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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.

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.

This one, drawn

Estimate an excited-state energyVariational 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 insidePenalty-constrained variational eigensolver · 3 parts inside — 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 — click the name to read about itPenalty termsMultistate 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 here

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

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Why this is a layer

Four families answer this and they do not pay in the same currency, which is why none of them has retired the others. Deflation reuses the ground-state machinery unchanged and adds a term that pushes the search away from the states already found — so it needs those states in hand, in order, and the k-th state costs k runs of the whole loop. Subspace search refuses that ordering: it sends several mutually orthogonal inputs through one circuit and lets unitarity keep the outputs orthogonal, buying the whole low-energy subspace in one optimisation and paying in a wider objective and in inputs that must be prepared orthogonal to begin with. Subspace expansion and the equation-of-motion route do not optimise a second time at all — they take the ground state as given, measure matrix elements between excitation operators, and hand a small generalised eigenvalue problem to a classical solver, which buys excitation energies directly as differences and pays in the number of matrix elements. Folded-spectrum aims the same single search at a chosen energy rather than at the bottom, which is the only way here to reach a state whose index you do not know, and pays in the squared Hamiltonian. Prior states, extra inputs, extra measurements, or a squared operator: a reader standing here is choosing which of those they can afford, and no theorem settles it for them.

Ways to do this

7 methods recorded

  • 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.

  • 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.

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In the Atlas

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