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arXiv:2303.11204

Ground state preparation with shallow variational warm-start

Youle Wang, Chenghong Zhu, Mingrui Jing, Xin Wang

Published Mar 20, 2023• 4 citations
arXivHTMLPDF

Abstract

Preparing the ground states of a many-body system is essential for evaluating physical quantities and determining the properties of materials. This work provides a quantum ground state preparation scheme with shallow variational warm-start to tackle the bottlenecks of current algorithms, i.e., demand for prior ground state energy information and lack of demonstration of efficient initial state preparation. Particularly, our methods would not experience the instability for small spectral gap $Δ$ during pre-encoding the phase factors since our methods involve only $\widetilde{O}(1)$ factors while $\widetilde{O}(Δ^{-1})$ is requested by the near-optimal methods. We demonstrate the effectiveness of our methods via extensive numerical simulations on spin-$1/2$ Heisenberg models. We also show that the shallow warm-start procedure can process chemical molecules by conducting numerical simulations on the hydrogen chain model. Moreover, we extend research on the Hubbard model, demonstrating superior performance compared to the prevalent variational quantum algorithms.

DOI
10.48550/arxiv.2303.11204

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Preparing the ground states of a many-body system is essential for evaluating physical quantities and determining the properties of materials. This work provides a quantum ground state preparation scheme with shallow variational warm-start to tackle the bottlenecks of current algorithms, i.e., demand for prior ground state energy information and lack of demonstration of efficient initial state preparation. Particularly, our methods would not experience the instability for small spectral gap ΔΔΔ during pre-encoding the phase factors since our methods involve only O~(1)\widetilde{O}(1)O(1) factors while O~(Δ−1)\widetilde{O}(Δ^{-1})O(Δ−1) is requested by the near-optimal methods. We demonstrate the effectiveness of our methods via extensive numerical simulations on spin-1/21/21/2 Heisenberg models. We also show that the shallow warm-start procedure can process chemical molecules by conducting numerical simulations on the hydrogen chain model. Moreover, we extend research on the Hubbard model, demonstrating superior performance compared to the prevalent variational quantum algorithms.
Reliable preparation of many-body ground states is an essential task in quantum computing, with applications spanning areas from chemistry and materials modeling to quantum optimization and benchmarking. A variety of approaches have been proposed to tackle this problem, including variational methods. However, variational training often struggle to navigate complex energy landscapes, frequently encountering suboptimal local minima or suffering from barren plateaus. In this work, we introduce an iterative strategy for ground-state preparation based on a stepwise (discretized) Hamiltonian deformation. By complementing the Variational Quantum Eigensolver (VQE) with adiabatic principles, we demonstrate that solving a sequence of intermediate problems facilitates tracking the ground-state manifold toward the target system, even as we scale the system size. We provide a rigorous theoretical foundation for this approach, proving a lower bound on the loss variance that suggests trainability throughout the deformation, provided the system remains away from gap closings. Numerical simulations, including the effects of shot noise, confirm that this path-dependent tracking consistently converges to the target ground state.
Quantum mechanical ground states of many-body systems can be important resources for various investigations: for quantum sensing, for benchmarking quantum hardware with classically solvable states, as the initial states for nonequilibrium quantum dynamics following quenches, the simulation of quantum processes that start by coupling systems in ground states, eg, could be a process in quantum chemistry, while their approximations are required as inputs to quantum phase estimation algorithm. However, preparing ground states can be challenging; for example, it may require adiabatic switching of Hamiltonian terms slower than an inverse gap, which can be time consuming and bring in decoherence. Here we investigate the possibility of preparing a many-body entangled ground state of a certain Hamiltonian, which can be called a quantum ``problem'' Hamiltonian, using the time evolution of an initial fiducial state by another time independent ``solver'' Hamiltonian with couplings up to unit strength for a very short fixed (unit) time: a ``time efficient'' ansatz. The parameters of the solver Hamiltonian are optimised classically minimising energy as the cost function. We present a study of up to n=14n=14n=14 qubit many-body states prepared using this methodology. Importantly, we find that a strategy of combining a warm start (an already prepared ground state of a n−1n-1n−1 qubit Hamiltonian) and incrementally adding extra couplings of a qubit is the best scaling strategy to prepare the ground state of a nnn-qubit Hamiltonian.
The widespread use of the noninteracting ground state as the initial state for the digital quantum simulation of the Fermi-Hubbard model is largely due to the scarcity of alternative easy-to-prepare approximations to the exact ground state in the literature. Exploiting the fact that the spin-12\frac{1}{2}21​ Heisenberg model is the effective low-energy theory of the Fermi-Hubbard model at half-filling in the strongly interacting limit, here we propose a three-step deterministic quantum routine to prepare an educated guess of the ground state of the Fermi-Hubbard model through a shallow circuit suitable for near-term quantum hardware. First, the ground state of the Heisenberg model is initialized via a hybrid variational method using an ansatz that explores only the correct symmetry subspace. Second, a general method is devised to convert a multi-spin-12\frac{1}{2}21​ wave function into its fermionic version. Third, taking inspiration from the Baeriswyl ansatz, a constant-depth single-parameter layer that adds doublon-holon pairs is applied to this fermionic state. Numerical simulations on chains and ladders with up to 12 sites confirm the improvement over the noninteracting ground state of the overlap with the exact ground state for the intermediate values of the interaction strength at which quantum simulation is found to be most relevant. More broadly, the general scheme to convert a multi-spin-12\frac{1}{2}21​ state into a half-filled fermionic state may bridge the gap between quantum spin models and lattice models of correlated fermions in the realm of digital quantum simulation.
Ground-state preparation, energy estimation, and property estimation are fundamental tasks in quantum simulation. Existing ground-state preparation algorithms achieve near-optimal complexity using block encodings or controlled Hamiltonian evolution, raising a natural question: is controlled Hamiltonian evolution necessary, or can the same scaling be attained in the weaker access model that provides access to only uncontrolled Hamiltonian evolution? We resolve this question by giving two algorithms that access the input Hamiltonian HHH only through queries to UHU_HUH​ and its inverse UH†U_H^\daggerUH†​, where ∥H∥≤αH\|H\|\leα_H∥H∥≤αH​ and τ=Θ(1/αH)τ=Θ(1/α_H)τ=Θ(1/αH​). Given an input state with ground-state weight at least ηηη and a spectral gap of HHH of at least γγγ, our first algorithm uses fresh copies of the input state and requires O~(αH/(γη))\widetilde O(α_H/(γη))O(αH​/(γη)) queries to UHU_HUH​ and UH†U_H^\daggerUH†​, while our second algorithm uses coherent access to the input-state preparation unitary and its inverse to improve this complexity to O~(αH/(γη))\widetilde O(α_H/(γ\sqrtη))O(αH​/(γη​)). The latter bound matches the known lower bound up to logarithmic factors, showing that algorithms with additional access to controlled Hamiltonian evolution cannot improve the query complexity for ground-state preparation beyond logarithmic factors. Both algorithms are based on a simple mechanism that we call the spectral descent principle, in which the algorithm starts from the input state ∣ψ⟩|ψ\rangle∣ψ⟩ and progressively descends through the energy spectrum, moving closer to the ground state until the desired fidelity is reached. We also extend this framework to ground-state property and energy estimation and give explicit complexity guarantees for both tasks.
In recent years, substantial research effort has been devoted to quantum algorithms for ground-state-energy estimation (GSEE) in chemistry and materials. Given the many heuristic and nonheuristic methods being developed, it is challenging to assess what combination of these methods will ultimately be used in practice. One important metric for assessing utility is the runtime, which depends on the ground-state preparation (GSP) for most GSEE algorithms. Towards assessing the utility of various combinations of GSEE and GSP methods, we asked under which conditions a GSP method should be accepted over a reference method, such as the Hartree-Fock method. We introduce a criterion for accepting or rejecting a GSP method for the purposes of GSEE. We consider different GSP methods ranging from heuristics to algorithms with provable performance guarantees and perform numerical simulations to benchmark their performance on different chemical systems, starting from small molecules like the hydrogen molecule to larger systems like jellium. In the future, this approach may be used to abandon certain variational quantum eigensolver (VQE) ansatzes and other heuristics. Yet our findings do not provide evidence against using VQE and more expensive heuristic methods, like the low-depth booster. This work sets a foundation from which to further explore the requirements to achieve quantum advantage in quantum chemistry.
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