pith. sign in

arxiv: 1906.08728 · v1 · pith:65ESAFO2new · submitted 2019-06-20 · 🪐 quant-ph

Hybrid Quantum/Classical Derivative Theory: Analytical Gradients and Excited-State Dynamics for the Multistate Contracted Variational Quantum Eigensolver

Robert M. Parrish , Edward G. Hohenstein , Peter L. McMahon , Todd J. Martinez This is my paper

Pith reviewed 2026-05-25 19:31 UTC · model grok-4.3

classification 🪐 quant-ph
keywords analytical derivativeshybrid quantum-classicalMC-VQEvariational quantum eigensolverexcited-state dynamicsLagrangianrelaxed density matricesab initio exciton model
0
0 comments X

The pith

Analytical nuclear gradients for MC-VQE are obtained via a Lagrangian using only quantum observables and relaxed density matrices.

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

The paper develops a Lagrangian-based approach to compute analytical first derivatives for hybrid quantum/classical methods, focusing on the nuclear energy gradient of the multistate contracted variational quantum eigensolver within the ab initio exciton model. It relies solely on observable quantities from the quantum circuit rather than explicit wavefunction access. Relaxed density matrices create a clean separation between quantum and classical contributions. The quantum response equations (CP-MC-VQE) decouple from classical wavefunction response and gradient perturbations. The approach is tested in quantum circuit simulations of gradients and adiabatic excited-state dynamics.

Core claim

By defining an appropriate set of relaxed density matrices, the nuclear energy gradient for MC-VQE+AIEM can be computed with a clean separation between the quantum and classical parts of the problem, where the coupled-perturbed MC-VQE equations are decoupled from the wavefunction response equations and gradient perturbations in the classical part of the algorithm.

What carries the argument

Lagrangian constructed from observable quantities and an appropriate set of relaxed density matrices, enabling analytical derivatives without direct wavefunction access.

If this is right

  • Hellmann-Feynman and response contributions to the gradients can be quantified separately in quantum circuit simulations of MC-VQE+AIEM.
  • Adiabatic excited state dynamics simulations become feasible with MC-VQE+AIEM on quantum circuit simulators.
  • The quantum and classical parts of the hybrid algorithm remain decoupled even when computing first derivatives.
  • Analytical gradients are available using only quantities directly measurable on the quantum device.

Where Pith is reading between the lines

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

  • The separation via relaxed density matrices may extend to derivative calculations in other variational quantum algorithms that lack explicit wavefunction access.
  • If sampling costs for the required density matrices remain modest, the method could support dynamics on larger molecular systems than pure classical approaches allow.
  • The decoupling suggests a route to port classical derivative efficiencies into hybrid quantum settings without redesigning the entire response theory.

Load-bearing premise

The quantum circuit can supply sufficiently accurate estimates of the relaxed density matrices and response quantities needed for the Lagrangian without prohibitive sampling overhead or noise that would invalidate the gradient.

What would settle it

Compare the MC-VQE+AIEM nuclear gradient computed on a quantum circuit simulator for a small molecule against the corresponding classical gradient, checking agreement within expected sampling error.

Figures

Figures reproduced from arXiv: 1906.08728 by Edward G. Hohenstein, Peter L. McMahon, Robert M. Parrish, Todd J. Martinez.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic of (A) BChl-a dimer system and (B) [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Visual comparison of nuclear gradients computed for [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Energy characteristics of [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
read the original abstract

The maturation of analytical derivative theory over the past few decades has enabled classical electronic structure theory to provide accurate and efficient predictions of a wide variety of observable properties. However, classical implementations of analytical derivative theory take advantage of explicit computational access to the approximate electronic wavefunctions in question, which is not possible for the emerging case of hybrid quantum/classical methods. Here, we develop an efficient Lagrangian-based approach for analytical first derivatives of hybrid quantum/classical methods using only observable quantities from the quantum portion of the algorithm. Specifically, we construct the key first-derivative property of the nuclear energy gradient for the recently-developed multistate, contracted variant of the variational quantum eigensolver (MC-VQE) within the context of the ab initio exciton model (AIEM). We show that a clean separation between the quantum and classical parts of the problem is enabled by the definition of an appropriate set of relaxed density matrices, and show how the wavefunction response equations in the quantum part of the algorithm (coupled-perturbed MC-VQE or CP-MC-VQE equations) are decoupled from the wavefunction response equations and and gradient perturbations in the classical part of the algorithm. We explore the magnitudes of the Hellmann-Feynman and response contributions to the gradients in quantum circuit simulations of MC-VQE+AIEM and demonstrate a quantum circuit simulator implementation of adiabatic excited state dynamics with MC-VQE+AIEM.

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 paper develops a Lagrangian-based analytical derivative theory for computing nuclear energy gradients in the hybrid quantum/classical multistate contracted variational quantum eigensolver (MC-VQE) combined with the ab initio exciton model (AIEM). It constructs relaxed density matrices from observable quantities to achieve a clean separation between quantum and classical parts of the problem, decoupling the coupled-perturbed MC-VQE (CP-MC-VQE) response equations from classical wavefunction response and gradient perturbations, and demonstrates the approach via quantum circuit simulations of gradient contributions and adiabatic excited-state dynamics.

Significance. If the formal separation via relaxed density matrices holds under the method's assumptions, this enables analytical gradients for hybrid VQE methods without explicit wavefunction access, a key requirement for excited-state dynamics on quantum hardware. The explicit construction from observables and the demonstration of decoupled response equations are strengths; the simulations provide concrete evidence on the relative sizes of Hellmann-Feynman and response terms.

major comments (2)
  1. [§3] §3 (Lagrangian construction and CP-MC-VQE equations): The decoupling between quantum and classical response equations is derived formally using the relaxed density matrices, but the manuscript does not provide an explicit error propagation analysis showing that finite-shot estimates of these matrices (as used in the §4 simulations) preserve the exact decoupling; any sampling bias could reintroduce effective coupling not present in the ideal Lagrangian.
  2. [§4] §4 (quantum circuit simulations): The reported gradients and dynamics use circuit simulations whose shot count and noise model are not specified in sufficient detail to confirm that the relaxed density matrix estimates support the claimed separation without post-hoc adjustments; this is load-bearing for the practical claim that the method works with observable quantities only.
minor comments (2)
  1. [§2] The notation for the relaxed density matrices (e.g., how they differ from standard 1-RDMs) should be defined more explicitly in the first appearance to aid readers unfamiliar with the AIEM context.
  2. [§5] Figure captions for the dynamics trajectories should include the number of time steps and the integrator used, as these details affect reproducibility of the excited-state dynamics results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive recommendation of minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Lagrangian construction and CP-MC-VQE equations): The decoupling between quantum and classical response equations is derived formally using the relaxed density matrices, but the manuscript does not provide an explicit error propagation analysis showing that finite-shot estimates of these matrices (as used in the §4 simulations) preserve the exact decoupling; any sampling bias could reintroduce effective coupling not present in the ideal Lagrangian.

    Authors: The decoupling is formally exact when the relaxed density matrices are obtained from exact expectation values. Finite-shot estimates are unbiased estimators of these quantities, so the separation between quantum and classical response equations holds in expectation; any re-coupling arises only from statistical fluctuations whose magnitude scales as 1/sqrt(N_shots). We agree that an explicit propagation of these fluctuations through the CP-MC-VQE equations would strengthen the practical claims. In the revised manuscript we will add a short discussion (with supporting estimates drawn from the existing §4 data) quantifying the size of the residual coupling under the shot counts employed. revision: yes

  2. Referee: [§4] §4 (quantum circuit simulations): The reported gradients and dynamics use circuit simulations whose shot count and noise model are not specified in sufficient detail to confirm that the relaxed density matrix estimates support the claimed separation without post-hoc adjustments; this is load-bearing for the practical claim that the method works with observable quantities only.

    Authors: We will expand the simulation description in the revised §4 to state the precise number of shots allocated to each observable, the absence (or explicit form) of any noise model, and confirmation that all reported quantities are obtained directly from the sampled observables without additional post-processing or fitting. These additions will make the numerical evidence for the observable-only workflow fully reproducible. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard Lagrangian and observable-based relaxed densities

full rationale

The paper derives analytical gradients for MC-VQE+AIEM via a Lagrangian constructed from observable quantities (relaxed density matrices) supplied by the quantum circuit. The abstract and provided text frame the decoupling of CP-MC-VQE response equations from classical perturbations as following directly from the definition of those matrices, without any reduction of the final gradient expression to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors. No load-bearing step is shown to be equivalent to its inputs by construction. This is the normal case of a self-contained formal derivation against external benchmarks (exact Lagrangian theory in quantum chemistry), warranting score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the existence of relaxed density matrices that can be measured on the quantum device and on the decoupling of CP-MC-VQE response equations from classical perturbations; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Relaxed density matrices from the quantum circuit suffice to construct the nuclear gradient without additional classical wavefunction information.
    Invoked to enable the clean separation between quantum and classical parts.

pith-pipeline@v0.9.0 · 5807 in / 1215 out tokens · 15221 ms · 2026-05-25T19:31:52.124430+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Variationally Compressing Quantum Circuits to Approximate Nonadiabatic Molecular Quantum Dynamics

    quant-ph 2026-05 unverdicted novelty 4.0

    Variational compression of Trotterized circuits preserves reaction rate coefficients in nonadiabatic dynamics simulations while reducing circuit depth.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    • I - Contracted reference state (CRS) configuration, e.g., configuration interaction singles (CIS) config- uration

    Indices The following index classes are used in this work, • A - Monomer. • I - Contracted reference state (CRS) configuration, e.g., configuration interaction singles (CIS) config- uration. • Ξ - CRS eigenstate. • Θ - MC-VQE eigenstate. • M - CIS quantum circuit angle. • g - VQE entangler quantum circuit angle. • ζ - Nuclear gradient perturbation. Primes ar...

    work page

  2. [2]

    • ϵA H: The energy of the singlet ground (hole) state of the monomer

    Monomer Properties To begin, for an ab initio exciton model (AIEM) with N neutral monomers, each with two relevant electronic states, and with restricted two-body interactions com- puted in the dipole-dipole approximation, compute the following quantities at the current nuclear positions{⃗ rζ}. • ϵA H: The energy of the singlet ground (hole) state of the ...

    work page

  3. [3]

    Dimer Interaction Matrix Elements In the present work, electrostatic interactions between monomers are computed in the dipole-dipole approxima- tion, vAA′ = ⃗ µA ·⃗ µA′ r3 AA′ − 3(⃗ µA ·⃗ rAA′)(⃗ µA′ ·⃗ rAA′) r5 AA′ (19) Here⃗ rAA′ ≡⃗ rA′ 0 −⃗ rA 0 , and rAA′ = √ |⃗ rAA′|2. The dipole moments run over the types of H, T, and P for monomers A andA′, leading...

    work page

  4. [4]

    Monomer-Basis AIEM Hamiltonian In the monomer basis, the AIEM Hamiltonian is suc- cinctly written as, ˆH ≡ ˆH(1) + ˆH(2) (20) = ∑ A ∑ p,q∈[0,1] (pA|ˆh|qA)|pA⟩⟨qA| +1 2 ∑ <A,A′> ∑ p,q,r,s∈[0,1] (pAqA|ˆh|rA′sA′)|pA⟩⟨qA|⊗|rA′⟩⟨sA′| Where now |0A⟩ refers to the monomer ground state (hole), |1A⟩ refers to the monomer excited state (par- ticle), and in chemist’...

    work page

  5. [5]

    ˆI operators correspond to S, ˆX operators correspond to T, and ˆZ operators correspond to D

    Pauli-Basis AIEM Hamiltonian After some straightforward algebra, the AIEM Hamil- tonian can be succinctly re-written in terms of Pauli op- erators, ˆH ≡ E + H(1) + H(2) = E ˆI + ∑ A ZA ˆZA + XA ˆXA +1 2 ∑ <A,A′> X XAA′ ˆXA ⊗ ˆXA′ + X ZAA′ ˆXA ⊗ ˆZA′ + ZXAA′ ˆZA ⊗ ˆXA′ + ZZAA′ ˆZA ⊗ ˆZA′ (33) The matrix elements are, E = ∑ A (ϵA H+ϵA P)/2+1 2 ∑ <A,A′> (vAA...

    work page

  6. [6]

    Formal Definition: MC-VQE Eigenstates MC-VQE approximates the exact diagonalization of the Pauli Hamiltonian ˆH by producing a number NΘ of MC-VQE approximate eigenstates |ΨΘ⟩. |ΨΘ⟩ ≡ ˆU(θg) ∑ Ξ |ΦΞ⟩VΞΘ (41) = ˆU(θg) ∑ I ∑ Ξ |I⟩CIΞVΞΘ (42) = ˆU(θg) ∑ I |I⟩ΓIΘ (43) 8 = ˆU(θg)|ΓΘ⟩ (44) The MC-VQE eigenstates are orthonormal, ⟨ΨΘ|ΨΘ′⟩ =δΘΘ′ (45) and within th...

    work page

  7. [7]

    contracted reference states

    Configuration Interaction Singles Contracted Reference States To begin, we classically determine a numberNΞ =NΘ of “contracted reference states” (CRS), {|ΦΞ⟩}, by solv- ing a polynomial-scaling electronic structure problem to “sketch out” the shapes of the desired electronic states. One particularly appealing choice for the contracted ref- erence states is...

    work page

  8. [8]

    ⟩, while the composite two-body gates between 9 each pair of wires control the amplitudes of each singly- excited ket |10

    Technical Detail: CIS State Preparation Circuit A quantum circuit to prepare the CIS state |Φ⟩ ≡∑ I |I⟩CI ≡ ˆUCIS|0⟩ is sketched for N = 4, |0A⟩ Ry(θ0) • |0B⟩ Ry(−θAB/2) • Ry(+θAB/2) • • |0C⟩ Ry(−θBC/2) • Ry(+θBC/2) • • |0D⟩ Ry(−θCD/2) • Ry(+θCD/2) • (53) The Ry(θ0) gate controls the amplitude of the reference ket |00... ⟩, while the composite two-body ga...

    work page

  9. [9]

    CIS Circuit Angles Using the recipe from the previous section, compute theN CIS circuit angles from the N + 1 CIS coefficients, for each state Ξ: θM[CIΞ] (63)

    work page

  10. [10]

    State-Averaged VQE A key step in MC-VQE is the use of a state-averaged VQE entangler operator ˆUVQE(θg) to maximally decou- ple the contracted reference states{|ΦΞ⟩} from the rest of 10 the Hilbert space, i.e., using the VQE entangler operator to maximally block-diagonalize the Hamiltonian. This VQE entangler operator acts over the full qubit Hilbert spac...

    work page

  11. [11]

    Technical Detail: VQE Entangler Parametrization The definition and parametrization of the VQE entan- gler circuit is something of an art. In any configuration- space basis, the adiabatic eigenfunctions of the real elec- tronic or ab initio exciton Hamiltonian can be written as real, orthonormal vectors with arbitrary total phase of ±1. Therefore, the VQE en...

    work page

  12. [12]

    interfering

    Subspace Eigenstates The final MC-VQE eigenstates are determined by solv- ing for the subspace eigenvectorsVΞΘ and eigenvaluesEΘ by classically solving the entangled, contracted subspace eigenproblem, ∑ Ξ′ HΞΞ′VΞ′Θ =VΞΘEΘ : ∑ Ξ VΞΘVΞΘ′ =δΘΘ′ (86) where the entangled, contracted subspace Hamiltonian is, HΞΞ′ ≡ ⟨ΦΞ| ˆU† VQE ˆH ˆUVQE|ΦΞ′⟩ (87) This can be eva...

    work page

  13. [13]

    generating states

    Generating State Coefficients For convenience, the MC-VQE states can be re- expressed in terms of “generating states” |ΓΘ⟩, |ΨΘ⟩ ≡ ˆUVQE|ΓΘ⟩ (94) 13 where, |ΓΘ⟩ ≡ ∑ I |I⟩ΓIΘ (95) where, ΓIΘ ≡ ∑ Ξ CIΞVΞΘ (96) Thus the generating states are rotations of the contracted reference states, and can also be expressed in CIS form. To utilize this expression, classic...

    work page

  14. [14]

    unrelaxed

    Formal Definition: MC-VQE Energy Gradient and Lagrangian Pauli-Basis Density Matrices: For a given MC-VQE wavefunction |ΨΘ⟩, the adiabatic state energy is, EΘ ≡ ⟨ΨΘ| ˆH|ΨΘ⟩ (100) and our objective in this section is to form the “relaxed” one- and two-particle density matrices in the Pauli basis, e.g., γΘ ZA ≡ dEΘ dZA , ∀ A (101) ΓΘ ZZAA′ ≡ dEΘ dZZAA′ , ∀ A...

    work page

  15. [15]

    However, it is still useful to explicitly demonstrate this, as nonzero responses arise from this term for other properties such as non-adiabatic coupling vectors

    Subspace Eigenstate Response (Analytical) The subspace eigenstate response equations have an analytical solution that yields a trivial result for the spe- cial case of energy gradients. However, it is still useful to explicitly demonstrate this, as nonzero responses arise from this term for other properties such as non-adiabatic coupling vectors. The subs...

    work page

  16. [16]

    Unrelaxed Pauli Density Matrices Form the unrelaxed Pauli density matrices, e.g., γΘ,0 ZA ≡ ∂EΘ ∂ZA =λZA[θg,θM[ΓIΘ]] (121) ΓΘ,0 ZZAB ≡ ∂EΘ ∂ZZAB = ΛZZAB[θg,θM[ΓIΘ]] (122)

    work page

  17. [17]

    CP-SA-VQE Response The coupled-perturbed state-averaged variational quantum eigensolver equations (CP-SA-VQE) are, ∂LΘ ∂θg = 0 ∀ g (123) Both the right- and left-hand-sides of the CP-SA-VQE equations involve quantities from quantum tomography measurements

    work page

  18. [18]

    CP-SA-VQE Response RHS Form the gradient of the energy with respect to VQE parameters, GΘ g ≡ ∂EΘ ∂θg =εθg+π/4[θg,θM[ΓIΘ]] (124) −εθg−π/4[θg,θM[ΓIΘ]]

    work page

  19. [19]

    CP-SA-VQE Response LHS Form the SA-VQE Hessian, including the diagonal, Hgg ≡ ∂2 ¯E ∂θ2g = 1 NΘ ∑ Θ εθg+π/2[θg,θM[CIΘ]] 15 +2εθg[θg,θM[CIΘ]] −εθg−π/2[θg,θM[CIΘ]] (125) and off-diagonal contributions, Hg̸=g′ ≡ ∂2 ¯E ∂θg∂θg′ = 1 NΘ ∑ Θ εθg+π/4,θg′+π/4[θg,θM[CIΘ]] −εθg+π/4,θg′−π/4[θg,θM[CIΘ]] −εθg−π/4,θg′+π/4[θg,θM[CIΘ]] +εθg−π/4,θg′−π/4[θg,θM[CIΘ]] (126)

    work page

  20. [20]

    CP-SA-VQE Response Equations Solve the CP-SA-VQE response equations, ∑ g′ Hgg ′ ˜θΘ g′ = −GΘ g (127)

    work page

  21. [21]

    CP-SA-VQE Response Contribution The CP-SA-VQE response contribution to the Pauli density matrix is, e.g., γΘ,VQE ZA = ∑ g ˜θg ∂2 ¯E ∂θg∂ZA = 1 NΘ ∑ Θ ∑ g ˜θg (128) [ λθg+π/4 ZA [θg,θM[CIΘ]] −λθg−π/4 ZA [θg,θM[CIΘ]] ]

    work page

  22. [22]

    CP-CRS Response The coupled-perturbed contracted reference state (CP- CRS) equations are, ∂LΘ ∂CIΞ = 0 ∀ I, Ξ (129) The right-hand-side of the CP-CRS equations involves quantities from quantum tomography measurements - the left-hand-side is a classical coupled-perturbed con- figuration interaction singles (CP-CIS) Hessian

    work page

  23. [23]

    CP-CRS Response RHS The CP-CRS RHS is defined as, GΘ IΞ ≡ ∂EΘ ∂CIΞ + ∂LVQE Θ ∂CIΞ (130)

    work page

  24. [24]

    CP-CRS Response RHS #1 The first contribution to the CP-CRS RHS involves the derivative of the state energy with respect to the CIS CRS coefficients through the generator state coefficients and angles, GΘ IΞ ← ∑ M ∂Eθ ∂θM[ΓIΘ] ∂θM[ΓIΘ] ∂ΓIΘ ∂ΓIΘ ∂CIΞ = ∑ M ∂ε[θg,θM[ΓIΘ]] ∂θM[ΓIΘ] ∂θM[ΓIΘ] ∂ΓIΘ VΞΘ (131) The gradients of the CIS circuit angles with respect to t...

    work page

  25. [25]

    CP-CRS Response RHS #2 The second contribution to the CP-CRS RHS involves the derivative of the SA-VQE Lagrangian energy with respect to the CIS CRS coefficients, GΘ IΞ ← ∑ g ∑ M ˜θΘ g ∂2 ¯E ∂θg∂θM[CIΞ] ∂θM[CIΞ] ∂CIΞ (134) 16 = 1 NΘ ∑ g ∑ M ˜θΘ g ∂εθg+π/4[θg,θM[CIΞ]] ∂θM[CIΞ] ∂θM[CIΞ] ∂CIΞ − 1 NΘ ∑ g ∑ M ˜θΘ g ∂εθg−π/4[θg,θM[CIΞ]] ∂θM[CIΞ] ∂θM[CIΞ] ∂CIΞ Thi...

    work page

  26. [26]

    CP-CRS Response LHS The CP-CRS Hessian is (block diagonal in CIS eigen- state Θ), HIΞ,I ′Ξ′ =δΞΞ′ [ HII ′ −ECIS Ξ (δII ′ + 2CIΞCI ′Ξ) ] (135)

    work page

  27. [27]

    CP-CRS Response Equations Solve the CP-CRS response equations (block diagonal in CIS eigenstate Ξ), HIΞ,I ′Ξ′ ˜ΞΘ I ′Ξ′ = −GΘ IΞ (136)

    work page

  28. [28]

    CP-CRS Response Contribution The CRS response contribution to the density matrix is, e.g., γΘ,CRS ZA ≡ ∂ ∂ZA ∑ IΞ ˜ΞΘ IΞRIΞ = ∑ IΞ ˜ΞΘ IΞ ∂ ∂ZA RIΞ = ∑ IΞ ∑ I ′I ′′ ˜ΞΘ IΞ ∂RIΞ ∂HI ′I ′′ ∂HI ′I ′′ ∂ZA = ∑ I ′I ′′ DΘ I ′I ′′ ∂HI ′I ′′ ∂ZA (137) where, DΘ I ′I ′′ = ∑ Ξ ˜ΞΘ I ′ΞCI ′′Ξ − ∑ Ξ [∑ I ˜ΞΘ IΞCIΞ ] CI ′ΞCI ′′Ξ (138) The final partials of the CIS Hami...

    work page

  29. [29]

    Classical AIEM Gradient Stage

    Relaxed Pauli Density Matrices The target relaxed Pauli density matrices are simply given as the sum of the unrelaxed density matrices plus the individual response contributions, e.g., γΘ ZA ≡ dEΘ dZA = ∂LΘ ∂ZA =γΘ,0 ZA +γΘ,VQE ZA +γΘ,CRS ZA (139) D. Classical AIEM Gradient Stage

    work page

  30. [30]

    Pauli Basis AIEM Gradient At present, we have the Lagrangian representation, EΘ = LΘ = E + ∑ A ZAγΘ ZA + XAγΘ XA +1 2 ∑ <A,A′> X XAA′ΓΘ XXAA′ + X ZAA′ΓΘ XZAA′ + ZXAA′ΓΘ ZXAA′ + ZZAA′ΓΘ ZZAA′ (140) But, due to the solution of the response equations above, the Lagrangian is fully relaxed in terms of MC-VQE pa- rameters. Therefore, dζEΘ =∂ζLΘ = Eζ + ∑ A Zζ A...

    work page

  31. [31]

    ] = [ ∂ ∂ϵA′′ H E ]    1/2 + ∑ A [ ∂ ∂ϵA′′ H ZA ]    δAA′′/2 γΘ ZA +

    Monomer Basis AIEM Gradient At this point it is advantageous to perform a linear transformation to a monomer-basis matrix element rep- resentation of the gradient, dζEΘ = ∑ A ϵA,ζ H γA,Θ H +ϵA,ζ P γA,Θ P +ϵA,ζ T 0 γA,Θ T +1 2 ∑ <A,A′> vAA′,ζ HH ΓAA′,Θ HH +vAA′,ζ HT ΓAA′,Θ HT +vAA′,ζ HP ΓAA′,Θ HP 17 +vAA′,ζ TH ΓAA′,Θ TH +vAA′,ζ TT ΓAA′,Θ TT +vAA′,ζ TP ...

    work page

  32. [32]

    Dimer Interaction Matrix Element Gradients At this point, it is advantageous to contract through partial derivatives of the dimer interaction matrix ele- ments to obtain a monomer-property-only representation of the analytical derivative. dζEΘ = ∑ A ϵA,ζ H γA,Θ H +ϵA,ζ P γA,Θ P +ϵA,ζ T 0 γA,Θ T +⃗ µA,ζ H ⃗ ηA,Θ H +⃗ µA,ζ P ⃗ ηA,Θ P +⃗ µA,ζ T ⃗ ηA,Θ T ...

    work page

  33. [33]

    Monomer Property Gradients Finally, the contraction of the monomer property den- sity matrices with the monomer property nuclear gradi- ents can be performed as indicated in Equation 144. For today’s exercise, this is a simple set of multiply-add op- erations with pre-supplied monomer property gradients (obtained with classical Lagrangian theory in method...

    work page

  34. [34]

    Pauli Density Matrices Table I compares the deviations of Pauli-basis den- sity matrices computed with different methods and gra- dient approaches for the ground state and first ex- cited states of BChl-a dimer within the AIEM based on ωPBE(ω = 0.3)/6-31G*-D3.106,107. The principal find- ing is that MC-VQE including all response terms [de- noted as VQE(Y,Y) ...

    work page

  35. [35]

    Monomer Property Density Matrices Table II shows the same analysis as Table I, but now for monomer-property density matrices. The findings are identical - the response-including MC-VQE gradients agree with finite difference to the same order as FCI or CIS, and the errors induced by neglecting response terms in MC-VQE gradients are of essentially the same ord...

    work page

  36. [36]

    OpenFermion: The Electronic Structure Package for Quantum Computers

    Full Nuclear Gradients Table III shows the same analysis as Table II, but now for the complete nuclear gradients of selected atoms. The very nature of this study highlights the imperative to have efficient analytical gradient methodology where only a minimal number of response equations are solved: ob- taining the finite difference gradient for the full syste...

    work page internal anchor Pith review Pith/arXiv arXiv 2014