[Paper Review] Towards Practical Quantum Variational Algorithms

Dave Wecker, Matthew B. Hastings+1 more|arXiv (Cornell University)|Jul 31, 2015

Quantum Computing Algorithms and Architecture10 citations

TL;DR

This paper proposes a Hamiltonian variational ansatz for quantum state preparation on near-term quantum computers, using short-depth circuits based on evolution under terms of the problem Hamiltonian. It achieves faster convergence and higher ground state overlap than unitary coupled cluster methods, especially for Hubbard models, making variational quantum algorithms potentially practical for strongly correlated systems despite high measurement overhead in quantum chemistry applications.

ABSTRACT

The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin-orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.

Motivation & Objective

• To develop a practical variational quantum algorithm for near-term quantum computers with low circuit depth and few parameters.
• To improve convergence speed and ground state fidelity compared to existing ansatzes like unitary coupled cluster (UCC) for strongly correlated systems.
• To assess the feasibility of variational quantum algorithms for quantum chemistry and condensed matter models under realistic noise and measurement constraints.
• To compare the efficiency of variational state preparation against adiabatic state preparation in terms of circuit depth and required parameters.
• To evaluate the measurement cost and scalability of the method for both quantum chemistry and Hubbard model applications.

Proposed method

• The method constructs variational states by evolving an initial state using unitary operations generated by individual terms in the problem Hamiltonian, inspired by adiabatic state preparation.
• Circuit depth is minimized by selecting only physically relevant terms (e.g., hopping and on-site interaction in the Hubbard model), avoiding full UCC excitation operators.
• Optimization is performed via classical black-box search over variational parameters, treating energy as a function of parameters without requiring gradient access.
• The approach uses second-order Trotter-Suzuki decomposition to simulate time evolution in discrete steps, enabling implementation on NISQ devices.
• For comparison, the paper also evaluates UCC and its Rxx variants with limited Trotter steps to reduce circuit depth.
• The method is tested numerically on Hubbard models (up to 12 sites, 24 spin-orbitals) and small molecules, with energy and overlap measured via quantum circuit sampling.

Experimental results

Research questions

• RQ1 Can a variational ansatz based on Hamiltonian evolution achieve faster convergence and higher ground state overlap than UCC for strongly correlated systems?
• RQ2 What is the required circuit depth and number of measurements for the variational method to achieve high-fidelity state preparation in quantum chemistry and Hubbard models?
• RQ3 How does the performance of the Hamiltonian variational method compare to adiabatic state preparation in terms of circuit depth and fidelity?
• RQ4 Can limited Trotter numbers in UCC-based methods reduce circuit depth without sacrificing accuracy?
• RQ5 Is the measurement cost for energy estimation in variational quantum algorithms feasible for near-term devices, especially in condensed matter models?

Key findings

• The Hamiltonian variational ansatz achieves higher ground state overlap than UCC and Rxx variants, particularly for larger Hubbard models with strong correlations.
• For the 12-site Hubbard model, the variational method required only S=19 circuit steps to achieve 0.9883 overlap, compared to 640 steps for adiabatic preparation.
• In contrast, the UCC method required significantly more circuit depth and evaluations, especially for larger molecules with strong electron correlation.
• The measurement cost for energy estimation remains astronomically high in quantum chemistry applications, making the method impractical there with current resources.
• For the Hubbard model, the reduced number of interaction terms and translation invariance lower the measurement overhead, making the approach potentially practical on small quantum computers.
• The method enables faster state preparation than adiabatic evolution, with a 32-fold reduction in circuit depth for the 12-site system, suggesting scalability advantages for larger systems.