Optimal Effective Hamiltonian for Quantum Computing and Simulation

TL;DR

The paper addresses a fundamental obstacle in quantum engineering: the gauge freedom arising from non-unique block-diagonalization when forming effective Hamiltonians. It introduces the Least Action Unitary Transformation (LAUT) to select a canonical, symmetry-preserving BD map by minimizing the geometric distance to the identity, and pairs it with the Bloch-Brandow perturbative framework to obtain nonperturbative and analytic EHs that respect underlying symmetries. Across superconducting hardware, LAUT-based methods (exact EBD-LAUT and perturbative PBD-BB) achieve superior spectral accuracy, faithfully reproduce experimental interaction rates in driven cross-resonance gates, and capture beyond-RWA renormalizations and genuine multi-body interactions such as $XZX+YZY$, improving calibration and Hamiltonian learning. The framework provides a principled, scalable route to high-precision Hamiltonian engineering, with a static soundness metric to certify model fidelity and potential extensions to non-Hermitian, Floquet, and many-body embedding contexts. Altogether, LAUT unifies geometry, variational optimality, and symmetry considerations into a practical toolkit for Hamiltonian learning and quantum simulation.

Abstract

The effective Hamiltonian serves as the conceptual pivot of quantum engineering, transforming physical complexity into programmable logic; yet, its construction remains compromised by the mathematical non-uniqueness of block diagonalization, which introduces an intrinsic "gauge freedom" that standard methods fail to resolve. We address this by establishing the Least Action Unitary Transformation (LAUT) as the fundamental principle for effective models. By minimizing geometric action, LAUT guarantees dynamical fidelity and inherently enforces the preservation of symmetries--properties frequently violated by conventional Schrieffer-Wolff and Givens rotation techniques. We identify the Bloch-Brandow formalism as the natural perturbative counterpart to this principle, yielding analytic expansions that preserve symmetries to high order. We validate this framework against experimental data from superconducting quantum processors, demonstrating that LAUT quantitatively reproduces interaction rates in driven entangling gates where standard approximations diverge. Furthermore, in tunable coupler architectures, we demonstrate that the LAUT approach captures essential non-rotating-wave contributions that standard models neglect; this inclusion is critical for quantitatively reproducing interaction rates and revealing physical multi-body interactions such as $XZX+YZY$, which are verified to be physical rather than gauge artifacts. By reconciling variational optimality with analytical tractability, this work provides a systematic, experimentally validated route for high-precision system learning and Hamiltonian engineering.

Figures (13)

• Figure 1: (Color online) Block diagonalization for effective Hamiltonian construction. The two representative cases introduced below exemplify how effective Hamiltonians reconcile microscopic complexity with low-dimensional physical intuition. (a) Decoupling ancillary subsystems: A physical quantum circuit with couplers (left) is mapped to an effective model (right) where computational states are dynamically isolated. (b) Transforming interband into intraband couplings: A multiband system with hybridization (left) is transformed into an effective theory with only intraband dynamics (right). Dashed lines indicate bare energies; pink solid lines indicate renormalized energies; blue arrows denote effective couplings.
• Figure 2: (Color online) Geometric interpretation of the Least Action Unitary Transformation (LAUT) principle. As described in the Lemma, the set of all unitaries that block-diagonalize the Hamiltonian forms a continuous manifold $\mathcal{T}_{BD}$ (blue surface). Standard methods (grey dashed line) traverse this manifold via unconstrained trajectories. In contrast, the LAUT criterion (Definition 1) uniquely identifies the transformation $T$ (red dot) that minimizes the geometric distance $||T-\mathcal{I}||_F$ to the identity (black dot), thereby fixing the gauge freedom with minimal structural deformation.
• Figure 3: Illustration of the LAUT-based fidelity bound in a minimal three-level model. The left- (blue squares) and right- (red line) hand sides of Eq. (\ref{['eq:key_bound']}) are shown as functions of the coupler frequency in a symmetric configuration ($\omega_1/2\pi=\omega_2/2\pi=4~\mathrm{GHz}$). The inequality holds for all parameters, and the two curves converge as $\|T-\mathcal{I}\|_F^2 \rightarrow 0$, reflecting the expected tightening of the bound in the small-rotation regime. The inset displays the point-wise difference (pink circles) between the two sides of the inequality; the fitted line (black) exhibits a clear quadratic trend.
• Figure 4: (Color online) Analysis of symmetry preservation and transformation fidelity. (a) The symmetry-breaking metric $\zeta$. LAUT maintains exact symmetry ($\zeta \approx 0$) by construction, while GR introduces spurious asymmetry. (b) LAUT yields superior fidelity in the near-resonant regime, strictly saturating the analytical lower bound of the Theorem. GR, lacking variational optimality, falls below this limit (red arrow). Data are shown versus the qubit--coupler detuning.
• Figure 5: (Color online) Effective coupling comparison for $\omega_1=\omega_2$ in the Qubit--Coupler--Qubit system. (a) Coupling $2\tilde{g}$ obtained via LAUT, GR, and FFT extraction. (b) Relative error $\eta_{g} = |(\tilde{g}-\tilde{g}_{\mathrm{FFT}})/\tilde{g}_{\mathrm{FFT}}|$. LAUT tracks the FFT benchmark closely across the full range, while GR shows large resonance-enhanced errors.

Theorems & Definitions (8)

• Definition 1
• Lemma : Gauge Freedom in Block Diagonalization
• proof
• Theorem : Geometric Lower Bound on Dynamical Fidelity
• Corollary 1: Symmetry Preservation under Least‑Action Gauge Fixing
• proof
• Corollary 2
• proof