Towards near-term quantum simulation of materials

Quantum simulation of materials is constrained on NISQ devices by limited qubit numbers and gate fidelities, making circuit depth a primary bottleneck for state preparation, ground-state property estimation, and dynamics. Many naive cost estimates do not exploit materials-specific structure such as locality, translational symmetry, or focus on local properties near equilibrium. This work addresses the question: to what extent can the required circuit depth and qubits for simulating realistic materials be reduced by incorporating physical structure and specialized encodings? The authors target two key algorithms—variational quantum eigensolver (VQE) with a Hamiltonian variational ansatz and time-dynamics simulation (TDS)—and aim to implement the short-time evolution subroutine U(δt) = e^{-iHδt} with minimal depth and qubits. The study leverages active space identification near the Fermi level, maximally localized Wannier functions to localize interactions, a hybrid fermion-to-qubit mapping tailored to the sparsity pattern, and custom fermionic swap networks, all compiled into low-depth circuits intended for near-term feasibility.

Prior work using Jordan–Wigner (JW) mappings and fermionic swap (fswap) protocols for materials Hamiltonians yields O(N_cells^2) scaling in gate counts and depth on nearest-neighbor devices (e.g., Kanno et al.). First-quantization plane-wave approaches with phase estimation achieve fault-tolerant Toffoli counts but are less aligned with near-term constraints (e.g., Delgado et al.). Other resource analyses focus on fault-tolerant regimes for molecules, jellium, and periodic systems, emphasizing T/Toffoli counts more than circuit depth. Active-space approaches (DFT+DMFT/DMET) and automated active space selection have proven effective for correlated systems. The present work departs by targeting near-term depth minimization, exploiting Wannier-localized Hamiltonians and introducing a hybrid encoding to attain O(N_cells) gate scaling with layer depth independent of N_cells, contrasting sharply with JW-based baselines.

The workflow comprises: (1) Active space identification via DFT Kohn–Sham (KS) band structures, selecting bands within an energy window around the Fermi level that capture dominant low-energy dynamics. (2) Wannierization: construction of maximally localized Wannier functions (MLWFs) from the chosen KS subspace (using Wannier90), yielding a real-space Hamiltonian with rapidly decaying intercell hopping and Coulomb terms that are predominantly onsite, nearest-neighbor, or next-nearest-neighbor. (3) Hamiltonian truncation: retain dominant localized terms to reduce Hamiltonian size while controlling accuracy. (4) Hybrid fermionic encoding: map densely interacting modes within each unit cell using a JW-style string, while connecting strings across sites using a compact encoding for sparse intersite interactions. This hybridization leverages low operator weight for intersite terms and manages unavoidable high-weight onsite terms with fswap. The encoding is expressed via Majorana-based edge and vertex operators E_jk = -iγ_jγ_k and V_j = -iγ̄_jγ_j and rules to translate Majorana monomials into Pauli strings. (5) Custom fermionic swap network: an algorithm that searches fswap layers tailored to the target Hamiltonian and encoding geometry, minimizing fswap depth by a steepest-descent heuristic over an l_p distance (empirically p = 0.5) that measures adjacency to ideal pairings for two-, three-, and four-mode interactions. Graph coloring (DSATUR heuristic) schedules simultaneously executable interaction terms acting on disjoint qubit sets per layer. (6) Compiler: end-to-end automation from atomic specification to circuits, including active space detection, Wannierization, Monte Carlo estimation of Hamiltonian coefficients, hybrid encoding selection, optional commuting-group decompositions to manage Trotter error, fswap protocol search, circuit decomposition and small-scale optimizations, and cost analysis. A tilable per-unit-cell circuit construction enables system-size-independent depth for a Trotter or VQE layer. (7) Measurement optimization: energy measurement schemes for VQE that group operators beyond qubitwise commuting (QWC), introducing non-crossing (NC) and commuting (COM) strategies with analytical bounds on measurement rounds; NC achieves fewer rounds than QWC while keeping constant-depth implementation. (8) Algorithmic primitives and cost model: with all-to-all qubit connectivity, two-qubit gates of unit cost, and free single-qubit gates. Implement e^{iθZ} on k qubits in depth 2⌈log2 k⌉−1 by parity tree with final two CNOTs and rotation combined. Fswap in 2D hybrid encoding uses weight-3 edge decomposition for depth 4; 3D uses naive decomposition. Givens rotations are 2-qubit for adjacent JW modes; across sites they require depth 4 (2D) or 6 (3D). State preparation: Fock states in depth ~12 (2D) or ~50 (3D); Gaussian states in depth ≤ 4.5M (2D) or 6.5M (3D) for M modes.

• Localized Wannier representations reduce Hamiltonian term counts from quadratic to linear in N_cells and align interactions to local, low-weight qubit operators. • Hybrid fermion-to-qubit encoding plus custom fswap networks enables O(N_cells) gate scaling and single-layer depth independent of system size for Trotter/VQE layers on all-to-all devices. • Depth reductions up to six orders of magnitude for SrVO3 Trotter layers versus standard JW-in-Bloch-basis baselines. Example resources for a single VQE layer (system sizes noted): - SrVO3 (3×3×3): 3 bands, 180 qubits, 7.5×10^3 two-qubit gates, depth 8.8×10^2; baseline 3.2×10^11 gates, depth 6.7×10^8. - GaAs (5×5×5): 4 bands, 1120 qubits, 4.1×10^5 gates, depth 7.9×10^3; baseline 3.0×10^12 gates, depth 3.5×10^9. - H3S (5×5×5): 7 bands, 1870 qubits, 2.4×10^6 gates, depth 3.7×10^4; baseline 3.0×10^12 gates, depth 3.5×10^9. - Li2CuO2 (5×3×3): 11 bands, 1024 qubits, 2.3×10^5 gates, depth 8.4×10^3; baseline 1.5×10^12 gates, depth 2.1×10^9. - Si (5×5×5): 4 bands, 1120 qubits, 4.5×10^5 gates, depth 8.5×10^3; baseline 1.8×10^11 gates, depth 4.3×10^8. • Custom fswap optimization substantially reduces onsite and nearest-neighbor compilation depth (e.g., for H2S, depth reduced from ~7000 to 1214 for those terms). • Measurement strategies yield analytical bounds on required rounds for M modes: QWC lower/upper bounds M/2 and M; NC M/2 and 5M/2; COM M/2 and 7M/2. • Practical viability estimate: with reported ~99.9% two-qubit gate fidelities and prior ~3000 two-qubit gates demonstrated for Hamiltonian simulation, a budget of ~2000–3000 two-qubit gates may be feasible. The SrVO3 layer at 7507 gates suggests a further ~3× gate-count reduction could make single-layer HVA VQE plausible for qualitative features.

The results directly address the core challenge of reducing depth and gate counts for realistic materials Hamiltonians on NISQ hardware by leveraging problem structure. Translational symmetry and Wannier localization restrict dominant interactions to local neighborhoods, enabling a hybrid encoding with sparse low-weight intersite operations and manageable onsite handling through optimized swap networks. This design transforms scaling characteristics: linear-in-size gate counts with system-size-independent per-layer depth. Empirical resource estimates across diverse materials demonstrate orders-of-magnitude improvements over JW-in-Bloch baselines, indicating that carefully tailored algorithms can move selected materials simulations towards near-term feasibility. Nonetheless, these are not yet quantum-advantage proposals; hardware-native gate sets, connectivity, and precise error models are not fully incorporated, and standard Trotter long-time dynamics likely remain out of reach without further reductions or alternative algorithms. Still, the framework provides a concrete path to prioritize materials whose physics (e.g., localized d-orbitals near the Fermi level) aligns with near-term constraints and to refine encodings, swap protocols, and measurement strategies accordingly.

By integrating active-space selection from DFT, maximally localized Wannier functions, a hybrid fermion-to-qubit mapping, custom fswap networks, and a specialized compiler with optimized measurement schemes, the study achieves single-layer circuit depths independent of system size and gate-count reductions by several orders of magnitude compared with naive baselines. This pushes certain materials simulations—especially for systems like SrVO3—closer to near-term viability for VQE layers and short-time dynamics. Future work should incorporate more accurate screened interactions and double-counting corrections, further optimize encodings and fswap protocols for specific interaction graphs, co-design with hardware-native gates and limited connectivity, refine Trotterization or alternative simulation methods, and apply high-throughput workflows to identify classes of materials best suited for NISQ-era quantum simulation.

Resource estimates assume all-to-all qubit connectivity, unit-cost two-qubit gates, and free single-qubit gates, which may not reflect hardware constraints. The Hamiltonians use unscreened Coulomb interactions, neglecting screening and double-counting corrections; this likely overestimates interaction strength and long-range terms, yielding conservative (upper-bound) circuit complexities but limiting quantitative realism. Active-space truncations and Wannierization introduce approximation errors whose impact on target properties and dynamics requires further validation. The work does not fully account for native gate sets, calibration, or noise beyond coarse gate-fidelity budgets; long-time Trotterized dynamics still demand many layers. State preparation and measurement overheads, while optimized, remain significant and problem-dependent.