Quantum simulations of strongly interacting fermionic systems, such as those described by the Hubbard model, are promising candidates for useful early fault-tolerant quantum computing applications. This paper presents tile Trotterization, a generalization of plaquette Trotterization, which uses a set of tiles to construct Trotter decompositions of arbitrary lattice Hubbard models. The tile Trotterization scheme also enables the simulation of more complex models, including the extended Hubbard model. We improve previous Hubbard-model commutator bounds, further provide tight commutator bounds for periodic extended Hubbard models, and demonstrate the use of tensor network methods for this task. We consider applications of tile Trotterization to simulate hexagonal lattice Hubbard models and compare the resource requirements of tile Trotterization for performing quantum phase estimation to a qubitization-based approach, demonstrating that tile Trotterization scales more efficiently with system size. These advancements significantly broaden the potential applications of early fault-tolerant quantum computers to models of practical interest in materials research and organic chemistry.
Clifford+T synthesis of rotation gates is an important routine in fault-tolerant quantum compilation. While Clifford+T synthesis is scalable, it has a high overhead of tens of T gates per rotation in practice, translating to high resource estimates for many fault-tolerant algorithms. However, these well-known results, including those using probabilistic mixtures [Quantum 7, 1208 (2023)], are independent of the rotation angle
θ, requiring
O(log1/δ) T gates. We show that it is possible to do much better for small angles, reducing the T cost to
O~(θ2/δ), and returning to existing
O(log1/δ) results in the worst case. This is particularly important since many algorithms, such as Trotterization, are dominated by small-angle rotations. Further, we perform a detailed theoretical and numerical study of quasi-probabilities, which can further reduce the total T cost of large circuits by orders of magnitude with only a small overhead in sample complexity. We also develop a scheme based on quasi-probability mixtures of Clifford+T fallback channels. We derive new
θ-dependent formulas that can be used for resource estimation of fault-tolerant quantum algorithms. As an application of our results, we show that the gate cost of Trotterization circuits compiled to a Clifford+T gate set is constant in the small Trotter step size limit, and can be reduced by orders of magnitude even for large step sizes. The cost of fault-tolerant Trotterization for a variety of applications should be re-examined in light of these results. Our work dispels the widely-stated claim that Clifford+T rotation synthesis has a high cost independent of
θ, and further develops a scalable quasi-probability method for rotation synthesis. We also expect our results to bring forward useful early fault-tolerant quantum computing by reducing required magic state resources.
Recent work has deployed linear combinations of unitaries techniques to reduce the cost of fault-tolerant quantum simulations of correlated electron models. Here, we show that one can sometimes improve upon those results with optimized implementations of Trotter-Suzuki-based product formulas. We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (e.g. energies per unit cell), as is often the goal for condensed-phase systems. In this context, simulations of the Hubbard and plane-wave electronic structure models with N < 10 5 fermionic modes can be performed with roughly O ( 1 ) and O ( N 2 ) T complexities. We perform numerics revealing tradeoffs between the error and gate complexity of a Trotter step; e.g., we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates and assuming error rates of one part per thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with a few hundred thousand physical qubits.
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
7.7×108 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.
Partially fault-tolerant quantum computing (FTQC) has recently emerged as a promising approach for the execution of megaquop-scale circuits with millions of logical operations. In this work, we demonstrate the strengths and the limitations of this approach by conducting quantum resource estimation (QRE) of the space--time-efficient analog rotation (STAR) architecture using realistic hardware specifications for superconducting processors, and compare it against the QRE of the full FTQC architecture. We show how the performance of the STAR architecture's protocols is affected by hardware improvements. We also reduce the space requirements for partial FTQC by developing a procedure leveraging code growth to decrease the size of a factory producing analog rotation states. Our results reveal a non-trivial dependence of the optimal pre-growth code distance on the rotation angle with respect to post-growth infidelity. Further, we analyze space--time trade-offs between the factory size and the error-mitigation overhead, and observe that in an application-agnostic setting, there is a Goldilocks zone for circuits in the regime of roughly
105--
106 small-angle rotation gates. We show that quantum simulation of 2D Fermi--Hubbard model systems is a particularly well-suited application for the STAR architecture, requiring only hundreds of thousands of physical qubits and runtimes on the order of minutes for modest system sizes. Due to its favourable algorithmic scaling to larger system sizes, utility-scale simulation of the 2D Fermi--Hubbard model could potentially be attained using partial FTQC.
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.