Feedback-Based Quantum Algorithm for Excited States Calculation

TL;DR

This work extends feedback-based quantum algorithms to the computation of excited states by encoding the target excited state as the ground state of an augmented operator $P=H_0+ extstyleigl( ext{sum of shifts}igr)$, under Lyapunov control. It introduces FQAE, a layer-wise, Lyapunov-guided circuit design, and presents two hybrid controller-evaluation strategies: an expectation/overlap method and a gradient-based method via the Lyapunov function gradient, with the latter using finite-difference or the parameter-shift-rule (PSR). The approach is demonstrated on a two-qubit Ising model and a hydrogen molecule Hamiltonian, including numerical simulations and an IBM quantum hardware experiment, showing monotonic Lyapunov decrease and increasing fidelity to the targeted excited state, while analyzing robustness to sampling noise. Overall, FQAE offers a promising near-term pathway to excited-state quantum simulations, with future work aimed at reducing circuit depth, incorporating warm starts, randomized constructions, fixed-time control, and mid-circuit measurements to enhance practicality.

Abstract

Recently, feedback-based quantum algorithms have been introduced to calculate the ground states of Hamiltonians, inspired by quantum Lyapunov control theory. This paper aims to generalize these algorithms to the problem of calculating an eigenstate of a given Hamiltonian, assuming that the lower energy eigenstates are known. To this aim, we propose a new design methodology that combines the layer-wise construction of the quantum circuit in feedback-based quantum algorithms with a new feedback law based on a new Lyapunov function to assign the quantum circuit parameters. We present two approaches for evaluating the circuit parameters: one based on the expectation and overlap estimation of the terms in the feedback law and another based on the gradient of the Lyapunov function. We demonstrate the algorithm through an illustrative example and through an application in quantum chemistry. To assess its performance, we conduct numerical simulations and execution on IBM's superconducting quantum computer.

Figures (14)

• Figure 1: Quantum circuit implementation of the Pauli gadget for implementing the unitary evolution $e^{-iO_{k} \Delta t}$, where $O_{k}=O_{k,1} \otimes O_{k,2} \otimes \cdots \otimes O_{k,n}$ is a Pauli string. and $U_{l}=\left\{R_y(-\pi/2) ,\text{ if } O_{k,l}=X,R_x(\pi/2),\text{ if } O_{k,l}=Y,I,\text{ if } O_{k,l}=Z. \right.$
• Figure 2: This figure is adapted from magann2022feedback. The figure demonstrates the procedural steps involved in FALQON. In the first step, the algorithm is fed with an initial guess of the value of the controller, the state $\ket{\psi_1}$ is prepared in the quantum computer and the controller for the next layer $u_2$ is calculated. Over the following $k$ iterations, the quantum circuit expands by adding a layer composed of $U_1(u_k)U_0$, with $u_k$ calculated from the preceding layer using Equation \ref{['PB:udis1']}. The process concludes upon reaching the maximum layer count $p-1$.
• Figure 3: Basic Hadamard test, where $a \in \{0, 1\}$, $H$ is the Hadamard gate, $S=e^{-i\pi Z/4}$ is the phase gate, and $U$ is the unitary we are interested in its expected value with respect to the state $\ket{\psi}$. For $a = 0$, $\expval{Z}$ corresponds to the real part of the inner product $\text{Re}\bra{\psi} U \ket{\psi}$, and for $a = 1$, $\expval{Z}$ corresponds to the imaginary part of the same inner product $\text{Im}\bra{\psi} U \ket{\psi}$.
• Figure 7: The gradient-based approach for calculating the controller using finite difference approximation to evaluate the gradient.
• Figure 8: The gradient-based approach for calculating the controller using PSR to evaluate the gradient.