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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.

Takes

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.

Returns

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.

Same contract as the slot it fills.

This one, drawn

Folded-spectrum variational eigensolverVariational quantum deflation · 3 parts insideVariational quantum deflation · 3 parts inside — click the line to open it hereSubspace-search variational eigensolver · 3 parts insideSubspace-search variational eigensolver · 3 parts inside — click the line to open it hereQuantum subspace expansionQuantum equation of motionFolded-spectrum variational eigensolver · 3 parts inside, openFolded-spectrum variational eigensolver · 3 parts inside, openFolded-spectrum variational eigensolver · 3 parts inside, open — click the line to close itChoose a parameterised trial state · 13 ways throughChoose a parameterised trial state · 13 ways through — click the line to open it hereMinimise the energy varianceMeasure commuting terms togetherPenalty-constrained variational eigensolver · 3 parts inside, openPenalty-constrained variational eigensolver · 3 parts inside, openPenalty-constrained variational eigensolver · 3 parts inside, open — click the line to close itChoose a parameterised trial state · 13 ways throughChoose a parameterised trial state · 13 ways through — click the line to open it hereMinimise the objective over the parameters · 6 ways throughMinimise the objective over the parameters · 6 ways through — click the line to open it hereEstimate an observable · 4 ways throughEstimate an observable · 4 ways through — click the line to open it hereMultistate contracted variational eigensolver · 3 parts insideMultistate contracted variational eigensolver · 3 parts inside — click the line to open it hereVariational quantum deflation · 3 parts inside — click the name to read about itVQDSubspace-search variational eigensolver · 3 parts inside — click the name to read about itSSVQEQuantum subspace expansion — click the name to read about itSubspace expansionQuantum equation of motion — click the name to read about itqEOMFolded-spectrum variational eigensolver · 3 parts inside, open — click the name to close itFolded spectrumChoose a parameterised trial state · 13 ways through — click the name to read about itChoose an ansatzMinimise the energy variance — click the name to read about itVariance objectiveMeasure commuting terms together — click the name to read about itGrouped measurementPenalty-constrained variational eigensolver · 3 parts inside, open — click the name to close itPenalty termsChoose a parameterised trial state · 13 ways through — click the name to read about itChoose an ansatzMinimise the objective over the parameters · 6 ways through — click the name to read about itMinimise the objectiveEstimate an observable · 4 ways through — click the name to read about itEstimate an observableMultistate contracted variational eigensolver · 3 parts inside — click the name to read about itMC-VQEHamiltonian whose excited state is wanted — you start hereNumber with an error bar — you finish hereParameterised circuit family — what one part hands to the nextState you can prepare — what one part hands to the nextParameterised circuit family — what one part hands to the nextState you can prepare — what one part hands to the next

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From Hamiltonian whose excited state is wanted to Number with an error bar

A circle is an object you are holding. This method is drawn heavier, opened into its own steps; the other lines between the same two ends are the alternatives recorded for the same slot. Circles are named on hover, and each one is a link.

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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

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.

Requires

Every step this method names moves its route along, so there is nothing it needs alongside them.

Example

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)

Cost, as the source states it

Cadi Tazi and Thom count the squared operator directly: forming (H^−ω)2(\hat{H}-\omega)^2 takes H2O\mathrm{H_2O} in STO-3G from 15781578 Pauli strings to 111615111615, which qubit-wise-commuting grouping under Jordan-Wigner returns to 2039320393 groups, one circuit evaluation each (Table II, section VIII B). That section calls the group count O(N6)O(N^6) in NN spin-orbitals, but the figure is their import of Suchsland et al.'s ungrouped string count: empirical, over molecules to 1818 qubits, no proved power law, and never established for groups. SPSA costs 22 evaluations per iteration (QASM simulations); for H2\mathrm{H_2}, whose 2424 strings fall into two general-commuting groups, one evaluation is two circuits. Each string needs O(1/ε2)O(1/\varepsilon^2) shots at fixed confidence and bounded variance, here 10001000 rising to 1000010000. UCCSD depth is O(N4)O(N^4). Nothing bounds the iterations: the loop stops on a 10−910^{-9} gradient threshold (section VIII A). The classical Pauli reduction and grouping underneath is NP-hard and, the authors say, not yet scalable.

Implementations

Nobody has written one up yet. That is a gap in this record, not a statement that the method has never been run — the paper register already records, per paper, which sources report numerics or a hardware run.

What it needs

  1. Choose a parameterised trial state 13 methods

    Fix the gate structure of a circuit family and leave its rotation angles open. What comes back is not a circuit but the set of states the later optimisation is allowed to search — which is why this is a slot of its own and not a paragraph in one method's write-up.

  2. Minimise the objective over the parameters 6 methods

    Search the parameters of a fixed circuit family for the ones that minimise a chosen objective, given that every evaluation of that objective is a noisy estimate someone paid shots for. The output is a routine that prepares one particular state — the family collapsed to a member.

  3. Estimate an observable 4 methods

    Given the ability to prepare ∣ψ⟩|ψ⟩ and a description of an observable OO, return a classical scalar within ε\varepsilon of ⟨O⟩⟨O⟩ at confidence 1−δ1−δ. The state is never returned; only the number is.

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.

  • Quantum equation of motion

    Ask for the gaps directly rather than for two energies to subtract. Measure the matrix elements the classical equation-of-motion formalism needs between excitation operators on the ground state, and solve its generalised eigenvalue problem classically; what comes back is an excitation energy, which is the quantity a spectrum is made of.

  • 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

  • Folded-spectrum VQE

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

Sources