Real time evolution for ultracompact Hamiltonian eigenstates on quantum hardware (2103.08563v3)

Abstract: In this work we present a detailed analysis of variational quantum phase estimation (VQPE), a method based on real-time evolution for ground and excited state estimation on near-term hardware. We derive the theoretical ground on which the approach stands, and demonstrate that it provides one of the most compact variational expansions to date for solving strongly correlated Hamiltonians. At the center of VQPE lies a set of equations, with a simple geometrical interpretation, which provides conditions for the time evolution grid in order to decouple eigenstates out of the set of time evolved expansion states, and connects the method to the classical filter diagonalization algorithm. Further, we introduce what we call the unitary formulation of VQPE, in which the number of matrix elements that need to be measured scales linearly with the number of expansion states, and we provide an analysis of the effects of noise which substantially improves previous considerations. The unitary formulation allows for a direct comparison to iterative phase estimation. Our results mark VQPE as both a natural and highly efficient quantum algorithm for ground and excited state calculations of general many-body systems. We demonstrate a hardware implementation of VQPE for the transverse field Ising model. Further, we illustrate its power on a paradigmatic example of strong correlation (Cr2 in the SVP basis set), and show that it is possible to reach chemical accuracy with as few as ~50 timesteps.

Real-Time Evolution for Hamiltonian Eigenstates on Quantum Hardware

In the pursuit of quantum computation, the estimation of ground and excited states for many-body systems has been a significant challenge. The paper authored by Katherine Klymko et al. introduces an innovative approach called variational quantum phase estimation (VQPE) that takes advantage of real-time evolution on quantum hardware to estimate these eigenstates efficiently. This method aims to build highly compact variational expansions for strongly correlated Hamiltonians, making it particularly suitable for the current noisy intermediate-scale quantum (NISQ) era.

Overview

Traditional methods for solving the eigenvalue problem in quantum systems often demand considerable quantum resources. For example, Quantum Phase Estimation (QPE) necessitates deep quantum circuits and numerous ancillary qubits, while Variational Quantum Eigensolver (VQE) relies heavily on classical optimization routines. VQPE circumvents these limitations through its basis formation from dynamically evolved states, thereby minimizing reliance on classical resources and reducing circuit depth.

Theoretical Foundations

VQPE is founded on exploiting the time-dependent Schrödinger equation, contrasting with the conventional focus on the time-independent variant. The authors introduce a set of expansion states derived from real-time propagation of a reference state, allowing the quantum system to evolve naturally. The paper delineates the conditions necessary for eigenstate extraction via phase cancellation, akin to techniques used in quantum Fourier analyses and classical signal processing algorithms.

Moreover, the paper unveils the unitary formulation of VQPE, highlighting its efficacy in reducing the required number of quantum measurements. By recognizing the Toeplitz matrix structure inherent in the overlap matrices, VQPE simplifies the measurement process to a linear scale—critical for practical deployment on quantum devices.

Noise and Robustness

Quantum computations are inherently susceptible to noise, and VQPE is no exception. The authors provide a detailed noise analysis, leveraging singular value decomposition (SVD) to mitigate numerical instability and measurement errors in the generalized eigenvalue problem. This technique ensures the robustness of VQPE, making it more viable for NISQ applications where precision is challenging.

Numerical Results

Simulations across various molecular systems, including weakly and strongly correlated examples, demonstrate the compactness and efficiency of VQPE. Remarkably, the algorithm achieves chemical accuracy using an order of magnitude fewer parameters than state-of-the-art classical methods. This advantage positions VQPE as a promising tool for tackling complex quantum systems beyond classical computational capabilities.

Noteworthy is the application to the Cr$_2$ dimer, which underscores the ability of VQPE to reach chemical accuracy with significantly fewer variational parameters. These results illustrate the algorithm's proficiency in adapting to different levels of electronic correlation, highlighting its versatility and potential impact in quantum simulations.

Practical and Theoretical Implications

The implications of VQPE are manifold. Practically, it offers a path forward for conducting accurate simulations on existing quantum hardware, crucially cutting down on resources while maintaining computational feasibility. Theoretically, the insights gained from the phase cancellation framework may further inform developments in quantum algorithms, possibly spawning new methodologies for eigenstate extraction across various domains.

Future Directions

VQPE sets the stage for future research in energy estimation algorithms, extending applicability to broader ranges of quantum models. Expanding the methodology to include multiple reference states and exploring its limitations in favor of adaptive strategies will further refine its scope.

In conclusion, the paper by Klymko et al. presents significant advancements in quantum computing approaches, offering a robust framework for extracting Hamiltonian eigenstates that could shape the future trajectory in this dynamic field.