Lee et al. define the ansatz by the distinction it removes: "Here the single and double 'excitation' terms do not distinguish between occupied and unoccupied orbitals and they are therefore called 'generalized' singles and doubles (GSD)", and name the result in the same section — "We shall term this ansatz UCCGSD". Its cost is stated beside its sibling's rather than alone: the paper's Table 1 puts UCCGSD at $O(N^4)$ gates and $O(N^3)$ depth against k-UpCCGSD's $O(kN^2)$ and $O(kN)$. The two are separate ansätze in this paper, compared against each other and against UCCSD, which is why `k-upccgsd-ansatz` and this node are separate rather than one node drawn twice.

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 familyexcite any orbital pair, not just occ to virt

   The ansatz keeps UCC's exponential form $|\psi\rangle = e^{\hat T - \hat T^\dagger}|\phi_0\rangle$, with the anti-Hermitian combination guaranteeing $e^{\hat T-\hat T^\dagger}$ is unitary, but redefines the cluster operator itself: $\hat T = \hat T_1 + \hat T_2 = \frac12\sum_{pq} t^q_p \hat a_q^\dagger \hat a_p + \frac14\sum_{pqrs} t^{rs}_{pq}\hat a_r^\dagger \hat a_s^\dagger \hat a_q \hat a_p$, where every index $p,q,r,s$ runs over the full set of spin-orbitals rather than being split into occupied indices $i,j$ and virtual indices $a,b$ as in UCCSD's $\hat T_1=\sum_{ia}t_i^a \hat a_a^\dagger \hat a_i$. The energy is evaluated variationally, $E=\langle\psi|\hat H|\psi\rangle/\langle\psi|\psi\rangle$, rather than by the projective CC route. approximation: on real hardware $e^{\hat T-\hat T^\dagger}$ is not applied exactly but Trotterized into a small number of steps, so the state actually prepared is not identical to the one the amplitudes were optimized for

   approximation

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  Hamiltonian H on N spin-orbitals,  eta electrons
       reference state |phi_0>  (an eta-electron Slater determinant, e.g. RHF;
           a multi-determinantal reference is also usable)

requires  a fermion-to-qubit encoding fixed ahead of time, one qubit per spin-orbital
    # qubit count is linear in N by construction of the mapping (Sec. Quantum
    #   Resource Requirements)

# --- cluster operator, over ALL spin-orbital pairs, not just occ-to-virt -------------
T1 = (1/2) sum_pq    t^q_p    a_q^dag a_p
T2 = (1/4) sum_pqrs  t^rs_pq  a_r^dag a_s^dag a_q a_p          (Eq. 11-12)
    # p,q,r,s each range over all N spin-orbitals -- occupied and unoccupied are
    #   NOT distinguished, unlike UCCSD's T1 = sum(i occ, a virt) t^a_i a_a^dag a_i
    #   (Eq. 3-5): "the single and double 'excitation' terms do not distinguish
    #   between occupied and unoccupied orbitals" -- hence 'generalized' (GSD)
T = T1 + T2
    # the free amplitudes {t^q_p, t^rs_pq} number as many as the parameters of the
    #   2nd-quantized Hamiltonian itself (Sec. Generalized CC, same T as Eq. 11)

# --- anti-Hermitize and exponentiate --------------------------------------------------
generator = T - T^dag
    # the anti-Hermitian combination is what makes exp(generator) unitary
U(theta) = exp(T - T^dag)                     (Eq. 13 -- the paper names this UCCGSD)
|psi(theta)> = U(theta) |phi_0>
    # theta is the amplitude vector {t^q_p, t^rs_pq}

# --- what a device actually runs -------------------------------------------------------
U(theta) implemented as  Trotter_m[ exp(T - T^dag) ],   m a small FIXED step count
    # "a practical implementation of UCC relies on approximating e^(T-T^dag) by a
    #   small number of Trotter steps" -- the circuit prepared is not exactly the
    #   |psi(theta)> above; it differs from the state the amplitudes were fit to
    #   (Sec. Quantum Resource Requirements)
    # this paper does not derive how large m must be for UCCGSD; it reports that
    #   even m=1 has been shown elsewhere to reach high accuracy for UCC ansatze
    #   in general (cited to Ref. 64) -- an external result, not proved here

# --- how the resulting family is scored ------------------------------------------------
E(theta) = <phi_0| U(theta)^dag H U(theta) |phi_0>  /  <phi_0| U(theta)^dag U(theta) |phi_0>
    # evaluated variationally (Eq. 9's form, carried over unchanged from the general
    #   UCC ansatz Eq. 8 to the UCCGSD ansatz Eq. 13, which only redefines T),
    #   NOT by the projective coupled-cluster equations (Eq. 6-7)

return  the parameterized circuit  U(theta) |phi_0>,  theta ranging over {t^q_p, t^rs_pq}
# a circuit FAMILY, not yet a converged state -- theta is fit by an outer classical
#   loop elsewhere in the pipeline

Lee, Huggins, Head-Gordon and Whaley cost UCCGSD in Table 1: $\mathcal{O}(N^4)$ gates and $\mathcal{O}(N^3)$ circuit depth in $N$ spin-orbitals, with no dependence on the electron number $\eta$ — against UCCSD's $\mathcal{O}((N-\eta)^2\eta^2)$ and $\mathcal{O}((N-\eta)^2\eta)$. Conditions: state preparation only; a fixed number of Trotter steps of $e^{\hat{T}-\hat{T}^{\dagger}}$ rather than the exact exponential; gate count taken as the two-qubit count; and a depth assuming maximum parallelization, no Jordan-Wigner overhead, and a linearly connected qubit array, the last resting on their Refs. 59 and 63, the latter unpublished. Qubits are linear in $N$. Table 9 reports amplitudes per benchmark case, not as a formula: 27 at $N=8$, 135 at $N=12$, 434 at $N=16$ for both $\eta=4$ and $\eta=10$; their classical code stops near 16 spin-orbitals. Nothing bounds the optimization — each one was rerun thirty to two hundred times from random starting points — and the paper gives no absolute shot or measurement count, deferring measurement to McClean et al.

• Generalized excitation VQE · Qiskit

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  Generalized singles and doubles relax occupied-to-virtual restrictions to enlarge the variational manifold.

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  Literature-backed method record; algorithmic scope and evidence boundary reviewed, with no benchmark run claimed.

• Generalized excitation VQE

  Generalized singles and doubles relax occupied-to-virtual restrictions to enlarge the variational manifold.