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.
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Main text: 18 pages, 9 figures; Supplementary Material: 17 pages
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.
Effective models make complex quantum dynamics tractable by retaining only the states, processes, and timescales relevant to a given physical question. Their predictive power, however, depends on treating the retained manifold, initial state, observables, micromotion, dissipative channels, and eliminated degrees of freedom consistently. We develop a unified framework for constructing and validating effective descriptions using projection methods, Schrieffer--Wolff transformations, Magnus expansions, adiabatic elimination, small rotations, time coarse graining, and Floquet theory. The quantum Rabi model and its dispersive limits serve as recurring benchmarks for identifying control parameters, virtual processes, spectral corrections, observable dressing, and breakdown. Higher-order resonances and applications to collective light--matter coupling, Raman processes, spin-chain buses, lossy mediators, and driven lattices illustrate how effective interactions can be interpreted and designed. A physical prediction is organized into three order-consistent steps: encoding into the retained description, evolution under the effective dynamics, and reconstruction of the physical observable. This viewpoint exposes failures caused by leakage, small denominators, neglected micromotion, inconsistent observables, memory effects, or incorrectly transformed dissipation, while linking derivation, physical interpretation, and mechanism-guided discovery. Companion notebooks provide QuTiP implementations, convergence tests, and microscopic-to-effective comparisons.
Quantum computers hold the promise of solving certain computational problems significantly faster than classical computers. Unlocking the full potential of quantum hardware depends on our ability to efficiently translate high-level quantum algorithms into executable sequences of quantum operations tailored to the specific constraints of a given device, a task known as quantum compilation. Among the most promising applications is the simulation of quantum systems, such as in quantum chemistry, where scalable, error resilient, and experimentally feasible Hamiltonian simulation techniques are essential. In the first part of this thesis, we present recent advances in the field of quantum compilation, emphasizing on efficient gate decomposition and circuit optimization. In particular, we analyze compilation techniques for Clifford circuits and general single-qubit rotations. Clifford operations play a central role in quantum error correction, the characterization of quantum devices and quantum cryptography. We explore asymptotically optimal decomposition methods, including constructions using either two-qubit gates or global entangling operations, the latter being especially well-suited for platforms like trapped ions. For single-qubit unitaries, we discuss approximation schemes that achieve the theoretically optimal scaling of gate count with respect to the desired precision. Beyond gate synthesis, we examine various hardware-aware optimization strategies aimed at reducing circuit depth and gate overhead. These include techniques for qubit routing in systems with limited connectivity, as well as methods for removing redundant operations. The second and primary, scientific contribution of this thesis focuses on implementation of effective many-body interactions on quantum hardware by combining the system’s native dynamics with control operations, which we call Hamiltonian engineering. Motivated by the central role of Hamiltonian simulation in quantum science, we introduce a general and efficient framework for synthesizing arbitrary target Hamiltonians under mild locality constraints. We propose a novel linear programming (LP) approach that computes pulse sequences consisting of single-qubit π or π/2 rotations applied at certain times during free evolution under the system’s native Hamiltonian. Our LP-based framework offers multiple advantages. First, its classical runtime depends only on the number and locality of the interactions in the system Hamiltonian, not directly on the overall system size. Second, we extend the method to include robustness against common error sources such as finite pulse time effects and systematic control errors, leveraging tools from average Hamiltonian theory and robust composite pulses. Our approach enables the efficient realization of high-fidelity Hamiltonian dynamics with minimal experimental overhead. The scalability of the approach are demonstrated through simulations, including the engineering of a random two-body Hamiltonian on a 225 qubit lattice in under a minute of classical preprocessing. In conclusion, this thesis contributes both theoretical insights and practical tools that bridge the gap between algorithm design and experimental implementation in quantum computing. The techniques developed here are directly applicable to a variety of platforms, including trapped ions, neutral atoms, and superconducting circuits. By advancing the frontiers of Hamiltonian engineering, this work enables more efficient, robust, and scalable quantum simulations on nearterm devices. Finally, it brings us closer to realizing quantum technologies, which are possibly capable of solving meaningful problems.
This thesis develops structure-preserving methods to control and compile quantum evolutions with provable guarantees. In quantum dynamics, a fundamental resource is coherence maintained within a designated subspace; leakage – population transfer out of that subspace – is a major source of error that demands robust methods for its suppression and control. For example, maintaining the evolution within the computational (logical) subspace in quantum computing is essential for practical quantum information processing. On the dynamics side, we study adiabatic evolution under weak perturbations through effective-Hamiltonian frames. Using Bloch wave operators and Schrieffer-Wolff constructions, we obtain block-diagonal generators and prove that their evolutions remain uniformly close to the true dynamics. Moreover, we establish time-independent (“eternal”) bounds on spectral leakage that scale as 𝑂(𝛾−1) in the weak-coupling regime, where 𝛾 is the dimensionless ratio between the strength of the strong system Hamiltonian and the weak drive. The analysis covers both finite-dimensional, time-dependent settings and a novel coarse-grained framework that accommodates continuous spectra and even unbounded Hamiltonians, enabling applications to banded tight-binding models and superconducting-qubit architectures. Optimising quantum algorithms for reduced depth or fewer non-error-corrected physical gates is also key to mitigating leakage. We therefore formulate quantum compiling as exact or approximate unitary synthesis up to a physically irrelevant global phase and solve it via mixed-integer linear programming (MILP). The formulation supports multiple objectives (depth, two-qubit gate count, non-Clifford cost) and provides theoretical guarantees of optimality. To the best of our knowledge, this is the first MILP formulation that targets circuit depth directly, rather than minimising gate counts and inferring depth retroactively. The model explores the algebraic structure of the problem in order to accelerate the solver through symmetry breaking, redundancy elimination, and other mathematically sound constraints. To scale beyond exact global optimisation, we introduce a rolling-horizon strategy that compiles circuits in overlapping windows. An open-source implementation, written in the Julia language and released as QuantumCircuitOpt.jl, provides these capabilities to facilitate further research. In both cases, similarity transformations and algebraic identities reduce complexity while focusing attention on the spectral/target content, and both yield quantitative guarantees. Collectively, these contributions confront leakage, the key threat to coherent, in-subspace computation, with the ultimate aim of getting error-resilient qubits to work.
Quantum simulators enable the exploration of complex quantum phenomena in condensed-matter systems by reproducing their dynamics on controllable quantum devices. However, experimental constraints often restrict the class of Hamiltonians that can be realized natively. Hamiltonian engineering addresses this limitation by expanding the set of accessible Hamiltonians beyond the fixed system Hamiltonian provided by the hardware. We introduce a framework for fermionic Hamiltonian engineering based on interleaving free evolution under the system Hamiltonian with a sequence of experimentally feasible local fermionic unitaries. The required sequences and free-evolution times are obtained efficiently through a linear program, enabling effective time evolution under a broad class of target Hamiltonians. In particular, our method realizes arbitrary complex tunnelling coefficients, constrained only by the connectivity of the underlying system Hamiltonian. We further show that symmetries of the Hamiltonian engineering task can be exploited to drastically reduce the cost of computing the control parameters, and that errors arising from finite pulse times can be compensated within the same framework through a simple modification of the linear program. To our knowledge, this is the first automated method to realize locally tunable artificial gauge fields in interacting fermionic lattice models using only site-resolved energy shifts as control. We demonstrate the method by engineering the dynamics of interacting Fermi-Hubbard chains with complex tunnelling coefficients and of the Harper-Hofstadter model on a 1088-mode lattice. For the latter, we exploit the translation symmetry of the model to determine a provably optimal six-pulse sequence that produces the desired dynamics independent of the lattice size.
Effective low-energy theories represent powerful theoretical tools to reduce the complexity in modeling interacting quantum many-particle systems. However, common theoretical methods rely on perturbation theory, which limits their applicability to weak interactions. Here we introduce the Variational Adiabatic Gauge Transformation (VAGT), a nonperturbative hybrid quantum algorithm that can use nowadays quantum computers to learn the variational parameters of the unitary circuit that brings the Hamiltonian to either its block-diagonal or full-diagonal form. If a Hamiltonian can be diagonalized via a shallow quantum circuit, then VAGT can learn the optimal parameters using a polynomial number of runs. The accuracy of VAGT is tested through numerical simulations, as well as simulations on Rigetti and IonQ quantum computers.
We present a numerical, non-perturbative framework for constructing effective Hamiltonians that describe the dynamics of low-energy degrees of freedom within a restricted Hilbert space in quantum many-body systems. The approach is based on block diagonalization guided by a minimal-deformation principle imposed within a selected target sector. The formulation is designed to remain compatible with the numerical linked-cluster expansion. For gapped systems, the relation between minimal deformation and cluster additivity requires careful treatment when excited eigenstates contain finite admixtures of the ground state. After establishing a cluster-additive basis that reproduces the Hörmann-Schmidt construction, the minimal-deformation criterion uniquely determines the effective Hamiltonian within each excitation sector. The same criterion also provides a practical numerical procedure for selecting relevant low-energy eigenstates, including regimes characterized by strong level mixing and avoided crossings. The framework is illustrated using two spin models: the one-dimensional transverse-field Ising model as a benchmark and the two-dimensional Shastry-Sutherland model with Dzyaloshinskii-Moriya interactions, relevant to SrCu _2 2 (BO _3 3 ) _2 2 . In both cases, the resulting effective Hamiltonians accurately capture the excitation dynamics and the associated band structures.