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arXiv:2103.02403 · Physical Review Research

Filter-function formalism and software package to compute quantum processes of gate sequences for classical non-Markovian noise

Tobias Hangleiter, Pascal Cerfontaine, Hendrik Bluhm

Published Oct 18, 2021• 17 citations
arXivHTMLPDF

Abstract

Correlated, non-Markovian noise is present in many solid-state systems employed as hosts for quantum information technologies, significantly complicating the realistic theoretical description of these systems. In this regime, the effects of noise on sequences of quantum gates cannot be described by concatenating isolated quantum operations if the environmental correlation times are on the scale of the typical gate durations. The filter-function formalism has been successful in characterizing the decay of coherence under the influence of such classical, non-Markovian environments, and here we show it can be applied to describe unital evolution within the quantum operations formalism. We find exact results for the quantum process and a simple composition rule for a sequence of operations. This enables the detailed study of effects of noise correlations on algorithms and periodically driven systems. Moreover, we point out the method's suitability for numerical applications and present the open-source Python software package filter_functions. Amongst other things, it facilitates computing the noise-averaged transfer matrix representation of a unital quantum operation in the presence of universal classical noise for arbitrary control sequences. We apply the presented methods to selected examples.

DOI
10.1103/physrevresearch.3.043047

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Correlated, non-Markovian noise is present in many solid-state systems employed as hosts for quantum information technologies, significantly complicating the realistic theoretical description of these systems. In this regime, the effects of noise on sequences of quantum gates cannot be described by concatenating isolated quantum operations if the environmental correlation times are on the scale of the typical gate durations. The filter-function formalism has been successful in characterizing the decay of coherence under the influence of such classical, non-Markovian environments, and here we show it can be applied to describe unital evolution within the quantum operations formalism. We find exact results for the quantum process and a simple composition rule for a sequence of operations. This enables the detailed study of effects of noise correlations on algorithms and periodically driven systems. Moreover, we point out the method's suitability for numerical applications and present the open-source Python software package filter_functions. Amongst other things, it facilitates computing the noise-averaged transfer matrix representation of a unital quantum operation in the presence of universal classical noise for arbitrary control sequences. We apply the presented methods to selected examples.
Many qubit implementations are afflicted by correlated noise not captured by standard theoretical tools that are based on Markov approximations. While independent gate operations are a key concept for quantum computing, it is actually not possible to fully describe noisy gates locally in time if noise is correlated on times longer than their duration. To address this issue, we develop a method based on the filter function formalism to perturbatively compute quantum processes in the presence of correlated classical noise. We derive a composition rule for the filter function of a sequence of gates in terms of those of the individual gates. The joint filter function allows us to efficiently compute the quantum process of the whole sequence. Moreover, we show that correlation terms arise which capture the effects of the concatenation and, thus, yield insight into the effect of noise correlations on gate sequences. Our generalization of the filter function formalism enables both qualitative and quantitative studies of algorithms and state-of-the-art tools widely used for the experimental verification of gate fidelities like randomized benchmarking, even in the presence of noise correlations.
Autocorrelated noise appears in many solid-state qubit systems and hence needs to be taken into account when developing gate operations for quantum information processing. However, explicitly simulating this kind of noise is often less efficient than approximate methods. Here, we focus on the filter function formalism, which allows the computation of gate fidelities in the presence of autocorrelated classical noise. Hence, this formalism can be combined with optimal control algorithms to design control pulses, which optimally implement quantum gates. To enable the use of gradient-based algorithms with fast convergence, we present analytically derived filter function gradients with respect to control pulse amplitudes, and analyze the computational complexity of our results. When comparing pulse optimization using our derivatives to a gradient-free approach, we find that the gradient-based method is roughly 2 orders of magnitude faster for our test cases. We also provide a modular computational implementation compatible with quantum optimal control packages.
Treating the effects of a time-dependent classical dephasing environment during quantum logic operations poses a theoretical challenge, as the application of noncommuting control operations gives rise to both dephasing and depolarization errors that must be accounted for in order to understand total average error rates. We develop a treatment based on effective Hamiltonian theory that allows us to efficiently model the effect of classical noise on nontrivial single-bit quantum logic operations composed of arbitrary control sequences. We present a general method to calculate the ensemble-averaged entanglement fidelity to arbitrary order in terms of noise filter functions, and provide explicit expressions to fourth order in the noise strength. In the weak noise limit we derive explicit filter functions for a broad class of piecewise-constant control sequences, and use them to study the performance of dynamically corrected gates, yielding good agreement with brute-force numerics.
The main obstacles to implementing ideal quantum operations are unwanted interactions of quantum systems with the environment and noise in control fields. This problem can be tackled by methods of quantum control. Among these methods are \ncomposite pulse (CP) sequences which have long been employed in nuclear magnetic resonance (NMR) to mitigate the effects of systematic errors in the control. CP sequences have been initially developed to correct for static but otherwise unknown \nerrors in the amplitude or frequency of the driving field. One of the challenges to the systematic incorporation of these control protocols into practical quantum information systems remains the limited understanding of CP performance in the presence \nof time-dependent noise. Treating the influence of time-dependent noise processes on quantum control operations has been facilitated by recent advances in dynamical \nerror suppression based on open-loop Hamiltonian engineering. These approaches provide a general framework for understanding and mitigating non-Markovian time- \ndependent noise in a finite-dimensional open quantum system. Particularly, arbitrary single-qubit control characteristics may be captured quantitatively in filter-transfer \nfunctions (FF) using methods of spectral overlap in the frequency domain. In this thesis work, we present a systematic study of control pulse sequences in the presence of time-dependent noise. We use a combination of analytic formulations based on FFs and numerical simulations to demonstrate that CPs are able to effectively suppress control errors caused by time-dependent processes possessing realistic noise \npower spectra. We provide a geometric interpretation of CP performance under time-dependent amplitude noise, further linking the FF formalism with known techniques in CP construction. We also develop new optimized pulse sequences that act as notch filters for time-dependent noise. These high-fidelity control protocols effectively sup- \npress errors from the noise sources with sharp features in spectral densities and can be used practically on various quantum architectures. We also present our work \non simulation of randomized benchmarking protocols and CPs that have been used experimentally by our collaborators to measure gate errors.
A major problem in exploiting microscopic systems for developing a new technology based on the principles of quantum information is the influence of noise, which tends to work against the quantum features of such systems. It thus becomes crucial to understand how noise affects the evolution of quantum circuits: several techniques have been proposed, among which stochastic differential equations (SDEs) can represent a very convenient tool. We show how SDEs naturally map any Markovian noise into a linear operator, which we call a noise gate, acting on the wave function describing the state of the circuit, and we discuss some examples. We show that these gates can be manipulated like any standard quantum gate, thus simplifying in certain circumstances the task of computing the overall effect of the noise at each stage of the protocol. This approach yields equivalent results to those derived from the Lindblad equation; yet, as we show, it represents a handy and fast tool for performing computations, and, moreover, it allows for fast numerical simulations and generalizations to non-Markovian noise. In detail we review the depolarizing channel and the generalized amplitude damping channel in terms of this noise gate formalism and show how these techniques can be applied to any quantum circuit.
In quantum dynamics, symmetries are vital for identifying and assessing conserved quantities that govern the evolution of a quantum system. When promoted to the open quantum system setting, dynamical symmetries can be negatively altered by system-environment interactions, thus, complicating their analysis. Previous work on noisy symmetric quantum dynamics has focused on the Markovian setting, despite the ubiquity of non-Markovian noise in a number of widely used quantum technologies. In this Letter, we develop a framework for quantifying the impact of non-Markovian noise on symmetric quantum evolution via root space decompositions and the filter function formalism. We demonstrate analytically that symmetry-preserving noise maintains the symmetric subspace, while nonsymmetric noise leads to highly specific leakage errors that are block diagonal in the symmetry representation. We support our findings with numerical studies of a transverse-field Ising model and a quantum error detecting code subject to spatiotemporally correlated multiaxis noise. Our results are broadly applicable, providing new analytic insights into the control and characterization of open quantum system dynamics.