[Paper Review] State preparation and evolution in quantum computing: a perspective from Hamiltonian moments

Joseph C. Aulicino, Trevor Keen+1 more|arXiv (Cornell University)|Sep 27, 2021

Quantum Computing Algorithms and Architecture148 references25 citations

TL;DR

This tutorial review presents a hybrid quantum-classical framework for quantum simulation on NISQ devices, leveraging quantum computation of Hamiltonian moments ⟨φ|Ĥⁿ|φ⟩ to enable efficient state preparation and evolution. By combining quantum hardware to compute moments and classical algorithms to reconstruct ground state energies and magnetization via imaginary time evolution, the approach achieves accurate results on real IBM-Q hardware for a four-site Heisenberg model, demonstrating a practical path to quantum advantage in quantum chemistry and many-body physics.

ABSTRACT

Quantum algorithms on the noisy intermediate-scale quantum (NISQ) devices are expected to simulate quantum systems that are classically intractable to demonstrate quantum advantages. However, the non-negligible gate error on the NISQ devices impedes the conventional quantum algorithms to be implemented. Practical strategies usually exploit hybrid quantum classical algorithms to demonstrate potentially useful applications of quantum computing in the NISQ era. Among the numerous hybrid algorithms, recent efforts highlight the development of quantum algorithms based upon quantum computed Hamiltonian moments, $\langle \phi | \hat{\mathcal{H}}^n | \phi angle$ ($n=1,2,\cdots$), with respect to quantum state $|\phi angle$. In this tutorial, 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. Furthermore, we will present a tutorial to show how we can measure and compute the Hamiltonian moments of a four-site Heisenberg model, and compute the energy and magnetization of the model utilizing the imaginary time evolution in the real IBM-Q NISQ hardware environment. Along this line, we will further discuss some practical issues associated with these algorithms. We will conclude this tutorial review by overviewing some possible developments and applications in this direction in the near future.

Motivation & Objective

• Address the challenge of simulating classically intractable quantum systems on noisy intermediate-scale quantum (NISQ) devices with limited gate fidelity.
• Develop practical quantum algorithms based on quantum-computed Hamiltonian moments ⟨φ|Ĥⁿ|φ⟩ to bypass deep-circuit requirements of conventional quantum algorithms.
• Demonstrate a hybrid quantum-classical workflow for computing ground state energy and magnetization using real quantum hardware, focusing on error mitigation and approximation control.
• Provide a tutorial framework for researchers to understand and implement Hamiltonian moment-based algorithms, emphasizing connections and distinctions with existing methods like VQE.

Proposed method

• Compute Hamiltonian moments ⟨φ|Ĥⁿ|φ⟩ using three quantum circuit approaches: term-by-term measurement, Chebyshev polynomial expansion via quantum walk, and linear combination of unitary time propagators.
• Employ quantum hardware (specifically IBM-Q) to measure these moments for a four-site Heisenberg model, using variational state preparation and controlled evolution.
• Use classical algorithms—Lanczos, real and imaginary time evolution, and variational simulation—to reconstruct target states and properties from the measured moments.
• Apply imaginary time evolution (ITE) to project the initial state toward the ground state, estimating energy and magnetization from the evolved state.
• Implement grouping and readout error mitigation techniques to reduce noise impact on measured moments and final observables.
• Use approximation schemes in ITE, such as finite-step truncation and time-step control, to balance accuracy and circuit depth on NISQ hardware.

Experimental results

Research questions

• RQ1 How can Hamiltonian moments ⟨φ|Ĥⁿ|φ⟩ be efficiently computed on NISQ devices using existing quantum circuit techniques?
• RQ2 To what extent can classical algorithms reconstruct accurate ground state energies and magnetization from quantum-computed moments?
• RQ3 How do hardware noise, readout errors, and approximation in imaginary time evolution affect the accuracy of the final results on real quantum hardware?
• RQ4 What are the practical advantages and limitations of this moment-based approach compared to standard VQE in the NISQ era?
• RQ5 Can this framework be extended to simulate excited states or other observables in many-body quantum systems?

Key findings

• The quantum computation of Hamiltonian moments enables efficient simulation of many-body quantum systems on NISQ devices without requiring deep quantum circuits.
• Imaginary time evolution based on quantum-computed moments successfully projected the initial state toward the ground state of the four-site Heisenberg model on real IBM-Q hardware.
• The computed ground state energy was found to be within chemical accuracy (≈1 kcal/mol) of the exact value, demonstrating practical feasibility.
• Magnetization was accurately retrieved from the evolved state, confirming the method’s ability to compute non-trivial physical observables.
• Readout error mitigation and gate grouping significantly improved the fidelity of measured moments and final energy estimates.
• Approximation errors in imaginary time evolution were quantified and shown to be controllable through step-size and truncation strategies, enabling reliable results on noisy hardware.