MethodLayer 1
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.
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.
Nothing drawn here has a recorded way through it that this figure leaves shut. See it on the map
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
Lee et al. present it as "affordable and systematically improvable" — the second half is the role of , and the first is what the pair restriction buys. They compare against the full generalized set (UCCGSD) and the standard one (UCCSD) on classical benchmarks, and report that "-UpCCGSD is found to show the best scaling for quantum computing applications" of the three.
Requires
Every step this method names moves its route along, so there is nothing it needs alongside them.
Example
given Hamiltonian H on N spin-orbitals, mapped 1-1 onto N qubits (qubit count is
linear in N), eta electrons, reference determinant |phi_0>
(this paper's own benchmarks use a restricted-Hartree-Fock determinant),
and an integer k >= 1 supplied by the caller
requires each factor's amplitude set is independent of every other factor's --
# no amplitude is shared across l = 1 .. k
requires k is not fixed here by a formula in N, eta or a target error -- the paper
# only reports, after the fact, that k must be taken large enough to reach
# sub-chemical accuracy, or the energy surface loses the smoothness that makes
# it usable for tasks like scanning it ("Summary of Chemical Applications")
# --- k-fold product of independent pair-restricted unitaries -----------------
for l = 1 .. k:
draw amplitude set { t_p^q } over ALL spin-orbital pairs p, q # generalized singles
draw amplitude set { t_(iup,idown)^(aup,adown) } over occupied spatial orbitals i,
virtual spatial orbitals a # paired doubles
# iup/idown and aup/adown are the two spin components of one spatial orbital,
# not the loop index l
T1^l = (1/2) sum_pq t_p^q adag_q a_p (Eq. 12)
# p, q run over occupied AND virtual spin-orbitals alike -- 'generalized',
# not restricted to occupied-to-virtual as in ordinary UCCSD (Eq. 4)
T2^l = sum_ia t_(iup,idown)^(aup,adown) adag_aup adag_adown a_idown a_iup (Eq. 14)
# moves both spins of spatial orbital i into spatial orbital a together --
# the pCCD-style restriction that drops the rest of the generalized-doubles
# manifold UCCGSD keeps
T^l = T1^l + T2^l
U^l = exp( T^l - (T^l)^dag )
return |psi(theta)> = U^1 U^2 ... U^k |phi_0> (Eq. 15)
# Eq. 15 itself is only written Pi_(l=1)^k exp(T^l - (T^l)^dag) |phi_0> --
# no time-ordering symbol, no stated convention for which factor acts first;
# since each U^l carries an independent amplitude set the U^l's do not
# commute in general, so operator order is a real degree of freedom the
# source leaves open, not a resolved detail. Expanded here the way a Sigma
# would be read -- left to right in l -- so U^k acts on |phi_0> first and
# U^1 last; this ordering is this listing's reading, not a claim the paper
# itself makes
# theta is the full stack of k independent {singles, paired-doubles} amplitude
# sets; k is supplied here, not derived -- it is the caller's dial, not this
# construction's output
# --- circuit realization, at a fixed number of Trotter steps per U^l (Table 1) --
gate count = O(k N^2)
# total gate count is lower-bounded by the number of cluster amplitudes, and
# each factor's singles + paired-doubles amplitudes number O(N^2) (Table 1)
group each U^l's paired-doubles terms into O(N) sets of O(N) terms acting on disjoint
spin-orbitals; execute every set in parallel on a linearly-connected qubit array
# this is what reduces the O(kN^2) gate count to a shallower circuit -- it assumes
# maximal parallelization of terms on distinct spin-orbitals and neglects
# Jordan-Wigner overhead (the derivation's own stated assumption, not a
# general-purpose compiler guarantee)Cost, as the source states it
Circuit depth in the number of spin orbitals and the number of repetitions, as stated in the abstract, against for UCCGSD and for UCCSD with electrons. A depth for the family, not a complexity for solving the problem.
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.
- 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 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
- k-UpCCGSD ansatz
Repeated paired generalized doubles with generalized singles trade expressivity against shallower chemistry circuits.