Quantum Circuits: Transport, anomalous relaxation, and complexity

Abstract

Gate-based quantum computers can implement many-body system dynamics through repeated gate cycles. This periodic driving discretises time and relaxes energy conservation, while locality and symmetries still constrain transport, relaxation, and complexity. This thesis focuses on Floquet-circuit diagnostics linking spectral structure, transport, and operator complexity. Part I introduces spectral and linear-response tools for Hamiltonian and Floquet systems, spectral projections of long-time limits, operator-resolved spectral densities (DOS/LDOS), and current spectral functions, and adapts them to stroboscopic dynamics with U(1) conservation. We implement a moment-expansion (KPM) method for spectral densities as a hybrid quantum-classical workflow, and validate it by extracting the interacting-spin DOS on a trapped-ion quantum computer with up to N = 18 spins. Using disordered discrete-time XXZ circuits with conserved charge, we benchmark ergodicity and transport. Because experiments access only finitely many cycles, we propose two finite-time transport estimators from Z-basis measurements: from the local-quench profile and its first circular moment we obtain drift and circular-variance measures that are more robust than conventional diagnostics. In this platform they reveal a long-lived near-SWAP prethermal swappy regime. To probe discrete-time transport spectroscopically, we reconstruct spectral densities of stroboscopic correlators and derive a current operator valid for arbitrary circuit geometries. This yields quasienergy-resolved transport diagnostics, showing that a single integrability-breaking impurity can induce a prethermal plateau when studying quasi-conserved charges while leaving the current spectral function unchanged. In parallel, we recast anomalous equilibration, including thermal and symmetry Mpemba effects, in resource-theoretic terms: monotone crossings are set by overlaps with slow, resourceful modes. The criterion extends to complexity-like resources and two-step Pontus-Mpemba protocols in unitary circuits, and the symmetry Mpemba effect can arise from a purely classical mechanism. Part II studies operator growth via Krylov complexity in Floquet circuits. We relate continuous-time Lanczos and discrete-time Arnoldi recursion through a spectral-aliasing, or folding, identity and show that defining an effective generator log U is a physical branch choice. We control this choice with a locality-guided adiabatic unfolding protocol that tracks eigenprojectors along a Floquet path, producing quasi-local effective Hamiltonians when they exist. Benchmarks across dual-unitary endpoint families separate bounded free-fermion dynamics from qualitatively different interacting behaviour, clarifying when Krylov diagnostics require care for nonlocal Floquet generators. Overall, the thesis provides experimentally native probes and reconstruction tools connecting discrete-time spectra, transport, and operator complexity to diagnose integrability breaking, prethermalisation, and anomalous relaxation in driven quantum matter.