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arxiv: 2404.05108 · v3 · pith:EVYAHYLAnew · submitted 2024-04-07 · 🪐 quant-ph · cs.IT· cs.LG· math.IT

Efficient Gradient Estimation for Parameterized Quantum Systems with Lie Algebraic Symmetries

Mohsen Heidari , Masih Mozakka , Wojciech Szpankowski This is my paper

Pith reviewed 2026-05-25 08:21 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITcs.LGmath.IT
keywords gradient estimationparameterized quantum circuitsLie algebraHadamard testshadow tomographyquantum optimizationvariational quantum algorithms
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The pith

A Lie algebraic structure in parameterized quantum circuits allows gradient estimation from a logarithmic number of measurement shots.

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

The paper develops a gradient estimator for parameterized quantum circuits that exploits an underlying Lie algebraic symmetry. The differential of the matrix exponential decomposes the gradient into a linear combination of Hadamard-test expectation values whose coefficients depend only on the circuit parameters. These coefficients are recovered by shadow tomography, so the total number of quantum measurements scales logarithmically with the number of parameters while classical and quantum runtimes stay polynomial. Standard finite-difference and parameter-shift estimators lack this scaling and require far more shots. A sympathetic reader would therefore expect the method to make training of high-dimensional variational quantum models practical on near-term hardware.

Core claim

By analyzing the differential of the matrix exponential for circuits whose generators close under a Lie algebra, the gradient of any observable expectation value is expressed as a linear combination of Hadamard-test expectation values. The numerical coefficients in this linear combination are independent of the quantum state and can be estimated to sufficient accuracy with classical shadow tomography, producing an overall procedure whose measurement cost grows only logarithmically with the number of parameters.

What carries the argument

The differential of the matrix exponential under Lie algebraic parameterization, which produces a linear combination of Hadamard-test observables whose coefficients are recovered by shadow tomography.

If this is right

  • Gradient estimation requires only logarithmically many shots in the number of parameters.
  • Both classical post-processing and quantum circuit execution remain polynomial in system size and parameter count.
  • Measurement cost is exponentially lower than finite-difference or parameter-shift estimators for the same accuracy.
  • Training of parameterized quantum circuits with hundreds of variational parameters becomes feasible on current devices.

Where Pith is reading between the lines

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

  • The same Lie-algebraic decomposition may be reusable for higher-order derivatives or for estimating the quantum Fisher information matrix.
  • Circuits whose generators only approximately close under a Lie algebra could still benefit from bounded-error versions of the estimator.
  • Integration with existing variational quantum algorithms would reduce the dominant shot overhead in hybrid optimization loops.

Load-bearing premise

The circuit parameterization must admit a Lie algebraic structure such that the gradient coefficients depend only on the parameters and can be recovered by shadow tomography.

What would settle it

A concrete parameterized circuit satisfying the Lie algebraic condition for which the true gradient cannot be recovered to constant accuracy with a number of shots that scales only logarithmically with the parameter count.

Figures

Figures reproduced from arXiv: 2404.05108 by Masih Mozakka, Mohsen Heidari, Wojciech Szpankowski.

Figure 1
Figure 1. Figure 1: Hadamard test with backpropagation for measuring the partial derivative of prod￾uct ans¨atze with respect to a parameter asl appearing at layer l. Here U≤l cor￾responds to the first l layers of the ansatz, and U>l to the remaining layers. in addition, X is the X-gate, H is the Hadamard gate, and Rsl is the controlled rotation around Pauli σ sl . either unknown or computationally intractable. Hence, one nee… view at source ↗
Figure 2
Figure 2. Figure 2: This figure illustrates the concept behind the subgroup gradient estimation al￾gorithm. The expression in Theorem 8 points to an alternative representation of the problem. Instead of picturing the gradient as a function in the space of the parameters (the left picture), we can view it as a vector in the landscape of Hadamard tests Ds. In other words, each Dsj is associated with the jth canonical basis vect… view at source ↗
Figure 3
Figure 3. Figure 3: Estimation error for gradient approximation as a function of the number of terms for various Hamiltonians with different numbers of qubits. 5.2.2 Unbiased Estimation via Randomization The infinite sum can be estimated via a randomization technique. We show that the partial derivative in Theorem 7 can be written as an expectation value of a function of the Poisson random variable. Let K be the Poisson rando… view at source ↗
read the original abstract

Gradient estimation is a central challenge in training parameterized quantum circuits (PQCs) for hybrid quantum-classical optimization and learning problems. This difficulty arises from several factors, including the exponential dimensionality of the Hilbert spaces and the information loss in quantum measurements. Existing estimators, such as finite difference and the parameter shift rule, often fail to adequately address these challenges for certain classes of PQCs. In this work, we propose a novel gradient estimation framework that leverages the underlying Lie algebraic structure of PQCs, combined with the Hadamard test. By analyzing the differential of the matrix exponential, we derive an expression for the gradient as a linear combination of expectation values obtained via Hadamard tests. The coefficients in this decomposition depend solely on the circuit's parameterization and can be estimated using state-of-the-art shadow tomography techniques. Hence, our approach enables efficient gradient estimation, requiring a number of measurement shots that scales logarithmically with the number of parameters, and with polynomial classical and quantum time. This is an exponential reduction in the measurement cost and a polynomial speed-up in time compared to existing works.

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

1 major / 0 minor

Summary. The paper claims to provide an efficient gradient estimation method for parameterized quantum circuits (PQCs) admitting Lie algebraic structure. By analyzing the differential of the matrix exponential, the gradient is expressed as a linear combination of Hadamard-test expectation values whose coefficients depend solely on the circuit parameterization and can be recovered via shadow tomography. This yields a logarithmic scaling of measurement shots with the number of parameters, together with polynomial classical and quantum runtime, constituting an exponential reduction in measurement cost relative to prior estimators such as finite differences or the parameter-shift rule.

Significance. If the decomposition is both state-independent in its coefficients and efficiently estimable, the result would materially improve the practicality of variational quantum algorithms on high-dimensional parameter spaces by reducing the dominant measurement overhead from linear or worse to logarithmic in the number of parameters.

major comments (1)
  1. [Abstract / central derivation] Abstract (and the central derivation): the claim that coefficients 'depend solely on the circuit's parameterization' yet 'can be estimated using state-of-the-art shadow tomography techniques' is ambiguous. If the coefficients are functions of θ alone they are classical; if shadow tomography is required then either the coefficients depend on the input state or the decomposition is not fully classical. Either reading directly affects whether the log(N_params) shot scaling follows from standard shadow bounds on a fixed set of observables. This is the load-bearing step for the exponential measurement reduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The central concern is an ambiguity in how the coefficients of the gradient decomposition are described. We clarify the point below and indicate the corresponding revision.

read point-by-point responses

  1. Referee: [Abstract / central derivation] Abstract (and the central derivation): the claim that coefficients 'depend solely on the circuit's parameterization' yet 'can be estimated using state-of-the-art shadow tomography techniques' is ambiguous. If the coefficients are functions of θ alone they are classical; if shadow tomography is required then either the coefficients depend on the input state or the decomposition is not fully classical. Either reading directly affects whether the log(N_params) shot scaling follows from standard shadow bounds on a fixed set of observables. This is the load-bearing step for the exponential measurement reduction.

    Authors: We agree that the original wording in the abstract is ambiguous and could be misread. The coefficients are functions of the parameterization θ alone and are independent of the input state; they arise from the differential of the matrix exponential in the Lie algebra and can be viewed as classical numbers once the parameterization is fixed. Direct classical evaluation of these coefficients, however, becomes intractable for high-dimensional Lie algebras. We therefore encode the parameterization into an auxiliary quantum state (independent of the data state) and apply shadow tomography to estimate the coefficients as expectation values of a fixed collection of observables whose number is polynomial in the number of parameters. Standard shadow bounds then yield an overall logarithmic shot cost in the number of parameters. We will revise the abstract and the derivation section to state explicitly that the coefficients are state-independent classical functions of θ, that they are recovered via shadow tomography on an auxiliary state, and that the logarithmic scaling follows directly from the standard shadow-tomography sample complexity for a polynomial number of fixed observables. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via Lie algebra and standard primitives

full rationale

The paper derives the gradient via the differential of the matrix exponential applied to the Lie algebraic structure of the PQC, yielding a linear combination of Hadamard-test expectations whose coefficients are stated to depend solely on the parameterization. This is a direct mathematical reduction using established operations (matrix exponential differential, Hadamard test) and external estimation techniques (shadow tomography), without any reduction of the claimed log(N) shot scaling or polynomial runtime to a fitted parameter or self-defined quantity. No self-citation is invoked as load-bearing for the central claim, and the result does not rename or smuggle an ansatz equivalent to the output. The derivation chain remains independent of the target efficiency claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the domain assumption that the circuit parameterization possesses exploitable Lie algebraic structure; no free parameters or invented entities are explicitly introduced in the provided text.

axioms (1)
  • domain assumption Parameterized quantum circuits possess Lie algebraic symmetries that permit expressing the gradient of the matrix exponential as a linear combination of Hadamard-test expectations.
    This structure is invoked as the foundation for the entire gradient decomposition described in the abstract.

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Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages · 2 internal anchors

  1. [1]

    Local, expressive, quantum-number-preserving vqe ansätze for fermionic systems

    Gian-Luca R Anselmetti, David Wierichs, Christian Gogolin, and Robert M Parrish. Local, expressive, quantum-number-preserving vqe ansätze for fermionic systems. New Journal of Physics , 23(11):113010, November 2021

    work page 2021

  2. [2]

    A new proof for the existence of mutually unbiased bases

    Somshubhro Bandyopadhyay, P. Oscar Boykin, Vwani Roychowdhury, and Farrokh Vatan. A new proof for the existence of mutually unbiased bases. quant-ph/0103162 , 2001

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  3. [3]

    Leonardo Banchi and Gavin E. Crooks. Measuring analytic gradients of general quantum evolution with the stochastic parameter shift rule. Quantum , 5:386, January 2021

    work page 2021

  4. [4]

    Parameterized quantum circuits as machine learning models

    Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini. Parameterized quantum circuits as machine learning models. Quantum Science and Technology , 4(4):043001, 2019

    work page 2019

  5. [5]

    Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C

    M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. Variational quantum algorithms. Nature Reviews Physics , 3(9):625--644, aug 2021

    work page 2021

  6. [6]

    A. R. Calderbank, E. M Rains, P. W. Shor, and N. J. A. Sloane. Quantum error correction via codes over gf(4). August 1996

    work page 1996

  7. [7]

    Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J

    M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature Communications , 12(1), March 2021

    work page 2021

  8. [8]

    Efficient quantum measurement of pauli operators in the presence of finite sampling error

    Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl Campbell, and Stephen Brierley. Efficient quantum measurement of pauli operators in the presence of finite sampling error. arXiv:1908.06942 , 2020

    work page arXiv 1908

  9. [9]

    Variational quantum algorithm for molecular geometry optimization

    Alain Delgado, Juan Miguel Arrazola, Soran Jahangiri, Zeyue Niu, Josh Izaac, Chase Roberts, and Nathan Killoran. Variational quantum algorithm for molecular geometry optimization. Physical Review A , 104(5):052402, November 2021

    work page 2021

  10. [10]

    DiVincenzo, D.W

    D.P. DiVincenzo, D.W. Leung, and B.M. Terhal. Quantum data hiding. IEEE Transactions on Information Theory , 48(3):580--598, March 2002

    work page 2002

  11. [11]

    A quantum approximate optimization algorithm, 2014

    Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm, 2014

    work page 2014

  12. [12]

    The adjoint is all you need: Characterizing barren plateaus in quantum ansätze

    Enrico Fontana, Dylan Herman, Shouvanik Chakrabarti, Niraj Kumar, Romina Yalovetzky, Jamie Heredge, Shree Hari Sureshbabu, and Marco Pistoia. The adjoint is all you need: Characterizing barren plateaus in quantum ansätze. September 2023

    work page 2023

  13. [13]

    Classification with quantum neural networks on near term processors

    Edward Farhi and Hartmut Neven. Classification with quantum neural networks on near term processors. February 2018

    work page 2018

  14. [14]

    Grimsley, Sophia E

    Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature Communications , 10(1), July 2019

    work page 2019

  15. [15]

    Stabilizer codes and quantum error correction

    Daniel Gottesman. Stabilizer codes and quantum error correction . California Institute of Technology, 1997

    work page 1997

  16. [16]

    Córcoles, Kristan Temme, Aram W

    Vojtěch Havlíček, Antonio D. Córcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta. Supervised learning with quantum-enhanced feature spaces. Nature , 567(7747):209--212, March 2019

    work page 2019

  17. [17]

    Grama, and Wojciech Szpankowski

    Mohsen Heidari, Ananth Y. Grama, and Wojciech Szpankowski. Toward physically realizable quantum neural networks. Association for the Advancement of Articial Intelligence (AAAI) , 2022

    work page 2022

  18. [18]

    Predicting many properties of a quantum system from very few measurements

    Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics 16, 1050--1057 (2020) , February 2020

    work page 2020

  19. [19]

    Harrow and John C

    Aram W. Harrow and John C. Napp. Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms. Physical Review Letters , 126(14):140502, apr 2021

    work page 2021

  20. [20]

    A theoretical framework for learning from quantum data

    Mohsen Heidari, Arun Padakandla, and Wojciech Szpankowski. A theoretical framework for learning from quantum data. In IEEE International Symposium on Information Theory ( ISIT ) , 2021

    work page 2021

  21. [21]

    Learning k-qubit quantum operators via pauli decomposition

    Mohsen Heidari and Wojciech Szpankowski. Learning k-qubit quantum operators via pauli decomposition. In Francisco Ruiz, Jennifer Dy, and Jan-Willem van de Meent, editors, Proceedings of The 26th International Conference on Artificial Intelligence and Statistics , volume 206 of Proceedings of Machine Learning Research . PMLR, 25--27 Apr 2023

    work page 2023

  22. [22]

    New bounds on quantum sample complexity of measurement classes

    Mohsen Heidari and Wojciech Szpankowski. New bounds on quantum sample complexity of measurement classes. In 2024 IEEE International Symposium on Information Theory (ISIT) , pages 1515--1520. IEEE, July 2024

    work page 2024

  23. [23]

    From the quantum approximate optimization algorithm to a quantum alternating operator ansatz

    Stuart Hadfield, Zhihui Wang, Bryan O’Gorman, Eleanor Rieffel, Davide Venturelli, and Rupak Biswas. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms , 12(2):34, February 2019

    work page 2019

  24. [24]

    Benjamin

    Tyson Jones, Suguru Endo, Sam McArdle, Xiao Yuan, and Simon C. Benjamin. Variational quantum algorithms for discovering hamiltonian spectra. Physical Review A , 99(6):062304, June 2019

    work page 2019

  25. [25]

    Chow, and Jay M

    Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature , 549(7671):242--246, sep 2017

    work page 2017

  26. [26]

    Mutually unbiased binary observable sets onNqubits

    Jay Lawrence, C aslav Brukner, and Anton Zeilinger. Mutually unbiased binary observable sets onNqubits . Physical Review A , 65(3), feb 2002

    work page 2002

  27. [27]

    Coles, and M

    Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo. Diagnosing barren plateaus with tools from quantum optimal control. Quantum , 6:824, September 2022

    work page 2022

  28. [28]

    Differentiable learning of quantum circuit born machines

    Jin-Guo Liu and Lei Wang. Differentiable learning of quantum circuit born machines. Physical Review A , 98(6):062324, December 2018

    work page 2018

  29. [29]

    How does noise help robustness? explanation and exploration under the neural sde framework

    Xuanqing Liu, Tesi Xiao, Si Si, Qin Cao, Sanjiv Kumar, and Cho-Jui Hsieh. How does noise help robustness? explanation and exploration under the neural sde framework. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) , pages 279--287, 2020

    work page 2020

  30. [30]

    McClean, Sergio Boixo, Vadim N

    Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature Communications , 9(1), November 2018

    work page 2018

  31. [31]

    Quantum analog-digital conversion

    Kosuke Mitarai, Masahiro Kitagawa, and Keisuke Fujii. Quantum analog-digital conversion. Physical Review A , 99(1):012301, January 2019

    work page 2019

  32. [32]

    Mitarai, M

    K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii. Quantum circuit learning. Physical Review A , 98(3):032309, sep 2018

    work page 2018

  33. [33]

    Ollitrault, and Ivano Tavernelli

    Alexander Miessen, Pauline J. Ollitrault, and Ivano Tavernelli. Quantum algorithms for quantum dynamics: A performance study on the spin-boson model. Physical Review Research , 3(4):043212, dec 2021

    work page 2021

  34. [34]

    Love, Al \' a n Aspuru-Guzik, and Jeremy L

    Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Al \' a n Aspuru-Guzik, and Jeremy L. O'Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications , 5(1), jul 2014

    work page 2014

  35. [35]

    Lie groups: An Introduction through Linear Groups

    Wulf Rossmann. Lie groups: An Introduction through Linear Groups . Number 5 in Oxford graduate texts in mathematics. Oxford University Press, Oxford [u.a.], 1. publ. in paperback edition, 2006. Reprinted 2011

    work page 2006

  36. [36]

    Evaluating analytic gradients on quantum hardware

    Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Phys. Rev. A 99, 032331 (2019) , November 2018

    work page 2019

  37. [37]

    Quantum machine learning in feature hilbert spaces

    Maria Schuld and Nathan Killoran. Quantum machine learning in feature hilbert spaces. Physical Review Letters , 122(4):040504, February 2019

    work page 2019

  38. [38]

    Sandeep Pradhan

    Mohammad Aamir Sohail, Mohsen Heidari Khoozani, and S. Sandeep Pradhan. Quantum natural stochastic pairwise coordinate descent. arXiv:2407.13858 , July 2024

    work page arXiv 2024

  39. [39]

    Identification of dynamical lie algebras for finite-level quantum control systems

    SG Schirmer, ICH Pullen, and AI Solomon. Identification of dynamical lie algebras for finite-level quantum control systems. Journal of Physics A: Mathematical and General , 35(9):2327, 2002

    work page 2002

  40. [40]

    Generalized trotter s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems

    Masuo Suzuki. Generalized trotter s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Communications in Mathematical Physics , 51(2):183--190, jun 1976

    work page 1976

  41. [41]

    Faehrmann, Barth \' e l \' e my Meynard-Piganeau, and Jens Eisert

    Ryan Sweke, Frederik Wilde, Johannes Meyer, Maria Schuld, Paul K. Faehrmann, Barth \' e l \' e my Meynard-Piganeau, and Jens Eisert. Stochastic gradient descent for hybrid quantum-classical optimization. Quantum , 4:314, aug 2020

    work page 2020

  42. [42]

    “proper” shift rules for derivatives of perturbed-parametric quantum evolutions

    Dirk Oliver Theis. “proper” shift rules for derivatives of perturbed-parametric quantum evolutions. Quantum , 7:1052, July 2023

    work page 2023

  43. [43]

    Calculus on parameterized quantum circuits

    Javier Gil Vidal and Dirk Oliver Theis. Calculus on parameterized quantum circuits. arXiv:1812.06323 , December 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  44. [44]

    Izmaylov

    Vladyslav Verteletskyi, Tzu-Ching Yen, and Artur F. Izmaylov. Measurement optimization in the variational quantum eigensolver using a minimum clique cover. The Journal of Chemical Physics , 152(12), March 2020

    work page 2020

  45. [45]

    Hastings, and Matthias Troyer

    Dave Wecker, Matthew B. Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A , 92(4):042303, October 2015

    work page 2015

  46. [46]

    General parameter-shift rules for quantum gradients

    David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yu Lin. General parameter-shift rules for quantum gradients. Quantum , 6:677, March 2022

    work page 2022

  47. [47]

    Kemper, and Bojko N

    Roeland Wiersema, Efekan Kökcü, Alexander F. Kemper, and Bojko N. Bakalov. Classification of dynamical lie algebras for translation-invariant 2-local spin systems in one dimension, 2023

    work page 2023

  48. [48]

    Here comes the su(n): multivariate quantum gates and gradients

    Roeland Wiersema, Dylan Lewis, David Wierichs, Juan Carrasquilla, and Nathan Killoran. Here comes the su(n): multivariate quantum gates and gradients. Quantum , 8:1275, March 2024

    work page 2024

  49. [49]

    Improved quantum computing with higher-order trotter decomposition

    Xiaodong Yang, Xinfang Nie, Yunlan Ji, Tao Xin, Dawei Lu, and Jun Li. Improved quantum computing with higher-order trotter decomposition. Physical Review A , 106(4):042401, oct 2022

    work page 2022