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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.

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

Multistate contracted 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 insideFolded-spectrum variational eigensolver · 3 parts inside — click the line to open it herePenalty-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 inside, openMultistate contracted variational eigensolver · 3 parts inside, openMultistate contracted 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 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 — click the name to read about itFolded spectrumPenalty-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, open — click the name to close itMC-VQEChoose 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 observableHamiltonian 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

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.

Requires

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

Example

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

Cost, as the source states it

Parrish, Hohenstein, McMahon and Martínez count MC-VQE in contracted Hamiltonian elements: optimising the state-averaged entangler "requires only the measurement of NΘN_\Theta diagonal matrix elements HΘΘH_{\Theta\Theta}", and the off-diagonal elements — written NΘ2N_\Theta^2, though only NΘ(NΘ−1)/2N_\Theta(N_\Theta-1)/2 are distinct — are determined afterwards, two interference-state circuits each (Eq. 5), from 1- and 2-body Paulis alone: 2N2N one-body terms plus four per monomer pair (Eq. 8), truncated to NN nearest-neighbour pairs in the demonstration, where NΘ=N+1N_\Theta = N+1 on NN qubits. Nothing bounds the outer loop: the demonstration reports 108 parameters — six per SO(4)\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 μ\mueV and ≪1%\ll 1\% oscillator strength of FCI on the N=18N=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 NN for exciton models "with local Hamiltonian connectivity", without derivation.

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.

  • 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.

In the Atlas

  • Multistate contracted VQE

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

Sources