[Paper Review] Adaptive Pauli Shadows for Energy Estimation

Charles Hadfield|arXiv (Cornell University)|May 25, 2021

Quantum Computing Algorithms and Architecture5 references28 citations

TL;DR

Adaptive Pauli Shadows (APS) augment locally-biased classical shadows with a small amount of classical computation to improve energy estimation of quantum Hamiltonians, achieving competitive or superior accuracy versus existing methods depending on encoding and system size.

ABSTRACT

Locally-biased classical shadows allow rapid estimation of energies of quantum Hamiltonians. Recently, derandomised classical shadows have emerged claiming to be even more accurate. This accuracy comes at a cost of introducing classical computing resources into the energy estimation procedure. This present note shows, by adding a fraction of this classical computing resource to the locally-biased classical shadows setting, that the modified algorithm, termed Adaptive Pauli Shadows is state-of-the-art for energy estimation.

Motivation & Objective

• Motivate efficient energy estimation for quantum Hamiltonians in variational quantum algorithms.
• Extend locally-biased classical shadows (LBCS) with a controlled amount of classical computation.
• Develop the Adaptive Pauli Shadows (APS) framework to optimize measurement choices.
• Benchmark APS against CS, LBCS, and derandomized shadows across small molecular Hamiltonians and common fermionic encodings.

Proposed method

• Represent the Hamiltonian H as a Pauli sum H = sum_P alpha_P P over n qubits.
• Use measurements in Pauli bases B = tensor_i B_i to estimate Tr(P rho) for covered Paulis P.
• Algorithm 1: iteratively update mu_P for each Pauli term P and return sum_P alpha_P mu_P as energy estimate.
• Algorithm 2: per-qubit basis selection by solving a convex optimization to minimize sum_B c_B / beta(B), yielding beta(B) = sqrt(c_B)/sum sqrt(c_B) when sum c_B > 0.
• Leverage a diagonal cost function to justify independent per-qubit basis distributions (product form).
• Analyze runtime: O(n_H * n) where n_H is the number of nonzero Pauli terms in H, improving over some derandomization approaches.

Experimental results

Research questions

• RQ1 How does adding a calibrated amount of classical computation to LBCS (APS) affect energy estimation accuracy for quantum Hamiltonians?
• RQ2 How does APS compare to CS, LBCS, and derandomized shadows across different molecular Hamiltonians and fermionic encodings (JW, Parity, BK)?
• RQ3 What is the computational cost and scalability of APS relative to competing methods?
• RQ4 Is the APS approach robust to noise similarly to standard classical shadows in the energy estimation setting?

Key findings

• APS provides state-of-the-art or competitive energy estimation accuracy in many tested cases when classical computation is augmented to LBCS.
• Across several molecules and encodings, APS often yields lower or comparable average errors to CS, LBCS, and sometimes improves on derandomized shadows under certain conditions.
• The reported results cover H2, LiH, BeH2, H2O, and NH3 mapped to 8–16 qubits with JW, Parity, and BK encodings, illustrating performance trends across system size and encoding.
• The approach achieves a balance between measurement-based quantum resources and classical computation, with runtime scaling favorable compared to some derandomization methods.