Universal Error Bound for Constrained Quantum Dynamics

It is well known in quantum mechanics that a large energy gap between a Hilbert subspace of specific interest and the remainder of the spectrum can suppress transitions from the quantum states inside the subspace to those outside due to additional couplings that mix these states, and thus approximately lead to a constrained dynamics within the subspace. While this statement has widely been used to approximate quantum dynamics in various contexts, a general and quantitative justification stays lacking. Here we establish an observable-based error bound for such a constrained-dynamics approximation in generic gapped quantum systems. This universal bound is a linear function of time that only involves the energy gap and coupling strength, provided that the latter is much smaller than the former. We demonstrate that either the intercept or the slope in the bound is asymptotically saturable by simple models. We generalize the result to quantum many-body systems with local interactions, for which the coupling strength diverges in the thermodynamic limit while the error is found to grow no faster than a power law $t^{d+1}$ in $d$ dimensions. Our work establishes a universal and rigorous result concerning nonequilibrium quantum dynamics.

Figure 1

(a) Schematic illustration of the setup. The original Hamiltonian $H_0$ has an isolated energy band onto which the projector is $P$, whose complement is $Q=1-P$. A general coupling can be decomposed into $V=V_P+V_Q+V_{\text{off}}$, where $V_P\equiv PVP$, $V_Q\equiv QVQ$, and $V_{\text{off}}\equiv PVQ+QVP$. The arrow within $P$ refers to a constrained dynamics. (b) Typical energy spectrum of $H_0$ with an isolated band ${\ell }_0$. The energy gap is defined as ${\mathrm{\Delta }}_0\equiv \min \{{\mathrm{\Delta }}_u,{\mathrm{\Delta }}_d\}$. (c) Error dynamics (2) in a random model with ${\mathrm{\Delta }}_0=10\parallel V\parallel$. Different colors correspond to different realizations. The rectangle and the arrow indicate the initial sudden jump and the subsequent linear growth, respectively. Inset: Enlargement of the initial jump.

Figure 2

“Worst” models which realize the saturation of (a) the intercept and (b) the slope in Eq. (3). (c) and (d) are the corresponding error dynamics (green solid curves) of (a) and (b) given in Eqs. (4) and (5) with ${\mathrm{\Delta }}_0=10\mathrm{\Omega }$. The black dotted line indicates the intercept and the red dashed lines are the asymptotic bound (3).

Figure 3

(a) An array of two-level atoms with interaction ${\mathrm{\Delta }}_0$ between adjacent excited states. Each atom is resonantly driven with an identical Rabi frequency $\mathrm{\Omega }$. (b) The quadratic growth of $\epsilon (t)$ concerning the error dynamics of $O={\sigma }_{j=1}^y$ in the parent Hamiltonian of the PXP model defined by Eq. (17) with ${\log }_{10}{\mathrm{\Delta }}_0=1.0$, 1.5, 2.0, 2.5. The best-fitting quadratic curves (gray dotted lines) are confirmed to be more accurate than the linear ones, while the growth becomes rather linear after the correlation spreads throughout the system at $\mathrm{\Omega }t\sim 12$. The Rabi frequency is $\mathrm{\Omega }=2$ and the system size is $L=12$. Inset: Rescaled error ${\mathrm{\Delta }}_0\epsilon (t)$, whose quasi-independence on ${\mathrm{\Delta }}_0$ is consistent with Eq. (15).