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Correlated Purification for Restoring $N$-Representability in Quantum Simulation

Yuchen Wang, Irma Avdic, Michael Rose, Lillian I. Payne Torres, Anna O. Schouten, Kevin J. Sung, David A. Mazziotti

TL;DR

Noisy measurements often yield $2$-RDMs that violate $N$-representability, leading to unphysical energies in quantum simulations. The authors introduce correlated purification, a semidefinite-programming framework that enforces $p$-positivity constraints while correcting the measured $2$-RDM $^2D_e$ toward a physically valid $^2D_p$ via a nuclear-norm–based correction $(^2D_p-^2D_e=E^+-E^-)$. The objective combines the energy term ${ m Tr}(^2K\,{}^2D_p)$ with a nuclear-norm penalty controlled by a weight $w$, yielding a robust correction that improves both energetic and $2$-RDM fidelity, and can be tuned to target ground, excited, or nonstationary states. Demonstrations on fermionic classical shadows of hydrogen chains show substantial performance gains over uncorrected data, with chemical accuracy across dissociation curves, and hardware experiments confirm the method’s ability to enforce positivity and reduce errors on real devices, highlighting its practical impact for scalable, physics-informed quantum tomography.

Abstract

Classical shadow tomography offers a scalable route to estimating properties of quantum states, but the resulting reduced density matrices (RDMs) often violate constraints that ensure they represent $N$-electron states -- known as $N$-representability conditions -- because of statistical and hardware noise. We present a correlated purification framework based on semidefinite programming to restore accuracy to these noisy, unphysical two-electron RDMs. The method performs a bi-objective optimization that minimizes both the many-electron energy and the nuclear norm of the change in the measured 2-RDM. The nuclear norm, often employed in matrix completion, promotes low-rank, physically meaningful corrections to the 2-RDM, while the energy term acts as a regularization term that can improve the purity of the ground state. While the method is particularly effective for the ground state, it can also be applied to excited and non-stationary states by decreasing the weight of the energy relative to the error norm. In an application to fermionic shadow tomography of large hydrogen chains, correlated purification yields substantial reductions in both energy and 2-RDM error, achieving chemical accuracy across dissociation curves. This framework provides a robust strategy for tomography in many-body quantum simulations.

Correlated Purification for Restoring $N$-Representability in Quantum Simulation

TL;DR

Noisy measurements often yield -RDMs that violate -representability, leading to unphysical energies in quantum simulations. The authors introduce correlated purification, a semidefinite-programming framework that enforces -positivity constraints while correcting the measured -RDM toward a physically valid via a nuclear-norm–based correction . The objective combines the energy term with a nuclear-norm penalty controlled by a weight , yielding a robust correction that improves both energetic and -RDM fidelity, and can be tuned to target ground, excited, or nonstationary states. Demonstrations on fermionic classical shadows of hydrogen chains show substantial performance gains over uncorrected data, with chemical accuracy across dissociation curves, and hardware experiments confirm the method’s ability to enforce positivity and reduce errors on real devices, highlighting its practical impact for scalable, physics-informed quantum tomography.

Abstract

Classical shadow tomography offers a scalable route to estimating properties of quantum states, but the resulting reduced density matrices (RDMs) often violate constraints that ensure they represent -electron states -- known as -representability conditions -- because of statistical and hardware noise. We present a correlated purification framework based on semidefinite programming to restore accuracy to these noisy, unphysical two-electron RDMs. The method performs a bi-objective optimization that minimizes both the many-electron energy and the nuclear norm of the change in the measured 2-RDM. The nuclear norm, often employed in matrix completion, promotes low-rank, physically meaningful corrections to the 2-RDM, while the energy term acts as a regularization term that can improve the purity of the ground state. While the method is particularly effective for the ground state, it can also be applied to excited and non-stationary states by decreasing the weight of the energy relative to the error norm. In an application to fermionic shadow tomography of large hydrogen chains, correlated purification yields substantial reductions in both energy and 2-RDM error, achieving chemical accuracy across dissociation curves. This framework provides a robust strategy for tomography in many-body quantum simulations.

Paper Structure

This paper contains 4 sections, 8 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Results
  4. Discussion and Conclusions

Figures (5)

  • Figure 1: Both (a) absolute energy error and (b) 2-RDM deviation as shown as a function of the weight $w$ for fermionic classical shadows (FCS), the variational two-electron reduced density matrix (v2RDM) method, and correlated purification (CP). For FCS, $10^{5}$ shots are used in the estimation.
  • Figure 2: Comparison of error metrics for hydrogen chains H$_{n}$ for $n=4$ through $n=10$ using $10^{6}$ shots and $w = 0.001$. The top panel (a) shows the absolute energy error, and the bottom panel (b) shows the 2-RDM deviation for FCS, v2RDM, and CP. Error bars denote 95% confidence intervals (approximately 2$\sigma$) computed from these measurements.
  • Figure 3: Energy of the 7th excited state of H$_6$ as a function of the weight parameter $w$. Each data point represents an average over 50 independent runs. At larger $w$, CP exhibits a smaller standard deviation than the FCS results, demonstrating improved statistical stability of the energy.
  • Figure 4: Dissociation curve and energy error for the H$_4$ molecule obtained on the IBM fez quantum device. (a) Potential energy curves from full configuration interaction (FCI), FCS, and CP ($w = 1$ and $w = 0.001$) as functions of bond distance. (b) Absolute energy error of FCS and CP ($w = 1$ and $w = 0.001$) relative to FCI on a logarithmic scale. CP maintains chemical accuracy across most bond distances, while the FCS results exhibit significantly larger deviations.
  • Figure 5: Lowest eigenvalues of the two-electron reduced density matrix (2-RDM) along the H$_4$ dissociation curve shown in Fig. \ref{['fig:h4_dissociation']}, obtained from FCS and CP. The FCS results exhibit unphysical negative eigenvalues, while CP enforces positivity under all conditions. Note that the lowest eigenvalue from CP with any $w$ is effectively zero, and hence, $w=1$ and $w=0.001$ are not shown separately.