Compile-once block encodings for masked similarity-transformed effective Hamiltonians

TL;DR

A mask-aware similarity-sandwich effective-Hamiltonian construction and benchmark stability under low-rank and second-order-perturation-guided screening under low-rank and second-order-perturation-guided screening are introduced.

Abstract

We present COMPOSER, a compile-once modular parametric oracle for similarity-encoded effective reduction of electronic-structure operators (e.g., Schrieffer-Wolff-type constructions). Low-rank factorizations compress Hamiltonians and anti-Hermitian generators into rank-one bilinear and projected-quadratic ladders with near-linear scaling at fixed thresholds; each ladder admits deterministic, number-conserving preparation and a block encoding using constant number of signal ancillas. A fixed PREP-SELECT-PREP template multiplexes these ladders, and one QSP polynomial performs the spectral transformation with degree set by operator norms. For a fixed orbital pool and qubit register, the two-qubit fabric is compiled once; geometry, active-space (mask) updates, and truncations are absorbed by re-dialed single-qubit rotations. We introduce a mask-aware similarity-sandwich effective-Hamiltonian construction and benchmark stability under low-rank and second-order-perturation-guided screening. COMPOSER is an execution architecture: algorithmic errors (block-encoding and QSP approximation) are tunable for any supplied parameters, while physical accuracy depends on how those parameters are obtained if not refined.

Figures (9)

• Figure 1: Conceptual motivation for operator-level structure control. Conventional workflows flatten second-quantized operators ($\hat{H}$, $\hat{T}$, $\hat{\sigma}$, …) into unstructured Pauli expansions, leading to expanding circuit constructions and repeated recompilation during iterative optimization or instance updates. In this flat execution paradigm, excitation hierarchy and chemically meaningful subspaces are obscured at the circuit level. In contrast, the COMPOSER architecture treats operator structure as the primary object of control. Nested low-rank factorizations yield structured blocks (e.g., active subspaces, low-rank ladders, residual sectors) that are embedded within a fixed algebraic topology. Instance dependence enters only through scalar parameter updates, enabling stable compilation and predictable resource scaling. This shift from gate-level optimization to operator-level control, implemented as a fixed-topology block-encoding architecture with streamed parameters, is guided by three motivating scientific constraints: preserving chemical meaning, ensuring predictable resource growth, and enabling systematic hierarchy refinement.
• Figure 2: Deterministic ladder circuit for one-electron state preparation. The ladder topology is fixed; only the angles $\{\theta_{pr}\}$ and phases $\{\phi_p\}$ depend on the data vector $u$ (see Appendix \ref{['app:ladders']}). The interleaved swaps indicate a connectivity-aware routing pattern; when the ladder is used as a basis rotation on general $N$-electron states, these swaps should be interpreted as fermionic swap-network primitives (or an equivalent orbital-routing compilation) Kivlichan2018FermionicSwap.
• Figure 3: Binary-multiplexed PREP-SELECT-PREP$^{\dagger}$ circuit realizing the unitary $W$. The SELECT stage implements the controlled table $W_{\texttt{sel}}=\sum_{s} \ket{s}\bra{s}\otimes (e^{i\phi_s}W_s)$, where phases $\phi_s$ absorb coefficient signs/phases. The blue dashed fan-out applies $W_s$ when the selector encodes $|s\rangle$. The workspace register has fixed width $t=\max_s t_s$ (including any adaptor-internal index/flag/signal qubits); adaptors with smaller workspace act trivially on unused ancillas. The $t$ oracle ancillas required by a single $W_s$ are reused across branches and projected to $|0^t\rangle$.
• Figure 4: Compile-once circuit for the similarity-sandwiched effective Hamiltonian. Changing the classical mask $\mathcal{M}^{(m)}$ only redials the rotation angles in $U_{\texttt{prep}}(\mathcal{M}^{(m)})$; every other gate, ancilla, and QSP phase is reused verbatim for every mask.
• Figure 5: Schematic workflow for quantum simulation using COMPOSER. The two-qubit circuit topology is fixed in a one-time synthesis pass; subsequent updates to molecular geometry, active space, or truncation mask re-dial only single-qubit rotations.

Theorems & Definitions (9)

• Definition 1: Rank-one bilinear operator
• Definition 2: Rank-one pair-excitation operator
• Definition 3: Projected quadratic rank-one operator
• Lemma 1: Single-ancilla dyad block encoding in the one-excitation register
• Lemma 2: Deterministic block encoding of a squared diagonalized Cholesky channel
• Theorem 1: Binary-multiplexed block encoding
• proof
• proof
• proof