Variational quantum algorithms : a noisy landscape

Vojtěch Novák, Ivan Zelinka, Václav Snåšel

Lavinia Stiliadou, Johanna Barzen, Frank Leymann +2 more

Theunissen, Thom M.

Pinaki Sen, Amandeep Singh Bhatia, Kamalpreet Singh Bhangu +1 more

In quantum computing, the variational quantum algorithms (VQAs) are well suited for finding optimal combinations of things in specific applications ranging from chemistry all the way to finance. The training of VQAs with gradient descent optimization algorithm has shown a good convergence. At an early stage, the simulation of variational quantum circuits on noisy intermediate-scale quantum (NISQ) devices suffers from noisy outputs. Just like classical deep learning, it also suffers from vanishing gradient problems. It is a realistic goal to study the topology of loss landscape, to visualize the curvature information and trainability of these circuits in the existence of vanishing gradients. In this paper, we calculate the Hessian and visualize the loss landscape of variational quantum classifiers at different points in parameter space. The curvature information of variational quantum classifiers (VQC) is interpreted and the loss function's convergence is shown. It helps us better understand the behavior of variational quantum circuits to tackle optimization problems efficiently. We investigated the variational quantum classifiers via Hessian on quantum computers, starting with a simple 4-bit parity problem to gain insight into the practical behavior of Hessian, then thoroughly analyzed the behavior of Hessian's eigenvalues on training the variational quantum classifier for the Diabetes dataset. Finally, we show how the adaptive Hessian learning rate can influence the convergence while training the variational circuits.

Michał Stęchły, Lanruo Gao, Boniface Yogendran +2 more

In this paper, we aim to expand the understanding of the relationship between the composition of the Hamiltonian in the Quantum Approximate Optimization Algorithm (QAOA) and the corresponding cost landscape characteristics. QAOA is a prominent example of a Variational Quantum Algorithm (VQA), which is most commonly used for combinatorial optimization. The success of QAOA heavily relies on parameter optimization, which is a great challenge, especially on scarce noisy quantum hardware. Thus understanding the cost function landscape can aid in designing better optimization heuristics and therefore potentially provide eventual value. We consider the case of 1-layer QAOA for Hamiltonians with up to 5-local terms and up to 20 qubits. In addition to visualizing the cost landscapes, we calculate their Fourier transform to study the relationship with the structure of the Hamiltonians from a complementary perspective. Furthermore, we introduce metrics to quantify the roughness of the landscape, which provide valuable insights into the nature of high-dimensional parametrized landscapes. While these techniques allow us to elucidate the role of Hamiltonian structure, order of the terms and their coefficients on the roughness of the optimization landscape, we also find that predicting the intricate landscapes of VQAs from first principles is very challenging and unlikely to be feasible in general.

Marco Cerezo, Andrew Arrasmith, Ryan Babbush +8 more