Quantum equation of motion

What it fills

• Estimate an excited-state energy

  Given a Hamiltonian you can query and a statement of which state above the lowest one is wanted, return an estimate of that state's energy — or of its distance from the ground state, which is the quantity an experiment usually measures. Whether the answer comes back as an absolute energy or as an excitation energy is a property of the method, and the two are not interchangeable.

When it applies

Ollitrault et al. name the gap they are filling — "there is currently a lack of such algorithms for correlated molecular systems that is amenable to near-term, noisy hardware" — and propose "an extension of the well-established classical equation of motion approach to a quantum algorithm for the calculation of molecular excitation energies on noisy quantum computers". This is the one route in this slot whose abstract reports a device run rather than a simulation: they "demonstrate the efficiency of this approach in the calculation of the excitation energies of the LiH molecule on an IBM Quantum computer". The abstract states no cost comparison against the other excited-state routes, and none is supplied here on its behalf.

Requires

These do not move the route along. The method needs each of them alongside its own work, and the cost of getting them is part of what the method costs.

• 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.

  qEOM does not solve this slot, it consumes it: one VQE run supplies the $|0\rangle$ every later expectation value is taken on, and the value returned is $\langle\Psi|\hat{H}|\Psi\rangle$. For the statevector dissociation profiles, Hartree-Fock/STO-3G integrals in the particle-hole framework with effective core potentials for the $1s$ cores of LiH and $\mathrm{H_2O}$, and a full-basis UCCSD trial state over all occupied and all virtual Hartree-Fock orbitals, optimised by L-BFGS-B. The hardware track instead simply freezes the core orbitals, shifting the absolute energies, and runs no optimiser: the single variable rotation $R_z(\theta)$ of a 6-CNOT circuit is swept and its optimum read off a quadratic fit to the mitigated sweep, the one excitation chosen by the largest MP2 coefficient, Eq. (S2), $C^{\mathrm{MP2}}_{ijlk}=(h_{ijkl}-h_{ijlk})/(e_i+e_j-e_k-e_l)$. approximation: LiH's 10 orbitals are cut to a 4-orbital active space and the UCC ansatz restricted to that one excitation; together these recover at least 56% of the correlation energy over the entire dissociation range considered, an error the paper reports as $\leq 7$ mHa assumption: the reduced ground state is taken "good enough to compute accurate excitation energies"

  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.

  What this route reads out is not an energy but the elements of four matrices, all on that same $|0\rangle$. The operators are singles and doubles, $\hat{E}^{(1)}_{\mu_1}=\hat{a}^\dagger_m\hat{a}_i$ and $\hat{E}^{(2)}_{\mu_2}=\hat{a}^\dagger_m\hat{a}^\dagger_n\hat{a}_i\hat{a}_j$, restricted to particle- and spin-conserving excitations, so the states the route returns are the lowest within that subspace, not strictly the lowest. Jordan-Wigner can be used to map the commutators into qubit space, giving $M_{\mu_\alpha\nu_\beta}=\langle 0|[(\hat{E}^{(\alpha)}_{\mu_\alpha})^\dagger,\hat{H},\hat{E}^{(\beta)}_{\nu_\beta}]|0\rangle$ and $V_{\mu_\alpha\nu_\beta}=\langle 0|[(\hat{E}^{(\alpha)}_{\mu_\alpha})^\dagger,\hat{E}^{(\beta)}_{\nu_\beta}]|0\rangle$, with $[\hat{A},\hat{B},\hat{C}]=\tfrac{1}{2}\{\big[ [\hat{A},\hat{B}],\hat{C}\big]+\big[\hat{A},[\hat{B},\hat{C}]\big]\}$ approximation: that symmetrised numerator equals the plain double commutator only "for the exact ground state", and the state measured here is not one. $Q$ and $W$ are the same brackets against $(\hat{E}^{(\beta)}_{\nu_\beta})^\dagger$, negated; they exist because this route keeps the de-excitation operators QSE neglects, which is why the problem is $2n \times 2n$ rather than $n \times n$. The commutator form keeps the operators Hermitian, which the paper says permits a systematic reduction of terms through the Pauli commutation relations. For LiH the 10-qubit Pauli strings are measured on 4 qubits by factorising $\langle\psi_{\mathrm{full}}|\hat{A}\hat{I}|\psi_{\mathrm{full}}\rangle=\langle\psi_{\mathrm{active}}|\hat{A}|\psi_{\mathrm{active}}\rangle\langle\psi_{\mathrm{inert}}|\hat{I}|\psi_{\mathrm{inert}}\rangle$ into active and inert factors: the inert one is $1$ if built from $Z$ and $I$, $0$ if any $X$ or $Y$ appears assumption: the two registers are uncorrelated, the inert qubits all in $|0\rangle$. Every element is then re-measured at stretch factors $c=1,1.25,1.5$ approximation: zero-noise estimates by linear extrapolation, under the assumption of time invariant noise

  approximationassumption

Example

given  electronic-structure Hamiltonian H; an approximate ground state |0>
       prepared on the device by a prior run -- "e.g., a VQE calculation"
       (its ansatz and optimizer are not part of this method)          (Sec. II)
       the exact EOM objects this method approximates:
           O_n^dagger = |n><0|      (excitation, |n> the n-th excited state)
           O_n        = |0><n|      (de-excitation)                    (Sec. II)
       single/double excitation operators over the active-space orbitals:
           E_mu^(1) = a_m^dagger a_i                  (m virtual, i occupied)
           E_mu^(2) = a_m^dagger a_n^dagger a_i a_j                    (Eq. 3)

requires  the excitation-operator basis truncated to alpha = 1, 2 -- singles and
    doubles only. Eq. 3 as written sums over every degree alpha; "in this work"
    the sum is restricted, which is this paper's choice, not a restriction of
    the EOM formalism itself                                           (Sec. II)
    # this truncation is what fixes n below: "the rank of all these matrices
    #   equals the number of possible single and double excitations included
    #   in the active space that defines the operators in Eq. (3)"     (Sec. II)
    # the paper does NOT use the label "qEOM-SD" for this truncation, and
    #   states no vacuum-annihilation / killer condition -- do not read one in

# --- stage 1: pose the excitation ansatz --------------------------------------
O_n^dagger = sum_alpha sum_{mu_alpha} [ X_mu^(alpha)(n) E_mu^(alpha)
                                       - Y_mu^(alpha)(n) (E_mu^(alpha))^dagger ]  (Eq. 3)
    # X, Y are the unknowns solved for below
    # the de-excitation (Y) terms are kept deliberately: the expansion "is
    #   general and explicitly treats the possible de-excitation of all states
    #   involved", which matters because a prepared ground state is in general
    #   a many-determinant wave function                                (Sec. II)

# --- stage 2: the parametric equation for the excitation energy ---------------
[H, O_n^dagger] |0>  =  E_0n  O_n^dagger |0>                            (Eq. 1)
    # E_0n is a gap, E_n - E_0, never an absolute energy                (Sec. II)
E_0n = <0| [O_n, [H, O_n^dagger]] |0>  /  <0| [O_n, O_n^dagger] |0>
     = <0| [O_n, H, O_n^dagger] |0>    /  <0| [O_n, O_n^dagger] |0>     (Eq. 2)
    # the symmetrized double commutator is [A,B,C] = 1/2( [[A,B],C] + [A,[B,C]] )
    # the second equality holds "for the exact ground state" ONLY; the state
    #   actually used here is an approximate prepared state, not an exact
    #   eigenvector of H, so this step is an approximation in practice  (Sec. II)
    # the symmetrized form is kept anyway for a separate reason: the plain
    #   nested commutator "may not be Hermitian", the double commutator is,
    #   which is what guarantees real-valued E_0n                (note [35], p.8)
insert Eq. 3 into Eq. 2 -> a parametric equation in X, Y
apply the variational principle  delta(E_0n) = 0
    in the parameter space spanned by X_mu^(alpha), Y_mu^(alpha)        (Sec. II)

# --- stage 3: the resulting secular equation ----------------------------------
for every pair of excitation indices (mu of type alpha, nu of type beta), with
    a = (E_mu^(alpha))^dagger  and  b = E_nu^(beta):
    M_{mu,nu} =  <0| [a, H, b]          |0>                             (Eq. 4)
    Q_{mu,nu} = -<0| [a, H, b^dagger]   |0>
    V_{mu,nu} =  <0| [a, b]             |0>
    W_{mu,nu} = -<0| [a, b^dagger]      |0>
    # V and W carry no H; they exist only because Eq. 3 keeps the Y terms
    # "the QSE approach introduced in [4] neglects the de-excitation operators
    #   (as in the Tamm-Dancoff approximation), which limits its application to
    #   systems for which this approximation is valid"                  (Sec. II)
    # that is Ollitrault et al.'s characterization of QSE, not verified here
    #   against McClean et al. directly. The paper states NO matrix dimension
    #   for QSE anywhere, main text or supplementary -- only this method's
    #   own problem is 2n x 2n (Eq. 4). Do not infer QSE's size from here

# --- stage 4: measure the four matrices on quantum hardware -------------------
map the four commutator strings above through the Jordan-Wigner transformation
    into the qubit space                                                (Sec. II)
evaluate each mapped string on the SAME prepared ground state |0>       (Sec. II)
    # "the measurement of the expectation values in a quantum computer scales
    #   with the number of terms in the Hamiltonian as O(N^4)"; classically the
    #   scaling of the same operation instead depends on the wave-function
    #   ansatz. This is the paper's claimed source of advantage         (Sec. II)
    # Sec. II gives no total shot count and no per-element error budget; shot
    #   counts appear only in the Supplementary Materials (hardware runs at
    #   N_S in {8192, 4096, 2048, 1024})

# --- stage 5: solve classically -----------------------------------------------
construct and solve the 2n x 2n generalised eigenvalue (secular) equation
    ( M    Q  ) (Xn)          (  V     W ) (Xn)
    ( Q*   M* ) (Yn)  =  E_0n ( -W*   -V* ) (Yn)                        (Eq. 4)
    # this runs on a classical computer, and the paper's justification is the
    #   small rank of the matrices, not an asymptotic argument          (Sec. II)

return  from the 2n eigenvalues, "the first n excitation (and corresponding
        de-excitation) energies" with their (Xn, Yn) coefficients       (Sec. II)
# the paper describes the 2n solutions as excitation/de-excitation PAIRS; it
#   never says the returned ones are the numerically lowest. Selecting "the n
#   lowest" eigenvalues would pick the de-excitation branch instead
# the prepared |0> and the run that produced it are NOT part of this method --
#   qEOM only measures matrix elements against a state handed to it
# solving directly for gaps rather than for two absolute energies is what the
#   paper calls size-intensive, "contrary to QSE, which is not size-intensive"
#                                                                       (Sec. II)
# no error bound on E_0n for an approximate |0> is derived; the paper reports
#   numerics only. Conclusions: simulated excitation energies "within chemical
#   accuracy (errors <= 1.5 mH)" (Sec. V) -- but Sec. III draws that band at
#   "< 0.015 Hartree" and reports H2O excited states "slightly above chemical
#   accuracy", and the paper's own Sec. I definition is 1 kcal/mol = 0.043 eV
#   (~1.6 mHa). The three figures are not mutually consistent in the source;
#   treat ~1.6 mHa as the definition and 1.5 mH as the reported simulation error
# the "<= 7 mHa" LiH figure is NOT an excitation-energy error: it is the
#   ground-state energy error of the reduced-active-space UCC circuit used for
#   the hardware run, quoted from its 56-87% correlation-energy recovery (Sec. IV)

Cost, as the source states it

Ollitrault et al. give a per-element cost and an instance size, never a total. Measuring each element of the qEOM matrices $M$, $Q$, $V$, $W$ scales with the number of Hamiltonian terms as $O(N^4)$ — their $N$ is glossed as electrons or basis functions — while the secular equation, Eq. (4), has dimension $2n \times 2n$, with $n$ counting the particle- and spin-conserving singles and doubles in the active space; its $2n$ eigenvalues are diagonalised classically. No formula counts those matrix elements, and no shot budget is tied to a target error. Reported qubit counts are 4, 10 and 12 for $\mathrm{H}_2$, LiH and $\mathrm{H}_2\mathrm{O}$. For LiH the secular problem is rebuilt over the full 10-qubit frozen-core basis, giving $n = 24$, while the ground state runs in a 4-orbital active space: UCCSD in full STO-3G is over 12000 CNOTs and 92 parameters, the conventional 4-orbital circuit is 48 CNOTs, and the circuit actually run is 6 CNOTs with one MP2-selected excitation — a UCC-inspired ansatz recovering 56% to 87% of the correlation energy, an error of $\leq 7$ mHa.

Implementations

• The statevector qEOM sweep of $\mathrm{H}_2$, LiH and $\mathrm{H}_2\mathrm{O}$ (Sec. III)

  • Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor

  Section III is the paper's own validation step, run before any hardware touches the algorithm: "To validate the performance and the accuracy of the qEOM approach, statevector-type simulations (where the exact unitary matrix representation of the circuit is applied on the state vector, no sampling or hardware noise is included) are performed. The algorithm is tested on three molecules, namely H2, LiH and H2O." The question it settles is not whether qEOM is cheaper than anything but whether the gaps it returns are the right ones when nothing is in the way. The Supplementary Materials then ask the question the hardware run depends on and state its difficulty plainly — "Assessing the error propagation from the ground state wave function calculation to the excited state energies is not straightforward" — and answer it by perturbing the ground state on purpose.

  The ground state is a VQE solution and the qEOM matrix elements are measured on it: "The qEOM operators are measured on a ground state obtained with VQE. The wave function is approximated with the UCCSD Ansatz where the indices for the excitations run over all occupied and all virtual Hartree-Fock orbitals for the given basis set. The variational parameters are optimized with the L-BFGS-B algorithm as implemented in SciPy." Only particle- and spin-conserving excitation operators are admitted, which is why the three states plotted "are not strictly the 3 lowest lying ones but rather the 3 lowest lying within this specific particle- and spin-number subspace". The Hamiltonian is mapped into the particle-hole framework, and effective core potentials replace the $1s$ core electrons of LiH and $\mathrm{H}_2\mathrm{O}$; that reduction is what fixes the register sizes, "resulting in 4, 10 and 12 qubits for H2, LiH and H2O, respectively". Two perturbations are then applied deliberately in the Supplementary Materials, both on $\mathrm{H}_2$. An ad-hoc error $\epsilon$ is added to every parameter defining the ground state wave function and $|\Delta E_i(\epsilon)| = |E_i(0) - E_i(\epsilon)|$ recorded for $i \in \{0, 1, 2, 3\}$; and a statistical error is introduced by "projecting the state and averaging over a given number of shots, $N_S \in \{8192, 4096, 2048, 1024\}$", with 100 computations of the excited-state energies for each choice of $N_S$ and the errors averaged over those 100 realizations.

  $\mathrm{H}_2$, LiH and $\mathrm{H}_2\mathrm{O}$ in the STO-3G basis, each along its dissociation profile. The integrals are classical and their source is named: "We prepare the Hamiltonians by computing the one- and two-electron integrals using a Hartree-Fock/STO-3G calculation performed on Gaussian09". The reference the errors in Fig. 1b are measured against is not an experimental value but the exact eigenenergies of the same Hamiltonian, obtained by diagonalising it.

  The paper names two classical tools for this run: Gaussian09, which supplies the one- and two-electron integrals, and SciPy, whose L-BFGS-B routine optimises the VQE parameters. The qEOM implementation the authors themselves wrote is recorded as a separate entry on this method.

  Fig. 1b draws the ground state and the first three excited states of all three molecules across the dissociation profile against exact diagonalisation, with the energy errors in the panels above. There is no device and no sampling here: the instrument is a statevector simulation, in the authors' own definition one "where the exact unitary matrix representation of the circuit is applied on the state vector, no sampling or hardware noise is included". Two molecules come out clean and one does not — "In the case of H2 and LiH the accuracy of the ground state is excellent over the entire dissociation curve (see upper panels in Fig. 1), allowing us to compute accurate excited state energies within chemical accuracy (with errors < 0.015 Hartree, shaded area) for all geometries. For H2O, the VQE results are less accurate, leading to excited states slightly above chemical accuracy." The Conclusions then state the three molecules together as "excitation energies within chemical accuracy (errors $\leq 1.5$ mH)", which does not sit with Sec. III's own verdict on $\mathrm{H}_2\mathrm{O}$; and Sec. III's parenthetical bound, "< 0.015 Hartree", is ten times the definition the paper gives in Sec. I, "chemical accuracy (errors less than 1 kcal/mol = 0.043 eV)". All three figures are quoted as the paper writes them rather than reconciled here. The perturbation studies report the direction that the hardware run then relies on, "that the qEOM energy gaps are less sensitive to an error in the ground state than the ground state energy itself", and under shot noise the excited-state error follows the error in the matrix norms — for $\mathrm{H}_2$ the $W$ matrix is null, the $Q$ coefficients stay "relatively small ($< 10$) across the whole range of internuclear distances" so $Q$ is weakly affected, and $V$ is the matrix mostly affected. One outright failure is stated rather than smoothed over: at $\epsilon = 10^{-1}$ and bond lengths larger than 1.4 Å "it was not possible to obtain energies for the first excited state", because there the ground and first excited state become almost degenerate and the system to solve becomes ill-conditioned.

• The LiH excitation-energy experiment on IBM Q Poughkeepsie (Sec. IV)

  • Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor

  Section IV is the device run, and its problem statement is a circuit-size wall: "The UCCSD circuit for the optimization of the LiH ground state using a STO-3G basis set comprises over 12000 CNOT gates and 92 variational parameters. Given the limitations of state-of-the-art quantum hardware, we reduce the active space from 10 to 4 orbitals." Everything else in the section follows from having to fit LiH onto four qubits of a 20-qubit superconducting processor and still read a spectrum off it. It is the run the abstract points at when it claims the excitation energies of LiH were computed "on an IBM Quantum computer".

  Two reductions, then a measurement trick, then mitigation. The active space is cut from 10 to 4 orbitals, the orbitals "selected according to their contribution to the CI expansion", and the UCC ansatz is restricted to a single excitation — one variational parameter — chosen by the largest MP2 coefficient of Eq. (S2), $C^{\mathrm{MP2}}_{ijlk} = (h_{ijkl} - h_{ijlk})/(e_i + e_j - e_k - e_l)$. The eight entangling blocks a double excitation normally needs are replaced "by a single one with variable pre/post rotations (in $U$) where the angles are optimized in simulation to best approximate the exact UCCSD results", and the Supplementary says which system that simulation ran on: the angles "are optimized on a small system (e.g., H2 is used to optimize the pre/post rotation angles of the double excitation blocks)", so the angles the LiH circuit executes were fitted on $\mathrm{H}_2$ and carried over. That takes the LiH circuit "from 48 to 6 CNOTs without loss of accuracy"; what is executed is "6 CNOTs, 8 fixed single qubit rotations and a single variable qubit rotation $R_z(\theta)$". The object measured is still the full 10-qubit LiH Hamiltonian: the 6-qubit inert register stays in $|0\rangle$, so Eq. (S3), $\langle\psi_{\mathrm{full}}|\hat{A}\hat{I}|\psi_{\mathrm{full}}\rangle = \langle\psi_{\mathrm{active}}|\hat{A}|\psi_{\mathrm{active}}\rangle\langle\psi_{\mathrm{inert}}|\hat{I}|\psi_{\mathrm{inert}}\rangle$, lets any Pauli string carrying an $X$ or $Y$ on the inert register be set to zero and the rest be read off the active register. Noise is then amplified rather than removed, under a scheme the paper takes from earlier work — "we use the error mitigation scheme previously implemented in [13]", reference 13 being Kandala et al., "Error mitigation extends the computational reach of a noisy quantum processor" — in which "expectation values of interest are re-measured under amplified noise strengths in order to then extrapolate to the zero-noise limit". Here that is achieved by stretching the gates in time at "stretch factors of $c = 1, 1.25, 1.5$" with linear extrapolation, a four-pulse echo sequence building the ZX90 primitive of the CNOT so the stretching does not introduce coherent errors, and every measured expectation value corrected for assignment infidelity from a readout calibration of all the basis states. At each internuclear distance $\theta$ is swept, the optimal $\theta$ taken from a quadratic fit to the mitigated sweep, and then "the error mitigation protocol is then extended to each matrix element of the qEOM and the resulting “mitigated” secular equation is then solved classically, leading to the excitation energies $E_{0n}$ with $n = 1, \ldots, 24$".

  LiH in the STO-3G basis. The paper states the orbital provenance once, and it states it in the Sec. III paragraph that sets up the Qiskit runs preparing this experiment rather than in Sec. IV itself: "Using Qiskit, the molecular orbitals are computed with Hartree-Fock/STO-3G on the PySCF classical code. In this case the core orbitals are simply frozen (note that the absolute energies are shifted in comparison to using ECPs)." What is unambiguously the experiment's own input is printed rather than described: Table II of the Supplementary Materials lists every Pauli operator and coefficient of the 4-qubit LiH Hamiltonian at the equilibrium bond length of 1.6 Å, "not taking into account for the energy shifts due to the frozen core orbitals and the Coulomb repulsion between nuclei". The geometries at which experimental errors are reported, panel by panel in Fig. S7, are 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0 Å.

  The authors "implement the qEOM algorithm in the Qiskit software library" in preparation for this experiment, and the paper attributes the noisy simulations to that implementation; it does not state how the device circuits themselves were submitted. The pre/post rotation angles the reduced circuit needs were optimised in simulation beforehand and are published as numbers in the caption of Fig. 2, $\Phi_0 = \{\pi/2, 0.0, 0.930\}$, $\Phi_1 = \{-\pi/2, \pi, -1.207\}$, $\Phi_2 = \{-\pi/2, -\pi, 1.310\}$ and $\Phi_3 = \{-\pi/2, 0.0, 1.877\}$. The released solver itself is recorded as a separate implementation of this method.

  The device is named and the slice of it used is small: "The experiments presented in this work used 4 superconducting qubits (Q0, Q1 ,Q2 and Q5)" of "the 20 qubit processor IBM Q Poughkeepsie", a 20-qubit superconducting processor whose qubit frequencies "are in the range 4.8-5 GHz, with relaxation and coherence times of $T_1$ and $T_{2,\mathrm{echo}} \sim 40 - 110\mu s$". The gates are all-microwave; "the shortest single qubit gates used for the experiments are of duration 103 ns, and the shortest gate times for $\mathrm{CNOT}_{21}$, $\mathrm{CNOT}_{10}$ and $\mathrm{CNOT}_{05}$ are 1278 ns, 1210 ns and 1448 ns respectively". Table I of the main text gives the randomized-benchmarking characterisation per stretch factor — $\mathrm{CNOT}_{0-5}$ reads $0.038 \pm 0.001$, $0.043 \pm 0.002$ and $0.048 \pm 0.002$ at $c = 1$, $1.25$ and $1.5$ — and "the average readout assignment errors for the four qubits were $r \sim 0.05$". What comes out is a spectrum, not a single number: Fig. 3c draws the "dissociation profile of the five lowest-lying electronic transition energies of the LiH molecule" with "the markers show the mitigated experimental results" against the statevector qEOM of the same reduced circuit, error bars "computed using 50 numerical experiments obtained by bootstrapping of the experimental data points" spanning the 1st to 3rd quantile. At the equilibrium bond length of 1.6 Å, which is the geometry Fig. 3b measures, mitigation buys "a gain in precision of approximately one order of magnitude in both the ground state and excitation energies"; the paper does not restate that factor for the other five geometries. The claim it leans on is that the gaps were already the robust quantity without mitigation: "we experimentally observe that the unmitigated results are more accurate for the excited states than the ground state (about 1e-2 Ha for the lowest energy states against an error superior to $\sim$1e-1 Ha for the ground state), from the runs at different stretch/noise amplification factors." The ceiling on the run is the reduced circuit rather than the device, and it is a simulated ceiling: the circuit "can recover at least 56% of the correlation energy (and up to 87%, see the details in the Supplementary Materials) over the entire dissociation range considered, which corresponds to an energy error $\leq 7$ mHa" — figures whose source is Table III, whose own caption puts them in statevector-type simulation with the COBYLA optimizer against diagonalization of the full HF/STO-3G Hamiltonian, over 1.0 to 2.5 Å. The experimental panels of Fig. S7 stop at 2.0 Å.

• The qEOM solver released in Qiskit Aqua and Qiskit Nature

  • Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor

  The authors state their own reason for writing the code, and it is the device: "In preparation of computing molecular excited states on a quantum hardware, we implement the qEOM algorithm in the Qiskit software library. This enables us to perform realistic noisy simulations and model the performance of the algorithm on the IBM Q Poughkeepsie device." The author contributions name who wrote it and where — "C.C., A.M., M.P. and P.J.O. implemented the algorithm in Qiskit Aqua". The artefact outlived the paper: a later port of the solver is still shipped under Qiskit Nature, and its class docstring still points back at this arXiv entry. What continues is the artefact rather than the 2019 code — the Qiskit Nature file is headed "(C) Copyright IBM 2020, 2026" over imports that postdate the paper, and no source names that port's authors.

  The commutators the secular equation needs are computed numerically on a ground state that a separate solver supplies, then Eq. (4) is diagonalised classically. The routine that fills the matrices says as much in its own docstring — "Numerically computes the commutator / double commutator between operators" — and what it acts on are `SparsePauliOp` objects. The released class keeps the paper's own notation on the surface: its result object exposes `m_matrix`, `v_matrix`, `q_matrix` and `w_matrix` together with their standard deviations, so what a caller receives back is the $M$, $Q$, $V$, $W$ of Eq. (4). Its documentation states the same dependence the paper's error-propagation study measured: "The precision of the ``GroundStateSolver.solve`` method for the ground state approximate directly affects the precision of the qEOM algorithm for the same problem." For the noisy runs reported in the paper the same circuit reduction as the hardware experiment is used, "leading to circuits parametrized with a single angle $\theta$ for both molecules", and $\theta$ is switched "within the range $[-\pi; \pi]$".

  $\mathrm{H}_2$ and LiH at their equilibrium distances, 0.75 Å and 1.6 Å. "Using Qiskit, the molecular orbitals of a H2 and LiH molecules are computed with Hartree-Fock/STO-3G and the PySCF classical code. The core orbitals are frozen."

  Python, Apache-2.0. The implementation the paper names is Qiskit Aqua, whose now-archived repository at https://github.com/qiskit-community/qiskit-aqua carries `qiskit/chemistry/algorithms/excited_states_solvers/qeom.py`, defining `class QEOM(ExcitedStatesSolver)` under an Apache-2.0 header. The same solver is maintained today in Qiskit Nature, https://github.com/qiskit-community/qiskit-nature, as `qiskit_nature/second_q/algorithms/excited_states_solvers/qeom.py` with `class QEOM(ExcitedStatesSolver)` and `class QEOMResult(EigensolverResult)`, alongside `qeom_electronic_ops_builder.py` and `qeom_vibrational_ops_builder.py`, which build what their module docstrings call the fermionic and vibrational hopping operators and what `build_electronic_ops` calls "the product of raising and lowering operators (basic excitation operators)"; the class docstring reads "For more details, please refer to https://arxiv.org/abs/1910.12890." It installs as the `qiskit-nature` package on PyPI, which lists version 0.8.0.

  The simulator is named and so is the noise it carries: "The simulations are done using the qasm simulator with the Poughkeepsie noise model and 100K shots." Fig. S3 scans $\theta$ for $\mathrm{H}_2$ and LiH at equilibrium and puts the noisy points against the noiseless curves. "Around the best $\theta$ value, the noise has the effect to shift up the computed ground state energy. On the other hand the qEOM is robust to the noise, leading to an error in the excitation energies (energy gaps) of about one order of magnitude smaller than for the ground state (see plots c and f of Fig. S3)." That scan is also what chose the window the hardware run then used: the region to inspect experimentally "was selected to be $\theta \in [-1; 1]$", on the stated criterion that in it "the energy change with $\theta$ should be higher than the fluctuations coming from the hardware noise but close enough to the bottom of the well such that the data can be fitted to an harmonic curve".

What it needs

Every step this method names is listed under Requires above. It walks its own span in one hop and calls out to the rest — that is a fact about the recorded route, not a claim that the span is simple.

Other ways to fill the same slot

Different approaches

• Variational quantum deflation

  Find the ground state first, then run the same variational search again with a term that punishes overlap with every state already found. Each state is reached by pushing the search off the ones below it, so they have to be found in order and each one costs another pass through the whole loop.

• Subspace-search variational eigensolver

  Send several mutually orthogonal input states through one parameterised circuit and minimise their energies together. A unitary keeps orthogonal inputs orthogonal, so the whole low-energy subspace comes out of a single optimisation — no earlier state to deflate, and no ancilla to test overlaps with.

• Quantum subspace expansion

  Stop optimising and start measuring: take the state the ground-state run already produced, measure matrix elements in a small space of operators applied to it, and let a classical solver diagonalise the little matrix that results. The extra accuracy is bought with classical work and more measurements, not with more coherence.

• Folded-spectrum variational eigensolver

  Point the same search somewhere other than the bottom. Minimising the variance around a chosen energy makes every eigenstate a minimum and the one nearest that energy the reachable one — so a state can be found without knowing its index, and the bill arrives as a squared Hamiltonian with far more terms to measure.

• Penalty-constrained variational eigensolver

  Add a term to the objective that punishes the trial state for leaving the symmetry sector you asked for, and the ordinary ground-state search returns that sector's lowest state — an excited state of the whole Hamiltonian whenever the sector is not the one the ground state lives in. Which penalty is used matters: one common form is proved not to work.

• Multistate contracted variational eigensolver

  Optimise one circuit for several states at once and read off the transitions between them — both the energy of each transition and the oscillator strength that says how strongly light drives it. The answer is a spectrum with intensities, which is what an absorption experiment actually produces.

In the Atlas

• Quantum equation-of-motion VQE

  Commutator matrix elements over a VQE reference produce excitation energies through an equation-of-motion problem.

Sources

• Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor↗Pauline J Ollitrault, Abhinav Kandala, Chun-Fu Chen, Panagiotis Kl Barkoutsos, Antonio Mezzacapo, Marco Pistoia, Sarah Sheldon, Stefan Woerner, Jay Gambetta, Ivano Tavernelli · 2019theory yes · simulation ? · hardware yes