Mizukami et al. state what is varied and what follows from it. OO-UCC "variationally determines the coupled cluster amplitudes and also molecular orbital coefficients", and "owing to its fully variational nature, first-order properties are readily available" — which they cash out immediately: "this feature allows the optimization of molecular structures in VQE without solving any additional equations". The resource claim is comparative and unquantified in the abstract: "the method requires smaller active space and shallower quantum circuit than UCC to achieve the same accuracy", with no number attached, so none is quoted here. Evidence is simulation, and it is worth naming what was simulated because it is a property calculation rather than a single energy: "numerical examples of OO-UCC using quantum simulators, which include the geometry optimization of the water and ammonia molecules using analytical first derivatives of the VQE".

The Hamiltonian whose ground state is wanted, together with whatever structure is to be respected — particle number, spin, point-group symmetry, a reference determinant — and the connectivity and native gate set of the device the family has to run on.

1. Hamiltonian whose eigenvalues are wanted → Parameterised circuit familyrotate orbitals via Newton-Raphson on Hk=-g

   The route starts from UCCSD, $|\Psi\rangle=e^{\hat A_1+\hat A_2}|0\rangle$ with $\hat A_n=\hat T_n-\hat T_n^\dagger$, and factors it as $|\Psi'\rangle=e^{\hat A_2}e^{\hat A_1}|0\rangle$. Because the singles generator $e^{\hat A_1}$ is identical to the orbital-rotation operator $e^{\hat\kappa}$, $\hat\kappa=\sum_{pq}\kappa_{pq}(\hat E_{pq}-\hat E_{qp})$, assumption: the singles/orbital part can be optimized on a classical computer from the 1- and 2-electron reduced density matrices VQE already measures, rather than varied on the circuit, collapsing the ansatz to OO-UCCD, $|\Psi_{OO-UCCD}\rangle=e^{\tilde A_2}|\tilde 0\rangle$, $|\tilde 0\rangle=e^{\hat A_1}|0\rangle$. Orbitals move by a Newton-Raphson step $H\kappa=-g$ built from the electronic Hessian and gradient $g_{pq}=\langle\Psi|[\hat H,\hat E_{pq}^-]|\Psi\rangle$, alternated with VQE until self-consistent. approximation: mapping $e^{\tilde A_2}$ onto the circuit truncates the Trotter expansion at its first step, $|\Psi\rangle=\prod_\mu e^{\tilde A_{2,\mu}}|\tilde 0\rangle$.

   approximationassumption

A circuit family with a fixed structure and free real parameters, together with the number of those parameters — which is the size of the classical search problem handed to the next layer.

given  molecular Hamiltonian H built from 1- and 2-electron integrals h_pq, h_pqrs
       in a chosen orbital basis;  reference determinant |0>  (conventionally Hartree-Fock)
       UCCSD ansatz  |Psi> = e^(A1+A2) |0>,   A_n = T_n - T_n^dagger        (Eq. 3, Eq. 9)
       a VQE circuit that already measures the 1- and 2-particle RDMs needed below
           # RDMs are "readily available in VQE, because it measures [them] to
           #   compute electronic energy in a given quantum circuit" -- no extra
           #   measurement is added for the orbital step

requires  no formal bound is stated on the outer loop below;
    # optimizing the singles/orbital part classically from the measured RDMs, rather
    #   than varying it on the circuit, is presented as a design choice (Sec. II),
    #   not as something the paper proves necessary

# --- decompose UCCSD into a classical orbital rotation and a Trotterized doubles part ---
model  |Psi'> = e^(A2) e^(A1) |0>                                          (Eq. 10)
    # a distinct wave-function model, not an exact rewrite of e^(A1+A2)|0>:
    #   A1 and A2 do not commute, so separating them "is different from the
    #   original ansatz" (Sec. II)

identify  e^(A1)  with the orbital-rotation operator  e^(kappa)             (Sec. II)
    kappa = sum_pq  kappa_pq (Epq - Eqp),        Epq^- := Epq - Eqp
    # "the singles part e^(A1) ... is identical to the orbital rotation unitary
    #   operator e^kappa" -- so it can be optimized on a classical computer via
    #   standard orbital-optimization technique instead of varied on the circuit

collapse to  |Psi_OOUCCD> = e^(A2~) |0~>,    |0~> = e^(A1) |0>             (Eq. 11)

# --- outer loop: alternate VQE with a classical Newton-Raphson orbital step ---
repeat
    run VQE, optimizing doubles amplitudes A2~ on the circuit  ->  E(A2~, kappa)
    from the 1RDM, 2RDM VQE just measured, form
        g_pq       = <Psi|[H, Epq^-]|Psi>                                  (Eq. 8)
        Hess_pq,rs = (1/2) ( <Psi|[[H,Epq^-],Ers^-]|Psi> + <Psi|[[H,Ers^-],Epq^-]|Psi> )   (Eq. 7)
    solve  Hess kappa_step = -g   for the orbital-rotation step             (Eq. 6, Newton-Raphson)
    kappa <- kappa + kappa_step
until self-consistent
    # "we repeatedly perform the VQE and the orbital optimization until
    #   convergence" -- no cycle-count bound is given (Sec. II);
    #   "UCCSD-level ... results ... at the cost of repeated VQE optimizations" (Sec. III)
    # at self-consistency the singles operator A1~ itself vanishes (the Brueckner
    #   condition) together with dE/dkappa = 0, simultaneously and by construction
    #   of the separation above -- "variational orbitals are Brueckner orbitals
    #   in this ansatz" (Sec. II, 'Trotterization and Brueckner orbitals')
    # the paper's own printed condition is the exponential  e^(A1~) = 0 -- which
    #   cannot literally hold for a unitary exponential of an anti-Hermitian
    #   operator;  read here, per that same sentence's prose ("the singles'
    #   contribution ... vanishes"), as the operator condition  A1~ = 0

# --- map the converged doubles generator onto the circuit ---
Trotterize  e^(A2~)  ~=  prod_mu e^(A2~_mu),   truncated at the first Trotter step   (Eq. 14)
    # this defines OO-UCCD1; a single Trotter step is a large truncation of the
    #   BCH-infinite UCC generator, and this paper proves no bound on the
    #   truncation error -- it cites Barkoutsos et al.'s empirical finding that a
    #   single step "reproduces ground-state energy accurately", and reports its own
    #   numerics (LiH, H2O, NH3) showing OO-UCCD1 tracks UCCSD1 closely (Table I, Fig. 1)

return  the OO-UCCD1 circuit family together with converged (theta, kappa)
        as the trial state for the eigenvalue problem
# because theta and kappa are BOTH variationally converged, the full chain rule
#   dE/dx = dE/dx|_x + (dE/dtheta)(dtheta/dx) + (dE/dkappa)(dkappa/dx)      (Eq. 12)
#   collapses to just the Hellmann-Feynman term:
    dE/dx = <Psi| dH/dx |Psi>                                    # consequence of Eq. 12
    # the d(E)/d(theta) . d(theta)/dx term vanishes because VQE has converged theta;
    #   the d(E)/d(kappa) . d(kappa)/dx term vanishes because the orbitals are optimized
# a non-orbital-optimized VQE keeps that second term and must instead solve a
#   coupled-perturbed Hartree-Fock (CPHF) equation for the orbital response (Eq. 13)

The paper's only resource figures are measured on one molecule, not derived. Mizukami, Mitarai, Nakagawa, Yamamoto, Yan and Ohnishi report, for ammonia in STO-3G with all electrons correlated and no active-space approximation, 120 VQE parameters $\theta$ and quantum-circuit depth 2720 for OO-UCCD truncated at the first Trotter step — the same pair they quote for plain UCCD at that truncation — against 135 and 2780 for UCCSD, likewise at one Trotter step. They call the reduction slight, and never define what the depth counts or in what gate set. It is paid for by an unbounded outer loop: singles are handled classically, so VQE and a classical Newton–Raphson orbital optimisation are repeated until convergence, UCCSD-level accuracy coming "at the cost of repeated VQE optimizations", with no bound and no measured count of cycles. Once both $\theta$ and $\kappa$ are converged, a first derivative costs only the expectation value of the derivative of the Hamiltonian; second and higher derivatives still need the circuit-parameter response. The abstract's other claim — smaller active space, hence fewer qubits — is nowhere quantified.

• Orbital-optimized VQE · Qiskit

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  Orbital rotations are optimized alongside circuit parameters to improve compact active-space descriptions.

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• Orbital-optimized VQE

  Orbital rotations are optimized alongside circuit parameters to improve compact active-space descriptions.