Qubit coupled-cluster 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

Ryabinkin et al. name two separate problems with the unitary coupled-cluster route, and the second is a hardware constraint rather than an accuracy one: the accuracy "depends on how many and what kind of terms are included", and there is "a growth of the number of simultaneously entangled qubits even at the fixed fermionic excitation rank", which "not all quantum computing architectures can cope with". Their method "starts directly in the qubit space and uses energy response estimates for ranking the importance of individual entanglers".

Requires

These do not move the route along. The method needs each of them alongside its own work, and the cost of getting them is part of what the method costs.

• Estimate an observable

  Given the ability to prepare $|ψ⟩$ and a description of an observable $O$, return a classical scalar within $\varepsilon$ of $⟨O⟩$ at confidence $1−δ$. The state is never returned; only the number is.

  QCC hands this ingredient a fixed, shallow preparation and asks for one scalar back. The state is the QMF product $|\Omega\rangle$ built from one-qubit rotations, followed by the surviving entanglers $\exp(-i\tau_k\hat{P}_k/2)$ in order; the paper's only device-level instance is Fig. 2, a Rigetti circuit for $\mathrm{H}_2$ that reaches it with $R_x$, $R_z$ and Hadamard gates and exactly two CNOT gates for its single entangler. What is estimated is the qubit-space $\hat{H}$ on that state — the similarity-transformed $\hat{H}[\tau;\hat{P}]$ that ranks entanglers is a classical expansion, not something handed to the estimator — one estimate per energy evaluation of the joint $(\tau,\Omega)$ minimization, with no grouping of $\hat{H}$'s Pauli words into commuting sets prescribed. assumption: $E(\tau,\Omega)$ is treated as an exact expectation value; the paper raises the "inherent quantum uncertainty" of a quantum computer's expectation values only when contrasting its numbers with a hardware run, and no shot budget enters its accounting

  assumption

Example

given  H = sum_I C_I P_I  on N_q qubits,  the JW/BK-transformed
           electronic Hamiltonian, C_I numerical, P_I Pauli words     (Eq. 4-5)
       pool of Pauli-word entanglers P_k,  |P_k| = 2 ... N_q,
           pool size  4^N_q - 3N_q - 1                                (Sec. II A)

requires  each P_k involutory,  P_k^2 = I                            (Sec. II A)
    # true of any Pauli word; this is what lets H[tau;P_k] below close
    #   after exactly two extra terms, not a truncated series

# --- QMF reference state ---------------------------------------------
|Omega> = product_i ( cos(theta_i/2)|alpha_i> + e^(i phi_i) sin(theta_i/2)|beta_i> )   (Eq. 16)
    # 2 Bloch angles (theta_i, phi_i) per qubit -- 2 N_q parameters here  (Eq. 15)
E_QMF = min_Omega  <Omega| H |Omega>                                   (Sec. II B, Eq. 22)

# --- rank the pool: cheap derivative tests before the costly one -----
for each P_k in pool:
    d1_k = <Omega_min| -i/2 [H, P_k] |Omega_min>   # dE/dtau at tau=0   (Eq. 25)
    if d1_k != 0:
        tier1 += P_k
    else:
        d2_k = <Omega_min| (1/2) P_k[H,P_k] |Omega_min>  -  c_k^T D^-1 c_k   (Eq. A.12-13)
            # relaxation-corrected second derivative -- c, D are the
            #   Bloch-angle cross term and Hessian at Omega_min (Eq. A.8-A.9);
            #   the correction accounts for Omega_min moving as tau moves off 0
        if d2_k is significantly negative:
            tier2 += P_k   # "significant negative" -- no numeric cutoff given

shortlist = top-ranked of tier1 (by |d1_k|)  union  top-ranked of tier2 (by |d2_k|)
    # "top entanglers in both tiers" is the paper's own phrase for this cut --
    #   how many survive it is left to inspection of the ranked list, not a
    #   fixed rule                                                      (Sec. II B)

# --- full re-optimisation, but only on that shortlist -----------------
for each P_k in shortlist:
    H[tau; P_k] = H - i sin(tau)/2 [H,P_k] + (1-cos(tau))/2 P_k[H,P_k]  (Eq. 20)
    dE[P_k] = min_tau ( min_Omega <Omega| H[tau; P_k] |Omega> )  -  E_QMF   (Eq. 21-22)
    # dE[P_k] <= 0 by construction

select  {P_1, ..., P_Nent} = the entanglers kept from the dE[P_k] ranking
    # how many are kept is likewise not fixed by the theory above -- the
    #   paper settles it molecule by molecule, e.g. all 6 energy-lowering
    #   H2 entanglers turn out redundant and only 1 is used in the end;
    #   for LiH near equilibrium (R~1.5 A) a few (6-7) already reach
    #   chemical accuracy THERE, but do not hold over the whole 0.5-5.0 A
    #   curve -- that takes a different, specific 7, chosen below at the
    #   symmetry-breaking geometry R=3.2 A                        (Sec. III C)
    # at that symmetry-breaking geometry the same first-derivative gradient
    #   test (not the full two-tier test -- the paper writes out only this
    #   half at the constrained reference) is rerun on a spin-singlet
    #   CONSTRAINED QMF reference instead, whose gradient favours
    #   entanglers that restore the broken symmetry                  (Eq. 33)

# --- assemble the ansatz -----------------------------------------------
U(tau) = product_{k=1..N_ent}  exp( -i tau_k P_k / 2 )                    (Eq. 17)
Psi(tau, Omega) = U(tau) |Omega>                                          (Eq. 14)

# if hardware caps entanglement at two qubits, factor any |P_k| >= 3 term:
# split P = P_1 w_k P_2, use [w_k', w_k''] = 2i w_k on the shared qubit,
#   giving exp(-i t P) as a product of ~log2|P_k| two-qubit exponentials   (Eq. 26-32)
# gate count only -- Psi and the parameter count below are unaffected;
#   e.g. LiH's 7 entanglers factor into 31 such two-qubit terms            (Sec. III C)

return  Psi(tau, Omega) = U(tau)|Omega>,  jointly variational in tau, Omega
        -- 2 N_q + N_ent parameters total                                 (Sec. II A)
# feeds observable-estimation: E(tau,Omega) = <Psi|H|Psi> is minimized
#   jointly over both parameter sets, one estimate per energy evaluation   (Eq. 18-19)

Cost, as the source states it

Ryabinkin, Yen, Genin and Izmaylov price the QCC ansatz at $2N_q + N_{\mathrm{ent}}$ variational parameters, two Bloch angles per qubit plus one amplitude per entangler (Sec. II A), with $N_{\mathrm{ent}}$ bounded only by the pool of Pauli words of length $2$ to $N_q$, $4^{N_q} - 3N_q - 1$. Ranking is two-stage (Sec. II B): over the whole pool, one commutator expectation per entangler, $\langle \Omega_{\min}| -\tfrac{i}{2}[\hat{H},\hat{P}_k] |\Omega_{\min}\rangle$ (Eq. 25), and where that vanishes a second derivative (Eq. A.12); then full re-optimisation of $\tau$ and all $2N_q$ Bloch angles, but only for the top entanglers of both tiers. Eq. (20) expands the Hamiltonian into three terms per entangler, so the exactly transformed $\tilde{H}$ holds $\sim 3^{N_{\mathrm{ent}}}$ operators, a classical expansion and not a gate count. Measured, not proved: LiH in STO-3G, reduced to a $4$-qubit, $100$-Pauli-term effective Hamiltonian, reaches $\le 1$ kcal/mol with $7$ entanglers, $15$ parameters and $31$ two-qubit factors, those $7$ chosen at $R = 3.2$ angstrom by an added spin-singlet constrained-QMF gradient (Eq. 33). No shot count, iteration count, optimiser or circuit depth is given.

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

Every step this method names is listed under Requires above. It walks its own span in one hop and calls out to the rest — that is a fact about the recorded route, not a claim that the span is simple.

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.

• 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.

• 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.

Narrower versions of this one

• 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.

In the Atlas

• Qubit coupled-cluster VQE

  Qubit coupled-cluster uses Pauli-word entanglers and a product-state reference directly in qubit space.

Sources

• Qubit coupled-cluster method: A systematic approach to quantum chemistry on a quantum computer↗Ilya G. Ryabinkin, Tzu-Ching Yen, Scott N. Genin, Artur F. Izmaylov · 2018theory yes · simulation yes · hardware no