MethodLayer 1
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.
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.
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.
Same contract as the slot it fills.
This one, drawn
From Hamiltonian whose eigenvalues are wanted to Parameterised circuit family
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
- Choose a parameterised trial state
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.
When it applies
Mizukami et al. state what is varied and what follows from it. OO-UCC "variationally determines the coupled cluster amplitudes and also molecular orbital coefficients", and "owing to its fully variational nature, first-order properties are readily available" — which they cash out immediately: "this feature allows the optimization of molecular structures in VQE without solving any additional equations". The resource claim is comparative and unquantified in the abstract: "the method requires smaller active space and shallower quantum circuit than UCC to achieve the same accuracy", with no number attached, so none is quoted here. Evidence is simulation, and it is worth naming what was simulated because it is a property calculation rather than a single energy: "numerical examples of OO-UCC using quantum simulators, which include the geometry optimization of the water and ammonia molecules using analytical first derivatives of the VQE".
Requires
Every step this method names moves its route along, so there is nothing it needs alongside them.
Example
given molecular Hamiltonian H built from 1- and 2-electron integrals h_pq, h_pqrs
in a chosen orbital basis; reference determinant |0> (conventionally Hartree-Fock)
UCCSD ansatz |Psi> = e^(A1+A2) |0>, A_n = T_n - T_n^dagger (Eq. 3, Eq. 9)
a VQE circuit that already measures the 1- and 2-particle RDMs needed below
# RDMs are "readily available in VQE, because it measures [them] to
# compute electronic energy in a given quantum circuit" -- no extra
# measurement is added for the orbital step
requires no formal bound is stated on the outer loop below;
# optimizing the singles/orbital part classically from the measured RDMs, rather
# than varying it on the circuit, is presented as a design choice (Sec. II),
# not as something the paper proves necessary
# --- decompose UCCSD into a classical orbital rotation and a Trotterized doubles part ---
model |Psi'> = e^(A2) e^(A1) |0> (Eq. 10)
# a distinct wave-function model, not an exact rewrite of e^(A1+A2)|0>:
# A1 and A2 do not commute, so separating them "is different from the
# original ansatz" (Sec. II)
identify e^(A1) with the orbital-rotation operator e^(kappa) (Sec. II)
kappa = sum_pq kappa_pq (Epq - Eqp), Epq^- := Epq - Eqp
# "the singles part e^(A1) ... is identical to the orbital rotation unitary
# operator e^kappa" -- so it can be optimized on a classical computer via
# standard orbital-optimization technique instead of varied on the circuit
collapse to |Psi_OOUCCD> = e^(A2~) |0~>, |0~> = e^(A1) |0> (Eq. 11)
# --- outer loop: alternate VQE with a classical Newton-Raphson orbital step ---
repeat
run VQE, optimizing doubles amplitudes A2~ on the circuit -> E(A2~, kappa)
from the 1RDM, 2RDM VQE just measured, form
g_pq = <Psi|[H, Epq^-]|Psi> (Eq. 8)
Hess_pq,rs = (1/2) ( <Psi|[[H,Epq^-],Ers^-]|Psi> + <Psi|[[H,Ers^-],Epq^-]|Psi> ) (Eq. 7)
solve Hess kappa_step = -g for the orbital-rotation step (Eq. 6, Newton-Raphson)
kappa <- kappa + kappa_step
until self-consistent
# "we repeatedly perform the VQE and the orbital optimization until
# convergence" -- no cycle-count bound is given (Sec. II);
# "UCCSD-level ... results ... at the cost of repeated VQE optimizations" (Sec. III)
# at self-consistency the singles operator A1~ itself vanishes (the Brueckner
# condition) together with dE/dkappa = 0, simultaneously and by construction
# of the separation above -- "variational orbitals are Brueckner orbitals
# in this ansatz" (Sec. II, 'Trotterization and Brueckner orbitals')
# the paper's own printed condition is the exponential e^(A1~) = 0 -- which
# cannot literally hold for a unitary exponential of an anti-Hermitian
# operator; read here, per that same sentence's prose ("the singles'
# contribution ... vanishes"), as the operator condition A1~ = 0
# --- map the converged doubles generator onto the circuit ---
Trotterize e^(A2~) ~= prod_mu e^(A2~_mu), truncated at the first Trotter step (Eq. 14)
# this defines OO-UCCD1; a single Trotter step is a large truncation of the
# BCH-infinite UCC generator, and this paper proves no bound on the
# truncation error -- it cites Barkoutsos et al.'s empirical finding that a
# single step "reproduces ground-state energy accurately", and reports its own
# numerics (LiH, H2O, NH3) showing OO-UCCD1 tracks UCCSD1 closely (Table I, Fig. 1)
return the OO-UCCD1 circuit family together with converged (theta, kappa)
as the trial state for the eigenvalue problem
# because theta and kappa are BOTH variationally converged, the full chain rule
# dE/dx = dE/dx|_x + (dE/dtheta)(dtheta/dx) + (dE/dkappa)(dkappa/dx) (Eq. 12)
# collapses to just the Hellmann-Feynman term:
dE/dx = <Psi| dH/dx |Psi> # consequence of Eq. 12
# the d(E)/d(theta) . d(theta)/dx term vanishes because VQE has converged theta;
# the d(E)/d(kappa) . d(kappa)/dx term vanishes because the orbitals are optimized
# a non-orbital-optimized VQE keeps that second term and must instead solve a
# coupled-perturbed Hartree-Fock (CPHF) equation for the orbital response (Eq. 13)Cost, as the source states it
The paper's only resource figures are measured on one molecule, not derived. Mizukami, Mitarai, Nakagawa, Yamamoto, Yan and Ohnishi report, for ammonia in STO-3G with all electrons correlated and no active-space approximation, 120 VQE parameters and quantum-circuit depth 2720 for OO-UCCD truncated at the first Trotter step — the same pair they quote for plain UCCD at that truncation — against 135 and 2780 for UCCSD, likewise at one Trotter step. They call the reduction slight, and never define what the depth counts or in what gate set. It is paid for by an unbounded outer loop: singles are handled classically, so VQE and a classical Newton–Raphson orbital optimisation are repeated until convergence, UCCSD-level accuracy coming "at the cost of repeated VQE optimizations", with no bound and no measured count of cycles. Once both and are converged, a first derivative costs only the expectation value of the derivative of the Hamiltonian; second and higher derivatives still need the circuit-parameter response. The abstract's other claim — smaller active space, hence fewer qubits — is nowhere quantified.
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
Nobody has taken this apart yet. That is a gap in this graph, not a claim that the method has no parts.
Other ways to fill the same slot
Different approaches
- 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 a narrower version of ADAPT-VQE adaptive 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.
- 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 a narrower version of ADAPT-VQE adaptive 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 a narrower version of Qubit coupled-cluster ansatz
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 a narrower version of ADAPT-VQE adaptive 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.
In the Atlas
- Orbital-optimized VQE
Orbital rotations are optimized alongside circuit parameters to improve compact active-space descriptions.