Random State Approach to Quantum Computation of Electronic-Structure Properties

TL;DR

The paper tackles the memory and scaling bottlenecks of classical electronic-structure calculations for large materials by introducing random-state quantum algorithms. It develops three methods—Q-TDPM, Q-KPM, and M-QPE—that use Haar-random state preparation and Hadamard-test measurements to extract density-of-states and real-space local density of states from real-time evolution, Chebyshev filtering, and modified quantum phase estimation. The authors validate these approaches on graphene, twisted bilayer graphene quasicrystals, and fractal lattices, showing accurate spectral features and spatial patterns while highlighting qubit efficiency (scaling as $1+\log_2 N$) and Trotter-depth as the main resource driver. This framework offers a practical path toward scalable electronic-property calculations on quantum hardware, enabling simulations of large-scale materials and informing hardware development for near-term devices.

Abstract

Classical computation of electronic properties in large-scale materials remains challenging. Quantum computation has the potential to offer advantages in memory footprint and computational scaling. However, general and practical quantum algorithms for simulating large-scale materials are still lacking. We propose and implement random-state quantum algorithms to calculate electronic-structure properties of real materials. Using a random state circuit with only a few qubits, we employ real-time evolution with first-order Trotter decomposition and Hadamard test to obtain electronic density of states, and we develop a modified quantum phase estimation algorithm to calculate real-space local density of states via direct quantum measurements. Furthermore, we validate these algorithms by numerically computing the density of states and spatial distributions of electronic states in graphene, twisted bilayer graphene quasicrystals, and fractal lattices, covering system sizes from hundreds to thousands of atoms. Our results manifest that the random-state quantum algorithms provide a general and qubit-efficient route to simulating electronic properties of large-scale periodic and aperiodic materials on quantum computers.

Figures (10)

• Figure 1: Random state circuit (RSC) for evaluating the time-correlation function $C^{p}(t)$ for the $p$-th stochastic sample using the Hadamard test. The ancilla qubit is initialized in the $\ket{0}$ state and transformed to the $\ket{+}$ state via a Hadamard gate. The data register is prepared in a normalized Haar-random state $\ket{\psi_0^{p}}$, representing the $p$-th sample in the ensemble, using the randomized state preparation module (yellow box). Controlled by the ancilla, the time evolution operator $e^{-iH\Delta t}$ is applied to the data qubits through a Trotterized approximation with a time step $\Delta t$ (green boxes). A second Hadamard gate is applied to the ancilla before measurement. The expectation value of the ancilla yields the real part $\Re[\bra{\psi_0^{p}}e^{-iHt}\ket{\psi_0^{p}}]$. To access the imaginary part $\Im[\bra{\psi_0^{p}}e^{-iHt}\ket{\psi_0^{p}}]$, a $S$ gate shown in dotted blue box, is inserted before the final Hadamard.
• Figure 2: Quantum circuit scheme for calculating quasi-eigenstates based on a modified quantum phase estimation (QPE) protocol combined with RSC. The purple block (labeled with $A$) initializes the ancilla qubits with a sequence of Hadamard and single-ubit controlled-phase gates with the energy level $\varepsilon$ attached. The orange block (denoteds as RSC) prepares the data register in a randomized initial state. The green block applies the set of controlled unitaries $U^j = e^{-iH 2^j \Delta t}$, each conditioned on the $j$th ancilla qubit, while the blue block performs the quantum Fourier transform (QFT) on the ancilla register. Finally, the red blocks represent the measurement stage, where the data qubits are projected only when the ancilla qubits collapse to $\ket{0}^{\otimes m}$, enabling the extraction of quasi-eigenstates at selected energy levels.
• Figure 3: Comparison of Q-TDPM (solid black line) and classical numerical (red circles) density of states (DOS) calculated via time evolution with 1000 random samples. (a) DOS for a 64×64 graphene lattice. (b)DOS for a 4 nm radius disk of 30°-twisted bilayer graphene (30°-tBG, 3828 atoms) . (c) DOS for a Sierpiński carpet fractal (256 atoms).
• Figure 4: Comparison of Q-KPM (solid black line) and classical numerical (red circles) density of states (DOS) calculated via the kernel polynomial method (KPM) with 1000 random samples. (a) DOS for a 64×64 graphene lattice. (b)DOS for a 4 nm radius disk of 30°-twisted bilayer graphene (30°-tBG, 3828 atoms) . (c) DOS for a Sierpiński carpet fractal (256 atoms).
• Figure 5: Comparison of quantum (left) and classical (right) quasi-eigenstates maps with 300 random samples. (a) Quasi-eigenstate maps at $0\ eV$ for an 8×8 graphene lattice with a single vacancy (127 atoms). (b) Quasi-eigenstate maps at minimum eigenvalue $-11.39\ eV$ for a 1.3 $nm$ radius disk of $30^\circ$-twisted bilayer graphene (408 atoms). (c) Quasi-eigenstate maps at $-3.41\ eV$ for a Sierpiński carpet fractal (256 atoms).