Towards near-term quantum simulation of materials

Understanding and designing chemicals and materials is crucial for scientific and industrial advancement. Classical computational methods struggle with accurately simulating electron-electron interactions, particularly in the strong-coupling regime relevant to many technological applications. Quantum computers offer a potential solution by natively simulating these interactions. However, near-term Noisy Intermediate-Scale Quantum (NISQ) devices are limited by low gate fidelities and small qubit numbers, restricting the applicability of algorithms. Current estimates for the circuit depth and qubit requirements of material simulations far exceed NISQ capabilities. This research aims to improve these estimates by exploiting the inherent features of material systems, such as computing local properties at equilibrium or simulating out-of-equilibrium dynamics.

Prior work on qubit and gate resource estimations for Trotterized Hamiltonian simulation algorithms has shown O(N_cells²) scaling of the gate count for Hamiltonians defined in N_cells unit cells, using a Jordan-Wigner (JW) transform and fermionic swaps. Other studies have analyzed quantum algorithm resources for battery materials, focusing on quantum phase estimation. These often focus on the fault-tolerant regime, counting Toffoli or T gates. However, near-term quantum computers are better evaluated based on circuit depth due to limitations imposed by decoherence and error propagation. Existing work on quantum simulation resource costs for various systems, including molecules, interacting electron gas (jellium), and periodic systems, often don't fully leverage material-specific characteristics like translational symmetry.

This work introduces several physics-based techniques to reduce the resource requirements of quantum algorithms for material simulation. The key innovations include: 1. **Identifying the active space:** This involves using density functional theory (DFT) to identify a subspace of the single-particle Hilbert space around the Fermi level, which captures the dominant dynamics. This reduces the size of the Hilbert space and the number of qubits needed. 2. **Localized Wannier representation:** Employing maximally localized Wannier functions in real space leads to highly localized interactions, further minimizing the number of dominant terms in the Hamiltonian. This linearizes the scaling of Hamiltonian terms with system size. 3. **Hybrid fermionic encoding and fermionic swap networks:** A hybrid encoding scheme combines the benefits of Jordan-Wigner (JW) and compact encodings. JW encoding handles dense short-range interactions within a unit cell, while compact encoding efficiently manages sparse long-range interactions between cells. A custom-designed fermionic swap (fswap) network optimization algorithm dynamically reorders fermionic modes to minimize the depth of the circuit. 4. **Compiler:** A compiler automates the process, incorporating the above techniques along with additional optimizations such as small-scale circuit optimizations and leveraging symmetries. It can handle finite lattices or produce infinitely tilable circuits, leading to a circuit depth independent of system size. The compiler also optimizes measurement protocols to reduce overheads. The compiler allows for prioritizing either minimizing Trotter error or circuit depth, and estimates the resulting Trotter error.

The combined methods significantly reduce the resources required for material simulations. Table 1 in the paper compares circuit depth obtained using the novel methods with a standard approach that doesn't exploit Hamiltonian structure. The materials analyzed (GaAs, H3S, Li2CuO2, Si, SrVO3) represent a range of systems with distinct behaviors. The results show substantial improvements in circuit depth using the proposed techniques. For SrVO3, a 3x3x3 supercell, the two-qubit gate count is reduced from approximately 3.2E+11 to 7.5E+03. The achieved circuit depth for a single Trotter step is independent of the system's size, a major improvement over the O(N_cells²) depth scaling of the standard Jordan-Wigner approach. The analysis considers the costs of various operations, including state preparation, time evolution, fermionic swaps, and measurements. Specific details of the depths of circuits and improvements are available in supplementary information.

The findings demonstrate that incorporating physical constraints into quantum algorithm design significantly reduces the resource requirements for simulating materials. The improvements bring material simulation within closer reach of near-term quantum computers. For example, considering reported two-qubit gate fidelities approaching 99.9%, the reduced gate count for SrVO3 suggests that simulating a single layer of the Hamiltonian variational ansatz (HVA) VQE might be feasible on near-term hardware. Although a single layer might reveal qualitative features, simulating long-timescale dynamics through Trotterization would still require many layers and remains a challenge. The results presented provide a crucial step towards practical quantum simulations of materials, but further improvements in both hardware and algorithms are still needed. Rigorous quantification of the trade-offs introduced by approximations in the simulation of large materials remains an open problem.

This research presents a framework for materials simulation on quantum computers that dramatically reduces circuit depth by orders of magnitude compared to naive estimates. The key is incorporating physically motivated structure into all aspects of the quantum algorithm design. This work identified materials, such as SrVO3, particularly suitable for near-term quantum simulation. Future work should focus on further improving error mitigation, exploring additional materials, and developing data-driven techniques to identify optimal candidates for quantum simulation. This combined algorithmic and materials-driven approach accelerates progress towards achieving quantum advantage in materials science.

The analysis assumes all-to-all qubit connectivity, which is not yet realized in current quantum hardware. Furthermore, while the methods address the issue of circuit depth, they do not fully account for noise and error mitigation strategies necessary for real-world quantum computers. The active space approximation and the use of unscreened Coulomb interactions introduce potential inaccuracies in the simulation, although these limitations are common in many current classical computational methods for simulating materials.