Experimental noise filtering by quantum control

Abstract

Extrinsic interference is routinely faced in systems engineering, and a common solution is to rely on a broad class of filtering techniques to a ord stability to intrinsically unstable systems or isolate particular signals from a noisy background. Experimentalists leading the development of a new generation of quantum-enabled technologies similarly encounter time-varying noise in realistic laboratory settings. They face substantial challenges in either suppressing such noise for high-fidelity quantum operations 1 or controllably exploiting it in quantum-enhanced sensing [2][3][4] or system identification tasks 5,6 , due to a lack of e cient, validated approaches to understanding and predicting quantum dynamics in the presence of realistic time-varying noise. In this work we use the theory of quantum control engineering . We demonstrate the utility of these constructs for directly predicting the evolution of a quantum state in a realistic noisy environment as well as for developing novel robust control and sensing protocols. These experiments provide a significant advance in our understanding of the physics underlying controlled quantum dynamics, and unlock new capabilities for the emerging field of quantum systems engineering.Time-varying noise coupled to quantum systems-typically qubits-generically results in decoherence, or a loss of 'quantumness' of the system. Broadly, one may think of the state of the quantum system becoming randomized through uncontrolled (and often uncontrollable) interactions with the environment during both idle periods and active control operations (Fig. 1a). Despite the ubiquity of this phenomenon, it is a challenging problem to predict the average evolution of a qubit state undergoing a specific, but arbitrary operation in the presence of realistic time-dependent noise-how much randomization does one expect and how well can one perform the target operation? Making such predictions accurately is precisely the capability that experimentalists require in realistic laboratory settings. Moreover, this capability is fundamental to the development of novel control techniques designed to modify or suppress decoherence as researchers attempt to build quantum-enabled technologies for applications such as quantum information and quantum sensing.These considerations motivate the development of novel engineering-inspired analytic tools enabling a user to accurately predict the behaviour of a controlled quantum system in realistic laboratory environments. Recent work has demonstrated that the average dynamics of a controlled qubit state evolution may be captured using filter-transfer functions (FFs) characterizing the control. The fidelity of an arbitrary operation over duration τ ,, is degraded owing to frequency-domain spectral overlap between noise in the environment given by a power spectrum S(ω), and the filter-transfer functions denoted F(ω) (Methods) [11][12][13][14] . The FF description of ensemble-average quantum dynamics tremendously simplifies the task of analysing the expected performa...

General Transfer-Function Approach to Noise Filtering in Open-Loop Quantum Control

“…We prove that cancellation and filtering are inequivalent notions, with highorder cancellation in the Magnus sense not implying highorder filtering in general, and with both notions being a priori equally significant for assessing the control performance. Our results provide a firm foundation for recent analyses where this inequivalence has manifested in the context of compositepulse and Walsh-modulated protocols [13,15], as well as a new perspective on dynamical error control strategies, with potential implications for quantum fault tolerance.Control-theoretic setting.-We consider a general finitedimensional open quantum system S coupled to an uncontrollable environment (bath) B, whose free evolution is described by a joint Hamiltonian of the form H(t) = H S + H SB (t), with respect to the interaction picture associated to the physical bath Hamiltonian H B . Open-loop control is introduced via a time-dependent Hamiltonian H ctrl (t) acting on S alone, with the controlled dynamics being represented in terms of an intended plus error component, namely, H(t) + H ctrl (t) ≡ arXiv:1408.3836v2 [quant-ph]…”

“…We prove that cancellation and filtering are inequivalent notions, with highorder cancellation in the Magnus sense not implying highorder filtering in general, and with both notions being a priori equally significant for assessing the control performance. Our results provide a firm foundation for recent analyses where this inequivalence has manifested in the context of compositepulse and Walsh-modulated protocols [13,15], as well as a new perspective on dynamical error control strategies, with potential implications for quantum fault tolerance.…”

“…In this context, generalized transfer filter function (FF) techniques motivated by control engineering are providing an increasingly important tool for understanding the dynamical response of the target system in Fourier space and for quantitatively analyzing the control performance [9][10][11][12]. In particular, this formalism has proved remarkably successful in predicting operational fidelities for a variety of control settings in recent trapped-ion experiments [13], as long as noise is sufficiently weak for low-order approximations to be viable.From a control-theory standpoint, a filtering approach to dynamical error suppression is desirable for a variety of reasons. Besides providing an open-loop counterpart to the transfer-function perspective that is central to both classical and quantum feedback networks [14], FF techniques allow, when available, for a substantially more efficient analysis of the underlying noisy dynamics than direct simulation [15,16].…”

“…In this context, generalized transfer filter function (FF) techniques motivated by control engineering are providing an increasingly important tool for understanding the dynamical response of the target system in Fourier space and for quantitatively analyzing the control performance [9][10][11][12]. In particular, this formalism has proved remarkably successful in predicting operational fidelities for a variety of control settings in recent trapped-ion experiments [13], as long as noise is sufficiently weak for low-order approximations to be viable.…”

“…Thus, explicit calculations have largely focused thus far on single-qubit controlled dynamics in the presence of classical noise, by truncating this recursion to the lowest order and additionally exploiting Gaussian noise statistics. As experiments are revealing the importance of higher-order terms outside the weak-noise regime [13], and are rapidly progressing toward coupled-qubit systems and/or more complex, non-Gaussian error models [18], overcoming these limitations becomes imperative for further progress. An additional shortcoming arises from the fact that only in very special situations where a single FF suffices to characterize the controlled dynamics, one may unambiguously identify the "order of error suppression" of the protocol, associated to the order of cancellation in the Magnus expansion, with its "filtering order", determined by the low-frequency behavior of the FF [4,12].…”