A variational eigenvalue solver on a quantum processor

TL;DR

An alternative approach that greatly reduces the requirements for coherent evolution is presented and is combined with a new approach to state preparation based on ansätze and classical optimization, which drastically reduces the coherence time requirements.

Abstract

Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the dimension of the problem space grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm can efficiently find the eigenvalue of a given eigenvector but requires fully coherent evolution. We present an alternative approach that greatly reduces the requirements for coherent evolution and we combine this method with a new approach to state preparation based on ansätze and classical optimization. We have implemented the algorithm by combining a small-scale 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+, to within chemical accuracy. The proposed approach, by drastically reducing the coherence time requirements, enhances the potential of the quantum resources available today and in the near future.

Figures (4)

• Figure 1: Architecture of the quantum-variational eigensolver. Algorithm 1: Quantum states that have been previously prepared, are fed into the quantum modules which compute $\mathinner{\langle{{\cal H}_i}\rangle}$, where ${\cal H}_i$ is any given term in the sum defining ${\cal H}$. The results are passed to the CPU which computes $\mathinner{\langle{{\cal H}}\rangle}$. Algorithm 2: The classical minimization algorithm, run on the CPU, takes $\mathinner{\langle{{\cal H}}\rangle}$ 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 $\langle \psi | \sigma_i \otimes \sigma_j | \psi \rangle$ are performed using a quantum photonic chip. Photon pairs, generated using spontaneous parametric down-conversion are injected into the waveguides encoding the $\mathinner{|{00}\rangle}$ state. The state $\mathinner{|{\psi}\rangle}$ is prepared using thermal phase shifters $\phi_{1-8}$ (orange rectangles) and one CNOT gate and measured using photon detectors. Coincidence count rates from the detectors $\hbox{D}_{1-4}$ 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 $\hbox{He-H}^+$ for a specific molecular separation, $R=90$ pm. (a) Experimentally computed energy $\mathinner{\langle{{\cal H}}\rangle}$ (colored dots) as a function of the optimization step $j$. The color represents the tangle (degree of entanglement) of the physical state, estimated directly from the state parameters $\{\phi_i^j\}$. The red lines indicate the energy levels of ${\cal H}(R)$. 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 $|\mathinner{\langle{\psi^{j} | \psi^{G}}\rangle}|$ between the experimentally computed state $\mathinner{|{\psi^{j}}\rangle}$ at each the optimization step $j$ and the theoretical ground state of ${\cal H}$, $\mathinner{|{\psi^G}\rangle}$. Further details are provided in the Appendix.
• Figure 4: Bond dissociation curve of the $\hbox{He-H}^+$ molecule. This curve is obtained by repeated computation of the ground state energy (as shown in Fig. \ref{['optimization']}) for several ${\cal H}(R)$. 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.