Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states

Using quantum devices supported by classical computational resources is a promising approach to quantum-enabled computation. One powerful example of such a hybrid quantum-classical approach optimized for classically intractable eigenvalue problems is the variational quantum eigensolver, built to utilize quantum resources for the solution of eigenvalue problems and optimizations with minimal coherence time requirements by leveraging classical computational resources. These algorithms have been placed as leaders among the candidates for the first to achieve supremacy over classical computation. Here, we provide evidence for the conjecture that variational approaches can automatically suppress even nonsystematic decoherence errors by introducing an exactly solvable channel model of variational state preparation. Moreover, we develop a more general hierarchy of measurement and classical computation that allows one to obtain increasingly accurate solutions by leveraging additional measurements and classical resources. We demonstrate numerically on a sample electronic system that this method both allows for the accurate determination of excited electronic states as well as reduces the impact of decoherence, without using any additional quantum coherence time or formal error-correction codes.

Figure 1

A cartoon schematic of the quantum subspace expansion proposed in this work. One prepares a quantum state $|\mathrm{\Psi }\rangle$ that is passed through a quantum channel. At the exit of the channel, partial tomography of the state is used to expand in a linear subspace around the resulting quantum state. This subspace is used to determine both the ground and excited states of a quantum Hamiltonian of interest while also potentially correcting for errors caused by the quantum channel.

Figure 2

The fidelity of the initial and final states as a function of internuclear distance ($R$) in ${\mathrm{H}}_2$ after being passed through three different quantum channels (Ph: dephasing, $AP$: amplitude and phasing damping; Depol: depolarizing) improves under variation in the presence of the channel (VCS solution, solid lines with markers) when compared to the exact less ground state as input (ground state of untransformed Hamiltonian, markers only). The dotted lines represent the expected value of $S^2$ for the same color line under the ideal solution with a channel (solid line). From this, one can see the kinks correspond precisely to discontinuous changes in the spin symmetry. The discontinuities are indicators of locating a decoherence-resistant state that is symmetry broken with respect to the exact, ideal solution. The channels are characterized by an experiment time $T_p$ relative to coherence time parameters $T_1=T_2=T_{\text{Depol}}$ time such that $T_p/T_1=0.05$.

Figure 3

The exact solution of the VCS model for the ground state of ${\mathrm{H}}_2$ is shown for a number of different quantum channels including amplitude and phase damping ($AP$), dephasing only (Ph), and depolarizing noise (Dep). These results are shown alongside both the exact (Ex) ground state, an antisymmetric product state approximation (RHF), and the best solution of a dephasing channel constrained to have correct spin $(S^2=0)$. It is seen that dephasing noise is sufficient to destroy the entanglement required to describe the dissociated limit, in that the solution without symmetry constraints obtains an accurate energy, but only by breaking spin symmetry in an unphysical way, as evidenced by the difference when compared to the optimal dephasing solution under symmetry constraints. Other types of noise raise the energy of the whole curve due to number symmetry breaking. The kink in the curves without symmetry enforcement results from a spin symmetry breaking in the variationally optimal solution in the presence of decohering noise.

Figure 4

A cartoon schematic of the basis hierarchy obtained from expanding about the VQE solution reference. At $k=1$ one has the linear response (LR) subspace and at $k=2$ one has the the quadratic response (QR) space continuing to $k=N_e$, where one spans the entire subspace corresponding to the particle number of the reference state.

Figure 5

The energy of different electronic states as a function of internuclear separation for the ${\mathrm{H}}_2$ molecule in a minimal STO-6G basis, encoded into qubits by the Jordan-Wigner mapping. Both the entire exact spectrum is shown with dotted-dashed lines, as well as the exact spectrum restricted to a neutral molecule $(N_e=2)$. The blue dotted lines depict only ionic $N_e\ne 2$ states that are not the focus in this study. This depicts the volume of states in a naive mapping that must be avoided to capture physical solutions. In this case, the linear response (LR) approximation from an exact reference is sufficient to capture exactly the excited states with the correct number symmetry, as the excitation operators conserve the number from the exact reference state.

Figure 6

In examining the energy as a function of separation for the ground state of ${\mathrm{H}}_2$ under a particular amplitude and phase damping channel ($AP$), one sees that in the linear response subspace ($AP$ LR), the solution quality in increased with respect to the optimal solution of the quantum channel model ($AP$). The qualitative kink in the approximate solution can be repaired by enforcing the correct spin symmetry $(S^2=0)$. This also demonstrates the effect on the energy between variational minimization with the channel ($AP$) and without (No Var).

Figure 7

The ground and first two excited states of ${\mathrm{H}}_2$ are plotted as a function of internuclear separation using both the zero approximation (ZA) and the zero approximation under the commutator construction (ZC). Both methods require only the original measurements used for the ground state to approximate the excited states, and we see here that ZC achieves an extremely high accuracy, while ZA is qualitatively correct in some cases but produces subvariational solutions in others.