How Variational Quantum Algorithms Work Part 1 (Future Technology) | by Monodeep Mukherjee | October 2022

picture by Jason Leung on Unsplash
  1. Fast gradient estimation for variational quantum algorithms(arXiv)

Author : Lennart Bittel, Jens Watty, Martin Kliesch

Summary : Many optimization methods for training variational quantum algorithms are based on estimating cost function gradients. Due to the statistical nature of quantum measurements, this estimation requires many circuit evaluations, which is a crucial bottleneck for the whole approach. We propose a novel gradient estimation method to alleviate this measurement challenge and reduce the required measurement cycles. In a Bayesian framework and based on the generalized parameter lag rule, we use prior circuit information to find an estimation strategy that simultaneously minimizes the expected statistical and systematic errors. We demonstrate that this approach can significantly outperform traditional gradient estimation methods, reducing the required measurement cycles by up to an order of magnitude for a common QAOA setup. Our analysis also shows that finite difference estimation can outperform the parameter shift rule in gradient accuracy for small and moderate measurement budgets.

2. A disruptive gadget to delay the appearance of sterile plateaus in variational quantum algorithms(arXiv)

Author : Simon Cichy, Paul K. Faehrmann, Sumeet Khatri, Jens Eisert

Summary : Variational quantum algorithms are explored as a promising approach to find useful applications for noisy intermediate-scale quantum computers. However, the cost functions corresponding to many problems of interest are inherently global, defined by Hamiltonians with many-body interactions. Consequently, the optimization landscape may exhibit exponential vanishing gradients, called barren plateaus, making optimal solutions difficult to find. Strategies to mitigate sterile plateaus are therefore needed to make variational quantum algorithms trainable and able to run on larger-scale quantum devices. In this work, we bring the toolbox of perturbative gadgets to the portfolio of methods explored with the aim of making noisy quantum devices at intermediate scale useful. Specifically, we introduce a new perturbative gadget, suitable for variational quantum algorithms, which can be used to avoid sterile plateaus. Our perturbative gadget encodes an arbitrary many-body Hamiltonian corresponding to a global cost function in the low-energy subspace of a three-body Hamiltonian. Our construction requires rk extra bits for a k-body Hamiltonian with r terms. We provide rigorous guarantees on the optimization of the local cost function defined by our Hamiltonian three-body gadget with respect to the original cost function, and we prove that this local cost function exhibits nonzero gradients, delaying thus the appearance of sterile trays. We then provide numerical demonstrations to show how our approach works.

3.Exploring the role of parameters in variational quantum algorithms (arXiv)

Author : Abhinav Anand, Sumner Alperin-Lea, Alexandre Choquette, Alan Aspuru-Guzik

Summary : In this work, we introduce a method inspired by quantum control for the characterization of variational quantum circuits using the rank of the dynamic Lie algebra associated with the Hermitian generator(s) of the individual layers. Layer-based architectures in variational algorithms for calculating the ground state energies of physical systems are considered the focus of this exploration. A promising connection is found between the Lie rank, the precision of the calculated energies and the depth required to reach the target states via a given circuit architecture, even using a large number of parameters which is significantly less than the number of separate terms in the generators. As the cost of calculating the dynamic Lie rank via an iterative process grows exponentially with the number of qubits in the circuit and thus quickly becomes prohibitive, reliable approximations of it are desirable. The rapidity of the dynamic Lie rank increase in the first iterations of the computation proves to be a viable proxy (lower bound) for the full computation, the balance between accuracy and computational expense. We therefore propose the dynamic Lie rank and its surrogates as a useful design metric for layer-structured quantum circuits in variational algorithms.

Sharon D. Cole