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Fermionic-Adapted Shadow Tomography for dynamical correlation functions

Taehee Ko, Mancheon Han, Hyowon Park, Sangkook Choi

Abstract

Dynamical correlation functions are essential for characterizing the response of the quantum many-body systems to the external perturbation. As their calculation is classically intractible in general, quantum algorithms are promising in this aspect, but most rely on brute force measurement strategies that evaluate one body observable pair per circuit. In this work, we introduce Fermionic-Adapted Shadow Tomography (FAST) protocols, a new framework for the efficient calculation of multiple dynamical correlation functions. The key idea is to reformulate these functions into forms that are compatible with shadow tomography techniques. The circuits in our protocols require at most two-copy measurements with uncontrolled Hamiltonian simulation. We show that the proposed protocols enhance sample efficiency and/or reduce the number of measurement circuits by an order of one or two with respect to the number of qubits across a range of scenarios.

Fermionic-Adapted Shadow Tomography for dynamical correlation functions

Abstract

Dynamical correlation functions are essential for characterizing the response of the quantum many-body systems to the external perturbation. As their calculation is classically intractible in general, quantum algorithms are promising in this aspect, but most rely on brute force measurement strategies that evaluate one body observable pair per circuit. In this work, we introduce Fermionic-Adapted Shadow Tomography (FAST) protocols, a new framework for the efficient calculation of multiple dynamical correlation functions. The key idea is to reformulate these functions into forms that are compatible with shadow tomography techniques. The circuits in our protocols require at most two-copy measurements with uncontrolled Hamiltonian simulation. We show that the proposed protocols enhance sample efficiency and/or reduce the number of measurement circuits by an order of one or two with respect to the number of qubits across a range of scenarios.

Paper Structure

This paper contains 1 section, 12 theorems, 42 equations, 2 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Numerical result

Key Result

Lemma 1

Given scheduled $N_s=\mathcal{O}(\frac{\log\frac{1}{\delta}}{\epsilon^2})$ circuit samplings, let $n_{+}$ and $n_{-}$ be the number of outcomes being $\ket{0}$ and $\ket{1}$ in the ancilla qubit, respectively. Then, the following holds with probability at least $1-\delta$, and either $\rho_{+}$ or $\rho_{-}$ is prepared with at least $\frac{N_s}{2}$ times. Consequently, in either of the fermionic

Figures (2)

  • Figure 1: Circuits for single-copy and two-copy measurements used in our protocols for estimating the dynamical correlation functions. The gate $B$ denotes a given Pauli string. In \ref{['fig: circuit a']}, a set of Clifford unitaries for mutually commuting observables, denoted as $U$, is used. In \ref{['fig: circuit b']}, \ref{['fig: circuit d']}, we use the dynamic circuit technique for random Pauli measurement kanno2025efficient. Following the same notation in kanno2025efficient, we denote by $G,F$, the operations for generating the probability distribution for which Pauli basis is sampled, and receiving the feedback for constructing the corresponding Pauli basis. In \ref{['fig: circuit e']}, we perform the Bell sampling to throw out observables of negligible expectation magnitudes huang2021informationking2025triply. In \ref{['fig: circuit f']}, we also apply Bell sampling in a certain case, while in the other, we use a variant that requires measurement in a problem-dependent basis, as will be shown in \ref{['lem: ours']}.
  • Figure 2: Simulation results that show the maximum of errors between exact and estimated correlation functions of the form in \ref{['eq: test example']}, which are obtained from the FAST and brute-force measurements. The parameters $(a,b)$ are set to $(1,2)$ and $(2,2)$ for the top and bottom panels, respectively. The y-axis indicates the maximum error between exact and estimated correlation functions.

Theorems & Definitions (23)

  • Example 1: Green's functions
  • Example 2: Density-Density Response
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 1
  • ...and 13 more