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

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

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AbstractWe 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.Featured image: Illustration of the stochastic protocol. Circuit representations of the external perturbation are inserted at different times during the system's evolution, represented here by the clock icons, and the resulting measurements provide entries of the retarded Green's function. The colored points show how information gathered at different times is assembled into the final time-dependent response curve. The central idea is to reconstruct dynamical response from quantum-circuit derivatives rather than by measuring the Green's function directly.Popular summaryUnderstanding how a quantum system responds to an external disturbance is a basic problem across condensed-matter physics, chemistry, and materials science. That response is often captured by quantities called Green's functions, which reveal how excitations move and how a system absorbs energy. These objects are extremely useful, but also notoriously hard to compute — especially on quantum hardware, where standard approaches can require cumbersome controlled operations and additional experimental overhead. Our work shows that there is a different way to think about this problem. Instead of measuring Green's functions directly, we recast them as derivatives of a quantum circuit with respect to inserted perturbations. This creates a bridge between quantum dynamics and circuit differentiation, an area where many efficient tools already exist. Using this viewpoint, we design protocols that can estimate dynamical response functions with modest circuit overhead, including a stochastic method that gathers information about many times in parallel. Benchmark calculations on spin and fermionic lattice models show that the method can recover accurate dynamical correlations and spectral features, even under realistic noise assumptions. The result is both practical and conceptual: practical because it offers new algorithms tailored to near-term devices, and conceptual because it places response-function calculations inside a broader framework of circuit derivatives. This perspective could help unify methods for quantum simulation today while also opening routes toward more powerful implementations on future fault-tolerant quantum computers.► BibTeX data@article{Piccinelli2026circuit, doi = {10.22331/q-2026-04-13-2060}, url = {https://doi.org/10.22331/q-2026-04-13-2060}, title = {A circuit-differentiation framework for {G}reen's functions on quantum computers}, author = {Piccinelli, Samuele and Tacchino, Francesco and Tavernelli, Ivano and Carleo, Giuseppe}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2060}, month = apr, year = {2026} }► References [1] Alexander Miessen, Pauline J. Ollitrault, Francesco Tacchino, and Ivano Tavernelli. ``Quantum algorithms for quantum dynamics''.

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Physical Review A 69, 010301 (2004). https://doi.org/10.1103/physreva.69.010301Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-13 17:39:00: Could not fetch cited-by data for 10.22331/q-2026-04-13-2060 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-13 17:39:00: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe 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.Featured image: Illustration of the stochastic protocol. Circuit representations of the external perturbation are inserted at different times during the system's evolution, represented here by the clock icons, and the resulting measurements provide entries of the retarded Green's function. The colored points show how information gathered at different times is assembled into the final time-dependent response curve. The central idea is to reconstruct dynamical response from quantum-circuit derivatives rather than by measuring the Green's function directly.Popular summaryUnderstanding how a quantum system responds to an external disturbance is a basic problem across condensed-matter physics, chemistry, and materials science. That response is often captured by quantities called Green's functions, which reveal how excitations move and how a system absorbs energy. These objects are extremely useful, but also notoriously hard to compute — especially on quantum hardware, where standard approaches can require cumbersome controlled operations and additional experimental overhead. Our work shows that there is a different way to think about this problem. Instead of measuring Green's functions directly, we recast them as derivatives of a quantum circuit with respect to inserted perturbations. This creates a bridge between quantum dynamics and circuit differentiation, an area where many efficient tools already exist. Using this viewpoint, we design protocols that can estimate dynamical response functions with modest circuit overhead, including a stochastic method that gathers information about many times in parallel. Benchmark calculations on spin and fermionic lattice models show that the method can recover accurate dynamical correlations and spectral features, even under realistic noise assumptions. The result is both practical and conceptual: practical because it offers new algorithms tailored to near-term devices, and conceptual because it places response-function calculations inside a broader framework of circuit derivatives. This perspective could help unify methods for quantum simulation today while also opening routes toward more powerful implementations on future fault-tolerant quantum computers.► BibTeX data@article{Piccinelli2026circuit, doi = {10.22331/q-2026-04-13-2060}, url = {https://doi.org/10.22331/q-2026-04-13-2060}, title = {A circuit-differentiation framework for {G}reen's functions on quantum computers}, author = {Piccinelli, Samuele and Tacchino, Francesco and Tavernelli, Ivano and Carleo, Giuseppe}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2060}, month = apr, year = {2026} }► References [1] Alexander Miessen, Pauline J. Ollitrault, Francesco Tacchino, and Ivano Tavernelli. ``Quantum algorithms for quantum dynamics''.

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

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