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SlotLayer 1

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.

Takes

A Hermitian HH reachable as a sum of terms, as sparse-access oracles or as a block-encoding; a way to prepare trial states, and — for the methods that need it — a trial state whose overlap with the ground state is not negligible; a target additive error ε\varepsilon and a confidence 1−δ1-\delta.

Returns

A scalar estimate of the lowest eigenvalue with a stated additive-error guarantee, plus the run or query budget it consumed. Whether that estimate is also a rigorous upper bound is a property of the method and is not promised by the slot.

This one, drawn

Estimate a Hamiltonian's ground-state energyVariational quantum eigensolver · 3 parts insideVariational quantum eigensolver · 3 parts inside — click the line to open it hereQuantum imaginary-time evolutionVariational imaginary-time evolution · 2 parts insideVariational imaginary-time evolution · 2 parts inside — click the line to open it hereGround-state energy by phase estimation · 2 parts insideGround-state energy by phase estimation · 2 parts inside — click the line to open it hereVariational quantum eigensolver · 3 parts inside — click the name to read about itVQEQuantum imaginary-time evolution — click the name to read about itQITEVariational imaginary-time evolution · 2 parts inside — click the name to read about itVariational ITEGround-state energy by phase estimation · 2 parts inside — click the name to read about itPEA energyHamiltonian whose ground state is wanted — you start hereNumber with an error bar — you finish here

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

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Why this is a layer

Four genuinely different families compete for this slot and none of them dominates, because they do not even promise the same kind of thing. A variational search returns a rigorous upper bound on the energy for any trial state it reaches — that much is the variational principle and needs no assumption — but nothing bounds how close to the true minimum it gets, or how many turns it takes to get there. Phase estimation returns an actual eigenvalue to a precision you can prove, and pays for it in coherent circuit depth far beyond what present hardware runs, plus a trial state whose overlap with the ground state is not negligible. Adiabatic preparation trades that depth for a runtime governed by the spectral gap, which is exactly the quantity nobody can bound in general. Imaginary-time methods converge without any optimiser at all, and pay in ancillas or in tomography of local domains. So the choice here is forced by which resource you are short of — depth, shots, coherence, or a proof — rather than settled by theory, and a reader standing on this slot is choosing between incomparable guarantees. That is the condition a layer has to meet. **Three of those four families are drawn here; adiabatic preparation is not, and its absence is a refusal rather than a gap.** W21 declined it a node on relevance: Farhi et al. prepare a ground state, they do not estimate an energy, so the contract does not meet — and stretching a contract to admit a family this paragraph names would be the easier and worse move. Until session 15 only two families were drawn while this paragraph named four, which is this test failing quietly: a justification describing a population the graph does not have is a sentence doing the work of methods that are not there.

Ways to do this

4 methods recorded

  • Variational quantum eigensolver

    Prepare a parameterised trial state on the quantum computer, measure the Hamiltonian's expectation value in it, and let a classical optimiser move the parameters. The quantum computer never runs a long coherent evolution; it runs a short one many times, and the loop closes through a classical number.

  • Quantum imaginary-time evolution

    Approximate evolution in imaginary time, which damps every excited state faster than the ground state, by a sequence of unitaries determined from measurements on a local domain. There is no variational ansatz and no classical optimiser: the method converges because imaginary time converges.

  • Variational imaginary-time evolution

    Keep the parameterised trial state, but stop treating the parameters as something to optimise: derive their equation of motion from a variational principle and integrate it in imaginary time. The parameters move because a differential equation says where they go, not because a search tried somewhere and liked the answer.

  • Ground-state energy by phase estimation

    Prepare a state that already overlaps the ground state, evolve it under the molecular Hamiltonian, and read the energy off the accumulated phase. Nothing is optimized and nothing is varied — the answer is a measured eigenvalue, and the whole difficulty moves into the starting state.

Routes that skip this layer

No recorded route avoids this step.

This is a step inside

  • 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.

  • 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.

In the Atlas

  • Coarse-grained variational quantum eigensolver for intermolecular interactions

    Determine the ground state of weakly-interacting, non-covalently bonded molecules — the weakly-bound intermolecular regime that variational quantum algorithms applied to strongly-bound, covalently-bonded systems with full molecular-orbital bases had left largely unexplored — using a coarse-grained representation of the electronic response suited to a VQA.

  • Projection-based embedding for VQE-in-DFT

    Simulate strongly correlated chemical systems on near-term quantum hardware, whose noise and limited size otherwise confine such simulations to small chemical systems, by embedding a quantum treatment of a strongly correlated fragment within a larger classical calculation.

  • Protein folding on a tetrahedral lattice by a variational quantum algorithm

    Predict the three-dimensional structure a protein takes from its primary sequence of amino acids, posed here on the model Hamiltonian the paper defines for a chain of N monomers placed on a tetrahedral lattice.

  • Double-bracket iterations for diagonalization

    Find quantum circuits that diagonalize a given input Hamiltonian, that is, approximate its eigenstates, without resorting to brute-force optimization of an unstructured variational circuit, which runs into barren plateaus.