Certified Fidelity Susceptibility from Classical Shadows

Technical Summary: Certified Fidelity Susceptibility from Classical Shadows

Problem Statement
Fidelity susceptibility ($\chi_F$) serves as a critical probe for quantum phase transitions and quantum metrology, offering detection capabilities without prior knowledge of order parameters. However, its direct evaluation on quantum devices is hindered by two primary challenges:

1. Spectral Requirements: Standard perturbative expansions require the full excited-state spectrum and transition matrix elements, which are inaccessible in typical ground-state preparation scenarios.
2. Noise Amplification: Finite-difference approaches, which estimate $\chi_F$ via fidelities between nearby ground states, amplify statistical noise.
   While classical shadow tomography offers a pathway to estimate many observables from few measurements, reconstructing $\chi_F$ directly from single-copy measurements of a reference ground state has remained an open problem. Existing methods, such as variational approaches or Quantum Singular Value Transformation (QSVT), often demand deep circuits or lack guarantees in practical settings.

Methodology
The author proposes a framework that integrates Krylov-subspace methods with a resolvent-based reformulation of fidelity susceptibility. The core strategy involves transforming the non-local operator problem into a classical moment-reconstruction problem.

1. Resolvent Reformulation:
   Instead of expanding $\chi_F$ over the full spectrum, the author defines a connected perturbation operator $B = H_I - \langle \Psi_0 | H_I | \Psi_0 \rangle I$ and a state $|\phi\rangle = B|\Psi_0\rangle$. The fidelity susceptibility is rewritten as:
   $$\chi_F = \langle \phi | (K^+)^2 | \phi \rangle$$
   where $K = H - E_0 I$ and $K^+$ is the Moore–Penrose pseudoinverse. This formulation avoids explicit dependence on excited states but retains the non-local operator $(K^+)^2$.

2. Krylov Subspace Approximation:
   To handle the non-local operator, the method approximates the vector $|y\rangle = K^+|\phi\rangle$ within an order-$n$ Krylov subspace $\mathcal{V}_n = \text{span}\{K|\phi\rangle, \dots, K^n|\phi\rangle\}$. The exact susceptibility is the squared norm $\|y\|^2$. The approximation $\chi_F^{(n)}$ is defined as the squared norm of the orthogonal projection of $|y\rangle$ onto $\mathcal{V}_n$.

   This leads to a monotone hierarchy of lower bounds:
   $$0 \le \chi_F^{(1)} \le \chi_F^{(2)} \le \dots \le \chi_F$$
   The $n$-th order approximation is determined entirely by the response moments $M_k = \langle \phi | K^k | \phi \rangle$. Specifically, $\chi_F^{(n)} = \mathbf{m}_n^\dagger \Gamma_n^{-1} \mathbf{m}_n$, where $\mathbf{m}_n$ is the moment vector and $\Gamma_n$ is the Gram (Hankel) matrix constructed from moments $M_0$ through $M_{2n}$.

3. Classical Shadow Estimation:
   Since each moment $M_k$ is a linear functional of the ground-state density matrix $\rho_0$, they can be estimated simultaneously from a single set of randomized single-copy measurements (classical shadows). The protocol uses independent random single-qubit Pauli measurements to generate snapshots.

   The author derives a certified lower bound $B^{(n)}$ that accounts for statistical errors in the estimated moments. If the estimated moments satisfy a uniform error bound $\eta$, the true fidelity susceptibility is guaranteed (with probability $1-\delta$) to satisfy:
   $$\chi_F \ge \Omega^{-2} \left[ \tilde{\chi}_F^{(n)} - L_n \eta - R_n \eta^2 \right]$$
   where $\tilde{\chi}_F^{(n)}$ is the reconstruction from noisy moments, and $L_n, R_n$ are penalty terms dependent on the condition number of the estimated Gram matrix and the error $\eta$.

Key Results and Theoretical Guarantees

• Monotonicity and Certification: The method provides a sequence of certified lower bounds that monotonically increase toward the exact value. The certification is rigorous, relying on the statistical properties of the shadow data.
• Geometric Convergence: The error $\chi_F - \chi_F^{(n)}$ converges geometrically with the Krylov order $n$. The convergence rate is governed by the condition number of the response spectrum ($\kappa_\phi$) rather than the full Hamiltonian spectrum, potentially allowing for rapid convergence in systems with spectrally sparse responses.
• Sample Complexity: The number of shots required to estimate $2n+1$ moments simultaneously scales logarithmically with the number of moments ($O(\log n)$), making the approach efficient for moderate Krylov orders.
• Robustness: The framework is extended to handle imperfect ground-state preparations, where the certification remains valid with a modified error budget.

Applications
The paper demonstrates two immediate applications:

1. Quantum Metrology: Since the Quantum Fisher Information (QFI) for a pure state is $F_\lambda = 4\chi_F$, the certified lower bound on $\chi_F$ directly yields a statistically certified lower bound on the QFI.
2. Linear Static Susceptibilities: The construction is generalized to linear static susceptibilities ($\chi_O$), which quantify the response to static external fields. This requires a shifted Krylov subspace and moments, yielding a similar monotone lower bound hierarchy.

Significance and Claims
The author claims that this work provides the first method to certify fidelity susceptibility using only randomized single-copy measurements of a ground state, eliminating the need for excited-state spectra or deep circuits.

• Operational Value: By providing a hierarchy of lower bounds that converge geometrically, the method offers a practical tool for detecting quantum phase transitions and assessing metrological utility on near-term quantum devices.
• Efficiency: The approach reduces the quantum estimation problem to a classical moment-reconstruction problem, leveraging the efficiency of classical shadow tomography.
• Future Directions: The author notes that while the current work focuses on specific functions ($1/x^2$ and $1/x$), the framework could be extended to general functions $f(x)$ and mixed states. They also highlight the trade-off between sample complexity and the choice of measurement ensemble (Pauli vs. Clifford) as a subject for future analysis.

The paper concludes that this integration of Krylov techniques with resolvent formulations and classical shadows offers a robust, scalable, and certifiable pathway for characterizing quantum systems on quantum hardware.