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Scalable, self-verifying variational quantum eigensolver using adiabatic warm starts

Bojan Žunkovič, Marco Ballarin, Lewis Wright, Michael Lubasch

TL;DR

The conditions under which gradient-based optimization successfully prepares the adiabatic ground states are derived and show that the barren plateau problem and local optima can be avoided.

Abstract

We study an adiabatic variant of the variational quantum eigensolver (VQE) in which VQE is performed iteratively for a sequence of Hamiltonians along an adiabatic path. We derive the conditions under which gradient-based optimization successfully prepares the adiabatic ground states. These conditions show that the barren plateau problem and local optima can be avoided. Additionally, we propose using energy-standard-deviation measurements at runtime to certify eigenstate accuracy and verify convergence to the global optimum.

Scalable, self-verifying variational quantum eigensolver using adiabatic warm starts

TL;DR

The conditions under which gradient-based optimization successfully prepares the adiabatic ground states are derived and show that the barren plateau problem and local optima can be avoided.

Abstract

We study an adiabatic variant of the variational quantum eigensolver (VQE) in which VQE is performed iteratively for a sequence of Hamiltonians along an adiabatic path. We derive the conditions under which gradient-based optimization successfully prepares the adiabatic ground states. These conditions show that the barren plateau problem and local optima can be avoided. Additionally, we propose using energy-standard-deviation measurements at runtime to certify eigenstate accuracy and verify convergence to the global optimum.
Paper Structure (2 theorems, 16 equations, 1 figure, 1 algorithm)

This paper contains 2 theorems, 16 equations, 1 figure, 1 algorithm.

Key Result

Theorem 1

Assume that Assumptions 1--3 hold and that the optimization protocol described above is used with the Pauli-rotation ansatz of Eq. eq:ansatz containing $M$ parameters. Then there exist universal numerical constants $c_0,c_1>0$ such that the following holds. If the adiabatic discretization satisfies and the number of optimization steps per slice satisfies then the optimised parameters $\bm{\theta

Figures (1)

  • Figure 1: AVQE optimization landscape --- Consider a Hamiltonian $H(\lambda)$ where $\lambda \in [0, 1]$ defines an adiabatic path, a variational ansatz parameterized by $\theta$, and the associated energy-based cost function $E_{\lambda}(\theta)$. Starting from the ground state at $\lambda = 0$, there exists a basin (blue) that follows the adiabatic path and guarantees convexity. The size of the convexity basin decreases at most linearly with the instantaneous gap $\Delta(\lambda)$ (black curve, right-hand 2D plot) which denotes the energy difference between the ground and first excited state at $\lambda$. Outside the convexity basin, the optimization landscape may exhibit barren plateaus (white) and local minima (red), which can hinder gradient-based optimization. In the verifiable region (green), which also scales linearly with $\Delta(\lambda)$, we cannot assume convexity, but we can check that the closest eigenstate is the ground state. This enables adaptation of the adiabatic step width $\delta\lambda$ to ensure provable convergence to the target ground state at $\lambda = 1$.

Theorems & Definitions (2)

  • Theorem 1: Adiabatic tracking
  • Theorem 2: Runtime verification