Takes

The Hamiltonian whose ground state is wanted, together with whatever structure is to be respected — particle number, spin, point-group symmetry, a reference determinant — and the connectivity and native gate set of the device the family has to run on.

Returns

A circuit family with a fixed structure and free real parameters, together with the number of those parameters — which is the size of the classical search problem handed to the next layer.

The ansatz is the one choice that bounds everything after it, and the families answering it fail in different ways rather than in the same way by different amounts. A chemically motivated family is built from excitations out of a reference determinant, so the state you want is in it by construction and the circuit is deep. A hardware-native family is built from the gates the machine actually has, so it is shallow and there is no argument that the state you want is in it at all. An adaptive family refuses to fix the structure in advance and grows it operator by operator from measured gradients, which buys a compact circuit and pays for it in measurements before every step. Expressibility, depth, and measurement overhead are three different currencies, no family is cheap in all three, and which one binds depends on the machine rather than on the molecule — so a reader has a real choice to make here and the literature has not made it for them.

• Unitary coupled-cluster singles and doubles

  Build the trial state from single and double excitations out of a reference determinant, exponentiated as a unitary. The family is chosen for chemistry rather than for the machine: the state you want is in it by construction, and the circuit that reaches it is deep.

• Hardware-efficient ansatz

  Build the trial state out of the gates and couplings the machine already has, and accept whatever states that reaches. The circuit is shallow because nothing in it was chosen for the chemistry; there is correspondingly no argument that the state you want is inside the family.

• ADAPT-VQE adaptive ansatz

  Refuse to fix the structure in advance. Start from nothing and add one operator at a time, choosing each from a pool according to what the molecule itself indicates, until the energy stops improving. The circuit ends up short because nothing was included that the problem did not ask for.

• qubit-ADAPT-VQE ansatz

  The same grow-it-one-operator-at-a-time construction, with the pool rebuilt out of qubit operators rather than fermionic excitations so that the circuits it produces are shallow enough for near-term hardware.

• k-UpCCGSD ansatz

  Take k repetitions of paired double excitations together with generalized singles, instead of the full set of doubles. The point of the restriction is that the depth then grows linearly in the number of orbitals rather than polynomially, and k is the dial that buys accuracy back.

• Qubit coupled-cluster ansatz

  Skip the fermionic layer and build the ansatz directly in qubit space, ranking candidate entangling operators by how much each would move the energy and keeping the ones that earn their place.

• Particle-hole coupled-cluster circuits

  Rewrite the Hamiltonian around the reference determinant so that what the circuit has to describe is excitations out of it, then build the family from gates that move an electron without creating or destroying one. Staying inside the right particle-number sector is a property of the gates, not something the optimiser has to discover.

• Orbital-optimized coupled-cluster circuits

  Let the orbitals move too. The usual family fixes a basis and varies the amplitudes; this one varies the molecular orbital coefficients alongside them, so the same accuracy is reachable from a smaller active space and a shallower circuit — and the energy becomes fully variational, which is what makes forces available.

• Symmetry-preserving state-preparation circuits

  Build the circuit so that it cannot leave the symmetry sector the chemistry lives in. Particle number, total spin, spin projection and time reversal are respected by the gate structure itself, so the search never spends parameters on states the answer cannot be in.

• TETRIS-ADAPT-VQE ansatz

  Keep ADAPT's habit of growing the ansatz from measured gradients, and stop adding one operator per round. Several operators acting on disjoint qubits can go in together, filling the same layer instead of stacking — the same circuit, packed rather than piled.

• Iterative qubit coupled cluster

  Stop growing the circuit and grow the Hamiltonian instead. Each round folds the entanglers found so far into the operator by a canonical transformation, so every round runs a circuit of the same size — the cost moves off the device and into the number of terms that have to be measured.

• Generalized singles and doubles ansatz

  Drop the rule that an excitation has to move an electron from an occupied orbital into an empty one. Every pair of orbitals may be coupled, so the circuit stops depending on which reference determinant it was built around — a wider variational manifold, paid for in parameters.

• Batched ADAPT-VQE ansatz

  Keep ADAPT's habit of growing the ansatz from measured gradients, and stop adding exactly one operator per round. Every operator whose gradient is close to the largest goes in together, so the ansatz reaches the same size in far fewer rounds — and it is the rounds, not the operators, that cost measurements.

• ADAPT-QAOA: an iterative, problem-tailored QAOA

  Find a better parameterized ansatz for the quantum approximate optimization algorithm (QAOA) applied to combinatorial optimization problems such as Max-Cut, where the standard, fixed-form QAOA ansatz is not known to be optimal and no systematic method exists for improving on it.

• Spin-adapted VQE ansatz

  Spin-complemented generators reduce leakage from a target total-spin sector.