Generalized singles and doubles ansatz

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. define the ansatz by the distinction it removes: "Here the single and double 'excitation' terms do not distinguish between occupied and unoccupied orbitals and they are therefore called 'generalized' singles and doubles (GSD)", and name the result in the same section — "We shall term this ansatz UCCGSD". Its cost is stated beside its sibling's rather than alone: the paper's Table 1 puts UCCGSD at $O(N^4)$ gates and $O(N^3)$ depth against k-UpCCGSD's $O(kN^2)$ and $O(kN)$. The two are separate ansätze in this paper, compared against each other and against UCCSD, which is why `k-upccgsd-ansatz` and this node are separate rather than one node drawn twice.

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,  eta electrons
       reference state |phi_0>  (an eta-electron Slater determinant, e.g. RHF;
           a multi-determinantal reference is also usable)

requires  a fermion-to-qubit encoding fixed ahead of time, one qubit per spin-orbital
    # qubit count is linear in N by construction of the mapping (Sec. Quantum
    #   Resource Requirements)

# --- cluster operator, over ALL spin-orbital pairs, not just occ-to-virt -------------
T1 = (1/2) sum_pq    t^q_p    a_q^dag a_p
T2 = (1/4) sum_pqrs  t^rs_pq  a_r^dag a_s^dag a_q a_p          (Eq. 11-12)
    # p,q,r,s each range over all N spin-orbitals -- occupied and unoccupied are
    #   NOT distinguished, unlike UCCSD's T1 = sum(i occ, a virt) t^a_i a_a^dag a_i
    #   (Eq. 3-5): "the single and double 'excitation' terms do not distinguish
    #   between occupied and unoccupied orbitals" -- hence 'generalized' (GSD)
T = T1 + T2
    # the free amplitudes {t^q_p, t^rs_pq} number as many as the parameters of the
    #   2nd-quantized Hamiltonian itself (Sec. Generalized CC, same T as Eq. 11)

# --- anti-Hermitize and exponentiate --------------------------------------------------
generator = T - T^dag
    # the anti-Hermitian combination is what makes exp(generator) unitary
U(theta) = exp(T - T^dag)                     (Eq. 13 -- the paper names this UCCGSD)
|psi(theta)> = U(theta) |phi_0>
    # theta is the amplitude vector {t^q_p, t^rs_pq}

# --- what a device actually runs -------------------------------------------------------
U(theta) implemented as  Trotter_m[ exp(T - T^dag) ],   m a small FIXED step count
    # "a practical implementation of UCC relies on approximating e^(T-T^dag) by a
    #   small number of Trotter steps" -- the circuit prepared is not exactly the
    #   |psi(theta)> above; it differs from the state the amplitudes were fit to
    #   (Sec. Quantum Resource Requirements)
    # this paper does not derive how large m must be for UCCGSD; it reports that
    #   even m=1 has been shown elsewhere to reach high accuracy for UCC ansatze
    #   in general (cited to Ref. 64) -- an external result, not proved here

# --- how the resulting family is scored ------------------------------------------------
E(theta) = <phi_0| U(theta)^dag H U(theta) |phi_0>  /  <phi_0| U(theta)^dag U(theta) |phi_0>
    # evaluated variationally (Eq. 9's form, carried over unchanged from the general
    #   UCC ansatz Eq. 8 to the UCCGSD ansatz Eq. 13, which only redefines T),
    #   NOT by the projective coupled-cluster equations (Eq. 6-7)

return  the parameterized circuit  U(theta) |phi_0>,  theta ranging over {t^q_p, t^rs_pq}
# a circuit FAMILY, not yet a converged state -- theta is fit by an outer classical
#   loop elsewhere in the pipeline

Cost, as the source states it

Lee, Huggins, Head-Gordon and Whaley cost UCCGSD in Table 1: $\mathcal{O}(N^4)$ gates and $\mathcal{O}(N^3)$ circuit depth in $N$ spin-orbitals, with no dependence on the electron number $\eta$ — against UCCSD's $\mathcal{O}((N-\eta)^2\eta^2)$ and $\mathcal{O}((N-\eta)^2\eta)$. Conditions: state preparation only; a fixed number of Trotter steps of $e^{\hat{T}-\hat{T}^{\dagger}}$ rather than the exact exponential; gate count taken as the two-qubit count; and a depth assuming maximum parallelization, no Jordan-Wigner overhead, and a linearly connected qubit array, the last resting on their Refs. 59 and 63, the latter unpublished. Qubits are linear in $N$. Table 9 reports amplitudes per benchmark case, not as a formula: 27 at $N=8$, 135 at $N=12$, 434 at $N=16$ for both $\eta=4$ and $\eta=10$; their classical code stops near 16 spin-orbitals. Nothing bounds the optimization — each one was rerun thirty to two hundred times from random starting points — and the paper gives no absolute shot or measurement count, deferring measurement to McClean et al.

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.

• 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.

• 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

• Generalized excitation VQE

  Generalized singles and doubles relax occupied-to-virtual restrictions to enlarge the variational manifold.

Sources

• Generalized Unitary Coupled Cluster Wavefunctions for Quantum Computation↗Joonho Lee, William J. Huggins, Martin Head-Gordon, K. Birgitta Whaley · 2018theory yes · simulation yes · hardware no