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MethodLayer 0

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.

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

Penalty-constrained 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 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 — 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 — 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 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

Kuroiwa and Nakagawa analyse the two penalty forms already in use and separate them: one "works properly in that eigenstates obtained by the VQE with the penalty term reside in the desired symmetry sector", and they "further give a convenient formula to determine the magnitude of the penalty term, which may lead to the faster convergence of the VQE". The other is not a weaker option but a wrong one — they "prove that the other type of penalty terms does not work for obtaining the target state with the desired symmetry in a rigorous sense and even gives completely wrong results in some cases". Two conditions follow. The target symmetry sector has to be known in advance, since it is what the penalty is written against; and this route reaches a symmetry-resolved spectrum rather than the k-th state as such, so it answers "the lowest state with these quantum numbers" and not "the third state overall". Their own framing is that the work "lay[s] the theoretical foundation for the use of the VQE with the penalty terms", validated by numerical simulation.

Requires

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

Example

given  Hamiltonian H on n qubits and a conserved-quantity observable C with
       [H, C] = 0 (particle number N, S^2, S_z, ...)                     (Sec. II C)
       target symmetry sector: an eigenvalue c of C, fixed before the run --
           c determines the sector, it is an input here, not an output   (Sec. II C)
       ordinary ansatz  |psi(theta)> = V(theta)|ref>,
           V(theta) = V_l(theta_l) ... V_1(theta_1)                      (Eq. 1-2)
           # |ref> is the paper's |psi_0> of Eq. 1, "some reference state" -- one
           #   FIXED state (|00001111> in the H4 runs, Sec. IV) reused unchanged by
           #   every round. Renamed here because Sec. II B writes |psi_0> for a
           #   different object: the converged ground state found earlier.
           # no symmetry structure is built into V(theta); the guarantees below
           #   are then derived assuming this ansatz can reach an ARBITRARY
           #   n-qubit state, |psi(theta)> = sum_i a_i |i>, because the paper's
           #   purpose is to study the two cost functions themselves, not a
           #   specific circuit's reach                                  (Sec. III, Eq. 8)
       for an excited state: eigenstates |psi_0>, ..., |psi_{k-1}> already
           obtained by earlier rounds of this same procedure             (Sec. II B)

requires  the target sector known in advance as the scalar c -- this route
    resolves a symmetry sector, so it answers "the lowest state with C = c",
    not "the k-th state overall" without one; denote that state's energy
    E_{i0}, meaning C_i != c for every i < i0 under E_0 <= ... <= E_{2^n-1}  (Sec. III)

# --- cost function: penalize deviation from the sector, don't build it in ---------
if excited state k:
    L(theta) = LVQD(theta) = <psi(theta)|H|psi(theta)>
                   + sum_{i=0}^{k-1} beta_i |<psi_i|psi(theta)>|^2        (Eq. 4)
        # equivalently plain VQE against H' = H + sum_i beta_i |psi_i><psi_i|,
        #   beta_i > 0 chosen only to enforce orthogonality against the
        #   already-found lower states                                   (Sec. II B)
else:
    L(theta) = LVQE(theta) = <psi(theta)|H|psi(theta)>                   (Eq. 3)

F(theta) = L(theta) + mu_C * <psi(theta)|(C - c)^2|psi(theta)>  = F^(1)   (Eq. 5)
    # the rival form F^(2) = L + mu_C*(<psi|C|psi> - c)^2 is NOT used here:
    #   proved to miss the target by O(1/mu_C) at every finite mu_C even in
    #   the best case, and when (c, E_{i0}) sits interior to the (C,E) convex
    #   envelope, no mu_C -- however large -- recovers it at all   (Sec. III B, Eq. 18-23)

# --- set the penalty weight --------------------------------------------------------
# proved sufficient, but needs the full spectrum so is not directly usable:
#   mu_C >= max_{i<i0} (E_{i0} - E_i) / (C_i - c)^2                       (Eq. 11)
C_min = smallest gap between distinct eigenvalues of C
    # 1 for the particle-number operator N; 3/4 for S^2; 1/2 for S_z      (Sec. III A)
mu_C_simple = (E_{i0} - E_0) / C_min^2
    # a looser but simpler bound than Eq. 11, obtained from E_i >= E_0
    #   for all i < i0 -- still proved sufficient                        (Eq. 12-13)
# E_{i0} - E_0 is itself unknown before the run; two ways to fill it in:
mu_C_ce    = (E_{i0}_classical - E_0_classical) / C_min^2   # e.g. Hartree-Fock
    #   or Moeller-Plesset estimates of the two energies                 (Eq. 14)
mu_C_rough = (2 / C_min^2) * sum_j |c_j|     # H = sum_j c_j P_j (Pauli decomposition)
    #   from the rigorous bound E_{i0} - E_0 <= 2||H|| <= 2 sum_j |c_j|  (Eq. 15)
mu_C := mu_C_rough      # pick exactly ONE of the three; F^(1) takes a single scalar
    # overestimating E_{i0} - E_0 still finds the target, only slower;
    #   UNDERestimating can move the minimum off the target -- so
    #   mu_C_rough (never an underestimate) is the safe default          (Sec. III A)
    # but the paper also says mu_C_rough "may be too large for the fast
    #   convergence", and that a large mu_C slows the optimizer, so
    #   mu_C := mu_C_ce is the faster choice where the classical estimate
    #   can be trusted not to underestimate the gap             (Sec. III A, Eq. 15)
    # several conserved quantities at once: one term and one mu_C^(l) per
    #   C^(l), each set by the same formula with its own C_min^(l)       (Eq. 16-17)

# --- optimize -----------------------------------------------------------------------
run a classical optimizer on F^(1)(theta) with that mu_C, theta -> theta*
    # the paper's own numerics use BFGS on noiseless simulation with exact
    #   expectation values -- no shot noise, no device noise              (Sec. IV)
while optimizing, track <psi(theta)|(C - c)^2|psi(theta)>
    # if this has not fallen near 0 by convergence, mu_C was too small and
    #   the run missed the sector -- raise mu_C and rerun                 (Sec. III A)

return  |psi(theta*)>, F^(1)(theta*) as the estimate of the sector's E_{i0}
# under a global depolarizing channel this minimizer is UNCHANGED -- the noisy
#   F^(1) equals (1-p) times F^(1) plus a theta-independent constant, so the
#   same theta* minimizes it; F^(2)'s minimizer instead shifts at O(p)  (Sec. III C, Eq. 26)

Cost, as the source states it

Kuroiwa and Nakagawa price penalty-constrained VQE in Pauli measurements — (cost-function evaluations) ×\times (Pauli operators per evaluation) — and bound neither factor: the working penalty F(1)F^{(1)} measures (C^−c)2(\hat{C}-c)^2 once per conserved quantity, more Pauli operators than C^\hat{C} alone but never counted, plus kk overlap terms for the kk-th state by VQD. Their figures are measured, not proved — 8-qubit H4\mathrm{H}_4 triplet, depth-12 symmetry-preserving ansatz, BFGS, noiseless, averaged over ten initial parameter sets. F(1)F^{(1)} takes 1.28×1071.28 \times 10^7 Pauli measurements at μC=0.01\mu_C = 0.01, rising to 1.18×1081.18 \times 10^8 at μC=100\mu_C = 100 where the residual degrades from 0.0000000.000000 to 0.0159190.015919 Ha. That cheapest accurate point still exceeds the 9.78×1069.78 \times 10^6 the unguaranteed F(2)F^{(2)} needs at μC=100\mu_C = 100, though the ordering reverses in their k=4k = 4 VQD run. On H4\mathrm{H}_4 with BFGS the classically estimated μC\mu_C needs 17,60417{,}604 evaluations against 147,063147{,}063 for the loose bound, which converged less accurately. Every guarantee assumes an ansatz spanning arbitrary nn-qubit states.

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.

  • 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

Sources