Cadi Tazi and Thom state what it buys and what it costs in the same breath: the method "provides the possibility of directly computing excited states around a selected target energy, using the same ansatz as for the ground state calculation", and, "inspired by the variance-based methods from the Quantum Monte Carlo literature", it "minimizes the energy variance, thus requiring a computationally expensive squared Hamiltonian". The squared operator is the binding cost and they treat it as one: "we alleviate this potentially poor scaling by employing a Pauli grouping procedure, identifying sets of commuting Pauli strings that can be evaluated simultaneously". The target energy is an input, not an output — a state is reached because it is near a number the user chose. Results are "all electronic excited states with chemical accuracy on ideal quantum simulators" for H2 and LiH; no device run is reported.

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

   FS-VQE keeps the ground-state circuit family and changes only what it acts on; an excited state adds Pauli operators and post-rotation gates, not a new ansatz. The trial state is Trotterised UCC truncated at rank $k$ ($k=2$ is UCCSD), eq. (7) in the paper's own indexing: $|\Psi_{\mathrm{UCC}}\rangle \approx \prod_{i=1}^{k} e^{\theta_i(\hat{T}_i-\hat{T}_i^{\dagger})}|\Psi_0\rangle$. Under Jordan-Wigner the compiled circuit carries one Pauli gadget and one $R_z$ angle per retained excitation: three for $\mathrm{H_2}$ in STO-3G, depth $71$, $44$ CNOTs. approximation: the truncation, not the Trotter split, is the accuracy limit; eq. (6) carries $+\mathcal{O}(1/p)$ but the implemented $p=1$ is claimed exact under Evangelista et al.'s ordering of excitations, which they adopt, while section VIII A concedes singles and doubles do not generally reach FCI accuracy beyond small systems What this route fixes differently is $|\Psi_0\rangle$: the Hartree-Fock determinant gives way to an excited determinant or a spin-symmetrised superposition, for $\mathrm{H_2}$ $(\sigma_g)(\sigma_u)$ for $T_1$ and $(\sigma_g)(\overline{\sigma_u})+(\overline{\sigma_g})(\sigma_u)$ for $S_1$ assumption: that reference overlaps the target wavefunction; strongly multi-reference molecules are left open

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

   Over the same angles as a ground-state run this route minimises $F(\theta)=\langle\Psi(\theta)|(\hat{H}-\omega)^{2}|\Psi(\theta)\rangle$ (eq. 14), not $\langle\hat{H}\rangle$, with $\omega$ an arbitrary target energy given as input. Eq. (13), $(\hat{H}-\omega)^{2}|\Psi\rangle=(E-\omega)^{2}|\Psi\rangle$ on every eigenpair, says the folded operator keeps $\hat{H}$'s eigenstates and folds the eigenvalues about $\omega$, so its lowest eigenstate is the one whose $E_i$ is nearest $\omega$; minimising $F\ge0$ variationally targets that state approximation: the state reached is an ansatz-limited approximation of that eigenstate. For QASM simulations SPSA does the walking: one random $\Delta\theta$ per iteration with a component in every dimension, $F$ sampled only at $\theta\pm\Delta\theta$, so $2$ evaluations per iteration whatever $\dim(\theta)$ approximation: every parameter's gradient is approximated from those two measurements alone. In all computations shots follow $s=s_{\max}-(s_{\max}-s_{\min})e^{-k\times\mathrm{iteration}}$, $k>0$, $1000$ to $10000$; the loop stops at a $10^{-9}$ cost-gradient threshold. $\theta_{\mathrm{final}}$ is post-processed, not the optimiser's output: refit to a parabola's vertex, then rounded to $0$, $\pi/2$ or $\pi$ where the cost improves. Along a surface $\omega_{k+1}=E_k$ and $\theta^{\mathrm{init}}_{k+1}=\theta^{\mathrm{opt}}_{k}$, with an $\eta|\theta_{k-1}-\theta_k|$ penalty, $\eta=0.1$, at isolated geometries where the state jumped.

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

   Inside the loop the scalar read back is $\langle(\hat{H}-\omega)^{2}\rangle$, not $\langle\hat{H}\rangle$: the folded operator is expanded as $\sum_i o_i\langle\hat{P}_i\rangle$ (eq. 15) and each string recovered from counts, $\langle\hat{P}_{\mathrm{diag}}\rangle=\sum_i\lambda_i|\alpha_i|^{2}$ with $\lambda_i=\pm1$ for $I/Z$ strings, and otherwise $\hat{P}=R^{\dagger}\tilde{P}R$ (eq. 24) with $\langle\hat{P}\rangle=\sum_i\tilde{\lambda}_i|\langle\Phi_i|\tilde{\Psi}\rangle|^{2}$ on $|\tilde{\Psi}\rangle=R|\Psi\rangle$, the post-rotation $R$ being a Hadamard on each $X$ qubit and an $R_x(\pi/2)$ on each $Y$ qubit. Strings sharing $R$ are read from one circuit. Squaring $\mathrm{H_2O}$'s $1578$-string Hamiltonian gives about $1578^{2}$ strings, cut by Pauli commutation and anti-commutation relations to $111615$, then by the qubit-wise grouping used here to $20393$ evaluations; $\mathrm{H_2}$'s $24$ folded strings make $9$ QWC groups, or the two general-commuting groups the error-mitigated run measures. Precision is $\epsilon\sim1/\sqrt{s}$ assumption: $\sigma^{2}$ bounded by a constant at a fixed failure probability, a Chebyshev worst case. The reported $\mathrm{H_2}$ QASM energy is a separate readout at $30000$ shots, $E=E_{\mathrm{nuc}}+\langle\Psi(\theta_{\mathrm{final}})|\hat{H}|\Psi(\theta_{\mathrm{final}})\rangle$ (figure 3).

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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  Hamiltonian H, as a weighted sum of Pauli strings h_i P_i
       an arbitrary target energy omega, chosen by the user -- not returned,
           not fitted, an input like any other
       the same Trotterized UCCSD ansatz used for a ground-state VQE run,
           applied to a reference determinant |Phi_ref> chosen for the
           symmetry sector of the wanted excited state
       a shot-budget scheduler, and the SPSA classical optimizer for the
           QASM-simulator runs                                    (Sec. VI B)
           # the LiH potential curves were run on a statevector simulator
           #   instead; SPSA is named only for the QASM runs      (Sec. VII B)

requires  omega closer to the wanted eigenvalue E than to any other -- when it
    is not, the loop still converges, just onto the neighbouring state; the
    paper reports this as expected behaviour of the method, observed as
    jumps between electronic states along a potential energy surface when
    two states are close in energy or gradients are large       (Sec. VI A 1)

# --- fold the spectrum about omega -------------------------------------------
(H - omega)^2 |Psi> = (E - omega)^2 |Psi>                              (Eq. 13)
    # true for every eigenstate |Psi> of H, for every omega -- proved from
    #   linearity of the Schrodinger equation, not ansatz- or method-specific
    # the folded operator keeps H's eigenstates but reorders the eigenvalues
    #   around omega, so its LOWEST eigenstate is the one whose E sits
    #   nearest omega                                     (Sec. III, Fig. 4)
    #   -- Section III has no lettered subsections; this is body text there

F(theta) = <Psi(theta)| (H - omega)^2 |Psi(theta)>                     (Eq. 14)
    # minimised in place of <H>; this swap is the only change from a
    #   ground-state VQE -- same ansatz, same reference-state machinery

# --- build the squared operator, then cut it back down ------------------------
square H to form (H - omega)^2                       # ~ (#Pauli strings in H)^2
    # "the number of Pauli strings primarily obtained is approximately the
    #   square of the number of terms in H"                     (Sec. IV B 1)
    # e.g. H2O/STO-3G: 1578 strings in H square to about 1578^2 before any
    #   reduction is applied                        (Table II, Appendix 2)
reduce via Pauli commutation/anticommutation relations           (Sec. IV B 1)
    # H2O: down to 111615 strings this way            (Table II, Appendix 2)
    # the paper reports this reduction gives an EMPIRICAL scaling below
    #   O(N^6) against the expected O(N^8), a result imported from a
    #   separate study of a different set of systems and only asserted, not
    #   derived, to extend to the folded operator here; the paper's own next
    #   sentence asks for "more formal analyses"                 (Sec. IV B 1)
partition the reduced strings into simultaneously-diagonalizable groups
    # qubit-wise commutativity (QWC): H2O -> 20393 groups under Jordan-
    #   Wigner, one circuit evaluation each; H2's 24 folded strings -> 9 QWC
    #   groups                                        (Table II, Appendix 2)
    # the coarser general-commutativity (GC) partition used for the
    #   error-mitigated run cuts H2's 24 strings to just 2 groups,
    #   G1 (16 strings) and G2 (8 strings)                   (Eq. 30, Sec. V C)
    #   -- Sec. V C is "Zero Noise Extrapolation", not the VI C
    #      post-processing subsections cited further down

# --- VQE loop: minimise F, not <H> --------------------------------------------
repeat until the cost gradient falls below 1e-9:                (Sec. VIII A)
    prepare |Psi(theta)> = UCCSD(theta) |Phi_ref>
    for each Pauli group:
        take s shots, s = smax - (smax - smin) * exp(-k * iteration),
            smin = 1000, smax = 10000, k > 0                       (Eq. 36)
            # used for ALL computations in the paper; precision only tightens
            #   as the run approaches convergence -- early, imprecise
            #   evaluations are tolerated by design              (Sec. VI B 2)
        recover the group's Pauli expectation values from one set of counts
    sum weighted group expectation values to get F(theta)
    theta <- SPSA_step(theta, F)
        # perturbs every theta_i at once; F is evaluated only at
        #   theta+delta and theta-delta -- 2 evaluations per iteration
        #   REGARDLESS of dim(theta), which is why SPSA is used here (Sec. VI B)

# --- optional: carry a state across a potential-energy-surface scan -----------
if scanning a PES, warm-start point k+1 from point k's converged result:
    omega_{k+1} <- E_k  ;  theta_init_{k+1} <- theta_opt_k         (Eq. 32, 33)
    # a continuity heuristic added on top of the method, not part of a
    #   single-geometry FS-VQE run                                 (Sec. VI A)
    # requires continuous MO coefficients too: RHF phase jumps are detected
    #   via P_{k,k+1} = C_k^dagger S_k C_{k+1} and rectified        (Eq. 34)
    at geometries observed to jump between states, add a penalty to F:
        F <- F + eta * |theta_{k-1} - theta_k|, eta = 0.1              (Eq. 35)
        # eta = 0.1 is an empirically chosen constant, applied only to "some
        #   isolated points" in the curve -- not a standing part of the
        #   loop                                                 (Sec. VI A 1)

# --- refine theta after the loop exits ----------------------------------------
sample F at a few points near theta_opt and fit a parabola;
    replace theta_opt with its vertex if F improves               (Sec. VI C 1)
    # rests on UCCSD eigenstates sitting at parabola vertices in the energy
    #   landscape; the paper calls the gain "usually very low" because the
    #   optimizer already locates the vertex well, with "a few tenths of a
    #   percent" gained only in some cases
round any theta_i landing close to 0 or to a fraction of pi;
    keep the rounding only if F improves                         (Sec. VI C 2)
    # runs AFTER the quadratic fit, on its output; "usually only improves the
    #   result very slightly", up to a few hundredths of a percent

take a final measurement at 30000 shots                          (Sec. VI B 2)

return  the energy at theta_opt (nuclear repulsion plus <H> in the optimised
        state) and the prepared state |Psi(theta_opt)>
# the exact form of that final-energy expression is read off Figure 3's
#   flowchart, whose symbols pdftotext does not extract -- inferred from
#   step 1 ("compute nuclear repulsion energy") and Sec. II's <H>, NOT
#   quoted; worth a visual check of the figure
# one run returns ONE state near one omega, not a spectrum -- "all excited
#   states ... may be obtained by modifying the parameter omega over a wide
#   enough range of energies", i.e. scanned by hand              (Sec. III)
# nothing in this loop certifies that the converged E is really nearest
#   omega rather than a local minimum of F reached by a limited ansatz
# no quantum-device run is reported: NISQ hardware is called "too limited"
#   and every result is a simulator result                        (Sec. VII)

Cadi Tazi and Thom count the squared operator directly: forming $(\hat{H}-\omega)^2$ takes $\mathrm{H_2O}$ in STO-3G from $1578$ Pauli strings to $111615$, which qubit-wise-commuting grouping under Jordan-Wigner returns to $20393$ groups, one circuit evaluation each (Table II, section VIII B). That section calls the group count $O(N^6)$ in $N$ spin-orbitals, but the figure is their import of Suchsland et al.'s ungrouped string count: empirical, over molecules to $18$ qubits, no proved power law, and never established for groups. SPSA costs $2$ evaluations per iteration (QASM simulations); for $\mathrm{H_2}$, whose $24$ strings fall into two general-commuting groups, one evaluation is two circuits. Each string needs $O(1/\varepsilon^2)$ shots at fixed confidence and bounded variance, here $1000$ rising to $10000$. UCCSD depth is $O(N^4)$. Nothing bounds the iterations: the loop stops on a $10^{-9}$ gradient threshold (section VIII A). The classical Pauli reduction and grouping underneath is NP-hard and, the authors say, not yet scalable.

• Folded-spectrum VQE · Qiskit

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  Minimizing the squared shifted Hamiltonian targets eigenstates near a chosen energy shift.

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

• Folded-spectrum VQE

  Minimizing the squared shifted Hamiltonian targets eigenstates near a chosen energy shift.