Minimise the objective over the parameters

Why this is a layer

Once the family is fixed, what is left is a non-convex minimisation in which every function evaluation is a noisy estimate bought with shots — so the classical optimiser is not an implementation detail, it is the thing that decides the total run count, and the run count is what the whole method is charged. Methods here differ in the currency they spend: a gradient-free search buys robustness to shot noise with many cheap evaluations, an analytic-gradient method buys direction with extra circuits per step, and a metric-aware method buys better steps with the cost of estimating the metric. The objective is a choice too, and a separate one: the plain energy expectation is not the only function whose minimiser is the answer, and changing it changes the landscape the optimiser is walking rather than the way it walks. Nothing about that is decided by the ansatz above or the readout below, which is what makes it a slot instead of a step in someone's write-up.

Records for what it handles

The catalogue's own entries for the objects on either side of this contract.

Ways to do this

6 methods recorded

• Conditional-value-at-risk objective

  Change what the classical loop is minimising rather than how it minimises. Instead of averaging every measurement outcome into an expectation value, keep only the best tail of them and average that — which is defensible precisely when the answer is a single good bitstring rather than a physical average.

• Analytic-gradient parameter search

  Get the gradient of the objective exactly, rather than by finite differences, by running the same circuit again at shifted parameter values. The direction is then not an estimate of a slope taken from two noisy numbers; it is the slope, estimated to whatever precision the shots allow.

• Minimise the energy variance

  Minimise how much the energy fluctuates rather than the energy itself. Any eigenstate has zero variance, so the objective's own value tells you whether you have arrived — which the energy never does, since a low number is only low relative to a minimum nobody knows.

• Follow the steepest descent in the state's own geometry

  Take the step that moves the state fastest, not the one that moves the parameters fastest. The two differ because equal changes in parameters do not make equal changes in the state, and the metric measuring that difference has to be estimated before every step.

• Grow the circuit a layer at a time while training it

  Do not settle the circuit before optimising it. Start shallow, train what is there, then hold most of it fixed and add the next layer on top — so every step of the search runs on a shallow circuit with few free parameters, which is where a gradient is still large enough to follow.

• Simultaneous-perturbation optimization

  Perturb every parameter at once, in one random direction, and take the difference of two objective evaluations as the gradient estimate. The estimate is bad in any single round and unbiased across rounds, so the cost of a step stops growing with the number of parameters.

Routes that skip this layer

No recorded route avoids this step.

This is a step inside

• 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.

• 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.

• 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.

In the Atlas

No record in the Atlas covers this yet. The catalogue is circuits and primitives; this part of the literature is not in it.