Witnessing eigenstates for quantum simulation of Hamiltonian spectra

Abstract

The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. We introduce the concept of an "eigenstate witness" and, through it, provide a new quantum approach that combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled unitary operations, and projective measurements. Both ground and excited states are experimentally found with fidelities >99%, and their eigenvalues are estimated with 32 bits of precision. We also investigate and discuss the scalability of the approach and study its performance through numerical simulations of more complex Hamiltonians. This result shows promising progress toward quantum chemistry on quantum computers.

Fig. 1. The WAVES protocol.

(A) Flowchart describing the protocol. The optimization of ${\mathcal{F}}_{\text{obj}}(\overrightarrow{\mathrm{\theta }})=\mathcal{E}+\mathit{T}\mathcal{S}$ using the circuit in (B) allows one to variationally find the ground state of the Hamiltonian, preparing trial states via the ansatz $\widehat{\mathit{A}}(\overrightarrow{\mathrm{\theta }})$ with no perturbation $({\widehat{\mathit{E}}}_{{\mathit{p}}_0}=\widehat{\mathit{I}})$. An initial guess for an excited state is given by a perturbation ${\widehat{\mathit{E}}}_{{\mathit{p}}_{\mathit{i}}}$ on the ground state and then refined using the same circuit by exploiting the eigenstate witness ${\mathcal{F}}_{\text{obj}}(\overrightarrow{\mathrm{\theta }})=\mathcal{S}(\text{high}‐\mathit{T} \text{limit})$. (C) For each target eigenstate found, the eigenvalues are precisely estimated via the IPEA using the quantum logic circuit, where H is the Hadamard gate. The color coding in (B) and (C), blue for the control and red for the target, refers to the difference in wavelength between the photon in the control qubit and the one in the target register in our experimental implementation. (D) Diagram schematically representing the intuition behind the proposed approach, where initial guesses of excited states are variationally refined using the witness and IPEA returns the eigenvalues.

Fig. 2. Silicon quantum photonic processor.

The quantum device enables one to produce maximally path-entangled photon states, perform arbitrary single-qubit state preparation and projective measurements, and, more importantly, perform any CÛ operation in the two-dimensional space. Photons are guided in the silicon waveguides and controlled by thermo-optical phase shifters. Photon pairs are directly generated inside the silicon spiral sources through SFWM, off-chip–filtered and postselected by AWG filters (not shown), and measured by SNSPDs. The generated signal (blue) and idler (red) photons are different in wavelength and form the control and target qubits, respectively. The quantum chip is interfaced with a classical computer. Inset: High-visibility quantum (blue) and classical (green) interference fringes obtained in the device using the photon sources part and configuring the top final interferometer. The high visibility is essential to verify the high-performance and correct characterization of the device.

Fig. 3. Experimental results.

A Hamiltonian representing a single-exciton transfer between two chlorophyll units is implemented on the silicon quantum photonic device for an experimental test of the protocol. (A and B) Color-coded evolution of the particle swarm for the WAVES search of the ground state (| − 〉) and excited state (| + 〉) shown on the Bloch spheres. Different colors correspond to different steps of the search protocol. For the ground and the excited state searches, we report the evolution of $\mathcal{F}$_obj in (C) and (D) and the fidelity (F = |〈Ψ|Ψ_ideal〉|²) versus search steps in (E) and (F), converging to a final value of 99.48 ± 0.28% and 99.95 ± 0.05%, respectively. Error bars are given by the variance of the particle distribution and photon Poissonian noise. Dashed lines are numerical simulations of the performance of the algorithm, averaged over 1000 runs, with shaded areas representing a 67.5% confidence interval. Insets: Behavior close to convergence. (G and H) Normalized photon coincidences used to calculate the 32 IPEA-estimated bits of the eigenphase for both eigenstates. The theoretical bit value is shown above each bar. Errors arising from Poissonian noise are shown as shaded areas on the bars.

Fig. 4. Numerical simulations for higher-dimensional Hamiltonians.

The cases studied refer to molecular hydrogen systems $({\mathrm{H}}_2,{\mathrm{H}}_3^{+},{\mathrm{H}}_3,{\mathrm{H}}_4)$ with the full PH ansatz. (A) Variational search for the ground state of each physical system. (B) Variational search for the targeted subspace of degenerate excited states with an initial excitation perturbation ${\widehat{\mathit{E}}}_{{\mathit{p}}_{\mathit{i}}}$. On the x axis, we refer to the cumulative number of trial states probed (that is, the number of particles in the swarm times the variational steps). For ease of comparison, the x-axis origin has been shifted in (A) for the various cases to have equivalent fidelity for the average initial guess. Dashed lines denote average fidelities, with the shaded areas indicating a 67.5% confidence interval. The average fidelities achieved by the particle swarm optimization for both ground and excited states are calculated for 100 independent runs of WAVES. In all simulations, a binomial noise model has been taken into account when performing projective measurements. Insets: Bar charts summarizing final fidelities obtained by each search. All the simulations converged to the same high fidelity within errors, as indicated by the dashed black line in the inset.