The LUCJ Ansatz

qc/algorithms qc/gates

The circuit that prepares $|\psi ⟩$ for SQD — Local Unitary Cluster Jastrow, a chemistry-motivated ansatz built specifically to be both physically meaningful and hardware-friendly.

From UCCSD to a hardware-native circuit

Start from Unitary Coupled-Cluster Singles and Doubles (UCCSD), the standard quantum-chemistry ansatz:

$|\Psi ⟩=e^{T-T^{†}}|{\Phi }_0⟩,T={\sum }_{ai}{t}_i^a{\hat{c}}_a^{†}{\hat{c}}_i+{\sum }_{abij}{t}_{ij}^{ab}{\hat{c}}_a^{†}{\hat{c}}_b^{†}{\hat{c}}_j{\hat{c}}_i$

(single and double fermionic excitation operators — see Hamiltonians and Encoding for Quantum Circuits for the underlying fermion encoding). This is accurate but expensive to implement directly. Jastrow-factorizing it gives an equivalent-in-spirit but circuit-friendlier form:

$T-T^{†}\simeq {\sum }_{\mu }{e}^{-K_{\mu }}i{J}_{\mu }{e}^{K_{\mu }}$

where $K_{\mu }$ is an orbital rotation generator and $J_{\mu }$ is a density-density Jastrow term ($\sum J_{pr}^{\sigma \tau }{\hat{n}}_{p\sigma }{\hat{n}}_{r\tau }$). LUCJ keeps this structure directly:

$|\Psi ⟩\simeq {\prod }_{\mu }{e}^{-K_{\mu }}{e}^{i{J}_{\mu }}{e}^{K_{\mu }}|{\Phi }_0⟩$

— a product of orbital-rotation → Jastrow → inverse-orbital-rotation layers, each implementable as a specific hardware-native gate block.

The tradeoff it resolves

	Hardware-efficient ansätze	Physically-motivated ansätze (e.g. plain UCCSD)
Compatible with real device gates/connectivity/depth	Yes	No — needs all-to-all connectivity, high depth
Easy to initialize/optimize	No — arbitrary parameters, no physical starting point	Yes — parameters come from classical calculations

Key insight: LUCJ is designed to get both — physical motivation (parameters derived from classical coupled-cluster theory, so optimization starts from a sensible point) and hardware compatibility (the gate sequence respects a specific device’s native connectivity).

Real-world evidence

LUCJ circuits are believed hard to classically simulate, have outperformed their classical coupled-cluster counterpart on real benchmarks, and are compatible with real hardware execution — demonstrated on an N₂ bond-breaking curve (matching HCI reference energies better than RHF/CISD/CCSD) and a 77-qubit Fe₄S₄ cluster calculation. Reference: Motta, Sung, Whaley, Head-Gordon, Shee, Chem. Sci. 14, 40, 11213–11227 (2023).

Concrete circuit construction (ffsim)

In practice, one LUCJ layer $\mu$ is built from two ingredients with a slightly different but equivalent notation to the derivation above:

$U_{\mu }=e^{i{\hat{K}}_{\mu }}{e}^{i{\sum }_{p<q}{J}_{pq}^{(\mu )}{\hat{n}}_p{\hat{n}}_q}{e}^{-i{\hat{K}}_{\mu }}$

— an orbital rotation $e^{i\hat{K}}$ (a one-body unitary that mixes molecular orbitals, i.e. “rearranges the seats”) composed with a Jastrow factor $e^{i{\sum }_{pq}{J}_{pq}{\hat{n}}_p{\hat{n}}_q}$ (a diagonal term that penalizes or rewards pairs of orbitals being simultaneously occupied), where ${\hat{n}}_p=a_p^{†}{a}_p$ is the occupation-number operator. ffsim.UCJOpSpinBalanced.from_t_amplitudes(t1, t2, n_reps, interaction_pairs) takes classically-computed CCSD amplitudes — $t_1[i,a]$ (single-electron jump amplitudes) and $t_2[i,j,a,b]$ (electron-pair scattering amplitudes) — and compiles them directly into the rotation angles $K_{\mu }$ and Jastrow couplings $J_{pq}^{(\mu )}$. This is the literal mechanism behind “parameters come from classical calculations” in the tradeoff table above: CCSD is not used as the final answer, just mined for its amplitudes as a high-quality parameter seed. “Spin-balanced” means the α and β sectors share the same orbital-rotation parameters, appropriate for closed-shell molecules like N₂ where both spin sectors are chemically identical.

Encoding onto qubits uses Jordan-Wigner: orbital $p$ occupied ⟺ qubit $p=|1⟩$, empty ⟺ $|0⟩$, with the qubit register split into a first block for spin-α orbitals and a second block for spin-β orbitals (PrepareHartreeFockJW sets the initial Hartree-Fock occupation pattern; UCJOpSpinBalancedJW applies the correlator in this qubit basis).

Hardware topology shapes the circuit, not just the ansatz math

The Jastrow term’s ${\hat{n}}_p{\hat{n}}_q$ couplings need physical qubit adjacency between orbital $p$ and orbital $q$ for the corresponding gate to be hardware-native — so which interaction_pairs are cheap to implement depends entirely on the target device’s connectivity graph, not on the chemistry. Two IBM topologies illustrate this concretely:

• Heron (ibm_kingston): the α and β qubit chains only touch at every 4th site — only 5 α–β crossing pairs are hardware-native.
• Nighthawk (ibm_miami): the α and β rails run adjacent to each other, so up to 24 α–β pairs can be made hardware-native — far denser cross-spin connectivity, at the cost of a harder initial_layout design problem (each same-spin chain must trace a connected path through the device graph and stay close to its α–β partners, occasionally requiring a few parallel swaps).

This is a concrete answer to “how does LUCJ perform on heavy-hex vs. a different connectivity graph”: more native cross-spin connectivity (Nighthawk) means fewer or shallower SWAP insertions are needed to realize the same Jastrow couplings, directly reducing two-qubit gate count for the identical ansatz.

Related

• Sample-Based Quantum Diagonalization (SQD)
• Hamiltonians and Encoding for Quantum Circuits
• Variational Quantum Eigensolver (VQE) — another parameterized-ansatz approach, different tradeoff
• SQD on N₂ (Worked Example) — the Heron-vs-Nighthawk mapping exercise and CCSD-seeded circuit construction described above, in full worked-example context

Self-Check

• Why does Jastrow-factorizing UCCSD make it more hardware-friendly?
• Could you explain the hardware-efficient vs. physically-motivated tradeoff, and how LUCJ tries to get both?
• Why does starting optimization from classically-derived parameters matter in practice?
• Why does the number of hardware-native α–β interaction pairs depend on the device topology rather than on the molecule being simulated?