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arXiv:2404.15223 · Open Systems & Information Dynamics

Effective Dynamics of Qubit Networks via Phase-Covariant Quantum Ensembles

Sean Prudhoe, Unnati Akhouri, Tommy Chin, Sarah E. Shandera

Published Dec 1, 2024• 1 citation
arXivHTMLPDF

Abstract

We derive a new constructive procedure to rapidly generate ensembles of phase-covariant dynamical maps that may be associated to the individual spins of a closed quantum system. We do this by first computing the single-spin dynamical maps in small XXZ networks and chains, specialized to the class of initial states that guarantees phase-covariant dynamics for each spin. Since the dynamics in any small, closed system contains oscillatory features associated to the system size, we define an averaging procedure to extract time-homogeneous dynamics. We use the the average map and the set of deviations from the average map in the exactly derived ensembles to motivate the form of distributional functions for map parameters. The distributions then straightforwardly generate arbitrary-sized ensembles of channels, constrained by a few global properties. This procedure can also generate ensembles where individual maps are not phase-covariant although the average map is, corresponding to realizations of disordered, or noisy, Hamiltonians. The construction procedure suggests new ways to realize random families of open-system dynamics, subject to constraints that require the ensemble to approximate a partition of a closed system.

DOI
10.1142/s1230161224500161

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We derive a new constructive procedure to rapidly generate ensembles of phase-covariant dynamical maps that may be associated to the individual spins of a closed quantum system. We do this by first computing the single-spin dynamical maps in small XXZ networks and chains, specialized to the class of initial states that guarantees phase-covariant dynamics for each spin. Since the dynamics in any small, closed system contains oscillatory features associated to the system size, we define an averaging procedure to extract time-homogeneous dynamics. We use the the average map and the set of deviations from the average map in the exactly derived ensembles to motivate the form of distributional functions for map parameters. The distributions then straightforwardly generate arbitrary-sized ensembles of channels, constrained by a few global properties. This procedure can also generate ensembles where individual maps are not phase-covariant although the average map is, corresponding to realizations of disordered, or noisy, Hamiltonians. The construction procedure suggests new ways to realize random families of open-system dynamics, subject to constraints that require the ensemble to approximate a partition of a closed system.
We design several examples of constrained, symmetric quantum circuit dynamics that generate nonequilibrium steady states. The qubit networks maintain local memory of the initial conditions and display inhomogeneous subsystem dynamics over long times, clearly distinguishable from approximately thermalizing networks of the same size. Each network can be described as an ensemble of open systems, a collection of qubits evolving with phase-covariant dynamics. Constraints from the conservation law and global unitary dynamics of the entire network bound the distribution of single-qubit dynamics in the ensemble, but different steady states are distinguishable by several measures. We quantify the distance of the steady states from the homogeneous steady state and further characterize them using the complexity of their mutual information networks, the volume of state space explored, a thermodynamic utility measure using extractable work, and correlated structure in the occurrence of noncompletely positive qubit propagator maps.
Understanding how physical systems are influenced by disorder is a fundamental challenge in quantum science. Addressing its effects often involves numerical averaging over a large number of samples, and it is not always easy to gain an analytical handle on exploring the effect of disorder. In this work, we derive exact solutions for disorder-averaged dynamics generated by any Hamiltonian that is a periodic matrix (potentially with non-trivial base, a property also called (p,qp,qp,q)-potency). Notably, this approach is independent of the initial state, exact for arbitrary evolution times, and it holds for Hermitian as well as non-Hermitian systems. The ensemble behavior resembles that of an open quantum system, whose decoherence function or rates are determined by the disorder distribution and the periodicity of the Hamiltonian. Depending on the underlying distribution, the dynamics can display non-Markovian characteristics detectable through non-Markovian witnesses. We illustrate the scheme for qubit and qudit systems described by (products of) spin 1/21/21/2, spin 111, and clock operators. Our methodology offers a framework to leverage disorder-averaged exact dynamics for a range of applications in quantum-information processing and beyond.
Symmetries are a key tool in understanding quantum systems, and, among many other things, can be exploited to increase the efficiency of numerical simulations of quantum dynamics. Disordered systems usually feature reduced symmetries and additionally require averaging over many realizations, making their numerical study computationally demanding. However, when studying quantities linear in the time-evolved state, i.e., expectation values of observables, one can apply the averaging procedure to the time evolution operator, resulting in an effective dynamical map, which restores symmetry at the level of superoperators. In this work, we develop schemes for efficiently constructing symmetric sectors of the disorder-averaged dynamical map using short-time and weak-disorder expansions. To benchmark the method, we apply it to an Ising model with random all-to-all interactions in the presence of a transverse field. After disorder averaging, this system becomes effectively permutation-invariant, and thus the size of the symmetric subspace scales polynomially in the number of spins, allowing for the simulation of large systems.
We obtain exact analytic expressions for a class of functions expressed as integrals over the Haar measure of the unitary group in d dimensions. Based on these general mathematical results, we investigate generic dynamical properties of complex open quantum systems, employing arguments from ensemble theory. We further generalize these results to arbitrary eigenvalue distributions, allowing a detailed comparison of typical regular and chaotic systems with the help of concepts from random matrix theory. To illustrate the physical relevance and the general applicability of our results we present a series of examples related to the fields of open quantum systems and nonequilibrium quantum thermodynamics. These include the effect of initial correlations, the average quantum dynamical maps, the generic dynamics of system-environment pure state entanglement and, finally, the equilibration of generic open and closed quantum systems.
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Abstract We develop a formalism for mapping the exact dynamics of an ensemble of disordered quantum systems onto the dynamics of a single particle propagating along a semi-infinite lattice, with interactions determined by the probability distribution of disorder realisations of the original heterogeneous quantum ensemble. This provides a geometric interpretation of the loss of coherence when averaging over the ensemble and allows computation of the observables of the ensemble in a single simulation. Conversely, the reverse map implies lattice particle dynamics to be equivalent to the dynamics of an ensemble of disordered quantum systems. The potential of the equivalence is demonstrated by computing the exact disorder-averaged density matrix dynamics for a qubit and a dimer ensemble, and interpreting dynamics of a tight-binding chain with constant and staggered hoppings in terms of an ensemble of disorder realisations.