Fault-tolerant quantum algorithms offer a promising pathway for estimating the ground-state energies of periodic materials that are beyond the practical reach of classical electronic-structure methods. A remaining challenge is finite-size mitigation: quantum algorithms evaluate a finite supercell or finite Brillouin-zone mesh, while materials properties are defined in the thermodynamic limit. In this work we develop a quantum post-processing strategy for the leading two-body finite-size correction. After one-body shell effects are reduced by twist averaging, the dominant residual error is controlled by long-wavelength density fluctuations, which are encoded in the small-momentum static structure factor $S(q)$. We formulate the corresponding operator in a Bloch-orbital basis, construct its block encoding through the density operator, and estimate its ground-state expectation value using an amplified Hadamard test. We also introduce adaptive global and local binary search procedures for identifying the infrared fitting window used to reconstruct the two-body finite size error correction. The resulting cost remains subleading relative to the main ground-state energy estimation routine: the structure-factor correction has leading $\tilde{O}(N_bN_k)^3$ dependence on the Bloch-orbital basis size, avoids the large plane-wave prefactor of full Hamiltonian simulation, and requires only $\tilde{O}(N_bN_k)$ logical qubits. This provides a fault-tolerant alternative to down-sampling, replacing repeated energy calculations on larger cells with targeted measurements of the infrared density correlations that control the finite-size effects.
Fault-tolerant quantum algorithms offer a promising pathway for estimating the ground-state energies of periodic materials that are beyond the practical reach of classical electronic-structure methods. A remaining challenge is finite-size mitigation: quantum algorithms evaluate a finite supercell or finite Brillouin-zone mesh, while materials properties are defined in the thermodynamic limit. In this work we develop a quantum post-processing strategy for the leading two-body finite-size correction. After one-body shell effects are reduced by twist averaging, the dominant residual error is controlled by long-wavelength density fluctuations, which are encoded in the small-momentum static structure factor S(q). We formulate the corresponding operator in a Bloch-orbital basis, construct its block encoding through the density operator, and estimate its ground-state expectation value using an amplified Hadamard test. We also introduce adaptive global and local binary search procedures for identifying the infrared fitting window used to reconstruct the two-body finite size error correction. The resulting cost remains subleading relative to the main ground-state energy estimation routine: the structure-factor correction has leading O~(NbNk)3 dependence on the Bloch-orbital basis size, avoids the large plane-wave prefactor of full Hamiltonian simulation, and requires only O~(NbNk) logical qubits. This provides a fault-tolerant alternative to down-sampling, replacing repeated energy calculations on larger cells with targeted measurements of the infrared density correlations that control the finite-size effects.
Strongly correlated materials are a natural target for fault-tolerant quantum computers, but they require tools beyond those developed for molecules. Electronic wavefunctions vary rapidly near nuclei yet remain delocalized across many unit cells, and bulk properties must be converged systematically with respect to finite-size errors. To resolve such issues, we present the Bloch--UPAW framework that combines Bloch-orbital k-space structure with unitary projector-augmented-wave (UPAW) augmentation. The UPAW Hamiltonian, expressed directly in the Bloch basis, retains explicit control of Brillouin-zone sampling, and incorporates near-nuclear physics through strictly local on-site corrections. The construction is independent of the underlying one-particle representation, so it applies to both plane-wave and localized bases, and it handles supercells for symmetry-breaking phenomena more efficiently. We derive a linear-combination-of-unitaries decomposition and a block-encoding circuit suitable for qubitization; UPAW augmentation adds one ancilla qubit and no Toffoli gates at leading order relative to a Bloch-only block encoding. Asymptotically, the Toffoli cost scales as O(Nk3) when refining the k-mesh and as O(Na3.5) when enlarging the supercell, enabling convergence to be steered by the most favorable route for a given material. Resource estimates for bulk diamond show approximately an order-of-magnitude reduction in Toffoli count relative to prior work on periodic solids.
Subspace eigenvalue problems arise ubiquitously in quantum chemistry and condensed-matter physics, where the relevant object is often a low-energy manifold rather than a single ground-state wavefunction. In this work, we propose a fault-tolerant quantum algorithm for this subspace-level task based on the Feshbach effective-Hamiltonian formalism. Given block-encoding access to the full Hamiltonian and a chosen d-dimensional reference subspace, the algorithm estimates eigenvalues of states with nonzero overlap with the reference subspace through a local secant fixed-point search. It then implements the associated wave operator and prepares an orthonormal basis whose span approximates the target invariant subspace. The construction combines projected block encodings with quantum singular value transformation (QSVT), which approximates the complementary-space resolvent and thereby provides both the self-energy used for eigenvalue estimation and the wave operator used for eigenstate reconstruction. For target accuracy ε, a single evaluation of the effective Hamiltonian has query complexity O(d3/(g2ε)), up to block-encoding normalization factors, where g is the distance between the target eigenvalue and the nearest pole of the effective Hamiltonian. Under the stated local regularity conditions, the secant search requires only O(loglog(1/ε)) effective-Hamiltonian evaluations to reach the working precision. Classical numerical emulations for an open 4×2 Fermi--Hubbard cluster, all-electron LiH bond stretching, and [Ru(bpy)3]2+ demonstrate the resolution and reconstruction of low-energy states and manifolds across spin-sector crossings, near-degeneracies, and dense excited-state spectra.
Quantum simulation of molecular electronic structure is one of the most promising applications of quantum computing. However, achieving chemically accurate predictions for strongly correlated systems requires quantum phase estimation (QPE) on fault-tolerant quantum computing (FTQC) devices. Existing resource estimates for typical FTQC architectures suggest that such calculations demand millions of physical qubits, thereby placing them beyond the reach of near-term devices. Here, we investigate the feasibility of performing QPE for chemically relevant molecular systems in an early-FTQC regime, characterized by partial fault tolerance, constrained qubit budgets, and limited circuit depth. Our framework is based on single-ancilla, Trotter-based QPE implementations combined with partially randomized time evolution. Within this framework, we develop a novel Hamiltonian optimization strategy, termed unitary weight concentration, that reduces algorithmic cost by reshaping linear-combination-of-unitaries representations. Applying this framework to active-space models of iron-sulfur clusters, cytochrome P450 active sites, and CO2-utilization catalysts, we perform end-to-end resource estimation using the latest version of the space-time efficient analog rotation (STAR) architecture. Our results indicate that ground-state energy estimation for active spaces of approximately 20 to 50 spatial orbitals, well beyond the reach of classical full configuration interaction, is achievable using ∼105 physical qubits, with runtimes on the order of days to weeks. These findings demonstrate that while full-fledged fault-tolerant quantum computers will be required for even larger molecular simulations, chemically meaningful quantum chemistry problems are already within reach in an experimentally relevant, early-FTQC regime.
We investigate the feasibility of early fault-tolerant quantum algorithms focusing on ground-state energy estimation problems. In particular, we examine the computation of the cumulative distribution function (CDF) of the spectral measure of a Hamiltonian and the identification of its discontinuities. Scaling these methods to larger system sizes reveals three key challenges: the smoothness of the CDF for large supports, the lack of tight lower bounds on the overlap with the true ground state, and the difficulty of preparing high-quality initial states. To address these challenges, we propose a signal processing approach to find these estimates automatically, in the regime where the quality of the initial state is unknown. Rather than aiming for exact ground-state energy, we advocate for improving classical estimates by targeting the low-energy support of the initial state. Additionally, we provide quantitative resource estimates, demonstrating a constant factor improvement in the number of samples required to detect a specified change in CDF. Our numerical experiments, conducted on a 26-qubit fully connected Heisenberg model, leverage a truncated density-matrix renormalization group (DMRG) initial state with a low bond dimension. The results show that the predictions from the quantum algorithm align closely with the DMRG-converged energies at larger bond dimensions while requiring several orders of magnitude fewer samples than theoretical estimates suggest. These findings underscore that CDF-based quantum algorithms are a practical and resource-efficient alternative to quantum phase estimation, particularly in resource-constrained scenarios.
Quantum computers hold the promise to solve some hard problems that are intractable for even the most powerful current supercomputers. One of the most famous examples is Shor's algorithm for factorizing large numbers, which has exponential speedup compared to its best classical counterparts. However, running such an algorithm will require to build a large-scale quantum computer consisting of thousands or even millions of qubits that include quantum error correction (QEC) and fault-tolerant (FT) mechanisms. Quantum computing is already a reality with the so-called Noisy Intermediate-Scale Quantum (NISQ) processors, some of them available in the cloud. Noisy refers to the imperfect control over the qubits and intermediate-scale to the relatively low number of quits (from fifty to a few hundred). Although current and near-term quantum devices will not have enough qubits for implementing large and fully corrected quantum computations, the use of small quantum error correction codes may extend the computation lifetime of NISQ devices. In this context and as a first step, it is important to test and demonstrate the fault-tolerance of these QEC codes. In this thesis, we explore the fault-tolerance of two small quantum correction codes that are good candidates to be applied to NISQ processors, the [[4,2,2]] code and the [[7,1,3]] Steane code. To this purpose, by following the FT criterion prosed by Daniel Gottesman in 2016, we tested both codes using two simulators, the stabilizer formalism simulator that includes quite simple error models and a full density matrix simulator called quantumsim, which includes more realistic noise. The simulations are performed under reasonable noise parameter values. For the [[4,2,2]] code, 235 circuits are tested based on the two simulators. The results show that in the stabilizer formalism simulation, the FT criterion is satised for all circuits, while not fully satisfied in the full density matrix simulation. For the [[7,1,3]] Steane code, we use a parallel-flag error correction implementation which is tested using the full density matrix simulator. Our results show that without applying any QEC cycle, for all circuits (84 circuits for 1 logical qubit simulation and 452 circuits for 2 logical qubits simulation), the error rate of the encoded circuits is lower than the unencoded ones. Adding a quantum error correction (QEC) cycle will in general increase the error rate of the computation.
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.