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.
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.
To maximize the value of fault-tolerant quantum computers, it is essential to develop concrete applications beyond well-established domains such as chemistry and condensed-matter physics. Here we construct and compile quantum algorithms to simulate the structure of atomic nuclei -- a topic that has received relatively little attention from the quantum computing community despite its similarities to the electronic structure problem in chemistry -- via effective shell-model Hamiltonians and no-core-shell-model Hamiltonians with three-body interactions derived from chiral effective field theory. Furthermore, we provide quantum resource estimates, in terms of Toffoli gate and qubit counts, for these algorithms, which, to our knowledge, are the first such estimates for fault-tolerant quantum simulation of atomic nuclei. Notably, the estimates for 32Mg and 219At shell-model Hamiltonians are comparable to recent estimates of Femoco simulations, a standard benchmark in chemistry. For no-core-shell-model Hamiltonians suitable for light nuclei (up to 40Ca or so), we find that resource requirements are significantly higher, suggesting that more bespoke strategies are required to make such simulations practicable. Throughout this work, we draw upon the similarities between nuclear and electronic structure problems, while also highlighting challenges that are specific to the former. We hope this work will spur long-term collaborations between the nuclear and quantum computing community with the ultimate goal of realizing useful nuclear simulations on quantum computers.
The first-principles quantum simulation of the full electronic Hamiltonian of a material model of realistic size appears daunting even in a future era of fully fault-tolerant quantum computers. In contrast, classical simulations of material excited states rely on heuristic quasiparticle frameworks, such as the GW approximation and Bethe-Salpeter equation, where only a few quasiparticles are explicitly considered. Here we give a detailed description of quasiparticle quantum simulation for multi-excitonic states within the BSE. We develop an explicit first-quantized, orbital-based block encoding of the multi-exciton BSE Hamiltonian for qubitization, and reduce its cost using integral factorization and crystal symmetries. For a materials model of size N, and a fixed number of quasiparticles, the resulting algorithms require O(N) Toffoli gates and O(logN) logical qubits, or, using space-time tradeoffs, O(N) Toffoli gates and O(N) logical qubits, achieving up to a sixth power reduction in Toffoli count, or up to exponential reduction in qubit count, versus standard full Hamiltonian simulation techniques that have been explored in materials simulation. Resource estimates for the classically challenging problem of triexcitonic eigenvalue estimation across a range of semiconductor models of a realistic physical size of O(103) atoms, show that our Toffoli-optimized implementation reduces the Toffoli count by 106−107 relative to full Hamiltonian simulation estimates, while a qubit-optimized implementation requires only a few hundred logical qubits. More broadly, these results show that classical heuristic frameworks will be essential in the design of future fault-tolerant quantum algorithms for materials.
Understanding the physics of strongly correlated materials is one of the grand challenge problems for physics today. A large class of scientifically interesting materials, from high-Tc superconductors to spin liquids, involve medium to strong correlations, and building a holistic understanding of these materials is critical. Doing so is hindered by the competition between the kinetic energy and Coulomb repulsion, which renders both analytic and numerical methods unsatisfactory for describing interacting materials. Fault-tolerant quantum computers have been proposed as a path forward to overcome these difficulties, but this potential capability has not yet been fully assessed. Here, using the multi-orbital Fermi-Hubbard model as a representative model and a source of scalable problem specifications, we estimate the resource costs needed to use fault-tolerant quantum computers for obtaining experimentally relevant quantities such as correlation function estimation. We find that advances in quantum algorithms and hardware will be needed in order to reduce quantum resources and feasibly address utility-scale problem instances.
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.
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.
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.