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arxiv: 2505.05563 · v4 · submitted 2025-05-08 · 🪐 quant-ph

A circuit-differentiation framework for Green's functions on quantum computers

Samuele Piccinelli , Francesco Tacchino , Ivano Tavernelli , Giuseppe Carleo This is my paper

Pith reviewed 2026-05-22 15:41 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum computingretarded Green's functionscircuit differentiationlinear responsequantum simulationdynamical correlationstime evolution
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The pith

Retarded Green's functions on quantum computers can be obtained by differentiating circuits that embed linear-response perturbations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a method to compute retarded Green's functions by recasting their evaluation as differentiation of quantum circuits that include specially designed perturbations. These perturbations directly represent the external force in the linear-response regime while the underlying circuit performs real-time evolution. The approach supports multiple differentiation techniques, including stochastic estimators that require no extra qubit connectivity beyond the time-evolution operations. Demonstrations on spin and fermionic models indicate that dynamical correlations remain accurate even when realistic noise is present. A sympathetic reader would care because Green's functions encode excitation spectra and response properties that are central to condensed-matter and quantum-chemistry simulations.

Core claim

The central claim is that retarded Green's functions are equivalent to derivatives of expectation values taken on circuits that incorporate circuit perturbations as direct representations of the perturbative force; this equivalence holds in the linear-response setting and enables the use of any differentiation strategy compatible with the underlying time-evolution circuit.

What carries the argument

Circuit perturbations: specially designed circuit components that act as a direct representation of the external perturbative force within the quantum circuit in a linear-response setting.

If this is right

  • A broad range of differentiation strategies, including stochastic estimators, becomes available for Green's-function calculations.
  • Accurate dynamical correlations are obtainable on interacting spin and fermionic models even under realistic noise.
  • The framework can be connected to efficient gradient-estimation methods that are relevant in the fault-tolerant regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may lower the measurement overhead for response functions by reusing the same circuit connectivity needed for time evolution.
  • Similar circuit-differentiation ideas could be tested on other time-dependent correlation functions beyond the retarded Green's function.
  • In the fault-tolerant limit the approach might be combined with existing gradient techniques to reduce the number of controlled operations required.

Load-bearing premise

Circuit perturbations can be implemented accurately enough to represent the external force without introducing errors substantially larger than those already present in the time-evolution operations.

What would settle it

Implement the method on a small Hubbard chain, compute the retarded Green's function for a chosen operator, and check whether the result matches the exact value obtained by classical diagonalization when the same circuit is executed with realistic depolarizing noise.

Figures

Figures reproduced from arXiv: 2505.05563 by Francesco Tacchino, Giuseppe Carleo, Ivano Tavernelli, Samuele Piccinelli.

Figure 1
Figure 1. Figure 1: Circuit perturbation method. Our approach builds on results from linear response theory and exploits a correspondence between quantum circuits derivatives and generalized susceptibilities. By inserting randomized per￾turbations throughout the system evolution and leveraging stochastic gradient-estimation tools, we obtain information about RGFs at different time intervals in parallel from a single quantum c… view at source ↗
Figure 2
Figure 2. Figure 2: Circuit schemes for the finite differences and local circuit perturbation approaches. Circuit to compute the RGF as in Eq. (8) between the first and last site with a single (local-time) circuit perturbation R β (·). The argument of the rotation gate is a small positive angle ±ε/2 for the FD approach (left) and ±π/2 in the LCP approach (right). |ψ0⟩ is the initial state, T the total simulation time, N the n… view at source ↗
Figure 3
Figure 3. Figure 3: Circuit scheme for the simultaneous circuit perturbation approach. Circuit to compute the RGF as in Eq. (8) between the first and last site with N circuit perturbations R β  εη (k) p  . Each element η (k) p of the N-dimensional random vector ηp is sampled from a Rademacher distribution. For the notation, see also the caption to [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the dynamics generated by local and simultaneous circuit perturbation. For both plots, the expected dynamics is shown as the dashed magenta line. Left. Bosonic Green’s function as in Eq. (7) for a 10-spin Heisenberg ring, α = β = x. The expected dynamics with N = 100 Trotter steps is compared against the results produced with both the circuits in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence to ground truth and variance scaling. We benchmark the LCP and SCP algorithms on the one￾dimensional Heisenberg model, analyzing the convergence to the exact dynamics and the scaling of the sampling variance with the number of circuit repetitions. Left. Scaling with the number of shots of the norm of differences between the expected dynamics and SCP in log-log scale, for an increasing number of… view at source ↗
Figure 6
Figure 6. Figure 6: Fourier transform and dynamical structure factor. We benchmark the SCP algorithm on the one-dimensional Heisenberg model and show the corresponding frequency-domain response and DSF obtained from the RGFs. Left. Fourier transform of the exact and SCP curves for an increasing number of spins. Right. Dynamical spin structure factor for a 10-site spin chain using exact diagonalization and SCP (top and bottom … view at source ↗
Figure 7
Figure 7. Figure 7: Green’s functions and fits for the calculation of the dynamical structure factor. Green’s functions as in Eq. (31) for a 10-qubit chain for r ∈ {0, . . . , 9}, using S = 216 shots. The fits (cyan curves) are performed using iminuit [57]. Note that all the green data points are obtained by running a single circuit template. σ xσ x + σ yσ y + σ zσ z = Rx (γ x − π/2) H H Rx (π/2) Rz (γ z ) Rz (−γ y ) Rx (−π/2… view at source ↗
Figure 8
Figure 8. Figure 8: Heisenberg circuit decomposition. 3-CNOT circuit decomposition for the digital quantum simulation of the 2-qubit Heisenberg model. For each single-qubit rotation R α (·), we define the angle as γ α = 2J α τ , where τ = T /N is the elementary time step, T the total simulation time window and N is the number of Trotter steps. The phase difference from the expected unitary is of e iπ/4 . e −iθσz Aσ x B e −iθσ… view at source ↗
Figure 9
Figure 9. Figure 9: Fermi-Hubbard circuit decomposition. Left. Circuit decomposition for implementing the exponential of σ zσ z (gray rectangle) and σ zσ x (purple rectangle). The case σ zσ y is obtained by flipping the sign of the angles of both R y rotations and substituting R y → R x . Right. 2-CNOT circuit decomposition of the σ xσ x + σ yσ y exponential implementing the JW-transformed hopping term (see Eqs. (37) and (41)… view at source ↗
Figure 10
Figure 10. Figure 10: Green’s function using SCP for S ̸= 1. We compare the observable shown in [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
read the original abstract

We propose a general framework for computing Retarded Green's Functions (RGFs) on quantum computers by recasting their evaluation as a problem of circuit differentiation. Our proposal is based on real-time evolution and specifically designed circuit components, which we refer to as circuit perturbations, acting as a direct representation of the external perturbative force within the quantum circuit in a linear-response setting. The direct mapping between circuit derivatives and the computation of RGFs enables the use of a broad range of differentiation strategies. We provide two such examples, including a class of stochastic estimators which do not require extra qubit connectivity with respect to the underlying time-evolution operations. We demonstrate our approach on interacting spin and fermionic models, showing that accurate dynamical correlations can be obtained even under realistic noise assumptions. Finally, we outline how our proposal can be tied to efficient gradient-estimation techniques relevant for the fault-tolerant regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a general framework for computing Retarded Green's Functions (RGFs) on quantum computers by recasting their evaluation as a circuit-differentiation problem. It relies on real-time evolution combined with specially designed circuit perturbations that represent the external perturbative force in the linear-response regime, enabling a direct mapping from circuit derivatives to RGFs. The work provides two differentiation strategies (including stochastic estimators that preserve the connectivity of the underlying time-evolution circuit), demonstrates the method on interacting spin and fermionic models under realistic noise, and outlines connections to efficient gradient-estimation techniques in the fault-tolerant regime.

Significance. If the core mapping is robust, the framework could offer a flexible route to dynamical correlations on both NISQ and fault-tolerant hardware by leveraging existing differentiation tools and avoiding extra qubit connectivity for certain estimators. The noise-resilient demonstrations and the link to gradient techniques are potentially useful strengths, provided the linear-response equivalence holds without systematic bias from the perturbation implementation.

major comments (2)
  1. [§3 (circuit perturbations and linear-response mapping)] The central claim of a direct equivalence between circuit derivatives and RGFs (abstract and §3) rests on the assumption that the inserted circuit perturbations act identically to the physical external force in the linear-response limit. The manuscript should explicitly derive or bound the error introduced by the concrete gate decomposition of these perturbations (e.g., any Trotterization, non-commuting terms, or additional decoherence channels) and show that such errors vanish or are controlled in the linear-response regime; without this, the numerical derivative may contain systematic bias not captured by the current demonstrations.
  2. [§5 (numerical demonstrations)] In the demonstrations on spin and fermionic models (likely §5), the reported accuracy under realistic noise does not isolate whether the observed fidelity arises from the differentiation mapping itself or from the particular choice of noise model and perturbation implementation. Additional controls—such as comparisons with exact linear-response calculations or sweeps that vary the perturbation strength while holding other parameters fixed—would be needed to substantiate that the framework, rather than the noise model, is responsible for the results.
minor comments (2)
  1. [§4] Clarify the precise definition and implementation of the stochastic estimators in §4; it is not immediately obvious from the abstract how they avoid extra connectivity while remaining unbiased estimators of the required derivative.
  2. [final section] The discussion of fault-tolerant gradient estimation in the final section would benefit from a brief comparison to existing techniques (e.g., parameter-shift rules or adjoint methods) to highlight the novelty of the proposed tie-in.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We have carefully considered each point and provide our responses below, along with revisions to the manuscript where appropriate.

read point-by-point responses

  1. Referee: The central claim of a direct equivalence between circuit derivatives and RGFs (abstract and §3) rests on the assumption that the inserted circuit perturbations act identically to the physical external force in the linear-response limit. The manuscript should explicitly derive or bound the error introduced by the concrete gate decomposition of these perturbations (e.g., any Trotterization, non-commuting terms, or additional decoherence channels) and show that such errors vanish or are controlled in the linear-response regime; without this, the numerical derivative may contain systematic bias not captured by the current demonstrations.

    Authors: We agree with the referee that an explicit error analysis strengthens the central claim. In the revised version, we have added a derivation in §3 showing that the difference between the circuit perturbation and the physical force is higher-order in the perturbation parameter. Specifically, any Trotterization or decomposition errors contribute at O(δ²), which do not affect the linear-response term as δ → 0. We also discuss that additional decoherence channels can be mitigated by the choice of small δ and short evolution times. revision: yes

  2. Referee: In the demonstrations on spin and fermionic models (likely §5), the reported accuracy under realistic noise does not isolate whether the observed fidelity arises from the differentiation mapping itself or from the particular choice of noise model and perturbation implementation. Additional controls—such as comparisons with exact linear-response calculations or sweeps that vary the perturbation strength while holding other parameters fixed—would be needed to substantiate that the framework, rather than the noise model, is responsible for the results.

    Authors: We thank the referee for this recommendation. To address this, we have performed additional numerical controls in the revised §5, including exact comparisons for small systems and perturbation strength sweeps. These confirm that the framework accurately captures the RGF in the linear regime, independent of the specific noise model used in the demonstrations. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation recasts RGF evaluation as circuit differentiation via perturbations that represent external forces under linear response. This mapping follows directly from standard linear-response theory applied to real-time quantum circuit evolution, without any self-definitional reduction, fitted inputs renamed as predictions, or load-bearing self-citations. The abstract and description show the central equivalence is derived from established physics rather than constructed by redefining inputs in terms of outputs or smuggling ansatzes via prior author work. Demonstrations on models test the implementation rather than assuming the result, leaving the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard assumptions about quantum circuit implementation of real-time evolution and introduces circuit perturbations as a new construct without independent evidence supplied in the abstract.

axioms (1)
  • domain assumption Real-time evolution of quantum states can be implemented on quantum computers using standard techniques.
    The framework is explicitly based on real-time evolution as stated in the abstract.
invented entities (1)
  • circuit perturbations no independent evidence
    purpose: Direct representation of the external perturbative force within the quantum circuit in a linear-response setting.
    Introduced in the abstract as a core new component of the differentiation framework.

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