- Research
- Open Access

# Walsh-synthesized noise filters for quantum logic

- Harrison Ball
^{1, 2}and - Michael J Biercuk
^{1, 2}Email author

**2**:11

https://doi.org/10.1140/epjqt/s40507-015-0022-4

© Ball and Biercuk; licensee Springer. 2015

**Received:**13 December 2014**Accepted:**19 March 2015**Published:**14 May 2015

## Abstract

We study a novel class of open-loop control protocols constructed to perform arbitrary nontrivial single-qubit logic operations robust against time-dependent non-Markovian noise. Amplitude and phase modulation protocols are crafted leveraging insights from functional synthesis and the basis set of Walsh functions. We employ the experimentally validated generalized filter-transfer function formalism in order to find optimized control protocols for target operations in SU(2) by defining a cost function for the filter-transfer function to be minimized through the applied modulation. Our work details the various techniques by which we define and then optimize the filter-synthesis process in the Walsh basis, including the definition of specific analytic design rules which serve to efficiently constrain the available synthesis space. This approach yields modulated-gate constructions consisting of chains of discrete pulse-segments of arbitrary form, whose modulation envelopes possess intrinsic compatibility with digital logic and clocking. We derive novel families of Walsh-modulated noise filters designed to suppress dephasing and coherent amplitude-damping noise, and describe how well-known sequences derived in NMR also fall within the Walsh-synthesis framework. Finally, our work considers the effects of realistic experimental constraints such as limited modulation bandwidth on achievable filter performance.

## Keywords

- decoherence suppression
- error correction
- open-loop control
- dynamic error suppression
- quantum control
- quantum logic
- qubit
- Walsh function
- functional analysis

## 1 Introduction

In realistic laboratory settings, decoherence in quantum systems is dominated by *time-dependent* non-Markovian noise processes with long correlations, frequently characterized by low-frequency dominated noise power spectra [1–5]. These may arise either from environmental fluctuations or - in the important case of *driven* quantum systems - from noise in the control device itself [6]. In either case, the result is a reduction in the fidelity of a target control operation, including both memory and nontrivial operations. These phenomena present a major challenge as quantum devices move from proof of principle demonstrations to realistic applications, where performance demands on the quantum devices are frequently extreme. Accordingly, finding ways to control quantum systems efficiently and effectively in the presence of noise is a central task in quantum control theory [7–10].

A range of techniques relying on both open- and closed-loop control have been devised to address this challenge [11–13] at various levels in a layered architecture for quantum computing [14]. In particular, open-loop dynamical error suppression strategies (without the need for measurement or feedback) such as dynamic decoupling (DD) [15–18], dynamically corrected gates (DCGs) [19–25], and composite pulsing [26–28], have emerged as *resource-efficient* approaches for physical-layer decoherence control. They are joined by a new class of continuously modulated (‘always-on’) dynamical decoupling and dynamically protected gate schemes [29–36] inspired by well established techniques in NMR.

These schemes all address the question of decoherence mitigation, but looking across their breadth, have both benefitted and suffered from reliance on a wide range of theoretical techniques. Unfortunately the analytic tools for crafting control protocols employed in any particular setting do not necessarily translate equivalently between approaches, nor do the methods generally employed for evaluating efficacy easily translate to experimentally measured characteristics of the environment. This is a major challenge for experimentalists or systems designers attempting to determine which of the many open-loop control schemes to employ in a particular experiment. As an example, the powerful group theoretic insights and consideration of time-varying environments that permit the construction of error-robust, bounded-strength SU(2) operations for quantum information in Viola’s DCG framework are quite different from the geometric considerations and quasi-static noise assumptions widely employed in NMR composite pulsing. This issue has been highlighted recently as new work has revealed striking differences between the time-domain noise sensitivity of control protocols as compared to longstanding notions of error cancellation in the Magnus expansion [37, 38].

A unified and experimentally relevant framework for *devising* and *evaluating* error-suppressing gates in realistic noise environments is therefore needed to secure the role of dynamical error suppression in systematic designs of quantum technologies including fault-tolerant quantum computers. Kurizki provided a promising path towards this end with his seminal work framing the problem of finding decoherence-suppressing control protocols by considering appropriate frequency-domain modification of the system-environment coupling [39, 40]. Residual errors could be calculated through overlap integrals of the power spectrum describing the environmental noise, and functions capturing the frequency-domain response of any applied control. This framework - effectively a quantum generalization of transfer functions widely used in control engineering [41] - provides a simple heuristic approach to understanding the performance of an arbitrary control protocol in an arbitrary noise environment. Stated simply, effective error-suppressing control protocols ‘filter’ the noise over a user-defined band, therefore mitigating decoherence in the quantum system [42].

Early demonstrations of this framework applied to the simple case of implementing the protected identity operator to qubits by dynamical decoupling [15, 43–47], where the filter functions could be calculated for pure dephasing in a straightforward manner using concepts of *linear control* [42]. Expanding significantly beyond this work, the challenge of crafting generalized analytic forms for the transfer functions describing arbitrary single-qubit control compatible with universal non-commuting noise (a problem in *nonlinear* control) has recently been addressed theoretically [28, 38, 48, 49], and validated in experiment [37]. Further theoretical extensions of filter-transfer functions to two-qubit gates highlight the breadth of applicability of this approach to quantum control [50–52].

Beyond its simple intuitive nature, the power of the filter transfer function approach comes from the fact that it can in principle be applied to studying dynamic-error-suppression control protocols derived through any manner of analytic approach. It permits the application of well tested engineering concepts for control systems design; the complex physics associated with quantum dynamics in time-dependent environments with non-commuting noise and control Hamiltonians is relegated to the calculation of the generalized filter transfer functions themselves, and once derived these may be deployed in block-diagram systems analyses [41].

With these significant advances and the promise of applying the suite of insights from control theory to the quantum regime, the noise-filtering approach to quantum control has leapt to the fore, providing a unifying framework applicable over a wide parameter range of interest to real experimental settings. Nonetheless, outstanding challenges remain in how to leverage the generalized filter-transfer-function framework [48, 49] to systematically craft effective error-suppressing gate constructions while also heeding realistic system constraints imposed by hardware systems. For instance, the presence of finite timing precision and limited classical communication bandwidth between the physical (quantum) layer and a classical controller [14] impose new constraints not generally captured when solely considering quantum dynamical evolution of an individual state.

We address this challenge, introducing a quantum control toolkit permitting the realization of physical-layer error-suppressing control protocols that are simultaneously effective in suppressing error and compatible with a variety of major hardware restrictions. We leverage the generalized filter-transfer function formalism as a unifying theoretical construct, and employ techniques from functional analysis in order to realize appropriate modulation protocols applied to a near-resonant carrier frequency for enacting high-fidelity quantum control operations on single qubits [30, 53]. Our work identifies the Walsh functions - square-wave analogues of the sines and cosines - as natural building blocks for constructing the modulation protocols designed to filter time-varying noise over a user defined band while enacting a nontrivial qubit rotation. The Walsh functions are defined in a uniform piecewise-constant fashion, building intrinsic compatibility with discrete clocking [54] and classical digital logic, and have previously been identified as providing a powerful mathematical framework in the context of quantum control sequencing [51]. Moreover, they may be arbitrarily combined using Fourier-like synthesis using techniques for arbitrary waveform generation well established in digital signal processing.

We treat a *Walsh-modulated* driven qubit system weakly interacting with both dephasing and coherent amplitude-damping noise processes. The task of finding Walsh-synthesized modulation patterns that produce effective filters is reduced to minimizing a cost function measuring the extent to which noise over a user-defined spectral band is filtered by the applied control. The performance of resulting control protocols is completely characterized by their *Walsh spectra*, facilitating intuitive analytic design rules based on symmetry and spectral properties of the Walsh basis. Our work details the various techniques and mathematical constructs through which we define and then optimize the filter-synthesis process in the Walsh basis, and considers the effects of realistic experimental constraints such as limited modulation bandwidth.

With these insights, we derive novel families of Walsh-modulated noise filters designed to suppress dephasing and coherent amplitude-damping noise, and describe their properties. Modulation protocols are tailored to a particular operation on SU(2), but are otherwise largely *model-robust* (being tailored to suppress noise over a frequency band rather than to a specific time-domain noise signal), and *portable* between different qubit technologies. Combined with the discovery, presented here, that several prominent composite pulse protocols derived in NMR actually fall within the Walsh-synthesis basis - mirroring similar insights in the context of dynamical decoupling [51] - this work positions the Walsh functions as a natural basis for crafting physical-layer error suppression strategies for scalable quantum technologies.

The remainder of this paper is organized as follows. In Section 2 we describe our model quantum system by defining relevant control and noise Hamiltonians. In Section 3 we review the generalized filter-transfer function formalism used to derive a spectral representation of the operational infidelity. Notation for defining and parameterizing the control space is introduced and explicit expressions for computing corresponding filter functions are presented. Section 4 provides a formal definition of a filter cost function used for optimizing operational fidelity over the control space and deriving useful filters. Performance characteristics of these filters are discussed and interpreted, with care taken to differentiate filter order from Magnus order. In Section 5 physically motivated constraints on the control space are established by synthesizing control waveforms as superpositions of functions in the Walsh basis, bounding the dimensionality of the filter-optimization task. Two useful representations of the Walsh basis - Paley ordering and the Hadamard representation - are introduced. We then develop a range of analytic filter-design rules for efficient filter construction based on the symmetry and spectral properties of the Walsh functions. In Sections 6-9 we apply the above framework to derive several novel families of noise filters implementing nontrivial logic gates. These include filters for dephasing and coherent amplitude-damping noise in addition to concatenated filters for universal noise. In Section 10 we study how relaxing the assumption of perfect square pulses reduces the performance of filters optimized in the Walsh basis, and demonstrate that these filter properties may be recovered in general by simply re-optimizing under the assumption of non-square pulses. We then close with a brief summary and outlook, followed by a number of appendices containing detailed derivations of relevant quantities used in the main text.

## 2 Physical setting

*τ*.

In the absence of noise the total Hamiltonian reduces to \(H(t) = {H}_{c} (t)\), in which case time-evolution is determined purely by control operations according to \(i\dot{U}_{c}(t,0) = H_{c}(t)U_{c}(t,0)\). An *intended* evolution path under ideal control is therefore described by the *control propagator*
\(U_{c} (t,0) = \mathcal {T}\exp (-i\int_{0}^{t}{H}_{c} (t')\,dt' )\), with \(\mathcal {T}\) denoting the time-ordering operator. For a single qubit the time-dependent control Hamiltonian may in general be written \({H}_{c} (t) = \Omega(t)\hat {\mathbf {n}}(t)\cdot \boldsymbol {\sigma }/2\). Here \(\hat {\mathbf {n}}(t)\cdot \boldsymbol {\sigma }\equiv n_{x}\hat {\sigma }_{x}+n_{y}\hat {\sigma }_{y}+n_{z}\hat {\sigma }_{z}\) is the rotation generator, \(\hat {\mathbf {n}}(t)\in\mathbb{R}\) is a unit vector defining the instantaneous axis of rotation, and \(\Omega(t)\) is the instantaneous rate of rotation (Rabi rate) for the Bloch vector.

In our model both \(\beta_{z} (t)\) and \(\beta_{{\Omega}}(t)\) are assumed to be classical random variables with zero mean and non-Markvovian power spectra. We also assume they are *wide sense stationary* and *independent*.^{1} The former implies the autocorrelation functions \(\langle\beta_{i}(t_{1})\beta_{i}(t_{2})\rangle\), \(i\in\{ z,\Omega\}\), depend only on the time *difference*
\(t_{1}-t_{2}\). The latter implies the cross-correlation functions vanish. That is, \(\langle\beta_{i}(t_{1})\beta_{j}(t_{2})\rangle= 0\) where \(i,j\in\{ z,\Omega|i\ne j\}\). The angle brackets denote a time average of the random variables. Finally, our model permits access to a wide range of parameter regimes, from quasistatic (noise slow compared to \({H}_{c} (t)\)) to the limit in which the noise fluctuates on timescales comparable to or faster than \({H}_{c} (t)\).

These noise Hamiltonians generate uncontrolled rotations in the qubit dynamics, leading to errors in the evolution path (and hence the final state) relative to the target transformation intended under \({H}_{c} (t)\). An estimate for this error is derived in the next section using the generalized filter-transfer function formalism.

## 3 Building noise filters

Overall, our objective is to craft control protocols such that the deleterious effects of time-dependent noise on the intended evolution of an arbitrary qubit transformation are suppressed - *filtered* by the control. Accordingly, we require a measure for the operational fidelity in the presence of both noise and the relevant control. For this we employ the method developed by *Green et al.* [49]. In this framework the error contributed by the noise fields over the duration of the control is approximated, to first order, via a truncated Magnus expansion. Each noise field then contributes a term to the gate infidelity in the spectral domain expressed as an overlap integral between the noise power spectrum and an appropriate generalized filter-transfer function. We describe this in detail below.

### 3.1 Calculating operational fidelity

In the absence of noise interactions, state evolution is determined by \(i\dot{U}_{c}(t)={H}_{c} (t)U_{c}(t)\) with \(U_{c}(t)\) the ideal evolution operator describing the target operation. Including the effects of noise, however, time evolution is determined by the operator \(U(t)\) satisfying \(i\dot{U}(t) = ({H}_{c} (t)+H_{0}^{(z)}(t)+H_{0}^{(\Omega )}(t))U(t)\). Our measure for operational fidelity is given by \(\mathcal{F}_{av}(\tau) = \frac {1}{4}\langle|\operatorname{Tr}(U_{c}^{\dagger}(\tau)U(\tau))|^{2}\rangle\), effectively measuring the extent to which the intended and realized operators ‘overlap’, as captured by the Hilbert-Schmidt inner product [55]. Computing the evolution dynamics, however, is very challenging since the control and noise Hamiltonians do not commute at different times; sequential application of the resulting time-dependent, non-commuting operations gives rise to both dephasing and depolarization errors, mandating approximation methods.

Our error model assumes non-dissipative qubit evolution with both control and noise interactions resulting in unitary rotations. Hence we approximate the evolution operator as a unitary using a Magnus expansion [56, 57]. This involves moving to a frame co-rotating with the control known as the *toggling frame*, originally appearing in the development of average Hamiltonian theory [58]. This approach allows us to separate the part of the system evolution due solely to the control from the part affected by environmental coupling, and is standard in the study of coherent control in NMR [58, 59] and quantum information.

Defining the *error propagator*
\(\tilde {U}(t)\equiv U_{c} ^{\dagger}(t)U (t)\), the total evolution operator is written \(U (t) = U_{c} (t)\tilde {U}(t)\). In this case the realized evolution operator approaches the target operation as \(\tilde {U}(\tau) \rightarrow \mathbf {I}\), establishing the condition for suppression of noisy evolution dynamics. However, moving to the toggling frame defined by *toggling frame Hamiltonian*
\(\tilde{H}_{0} (t) \equiv U_{c}^{\dagger}(t){H}_{0} (t) U_{c} (t)\), the error propagator satisfies the Schrodinger equation \(i\dot{\tilde {U}}(t) = \tilde{H}_{0} (t)\tilde {U}(t)\). Performing a Magnus expansion in this frame - assuming convergence of the series [57] - we may write \(\tilde {U}(\tau) = \exp [-i\sum_{\mu= 1}^{\infty} \mathbf {a}_{\mu}(\tau )\cdot \boldsymbol {\sigma }]\) where the *error vectors*
\(\mathbf {a}_{\mu}(\tau )\) determine expansion coefficients of the Magnus series operators \(\Phi_{\mu}(\tau)\) expressed in the basis of Pauli matrices (see Appendix A). We may then in principle approximate \(\tilde{U}(t)\) to arbitrary accuracy by truncating the infinite series at an appropriate order.

The operational fidelity \(\mathcal{F}_{av}(\tau)=\frac{1}{4}\langle |\operatorname{Tr}(\tilde {U}(\tau))|^{2}\rangle\) may now be fully expressed as an infinite power series over the ensemble-averaged magnitudes of the expansion vectors \(\mathbf {a}_{\mu}(\tau)\). In the limit of sufficiently weak noise,^{2} however, it is appropriate to truncate the expansion to first-order yielding \(\mathcal{F}_{av}(\tau) \approx1 - \langle a ^{2}_{1}\rangle\) with \(\langle a ^{2}_{1}\rangle \equiv\langle \mathbf {a}_{1}(\tau)\mathbf {a}_{1}^{T}(\tau)\rangle\) defining the *first order infidelity*. Now, as set out in Appendix A the first-order error vector is related to the first-order Magnus term according to Eq. (70), yielding \(\mathbf {a}_{1}(\tau)\cdot \boldsymbol {\sigma }= \Phi_{1}(\tau) = \int_{0}^{\tau}dt \tilde{H}_{0} (t)\). That is, the first-order infidelity \(\langle a^{2}_{1}\rangle\) is associated with the time-average of the toggling frame Hamiltonian over the total sequence duration.

### 3.2 Defining the control space

In order to realize specific noise filters, characterized by the filter-transfer functions introduced above, we require a simple framework to define the time-domain control operations that can be applied to the qubit. Representing the qubit state on the Bloch sphere, state manipulation maps to a rotation in \(\mathbb{R}^{3}\) of the Bloch vector associated with a unitary transformation \(\mathcal {U}(\theta, \hat {\sigma }_{\hat {\mathbf {n}}})\equiv\exp [-i (\boldsymbol {\sigma }\cdot \hat {\mathbf {n}})\theta/2 ]\), reflecting the homeomorphism between \(SU(2)\) and \(SO(3)\). The rotation generator \(\hat {\sigma }_{\hat {\mathbf {n}}} \equiv \hat {\mathbf {n}}\cdot \boldsymbol {\sigma }\equiv n_{x}\hat {\sigma }_{x}+n_{y}\hat {\sigma }_{y}+n_{z}\hat {\sigma }_{z}\) produces a rotation though an angle *θ* about the axis defined by the unit vector \(\hat {\mathbf {n}}\in\mathbb{R}^{3}\).

*n-segment sequence*of such unitaries, executed over the time period \([0, \tau ]\). This implies a natural partition of the total sequence duration

*τ*into

*n*subintervals \(I_{l} = [t_{l-1},t_{l}]\), \(l\in\{1,\ldots ,n\} \), such that the

*l*th control unitary has duration \(\tau_{l} = t_{l}-t_{l-1}\). Here \(t_{l-1}\) and \(t_{l}\) are the start and end times of the

*l*th rotation respectively, and we define \(t_{0} \equiv0\) and \(t_{n} \equiv\tau\). In particular we consider control unitaries of the form

*l*th time interval \([t_{l-1},t_{l}]\), and is assumed constant over the duration \(\tau_{l}\) of the associated control interaction. During this interaction the rotation generator \(\hat {\sigma }_{\phi _{l}}\), parameterized by \(\phi _{l}\in[0,2\pi]\), thus generates a rotation of the Bloch vector through an angle \(\theta_{l} \equiv\Omega_{l}\tau_{l}\) about the axis \(\hat {\mathbf {n}}_{l} \equiv (\cos(\phi_{l}),\sin(\phi_{l}),0 )\) in the

*xy*plane.

^{3}The control Hamiltonian associated with this

*n*-segment sequence takes the form

*during*the

*l*th time interval is, under this Hamiltonian, consequently described by the unitary

*l*th completed rotation is equivalently denoted by the operator \(P_{l} = U_{c} (t_{l},t_{l-1})\). For compactness we define the

*cumulative operator*

*l*sequentially competed rotations. Hence the control propagator at any time

*t*may be written

*n*triples \(\{(\Omega_{l},\tau_{l},\phi_{l})\}_{l=1}^{n}\), and each control operation is completely parameterized by the control variables according to \(P_{l} = P_{l}(\theta_{l},\Omega_{l},\tau_{l},\phi_{l})\). Although not strictly an independent parameter it is useful to include \(\theta_{l} = \Omega _{l}\tau_{l}\) in the argument to distinguish different

*realizations*of the same net rotation for different choices of \(\Omega_{l}\tau_{l}\). We define the \((n\times4 )\)

*n*-segment matrix

*n*-segment unitary control sequence (see Figure 1). The entire space \(\mathfrak {C}_{n}\) of such control forms, referred to the

*n*-

*segment control space*, and written formally

*composite-pulse sequences*in NMR and DCGs in quantum information. We use the more general control space, however, to construct novel qubit gates specifically designed to filter non-Markovian noise.

### 3.3 Generalized filter-transfer functions

*n*-segment control protocols implemented by Eq. (7). As outlined above, the filter-transfer functions are completely parametrized by the control variables \(\{(\Omega_{l},\tau_{l},\phi_{l})\}_{l=1}^{n}\cong \boldsymbol {\Gamma }_{n}\). Here we only provide a summary of the relevant computational quantities, leaving the major derivations and full explanation to Appendix A. We start by writing

*l*th unitary control segment. This takes the form (see Appendix B)

*lth-Segment Projection Vector*

*xy*plane of the Bloch sphere - associated with evolution during the

*l*th unitary. In fact, inspection of Eqs. (15) and (16) reveals that \(\vec {{\mathbb{T}}}^{(l)} \) is the computational analogue of \(\mathbf{R}^{P_{l}}_{z}\) for the amplitude noise quadrature. The simpler dependence of \(\vec{ {\mathbb{T}}}^{(l)} \) on the control variables, however, reflects the fact that amplitude noise in our model is always coaxial, and hence commutes with, the control.

*Control History Matrix*\(\boldsymbol{\Lambda}^{(l-1)}\), defined by

*completed*unitaries, implemented via the cumulative operator \(Q_{l-1}\).

## 4 Characteristics of noise filters

The power of the noise filtering formalism lies in the simple interpretation of the filter-transfer functions \(F_{i}(\omega)\), which may be characterized in a standard engineering approach, considering passbands, stopbands, and filter order [42, 48, 49, 60]. In particular, error suppression corresponds to minimizing \(F_{i}(\omega )\), \(i\in\{z,\Omega\}\) in the spectral region where the corresponding PSDs are non-negligible. This can, in principle, be achieved by judicious construction of the control sequence since the filter-transfer functions are completely parametrized in variables describing the time-domain control applied to the qubit.

We are now in a position to examine the characteristics of the filter-transfer functions for an arbitrary control sequence \(\boldsymbol {\Gamma }_{n}\), formally indicating the functional dependence of the filter-transfer functions on the control variables by writing \(F_{i}(\omega\tau) = F_{i}(\omega\tau; \boldsymbol {\Gamma }_{n})\), \(i\in\{ z,\Omega\}\). Inversely, we may commence a study of filter design based on constructing control sequences satisfying some *desired* filter property - our main goal. We now advance the main mathematical framework used in this paper to study filter design, pulling together the ideas introduced in the previous sections.

### 4.1 The filter cost function

A definition of the cost function associated with filter performance - captured through the *filter order* - leads us naturally to the imposition of constraints on the available space of controls. This cost function therefore lies at the heart of our attempts to craft control protocols appropriate for a given noise environment.

*n*, we may in principle construct a variational procedure over this control space to derive minimizing ‘values’ of \(\boldsymbol {\Gamma }_{n}\) satisfying a given cost function. In effect, the problem involves solving for paths in the control space over which the functional \(A_{i}(\boldsymbol {\Gamma }_{n})\) is minimized (up to some order).

Typically one would define the band \([\omega_{L},\omega_{c}]\) over which the cost function is defined to fall within the *stopband* of \(F_{i}(\omega\tau)\), below which filtering generally takes place. In general the band \([\omega_{L},\omega_{c}]\) may be tailored to target specific spectral regions in the noise PSD. Doing so may produce highly effective filtering over this narrow spectral region, though out-of-band behaviour can be quite poor if not specifically optimized.^{4}

### 4.2 The filter order

*filter order*which will play a central role in efficiently realizing effective noise filters. We will mainly consider high-pass filters for low-frequency noise, setting \(\omega_{L} = 0\) such that filtering takes place in the stopband up to the cutoff \(\omega_{c}\). In this case it is useful to perform the Taylor expansion of the filter-transfer function about \(\omega= 0\), written

*ωτ*due to the evenness of \(F_{i}(\omega\tau)\). Assuming sufficiently low-frequency noise (\(\omega_{c}<1/\tau\)), the approximation \(F(\omega\tau)\propto(\omega\tau)^{2p}\) holds for some

*p*associated with the most significant power law expansion term. This defines a high-pass filter with

*filter order*(determined by

*p*) visualized as the slope in the stopband on a log-log plot.

^{5}

*concurrent zero*of the first \((p-1)\) Taylor coefficients. That is, if \(C^{(i)}_{2}(\boldsymbol {\Gamma }_{n}) = C^{(i)}_{4}(\boldsymbol {\Gamma }_{n}) = \cdots= C^{(i)}_{2(p-1)}(\boldsymbol {\Gamma }_{n}) = 0\). In this case we approximate \(F_{i}(\omega\tau; \boldsymbol {\Gamma }_{n})\approx C^{(i)}_{2p}(\boldsymbol {\Gamma }_{n})(\omega\tau)^{2p}\) and consequently \(A_{i}(\boldsymbol {\Gamma }_{n})\approx C^{(i)}_{2p}(\boldsymbol {\Gamma }_{n})\frac{(\omega_{c}\tau)^{2p+1}}{2p+1}\). Thus \(\boldsymbol {\Gamma }_{n}\) is a \((p-1)\)-order (high-pass) filter in the

*i*th noise quadrature if the following equivalent conditions are satisfied

It is important to disambiguate the asymptotic filter order \((p-1)\), introduced above for characterizing the behaviour near zero frequency, from a more general metric capable of describing filter performance over an arbitrary spectral band. For this we introduce the *local filter order*
\((p^{*}-1)\) by the property that, over the band \([\omega _{L},\omega_{c}]\) the filter-transfer function is well approximated by \(F_{i}\propto(\omega\tau)^{2p^{*}}\). One may take the limit that \(\omega _{L}\rightarrow\omega_{c}\rightarrow\omega^{*}\) and thereby obtain the *instantaneous filter order*, effectively measuring the power-law behaviour at \(\omega^{*}\). Both local and instantaneous filter order reduce to the asymptotic filter order over the stopband if over this region \(F_{i}\) is well-characterized by its the zero-frequency behaviour.

### 4.3 Time-domain filter order vs. Magnus order

Both the asymptotic and instantaneous filter orders defined above for time-domain noise must be distinguished from the *Magnus order* of error cancellation. The latter is familiar from work in NMR in which quasi-static errors can be cancelled by suitable composite pulse sequence design. The regime of quasistatic errors coincides with the DC limit for the time-dependent noise fields introduced in Section 2. That is, the time-dependent noise fields reduce to scalar constants \(\beta_{z} (\beta_{{\Omega}})\). The Magnus expansion terms in (70), now denoted \(\Phi^{(\mathrm{DC})}_{\mu}\), are then evaluated strictly as time integrals over *ideal* control operations, scaled by factors \(\beta_{z} ^{\mu}(\beta_{{\Omega}}^{\mu})\) specifying the power law dependence on the magnitude of these static offsets errors. A pulse sequence for which \(\Phi^{(\mathrm{DC})}_{1} = \cdots= \Phi^{(\mathrm{DC})}_{\mu -1} = 0\) is then said to compensate offset errors to Magnus order \((\mu -1)\). In this case the total error operator satisfies \(\Phi^{(\mathrm{DC})}(\tau)=\mathcal{O}(\Phi^{(\mathrm{DC})}_{\mu})\) and is dominated by the residual error proportional to the *μ*th power in the magnitude of the error.

This is quite distinct from time-dependent noise where the error expansion used to calculate the fidelity contains terms of various Magnus order but equivalent time-dependent error norm in the ensemble average (see, *e.g.* Eq. 1 in Ref. [48]). The net result is the observation that *high-order error suppression in the Magnus expansion does not imply high-order time-domain noise filtering*. This has been validated using experiments on trapped ions [37], and formalized rigorously in Ref. [38], where it has been shown that \(p\leq\mu\), but \(p^{*}\) over a user-defined band is unrelated to *μ*. Our focus throughout this work will be on crafting efficient noise filters rather than high-order error suppressing gates.

## 5 Filter design by Walsh synthesis

Even with the general insights into the appropriate modulation protocols outlined above, it is desirable to bound the dimensionality of the control space, and hence the complexity of the filter-design task, by imposing physically motivated constraints on the form of \(\boldsymbol {\Gamma }_{n}\). In practice the achievable filter order is typically limited by the number of unitary operations in the control sequence; one may increase \((p-1)\) at the cost of increasing *n*. From an experimental standpoint, faced with the physical limitation set by a maximum achievable Rabi rate, this cost manifests as a longer total sequence duration \(\tau= \hat {\sigma }_{l}^{n}\tau_{l}\). This may offset the proposed benefit of the higher-order filter due to a longer noise interaction time. From a theoretical standpoint the cost is in the greater complexity of the variational search; the number of (free) variational parameters in \(\boldsymbol {\Gamma }_{n}\) grow as 3*n* and the number of matrix products in Eqs. (15) and (16) grows as *n*.

*synthesizing*relevant time-domain control fields in the basis set of Walsh functions - square wave analogues of the sines and cosines [51, 61] - using the concept of functional analysis. Walsh functions are defined in a uniform piecewise-constant fashion (Figure 2), building intrinsic compatibility with discrete clocking [54] and classical digital logic. Since their formulation in the first half of the twentieth century [62] they have played an important role in scientific and engineering applications. Their development and utilization has been strongly influenced by parallel developments in digital electronics and computer science since the 1960s, with Walsh-type transforms replacing Fourier transforms in a range of engineering applications such as communication, signal processing, pattern recognition, noise filtering and so forth [61, 63].

More recently the Walsh functions have been identified as an attractive resource in quantum information, with applications in time-resolved magnetometry using nitrogen-vacancy centres in diamond [64] and in DD for digital-efficient pulse sequencing [51]. Notably, in the latter scheme the decoupling performance was found to be determined by the distinct symmetry and spectral properties of the Walsh basis. These properties enable us to establish *analytic design rules* (see Section 5.3) to further streamline Walsh-synthesized filter construction.

We begin by reviewing the relevant mathematical details of the Walsh basis. Two equivalent representations are introduced, *Paley ordering* and the *Hadamard representation*, which shall be used throughout this paper.

### 5.1 The Paley and Hadamard representations

*frequency*. A number of different orderings exist [61, 65, 66] due to the different ways in which the basis elements may be defined. We employ the

*Paley ordering*[67] in which basis functions are generated from products of

*Rademacher functions*[68], defined by

*j*th Rademacher function \(R_{j}(x)\) is thus a periodic square wave switching \(2^{j-1}\) times between ±1 over the interval \([0,1]\). The Walsh function of Paley order

*k*, here denoted \(\operatorname {PAL}_{k}(x)\), is then defined by

*k*. That is, \(k = b_{m}2^{m-1}+b_{m-1}2^{m-2}+\cdots+b_{1}2^{0}\), where \(m(k)\) indexes the most significant binary digit, having defined \(b_{m}\equiv 1\). Consequently, \(\operatorname {PAL}_{k}(x)\) has factors \(R_{j}(x)\) whenever \(b_{j}\) is a nonzero binary digit of

*k*; the total number of Rademacher functions in the construction of \(\operatorname {PAL}_{k}(x)\) is thus given by the number of nonzero \(b_{j}\)’s in

*k*- namely, the

*Hamming weight*denoted \(r(k)\). For a given value of \(m(k)\), the maximum number Rademacher functions therefore occur for \(\operatorname {PAL}_{2^{m(k)}-1}(x)\). For example, setting \(m(k) = 3\), a maximum of three Rademacher functions are used to construct \(\operatorname {PAL}_{7}(x) = R_{3}(x)R_{2}(x)R_{1}(x)\), corresponding to the three nonzero digits \(b_{1,2,3}\) in the binary expansion \(k = 7 = (1,1,1)_{2}\). For illustration, the first 32 Walsh functions in the Paley ordering are shown in Figure 2.

The discrete-timestep properties of these basis functions produce, under linear superposition, piecewise-constant waveforms with digitized segment lengths. In our framework these segments are used to specify the a modulation of the control field, ultimately defining a piecewise-constant sequence of unitaries. We therefore require a straightforward expression for the envelope of an arbitrary synthesis \(\sum_{k = 0}^{N} X_{k}\operatorname {PAL}_{k}(x)\). Due to the aperiodicity of the Walsh functions, however, a general expression in Paley ordering is difficult. To overcome this it is convenient to use the Hadamard representation.

*j*th element, \(P^{(k)}_{j}\in\{\pm1\}\), defined by the value of \(\operatorname {PAL}_{k}(x)\) in the

*j*th bin. That is, \(\operatorname {PAL}_{k}(x)\) is isomorphic to the discrete digital vector written

*discrete*Walsh basis spanning \(\mathbb{R}^{2^{n}}\).

*Sylvester construction*[69] the \(2^{n}\)-dimensional Hadamard matrix \(H_{2^{n}}\) is generated recursively by

*S*is the Sylvester matrix, and ⊗

*n*denotes \(n\ge1\) applications of the Kronecker product. In this construction \(\mathbf{P}^{(k)}_{2^{n}}\) defines the \(i(k)= 1+\sum_{j=1}^{m(k)}b_{j}2^{n-j}\) column (row) of \(H_{2^{n}}\). The orthogonality of the Walsh basis is thereby reflected in the familiar property that \(H_{2^{n}}H_{2^{n}}^{T} = 2^{n}I\), implying the orthogonality of the Hadamard matrices.

The Hadamard representation of the Walsh functions has the distinct advantage of naturally specifying the piecewise-constant structure of time domain sequences constructed via linear combinations of Walsh functions. Any function synthesized in the Paley-ordered Walsh basis, \(f(x) = \sum_{k = 0}^{2^{n}-1} q_{k}\operatorname {PAL}_{k}(x)\), has a vector representation in the column space of \(H^{2^{n}}\). In this section we will use this observation to efficiently construct Walsh-synthesized filters, whose properties map compactly onto the Walsh spectrum.

### 5.2 Walsh-synthesized filters

*single-axis amplitude-modulated filters*(AMFs) and

*constant-amplitude phase-modulated filters*(PMFs). These constrained forms may be used to design filters for dephasing and amplitude noise

*separately*using minimal control resources. For \(\hat {\sigma }_{z}\) (dephasing) noise it is sufficient to employ rotations about a single (orthogonal) axis in the

*xy*plane and therefore restrict attention to AMFs. On the other hand, unless implementing the trivial identity gate such that the total gate rotation angle \(\Theta\equiv \sum_{l = 1}^{\mathcal {M}}\theta_{l}=0\), strict amplitude modulation is insufficient for filtering amplitude noise.

^{6}For nontrivial gates, amplitude noise filters generally require control over the rotation axis, and for this purpose we employ PMFs.

*Walsh amplitude*- (WAMF) and

*phase-*(WPMF)

*modulated filters*. To compactly express these modulated control forms as sequences of unitaries we now employ the Hadamard representation.

*N*. From Section 5.1 all basis functions in this synthesis projected onto a Hadamard matrix of (minimum) dimension \(\mathcal {M}(N)\equiv2^{m(N)}\). A discrete representation of the function \(f(x)\) therefore exists as a projection onto the column space of \(H_{\mathcal {M}}\) by writing

*l*th timestep. In this case \(\tau_{l}\) takes a fixed discrete value, though consecutive segments with the same values of \(\Omega_{l}\) and \(\phi_{l}\) may be combined sequentially to form effective operations of longer duration. The remaining degrees of freedom reside in the functional dependencies of \(\Omega_{l}(\mathbf{X})\) and \(\phi_{l}(\mathbf{Y})\) on the Walsh spectra,

^{7}the explicit forms of which are determined by the Hadamard matrix equations above.

The reduced control space, now compactly parameterized by the Walsh spectra, thus consists of bounded-strength unitary sequences inheriting the discrete timing properties of the Walsh basis. This contrasts with similar composite pulse methods in NMR and quantum information [20, 27, 70] which generally rely on structures defined in continuous time.^{8} In the next section we identify useful properties of the Walsh basis which capture filter performance and hence inform effective filter design.

### 5.3 Analytic filter-design rules

From Eqs. (31) and (32) the WAMF (WPMF) constructs are completely parameterized by the Walsh spectra \(\mathbf{X}^{(i)}\), \(i\in\{z,\Omega\}\). Here, for compactness, we denote \(\mathbf{X}^{(z)}(\mathbf{X}^{(\Omega)}) = \mathbf{X}( \mathbf{Y})\). Filter properties and gate characteristics consequently map onto the basis functions in the synthesis.

*variational*(\(\mathbf{X}^{(i)}_{\nu}\)) and

*fixed parameters*(\(\mathbf{X}^{(i)}_{\rho}\)). Making the formal substitution \(\boldsymbol {\Gamma }_{\mathcal {M}}\rightarrow\mathbf{X}^{(i)}\), the cost function in Section 4 is consequently re-expressed

- (1)
Alternate modulation quadratures for dephasing or amplitude noise.

- (2)
Restricting Walsh synthesis by symmetry considerations.

- (3)
Constraining Walsh spectra for target gate angle.

- (4)
Achievable filtering characteristics determined by \(m(k)\) and \(r(k)\).

(i) *Alternate modulation quadratures for dephasing or amplitude noise* - As the most basic element of design, we first reiterate the statements made above establishing WAMFs (WPMFs) as useful for filtering dephasing (amplitude) noise *separately*. In Section 9, however, we derive universal noise filters by concatenating these two filter constructs.

(ii) *Restricting Walsh synthesis by symmetry considerations* - As with the cosines (sines) constituting the Fourier basis, the Walsh basis separates into so-called CAL (SAL) functions with even (odd) parity. Restricting the synthesis to the CAL subset ensures the modulated waveform has time-reversal symmetry about the sequence midpoint \(\tau/2\). This can be a convenient and effective method in filter design, in line with the observation in dynamic decoupling literature [13, 71] that sequence performance is often improved using time-symmetric over -asymmetric building blocks.^{9}

*Constraining Walsh spectra for target gate angle*- Imposing desired physical properties on a candidate control sequence may generally be achieved by holding some subset \(\mathbf{X}^{(i)}_{\rho}\) of the Walsh-spectral amplitudes constant. For example, we may fix the total rotation angle of the Bloch vector in order to implement a target logic operation. For WAMFs this involves a very straightforward constraint on the Walsh spectrum: the total rotation angle depends only on \(X_{0}\). This can be seen as follows. First observe for Paley orders \(k\ge1\) the Walsh functions are balanced in the sense that \(\int_{0}^{1}\operatorname {PAL}_{k}(x)\,dx = \delta_{0k}\), where \(\delta _{ij}\) denotes the Kronecker delta. For WAMFs the total gate angle \(\Theta\equiv\int_{0}^{\tau}dt \Omega(t)\) therefore takes the form

*θ*.

(iv) *Achievable filtering characteristics determined by*
\(m(k)\)
*and*
\(r(k)\) - The achievable filter order over the entire stopband is essentially limited by the number of constituent control operations: one may achieve higher *p* at the cost of higher *n*. For the Walsh-synthesized filters in Eqs. (31) and (32), with *N* the highest-order basis function, \(n \equiv2^{m(N)}\). Hence higher-order Walsh functions generally produce higher-order filters.

*x*to the (non-dimensional) angular frequency variable

*ωτ*in Fourier space. Increasing the low-frequency

*roll-off*is therefore associated with maximizing \(r(k)\) for a given number of control operations \(n = 2^{m(k)}\). This corresponds to maximizing the number of Rademacher functions in the construction

^{10}and immediately identifies Paley orders \(k = 2^{\alpha}-1\), \(\alpha\in\mathbb{N}\), (see Figure 2) as key design resources.

## 6 Walsh Amplitude Modulated Filters (WAMFs)

Having introduced the basic physical picture and mathematical basis for Walsh filter synthesis, we move on to demonstrate explicit realizations of WAMFs for dephasing noise. Both first and second-order filters with high-pass filter characteristics are constructed.

### 6.1 First-order WAMFs

We begin by considering first-order filters for dephasing noise implementing target single-qubit rotations. Construction begins by considering the design rules (i)-(iv) outlined in Section 5.3. For filtering noise in this quadrature (i) implies we should employ the WAMF construction (Eq. (31)). In this case, invoking (iii), the requirement of implementing nontrivial gates dictates we include Paley order \(k = 0\) in the synthesis. The average Rabi rate (and hence rotation angle) is then determined by \(X_{0}\), the spectral amplitude of \(\operatorname {PAL}_{0}(t/\tau)\). The remaining synthesis choices include basis functions of Paley order \(k>0\) and are in principle unbounded.

As a first application, we pursue the construction minimizing the number \(n = 2^{m(N)}\) of unitary operations in the synthesized sequence such that error suppression is still attainable. In line with design rules (ii) and (iv), time-reversal symmetry is ensured and the number of Rademacher functions is maximized by reducing the remaining synthesis choices to the single basis function \(\operatorname {PAL}_{3}(t/\tau)\) (Figure 2). Hence, in this simple example, \(N = 3\), and \(\mathcal {M}(N) = 2^{m(N)} = 2^{2}\), yielding 4-segment gates with segment lengths \(\tau_{\mathcal {M}}= \tau/4\). These represent the lowest-order constructions with error suppression capabilites.

*fixed*parameter (see Eq. (36)) while \(X_{3}\) is treated as a

*variational*parameter by which to optimize the (dephasing) cost function (Eq. (34)). Thus, values of \(X_{3}\) for which \(A_{z}(X_{3};X_{0})\) is minimized specify the optimum modulation depths for an effective filter.

Figure 3(a) shows a two-dimensional representation of \(A_{z}(X_{3};X_{0})\) integrated over the stopband \(\omega\in [10^{-9}, 10^{-1}]\tau^{-1}\). The value of \(\operatorname{Log}_{10} [A_{z}(X_{3};X_{0}) ]\) is indicated by the color scale. Total sequence length is normalized to \(\tau=1\) in this data, so the total gate rotation angle \(\Theta\equiv X_{0}\) is given directly by the \(X_{0}\)-axis. As can be seen, for any fixed \(X_{0}\) there exist quasi-periodic tunings of \(X_{3}\) which minimize the cost function. In other words, we have a prescription for synthesizing *spectrally-optimized dephasing filters which implement arbitrary rotation angles*. Interestingly, the point \((X_{0},X_{3})\equiv( 3\pi,\pi)\) reproduces the previously derived first-order DCG NOT construction [19].

*first-order*filters for low-frequency noise due to the restrictions placed on the synthesis space.

^{11}To demonstrate that these optimized \(\mathrm{WAMF}^{(1)}_{0,3}\) gates perform as first-order filters we Taylor expand \(F_{z} (\omega; X_{3};X_{0})\) as in Eq. (35), and derive an easy analytic expression for the first order coefficient

The corresponding dephasing filter-transfer functions for these three optimized gates are shown in Figure 3(c). As expected from Eq. (21), with \(C_{2}^{(z)} = 0\), these approximately satisfy \(F_{z} \propto(\omega\tau)^{4}\) in the stopband, producing first-order filters with \((p-1) = 1\). For comparison we include the dephasing filter-transfer function for a primitive *π* rotation where \(F_{z} \propto(\omega\tau)^{2}\), implying \((p-1)=0\). The steeper slopes, or *roll-offs*, for the optimized \(\mathrm{WAMF}_{0,3}^{(1)}\) gates captures this difference. This filter design method, and the performance of the \(\mathrm{WAMF}^{(1)}_{0,3}\) filters, has recently been experimentally validated by our group [37].

### 6.2 Second-order WAMFs

We now consider higher-order dephasing filters by increasing the number *n* of segments in the sequence. In particular we consider 8-segment gates. Construction again begins by considering the design rules (i)-(iv) outlined in Section 5.3.

Using (i) and (iii) we employ the WAMF construction and include Paley order \(k=0\) to ensure nontrivial rotation angles. Extending to 8-segments, however, increases the accessible range of Walsh functions in the synthesis as identified in design rule (iv). Specifically we extend the synthesis to Paley orders \(k\le7\) corresponding to the complexity class \(m(k)\le3\), implying a \(2^{3} = 8\) segments construction in the Hadamard representation. We denote these constructions by \(\mathrm{WAMF}_{0:7}^{(2)}\) where the superscript indicates second-order filtering capabilities, as will be shown. Imposing time-reversal symmetry about \(\tau/2\) further restricts the synthesis space to \(k\in\{3,5,6\}\), corresponding to CAL functions referenced in design rule (ii). We therefore study synthesized filters with spectral amplitudes partitioned into fixed \(\mathbf{X}_{\rho}= X_{0}\) and variational \(\mathbf{X}_{\nu}= (X_{3}, X_{5}, X_{6})\) classes.

As a representative example we set \(X_{0} = 3\pi\) and restrict attention to filters implementing a net *π* rotation (\(\tau\equiv 1\)). Our cost function consequently takes the form \(A_{z}(X_{3},X_{5},X_{6};3\pi)=\int_{\omega_{L}}^{\omega_{c}} F_{z} (\omega\tau; X_{3},X_{5},X_{6};3\pi)\,d\omega\), implying a three-dimensional *variational* control space over which to derive spectrally-optimized filters. We accomplish this using a Nelder-Mead search to minimize \(A_{z}(\mathbf{X}_{\nu};3\pi )\) over the \(\mathbf{X}_{\nu}\)-domain.

*instantaneous filter order*ranging between \(2<(p^{*}-1)<3.8\) at various points.

^{12}For comparison we also plot the dephasing filter-transfer function for a primitive

*π*rotation (black dashed trace) and an optimized \(\mathrm{WAMF}_{0,3}^{(1)}\) gate (yellow dashed trace). These respectively satisfy \(F_{z} \propto(\omega\tau)^{2,4}\) over the whole stopband and are well characterized by the (asymptotic) filter orders \((p-1)=0,1\).

Figure 4(a) shows a two-dimensional representation of \(A_{z}(X_{5},X_{6})\) defined on a two-dimensional cross-section of the \(\mathbf{X}_{\nu}\)-domain, holding \(X_{3}\) fixed, and integrated over the stopband \(\omega\in [10^{-9}, 10^{-1}]\tau^{-1}\). The value of \(\operatorname{Log}_{10} [A_{z}(X_{5},X_{6}) ]\) is indicated by the color scale. Areas in blue indicate minima in \(A_{z}(X_{5},X_{6})\), indicating optimized paths in \(X_{5}X_{6}\) plane the over which effective filters may be found. Notably, it is possible to find ‘cross-regions’ (circled) in which the spectral amplitudes \(X_{5}\) and \(X_{6}\) may independently be varied substantially without the cost function moving off a local minima. This potentially indicates the existence of classes of WAMFs which may be robust to errors in the Walsh spectrum itself.

## 7 Walsh Phase Modulated Filters (WPMFs)

We now turn to filters for amplitude-damping noise constructed via phase modulation using the WPMF construction set out in Eq. (32). Following the same procedure described above for WAMFs, one can implement a Nelder-Mead search to derive spectrally-optimized WPMFs which implement nontrivial rotations. For these constructions, however, the target rotation angle is dependent on both the Rabi rate and the sequence of phase modulations. Consequently it is less straightforward to impose a constraint during the optimization procedure to ensure a particular target rotation. Although we do not pursue the general problem in detail in this paper, we demonstrate the approach in this and the following sections, deriving a family of WPMFs in which the synthesis space is limited to a variety of simple combinations of Walsh function.

In the remainder of this section we study a variation on the strict WPMF structure which resolves the difficulty of imposing a target rotation and enables us to make some useful connections with existing composite pulse sequences in NMR. This variation involves partitioning the control modulation into a target rotation \(P(\theta,0)\) followed by a sequence of phase-modulated *identity* operations \(\prod_{l = 1}^{\mathcal {M}} P(2\pi,\phi_{l})\) with the \(\phi_{l}\) chosen such that amplitude noise is filtered to some order. Here the operator \(P(\theta ,\phi)\) denotes the rotation through angle *θ* about \(\hat {\sigma }_{\phi}\). By insisting these \(\mathcal {M}\) ‘correction’ segments are all identity operations, the phase modulations do not produce complicated rotation paths and the net rotation is determined simply by the initial target pulse.

*c*) and write the vectorized phase \(\vec{\boldsymbol{\phi}}_{c}\) to indicate Walsh modulation during the ‘correction stage’ of the sequence, disambiguating this from the strict WPMF structure. For a given

*θ*we may now treat \(Y_{k}\) as a tuning parameter which may be optimized by minimizing the cost function \(A_{\Omega}(Y_{k};\theta)\equiv \int_{0}^{\omega_{c}}d\omega F_{{\Omega }}(\omega\tau;Y_{k};\theta)\). The optimized \(Y_{k}\) are thereby defined as an implicit function of

*θ*.

*θ*) sequence [72, 73] and the second-order Wimperis passband P2(

*θ*) sequence [27] by setting \(k = 1,3\) respectively. Hence these well-known NMR sequences, originally designed to compensate for

*static*amplitude errors to first and second

*Magnus order*respectively (see Section 4.3), appear in the Walsh filter space as phase-modulated filters for non-Markovian amplitude noise. Table 1 summarizes this.

**Filter characteristics of**
\(\pmb{\mathrm{WPMF}^{(c)}_{k}}\)
**constructions corresponding to well-known NMR sequences, SK1 and P2, originally designed to compensate for**
static
**amplitude errors to first and second**
Magnus order
**respectively (Section **
**4.3**
**)**

\(\mathrm{WPMF}^{(c)}_{k}\) construction | Amplitude errors | |||||
---|---|---|---|---|---|---|

| \(\mathcal {M}(k)\) | \(\vec{\boldsymbol{\phi}}\) | \(Y_{k}(\theta)\) | ( | ( | |

SK1( | 1 | 2 | \(Y_{1}(\theta )\operatorname {PAL}_{1}\) | cos | 1 | 1 |

P2( | 3 | 4 | \(Y_{3}(\theta )\operatorname {PAL}_{3}\) | cos | 2 | 1 |

_{0}and PAL

_{3}, setting \(\tilde{\mathbf{Y}} \equiv (Y_{0},0,0,Y_{3} )^{T}\) in analogy with the Walsh spectrum defining amplitude modulation in the \(\mathrm{WAMF}_{0,3}^{(1)}\) construction. The first-order filtering condition \(C_{2}^{(\Omega )}(Y_{0},Y_{3},\Omega_{0};\theta) = 0\) then implies solutions

*static*amplitude errors to second

*Magnus order*, it only provides

*first-order*noise filtering.

## 8 Walsh Rotary Spin Echo (WRSE)

In this section we treat a sub-class of Walsh modulated filters which may be described either in terms of phase- or amplitude-modulation. The phase-modulation consists of applying a sequence of *π*-phase shifts, relative to some offset \(\phi_{0}\), on the driving field with a constant amplitude \(\Omega _{0}\). This construction generalizes the rotary spin echo (RSE) sequence from NMR, analogous to the Hahn-echo sequence for driven systems, consisting of a single *π*-phase shift applied at the sequence midpoint \(\tau/2\). In quantum information RSE has been employed, for example, in relaxation noise spectroscopy [75] and in mitigating low-frequency off-resonant noise [76] in driven superconducting flux qubits. In contrast with previous approaches, our generalization permits higher-order filter performance in both amplitude and dephasing quadratures and may prove of significant use.

*switching function*defined to change sign at the application of each

*π*-shift. Specifically, we consider sequences based on Walsh functions by defining the switching function

*π*shift, and since any Walsh function of Paley order \(k>0\) is equally distributed between values ±1 over the domain these rotations perfectly cancel, yielding zero total rotation. This is formally equivalent to modulating the

*sign*of the Rabi rate and holding the phase \(\phi_{0}\) constant (see Appendix C), as schematically illustrated in Figure 5(a). These sequences, referred to as

*Walsh rotary spin echo*order

*k*(\(\mathrm{WRSE}_{k}\)), thus take the form

### 8.1 WRSE as amplitude filters

### 8.2 WRSE as dephasing filters

*k*, it follows the Taylor coefficients are functions only of \(\Omega _{0}\). Filtering to order \((p-1)\) then corresponds to the condition

*η*denotes some value of \(\Omega_{0}\) for which the above coefficients are concurrently zero. Here we have included the Paley order as a parameter of the coefficients. Analysis shows, however, concurrent zeros exist only for \(j\in\{1,2\}\). We effectively obtain the following ‘no-go theorem’: \(\mathrm{WRSE}_{k}\)

*sequences perform as (high-pass) dephasing noise filters up to (but not beyond) second order*. This result may be of use to characterize the relevant quadrature of an unknown noise source, by probing with higher-order \(\mathrm{WRSE}_{k}\) sequences and determining the resulting filtering properties.

*π*, and concurrent zeros of \(C^{(z)}_{2,4}\) occur at multiples of 8

*π*. However, since \(C^{(z)}_{6}\) is never zero at multiples of 8

*π*(see inset to Figure 6(a)), it follows we can never achieve higher than second-order filtering. We verify this by examining the slope of \(F_{z} (\omega;\Omega_{0})\) for the values \(\Omega_{0} = 2\pi q\), \(q\in\{1,\ldots,8\}\) (see Figure 6(b)). Similar considerations for other values of

*k*generalize the result.

## 9 Universal Walsh Modulated Filters (UWMFs)

In the previous sections we have considered WAMF and WPMF gates which implement target qubit rotations while filtering, to some order, *either* dephasing or amplitude noise respectively. In this section we derive filters for universal noise by concatenating both filter types into a composite structure that filters both noise quadratures simultaneously, while still implementing a target qubit rotation. We refer to such constructions as *universal Walsh modulated filters* (UWMFs).

*π*rotation) are shown in Figure 7(d), indicating effective filtering of both amplitude and dephasing noise. Below we detail two alternative constructions for realizing the UWMF structure.

### 9.1 Concatenation Method 1: Constrain Sequencing of WAMF Envelope

*not*, however, correspond directly to the optimum Walsh coefficients found for simple \(\mathrm{WAMF}_{0,3}^{(1)}\) construction shown in Figure 3(a). Rather, an equivalent tuning plot may be generated over the \(X_{0}X_{3}\) domain, essentially identical to Figure 3(a) but with minima shifted by a constant factor. The second method, detailed below, involves a slightly different construction in which we recover the original \(\mathrm{WAMF}_{0,3}^{(1)}\) tuning plot.

### 9.2 Concatenation Method 2: Constrain Sequencing of Target SK1 Rotations

In the second construction we impose the constraint that \(\tau_{1}:\tau _{4}:\tau_{7} = 1:2:1\). That is, the *target rotations* within the three successive SK1 sequences follow the same timing sequence as the three constant-amplitude pulses being replaced, previously constituting the amplitude-modulated \(\mathrm{WAMG}_{1}\) envelope. Thus we write \((\tau_{1},\tau_{4},\tau_{7} ) = \nu (1,2,1)\) where *ν* is some fraction of the total duration *τ* of the composite structure to be determined.

*ν*,

*concatenated*structure, yielding a desired net rotation (dictated by \(X_{0}\)) which filters both amplitude and dephasing noise to first order simultaneously.

## 10 Effect of bandwidth limits on Walsh filters

In the preceding sections of this paper filter design is based on optimizing the Walsh spectrum from which the relevant control structures are synthesized. This necessarily assumes perfectly square waveforms. Real control hardware, however, may suffer from bandwidth limitations which ‘smooth out’ the squareness of the pulse on the timescale of the application, leading to reduced filter performance. Here we show the assumption of perfect square pulses may be readily relaxed, with useful filter construction a simple matter of re-optimization in the Walsh-synthesis framework.

To illustrate the general procedure we consider the \(\mathcal {M}\)-segment WAMF. Each segment implements a rotation through angle \(\theta_{l} = \Omega_{l}\tau_{l}\) over duration \(\tau_{l}=\tau/\mathcal {M}\) and with constant Rabi rate \(\Omega_{l}\), \(l\in\{1,\ldots, \mathcal {M}\}\). The squareness of the resulting amplitude-modulated waveform may be relaxed by replacing the constant value \(\Omega_{l}\) with an arbitrarily varying function of time in each segment.

*rotation angle*implemented in a single segment rather than the Rabi rate. That is, we write \(\theta_{l} = \theta_{l}(X_{0}, X_{1},\ldots, X_{N})\) with the dependence on the Walsh spectra defined by the Hadamard-matrix equation

_{0}. The symmetry-based design rules similarly carry over and filter optimization proceeds in the same manner as for ordinary Walsh-modulated control by minimizing the filter cost function with respect to the Walsh spectrum.

*g*of the segment duration. The normalizing factor

*l*th segment is given by \(\int_{t_{l}}^{t_{l-1}} G_{l}(t;\mu _{l},\sigma_{l})\,dt = \theta_{l}\). We now impose the same structure on the segment rotations \(\theta_{l}\) as done for WAMFs in the Walsh-synthesis framework, such that the smooth-pulse sequence remains strictly parametrized in the Walsh spectrum

**X**.

Comparing with Figure 3 we conclude useful filter construction using Gaussian pulses is a simple matter of re-optimization in the Walsh-synthesis framework. This is readily achieved using a Nelder-Mead optimization of \(A_{z}(X_{3};X_{0})\) for any particular choice of *g*, \(\omega_{L}\), \(\omega_{c}\), \(X_{0}\) or \(N_{s}\). Minor changes in the filter performance and optimal constructions arise with changes in Gaussian pulse parameters such as *g*. Comparison with pulses constructed using a trapezoidal form (Figure 8(e)) we find a different optimization outcome that more closely approximates standard square pulses. Nonetheless, these results show that, irrespective the specific pulse form, re-optimization over the Walsh coefficients remains a direct method to construct useful filters. In cases where unknown waveform distortion is likely in hardware, it is possible to implement automated feedback mechanisms, as has previously been demonstrated in dynamical decoupling experiments [15].

We may also explore the impact of finite modulation bandwidth on the application of square pulses if re-optimization of the waveform is, for some reason, not possible. In order to explore these effects we systematically relax the infinite-modulation-bandwidth assumption underlying any square-pulse approximation by processing the ideal time-domain profile through a bandlimited digital filter with a user-defined cutoff. This results in an imperfect (bandlimited) profile envelope, effectively due to a reconstruction based on a truncated Fourier series. These profile distortions, manifesting as implementation errors, reduce filter performance, quantified by an increase in the area under the corresponding filter-transfer function.

## 11 Conclusion

As the size and complexity of quantum information processing technologies increase, resource-efficiency will play a vital role in selecting methods designed to reduce errors in quantum coherent hardware systems. The pressure to minimize quantum-hardware overhead is likely to make *open-loop control* protocols a key element in the design of error-robust quantum information systems [14]. For these to be practically useful, however, it is important to move toward realistic control and noise models.

Decoherence in real driven systems is predominantly due to low-frequency *correlated* noise environments. This strongly motivates our study of bounded-strength control as a noise filtering problem using time-dependent, non-Markovian error models. Moreover, in contrast with traditional DD schemes, the added complications of treating bounded-strength control - due to the continual presence of noise interactions during control operations and the resulting nonlinear dynamics - necessitates a streamlined approach to the design of noise-filtering control. The generalized filter-transfer function framework we employ takes as input experimentally measurable characteristics of the environment - namely noise power spectra - and provides a simple framework for both control construction and the calculation of predicted operational fidelities. It also efficiently captures the control nonlinearities implicit in situations where control and noise Hamiltonians do not commute. We have exploited these strengths to pursue a simple variational procedure for filter design by minimizing a cost function over the relevant control space.

A key strength of the method we have presented is derived from our use of functional analysis for the crafting of effective noise-filtering control protocols. In particular, employing the Walsh basis brings an intuitive set of analytic design rules for filter construction that further constrain the possible filter-construction space [51]. For instance, a user-imposed limit on the acceptable number of pulse segments in a filter construction impose additional constraints due to Walsh-function symmetry, spectral properties, and the level of recursiveness of the Walsh functions (measured by the Hamming weight of the Walsh Paley-ordered index).

In addition to efficiency of synthesis is the intrinsic compatibility with hardware controllers that comes with the selection of Walsh functions as our basis set. This is particularly important in the layered architecture for quantum information systems mentioned throughout this paper. In such a setting, it rapidly becomes undesirable to mandate a significant amount of communication between the physical qubit layer and hardware at the highest levels of system abstraction. This suggests that controllers implementing dynamic error suppression protocols (here producing noise filters for arbitrary driven operations) should be reasonably simple to implement in standard digital hardware and should require only limited communication bandwidth to higher levels of the system.

These considerations are explicitly met in crafting control solutions from the Walsh basis. First, the Walsh functions are defined using integer multiples of a fundamental clock period, meaning that limitations of finite timing precision in the definition of a control protocol are automatically inbuilt. Further, given a particular Walsh-modulated control protocol is entirely defined by its Walsh spectrum, programming of the controller can in principle be reduced to a simple vector of numbers representing the Walsh spectrum and minimum timestep. All other information *e.g.* the total time, total number of timesteps, etc., is carried implicitly in the spectrum. Moreover, the actual Walsh-function *generation* is compatible with simple hardware systems (adding of various harmonics of a fundamental square-wave clock) and when Walsh synthesis is performed at the level of the controller hardware, this may provide a path to on-the-fly synthesis of the required modulation waveform. Such capabilities also reduce the complexity of running automated hardware-driven optimization procedures for finding relevant control waveforms [15], by allowing efficient generation of many trial waveforms without the need for large memory stores at the local controller.

Synthesizing all of these considerations the Walsh-modulated noise filters we have developed in this work provide one of the first solutions for error suppression at the physical-qubit level simultaneously meeting the physical and engineering requirements we outline above for scalable control solutions. Using this framework we have derived a range of novel filters, chiefly WAMFs for dephasing noise and WPMFs for amplitude noise. Both are capable of spectral optimization subject to physically motivated constraints such as implementing target qubit rotations. These design forms are also compatible with concatenation for filtering universal noise. Interestingly, our approach unifies a number of existing composite pulse sequencing schemes; we have revealed how Walsh-modulated filter construction naturally incorporates familiar sequences (e.g., DCG, SK1, P2, BB1) in a non-Markovian time-dependent noise context. This potential to incorporate other approaches may prove useful in building a consistent picture of the scope and applicability of the many and varied schemes that continue to be developed by the quantum control community. These considerations make the Walsh basis an attractive design platform and we believe this simple framework will provide a straightforward path for the development of improved quantum control techniques.

The assumption of independence is reasonable, for instance, in the case of a driving field where random fluctuations in frequency and amplitude arise from different physical processes. A general model including correlations between noise processes is possible, however, following the approach outlined by *Green et al.* [49].

The first-order approximation has recently been experimentally tested and demonstrated to produce good agreement in the weak noise limit [37]. For the noise field \(\beta (t)\), this regime is sufficiently characterized by requiring \(\xi^{2}\ll 1\), where the *smallness parameter* is defined by \(\xi^{2}\equiv \langle\beta^{2}(t)\rangle\tau^{2}\equiv\tau^{2}\int_{-\infty }^{+\infty}d\omega S_{\beta}(\omega)\) [48]. The condition \(\xi^{2}<1\) is also required for the Magnus series to formally converge.

For a resonantly driven qubit \(\phi_{l}\) is the phase of the driving field and \(\Omega_{l}\) is linearly proportional to the driving amplitude.

This effect is captured by the multiple slopes in Figure 4(h) which clearly show the difference between the asymptotic zero-frequency *roll-off* and the local slope over targeted regions \([\omega_{L},\omega_{c}]\) in the stopband.

All stopbands ‘turn on’ with a finite response, the functional form of which determines the filter order and the effectiveness of noise suppression. In the stopband this is quantified by the slope, or *roll off* in the language of filter design.

This can be shown by Taylor expanding the amplitude noise filter function \(F_{{\Omega }}(\omega\tau; \boldsymbol {\Gamma }^{\mathrm{AMF}}_{n})\) and deriving the result that \(C^{(\Omega)}_{2}(\boldsymbol {\Gamma }^{\mathrm{AMF}}_{n}) = \frac{1}{4} (\sum_{l = 1}^{n}\theta_{l} )^{2}\).

We use the vectors \(\mathbf{X} \equiv(X_{0}, X_{1},\ldots, X_{N})\) and \(\mathbf{Y} \equiv(Y_{0}, Y_{1},\ldots, Y_{N})\) to compactly write the Paley ordered Walsh spectra implied by Eq. (29) in synthesizing \(\Omega(t)\) and \(\phi(t)\).

Pulse periods taking non-integer multiples values of \(\tau_{\mathrm{min}}\) then have intrinsic conflict with implementation in discretized time via digital control, giving rise to residual errors.

Our studies have not produced proof that this symmetry is strictly necessary. In fact for WPMFs it is not required. However WAMF constructions possessing time-reversal symmetry do appear to yield results more readily, and all WAMFs we have discovered have this property.

Maximizing the number of Rademacher functions does *not* correspond to maximizing the switching rate of \(\operatorname {PAL}_{k}(x)\). In fact, for a given \(m(k)\) the maximum switching rate for \(\operatorname {PAL}_{k}(x)\) corresponds to \(k = 2^{m(k)-1}\), which consists of the single Rademacher function \(R_{m(k)-1}(x)\).

These points may also be derived using Nelder-Mead optimization of \(A_{z}(X_{0};X_{3})\) over the two-dimensional domain. This method is useful for more complex constructions (see Section 6.2) where spectral optimization becomes a more multi-dimensional task.

Since the \(\mathrm{WAMF}_{0:7}^{(2)}\) gates in Figure 4 were derived by optimizing the cost function over local regions in the stopband, the asymptotic filter order \((p-1)\) associated with Taylor expanding \(F_{z} (\omega)\) about \(\omega= 0\) is not a meaningful descriptor of these filters. Hence we do not expect \(C^{(z)}_{2,4} = 0\) and do not pursue such a calculation. Instead the instantaneous filter order is used.

This is valid since \(\Phi(\tau)\) belongs to the Lie algebra of \(SU(2)\), inheriting this property from the toggling frame Hamiltonians from which it is derived.

The higher-order \(C^{(k)}_{2j}\) involve terms oscillating at multiple frequencies and have nontrivial dependencies on \(\Omega _{0}\). Their zeros must in general be determined numerically.

## Declarations

### Acknowledgements

We thank K Brown, JT Merrill and L Viola for useful discussions. This work partially supported by the US Army Research Office under Contract Number W911NF-11-1-0068, and the Australian Research Council Centre of Excellence for Engineered Quantum Systems CE110001013.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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