We discuss a few mathematical aspects of random dynamical decoupling, a key tool procedure in quantum information theory. In particular, we place it in the context of discrete stochastic processes, limit theorems and CPT semigroups on matrix algebras. We obtain precise analytical expressions for expectation and variance of the density matrix and fidelity over time in the continuum-time limit depending on the system Lindbladian, which then lead to rough short-time estimates depending only on certain coupling strengths. We prove that dynamical decoupling does not work in the case of intrinsic (i.e., not environment-induced) decoherence, and together with the above estimates this yields a novel method of partially identifying intrinsic decoherence.We discuss a few mathematical aspects of random dynamical decoupling, a key tool procedure in quantum information theory. In particular, we place it in the context of discrete stochastic processes, limit theorems and CPT semigroups on matrix algebras. We obtain precise analytical expressions for expectation and variance of the density matrix and fidelity over time in the continuum-time limit depending on the system Lindbladian, which then lead to rough short-time estimates depending only on certain coupling strengths. We prove that dynamical decoupling does not work in the case of intrinsic (i.e., not environment-induced) decoherence, and together with the above estimates this yields a novel method of partially identifying intrinsic decoherence.

Apr 17, 20156 citations

Effects of stochastic noise on dynamical decoupling procedures

József Zsolt Bernád, Holger Frydrych

Dynamical decoupling is an important tool to counter decoherence and dissipation effects in quantum systems originating from environmental interactions. It has been used successfully in many experiments; however, there is still a gap between fidelity improvements achieved in practice compared to theoretical predictions. We propose a model for imperfect dynamical decoupling based on a stochastic Ito differential equation which could explain the observed gap. We discuss the impact of our model on the time evolution of various quantum systems in finite- and infinite-dimensional Hilbert spaces. Analytical results are given for the limit of continuous control, whereas we present numerical simulations and upper bounds for the case of finite control.Dynamical decoupling is an important tool to counter decoherence and dissipation effects in quantum systems originating from environmental interactions. It has been used successfully in many experiments; however, there is still a gap between fidelity improvements achieved in practice compared to theoretical predictions. We propose a model for imperfect dynamical decoupling based on a stochastic Ito differential equation which could explain the observed gap. We discuss the impact of our model on the time evolution of various quantum systems in finite- and infinite-dimensional Hilbert spaces. Analytical results are given for the limit of continuous control, whereas we present numerical simulations and upper bounds for the case of finite control.

arXiv:1405.1852Physical Review AJun 23, 20143 citations

Mixing properties of stochastic quantum Hamiltonians

Emilio Onorati, Oliver Buerschaper, Martin Kliesch +3 more

Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications.Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications.

Jan 1, 201741 citations

Stochasticity, decoherence and an arrow of time from the discretization of time?

M. C. Valsakumar

Certain intriguing consequences of the discreteness of time on the time evolution of dynamical systems are discussed. In the discrete-time classical mechanics proposed here, there is an {\\it arrow of time} that follows from the fact that the replacement of the time derivative by the backward difference operator alone can preserve the non-negativity of the phase space density. It is seen that, even for free particles, all the degrees of freedom are {\\it correlated} in principle. The forward evolution of functions of phase space variables by a finite number of time steps, in this discrete-time mechanics, depends on the entire continuous-time history in the interval $[0, \\infty]$ . In this sense, discrete time evolution is {\\it nonlocal} in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible {\\it via} the density matrix approach. The interference between non-degenerate quantum mechanical states decays exponentially. This {\\it decoherence} is present, in principle, for all systems; however, it is of practical importance only in macroscopic systems, or in processes involving large energy changes.Certain intriguing consequences of the discreteness of time on the time evolution of dynamical systems are discussed. In the discrete-time classical mechanics proposed here, there is an {\\it arrow of time} that follows from the fact that the replacement of the time derivative by the backward difference operator alone can preserve the non-negativity of the phase space density. It is seen that, even for free particles, all the degrees of freedom are {\\it correlated} in principle. The forward evolution of functions of phase space variables by a finite number of time steps, in this discrete-time mechanics, depends on the entire continuous-time history in the interval $[0, \\infty]$. In this sense, discrete time evolution is {\\it nonlocal} in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible {\\it via} the density matrix approach. The interference between non-degenerate quantum mechanical states decays exponentially. This {\\it decoherence} is present, in principle, for all systems; however, it is of practical importance only in macroscopic systems, or in processes involving large energy changes.

May 7, 20053 citations

Random Processes over the Unitary Group

Emilio Onorati

According to one of the fundamental axioms of quantum mechanics, unitary operators rule the evolution of any quantum system; it is thus of prominent importance to investigate the corresponding group and discover its properties. Random processes over the unitary group have indeed a wide range of applications in the context of quantum information; in particular, they are involved in the construction of peculiar distributions called unitary designs which mimic the uniform Haar measure by matching its moments and are thus deeply connected to the description of phenomena such as quantum tomography, equilibration, thermalization, encryption and scrambling. Previous approaches have shown that unitary designs are efficiently approximated by random quantum circuits with local unitary operators. Moreover, these circuits are used to characterize the precision of experimental implementations of unitary gates via randomized benchmarking protocols and to rapidly decouple a system from the environment. The main novel contribution of this work is to extend these mixing properties to a continuous-time framework, namely, Brownian motion over the unitary group induced by stochastic local Hamiltonians. In order to achieve these new results, on the one hand we move to a representation theoretic formulation and make use of its tools to establish the gap of local generators linked to the moments of the distribution induced by the diffusion process; from this we then derive an expression for the length of time it takes to ensure convergence toward the moments of the Haar measure. On the other hand we project the stochastic evolution onto a random walk on Pauli matrices and tie this description to the analogous one for random quantum circuits to achieve decoupling in a run time scaling almost linearly with respect to the system size. In addition to providing a unifying framework for random processes over the unitary group, we hence aim at presenting new mathematical results and techniques for quantum information. We furthermore discuss applications to black holes dynamics in the perspective of the information paradox. As an additional novel result not related to Brownian motion, we propose a randomized benchmarking protocol exploiting the symmetry of the gate whose accuracy has to be estimated, in order to overcome current shortcomings afflicting the known schemes. We again rely on representation theory to reduce the computational effort and the amount of employed quantum resources.According to one of the fundamental axioms of quantum mechanics, unitary operators rule the evolution of any quantum system; it is thus of prominent importance to investigate the corresponding group and discover its properties. Random processes over the unitary group have indeed a wide range of applications in the context of quantum information; in particular, they are involved in the construction of peculiar distributions called unitary designs which mimic the uniform Haar measure by matching its moments and are thus deeply connected to the description of phenomena such as quantum tomography, equilibration, thermalization, encryption and scrambling. Previous approaches have shown that unitary designs are efficiently approximated by random quantum circuits with local unitary operators. Moreover, these circuits are used to characterize the precision of experimental implementations of unitary gates via randomized benchmarking protocols and to rapidly decouple a system from the environment. The main novel contribution of this work is to extend these mixing properties to a continuous-time framework, namely, Brownian motion over the unitary group induced by stochastic local Hamiltonians. In order to achieve these new results, on the one hand we move to a representation theoretic formulation and make use of its tools to establish the gap of local generators linked to the moments of the distribution induced by the diffusion process; from this we then derive an expression for the length of time it takes to ensure convergence toward the moments of the Haar measure. On the other hand we project the stochastic evolution onto a random walk on Pauli matrices and tie this description to the analogous one for random quantum circuits to achieve decoupling in a run time scaling almost linearly with respect to the system size. In addition to providing a unifying framework for random processes over the unitary group, we hence aim at presenting new mathematical results and techniques for quantum information. We furthermore discuss applications to black holes dynamics in the perspective of the information paradox. As an additional novel result not related to Brownian motion, we propose a randomized benchmarking protocol exploiting the symmetry of the gate whose accuracy has to be estimated, in order to overcome current shortcomings afflicting the known schemes. We again rely on representation theory to reduce the computational effort and the amount of employed quantum resources.

Jan 1, 20190 citations

Can quantum Markov evolutions ever be dynamically decoupled?

John Edward Gough, Hendra I. Nurdin

We consider the class of quantum stochastic evolutions (SLH-models) leading to a quantum dynamical semigroup over a fixed quantum mechanical system (taken to be finite-dimensional). We show that if the semigroup is dissipative, that is, the coupling operators are non-zero, then a dynamical decoupling scheme based on unitary rotations on the system space cannot suppress decoherence even in the limit where the period between pulses vanishes. We emphasize the role of the Fock space dilation used here to construct a quantum stochastic model, as there are often dilations of the same semigroup using an environmental noise model of lower level of chaoticity for which dynamical decoupling is effective. We show that the Chebotarev-Gregoratti Hamiltonian behind a quantum stochastic evolution is an example of a Hamiltonian dynamics on a joint system-environment that cannot be dynamically decoupled in this way.We consider the class of quantum stochastic evolutions (SLH-models) leading to a quantum dynamical semigroup over a fixed quantum mechanical system (taken to be finite-dimensional). We show that if the semigroup is dissipative, that is, the coupling operators are non-zero, then a dynamical decoupling scheme based on unitary rotations on the system space cannot suppress decoherence even in the limit where the period between pulses vanishes. We emphasize the role of the Fock space dilation used here to construct a quantum stochastic model, as there are often dilations of the same semigroup using an environmental noise model of lower level of chaoticity for which dynamical decoupling is effective. We show that the Chebotarev-Gregoratti Hamiltonian behind a quantum stochastic evolution is an example of a Hamiltonian dynamics on a joint system-environment that cannot be dynamically decoupled in this way.

arXiv:1710.11133IEEE Conference on Decision and Control (CDC)Dec 1, 20178 citations

[0, \\infty]

. In this sense, discrete time evolution is {\\it nonlocal} in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible {\\it via} the density matrix approach. The interference between non-degenerate quantum mechanical states decays exponentially. This {\\it decoherence} is present, in principle, for all systems; however, it is of practical importance only in macroscopic systems, or in processes involving large energy changes.