The theory of variational hybrid quantum-classical algorithms

TL;DR

This work addresses the challenge of performing meaningful eigenvalue and optimization tasks on near-term quantum hardware by strengthening the theory and practicality of the variational quantum eigensolver (VQE). It introduces variational adiabatic and unitary coupled cluster (UCC) ans"atze, elucidates a connection from second-order UCC to universal gate sets via relaxed exponential splitting, and proposes quantum error suppression to harness pre-threshold devices. The paper also advances Hamiltonian averaging with Bayesian and frequentist perspectives, proposes term truncation and correlated sampling to reduce measurement cost, and demonstrates that derivative-free optimization methods can dramatically lower the number of energy evaluations required. Together, these contributions push VQE toward robust, scalable performance on imperfect quantum devices and broaden its applicability to general observables beyond energy. Overall, the framework provides practical strategies for achieving quantum advantage in hybrid quantum-classical computations and informs future implementation on pre-threshold hardware.

Abstract

Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as "the quantum variational eigensolver" was developed with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through relaxation of exponential splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.

Figures (6)

• Figure 1: The ground and first excited state eigenvalues of the schedule Hamiltonian $H(s)$ as a function of the annealing path $A(s)$. This shows the avoided crossing that occurs at $A(s)=1/2$, the size of which is controlled by the perturbation parameters $\epsilon$ in the Hamiltonian, which in our example is set to a value of $\epsilon=0.1$.
• Figure 2: A comparison of the standard linear path $A(s)$ versus the two-parameter split path that is variationally optimal with respect to the expectation value of the Hamiltonian at the final point $H(1)$. The path naturally slows the evolution near the location of the avoided crossing, but is otherwise only slightly distorted from a standard linear path.
• Figure 3: The squared overlap of the system state $\mathinner{|{\Psi(s)}\rangle}$ at parameter value $s$ with the ground state at $H(1)$, $\mathinner{|{\Psi_f}\rangle}$ is show for both the standard Linear (Lin) schedule as well as the variationally optimal spline schedule for different total evolution times $\tau$. It can be seen here that the performance of the variational schedule offers similar performance to a linear schedule roughly $10$ times as long, indicating an order of magnitude reduction in the quantum evolution time required for the variationally optimal schedule.
• Figure 4: A cartoon depicting the concept of variationally suppressible errors on energy contours. Dotted lines represent errors that move the state away from the variational minimum, and solid lines characterize a shift of the ansatz parameters that can return the state to the minimum. In this case the vertical axis is within the manifold of the ansatz parameters, while the horizontal axis is not, as indicated by the cross in the line returning along that axis. However by adding additional operators, represented by the diagonal dashed line, it becomes possible to suppress these errors variationally.
• Figure 5: The accuracy of the final energy of the optimized wavefunction at convergence compared to the known exact solution, as a function of the precision in the function value in the optimizer for different methods ($\epsilon$). The values are averaged over 20 repetitions and the error bars indicate 1 standard deviation of the measured data. The TOMLAB methods provide dramatically superior performance at essentially all levels of measurement precision above $\epsilon=.1$.