The fidelity susceptibility serves as a universal probe for quantum phase transitions, offering an order-parameter-free metric that captures ground-state sensitivity to Hamiltonian perturbations and exhibits critical scaling. Classical computation of this quantity, however, is limited by exponential Hilbert space growth and correlation divergence near criticality, restricting analyses to small or specialized systems. Here, we present a quantum algorithm that achieves efficient and Heisenberg-limited estimation of fidelity susceptibility through a novel resolvent reformulation and pseudoinverse block encoding leveraging quantum singular value transformation and amplitude estimation for norm evaluation. This constitutes the first quantum algorithm for fidelity susceptibility with optimal precision scaling. Moreover, for frustration-free Hamiltonians, we show that the resolvent can be approximated with a further quadratic speedup. Beyond fidelity susceptibility, our resolvent-based framework applies broadly to observables expressible through spectral sums or linear response functions, suggesting a systematic approach to quantum algorithm design for many-body physics. Our Letter bridges quantum many-body physics and algorithmic design, enabling scalable exploration of quantum criticality with applications in materials simulation and metrology on fault-tolerant quantum platforms.The fidelity susceptibility serves as a universal probe for quantum phase transitions, offering an order-parameter-free metric that captures ground-state sensitivity to Hamiltonian perturbations and exhibits critical scaling. Classical computation of this quantity, however, is limited by exponential Hilbert space growth and correlation divergence near criticality, restricting analyses to small or specialized systems. Here, we present a quantum algorithm that achieves efficient and Heisenberg-limited estimation of fidelity susceptibility through a novel resolvent reformulation and pseudoinverse block encoding leveraging quantum singular value transformation and amplitude estimation for norm evaluation. This constitutes the first quantum algorithm for fidelity susceptibility with optimal precision scaling. Moreover, for frustration-free Hamiltonians, we show that the resolvent can be approximated with a further quadratic speedup. Beyond fidelity susceptibility, our resolvent-based framework applies broadly to observables expressible through spectral sums or linear response functions, suggesting a systematic approach to quantum algorithm design for many-body physics. Our Letter bridges quantum many-body physics and algorithmic design, enabling scalable exploration of quantum criticality with applications in materials simulation and metrology on fault-tolerant quantum platforms.

Physical Review LettersFeb 4, 20260 citations

Beyond Heisenberg Limit Quantum Metrology through Quantum Signal Processing

Yulong Dong, Jonathan A. Gross, Murphy Yuezhen Niu

Leveraging quantum effects in metrology such as entanglement and coherence allows one to measure parameters with enhanced sensitivity. However, time-dependent noise can disrupt such Heisenberg-limited amplification. We propose a quantum-metrology method based on the quantum-signal-processing framework to overcome these realistic noise-induced limitations in practical quantum metrology. Our algorithm separates the gate parameter $φ$ ~(single-qubit Z phase) that is susceptible to time-dependent error from the target gate parameter $θ$ ~(swap-angle between |10> and |01> states) that is largely free of time-dependent error. Our method achieves an accuracy of $10^{-4}$ radians in standard deviation for learning $θ$ in superconducting-qubit experiments, outperforming existing alternative schemes by two orders of magnitude. We also demonstrate the increased robustness in learning time-dependent gate parameters through fast Fourier transformation and sequential phase difference. We show both theoretically and numerically that there is an interesting transition of the optimal metrology variance scaling as a function of circuit depth $d$ from the pre-asymptotic regime $d \ll 1/θ$ to Heisenberg limit $d \to \infty$ . Remarkably, in the pre-asymptotic regime our method's estimation variance on time-sensitive parameter $φ$ scales faster than the asymptotic Heisenberg limit as a function of depth, $\text{Var}(\hatφ)\approx 1/d^4$ . Our work is the first quantum-signal-processing algorithm that demonstrates practical application in laboratory quantum computers.Leveraging quantum effects in metrology such as entanglement and coherence allows one to measure parameters with enhanced sensitivity. However, time-dependent noise can disrupt such Heisenberg-limited amplification. We propose a quantum-metrology method based on the quantum-signal-processing framework to overcome these realistic noise-induced limitations in practical quantum metrology. Our algorithm separates the gate parameter $φ$~(single-qubit Z phase) that is susceptible to time-dependent error from the target gate parameter $θ$~(swap-angle between |10> and |01> states) that is largely free of time-dependent error. Our method achieves an accuracy of $10^{-4}$ radians in standard deviation for learning $θ$ in superconducting-qubit experiments, outperforming existing alternative schemes by two orders of magnitude. We also demonstrate the increased robustness in learning time-dependent gate parameters through fast Fourier transformation and sequential phase difference. We show both theoretically and numerically that there is an interesting transition of the optimal metrology variance scaling as a function of circuit depth $d$ from the pre-asymptotic regime $d \ll 1/θ$ to Heisenberg limit $d \to \infty$. Remarkably, in the pre-asymptotic regime our method's estimation variance on time-sensitive parameter $φ$ scales faster than the asymptotic Heisenberg limit as a function of depth, $\text{Var}(\hatφ)\approx 1/d^4$. Our work is the first quantum-signal-processing algorithm that demonstrates practical application in laboratory quantum computers.

arXiv:2209.11207Sep 22, 20221 citation

Certified Fidelity Susceptibility from Classical Shadows

Kiran Adhikari

Fidelity susceptibility is a useful probe of quantum phase transitions and quantum metrology, but its direct evaluation on a quantum device is challenging. By integrating Krylov-subspace methods with a resolvent-based reformulation, we present a method for certifying fidelity susceptibility using randomized single-copy measurements across repeated ground-state preparations. The resulting approximations form monotone lower bounds to the exact fidelity susceptibility and converge geometrically. We also present applications to metrology and linear static susceptibilities.Fidelity susceptibility is a useful probe of quantum phase transitions and quantum metrology, but its direct evaluation on a quantum device is challenging. By integrating Krylov-subspace methods with a resolvent-based reformulation, we present a method for certifying fidelity susceptibility using randomized single-copy measurements across repeated ground-state preparations. The resulting approximations form monotone lower bounds to the exact fidelity susceptibility and converge geometrically. We also present applications to metrology and linear static susceptibilities.

quant-pharXiv:2608.24732Aug 26, 20260 citations

Quantum Monte Carlo Simulations of Fidelity at Magnetic Quantum Phase Transitions

David R. Schwandt, Fabien Alet, Sylvain Capponi

When a system undergoes a quantum phase transition, the ground-state wave function shows a change of nature, which can be monitored using the fidelity concept. We introduce two quantum Monte Carlo schemes that allow the computation of fidelity and its susceptibility for large interacting many-body systems. These methods are illustrated on a two-dimensional Heisenberg model, where fidelity estimators show marked behavior at two successive quantum phase transitions. We also develop a scaling theory which relates the divergence of the fidelity susceptibility to the critical exponent of the correlation length. A good agreement is found with the numerical results.When a system undergoes a quantum phase transition, the ground-state wave function shows a change of nature, which can be monitored using the fidelity concept. We introduce two quantum Monte Carlo schemes that allow the computation of fidelity and its susceptibility for large interacting many-body systems. These methods are illustrated on a two-dimensional Heisenberg model, where fidelity estimators show marked behavior at two successive quantum phase transitions. We also develop a scaling theory which relates the divergence of the fidelity susceptibility to the critical exponent of the correlation length. A good agreement is found with the numerical results.

arXiv:0907.0191Physical Review LettersOct 19, 2009134 citations

Quantum critical scaling of fidelity susceptibility

A. Fabricio Albuquerque, Fabien Alet, Clément Sire +1 more

The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.

arXiv:0912.2689Physical Review BFeb 18, 2010211 citations

Provable Quantum Advantage for Dynamical Phase Transition

Jue Xu, Xiao Yuan, Qi Zhao

The universal scaling of critical behavior in phase transitions is a cornerstone of physics. Dynamical quantum phase transitions (DQPTs) are their nonequilibrium analogues: abrupt nonanalyticities that emerge as a quantum system evolves in time. Yet the hardness and cost of detecting this phenomenon remain largely unexplored. We prove that estimating DQPT to a certain precision is intractable even for quantum computers, whereas deciding a subsystem variant of DQPT is as hard as simulating generic quantum circuits, implying a provable exponential quantum advantage. Furthermore, to search for critical times of local DQPTs, we show a quadratically faster quantum algorithm that estimates observables of Hamiltonian dynamics at multiple time points with Heisenberg-limited precision and sublinear scaling in the number of time points. Moreover, through encoding classical evolution into quantum dynamics, our framework enables broader quantum speedups for detecting anomalous phenomena in classical systems.The universal scaling of critical behavior in phase transitions is a cornerstone of physics. Dynamical quantum phase transitions (DQPTs) are their nonequilibrium analogues: abrupt nonanalyticities that emerge as a quantum system evolves in time. Yet the hardness and cost of detecting this phenomenon remain largely unexplored. We prove that estimating DQPT to a certain precision is intractable even for quantum computers, whereas deciding a subsystem variant of DQPT is as hard as simulating generic quantum circuits, implying a provable exponential quantum advantage. Furthermore, to search for critical times of local DQPTs, we show a quadratically faster quantum algorithm that estimates observables of Hamiltonian dynamics at multiple time points with Heisenberg-limited precision and sublinear scaling in the number of time points. Moreover, through encoding classical evolution into quantum dynamics, our framework enables broader quantum speedups for detecting anomalous phenomena in classical systems.

quant-pharXiv:2606.30396Jun 30, 20260 citations

φ

~(single-qubit Z phase) that is susceptible to time-dependent error from the target gate parameter

θ

~(swap-angle between |10> and |01> states) that is largely free of time-dependent error. Our method achieves an accuracy of

10^{-4}

radians in standard deviation for learning

θ

in superconducting-qubit experiments, outperforming existing alternative schemes by two orders of magnitude. We also demonstrate the increased robustness in learning time-dependent gate parameters through fast Fourier transformation and sequential phase difference. We show both theoretically and numerically that there is an interesting transition of the optimal metrology variance scaling as a function of circuit depth

d

from the pre-asymptotic regime

d \ll 1/θ

to Heisenberg limit

d \to \infty

. Remarkably, in the pre-asymptotic regime our method's estimation variance on time-sensitive parameter

φ

scales faster than the asymptotic Heisenberg limit as a function of depth,

\text{Var}(\hatφ)\approx 1/d^4

. Our work is the first quantum-signal-processing algorithm that demonstrates practical application in laboratory quantum computers.