| name | vqe-circuits |
| description | How the eigenstate-preparation pipeline is built from circuits: gate-based statevector simulation, the Pauli-rotation primitive, gate-level UCCSD and the hardware-efficient ansatz, the prolongation operator, adiabatic resolution refinement, and the rodeo algorithm. Use this skill WHENEVER the task involves VQE, UCCSD/HEA circuits, prolongation, resolution refinement, the adiabatic path, OR the rodeo algorithm (energy scans, eigenstate preparation, the cos^2 success probability, fidelity-vs-M, acceptance). Read `references/rodeo.md` before writing or modifying any rodeo code so the formulas and conventions match the published algorithm.
|
VQE circuits, resolution refinement, and the rodeo algorithm
The four pipeline stages, all implemented in pairinglib and reused (never
re-implemented) by the notebook and analysis scripts.
1. Coarse-state preparation (gate-level UCCSD-VQE)
- Trotterised product ansatz
prod_mu exp(theta_mu (tau_mu - tau_mu^dag)) |HF>.
- Each excitation compiles to commuting Pauli rotations via Jordan-Wigner: a
single -> 2 Pauli strings, a double -> 8. Use
gate_uccsd_setup/_vqe/_state.
- Optimise with the EXACT analytic product gradient. The naive two-term pi/4
parameter shift is WRONG beyond the first factor (later factors gain
frequency-1 cross terms; the spectrum is {1,2}). This is a known trap — always
use the analytic gradient already implemented in the library.
- Singles vanish for the pairing force (seniority): expect a doubles theory.
2. Prolongation
prolong_matrix(k_low, k_high, N) embeds every low-res basis state into the
same occupation in the larger space, leaving new levels empty: gate-free for
basis refinement, O(N_sites) two-qubit gates per dimension for lattices.- The prolonged state satisfies
<Phi0|H_high|Phi0> = E_low exactly.
3. Resolution refinement (adiabatic phase)
- Path
H(s) = cos^2(pi s/2) P (H_low - mu) P^dag + sin^2(pi s/2)(H_high - mu),
s = t/T, with shift mu = E_low + 0.6 to keep tracked states below the large
null space of P^dag. Use refine_state / refine_NK.
- The path stays gapped, so
T ~ 1/Delta E (not 1/Delta E_min^2). Overlap
rises toward 1; energy falls to FCI from above.
4. Rodeo algorithm (final filter) — see references/rodeo.md
- One cycle: ancilla Hadamard, controlled
e^{-iHt_k}, phase P(E t_k),
Hadamard, measure; post-select all-zero. The post-selected map is
(I + e^{iEt} e^{-iHt})/2 (rodeo_post0), equal to the explicit ancilla
circuit (rodeo_cycle_ancilla).
- Feed the rodeo the RESOLUTION-REFINED state (high overlap p): that is the
whole point — cost ~ 1/p and a few cycles when p ~ 1.
Validation anchors (assert these when you touch the code)
rodeo_cycle_ancilla == rodeo_post0 to ~1e-16.- Eigenstate input reproduces
prod cos^2(t_k (E_obj-E)/2) (Eq. 1); Gaussian
average reproduces [(1+e^{-(E_obj-E)^2 sigma^2/2})/2]^M (Eq. 2).
- Refined input (N=4, k=4, p~0.998) -> fidelity ~ 0.999995 within ~2 cycles,
acceptance -> p; off-target background ~
2^-M.
