Pith. sign in

REVIEW 4 major objections 5 minor 71 references

Quantum Many-body Simulations from a Reinforcement-Learned Exponential Ansatz

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that a reinforcement-learned agent can select the few two-body exponential transformations that reach chemical accuracy for H3 and H4, using five to ten actions where a filtered standard solver needs over twenty.

desk verdict A credible, useful incremental step that compresses CQE circuits with RL on small molecules, but the paper overstates the Markovian state claim and needs statistical and baseline strengthening before publication. read the letter →

arxiv 2505.01935 v1 pith:HEOZHE7K submitted 2025-05-03 quant-ph physics.chem-phphysics.comp-ph

Yuchen Wang , David A. Mazziotti This is my paper · ORCID
classification quant-phphysics.chem-phphysics.comp-ph
keywords reinforcementlearningcontractedquantumeigensolverSchrödingerequationtwo-bodyexponentialansatzsimulationcircuitdepthoptimizationmolecularelectronicstructuredeepQ-network
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that reinforcement learning (RL) can choose, step by step, which two-body exponential transformations to apply to a molecular wavefunction, producing a quantum circuit that reaches chemical accuracy with far fewer gates than the standard contracted quantum eigensolver (CQE). On the linear molecules H3 and H4 in a minimal basis, an agent allowed only five to ten actions matches exact-diagonalization energies to within a few milli-Hartree, while the conventional residual-filtered CQE needs more than twenty exponential steps. This matters because quantum hardware is noisy: shorter ansatz circuits mean less error accumulation and lower measurement cost for quantum many-body simulation. The authors further show that a policy trained on a distribution of H3 geometries transfers to unseen bond lengths with validation errors around 1.2–2.4 mHa.

What carries the argument

The load-bearing machinery is the Markovian decision process whose state is the CSE residual operator $\hat{R} = \sum_{ijkl} {}^2R^{ij}_{kl}\hat{\Gamma}^{ij}_{kl}$ and whose actions are the individual two-body exponential factors $e^{\theta \hat{\Gamma}_n}$ from the Trotterized product ansatz. The agent is a dueling double deep Q-network: two streams estimate state value and per-action advantage, trained with prioritized experience replay, a reuse penalty to discourage repeating operators, and a line-search over the continuous $\theta$ embedded in the discrete action. The paper's key comparison is against filtered CQE, which keeps only the five largest residual coefficients per iteration; the RL agent outperforms it because it can merge or reorder effectively commuting operators that filter-based selection applies redundantly.

What would settle it

Train the same RL-CQE protocol on a molecule whose ground state is strongly correlated beyond two-body information—for example, a stretched chain of four or six hydrogen atoms—and check whether a five-to-ten-action policy still reaches chemical accuracy. If the optimal next operator depends on information not captured by the two-electron residual, the validation error will rise above the ~1.6 mHa chemical threshold, directly testing the Markovian state assumption.

Watch

Extended reading notes

Core claim

The central claim is that the contracted Schrödinger equation (CSE) residual—the two-electron projection of the Schrödinger equation at the current state—is a complete state description for choosing the next wavefunction update, making ansatz construction a Markovian decision process. In this picture the agent's action is a single two-body exponential factor $e^{\theta \hat{\Gamma}_n}$ from the exact universal two-body exponential ansatz, and its reward is the negative of energy plus a weighted residual norm, $Q = -(E + \lambda \|R\|)$. A dueling double deep Q-network learns to select actions from the current residual; because the residual already encodes the Hamiltonian through commutation, the learned policy transfers across molecular geometries. The demonstrated consequence is that near-optimal ansätze of five to ten exponentials achieve chemical accuracy for H3 and H4, outperforming the filtered CQE baseline and suggesting a practical strategy for resource-limited quantum simulation.

Load-bearing premise

The load-bearing premise is that the two-electron CSE residual at the current wavefunction contains enough information to pick the optimal next exponential transformation; if deciding that step ever requires higher-order reduced density matrices or a memory of past actions, the Markovian state is incomplete and the learned compaction could fail on larger molecules.

Editorial extensions

If this is right

  • Chemical accuracy for H3 and H4 with five to ten two-body exponential transformations, so RL-CQE circuits are roughly two to four times shallower than filtered CQE.
  • Policies trained on a distribution of molecular geometries transfer to unseen geometries, with validation energy errors of 1.2–2.4 mHa, suggesting geometry-robust ansatz generation.
  • Because the state is the CSE residual, the learned policy may transfer across Hamiltonians and devices, including training on noisy quantum hardware where depth constraints are strict.
  • The RL formulation extends beyond CQE: the same Markovian action framework can be applied to other parameterized quantum eigensolvers such as VQE to optimize circuit architecture.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the Markovian assumption survives scaling, RL-CQE could become an automated ansatz designer: training once on a distribution of Hamiltonians and then deploying the policy on a quantum device would eliminate per-molecule circuit design and classical optimization loops.
  • The reuse penalty hints that the agent learns to avoid repeated operators, effectively discovering commutativity relations; a natural test is whether policies trained on H3 transfer to H4 or to larger basis sets, which the paper does not report.
  • Pairing the CSE residual state with a transformer-based policy (as the authors mention in outlook) would let the agent see operator histories, potentially correcting any non-Markovian effects and extending the method to molecules with stronger multi-reference character.
  • The depth–accuracy trade-off exposed by varying allowed actions (3, 5, 10) could be mapped against noise models, letting users pick the minimal circuit that achieves a target accuracy on a specific device — a resource-allocation tool for near-term quantum simulation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript introduces a reinforcement learning (RL) treatment of the contracted quantum eigensolver (CQE). The central idea is to formulate the iterative CQE wavefunction update as a Markov decision process in which the state is the two-electron contracted Schrödinger equation (CSE) residual, the actions are two-body exponential transformations of the wavefunction, and a dueling double deep Q-network is trained to select actions that minimize both energy and residual norm. The authors report that for H3 and H4 in the STO-3G basis, the RL-optimized ansatz reaches chemical accuracy with 5–10 exponential transformations, whereas filtered CQE requires more than 20, and that a policy trained on a grid of H3 geometries transfers to held-out geometries with energy errors of 1.2–2.4 mHa. They also discuss transfer between classical simulation and quantum hardware and provide open-source code.

Significance. If the claims hold, the paper makes a useful contribution: it connects CQE's exact two-body exponential ansatz with modern deep RL, provides an open-source implementation, and proposes a concrete accuracy-versus-circuit-depth benchmark. The empirical results are suggestive but preliminary: they involve two small molecules, single runs without statistical error bars, and validation errors that exceed the standard chemical-accuracy threshold for most folds. The main theoretical premise—that the CSE residual is a Markovian and sufficient state for action selection—is asserted rather than proved and is questionable in light of Eq. (6). The broader transferability claims therefore need additional empirical or theoretical support.

major comments (4)
  1. [Sec. II.B, item (1) and Eq. (6)] The paper asserts that the CSE residual provides a two-electron description of the current state and that the update from S_t(^2D) to S_{t+1}(^2D) is Markovian. However, Eq. (6) defines 2R^{ij}_{kl} = <Ψ|Γ^{ij}_{kl}(H−E)|Ψ>. Because H contains one- and two-body terms and Γ contains four fermionic operators, this expectation value depends on the 3- and 4-RDMs, not only on the 2-RDM. Two wavefunctions with identical 2-RDMs can therefore have different residuals and, consequently, different optimal next exponential actions. The current residual does not determine the next residual after an action, so the MDP state is not Markovian as stated. This is load-bearing for the claim that the learned policy transfers to quantum-hardware settings where only residual measurements are available. Please either prove the Markovian property under the exact classical update, or reframe the state as a heuristic sufficient statistic and provide empirical tests (for example, compare policies trained with residual-only inputs against policies trained with residual plus action history or additional RDM information). The authors' own Sec. IV remark that moving beyond the Markovian process is future work is consistent with this gap.
  2. [Sec. III.B, Fig. 4] The key quantitative comparison between RL-optimized and filtered CQE ansätze is presented as single convergence curves without error bars or multiple random seeds. As a result, the claim that RL reaches chemical accuracy with 5–10 actions while filtered CQE requires more than 20 is not statistically substantiated. Please provide means and standard deviations over at least 5–10 independent training runs, and explicitly state the chemical-accuracy threshold (presumably 1.6 mHa) used to define 'chemical accuracy' in Fig. 4.
  3. [Sec. III.B, second paragraph] The text notes that for H4 with five actions the agent can only converge the residual to less than 0.007, suggesting that the solution may not be at chemical accuracy. This is important for the interpretation of Fig. 2 and the depth-versus-accuracy trade-off. Please report the corresponding energy errors for the five-action H4 runs and clarify how the residual norm relates to the energy accuracy criterion.
  4. [Table I and Sec. III.C] The validation energy differences in Table I range from 1.21 to 2.44 mHa, i.e., above the standard 1.6 mHa chemical-accuracy threshold for most folds. The statement that the agent 'performs well in predicting the validation set' therefore needs qualification. Please report the fraction of held-out geometries within chemical accuracy, include standard deviations across the five folds, and provide a comparison with unfiltered CQE and with a standard ADAPT-VQE construction using the same operator pool to support the claimed advantage over conventional ansätze mentioned in Sec. III.C.
minor comments (5)
  1. The reward function in Eq. (8) does not include the penalty for reused actions described in the text; please define the full reward including the reuse penalty.
  2. The product over non-commuting exponentials in Eq. (7) is not specified; please state the ordering convention (e.g., right-to-left time ordering or a fixed sequence).
  3. Please provide the full set of DQN hyperparameters (number of episodes, replay buffer size, batch size, target-network update period, epsilon schedule, and any reward shaping constants) in a table or appendix for reproducibility; the text currently lists only some of them.
  4. The claim that using the residual as the state 'greatly expands the model's transferability' is not tested against an alternative state representation (e.g., the raw 2-RDM or energy plus 2-RDM); an ablation would strengthen this claim.
  5. The phrase 'we reformulate the wavefunction update as a Markovian decision process' is stronger than what is demonstrated; consider changing to 'treat as' or providing a proof of the Markov property.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RL-CQE results are empirical benchmarks and the ansatz exactness is imported from independent published proofs, not from the present fitting procedure.

full rationale

The central claims—that RL can select compact sequences of two-body exponentials and that the trained policy transfers to unseen geometries—are empirical results benchmarked against exact diagonalization, not outputs of the ansatz exactness theorems. The exactness of the two-body exponential ansatz and the CSE equivalence are imported from published proofs (Refs. [23,37,38,60,61]), including independent proof by Evangelista et al. (Ref. [61]); these are parameter-free mathematical results with stated assumptions that do not include the RL target, so they are independent support rather than self-citation circularity. The reward is defined directly from energy and residual norm and is minimized against exact answers; no fitted parameter is renamed as a prediction. Transfer experiments use held-out geometries, so validation errors are genuine out-of-sample predictions. The one asserted-but-unproven premise is the Markovian sufficiency of the CSE residual as an RL state (Sec. II.B). Equation (6) shows the residual tensor depends on higher-body reduced density matrices, so the claim that the update from the 2-RDM alone is Markovian is not established; however, this is an unsupported assumption or correctness risk, not a reduction of a prediction to its input, and the paper itself flags moving beyond the Markovian process as future work in Sec. IV. No circular step is exhibited.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the exactness of the two-body exponential ansatz and the CSE equivalence (cited prior results), plus the paper-specific assumption that the CSE residual is a Markovian state. The only ad hoc choices are the reward weight and training hyperparameters; no new physical entities are introduced.

free parameters (6)
  • reward weight lambda = 0.2
    Balances energy and residual norm in Eq. (8); chosen by hand; directly shapes the learned policy.
  • maximum number of actions per episode = 3, 5, 10, 20 (tested)
    A user-set resource constraint; the paper reports results for 5 actions in Table I and for 3, 5, 10, 20 in Figs. 3 and 4.
  • DQN discount factor = 0.99
    Training hyperparameter; affects credit assignment in the MDP.
  • replay buffer priority exponent alpha = 0.6
    Prioritized experience replay hyperparameter; affects sample weighting.
  • epsilon-greedy decay rate = 0.999 per episode
    Controls exploration-exploitation schedule; decays from 1.0 to 0.01.
  • target network update period = 1000 episodes
    Stabilizes double DQN training.
assumptions (5)
  • domain assumption A wavefunction satisfies the CSE if and only if it satisfies the Schrödinger equation.
    Invoked in Sec. II.A to justify solving the CSE; proven in Refs. [23,60].
  • domain assumption Any N-electron wavefunction can be represented exactly by a product of two-body exponential transformations.
    Invoked in Sec. II.B for the action pool; from Refs. [37,38,61].
  • domain assumption The CSE residual at the current iteration is a sufficient, Markovian state representation for choosing the next exponential transformation.
    Stated in Sec. II.B; not proven in this paper.
  • ad hoc to paper The reward Q = -(E + lambda*||R||) is an adequate proxy for accuracy and circuit-depth trade-off.
    Defined in Eq. (8); the weighting lambda is chosen by hand.
  • domain assumption Trotterization error of the product of exponentials is negligible for the convergence claims.
    Eq. (7) uses first-order Trotterization; the paper does not analyze Trotter error, only notes it complicates merging.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Many-body Simulations from a Reinforcement-Learned Exponential Ansatz." pith.science (2026). https://pith.science/paper/HEOZHE7K

@misc{pith2026250501935,
  author       = {Pith},
  title        = {Pith review of: Quantum Many-body Simulations from a Reinforcement-Learned Exponential Ansatz},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HEOZHE7K}},
  note         = {Machine review of arXiv:2505.01935}
}
read the original abstract

Solving for the many-body wavefunction represents a significant challenge on both classical and quantum devices because of the exponential scaling of the Hilbert space with system size. While the complexity of the wavefunction can be reduced through conventional ans\"{a}tze (e.g., the coupled cluster ansatz), it can still grow rapidly with system size even on quantum devices. An exact, universal two-body exponential ansatz for the many-body wavefunction has been shown to be generated from the solution of the contracted Schr\"odinger equation (CSE), and recently, this ansatz has been implemented without classical approximation on quantum simulators and devices for the scalable simulation of many-body quantum systems. Here we combine the solution of the CSE with a form of artificial intelligence known as reinforcement learning (RL) to generate highly compact circuits that implement this ansatz without sacrificing accuracy. As a natural extension of CSE, we reformulate the wavefunction update as a Markovian decision process and train the agent to select the optimal actions at each iteration based upon only the current CSE residual. Compact circuits with high accuracy are achieved for H3 and H4 molecules over a range of molecular geometries.

Figures

Figures reproduced from arXiv: 2505.01935 by the authors.

Figure 1
Figure 1. FIG. 1. A schematic overview of original CQE algorithm and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Training metric (a) Ground-state energy estimate, (b) Norm of the CSE residual, and (c) Average reward per episode [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Reward convergence with different numbers of actions [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Accuracy and circuit depth for RL optimized ansatz [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 49 canonical work pages

  1. [1]

    B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, et al. , Towards quantum chemistry on a quantum computer, Nat. Chem. 2, 106 (2010)

    work page 2010

  2. [2]

    P. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, et al. , Scalable quantum simulation of molecular energies, Phys. Rev. X 6, 031007 (2016)

    work page 2016

  3. [3]

    McArdle, S

    S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Quantum computational chemistry, Rev. Mod. Phys. 92, 015003 (2020)

    2020

  4. [4]

    Dutta, D

    R. Dutta, D. G. Cabral, N. Lyu, N. P. Vu, Y. Wang, B. Allen, X. Dan, R. G. Corti˜ nas, P. Khazaei, M. Scha¨fer, et al. , Simulating chemistry on bosonic quantum devices, J. Chem. Theory Comput. 20, 6426 (2024)

    work page 2024

  5. [5]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al. , Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021)

    2021

  6. [6]

    Kandala, A

    A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware- efficient variational quantum eigensolver for small molecules and quantum magnets, nature549, 242 (2017)

    2017

  7. [7]

    Tilly, H

    J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, et al. , The variational quantum eigensolver: a review of meth- ods and best practices, Phys. Rep. 986, 1 (2022)

    work page 2022

  8. [8]

    Wecker, M

    D. Wecker, M. B. Hastings, and M. Troyer, Progress to- wards practical quantum variational algorithms, Phys. Rev. A 92, 042303 (2015)

    2015

Show all 71 references
  1. [9]

    D’Cunha, T

    R. D’Cunha, T. D. Crawford, M. Motta, and J. E. Rice, Challenges in the use of quantum computing hardware- efficient ansa¨tze in electronic structure theory, J. Phys. Chem. A 127, 3437 (2023)

    2023

  2. [10]

    Larocca, S

    M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Bia- monte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, A review of barren plateaus in variational quantum computing, arXiv preprint arXiv:2405.00781 10.48550/arXiv.2405.00781 (2024)

    2024 doi arXiv

  3. [11]

    S. E. Smart and D. A. Mazziotti, Quantum solver of con- tracted eigenvalue equations for scalable molecular simu- lations on quantum computing devices, Phys. Rev. Lett. 126, 070504 (2021)

    2021

  4. [12]

    S. E. Smart and D. A. Mazziotti, Accelerated con- vergence of contracted quantum eigensolvers through a quasi-second-order, locally parameterized optimization, J. Chem. Theory Comput. 18, 5286 (2022)

    2022

  5. [13]

    J.-N. Boyn, A. O. Lykhin, S. E. Smart, L. Gagliardi, and D. A. Mazziotti, Quantum-classical hybrid algorithm for the simulation of all-electron correlation, J. Chem. Phys. 155, 244106 (2021)

    2021

  6. [14]

    S. E. Smart, J.-N. Boyn, and D. A. Mazziotti, Resolving correlated states of benzyne with an error-mitigated con- tracted quantum eigensolver, Phys. Rev. A 105, 022405 (2022)

    2022

  7. [15]

    Y. Wang, L. M. Smith, and D. A. Mazziotti, Quantum simulation of bosons with the contracted quantum eigen- solver, New J. Phys. 25, 103005 (2023)

    2023

  8. [16]

    Wang and D

    Y. Wang and D. A. Mazziotti, Electronic excited states from a variance-based contracted quantum eigensolver, Phys. Rev. A 108, 022814 (2023)

    2023

  9. [17]

    C. L. Benavides-Riveros, Y. Wang, S. Warren, and D. A. Mazziotti, Quantum simulation of excited states from parallel contracted quantum eigensolvers, New J. Phys. 26, 033020 (2024)

    2024

  10. [18]

    S. E. Smart and D. A. Mazziotti, Verifiably exact solu- tion of the electronic Schr¨ odinger equation on quantum devices, Phys. Rev. A 109, 022802 (2024)

    2024

  11. [19]

    S. E. Smart, D. M. Welakuh, and P. Narang, Many-Body Excited States with a Contracted Quantum Eigensolver, J. Chem. Theory Comput. 20, 3580 (2024), 2305.09653

    2024 arXiv

  12. [20]

    Warren, Y

    S. Warren, Y. Wang, C. L. Benavides-Riveros, and D. A. Mazziotti, Exact ansatz of fermion-boson systems for a quantum device, Phys. Rev. Lett. 133, 080202 (2024)

    2024

  13. [21]

    Warren, Y

    S. Warren, Y. Wang, C. L. Benavides-Riveros, and D. A. Mazziotti, Quantum algorithm for polaritonic chemistry based on an exact ansatz, Quantum Sci. Technol. 10, 02LT02 (2025)

    2025

  14. [22]

    Y. Wang, C. Cianci, I. Avdic, R. Dutta, S. Warren, B. Allen, N. P. Vu, L. F. Santos, V. S. Batista, and D. A. Mazziotti, Characterizing conical intersections of nucle- obases on quantum computers, J. Chem. Theory Com- put. 10.1021/acs.jctc.4c01434

    doi

  15. [23]

    D. A. Mazziotti, Contracted Schr¨ odinger equation: De- termining quantum energies and two-particle density ma- trices without wave functions, Phys. Rev. A 57, 4219 (1998)

    1998

  16. [24]

    Colmenero and C

    F. Colmenero and C. Valdemoro, Approximating q-order reduced density matrices in terms of the lower-order ones. II. Applications, Phys. Rev. A 47, 979 (1993)

    1993

  17. [25]

    Nakatsuji and K

    H. Nakatsuji and K. Yasuda, Direct Determination of the Quantum-Mechanical Density Matrix Using the Density Equation, Phys. Rev. Lett. 76, 1039 (1996)

    1996

  18. [26]

    D. A. Mazziotti, Comparison of contracted Schr¨ odinger and coupled-cluster theories, Phys. Rev. A 60, 4396 (1999). 7

    1999

  19. [27]

    Mukherjee and W

    D. Mukherjee and W. Kutzelnigg, Irreducible Brillouin conditions and contracted Schr¨ odinger equations for n- electron systems. I. The equations satisfied by the density cumulants, J. Chem. Phys. 114, 2047 (2001)

    2001

  20. [28]

    Yasuda, Uniqueness of the solution of the contracted Schr¨ odinger equation, Phys

    K. Yasuda, Uniqueness of the solution of the contracted Schr¨ odinger equation, Phys. Rev. A65, 052121 (2002)

    2002

  21. [29]

    D. A. Mazziotti, Variational method for solving the con- tracted Schr¨ odinger equation through a projection of the N-particle power method onto the two-particle space, J. Chem. Phys. 116, 1239 (2002)

    2002

  22. [30]

    Cohen and C

    L. Cohen and C. Frishberg, Hierarchy Equations for Re- duced Density Matrices, Phys. Rev. A 13, 927 (1976)

    1976

  23. [31]

    Valdemoro, L

    C. Valdemoro, L. Tel, D. Alcoba, and E. P´ erez-Rome- ro, The contracted Schr¨ odinger equation methodology: study of the third-order correlation effects, Theor. Chem. Account. 118, 503 (2007)

    2007

  24. [32]

    D. A. Mazziotti, Anti-Hermitian Contracted Schr¨ odin- ger Equation: Direct Determination of the Two-Electron Reduced Density Matrices of Many-Electron Molecules, Phys. Rev. Lett. 97, 143002 (2006)

    2006

  25. [33]

    D. A. Mazziotti, Anti-Hermitian part of the contracted Schr¨ odinger equation for the direct calculation of two- electron reduced density matrices, Phys. Rev. A 75, 022505 (2007)

    2007

  26. [34]

    D. A. Mazziotti, Multireference many-electron correla- tion energies from two-electron reduced density matri- ces computed by solving the anti-Hermitian contracted Schr¨ odinger equation, Phys. Rev. A76, 052502 (2007)

    2007

  27. [35]

    Boyn and D

    J.-N. Boyn and D. A. Mazziotti, Accurate singlet–triplet gaps in biradicals via the spin averaged anti-Hermitian contracted Schr¨ odinger equation, J. Chem. Phys. 154, 134103 (2021)

    2021

  28. [36]

    Bonet-Monroig, R

    X. Bonet-Monroig, R. Babbush, and T. E. O’Brien, Nearly optimal measurement scheduling for partial to- mography of quantum states, Phys. Rev. X 10, 031064 (2020)

    2020

  29. [37]

    D. A. Mazziotti, Exactness of wave functions from two- body exponential transformations in many-body quan- tum theory, Phys. Rev. A 69, 012507 (2004)

    2004

  30. [38]

    D. A. Mazziotti, Exact two-body expansion of the many- particle wave function, Phys. Rev. A 102, 030802 (2020)

    2020

  31. [39]

    Gidofalvi and D

    G. Gidofalvi and D. A. Mazziotti, Direct calculation of excited-state electronic energies and two-electron re- duced density matrices from the anti-Hermitian con- tracted Schr¨ odinger equation, Phys. Rev. A 80, 022507 (2009)

    2009

  32. [40]

    A. E. Rothman, J. J. Foley, and D. A. Mazziotti, Open- shell energies and two-electron reduced density matrices from the anti-Hermitian contracted Schr¨ odinger equa- tion: A spin-coupled approach, Phys. Rev. A 80, 052508 (2009)

    2009

  33. [41]

    J. W. Snyder and D. A. Mazziotti, Photoexcited con- version of gauche-1,3-butadiene to bicyclobutane via a conical intersection: Energies and reduced density matri- ces from the anti-Hermitian contracted Schr¨ odinger equa- tion, J. Chem. Phys. 135, 024107 (2011)

    2011

  34. [42]

    A. M. Sand and D. A. Mazziotti, Enhanced computa- tional efficiency in the direct determination of the two- electron reduced density matrix from the anti-Hermitian contracted Schr¨ odinger equation with application to ground and excited states of conjugated π-systems, J. Chem....

    2015

  35. [43]

    L. P. Kaelbling, M. L. Littman, and A. W. Moore, Re- inforcement Learning: A Survey, J. Artif. Intell. Res. 4, 237 (1996)

    1996

  36. [44]

    H. V. Hasselt, A. Guez, and D. Silver, Deep Reinforce- ment Learning with Double Q-Learning, Proceedings of the AAAI Conference on Artificial Intelligence 30, 10.1609/aaai.v30i1.10295 (2016)

    2016 doi

  37. [45]

    Silver, T

    D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Kumaran, T. Graepel, T. Lillicrap, K. Simonyan, and D. Hassabis, A general reinforcement learning algorithm that masters chess, shogi, and Go through self-play., Science (New York,...

    2018

  38. [46]

    Vinyals, I

    O. Vinyals, I. Babuschkin, W. M. Czarnecki, M. Mathieu, A. Dudzik, J. Chung, D. H. Choi, R. Powell, T. Ewalds, P. Georgiev, J. Oh, D. Horgan, M. Kroiss, I. Danihelka, A. Huang, L. Sifre, T. Cai, J. P. Agapiou, M. Jaderberg, A. S. Vezhnevets, R. Leblond, T. Pohlen, V. Dalibard,...

    2019

  39. [47]

    M. Y. Niu, S. Boixo, V. N. Smelyanskiy, and H. Neven, Universal quantum control through deep reinforcement learning, npj Quantum Inf. 5, 33 (2019)

    2019

  40. [48]

    Liang, L

    S. Liang, L. Zhu, X. Liu, C. Yang, and X. Li, Artificial- intelligence-driven shot reduction in quantum measure- ment, Chem. Phys. Rev. 5, 10.1063/5.0219663 (2024)

    2024 doi

  41. [49]

    Ostaszewski, L

    M. Ostaszewski, L. M. Trenkwalder, W. Masarczyk, E. Scerri, and V. Dunjko, Reinforcement learning for optimization of variational quantum circuit architec- tures, in Advances in Neural Information Processing Sys- tems, Vol. 34, edited by M. Ranzato, A. Beygelzimer, Y. Dauphin,...

    2021

  42. [50]

    Zhang, P.-L

    Y.-H. Zhang, P.-L. Zheng, Y. Zhang, and D.-L. Deng, Topological quantum compiling with reinforce- ment learning, Phys. Rev. Lett. 125, 170501 (2020)

    2020

  43. [51]

    H. P. Nautrup, N. Delfosse, V. Dunjko, H. J. Briegel, and N. Friis, Optimizing quantum error correction codes with reinforcement learning, Quantum 3, 215 (2019)

    2019

  44. [52]

    Andreasson, J

    P. Andreasson, J. Johansson, S. Liljestrand, and M. Granath, Quantum error correction for the toric code using deep reinforcement learning, Quantum 3, 183 (2019)

    2019

  45. [53]

    J. Olle, R. Zen, M. Puviani, and F. Marquardt, Si- multaneous discovery of quantum error correction codes and encoders with a noise-aware reinforcement learning agent, Npj Quantum Inf. 10, 1 (2024)

    2024

  46. [54]

    Fallani, M

    A. Fallani, M. A. Rossi, D. Tamascelli, and M. G. Genoni, Learning feedback control strategies for quantum metrol- ogy, PRX Quantum 3, 020310 (2022)

    2022

  47. [55]

    Belliardo, F

    F. Belliardo, F. Zoratti, F. Marquardt, and V. Gio- vannetti, Model-aware reinforcement learning for high- performance bayesian experimental design in quantum metrology, Quantum 8, 1555 (2024)

    2024

  48. [56]

    A. J. Coleman, Structure of Fermion Density Matrices, Rev. Mod. Phys. 35, 668 (1963)

    1963

  49. [57]

    Yokonuma, Tensor Spaces and Exterior Alge- bra, Translations of Mathematical Monographs 10.1090/mmono/108 (1992)

    T. Yokonuma, Tensor Spaces and Exterior Alge- bra, Translations of Mathematical Monographs 10.1090/mmono/108 (1992)

    1992 doi

  50. [58]

    D. A. Mazziotti, Quantum Chemistry without Wave Functions: Two-Electron Reduced Density Matrices, Acc. Chem. Res. 39, 207 (2006). 8

    2006

  51. [59]

    D. A. Mazziotti, Contracted Schr¨ odinger Equation, in Reduced-Density-Matrix Mechanics: With Application to Many-Electron Atoms and Molecules (John Wiley & Sons, Ltd, 2007) Chap. 8, pp. 165–203

    2007

  52. [60]

    Nakatsuji, Equation for the Direct Determination of the Density Matrix, Phys

    H. Nakatsuji, Equation for the Direct Determination of the Density Matrix, Phys. Rev. A 14, 41 (1976)

    1976

  53. [61]

    F. A. Evangelista, G. K. Chan, and G. E. Scuseria, Exact parameterization of fermionic wave functions via unitary coupled cluster theory, J. Chem. Phys. 151, 10.1063/1.5133059 (2019)

    2019 doi

  54. [62]

    Z. Wang, T. Schaul, M. Hessel, H. Hasselt, M. Lanctot, and N. Freitas, Dueling network architectures for deep re- inforcement learning, in International conference on ma- chine learning (PMLR, 2016) pp. 1995–2003

    2016

  55. [63]

    W. J. Hehre, R. F. Stewart, and J. A. Pople, Self- consistent molecular-orbital methods. i. use of gaussian expansions of slater-type atomic orbitals, J. Chem. Phys. 51, 2657 (1969)

    1969

  56. [64]

    Jordan and E

    P. Jordan and E. Wigner, ¨Uber das Paulische ¨Aquivalenzverbot, Z. Physik 47, 631 (1928)

    1928

  57. [65]

    RDMChem, Quantum Chemistry Toolbox for Maple (Maplesoft, Waterloo, 2025)

    2025

  58. [66]

    Maplesoft, Maple (Maplesoft, Waterloo, 2025)

    2025

  59. [67]

    Wang and D

    Y. Wang and D. A. Mazziotti, RL-CQE: A reinforce- ment learning approach to the contracted quantum eigen- solver, https://github.com/damazz/ML-CQE (2025), ac- cessed: May 3, 2025

    2025

  60. [68]

    Y. Wang, I. Avdic, and D. A. Mazziotti, Shadow ansatz for the many-fermion wave function in scalable molecular simulations on quantum com- puting devices, arXiv preprint arXiv:2408.11026 doi.org/10.48550/arXiv.2408.11026 (2024)

    2024 doi arXiv

  61. [69]

    Nakaji, L

    K. Nakaji, L. B. Kristensen, J. A. Campos-Gonzalez- Angulo, M. G. Vakili, H. Huang, M. Bagherimehrab, C. Gorgulla, F. Wong, A. McCaskey, J.-S. Kim, et al. , The generative quantum eigensolver (gqe) and its application for ground state search, arXiv preprint arXiv:2401.09253 10...

    2024 doi

  62. [70]

    W. Zeng, J. Zhang, L. Chen, and C. L. Benavides- Riveros, Simulating quantum many-body states with neural-network exponential ansatz, arXiv preprint arXiv:2411.07886 10.48550/arXiv.2411.07886 (2024)

    2024 doi arXiv

  63. [71]

    Halder, K

    S. Halder, K. Anand, and R. Maitra, Construction of chemistry inspired dynamic ansatz utilizing genera- tive machine learning, arXiv preprint arXiv:2503.08149 10.48550/arXiv.2503.08149 (2025)

    2025 doi arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.