Parrish et al. describe the extension by what it returns: MC-VQE "allows for the efficient computation of the transition energies between the ground state and several low-lying excited states of a molecule, as well as the oscillator strengths associated with these transitions". The oscillator strengths are the part no other route in this slot claims, and they are what turns a list of energies into a spectrum that can be compared with an experiment. Evidence is a simulation at a size worth stating: they "numerically simulate MC-VQE by computing the absorption spectrum of an ab initio exciton model of an 18-chromophore light-harvesting complex from purple photosynthetic bacteria". The abstract quantifies no circuit count, measurement count or scaling, and "efficient" is its word for the approach rather than a bound anyone proved here.

A Hermitian H reachable as a sum of terms that can be measured separately; a statement of which state is wanted — an index k, a symmetry sector, or a target energy to sit nearest; for most of the methods here, the ground state already solved, and for the deflation route every lower state as well; a target additive error and a shot budget.

1. Hamiltonian whose excited state is wanted → Parameterised circuit familyChoose a parameterised trial state

   MC-VQE splits this slot: only the second half is variational. A classical CIS solve fixes the contracted, orthonormal reference states $|\Phi_\Theta\rangle$ on $N$ qubits, one per chromophore; the demonstration keeps that whole manifold, so $N_\Theta = N+1$ there assumption: each monomer carries only a ground and a first excited state, a restriction adopted "to facilitate ease of mapping to qubits". Their amplitudes load through a fixed matryoshka ladder: an $R_y(\theta_0)$ pump, then controlled $F_y$ gates, with angles read recursively off the CIS coefficients, $\theta_0 = \cos^{-1}(\mu)$, $\theta_{AB} = \cos^{-1}(\alpha/\sin(\theta_0))$, the last resolved only up to a $\pm 1$ phase that a separate check on the forward formula fixes; none is free. All variational freedom lives in the entangler $\hat{U}$, restrictable to $\mathrm{SO}(2^N)$ without loss because the real exciton Hamiltonian's adiabatic eigenfunctions can be written as real, orthonormal vectors with arbitrary total phase $\pm 1$, and built as one six-parameter $\mathrm{SO}(4)$ block $\hat{U}_2 = \exp(-i\theta_{XY}\hat{X}_A \otimes \hat{Y}_B - \ldots - i\theta_{IY}\hat{I}_A \otimes \hat{Y}_B)$ per two-body interaction site. The ansatz is $|\Psi_\Theta\rangle \equiv \hat{U}\sum_{\Theta'}|\Phi_{\Theta'}\rangle V_{\Theta'\Theta}$.

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2. Parameterised circuit family → State you can prepareMinimise the objective over the parameters

   The entangler is tuned to "maximally decouple" the $N_\Theta$ approximate eigenstates from the rest of the Hilbert space — to block-diagonalise $\hat{H}$ approximation: that decoupling is only approximate, and only in a 2-norm sense in the Hamiltonian: the state average $\bar{E} = \frac{1}{N_\Theta}\sum_\Theta E_\Theta = \frac{1}{N_\Theta}\sum_\Theta H_{\Theta\Theta}$, the second equality following "from the definition of the trace". No penalty term and no projection against lower roots: one scalar covers every state, so "all states are computed to approximately equal accuracy". Only the trace enters, so the loop measures only the $N_\Theta$ diagonal elements $H_{\Theta\Theta} = \langle\Phi_\Theta|\hat{U}^\dagger\hat{H}\hat{U}|\Phi_\Theta\rangle$; off-diagonals wait until the angles converge. The demonstration ran L-BFGS on 108 parameters from a zero-entanglement guess of all-zero angles, converging in 14 iterations approximation: gradients are second-order symmetric finite differences with $\Delta\theta = 0.01$, not analytic assumption: barren plateaus are sidestepped because that solution lay "directly downhill" from the guess — one system, not a proof; another $\mathrm{SO}(4)$ parametrisation needs about twice the L-BFGS iterations to reach it, and a qualitatively different solution under steepest descent.

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3. State you can prepare → Number with an error barEstimate an observable

   Partial tomography of 1- and 2-body Paulis gives the diagonal $H_{\Theta\Theta}$; each off-diagonal element costs two further interference-state preparations, each measured over that same Pauli set, through $2H_{\Theta\neq\Theta'} = (\langle\Phi_\Theta| + \langle\Phi_{\Theta'}|)\hat{U}^\dagger\hat{H}\hat{U}(|\Phi_\Theta\rangle + |\Phi_{\Theta'}\rangle)/2 - (\langle\Phi_\Theta| - \langle\Phi_{\Theta'}|)\hat{U}^\dagger\hat{H}\hat{U}(|\Phi_\Theta\rangle - |\Phi_{\Theta'}\rangle)/2$, built by the same ladder with coefficients $\mu^{\pm} = (\mu^\Theta \pm \mu^{\Theta'})/\sqrt{2}$. Classical diagonalisation $H_{\Theta''\Theta'}V_{\Theta'\Theta} = V_{\Theta''\Theta}E_\Theta$ with $V_{\Theta'\Theta}V_{\Theta'\Theta''} = \delta_{\Theta\Theta''}$ returns Ritz eigenvalues and rotated reference states $|\Gamma_\Theta\rangle \equiv \sum_{\Theta'}|\Phi_{\Theta'}\rangle V_{\Theta'\Theta}$, with $|\Psi_\Theta\rangle \equiv \hat{U}|\Gamma_\Theta\rangle$. Substituting the desired operator for $\hat{H}$ in that formula, on the rotated states, gives transition properties — finally the oscillator strength $O_{\Theta\Theta'} = \frac{2}{3}(E_{\Theta'} - E_\Theta)\langle\Psi_\Theta|\hat{\mu}|\Psi_{\Theta'}\rangle^2$, the square running over Cartesian components of the vector-valued $\hat{\mu} \equiv \sum_A \vec{\mu}^A_{\hat{I}}\hat{I}_A + \vec{\mu}^A_{\hat{Z}}\hat{Z}_A + \vec{\mu}^A_{\hat{X}}\hat{X}_A$ — one-body Paulis only. MC-VQE needs higher-order density matrices at no stage assumption: shot noise and device error are absent — the demonstration's expectation values were contractions of simulator wavefunction amplitudes, "equivalent to infinite averaging of discrete Pauli measurements", with noise and error channels not modelled, so no shot count is charged.

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A scalar estimate — of that eigenvalue, or of the gap between it and the ground state — together with the run budget it consumed. Some methods return a whole low-energy subspace at once and others return one state per run, and that difference is a cost, not a convenience.

given  N chromophore monomers, each carrying two photoactive states |0_A>
       (ground) and |1_A> (excited) -- a restriction adopted "to facilitate
       ease of mapping to qubits", not a general limit of the method
       an ab initio exciton Hamiltonian H = E*I + H1 + H2 on N qubits,
       written in 1- and 2-body Pauli operators                       (Eq. 8)
       N_Theta = N+1 target states: the CIS manifold -- 1 ground
       configuration plus N singly-excited configurations, all allowed
       to mix                                            (Supp., CIS State Preparation)

requires  an efficient circuit to prepare each CIS reference state |Phi_Theta>
    and each interference state (|Phi_Theta> +/- |Phi_Theta'>)/sqrt(2) -- a
    condition specific to this construction, met here by a "matryoshka" ladder
    of one Ry pump gate on |0_A> and N-1 controlled Fy gates on the rest,
    with angles read recursively off the classical CIS coefficients
    mu, alpha, beta, ...                                    (Supp. Eqs. 42-55)

# --- stage 1: classical reference states ------------------------------------
solve CIS (or another polynomial-scaling classical method) for H
    -> N_Theta contracted, orthonormal reference states {|Phi_Theta>}
    # this only "sketches out" the shape of each state; nothing here is
    #   variational, and nothing here is proved accurate

# --- stage 2: state-averaged optimisation of the entangler ------------------
U(eta) = one SO(4) two-body entangler block per two-body interaction site
    of the exciton Hamiltonian, eta its 6 real angles              (Eq. 67, Supp.)
    # TWO separate restrictions here, and only the first is proved lossless:
    #   (a) restricting U to SO(2^N) is "without loss" -- the real exciton
    #       Hamiltonian's adiabatic eigenfunctions can be written as real,
    #       orthonormal vectors with only an overall +/-1 phase free
    #                                        (Supp., VQE Global Entangler Matrix)
    #   (b) building that SO(2^N) entangler out of ONE layer of local, per-site
    #       SO(4) blocks is a design choice, NOT covered by (a): the paper's
    #       own words are "we have elected to construct", with no completeness
    #       claim, and it offers further layers or entanglers extended "to the
    #       next layer(s) of nearest neighbors" when "additional variational
    #       flexibility in the ansatz is desired"          (Supp., after Eq. 60)

loop, varying eta, until the state-averaged energy stops decreasing:
    # the paper states no general stopping rule for this loop -- only what
    #   happened in its one demonstration; the phrasing here is a gloss
    for each Theta in [0, N_Theta):
        prepare |Phi_Theta>, apply U(eta)
        measure the diagonal element only
            H_ThetaTheta = <Phi_Theta| U(eta)^dagger H U(eta) |Phi_Theta>   (Eq. 4)
    Ebar(eta) = (1/N_Theta) * sum_Theta H_ThetaTheta                       (Eq. 6)
        # equals the trace of the contracted Hamiltonian in the {Phi_Theta}
        #   basis "from the definition of the trace" -- so minimising Ebar IS
        #   minimising the sum of these diagonal elements, nothing more
    step eta to reduce Ebar
        # demonstration: L-BFGS from an all-zero initial guess, second-order
        #   symmetric finite-difference gradients (delta_theta = 0.01),
        #   converged in 14 iterations on the N=18 system -- offered as an
        #   empirical route around barren plateaus (the minimum found lay
        #   "directly downhill" from the guess), not a proven one; a
        #   different logical SO(4) parametrisation of the same block took
        #   about twice as many L-BFGS iterations to the same solution
U_fixed <- U(eta) at the stopping point
    # one scalar objective covers every state, so all N_Theta states are
    #   "computed to approximately equal accuracy" -- no penalty term or
    #   projection singles out the ground state, unlike orthogonality-
    #   constrained VQE

# --- stage 3: off-diagonal elements, entangler now fixed --------------------
for each pair Theta, Theta' with Theta != Theta':
    prepare (|Phi_Theta> + |Phi_Theta'>)/sqrt(2), apply U_fixed, measure H
    prepare (|Phi_Theta> - |Phi_Theta'>)/sqrt(2), apply U_fixed, measure H
    H_ThetaTheta' = [ (sum-state expectation) - (difference-state expectation) ] / 2
                                                                           (Eq. 5)
    # two further circuits per off-diagonal element, this pair costed
    #   separately AFTER eta has converged -- not folded into stage 2
    # only 1- and 2-body Pauli expectation values are ever measured here;
    #   no 3- or 4-particle density matrix is required anywhere in MC-VQE

# --- stage 4: classical diagonalisation of the N_Theta x N_Theta problem ----
diagonalize H_ThetaTheta':
    H_Theta''Theta' V_Theta'Theta = V_Theta''Theta E_Theta,
    V_Theta'Theta V_Theta'Theta'' = delta(Theta, Theta'')                  (Eq. 2)
    # E_Theta are RITZ approximations to the exact eigenvalues, not the
    #   eigenvalues themselves
|Gamma_Theta> = sum_Theta'  |Phi_Theta'> V_Theta'Theta        # rotated references
|Psi_Theta>   = U_fixed |Gamma_Theta>                           # approximate eigenstates

# --- transition properties: stage 3's formula, mu_hat standing in for H ----
mu_hat = sum_A  mu_I^A * I_A + mu_Z^A * Z_A + mu_X^A * X_A
    # one-body Paulis only; obtained by substituting the desired operator
    #   for H in the stage-3 formula                                (Eq. 5, 77)
for each pair Theta, Theta':
    O_ThetaTheta' = (2/3) * (E_Theta' - E_Theta) * <Psi_Theta| mu_hat |Psi_Theta'>^2
        # sum over the Cartesian components of the vector-valued mu_hat  (Eq. 76)

return  {E_Theta} and {O_ThetaTheta'} over the CIS manifold
    -> an excitation-energy and oscillator-strength spectrum, e.g. an
       absorption spectrum built by broadening each transition
# demonstrated noiseless, at infinite shots: 1- and 2-body expectation values
#   were contractions of simulator wavefunction amplitudes, "equivalent to
#   infinite averaging of discrete Pauli measurements", with no noise or
#   error channel modelled and so no shot count charged; N=18, a SINGLE
#   entangler layer, 108 parameters, matched FCI to tens of microeV in
#   energy and well under 1% in oscillator strength -- one system, not proved
# depth linear and gate count quadratic in N is claimed only for exciton
#   models "with local Hamiltonian connectivity", in the Outlook, and without
#   derivation -- do not read it as proved or as general to other Hamiltonians

Parrish, Hohenstein, McMahon and Martínez count MC-VQE in contracted Hamiltonian elements: optimising the state-averaged entangler "requires only the measurement of $N_\Theta$ diagonal matrix elements $H_{\Theta\Theta}$", and the off-diagonal elements — written $N_\Theta^2$, though only $N_\Theta(N_\Theta-1)/2$ are distinct — are determined afterwards, two interference-state circuits each (Eq. 5), from 1- and 2-body Paulis alone: $2N$ one-body terms plus four per monomer pair (Eq. 8), truncated to $N$ nearest-neighbour pairs in the demonstration, where $N_\Theta = N+1$ on $N$ qubits. Nothing bounds the outer loop: the demonstration reports 108 parameters — six per $\mathrm{SO}(4)$ entangler, one entangler per interaction site, a single layer — converging in "14 L-BFGS iterations" of second-order symmetric finite differences, to within tens of $\mu$eV and $\ll 1\%$ oscillator strength of FCI on the $N=18$ LH2 B850 ring; one noiseless, infinite-shot simulation, not proved, and a different parametrisation "takes approximately twice as many iterations". The Outlook claims depth linear and gate count quadratic in $N$ for exciton models "with local Hamiltonian connectivity", without derivation.

• Multistate contracted VQE · Qiskit

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  A contracted reference subspace is jointly entangled before a small effective Hamiltonian is diagonalized.

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• Multistate contracted VQE

  A contracted reference subspace is jointly entangled before a small effective Hamiltonian is diagonalized.