Exactly solvable many-body dynamics from space-time duality

TL;DR

$Dual-unitary circuits provide exact analytic control of quantum many-body dynamics by enforcing unitarity in both time and space, enabling complete characterization of light-cone correlations, scrambling, and entanglement growth. The review surveys diagrammatic formalisms, parameterisations, ergodicity classifications, spectral statistics, solvable states, and connections to quantum computation, with a focal example being the self-dual kicked Ising model. It shows that correlations outside the light cone vanish, light-cone correlations map to quantum channels $\mathcal{M}_+$ and $\mathcal{M}_-$, and maximally ergodic gates achieve maximal entanglement velocity and chaotic spectra, while nonergodic circuits host solitons and generalized Gibbs ensembles. The space-time duality framework extends to open dynamics, higher dimensions, and biunitary constructions, offering insights into deep thermalisation, state designs, and experimentally realizable dynamics.$

Abstract

Recent years have seen significant advances, both theoretical and experimental, in our understanding of quantum many-body dynamics. Given this problem's high complexity, it is surprising that a substantial amount of this progress can be ascribed to exact analytical results. Here we review dual-unitary circuits as a particular setting leading to exact results in quantum many-body dynamics. Dual-unitary circuits constitute minimal models in which space and time are treated on an equal footings, yielding exactly solvable yet possibly chaotic evolution. They were the first in which current notions of quantum chaos could be analytically quantified, allow for a full characterisation of the dynamics of thermalisation, scrambling, and entanglement (among others), and can be experimentally realised in current quantum simulators. Dual-unitarity is a specific fruitful implementation of the more general idea of space-time duality in which the roles of space and time are exchanged to access relevant dynamical properties of quantum many-body systems.

Figures (20)

• Figure 1: Dynamical correlations vanish outside a causal light cone along the vertical direction (orange) due to unitarity in time, and outside a causal light cone along the horizontal direction (green) due to unitarity in space, such that ultralocal correlations in dual-unitary circuits can only be nonzero on the overlapping edges of both causal light cones.
• Figure 2: Evolution of correlation functions $c_{\alpha \beta}(t,t)$, where $\sigma_{\alpha,\beta} \in \mathbbm{C}^{q \times q}$ are randomly generated matrices with $\tr(\sigma_{\alpha})=1$ and $\tr(\sigma_{\beta})=0$ leading to a thermal value $c_{\alpha \beta}(t,t) \to 0$. Local Hilbert space dimension $q=4$ and $4$ different operators are considered. Insets show the eigenspectrum of $\mathcal{M}_+$ on the unit disc. Based on a similar figure in claeys_ergodic_2021.
• Figure 3: 'Boundary Chaos': Equivalence between a boundary-perturbed SWAP circuit of size $L$ and duration $t$ and a $2$-dimensional network on a helix of size $t/L \times L$. We show an example of a 2-point time correlator between local boundary observables $a$ and $b$ in the operator-space/folded representation. Adapted from fritzsch_boundary_2021.
• Figure 4: Spectral function of an operator $A$ as an extensive sum of 1-local observables (cf. Eq. \ref{['eq:extensive_sum']}) for five representative realizations of a dual-unitary circuit. The corresponding asymptotic expressions (cf. Eq. \ref{['eq:spectralfunction_asymptotic']}) are depicted as dashed black lines. Adapted from fritzsch_eigenstate_2021.
• Figure 5: Operator Rényi entropies $S^{(\alpha)}_{\rm Op}(A, t)$ of the reduced density matrix $\rho_A(t)$ after a quench from the Bell state in Eq. \ref{['eq:Bell']}. In Region I, ${t \leq (\ell-x)/2}$, the evolution is universal. In Region II, ${(\ell-x)/2\leq t \leq \ell/2}$, the behaviour depends on the dual-unitary gate and Rényi index $\alpha$. Adapted from reid2021entanglement.

Theorems & Definitions (5)

• Theorem : Equivalent MPS piroli_exact_2020
• Theorem : Theorem 1 bertini2021random
• Lemma : Leading Eigenvalues bertini2021random
• Lemma : Algebraic Reformulation bertini2021random
• Theorem : Theorem 2, ho_exact_2022