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arXiv:2610.08667

Fault-tolerant resource estimation for ground-state preparation via Lindblad simulation

Marius Bothe, Patrick Schoepf, Nick S. Blunt

quant-phSubmitted Oct 6, 2026• Announced Oct 7, 2026• 0 citations
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Abstract

Recent advances in algorithms for simulating Lindblad dynamics have clarified their theoretical potential for state preparation, but their practicality in the early fault-tolerant regime and beyond remains uncertain. In this work, we address this by investigating the cost of preparing ground states of the fermionic Hubbard model, following the single-ancilla approach of [Phys. Rev. Research 6, 033147 (2024)]. We derive rigorous error bounds including constant prefactors, and compare to empirical error behavior and practical convergence parameters obtained from circuit-level simulations. We find that empirically chosen parameters can reduce the required resources by orders of magnitude, resulting in an estimated $7.7 \times 10^8$ T gates to perform one unit of time evolution targeting the low-energy subspace of a 36-site fermionic Hubbard model. We identify the accurate filtering of energy transitions as the main source of this cost.

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Recent advances in algorithms for simulating Lindblad dynamics have clarified their theoretical potential for state preparation, but their practicality in the early fault-tolerant regime and beyond remains uncertain. In this work, we address this by investigating the cost of preparing ground states of the fermionic Hubbard model, following the single-ancilla approach of [Phys. Rev. Research 6, 033147 (2024)]. We derive rigorous error bounds including constant prefactors, and compare to empirical error behavior and practical convergence parameters obtained from circuit-level simulations. We find that empirically chosen parameters can reduce the required resources by orders of magnitude, resulting in an estimated 7.7×1087.7 \times 10^87.7×108 T gates to perform one unit of time evolution targeting the low-energy subspace of a 36-site fermionic Hubbard model. We identify the accurate filtering of energy transitions as the main source of this cost.
Abstract Simulation of the Hubbard model is a leading candidate for the first useful applications of a fault-tolerant quantum computer. A recent study of quantum algorithms for early simulations of the Hubbard model [Kivlichan et al 2019 Quantum 4 296] found that the lowest resource costs were achieved by split-operator Trotterization combined with the fast-fermionic Fourier transform (FFFT) on an L × L lattice with length L = 2 k . On lattices with length L ≠ 2 k , Givens rotations can be used instead of the FFFT but lead to considerably higher resource costs. We present a new analytic approach to bounding the simulation error due to Trotterization that provides much tighter bounds for the split-operator FFFT method, leading to 16× improvement in error bounds. Furthermore, we introduce plaquette Trotterization that works on any size lattice and apply our improved error bound analysis to show competitive resource costs. We consider a phase estimation task and show plaquette Trotterization reduces the number of non-Clifford gates by a factor 5.5× to 9× (depending on the parameter regime) over the best previous estimates for 8 × 8 and 16 × 16 lattices and a much larger factor for other lattice sizes not of the form L = 2 k . In conclusion, we find there is a potentially useful application for fault-tolerant quantum computers using around one million Toffoli gates.
We design a quantum algorithm for ground state preparation in the early fault tolerant regime. As a Monte Carlo style quantum algorithm, our method features a Lindbladian where the target state is stationary. The construction of this Lindbladian is algorithmic and should not be seen as a specific approximation to some weakly coupled system-bath dynamics in nature. Our algorithm can be implemented using just one ancilla qubit and efficiently simulated on a quantum computer. It can prepare the ground state even when the initial state has zero overlap with the ground state, bypassing the most significant limitation of methods like quantum phase estimation. As a variant, we also propose a discrete-time algorithm, demonstrating even better efficiency and providing a near-optimal simulation cost depending on the desired evolution time and precision. Numerical simulations using Ising and Hubbard models demonstrate the efficacy and applicability of our method. Published by the American Physical Society 2024
Abstract Exploring low-cost applications is paramount to creating value in early fault-tolerant quantum computers. Here, we optimize both gate and qubit counts of recent algorithms for simulating the Fermi-Hubbard model. We further devise and compile algorithms to simulate established models of cuprate and pnictide high-temperature superconductors, which include beyond-nearest-neighbor hopping terms and multi-orbital interactions that are absent in the Fermi-Hubbard model. We show that simulations of these more realistic models of high-temperature superconductors require only an order of magnitude or so more Toffoli gates than a simulation of the Fermi-Hubbard model. Furthermore, we find plenty classically difficult instances with Toffoli and qubit counts that are far lower than commonly considered quantum phase estimation circuits for electronic structure problems in quantum chemistry. We believe our results pave the way towards studying high-temperature superconductors on early fault-tolerant quantum computers.
The ability to simulate a fermionic system on a quantum computer is expected to revolutionize chemical engineering, materials design, nuclear physics, to name a few. Thus, optimizing the simulation circuits is of significance in harnessing the power of quantum computers. Here, we address this problem in two aspects. In the fault-tolerant regime, we optimize the Rz and T gate counts along with the ancilla qubit counts required, assuming the use of a product-formula algorithm for implementation. We obtain a savings ratio of two in the gate counts and a savings ratio of eleven in the number of ancilla qubits required over the state of the art. In the pre-fault tolerant regime, we optimize the two-qubit gate counts, assuming the use of the variational quantum eigensolver (VQE) approach. Specific to the latter, we present a framework that enables bootstrapping the VQE progression towards the convergence of the ground-state energy of the fermionic system. This framework, based on perturbation theory, is capable of improving the energy estimate at each cycle of the VQE progression, by about a factor of three closer to the known ground-state energy compared to the standard VQE approach in the test-bed, classically-accessible system of the water molecule. The improved energy estimate in turn results in a commensurate level of savings of quantum resources, such as the number of qubits and quantum gates, required to be within a pre-specified tolerance from the known ground-state energy. We also explore a suite of generalized transformations of fermion to qubit operators and show that resource-requirement savings of up to more than 20% , in small instances, is possible.
Quantum algorithms for ground-state energy estimation require initial states with non-negligible fidelity to the ground state. Guided by classical simulations, we design shallow circuits for the two-dimensional half-filled Hubbard model by first preparing an approximate Heisenberg ground state and then applying charge-fluctuation gates derived directly from the Schrieffer-Wolff (SW) generator. For the Heisenberg model, we demonstrate effective parameter transfer from a 4×44\times44×4 lattice to lattices up to 10×1010\times1010×10 without further optimization. We obtain promising lower bounds on the ground-state fidelity using tensor-network simulations, variational Monte Carlo reference states, and estimates of the ground-state energy and singlet gap. Exact 4×44\times44×4 calculations show that the SW gates substantially improve the Hubbard ground-state fidelity of the embedded Heisenberg state. The successful Heisenberg parameter transfer supports the use of these small-lattice results to design shallow Hubbard circuits on larger lattices beyond the reach of classical simulations. For a 10×1010\times1010×10 lattice, the estimated preparation cost is on the order of 10510^5105 TTT gates, including Clifford+TTT synthesis, which is well within the megaquop regime. These results offer a route to low-cost initial-state preparation for ground-starongly correlated systems.
Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] can estimate the ground state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block encoding input model of the Hamiltonian, whose implementation is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QET-U), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit and no multi-qubit control operations, and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anti-commutation relations and further removes the need of implementing the controlled Hamiltonian evolution. Coupled with Trotter based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM Qiskit for the transverse field Ising model. If we are further allowed to use multi-qubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground state preparation and energy estimating using a constant number of ancilla qubits (no more than 3).