McClean et al. state the shape of the guarantee rather than a bound: variational approaches "fit in a more general hierarchy of measurement and classical computation that allows one to obtain increasingly accurate solutions with additional classical resources", and they "demonstrate numerically on a sample electronic system that this method both allows for the accurate determination of excited electronic states as well as reduces the impact of decoherence, without using any additional quantum coherence time or formal error correction codes". The decoherence claim is separate and is argued from an "exactly solvable channel model of variational state preparation". What the abstract does NOT give is the mechanism at the level a reader would need to implement it — which operators span the expansion, and which matrix elements have to be measured — and neither is invented here; both sit in the body of the paper, and the honest summary of the abstract is the trade it names, not a construction it does not state.

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 → Number with an error barreuse the eigenvectors to weigh any operator

   Once the reference state $|\Psi\rangle$ is prepared, QSE builds a linear subspace spanned by particle-hole excitations $a_i^\dagger a_j|\Psi\rangle$ approximation: truncated to this first order (k=1, linear response) of a hierarchy that would reach exact diagonalization only at k=Ne, exponential cost and diagonalizes it as the generalized eigenvalue problem $H_{LR}C=S_{LR}CE$, whose overlap $S^{ij}_{kl}=\delta_{ik}\,{}^1D^{j}_{l}-2\,{}^2D^{jk}_{li}$ is fixed by the reference state's 1- and 2-RDM alone, while its Hamiltonian entries $H^{ij}_{kl}=\langle\Psi|(a_i^\dagger a_j)^\dagger H a_k^\dagger a_l|\Psi\rangle$ reach the 3- and 4-RDM, assumption: which McClean et al. take to cost only a polynomial number of additional measurements on the same reference state. Both are estimated by measurement, giving eigenvectors $C$ and energies $E$. The same construction, $O^{ij}_{kl}=\langle\Psi|(a_i^\dagger a_j)^\dagger O a_k^\dagger a_l|\Psi\rangle$, holds for any operator $O$ in place of $H$, so the eigenvectors already obtained can weigh that operator's matrix to read off its value for the ground or an excited state without a new optimization.

   approximationassumption

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.

• Estimate a Hamiltonian's ground-state energy

  Given a Hamiltonian you can query and some way of preparing trial states, return an estimate of its lowest eigenvalue to a stated additive error. The state achieving that energy may or may not come back with the number; almost every application wants the number.

  The route enters this slot with an ordinary VQE loop. A parameterised $|\Psi(\vec{\theta})\rangle$ is prepared and its parameters driven to the Rayleigh-Ritz minimum $\min_{\vec{\theta}} \langle H \rangle(\vec{\theta}) = \langle \Psi(\vec{\theta})|H|\Psi(\vec{\theta})\rangle / \langle \Psi(\vec{\theta})|\Psi(\vec{\theta})\rangle$, the energy read out by repeated state preparation and partial tomography as the contraction $\langle H \rangle = \sum_{ik} h_{ik}\,{}^1D^i_k + \sum_{ijkl} h_{ijkl}\,{}^2D^{ij}_{lk}$. McClean et al. model the preparation as a channel rather than a unitary: minimising $\mathrm{Tr}[(\sum_i K_i |\Psi\rangle\langle\Psi| K_i^\dagger) H]$ is equivalent to solving $H'|\Psi\rangle = E|\Psi\rangle$ on the transformed $H' = \sum_i K_i^\dagger H K_i$ for the lowest eigenvalue and eigenvector pair. assumption: the model prepares an arbitrary pure state, so it fixes the optimal possible performance separately from ansatz choice or experimental protocol Symmetry can be imposed by a penalty $H \to H + \sum_i \lambda_i (O_i - o_i)^2$ approximation: exact only as $\lambda_i \to \infty$, and only if the parameterisation can satisfy the symmetry at all; the authors call it prohibitively expensive and warn it may not generalise to excited states of a different symmetry. More than the energy leaves this slot: the 1- and 2-RDM already measured are the reference the next hop expands about, which then buys the 3- and 4-RDM with a polynomial number of additional measurements.

  approximationassumption

• Estimate an observable

  Given the ability to prepare $|ψ⟩$ and a description of an observable $O$, return a classical scalar within $\varepsilon$ of $⟨O⟩$ at confidence $1−δ$. The state is never returned; only the number is.

  Here the route stops optimising. It expands about $\rho$, the state at the exit of the channel, in the vectors $a_i^\dagger a_j |\Psi\rangle$. Partial tomography supplies the elements within that expansion, $O^{ij}_{kl} = \mathrm{Tr}[(a_i^\dagger a_j)^\dagger O\, a_k^\dagger a_l\, \rho]$ with metric $S^{ij}_{kl} = \delta_{ik}\,{}^1D^j_l - 2\,{}^2D^{jk}_{li}$, and those coupling it to the reference, $O^{ij}_g = \mathrm{Tr}[(a_i^\dagger a_j)^\dagger O \rho]$ with $S^{ij}_g = {}^1D^j_i$; that $g$ row and column border the matrices, so the reference sits inside the diagonalised space. A one-body operator reaches at most the 3-RDM, a two-body $V = \sum_{pqrs} a_p^\dagger a_q^\dagger a_r a_s$ the 4-RDM through a $-24\,{}^4D^{jkpq}_{srli}$ term. The classical step solves $H_{\mathrm{LR}} C = S_{\mathrm{LR}} C E$ for the ground and excited states; symmetry is imposed here with no $\lambda_i \to \infty$ limit, by building $O_{\mathrm{LR}}$ in the same subspace and projecting onto its eigenvectors of the wanted eigenvalue. approximation: truncating the hierarchy $\mathcal{B}^k_f$ at $k=1$ spans only the linear-response space, justified as the dominant contribution in a linear response theory of local time-dependent perturbations approximation: the commutator form $H^{ij}_{kl} = \langle \Psi|(a_i^\dagger a_j)^\dagger [H, a_k^\dagger a_l]|\Psi\rangle + E_g \langle \Psi|(a_i^\dagger a_j)^\dagger a_k^\dagger a_l|\Psi\rangle$ removes the 4-RDM at $O(M^6)$ measured terms and is exact only when $|\Psi\rangle$ is exact; ZC further sets the 3-particle cumulant ${}^3\Delta = 0$ approximation: ZA instead starts from the un-reduced 4-RDM and reconstructs it from the 2-RDM alone, setting ${}^4\Delta = {}^3\Delta = 0$

  approximation

given  a normalized reference state |Psi> already produced by a prior VQE run
       that minimized <H>(theta) over the ansatz parameters (Eq. 18) -- this
       method does not run that optimization, it consumes its output
       the reference's 1- and 2-RDM, 1D^i_k and 2D^ij_kl (Eq. 3-4), already
       measured as part of that VQE run
       M spin-orbitals (JW-encoded qubits), so excitation indices i,j range
       over [1, M]
       optionally, an operator O (a symmetry operator, or any observable
       whose value is wanted in the excited states) expressed as a sum of
       one- and two-body fermion operators, as H itself is (Eq. 10)

requires  |Psi> is the single state the subspace is built ABOUT, not a family
    of trial states -- the paper demonstrates the k=1 truncation's accuracy
    from an exact ground-state reference (Fig. 4) and separately from an
    approximate, channel-degraded one (Fig. 5, amplitude/phase damping); it
    states no general bound on truncation error for an arbitrary reference

# --- build the k=1 excitation subspace -------------------------------------
Bf1 = { a_i^dagger a_j |Psi>  :  i, j in [1, M] }                (Eq. 6, Eq. 8 at k=1)
    # justified only as the dominant contribution in a linear-response theory
    #   of local time-dependent perturbations to the system -- that result is
    #   cited to a separate reference, not derived in this paper       (ref [28])
    # Bf1 is level k=1 of a hierarchy Bf^k; k=Ne would span the whole
    #   Ne-fermion space and make the classical step exact diagonalization,
    #   but its cost then scales exponentially in system size      (text after Eq. 8)

# --- measure the subspace's Hamiltonian and overlap by partial tomography --
for each (i,j), (k,l) in [1,M] x [1,M]:
    H^ij_kl = <Psi| (a_i^dagger a_j)^dagger H a_k^dagger a_l |Psi>          (Eq. 45, O = H)
    S^ij_kl = delta_ik * 1D^j_l  -  2 * 2D^jk_li                            (Eq. 47)
    # S is the overlap/metric of the (non-orthogonal) excitation vectors --
    #   it is NOT the identity, which is exactly why the next step is a
    #   GENERALIZED eigenvalue problem, not a plain diagonalization
    # S alone is fixed by the 1- and 2-RDM (Eq. 47); H is NOT -- a one-body
    #   term of H reaches the 3-RDM, a two-body term the 4-RDM (matrix
    #   elements given explicitly, Eq. 48-51), and determining these costs a
    #   polynomial number of measurements beyond what the ground-state VQE
    #   run already took                                        (text after Eq. 5)

# --- solve the subspace problem classically ---------------------------------
solve  H_LR C = S_LR C E   for eigenpairs (C, E)                    (Eq. 7 / Eq. 21)
    # E's lowest eigenvalue approximates the ground state within Bf1; its
    #   higher eigenvalues are the excited-state estimates this method is for

# --- optional: enforce an exact symmetry in the subspace --------------------
if a symmetry operator O with target eigenvalue o is supplied:
    build  O^ij_kl  the same way as H^ij_kl above
    project onto the eigenvectors of O_LR carrying eigenvalue o
    diagonalize H_LR restricted to that projected subspace          (Supp. Sec. C)
    # exact for any finite subspace here -- unlike the penalty
    #   H -> H + sum_i lambda_i (O_i - o_i)^2 used to impose a symmetry on
    #   the reference state itself, which the paper notes is only exact as
    #   lambda_i -> infinity and "may not generalize well to excited states
    #   if they are of a different symmetry than the ground state"   (main text, Eq. 20)
    # the paper gives this projection only in prose, not as an explicit
    #   projector formula                                            (Supp. Sec. C)

# --- optional variants that avoid measuring the 4-RDM -----------------------
# the fermionic QSE above is exact within Bf1 but, in the paper's own words,
#   "can still be prohibitively expensive in the number of measurements
#   required"; both variants below are approximations aimed at that   (Supp. Sec. I)
# ZC: H^ij_kl ~= <Psi|(a_i^dagger a_j)^dagger [H, a_k^dagger a_l]|Psi>
#                + Eg <Psi|(a_i^dagger a_j)^dagger a_k^dagger a_l|Psi>        (Eq. 59)
#     exact only when |Psi> is the exact ground state (Eg its eigenvalue);
#     the commutator drops the expression to at most the 3-RDM, O(M^6)
#     terms; setting the irreducible 3-particle cumulant 3Delta = 0 removes
#     even that, needing nothing beyond the ORIGINAL ground-state 2-RDM
# ZA: skips the commutator and reconstructs the 4-RDM from the 2-RDM alone,
#     assuming 4Delta = 3Delta = 0
# the two differ in ACCURACY, not in measurement cost: the paper states
#   "both methods require only the original measurements used for the ground
#   state", with ZC at "extremely high accuracy" while ZA is "qualitatively
#   correct in some cases but produces sub-variational solutions in others";
#   the extra structure of ZC is what buys the accuracy         (Fig. 7 caption)
# both accuracy comparisons are a single numerical demonstration on H2, not
#   a bound                                                            (Fig. 7)

return  the eigenpairs (C, E) -- E holds ground- and excited-state energy
    estimates within Bf1, C the coefficients of each state in the
    a_i^dagger a_j |Psi> basis
# the identical C, with O^ij_kl computed in place of H^ij_kl, reads off <O>
#   for any of these states with no new optimization -- the mechanism this
#   method is reused for whenever the wanted quantity is not the energy

McClean, Schwartz, Carter and de Jong buy excited states with measurements, not coherence. At the linear-response level $k=1$ of the hierarchy of Eq. (8), whose $k$ index pairs each range over $[1,M]$ spin-orbitals — the paper writes the index ranges, not the counts — that is $M^2$ vectors $a_i^\dagger a_j |\Psi\rangle$, $M^4$ labelled matrix elements $O^{ij}_{kl}$, and one generalized eigenvalue problem $H_{LR}C = S_{LR}CE$ (Eq. 7) solved classically. The 1- and 2-RDM come from the ground-state run; for the extra 3- and 4-RDM the authors claim only "a polynomial number of additional measurements", never an exponent. Their one explicit count is the commutator form of Eq. (59), which "depends at most on the 3-RDM requiring only $O(M^6)$ terms to be measured", and is exact only when $|\Psi\rangle$ is exact; the ZC and ZA cumulant approximations need nothing beyond the ground-state 2-RDM. No shot count or precision bound is given. Truncated levels the authors call "efficient"; at $k=N_e$ the classical diagonalization "scales exponentially in the size of the system".

• Quantum subspace expansion · Qiskit

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  Measured response operators around a VQE state define a generalized eigenproblem for excitations and mitigation.

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• Quantum subspace expansion

  Measured response operators around a VQE state define a generalized eigenproblem for excitations and mitigation.