU. The CPU uses these samples together with a parameter vector ${\mathit{\theta }}_k$ to compute derivatives of the energy $E\left(\mathit{\theta }\right)$ of the trial state $|\mathrm{\Psi }\left(\mathit{\theta }\right)\rangle$ at $\mathit{\theta }={\mathit{\theta }}_k$ by obtaining sample means for $\mathrm{{\rm Y}}\left({\mathit{\theta }}_k\right), \mathrm{\Lambda }\left({\mathit{\theta }}_k\right), \mathbf{\nabla }\mathrm{{\rm Y}}\left({\mathit{\theta }}_k\right)$, and $\mathbf{\nabla }\mathrm{\Lambda }\left({\mathit{\theta }}_k\right)$ in Eqs. (14) and (21). These derivatives are then used to generate a new parameter vector ${\mathit{\theta }}_{k+1}$, using some optimization method $f[\mathbf{\nabla }E\left({\mathit{\theta }}_k\right),\cdots ]$, and the process is repeated until the optimization has been completed and the sought minimum energy obtained.

Figure 2

The energy $E(\phi ,\varphi )$ of the singlet two-electron trial state $|\mathrm{\Psi }(\phi ,\varphi )\rangle$ of the two-site Hubbard model with the Hamiltonian $\widehat{H}=t{\sum }_{\sigma }(c_{0\sigma }^{†}{c}_{1\sigma }^{\phantom{(}}+\text{H.c.})+U{\sum }_i{c}_{i\uparrow }^{†}{c}_{i\downarrow }^{†}{c}_{i\downarrow }^{\phantom{(}}{c}_{i\uparrow }^{\phantom{(}}$, where $t$ and $U$ are coefficients and $i\in \{0,1\}$ and $\sigma \in \{\uparrow ,\downarrow \}$ are site and spin indices, respectively, so that all index pairs are elements in the one-electron index set $(0\uparrow ,0\downarrow ,1\uparrow ,1\downarrow )$. The parameter space has been restricted by letting ${\lambda }_n\to i\infty$ for all families $n$ of occupation numbers, except for those associated with two electrons with a zero $z$ component of the total spin. Permutation symmetry requires that singlet states transform as $A_{1g}$ in the point group $D_{\infty h}$, which means that $|\mathrm{\Psi }(\phi ,\varphi )\rangle$ is a linear combination of $|{\mathrm{\Psi }}_1\rangle =(|0110\rangle +|1001\rangle )/\sqrt{2}$ and $|{\mathrm{\Psi }}_2\rangle =(|0011\rangle +|1100\rangle )/\sqrt{2}$. This requirement is imposed by the symmetry constraints ${\lambda }_{0110}={\lambda }_{1001}=\lambda$ and ${\lambda }_{0011}={\lambda }_{1100}=-\lambda$. The parametric equation $\lambda =\lambda (\phi ,\varphi )=\varphi /2-ilntan(\frac{\pi }{4}+\frac{\phi }{2})/2$ is defined on the parameter space $\left\{\right(\phi ,\varphi ):\phi \in (-\pi /2,\pi /2),\varphi \in (-\pi ,\pi \left]\right\}$, where $\phi$ and $\varphi$ represent the latitude and longitude, respectively, on a sphere. The energy $E(\phi ,\varphi )$ for $t/U=-0.158$ is shown in color in (a) with the colorbar in (b). The red curves trace the gradient-descent path along $\varphi =0$ from the initial guiding state $|{\mathrm{\Psi }}_0\rangle =(|{\mathrm{\Psi }}_1\rangle +|{\mathrm{\Psi }}_2\rangle )/\sqrt{2}$ to the ground state at $\phi \approx 1$ rad.