A variational eigenvalue solver on a photonic quantum processor

Abstract

Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the physical dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state preparation based on ansätze and classical optimization. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. We experimentally demonstrate the feasibility of this approach with an example from quantum chemistry--calculating the ground-state molecular energy for He-H(+). The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future.

Figure 1. Architecture of the quantum-variational eigensolver.

In QEE, quantum states that have been previously prepared are fed into the quantum modules, which compute ‹_i›, where _i is any given term in the sum defining . The results are passed to the CPU, which computes ‹›. In the quantum variational eigensolver, the classical minimization algorithm, run on the CPU, takes ‹› and determines the new state parameters, which are then fed back to the QPU.

Figure 2. Experimental implementation of our scheme.

(a) Quantum-state preparation and measurement of the expectation values ‹ψ|σ_i⊗σ_j|ψ› are performed using a quantum photonic chip. Photon pairs, generated using spontaneous parametric downconversion, are injected into the waveguides encoding the |00› state. The state |ψ› is prepared using thermal phase shifters φ₁₋₈ (orange rectangles) and one CNOT gate and measured using photon detectors. dc_{{1–4,9–13}} (dc_5–7) are 50% (30%) reflectivity directional couplers. Coincidence count rates from the detectors D_1–4 are passed to the CPU running the optimization algorithm. This computes the set of parameters for the next state and writes them to the quantum device. (b) A photograph of the QPU.

Figure 3. Finding the ground state of He–H⁺ for a specific molecular separation R=90 pm.

(a) Experimentally computed energy ‹› (coloured dots) as a function of the optimization step j. The colour represents the tangle (degree of entanglement) of the physical state, estimated directly from the state parameters . The red lines indicate the energy levels of (R). The optimization algorithm clearly converges to the ground state of the molecule, which has small but non-zero tangle. The crosses show the energy calculated at each experimental step, assuming an ideal quantum device. (b) Overlap |‹ψ^j|ψ^G› between the experimentally computed state |ψ^j› at each optimization step j and the theoretical ground state of , |ψ^G›. Error bars are smaller than the data points. Further details are provided in the Methods section, Supplementary Table 1 and Supplementary Methods.

Figure 4. Bond dissociation curve of the He–H⁺ molecule.

This curve is obtained by repeated computation of the ground-state energy (as shown in Fig. 3) for several (R) values. The magnified plot shows that after correction for the measured systematic error the data overlap with the theoretical energy curve, and, importantly, we can resolve the molecular separation of minimal energy. Error bars show the standard deviation of the computed energy, as described in the Methods section.