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arXiv:2001.03421 · Physical Review A

Error bounds for constrained dynamics in gapped quantum systems: Rigorous results and generalizations

Zongping Gong, Nobuyuki Yoshioka, Naoyuki Shibata, Ryusuke Hamazaki

Published May 27, 2020• 16 citations
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Abstract

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.

DOI
10.1103/physreva.101.052122

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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.
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.
We present perturbative error bounds for the non-Markovian dynamics of observables in open quantum systems interacting with Gaussian environments, governed by a general Liouville dynamics. This extends the work of [Mascherpa et al., Phys. Rev. Lett. 118, 100401, 2017], which demonstrated qualitatively tighter bounds over the standard Grönwall-type inequality for unitary system-bath evolution. Our results apply to systems with both bosonic and fermionic environments. Our approach utilizes a superoperator formalism, which avoids the need for formal coherent state path integral calculations, or the dilation of Lindblad dynamics into an equivalent unitary framework with infinitely many degrees of freedom. This enables a unified treatment of a wide range of open quantum systems. These findings provide a solid theoretical basis for various recently developed pseudomode methods in simulating open quantum system dynamics.
A Dissertation on open quantum systems, containing a general review of the current standing of the field, in particular regarding systems interacting with Gaussian environments, as well as original analytical and numerical results. The problem of the impact of a variation in the environmental properties on the reduced dynamics of an open system is investigated analytically, and a new computational technique for the simulation of non-perturbative, non-Markovian open quantum system dynamics based on devising error-certified effective environments is presented.
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.
Within quantum information, many methods have been proposed to avoid or correct the deleterious effects of the environment on a system of interest. In this work, expanding on our earlier paper (Paz-Silva et al 2012 Phys. Rev. Lett. 108 080501), we evaluate the applicability of the quantum Zeno effect as one such method. Using the algebraic structure of stabilizer quantum error correction codes as a unifying framework, two open-loop protocols are described which involve frequent non-projective (i.e. weak) measurement of either the full stabilizer group or a minimal generating set thereof. The effectiveness of the protocols is measured by the distance between the final state under the protocol and the final state of an idealized evolution in which system and environment do not interact. Rigorous bounds on this metric are derived which demonstrate that, under certain assumptions, a Zeno effect may be realized with arbitrarily weak measurements, and that this effect can protect an arbitrary, unknown encoded state against the environment arbitrarily well.
Within quantum information, many methods have been proposed to avoid or correct the deleterious effects of the environment on a system of interest. In this work, expanding on our earlier paper (Paz-Silva et al 2012 Phys. Rev. Lett. 108 080501), we evaluate the applicability of the quantum Zeno effect as one such method. Using the algebraic structure of stabilizer quantum error correction codes as a unifying framework, two open-loop protocols are described which involve frequent non-projective (i.e. weak) measurement of either the full stabilizer group or a minimal generating set thereof. The effectiveness of the protocols is measured by the distance between the final state under the protocol and the final state of an idealized evolution in which system and environment do not interact. Rigorous bounds on this metric are derived which demonstrate that, under certain assumptions, a Zeno effect may be realized with arbitrarily weak measurements, and that this effect can protect an arbitrary, unknown encoded state against the environment arbitrarily well.