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.
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.
In the accompanying letter [Gong et al., Phys. Rev. Lett. 124, 210606 (2020)], we introduced a universal error bound for constrained unitary dynamics within a well-gapped energy band of an isolated quantum system. Here, we provide the full details on the derivation of the bound. In addition, we generalize the result to isolated quantum many-body systems by employing the local Schrieffer-Wolff transformation, obtaining an error bound that grows polynomially in time. We also generalize the result to Markovian open quantum systems and quantitatively explain the quantum Zeno effect.
Finite-size errors (FSEs), the discrepancies between an observable in a finite system and in the thermodynamic limit, are ubiquitous in numerical simulations of quantum many-body systems.Although a rough estimate of these errors can be obtained from a sequence of finite-size results, a strict, quantitative bound on the magnitude of FSE is still missing.Here we derive rigorous upper bounds on the FSE of local observables in real-time quantum dynamics simulations initialized from a product state.In d-dimensional locally interacting systems with a finite local Hilbert space, our bound implies | Ŝ(t ) L -Ŝ(t ) ∞ | C(2vt/L) cL-μ , with v, C, c, μ constants independent of L and t, which we compute explicitly.For periodic boundary conditions (PBCs), the constant c is twice as large as that for open boundary conditions (OBCs), suggesting that PBCs have smaller FSEs than OBCs at early times.The bound can be generalized to a large class of correlated initial states as well.As a byproduct, we prove that the FSE of local observables in ground-state simulations decays exponentially with L under a suitable spectral gap condition.Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional (1D) quantum Ising and Fermi-Hubbard models.
We bound the error in ground-state expectations of local observables caused by truncating the spatial tails of a gapped quantum lattice Hamiltonian. We show that the error is controlled by the interaction strength discarded near each site, rather than by the extensive norm of the omitted Hamiltonian. If the gap remains open along a path that removes the interaction tail, the resulting bounds are uniform in system size and extend, under suitable assumptions, to thermodynamic-limit ground states. The convergence rate reflects the decay of the interaction tail: it is algebraic for power-law interactions, superpolynomial for superpolynomial interactions, and exponential at any strictly smaller rate for exponentially decaying interactions. For superpolynomial interactions, we also derive a direct infinite-volume estimate using automorphic equivalence. The results extend to isolated low-energy sectors and parity-even fermionic systems. For two-body interactions decaying as r−p in d dimensions, we prove an error bound O(R−(p−d)) for p>2d, where R is the truncation range. We construct a gapped non-translation-invariant example that saturates this scaling, showing that the bound is optimal for the general class considered. Our results quantify when finite-range truncations faithfully reproduce the local physics of gapped long-range systems.
One of the main methods for protecting quantum information against decoherence is to encode information in the ground subspace (or the low energy sector) of a Hamiltonian with a large energy gap which penalizes errors from environment. The protecting Hamiltonian is chosen such that its degenerate ground subspace is an error detecting code for the errors caused by the interaction with the environment. We consider environments with arbitrary number of local sites, e.g. spins, whose interactions among themselves are local and bounded. Then, assuming the system is interacting with a finite number of sites in the environment, we prove that, up to second order with respect to the coupling constant, decoherence and relaxation are suppressed by a factor which grows exponentially fast with the ratio of energy penalty to the norm of local interactions in the environment. The state may, however, still evolve unitarily inside the code subspace due to the Lamb shift effect. In the context of adiabatic quantum computation, this means that the evolution inside the code subspace is effectively governed by a renormalized Hamiltonian. The result is derived from first principles, without use of master equations or their assumptions, and holds even in the infinite temperature limit. We also prove that unbounded or non-local interactions in the environment at sites far from the system do not considerably modify the exponential suppression. Our main technical tool is a new bound on the decay of power spectral density at high frequencies for local observables and many-body Hamiltonians with bounded and local interactions in a neighborhood around the support of the observable.
It is a central problem in various fields of physics to elucidate the behavior of quantum many-body systems subjected to bulk dissipation. In this context, several microscopic derivations of the Lindblad quantum master equation for many-body systems have been proposed so far. Among them, the universal Lindblad equation derived by Nathan and Rudner is fascinating because it has the desired locality and its derivation seems to rely solely on the assumption that the bath correlation time is much shorter than the dissipation time, which is the case in the weak-coupling limit or the singular-coupling limit. However, it remains elusive whether errors from several approximations in deriving the universal Lindblad equation remain small during the time evolution in the thermodynamic limit. Here, rigorous error bounds on the time evolution of a local quantity are given, and it is shown that, under the assumption of accelerated dissipation in bulk-dissipated systems, those errors vanish in the weak-coupling limit or the singular-coupling limit after taking the thermodynamic limit.
Many physical systems of interest that are encountered in practice are input-output open quantum systems described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with the error in the finite-dimensional approximation of input-output open quantum systems defined on an infinite-dimensional underlying Hilbert space. We present explicit error bounds between the time evolution of the state of a class of infinite-dimensional quantum systems and its approximation on a finite-dimensional subspace of the original, when both are initialized in the latter subspace. Application to a physical example drawn from the literature is provided to illustrate our results.