State preparation and evolution in quantum computing: A perspective from Hamiltonian moments

Abstract

Quantum algorithms on noisy intermediate-scale quantum (NISQ) devices are expected to soon simulate quantum systems that are classically intractable. However, the non-negligible gate error present on NISQ devices impedes the implementation of many purely quantum algorithms, necessitating the use of hybrid quantum-classical algorithms. One such hybrid quantum-classical algorithm, is based upon quantum computed Hamiltonian moments ϕj Ĥn jϕ D E n ¼ 1,2,ÁÁÁ ð Þ , with respect to quantum state j ϕi. In this tutorial review, we will give a brief review of these quantum algorithms with focuses on the typical ways of computing Hamiltonian moments using quantum hardware and improving the accuracy of the estimated state energies based on the quantum computed moments. We also present a computation of the Hamiltonian moments of a four-site Heisenberg model and compute the energy and magnetization of the model utilizing the imaginary time evolution on current IBM-Q hardware. Finally, we discuss some possible developments and applications of Hamiltonian moment methods.

Improved Iterative Quantum Algorithm for Ground‐State Preparation

“…ITE is an iterative computational method to solve the ground‐state of many‐body quantum systems. [ 51 ] Consider the time‐independent Schrödinger equation in imaginary time ($$t\rightarrow -\iota t$$) $$$$\begin{align} \frac{\partial |\phi (t)\rangle }{\partial t}=-\hat{\mathcal {H}}|\phi (t)\rangle \end{align}$$$$where $$|\phi (t)\rangle$$ is a quantum state at time t and $$\hat{\mathcal {H}}$$ is the Hamiltonian. The formal solution of Equation (20) can be expressed as $$|\phi (t)\rangle =e^{-\hat{\mathcal {H}}t}|\phi (0)\rangle$$, where $$|\phi (0)\rangle$$ is an initial state at time $$t=0$$.…”

“…ITE is an iterative computational method to solve the groundstate of many-body quantum systems. [51] Consider the timeindependent Schrödinger equation in imaginary time (t → −𝜄t)…”

The basics of quantum computing for chemists

“…In a similar vein, quantum computers can be employed to prepare entangled states with sizable overlap with the true ground state and carry out the subsequent measurements in an efficient fashion. This can pave the way for widespread adoption of methodologies which estimate the ground state energy based on moments of the Hamiltonian [113][114][115][116][117], which share some commonalities with ITE [118], and real-time evolution on the grounds of Krylov basis ideas [66].…”

Improved iterative quantum algorithm for ground-state preparation

“…ITE is an iterative computational method to solve the ground-state of many-body quantum systems [51]. Consider the time-independent Schrödinger equation in imaginary time (t → −ιt),…”