MethodLayer 1
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.
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
Barkoutsos et al. put the transformation before the circuit: they "propose a transformation of the electronic structure Hamiltonian in the second quantization framework into the particle-hole (p/h) picture, which offers a better starting point for the expansion of the trial wavefunction", so that the state "is parametrized in a way to efficiently explore the sector of the molecular Fock space that contains the desired solution". The circuit family follows from that choice — "a new family of quantum circuits based on exchange-type gates that enable accurate calculations while keeping the gate count (i.e., the circuit depth) low" — and the resulting method is named in the paper: "the particle-hole implementation of the Unitary Coupled Cluster (UCC) method within the Variational Quantum Eigensolver approach ... named q-UCC". The claim to read carefully is the last one, because it is about Trotterisation rather than about accuracy in general: they "show how a single Trotter step can accurately and efficiently reproduce the ground state energies of simple molecular systems", and *simple* is the paper's own word. No hardware run is reported.
Requires
Every step this method names moves its route along, so there is nothing it needs alongside them.
Example
given electronic Hamiltonian in second quantization, HF orbital basis
H_el = sum_ij h_ij a_i^dag a_j + sum_ijkl g_ijkl a_i^dag a_j^dag a_l a_k (Eq. 1)
HF reference determinant |Phi_0> = prod_{i=1}^N a_i^dag |vac> (Eq. 4)
equivalently the bit-string ket |1...1 0...0>, N occupied orbitals set to 1
(unlabeled form, Sec. III, step iii)
index convention: i,j,k,l occupied; m,n,p,q virtual; r,s,t,u either
fermion-to-qubit map: Jordan-Wigner (Sec. III, step ii)
requires cluster operator truncated at singles+doubles, T = T1 + T2
# higher T_n need only longer circuits, not new theory -- the paper restricts
# its own implementation to UCCSD as "unpractical" beyond that (Sec. II B 1)
# --- rewrite the Hamiltonian in the particle-hole picture ----------------------
redefine b_i^dag = a_i, b_i = a_i^dag (hole creation / annihilation) (Eqs. 5, 7)
b_m^dag = a_m^dag, b_m = a_m (particle creation / annihilation) (Eqs. 6, 8)
# b's still obey fermionic anti-commutation -- N_b[.] then defines a valid
# quasiparticle vacuum at |Phi_0>
normal-order N_b[.] against |Phi_0> to get
H_p/h = E_HF + sum_rs <r|F|s> N_b[a_r^dag a_s]
+ (1/2) sum_rstu <rs|g|tu> N_b[a_r^dag a_s^dag a_u a_t] (Eq. 9)
# this is the operator the device measures -- the qubit register itself is
# not re-encoded, only the Hamiltonian is rewritten
# --- build the cluster operator --------------------------------------------------
T1 = sum_{i;m} theta_i^m a_m^dag a_i (Eq. 13)
T2 = (1/2) sum_{i,j;m,n} theta_{i,j}^{m,n} a_n^dag a_m^dag a_j a_i (Eq. 14)
# sums run over ALL occupied/virtual pairs -- an active-space restriction to
# primed subsets i',m' shrinks them directly, same functional form (Eqs. 23-24)
|Psi(theta)> = exp( (T1 - T1^dag) + (T2 - T2^dag) ) |Phi_0> (Eq. 12)
# --- exponentiate by first-order Trotter splitting, n steps ----------------------
exp( (T1-T1^dag) + (T2-T2^dag) )
~= [ exp((T1-T1^dag)/n) exp((T2-T2^dag)/n) ]^n (Eq. 25)
# exact only as n -> infinity, general two-term case (Eq. 16)
# each factor realised by the excitation circuits of Fig. 1;
# the underlying Pauli-string form is given in App. A (Eq. AA.6)
# --- optimise theta by VQE, at the chosen n --------------------------------------
loop
prepare |Psi(theta)> on the device via the n-step circuit above
measure <Psi(theta)| H_p/h |Psi(theta)>
theta <- BFGS step on the measured energy (Sec. III)
until convergence
return |Psi(theta_min)>, E_GS = E_HF + E_corr_UCCSD(theta_min) (Eq. 15)
# n=1, WITH theta fully re-optimised by VQE AT that n, measures error < 1e-10 Ha
# at equilibrium H2 (Eq. 27; Fig. 5 red triangles) -- a DIFFERENT, larger
# 'Trotter error' (Eq. 26, amplitudes fixed at their exact-exponentiation
# values) is plotted in the same figure for comparison and is NOT the
# quantity claimed small here. The paper calls the n=1 result "surprising",
# explained by VQE's flexibility 'absorbing' the error, and explicitly not
# a proven bound (Sec. IV B 2; Sec. V)Cost, as the source states it
Barkoutsos et al. give one scaling: q-UCCSD circuit depth as — binomial coefficients — for the cluster operator truncated at , where and count only orbitals taking part in the excitations, so an active space shrinks it directly. Everything else is measured on in 6-31G (8 qubits, no Hamiltonian reduction), in simulation, with nearest-neighbour connectivity and exchange gates counted as one native two-qubit operation. q-UCCSD at a single Trotter step: 15 parameters, 708 one-qubit and 608 two-qubit gates, identical under both Hamiltonians — the particle-hole rewrite buys optimiser work, BFGS iterations falling 53 to 27 and Pauli-string evaluations to with never fixed, an overall speed-up of "about a factor 2 to 3". For the exchange-gate heuristic ansatz, reaching Ha at equilibrium takes blocks (84 parameters, 42 two-qubit gates, no one-qubit gates) against (112, 56); demanding chemical accuracy ( Ha) along the whole dissociation path erases that saving — , 140, 70 under both. Nothing bounds the VQE iteration count; is fixed empirically. No hardware run.
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.
- 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
- Particle-conserving VQE circuits
Givens-style or excitation-preserving blocks keep evolution inside a fixed-particle-number sector.