Excitation Amplitude Sampling for Low Variance Electronic Structure on Quantum Computers

TL;DR

The paper tackles the high shot overhead and noise in extracting observables from ab initio Hamiltonians on quantum devices by proposing a mixed energy estimator that uses a classical trial state $|\Phi\rangle$ and a polynomial set of excitation amplitudes, obtained via shadow tomography, to compute $E = E_{\text{HF}} + \frac{1}{c_0} \sum_{ijab} c_{ij}^{ab}(2 v_{ijab} - v_{ijba})$. It shows how the same amplitudes can map to a CCSD surrogate for non-energetic properties and demonstrates integration into quantum embedding to tackle large, realistic materials, with NiO and cubic-BN as representative cases. Numerical results reveal dramatic shot reductions (up to approximately two orders of magnitude) and resilience to noise, though asymptotic scaling can become exponential if the trial overlap decays; the framework remains practically advantageous for near-term devices and large-scale materials modeling. The work suggests avenues for improvement via better classical states (e.g., tensor networks), refined shadow protocols, and broader embedding strategies to extend quantum-accelerated electronic structure to realistic systems.

Abstract

We combine classical heuristics with partial shadow tomography to enable efficient protocols for extracting information from correlated ab initio electronic systems encoded on quantum devices. By proposing the use of a correlation energy functional and sampling of a polynomial set of excitation amplitudes of the quantum state, we can demonstrate an almost two order of magnitude reduction in required number of shots for a given statistical error in the energy estimate, as well as observing a linear scaling to accessible system sizes. Furthermore, we find a high-degree of noise resilience of these estimators on real quantum devices, with up to an order of magnitude increase in the tolerated noise compared to traditional techniques. While these approaches are expected to break down asymptotically, we find strong evidence that these large system arguments do not prevent algorithmic advantage from these simple protocols in many systems of interest. We further extend this to consider the extraction of beyond-energetic properties by mapping to a coupled cluster surrogate model, as well as a natural combination within a quantum embedding framework. This embedding framework avoids the unstable self-consistent requirements of previous approaches, enabling application of quantum solvers to realistic correlated materials science, where we demonstrate the volume-dependence of the spin gap of Nickel Oxide.

Figures (8)

• Figure 1: Variance of mixed and Pauli energy estimators for Hydrogen chains over changing correlation strengths and system size. (a) Mean and standard deviation of the energy estimate distributions of the Pauli and mixed estimators sampled with 1000 shots, for a ten atom symmetrically stretched Hydrogen system in an STO-3G basis. Also included is the exact solution (FCI, red) and coupled-cluster with singles and doubles (CCSD, navy) results, which failed to converge for distances above $1.5$Å. Both the Pauli and mixed estimators are obtained from samples of the exact ground statevector, with the Pauli estimate fully grouping the Pauli operators into commuting sets using a graph coloring algorithm, and the mixed estimate employing the Clifford shadow protocol as described in Sec. \ref{['sec:shadowsampling']}. (b) Standard deviations of the Pauli (dashed) and mixed (solid) energy estimator distributions as the Hydrogen chain system size increases for a fixed interatomic distance of $1.5$Å, and a fixed 1000 shots per sample of the distribution. The black line again shows the Pauli operators fully grouped into as few commuting sets as possible, while the purple results group Pauli operators based on qubitwise commutivity. The orange line represents the standard deviations of the mixed estimator with the Hartree--Fock trial state using the Clifford shadows method, and is shown to exhibit approximately linear scaling of the error with system size.
• Figure 2: The effect of noise on the energy estimators for a fixed, optimized VQE ansatz, showing improved resilience of the mixed estimator. (a) Energies for a four atom Hydrogen chain, computed via the Pauli (purple) and mixed (orange) estimators for different noise levels, each using eight sets of 1000 shots. The points are the mean energy over these sets, characterizing the bias in the estimators due to the noise and inexact state, while the error bars characterize the standard error over these sets for the different estimators, indicating the statistical variance of the results due to finite shot numbers. The green points represent execution on the IBM Brisbane quantum device with the same number of shots, somewhat validating the performance of the simulated results at $\alpha=1$. The exact energy (FCI, dashed red) and Hartree--Fock energy (dashed light blue) are shown for comparison. The noise model for a given $\alpha$ is shown in Eq. \ref{['eq:noise_channel_interpolation']}, with $\alpha=0$ a noiseless simulation, and $\alpha=1$ approximating the noise of the real quantum device. (b) The mean-squared error (MSE) for the different estimators compared to the exact energy, incorporating both the bias and variance of the estimators in a single metric.
• Figure 3: Systematic and random errors in the magnetic correlations of Hydrogen chains (a) The spin-spin correlation function of an eight atom hydrogen chain (symmetrically stretched to $1.5$Å) between the first atom and its $i^\text{th}$ nearest neighbor in a STO-3G basis. Hartree--Fock (light blue), FCI (red), and CCSD (grey) are shown as classical methods, while ten sets of 40,960 shots are used to estimate the mean and error of the expectation values from the exact statevector, comparing a direct sampling of qubitwise grouped Pauli strings (purple) and a mapping of the shadow state to a CCSD ansatz (orange). (b) The local (same-atom) spin-spin correlation function for an N$_{\mathrm{atom}}$ hydrogen chain, showing the standard deviation of the distribution of expectation values with 40,960 shots for the two approaches, with the navy dashed line also showing the systematic error in the CCSD model for this property when the exact coefficients are used, quantifying the bias introduced by this CCSD model parameterization.
• Figure 4: Smooth and statistically resolved equations of state for Nickel Oxide phases, with shadow sampling of quantum solvers for each fragment demonstrating robustness of quantum embedding approach. Equation of state for the ferromagnetic and antiferromagnetic phases of NiO, employing an excitation amplitude based quantum embedding to fragment the four-atom unit cell into ten fragments. Compared are exact solutions for each cluster (embedded FCI) and a classical shadow sampling procedure for the excitation amplitudes (shadow configurational sampling) from the exact statevector. Each cluster solution was sampled with 10,000 shots, repeated 12 times to estimate the mean and error of the total energy. Also included is the Hartree--Fock energy for comparison. A $2\times2\times2$$k$-point grid was used, with the gth-szv-molopt-sr basis and associated gth-pade pseudopotential.
• Figure 5: Equation of state of $3\times3\times3$ cubic-BN, fragmenting the system into atomic valence orbital sets with VQE solvers. Each resulting cluster (with a maximum of 12 qubits) was solved with a UCCSD ansatz, before the excitation amplitudes sampled with 4,000 shots to obtain a total energy at each cell size. This was repeated 12 times to estimate the mean and error of the total energy. Also compared is a FCI solution to the same quantum embedding, giving excellent agreement, and the Hartree--Fock result for the system. A gth-dzvp basis was used with gth-pade pseudopotential.