Abstract
Extrinsic interference is routinely faced in systems engineering, and a common solution is to rely on a broad class of filtering techniques to afford stability to intrinsically unstable systems or isolate particular signals from a noisy background. Experimentalists leading the development of a new generation of quantumenabled technologies similarly encounter timevarying noise in realistic laboratory settings. They face substantial challenges in either suppressing such noise for highfidelity quantum operations^{1} or controllably exploiting it in quantumenhanced sensing^{2,3,4} or system identification tasks ^{5,6}, due to a lack of efficient, validated approaches to understanding and predicting quantum dynamics in the presence of realistic timevarying noise. In this work we use the theory of quantum control engineering^{7,8} and experiments with trapped ^{171}Yb^{+} ions to study the dynamics of controlled quantum systems. Our results provide the first experimental validation of generalized filtertransfer functions casting arbitrary quantum control operations on qubits as noise spectral filters^{9,10}. 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.
Main
Timevarying 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 timedependent 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 quantumenabled technologies for applications such as quantum information and quantum sensing.
These considerations motivate the development of novel engineeringinspired 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 filtertransfer functions (FFs) characterizing the control. The fidelity of an arbitrary operation over duration τ, , is degraded owing to frequencydomain spectral overlap between noise in the environment given by a power spectrum S(ω), and the filtertransfer functions denoted F(ω) (Methods)^{11,12,13,14}.
The FF description of ensembleaverage quantum dynamics tremendously simplifies the task of analysing the expected performance of a control protocol in a noisy environment as it permits consideration of control as noise spectral filtering. The FFs themselves may be described using familiar concepts such as frequency passbands, stopbands and filter order, enabling a simple graphical representation of otherwise complex concepts in the dynamics of controlled quantum systems (Fig. 1b). Noise filtering, in practice, is achieved through construction of a control protocol (Fig. 1a) which modifies the controllability of the quantum system by the noisy environment over a defined frequency band. Adjusting F(ω) and changing its overlap with the noise spectrum thus allows a user to change the average dynamics of the system in a predictable way.
To see the importance of this capability we may consider the various tasks that might be of interest in experimental quantum engineering and the role of noise spectral filtering in these applications. In quantum information an experimentalist may aim to suppress broadband lowfrequency noise to maximize the fidelity of a boundedstrength quantum logic operation (Fig. 1b, upper trace), and then calculate the residual error. Alternatively, in quantumenabled sensing or system identification he or she may perform narrowband spectral characterization of a given operation (Fig. 1b, lower trace), where any change in the measured fidelity under filter application represents the signal of interest^{4,6}.
The intuitive nature of this framework is belied by the challenge of calculating FFs for arbitrary control protocols, generally involving timedomain modulation of control parameters such as the frequency and amplitude of a driving field. The nature of quantum dynamics means that the implemented control framework is generally nonlinear; for instance, one finds complex dynamics in circumstances where the noise and control operations do not commute, such as a driven operation (∝ σ_{x}) in the presence of dephasing noise (∝ σ_{z}). Recent theoretical effort has allowed calculation of FFs for arbitrary singlequbit control and arbitrary universal classical noise^{9,10}, expanding significantly beyond previous demonstrations restricted to the identity operator in puredephasing environments^{15}. It is this more general case where the impact of noise filtering and the FFs may have the most significant impact on the quantum engineering community, and where experimental tests are vital.
In our experimental system, based on the 12.6 GHz qubit transition in ^{171}Yb^{+} (Supplementary Methods), we are able to perform quantitative tests of operational fidelity for arbitrary control operations; these may then be compared against calculations of as a fundamental test of FF validity. A key tool in our studies is bath engineering^{16}, in which we add noise with userdefined spectral characteristics to the control system, producing wellcontrolled unitary dephasing or depolarization.
As a first example (Fig. 1c), experimental measurements of operational fidelity for a π_{x}pulse driving qubit population from the dark state to the bright state, 0> → 1>, in the presence of engineered timedependent dephasing noise give good agreement with analytic calculation of using the noise power spectrum and analytic FFs (ref. 10) with no free parameters (Methods). This approach therefore immediately demonstrates the predictive power of the FF formalism.
The FFs for much more complex control, such as compensating composite pulses^{17,18}, can be calculated and experimentally validated as well (Fig. 1e). These protocols are commonly used in nuclear magnetic resonance and electron spin resonance in attempting to suppress static offsets in control parameters such as the frequency of the drive inducing spin rotations. Calculating the FFs for these protocols now reveals their sensitivity to timedependent noise—an important characteristic for deployment in realistic quantum information settings^{19}. We experimentally demonstrate a form of quantum system identification (Methods), effectively reconstructing the amplitudenoise filter functions, F_{Ω}(ω), for two wellknown compensating pulse sequences known by the shorthand designations SK1 and BB1 (Supplementary Methods). Again, calculations of match data well over the entire band in the weaknoise limit (Fig. 1f) with no free parameters.
Our choice of characterizing these compensating pulse sequences highlights an important issue in the prediction of ensembleaverage dynamics of controlled quantum systems. Ultimately, the underlying physical principles giving rise to the analytic form of F(ω) are based on the welltested average Hamiltonian theory^{20} exploited in crafting these pulses. Despite this shared theoretical foundation, the calculation of spectral filtering properties is quite distinct from calculation of quasistatic error terms in a Magnus expansion, with important consequences for average quantum dynamics in realistic timevarying noise environments^{10}.
Accordingly, our tests of the FF formalism reveal that compensating pulses designed to suppress errors to high order in a Magnusexpansion framework need not be efficient noise spectral filters (Supplementary Methods and ref. 19). Despite significant differences in their construction—the BB1 protocol is designed to provide higherorder cancellation of Magnus terms than SK1—both of the selected composite pulses provide similar filtering of timedependent noise, given by the filter order (slope of the FF in Fig. 1e). In the weaknoise limit frequencydomain characteristics are captured accurately through the FF across frequencies ranging from quasistatic to rapidly fluctuating on the timescale of the pulse (Fig. 1f). Performance deviations between the pulses arise and the FF approximation breaks down as the noise strength is increased and higherorder terms in the Magnus series become important, but only at low frequencies (Fig. 1g). At frequencies fast relative to the control the FF again accurately predicts the relevant quantum dynamics even in the strongnoise limit. This is the first direct manifestation of the difference between studying quantum dynamics in terms of frequencydomain noise filtering and calculation of error contributions in a Magnus expansion as is appropriate in the quasistatic limit.
These simple but powerful validations of the predictive power of the generalized FF formalism now open the possibility of demonstrating the construction of noise filters with a specified spectral response, employing the filtertransfer functions as key analytic tools. Filters may take a wide variety of forms as needed by users—including highpass filters for broadband noise suppression and bandstop filters useful for narrowband noise characterization (Fig. 1b).
In the discussion that follows, we focus on a common setting in which we aim to improve operational fidelity by reducing the influence of broadband nonMarkovian noise on a target state transformation. Filters are realized as nstep sequences of timedomain control operations with tunable pulse amplitude and phase, similar in spirit to compensating composite pulses in NMR (refs 17, 18, 19), dynamically corrected gates (DCGs) in quantum information^{21,22}, and openloop modulated pulses in quantum control^{23,24}. However, recalling the difference between Magnus cancellation order and filtering order described above, in this setting we wish to synthesize a filter with arbitrary, userdefined spectral characteristics captured by a cost function, A(Γ_{n}), to be minimized for a filter represented by Γ_{n}(θ_{l}, τ_{l}, ϕ_{l}) (Fig. 2b, c and Methods).
To provide efficient solutions to filter design we restrict our control space and focus on constructions synthesized using concepts from functional analysis in the basis set of Walsh functions—squarewave analogues of the sines and cosines^{4,25,26} (Fig. 2a). This approach provides significant benefits for our problem^{26}, but is by no means the only basis set for composite filter construction^{27,28}.
As an example we synthesize noise filters via weighted linear combinations of Walsh functions, PAL_{k}(x) denoted by the Paleyordered index, k. These filters are designed to suppress timevarying dephasing noise over a lowfrequency stopband while implementing a boundedstrength driven rotation about the x axis on the Bloch sphere (Supplementary Methods). In this case the Walshsynthesized waveform dictates an amplitude modulation pattern for the control field over discrete time segments. Importantly, Walsh filter synthesis is compatible with pulse segments possessing arbitrary pulse envelopes, including sequences of, for example, square (used here) or Gaussian pulse segments (Fig. 2b).
Analytic design rules provide simple insights into how one may craft effective modulation protocols, and a Nelder–Mead simplex optimization is used to find highperforming operations as defined by our cost function. Relative to an unfiltered primitive gate, the dephasing filter function, F_{z}(ω), for the simplest fourpulse construction W1 shows increased steepness in the stopband (Fig. 2c, red), reducing A(Γ_{4}) (here the gate performs θ = π). This measure of filter order may be further increased via construction W2, in turn reducing the cost function for optimization (blue shaded area in Fig. 2c). Relating back to earlier demonstrations of filter order in compensating pulses, W2 presents an interesting case of a highorder noise filter over the target band which provides only firstorder Magnus cancellation.
Filters W1 and W2 are representative, rather than unique solutions. In Fig. 3b we show the calculated cost function, A(Γ_{4}), as a function of the Walsh coefficients used in constructing W1, X_{0} and X_{3}, giving the modulation profile indicated in Fig. 3a. Blue areas meet our minimized target, indicating useful filters, revealing a wide variety of possible constructions with favourable characteristics. Experimental tests of these protocols reveal that Walshmodulated waveforms minimizing A(Γ_{4}) effectively suppress noise in the designated stopband for arbitrary rotation angles (Fig. 3c–e), and outperform standard pulses in the smallerror limit germane to quantum information (Fig. 3f). See Methods.
Our focus has been on providing a validated framework for the vital task of predicting quantum dynamics in realistic environments and demonstrating the relevant physics through construction of noise spectral filters. The Walshmodulated filters presented here—based on the achievable frequencydomain filter order—complement existing techniques rather than attempting to provide optimalperformance errorrobust gates. Our results on highpass noise filters, for instance, add to existing compensating pulse sequences designed for quasistatic noise, as well as gate constructions with interleaved dynamical decoupling that seek to periodically ‘refocus’ quantum evolution^{29,30,31,32}.
Importantly, recent work has demonstrated that the filtertransfer function formalism is applicable to multiqubit settings where dynamics may be considerably more complex than the singlequbit case^{33,34,35}. In addition, ongoing efforts suggest there exists a path towards further extension of the generalized filtertransfer function and noise filtering formalisms to arbitrary control settings involving multiple qubits subject to general noise from nonMarkovian classical and/or quantum mechanical environments. We believe that with the validations provided here, this simple extensible framework with precise predictive power will provide a path for experimentalists to characterize and suppress the effects of noise in generic quantum coherent technologies, ultimately enabling a new generation of engineered quantum systems.
Methods
The fidelity of a control operation for a single qubit in the presence of a timedependent environment is reduced as , where and τ is the total duration of the operation. In this expression for fidelity, the integral considers contributions from independent noise processes through their frequencydomain power spectra S_{i}(ω), i ∈ {z, Ω}, capturing dephasing along and amplitude noise corotating with a resonant drive field (Supplementary Methods). We employ here the socalled modified filtertransfer function, which subsumes a factor of ω^{−2} into the definition of F_{i}(ω). See refs 6, 14 for details.
Experimental measurements involve state initialization in 0> followed by a control operation—or series of control operations—designed to drive qubit population from the dark state to the bright state, 0> → 1> . For instance, tests of filters used for rotations θ < π are repeated sequentially such that the net rotation enacts 0> → 1> (Fig. 3d, e). The operational fidelity is measured as the probability that the qubit is in the bright state over an ensemble of measurements. Typical experimental uncertainties are limited by measurement fidelity (∼98.5%) and quantum projection noise with maximum value comparable to the measurement infidelity for qubit states near the equatorial plane of the Bloch sphere. In general, a nonMarkovian noise bath is engineered with specific properties of interest (see Supplementary Methods for full details). Additional measurement uncertainty of order ∼3–5% is added through finite sampling of the infinite ensemble of possible noise realizations. This is visible as fluctuations between neighbouring points in, for example, Fig. 3c–f.
Measurements in Fig. 1c are conducted for a simple π_{x} enacted while varying the highfrequency cutoff, ω_{c}, of a flattop engineered nonMarkovian dephasing bath (Fig. 1d). As the highfrequency cutoff of the noise is increased and fluctuations fast relative to the control (τ_{π}) are added to the noise power spectrum, S_{z}(ω), errors accumulate, reducing the measured fidelity. For ω_{c}/2π = 1 the highest frequency contribution to S_{z}(ω) undergoes a complete cycle of oscillation over τ_{π}, indicating that the noise is timedependent on the scale of a single experiment even for ω_{c}/2π ≪ 1. We calculate using the form of the noise and the analytic FF for a driven primitive gate under dephasing^{10}, finding good agreement with experimental measurements using no free parameters.
Measurements in Fig. 1f, g employ a narrowband ‘deltafunction’ noise power spectrum swept as an experimental variable, ω_{t}. Injected noise takes the form of fixedfrequency amplitude modulation of the nearresonant driving field during application of a control pulse, with strength (modulation depth) parameterized in terms of Ω, the Rabi rate of the drive. The form of demonstrates that the calculated fidelity involves an exponentiated value of the FF at frequency ω_{t}, meaning that fidelity measurements effectively reconstruct the filter functions. Key features in the data, such as performancecrossover frequencies between primitive and compensating gates and deep notches in the filter at high frequency, are quantitatively reproduced in experimental measurements.
Filter construction presented in Figs 2 and 3 is parameterized as a function of controllable properties of a nearresonant carrier frequency enacting driven operations. An arbitrary nsegment filter is represented over successive timesteps through the matrix quantity Γ_{n}(θ_{l}, τ_{l}, ϕ_{l}) (Fig. 2b); in each segment of duration τ_{l} we perform a driven operation generating a rotation through an angle about the axis r_{l} = (cos(ϕ_{l}), sin(ϕ_{l}), 0), with Ω_{l}(t) the Rabi rate over the lth pulse segment.
The value of n is chosen to be a power of two, compatible with synthesis over discretetime Walsh functions. The Walsh functions are piecewiseconstant over segments which are all integer multiples of base period τ_{l}. This approach brings benefits for the current setting^{26}; for instance, their piecewiseconstant construction builds intrinsic compatibility with discrete clocking and classical digital logic, while the wellcharacterized mathematical properties of the Walsh functions provide a basis for establishing simple analytic filterdesign rules, and flexibility in realizing a wide variety of filter forms.
For the filters W1 and W2 presented in the main text, Walshsynthesis design rules dictate that we implement our filtered rotation by θ_{x} over a minimum of four discrete steps, permitting synthesis over PAL_{0} to PAL_{3}. Within this small set, the coefficient of PAL_{0}, denoted X_{0}, sets the total rotation angle θ mod 2π for the modulated driven evolution, and only nonzero X_{3} preserves symmetry. We experimentally test the performance of foursegment amplitudemodulated filters by scanning over X_{3} for fixed X_{0} (denoted by white dotted lines in Fig. 3b). Values of X_{3} minimizing A(Γ_{4}) (dips in the dashed trace, right axis) also minimize the experimentally measured infidelity in the presence of engineered lowfrequency noise (open circles, left axis). This behaviour is observed for various target rotation angles of interest (Fig. 3c–e), with predicted shifts in the optimal values of X_{3} with changes in X_{0} borne out through experiment. Filter W2 is constructed over PAL_{0} to PAL_{7}, and has twice as many timesteps as W1. Interestingly, W1 is a special case of an analytically constructed dynamically corrected NOT gate (a πrotation)^{21}. For details of the Walsh functions, Walsh synthesis and Walshbasis analytic design rules see Supplementary Methods.
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Acknowledgements
We thank P. Fisk, M. Lawn, M. Wouters and B. Warrington for technical assistance and K. Brown, J. T. Merrill and L. Viola for useful discussions. This work partially supported by the US Army Research Office under Contract Number W911NF1110068, the Australian Research Council Centre of Excellence for Engineered Quantum Systems CE110001013, the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office, and the Lockheed Martin Corporation. X.Z. acknowledges useful discussions with G. L. Long and support from the National Natural Science Foundation of China (Grants No. 11175094 and No. 91221205) and the National Basic Research Program of China (No. 2011CB9216002).
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A.S., H.B., D.H. and M.J.B. conceived and performed the experiments, built experimental apparatus, contributed to data analysis and wrote the manuscript. T.J.G. conceived the relevant theoretical constructs. J.S., M.C.J. and X.Z. assisted with development of the experimental system and data collection. J.J.M. assisted with data collection.
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Soare, A., Ball, H., Hayes, D. et al. Experimental noise filtering by quantum control. Nature Phys 10, 825–829 (2014). https://doi.org/10.1038/nphys3115
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DOI: https://doi.org/10.1038/nphys3115
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