Non-Equilibrium Quantum Many-Body Physics with Quantum Circuits

These are the notes for the 4.5-hour course with the same title that I delivered in August 2025 at the Les Houches summer school ``Exact Solvability and Quantum Information’’. In these notes I pedagogically introduce the setting of brickwork quantum circuits and show that it provides a useful framework to study non-equilibrium quantum many-body dynamics in the presence of local interactions. I first show that brickwork quantum circuits evolve quantum correlations in a way that is fundamentally similar to local Hamiltonians, and then present examples of brickwork quantum circuits where, surprisingly, one can compute exactly several relevant dynamical and spectral properties in the presence of non-trivial interactions.

💡 Research Summary

These lecture notes present a comprehensive introduction to brickwork quantum circuits (BQCs) as a versatile framework for studying non‑equilibrium dynamics of interacting quantum many‑body systems. The author begins by motivating the need for emergent, coarse‑grained descriptions of quantum quenches, emphasizing that exact microscopic evolution is computationally intractable for large systems. BQCs are defined as one‑dimensional arrays of 2L qudits (d ≥ 2) acted upon by alternating layers of nearest‑neighbour two‑site unitary gates arranged in a brick‑wall pattern. Diagrammatic tensor‑network notation is introduced to visualize states, operators, and the causal structure of the circuit.

A central theme is the close analogy between BQC dynamics and those generated by local Hamiltonians. Both propagate correlations within a strict linear light‑cone, as formalized by the Lieb‑Robinson bound. Two explicit constructions that map a continuous‑time Hamiltonian evolution onto a discrete BQC are discussed. The first uses the standard Suzuki‑Trotter decomposition: by splitting the Hamiltonian into even and odd bonds and applying the corresponding exponentials in alternating steps, one approximates e^{-iHt} with (U_eU_o)^n. The error scales as O(1/n) (or O(1/n²) with higher‑order formulas), but achieving a fixed precision for a system of size L requires a circuit depth proportional to L, making the light‑cone of the circuit artificially narrow for finite n.

The second, more refined, construction follows Osborne’s block‑renormalization approach. Sites are grouped into blocks of size Ω = O(log L); within each block a strong two‑site unitary W(ℓ) is defined. The full evolution over a time τ is then expressed as a product of alternating even and odd layers of these block gates, with an error bounded by ∥ε∥ ≤ a L Ω e^{bτ−cΩ}. Choosing Ω logarithmic in L makes the error exponentially small while keeping the gate depth modest, and the circuit’s light‑cone now matches the Lieb‑Robinson cone of the original Hamiltonian.

The notes then explore dynamical properties of specific BQC families. Random unitary circuits (RUCs), where each gate is drawn independently from the Haar measure, exhibit linear growth of Rényi entropies with a rate set by the local Hilbert space dimension. Dual‑unitary circuits (DUCs), a special subclass satisfying a space‑time duality condition, allow exact analytical treatment of operator entanglement, two‑point functions, and entanglement growth. In DUCs, the entanglement entropy saturates to its maximal value after a few time steps, independent of the initial state, and the dynamics can be reduced to a simple transfer‑matrix problem.

Spectral properties are addressed via the spectral form factor K(t)=|Tr U^t|²/N. For DUCs, K(t) follows the universal random‑matrix theory (RMT) prediction, indicating fully chaotic quantum dynamics. By contrast, circuits with constrained gate sets or additional symmetries display deviations from RMT, revealing semi‑integrable behavior. This provides a concrete diagnostic for quantum chaos versus integrability within the circuit framework.

Overall, the author argues that BQCs serve as a powerful, analytically tractable, and numerically efficient platform bridging quantum information, statistical mechanics, and condensed‑matter physics. They retain the essential locality and causal structure of Hamiltonian systems while permitting exact calculations in non‑trivial interacting settings. The notes conclude with outlooks on higher‑dimensional generalizations, hybrid circuits involving measurements, and experimental realizations on near‑term quantum processors.