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arXiv:2609.37527

Digital quantum state preparation by nucleation

Jean Paul Sadia, Morten Hjorth-Jensen, Dean Lee, Ryan LaRose

quant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

Quantum state preparation is crucial for quantum algorithms, and improved state preparation methods can reduce resource requirements by orders of magnitude. In this work, we introduce a state preparation method for lattice Hamiltonians on digital quantum computers inspired by nucleation. In our method, the exact ground state of a small lattice Hamiltonian is prepared in a quantum circuit, then a series of growth stages are performed via Trotterized adiabatic evolution to increase the size of the lattice to the desired model. Using the adiabatic theorem and Trotter error bounds, we provide sufficient conditions on the total evolution time and number of Trotter steps required for nucleation to prepare an initial state with desired fidelity. In addition, we introduce optimized nucleation which considers variational optimization on top of the nucleation structure. For a two-dimensional Ising model, we show that optimized nucleation can outperform a current leading method for state preparation on digital quantum computers.

Comments
10 pages, 7 figures

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Quantum state preparation is crucial for quantum algorithms, and improved state preparation methods can reduce resource requirements by orders of magnitude. In this work, we introduce a state preparation method for lattice Hamiltonians on digital quantum computers inspired by nucleation. In our method, the exact ground state of a small lattice Hamiltonian is prepared in a quantum circuit, then a series of growth stages are performed via Trotterized adiabatic evolution to increase the size of the lattice to the desired model. Using the adiabatic theorem and Trotter error bounds, we provide sufficient conditions on the total evolution time and number of Trotter steps required for nucleation to prepare an initial state with desired fidelity. In addition, we introduce optimized nucleation which considers variational optimization on top of the nucleation structure. For a two-dimensional Ising model, we show that optimized nucleation can outperform a current leading method for state preparation on digital quantum computers.
The preparation of a given quantum state on a quantum computing register is a typically demanding operation, requiring a number of elementary gates that scales exponentially with the size of the problem. Using the adiabatic theorem for state preparation, whose error decreases exponentially as a function of the preparation time, we derive an explicit analytic expression for the dependence of the characteristic time on the Hamiltonian used in the adiabatic evolution. Exploiting this knowledge, we then design a preconditioning term that modifies the adiabatic preparation, thus reducing its characteristic time and hence giving an exponential advantage in state preparation. We prove the efficiency of our method with extensive numerical experiments on prototypical spin models, which gives a promising strategy to perform quantum simulations of manybody models via Trotter evolution on near-term quantum processors.
Adiabatic quantum computing is a general framework for preparing eigenstates of Hamiltonians on quantum devices. However, its digital implementation requires an efficient Hamiltonian simulation subroutine, which may introduce extra computational overhead or complicated quantum control logic. In this work, we show that the time-step sizes in time discretization can be much larger than expected, and the overall complexity is greatly reduced. Remarkably, regardless of the general convergence order of the numerical method, we can choose a uniform time-step size independent of tolerated error and evolution time for sufficiently accurate simulation. Furthermore, with the boundary cancellation condition where the continuous diabatic errors are exponentially suppressed, we provide strong evidence for a superpolynomial convergence of first-order Trotter with uniform time-step size. We apply our analysis to the example of adiabatic unstructured search and show several preferable features of the Trotterized adiabatic approach: It can match the Grover lower bound, it does not require knowledge of the number of marked states, and its performance can be asymptotically comparable with that of the quantum approximate optimization algorithm.
We describe tensor network algorithms to optimize quantum circuits for adiabatic quantum computing. To suppress diabatic transitions, we include counterdiabatic driving in the optimization and utilize variational matrix product operators to represent adiabatic gauge potentials. Traditionally, Trotter product formulas are used to turn adiabatic time evolution into quantum circuits and the addition of counterdiabatic driving increases the circuit depth per time step. Instead, we classically optimize a parameterized quantum circuit of fixed depth to simultaneously capture adiabatic evolution together with counterdiabatic driving over many time steps. The methods are applied to the ground-state preparation of quantum Ising chains with transverse and longitudinal fields. We show that the classically optimized circuits can significantly outperform Trotter product formulas. Additionally, we discuss how the approach can be used for combinatorial optimization. Published by the American Physical Society 2024
One of the key applications for the emerging quantum simulators is to emulate the ground state of many-body systems, as it is of great interest in various fields from condensed matter physics to material science. Traditionally, in an analog sense, adiabatic evolution has been proposed to slowly evolve a simple Hamiltonian, initialized in its ground state, to the Hamiltonian of interest such that the final state becomes the desired ground state. Recently, variational methods have also been proposed and realized in quantum simulators for emulating the ground state of many-body systems. Here, we first provide a quantitative comparison between the adiabatic and variational methods with respect to required quantum resources on digital quantum simulators, namely the depth of the circuit and the number of two-qubit quantum gates. Our results show that the variational methods are less demanding with respect to these resources. However, they need to be hybridized with a classical optimization which can converge slowly. Therefore, as the second result of the paper, we provide two different approaches for speeding the convergence of the classical optimizer by taking a good initial guess for the parameters of the variational circuit. We show that these approaches are applicable to a wide range of Hamiltonian and provide significant improvement in the optimization procedure.
Adiabatic time evolution can be used to prepare a complicated quantum many-body state from one that is easier to synthesize and Trotterization can be used to implement such an evolution digitally. The complex interplay between nonadiabaticity and digitization influences the infidelity of this process. We prove that the first-order Trotterization of a complete adiabatic evolution has a cumulative infidelity that scales as O(T^{-2}δt^{2}) instead of O(T^{2}δt^{2}) expected from general Trotter error bounds, where δt is the time step and T is the total time. This result suggests a self-healing mechanism and explains why, despite increasing T, infidelities for fixed-δt digitized evolutions still decrease for a wide variety of Hamiltonians. It also establishes a correspondence between the quantum approximate optimization algorithm and digitized quantum annealing.
Digitizing an adiabatic evolution is a strategy able to combine the good performance of gate-based quantum processors with the advantages of adiabatic algorithms, thus providing a hybrid model for efficient quantum information processing. In this work we develop validity conditions for high-fidelity digital adiabatic tasks. To this end, we assume a digitizing process based on the Suzuki-Trotter decomposition, which allows us to introduce a digitized adiabatic theorem. As a consequence of this theorem, we show that the performance of such a hybrid model is limited by the fundamental constraints on the adiabatic theorem validity, even in ideal quantum processors. We argue how our approach predicts the existence of intrinsic nonadiabatic errors reported by Barends et al. [Nature (London) 534, 222 (2016)] through an empirical study of digital annealing. In addition, our approach allows us to explain the existence of a scaling of the number of Suzuki-Trotter blocks for the optimal digital circuit with respect to the optimal adiabatic total evolution time, as reported by Mbeng et al. [Phys. Rev. B 100, 224201 (2019)] through robust numerical analysis of digital annealing. We illustrate our results through two examples of digitized adiabatic algorithms, namely, the two-qubit exact-cover problem and the three-qubit adiabatic factorization of the number 21.