The Efficiency Frontier: Classical Shadows versus Direct Quantum Measurement

Abstract

Interfacing quantum and classical processors is an important subroutine in full-stack quantum algorithms. The so-called ``classical shadow'' method efficiently extracts essential classical information from quantum states, enabling the prediction of many properties of a quantum system from only a few measurements. However, for a small number of highly non-local observables, or when classical post-processing power is limited, the classical shadow method is not always the most efficient choice. Here, we address this issue quantitatively by performing a full-stack resource analysis that compares classical shadows with direct quantum measurement. Under certain assumptions, our analysis illustrates an efficiency frontier between classical shadows and direct quantum measurement in the information-extraction stage. For observables expressed as linear combinations of Pauli matrices, the classical shadow method outperforms direct measurement when the number of observables is large and the Pauli weight is small. For observables in the form of large Hermitian sparse matrices, the classical shadow method shows an advantage when the number of observables, the sparsity of the matrix, and the number of qubits fall within a certain range. The key parameters influencing this behavior include the number of qubits $n$, observables $M$, sparsity $k$, Pauli weight $w$, accuracy requirement $ε$, and failure tolerance $δ$. We also compare the resource consumption of the two methods on different types of quantum computers and identify break-even points where the classical shadow method becomes more efficient, which vary depending on the hardware. This paper opens a new avenue for quantitatively designing optimal strategies for hybrid quantum-classical tomography and provides practical insights for selecting the most suitable quantum measurement approach in real-world applications.