Quantum Computing — A Full Method Set, on a Simulator and Real Gate Circuits

Classical electronic structure (PySCF)

• Engine: PySCF — a single entry point for HF, DFT (any libxc functional), CASSCF, and post-HF (MP2, CCSD, CCSD(T), FCI) on an HF reference — MP2 and CCSD include geometry optimization on analytic gradients.
• Calc types: single-point, geometry optimization, frequencies (IR/Raman), transition-state search, IRC, MD, excited-state optimization (TD-DFT + geomeTRIC), and PES/relaxed scans.
• Implicit solvation (C-PCM, IEF-PCM, SMD, COSMO) and Grimme dispersion (D3, D3(BJ), D4) applied before the SCF kernel so forces and Hessians stay exact.
• Open-shell handling (UHF/UKS/ROHF/ROKS chosen automatically), structured JSON output (result.json, no regex scraping), and units in Hartree, Hartree/Bohr, and kcal/mol (force field).

The quantum methods (in Configure)

Choose Quantum as the compute backend and a quantum panel replaces the classical method family. The qubit Hamiltonian comes from a PySCF restricted reference — RHF for closed shells, ROHF for open shells (radicals, triplets) — mapped through OpenFermion (Jordan–Wigner or Bravyi–Kitaev), with an optional frozen core and explicit active space to shrink it. Beyond ground-state energy, two calc types add vibrational structure and open-system dynamics. Each method is described in detail below, with its cautions.

• Variational: VQE (UCCSD ansatz), ADAPT-VQE (an ansatz grown to convergence from a UCCSD operator pool), and CQE (contracted quantum eigensolver / ACSE — a two-body anti-Hermitian iteration to the exact ground state).
• Sample-based diagonalization: SQD — samples electron configurations from a LUCJ ansatz prepared on the Hartree–Fock reference, then runs self-consistent configuration recovery and a selected-CI diagonalization (the IBM first-party qiskit-addon-sqd / ffsim workflow). Classical-heavy, hardware-light — one sampling job, no variational loop.
• Phase estimation: QPE (controlled time-evolution + inverse QFT) plus a full short-depth family that reads the energy from the time-domain signal g(t) = ⟨φ|e^{−iHt}|φ⟩ — QCELS, Robust PE, QMEGS (multi-eigenvalue spectrum), and Bayesian PE.
• Spectra and gaps: QMEGS resolves several eigenvalues at once; BPDE (Bayesian phase-difference estimation) returns the S₀→S₁ energy gap directly; QKD (quantum Krylov diagonalization) gives the ground and low-lying states from a real-time Krylov space.
• Amplitude amplification: QAAE — a non-variational amplify-learn eigensolver built on a spectral filter (open Grover/Brassard amplification, deliberately not the patented amplitude-estimation route).
• Vibrational structure (quantum): harmonic frequencies → a boson-to-qubit vibrational Hamiltonian (binary/gray/unary encodings) solved exactly or by VQE, for the vibrational levels and fundamentals.
• Open-system dynamics: Lindblad relaxation and dephasing of the molecular qubit register — the dissipative master equation, reporting the population, coherence and purity trajectory and the steady state.

VQE — variational quantum eigensolver (UCCSD)

The workhorse: a classical optimizer minimizes ⟨ψ(θ)|H|ψ(θ)⟩ over a Trotterized unitary coupled-cluster singles-and-doubles circuit prepared on the Hartree–Fock determinant [1]. Closed shells use the spin-adapted singlet UCCSD parameterization; open shells switch to an unrestricted UCCSD with one amplitude per spin-conserving excitation, so the singly-occupied orbital both relaxes and correlates. The result is a variational upper bound to the true ground-state energy — in our benchmarks it reaches the exact (FCI) answer for every full-space test system.

• Use it when: you want a robust, gate-circuit-native ground-state energy for a small active space, or a baseline to compare the other methods against.
• Caution — optimizer, not oracle: COBYLA (default) can stall in a local minimum or hit the iteration cap before converging; the result reports the convergence flag — treat an unconverged energy as an upper bound only. SLSQP/L-BFGS-B converge faster on smooth landscapes; SPSA tolerates noise.
• Caution — cost climbs steeply: parameters and circuit depth grow roughly with (occupied × virtual)²; past ~10 spatial orbitals (20 qubits) the local simulator refuses. Use frozen core and an active space first.
• Caution — on IBM hardware VQE runs a hardware-efficient EfficientSU2 ansatz (not UCCSD): shallower and noise-friendlier, but with no chemical structure — expect qualitatively rougher energies than the local UCCSD result.
• Caution — the deep UCCSD gate circuit is the most noise-sensitive method in our Aer noise study (chemical accuracy needs CX error below ~10⁻⁶ for H₂); trust the exact local path, export the circuit for study.

ADAPT-VQE — a grown, problem-shaped ansatz

Instead of a fixed UCCSD circuit, ADAPT-VQE grows the ansatz one operator at a time: at each step it measures the energy gradient of every excitation in a pool, appends the largest, and re-optimizes — stopping when the gradient norm falls below threshold [2]. The circuits are typically much shallower than full UCCSD for the same accuracy. Open shells use per-spin occupied windows in the pool, so excitations out of the singly-occupied orbital are present.

• Use it when: full UCCSD is too deep, or you want circuit depth spent only where the chemistry needs it (stretched bonds, multireference character).
• Caution — the gradient threshold is the accuracy dial: too loose and it stops early with a shallow but biased ansatz; too tight and late iterations churn on numerically tiny gradients. The iteration history is reported — check the energy plateaued.
• Caution — each growth step re-runs a full VQE optimization, so wall-clock can exceed plain VQE on small systems even though the final circuit is shallower.
• Caution — same IBM-hardware substitution as VQE (EfficientSU2, flagged with a warning glyph): the adaptive protocol itself runs only on the local simulator.

CQE — contracted quantum eigensolver (ACSE)

A non-variational iteration on the anti-Hermitian contracted Schrödinger equation: each step measures the two-body residual of the current state and applies the corresponding generalized two-body rotation, walking the state toward the exact ground state without a global parameter optimizer [3]. Because the pool is the full generalized (not just occupied→virtual) operator set, CQE can reach FCI where plain UCCSD stalls at an ACSE stationary point.

• Use it when: you want systematic convergence toward FCI without trusting a black-box optimizer, especially for stronger correlation.
• Caution — the generalized pool scales as O(N⁴) in spin orbitals: hundreds of operators at 8 qubits, thousands at 12. Small systems only.
• Caution — the generalized pool contains spin-flip generators and there is no Sz penalty: the spin sector is held only by the reference and the Hamiltonian's symmetry. For open shells, check the reported energy against the expected sector (compare with the classical FCI action).
• Caution — local simulator only for the true protocol; the IBM target substitutes EfficientSU2 (warned in Configure).

SQD — sample-based quantum diagonalization

The quantum-centric supercomputing workflow used in IBM's large-scale chemistry demonstrations [4]: prepare a LUCJ (local unitary cluster Jastrow) circuit seeded from classical CCSD amplitudes [5], sample electron configurations from it in the computational basis, run self-consistent configuration recovery, and diagonalize the Hamiltonian in the sampled determinant subspace classically. One sampling job, no variational loop — the heavy lifting is classical selected-CI. Open shells sample a spin-unbalanced LUCJ built from ROHF-CCSD amplitudes; with an explicit active space, the amplitudes are computed with all orbitals outside the window frozen.

• Use it when: you want the hardware-realistic workflow (it is what runs on IBM devices at scale), or a correlation estimate whose quality degrades gracefully with shot count.
• Caution — stochastic: the energy depends on the shot count, batch settings, and RNG seed (seeded and reproducible here, but a different seed gives a slightly different subspace). It is not a variational optimization — the answer lands between FCI and Hartree–Fock, wherever the sampled subspace reaches.
• Caution — weak correlation starves the sampler: when the LUCJ rotation is small (e.g. a compact active space near equilibrium), low shot counts may sample only the Hartree–Fock determinant and the energy comes back exactly at the reference. Raise shots / samples-per-batch, or read the subspace-dimension note in the result.
• Caution — CCSD amplitudes must converge for the LUCJ seed; when they fail the run degrades to MP2 amplitudes and then to bare-HF sampling, each step recorded in the result warnings — never silently.

QPE — quantum phase estimation

The textbook fault-tolerant algorithm [6]: controlled time evolutions e^{−iHt} against an ancilla register, followed by an inverse quantum Fourier transform, read the ground-state energy out of the accumulated phase [7]. On the local simulator the evolution is exact, so the only approximation is the phase grid itself.

• Use it when: you want the canonical reference algorithm, or to study how ancilla count and Trotterization trade against accuracy.
• Caution — the ancilla register quantizes the answer: resolution is (spectral range)/2ᵇ for b ancilla bits, and the returned energy snaps to the nearest grid point — it can land above or below the true value by up to half a bin. Benchmarks: 34–51 mHartree deviations at the default 6 bits, ~3 mHartree at 10 bits. Raise 'ancilla bits' for tighter answers (cost doubles per bit).
• Caution — the reference state must overlap the ground state: QPE returns the eigenvalue its input state projects onto most strongly. A poor Hartree–Fock reference (strong correlation) can lock onto an excited state.
• Caution — the gate-circuit path adds Trotter error on top of the grid, and the controlled-evolution depth is far beyond present-day device coherence: QPE stays local-simulator-only by design.

QCELS — quantum complex-exponential least squares

The first of the short-depth signal methods: sample the time signal g(t) = ⟨φ|e^{−iHt}|φ⟩ at a grid of times (each sample is one shallow Hadamard-test circuit) and fit its dominant complex exponential — the frequency is the ground-state energy [8]. Reaches Heisenberg-limited precision with circuits dramatically shallower than QPE.

• Use it when: you want near-QPE accuracy at NISQ-era depth — QCELS is the most hardware-friendly energy estimator here and one of the six methods with a faithful IBM path.
• Caution — the initial-state overlap is the whole game: the fit finds the dominant frequency, so the reference must overlap the ground state by more than any other single state (ideally |⟨φ|ψ₀⟩|² > 0.5). Strongly correlated references can flip it onto an excited state.
• Caution — accuracy is bounded by the time grid: the maximum evolution time sets the frequency resolution, samples-per-time set the noise floor on hardware. Local (exact-signal) runs land within ~10⁻⁵ Hartree in the anchors.

RPE — robust phase estimation

Estimates the phase of e^{−iHt} from Hadamard-test samples at iteratively doubled evolution times, with a bounded-error update at each doubling — a protocol originally built for gate calibration that transfers directly to energy estimation [9]. No ancilla register, no QFT, and provable robustness to bounded sampling noise.

• Use it when: you want Heisenberg scaling with the simplest possible circuits and an explicit noise-robustness guarantee; also IBM-capable.
• Caution — same overlap requirement as QCELS: the phase being tracked belongs to the state the reference overlaps most.
• Caution — robustness is to bounded errors: if noise at some doubling exceeds the bound, the wrap-around error is not recoverable downstream — on hardware, keep shots high enough per round.

QMEGS — multi-eigenvalue Gaussian-filtered search

Applies Gaussian filters to the same g(t) signal and searches the result for peaks, returning several dominant eigenvalues at once — the first short-depth method with Heisenberg-limited scaling and no spectral-gap assumption [10]. Useful when you want a low-lying spectrum, not just the ground state.

• Use it when: several states carry weight in the reference (near-degeneracy, multireference character) and you want them resolved individually; IBM-capable.
• Caution — a state only appears if the reference overlaps it: peaks below the overlap threshold are simply absent from the output, not flagged.
• Caution — peak width (resolution) is set by the maximum evolution time; closely-spaced states below that resolution merge into one peak.

BPE — Bayesian phase estimation

Maintains a posterior distribution over the eigenphase and updates it after every Hadamard-test sample, choosing each next (time, phase-shift) measurement to be maximally informative [11]. Extremely sample-efficient in the low-noise regime.

• Use it when: you want an uncertainty estimate along with the energy — the posterior width is part of the result.
• Caution — the adaptive schedule requires round-trip control of the next measurement, so BPE runs on the local simulator only (no IBM path; flagged in Configure).
• Caution — a badly mis-set prior (center or width) slows convergence or locks onto the wrong mode when the reference overlaps several states.

BPDE — Bayesian phase-difference estimation (S₀→S₁ gaps)

Runs the Bayesian protocol on a superposition of two states so the measured phase is directly the energy DIFFERENCE — the vertical S₀→S₁ gap in one calculation, without resolving either absolute energy [12]. The two reference states are prepared by VQD (variational quantum deflation) on the UCCSD ansatz.

• Use it when: the observable you care about is the gap itself (spectroscopy) — differences converge faster than two absolute energies subtracted.
• Caution — the answer is only as good as the two VQD reference states: a poorly converged excited-state ansatz biases the gap. Check the reported VQD convergence for both states.
• Caution — inherits the UCCSD ansatz (and its open-shell/unrestricted variant), so ansatz-expressivity limits apply to the excited state especially; local simulator only.

QKD — quantum Krylov diagonalization

Builds a Krylov subspace from real-time evolutions of the reference, {|φ⟩, e^{−iΔtH}|φ⟩, e^{−2iΔtH}|φ⟩, …}, measures the Hamiltonian and overlap matrices in that basis, and solves the small generalized eigenproblem classically [13]. Converges toward FCI within a handful of Krylov vectors and returns low-lying excited states from the same run.

• Use it when: you want near-exact ground (and low excited) states from very few, shallow quantum measurements; IBM-capable.
• Caution — the Krylov overlap matrix goes ill-conditioned as the basis grows: the solver regularizes by discarding small singular values, which caps the usable subspace size. More Krylov vectors is not always better.
• Caution — on hardware the matrix elements take a finite-difference derivative of g(t) that amplifies shot noise: the IBM path widens the time step and raises the shot floor (with a warning). QCELS/RPE are more shot-efficient if you only need the ground state.

QAAE — amplitude-amplification eigensolver

A non-variational amplify-learn loop: a spectral filter (a contraction built from the Hamiltonian and a Gershgorin bound on its spectrum) is applied repeatedly, amplifying the ground-state component of the trial state — a power method on the filter — while the Rayleigh quotient converges to the ground-state energy. The hardware form realizes each filter step as the good branch of a Sz.-Nagy dilation with Grover/Brassard-style amplitude amplification boosting post-selection; the implementation deliberately avoids any amplitude-estimation subroutine.

• Use it when: you want a deterministic, optimizer-free convergence trajectory (each round provably increases ground-state weight).
• Caution — the convergence rate is set by the filter's spectral gap: near-degenerate ground states converge slowly, with many rounds and little energy movement per round.
• Caution — the dilation-based gate path compiles the filter exactly via Quantum Shannon Decomposition, which caps the exportable circuit at ≈8 qubits; local simulator recommended.
• Caution — like every projection method, it amplifies the lowest state the trial overlaps: a zero-overlap ground state is invisible.

Vibrational structure (quantum calc type)

After a classical Hessian, the harmonic modes define a bosonic vibrational Hamiltonian that is encoded to qubits (unary, binary, or Gray encodings) and solved exactly or by VQE, giving vibrational levels and fundamentals in the quantum representation [14].

• Use it when: exploring quantum algorithms for vibrational problems — the boson-to-qubit encodings and their qubit costs are the point of the exercise.
• Caution — the physics is harmonic: anharmonicity enters only through the truncated mode basis, so do not read the fundamentals as an anharmonic prediction (classical VPT2-grade results need a classical anharmonic treatment).
• Caution — per-mode boson truncation caps the levels representable; more levels per mode means more qubits fast. This calc type has no exportable gate-circuit toggle.

Open-system dynamics (quantum calc type)

Evolves the molecular qubit register under a Lindblad master equation with relaxation and dephasing channels, reporting population, coherence, purity trajectories and the steady state. The exact path integrates the dense Liouvillian; the gate path realizes each Kraus step as a Sz.-Nagy unitary dilation [15].

• Use it when: studying decoherence of a molecular register or dissipative state preparation on a small system.
• Caution — dense Liouvillian scaling caps the system at ≤5 system qubits; the dilation gate path is depth-heavy and export-limited like QAAE's.
• Caution — relaxation drives toward the global ground state of the full Fock space, which need not share the particle number or spin of your reference: for an open-shell register, relaxation implies spin-flipping jumps across sectors. That is the model's physics, not an error — but read the trajectory accordingly.
• Caution — the relaxation/dephasing rates are user-set model parameters, not derived from the molecule.

Two ways to run a method — exact, or a real gate circuit

Most quantum methods offer two execution paths. The default is the exact path (fast, no shot noise — the value to trust). An opt-in toggle instead builds the genuine universal-gate circuit, the same form that runs on hardware. (VQE and ADAPT-VQE are already gate circuits; the vibrational method does not offer the exportable gate-circuit toggle.)

• Exact (default): the engine's statevector / linear-algebra path runs each method exactly — no Trotter or sampling error, computed without Qiskit. This is what the regression anchors pin (H₂ STO-3G VQE = −1.1372838345 Hartree).
• Gate circuit (opt-in): the method is compiled to universal {RY, RZ, CNOT} gates and simulated gate-by-gate (still without Qiskit); the resulting circuit can be exported to Qiskit, with the optional Qiskit components, to send to hardware. The signal/Krylov family uses the Hadamard test; QPE uses QFT + controlled-Uᵏ; VQE and CQE use Trotterized Pauli-exponential ansätze; QAAE and the open-system step compile their dilation exactly via the Quantum Shannon Decomposition.
• The gate path is Trotter-approximate (and depth-limited) where exact synthesis is impractical; it is the on-hardware form, not a claim of higher accuracy. The exact path stays the default and the recommended result.
• A live circuit estimate shows the qubit width and an approximate depth as you choose options; after the run, Analyze → Overview reports the actual circuit and ancilla count the engine used.

How the quantum runs execute

• Local (default): the engine's statevector simulator builds the molecular Hamiltonian from PySCF integrals (via OpenFermion), maps it to qubits, and runs every method exactly — no Qiskit required. Shots do not change a local result.
• IBM Quantum (Run → backend picker): six methods dispatch to real hardware via the optional Qiskit Runtime components (the app fails loud with an install hint if they are absent) — VQE (EstimatorV2, a hardware-efficient EfficientSU2 ansatz), SQD (SamplerV2 configuration sampling), and the signal/Krylov methods QCELS, RPE, QMEGS and QKD (their shared Hadamard-test signal g(t) sampled on SamplerV2). Your shot count applies to all of them.
• QKD on hardware is shot-hungry: its Krylov matrix takes a finite-difference derivative of g(t) that amplifies shot noise, so the IBM path widens the step and raises the shot floor (with a warning) — QCELS and RPE, which take no derivative, are more shot-efficient.
• Stay on the local simulator: QPE (deep controlled-evolution beyond device coherence) and BPE / BPDE (adaptive schedules / arbitrary excited-state preparation). ADAPT-VQE, CQE and QAAE ride method VQE, so on the IBM target they run as the EfficientSU2 ansatz, not the chosen algorithm — the Configure screen flags every one of these with a warning glyph and tooltip. See the noise study below.
• Tested under realistic device noise: we ran the gate circuits on Qiskit-Aer with an IBM-device-like noise model — see 'Tested on an IBM noisy simulator'.
• Regression anchors (RNG-pinned, reproducible): H₂ STO-3G VQE = −1.1372838345 Hartree; QPE ≈ −1.1385 Hartree; QKD/CQE/QAAE reach the FCI ground state; QCELS/RPE land within ~1e-5. Open-shell anchors: H₃ doublet and H₄ triplet VQE/ADAPT reproduce the sector-exact FCI energy, and an OH-radical active space matches classical CASCI — full tables in 'Benchmark results' below.