Proof of the Error Scaling for Universally Robust Dynamical Decoupling Sequences

Daniel Lidar; Domenico D'Alessandro; Phattharaporn Singkanipa

arxiv: 2604.25807 · v1 · submitted 2026-04-28 · 🪐 quant-ph · cs.IT· math.IT

Proof of the Error Scaling for Universally Robust Dynamical Decoupling Sequences

Domenico D'Alessandro , Phattharaporn Singkanipa , Daniel Lidar This is my paper

Pith reviewed 2026-05-07 16:34 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT

keywords dynamical decouplinguniversal robustnesserror scalingpulse imperfectionsquantum controlfidelity expansionphase prescriptionerror suppression

0 comments

The pith

URn dynamical decoupling sequences cancel error coefficients in the fidelity expansion up to order n for even n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a rigorous proof that the phase prescription in universally robust dynamical decoupling sequences of even order n meets necessary and sufficient conditions to nullify the low-order terms in a series expansion whose modulus gives the fidelity. This matters for quantum control because it confirms analytically that these sequences suppress arbitrary pulse imperfections to high order while using only a linear number of pulses. A sympathetic reader cares because prior support came from numerics and experiments; the proof now places the construction on exact mathematical footing. If correct, it shows precisely how the chosen phases force cancellation without relying on specific error forms.

Core claim

Using a series expansion of a complex quantity whose modulus is the fidelity F, the authors derive necessary and sufficient conditions for cancellation of its coefficients through order n-1. They then verify that the URn phase prescription for even n satisfies these conditions exactly, which directly implies that the infidelity satisfies 1-F = O(ε^n). The argument holds for arbitrary experimental parameters that produce pulse imperfections.

What carries the argument

The URn phase prescription that assigns specific phases to the pulses in the decoupling sequence, together with the Taylor coefficients of the complex amplitude whose modulus is the fidelity.

If this is right

For even n the infidelity scales exactly as O(ε^n) under the URn construction.
The sequences remain robust against completely arbitrary pulse errors parameterized by ε.
The pulse count grows only linearly with the target suppression order n.
The same phase rules that satisfy the derived cancellation conditions can be used to build sequences of any even order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The proof structure may suggest how to adapt the cancellation conditions for odd n or for continuous-time control.
Because the conditions are necessary as well as sufficient, any sequence that achieves the same scaling must share the same algebraic structure on its phases.
The result clarifies why linear pulse overhead suffices for high-order robustness, which could inform resource estimates in quantum sensing or computing protocols.

Load-bearing premise

The fidelity can be represented by the modulus of one complex quantity whose perturbative coefficients are independently cancellable by phase choices without interference from higher-order terms or non-perturbative effects.

What would settle it

An explicit computation of the fidelity series for a concrete even n and URn sequence showing that all coefficients up to order n-1 vanish while the order-n term remains nonzero, or a simulation demonstrating that altering the phases immediately populates a lower-order term.

read the original abstract

Universally robust dynamical decoupling (UR$n$) sequences were proposed to compensate pulse imperfections arising from arbitrary experimental parameters while achieving high-order error suppression with only a linear increase in the number of pulses. Although their performance was supported by analytical arguments, numerical simulations, and experiments, a complete mathematical proof of the claimed order of error compensation has been absent. In this work, we present a rigorous proof for UR$n$ DD sequences with even $n$. Using a series expansion of a quantity whose modulus is the fidelity $F$, we derive necessary and sufficient conditions for the cancellation of its coefficients up to, but not including, order $n$. The UR$n$ phase prescription satisfies these conditions, and therefore $1-F=O(\epsilon^n)$. Our results establish the UR$n$ construction on firm analytical grounds and clarify the structure responsible for its high-order robustness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper supplies the missing rigorous proof that URn phases cancel lower-order coefficients in the Q expansion for even n, yielding the claimed 1-F=O(ε^n) scaling.

read the letter

The main takeaway is that this work finally delivers the complete mathematical proof for the error scaling of the URn dynamical decoupling sequences. For even n the phases are shown to meet the necessary and sufficient conditions that make the coefficients of Q through order n-1 vanish, so the fidelity error scales as claimed. That proof was explicitly absent before, and the paper supplies it using a series expansion of the auxiliary quantity Q whose modulus is the fidelity F. The derivation is direct: they obtain the cancellation conditions independently of the specific error values and then verify that the URn prescription satisfies them exactly. This puts the earlier numerical and experimental support on firmer analytical footing and clarifies why the robustness holds with only linear growth in pulse number. The approach is clean and the verification step is explicit, which is the real contribution here. The conditions being independent of the pulse-error Hamiltonian is a strong point, since that is what makes the robustness universal. One soft spot worth checking is the move from Q = O(ε^n) to 1-F = O(ε^n). Because the fidelity is obtained via the modulus, higher-order terms could in principle mix into lower orders once the leading coefficients are zeroed out. The paper should confirm that this mixing does not occur under the error model, or that the expansion remains valid without additional assumptions. The abstract states the result holds for arbitrary imperfections, so any hidden dependence would matter, but the derivation as described avoids that. The restriction to even n is also noted; odd n may require separate handling. This is useful reading for anyone working on dynamical decoupling or quantum control who needs the analytical guarantee rather than just simulations. It closes a clear gap in the literature, so it deserves a serious referee to verify the expansion details and the modulus step.

Referee Report

0 major / 3 minor

Summary. The paper presents a rigorous proof for the error scaling of Universally Robust Dynamical Decoupling (URn) sequences with even n. It introduces a complex quantity Q whose modulus equals the fidelity F, expands Q in a power series in the error parameter ε, derives necessary and sufficient conditions for the vanishing of coefficients up to order n-1, and demonstrates that the URn phase prescription satisfies these conditions, thereby establishing 1-F = O(ε^n).

Significance. This work fills a critical gap by providing the first complete mathematical proof of the claimed high-order error suppression in URn sequences. The derivation of independent cancellation conditions from the series expansion of Q is a clear strength, as it is non-circular and parameter-free. If the steps hold, the result places the URn construction on firm analytical grounds, confirming prior numerical and experimental support while clarifying the structure enabling linear pulse overhead for order-n robustness. This is significant for quantum control and dynamical decoupling techniques.

minor comments (3)

The abstract and introduction should explicitly define the complex quantity Q (including its relation to the evolution operator) at the outset rather than assuming familiarity with the error model.
Clarify whether the claimed scaling 1-F=O(ε^n) is the tight bound or a conservative one, given that the modulus |Q| may yield higher-order suppression (e.g., O(ε^{2n})) when the leading coefficient of Q is purely imaginary.
The manuscript states the result holds for even n; a brief remark on the status for odd n (or why the proof technique does not extend) would improve completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation and recommendation of minor revision. The report accurately captures our main result: a non-circular derivation of necessary and sufficient cancellation conditions from the series expansion of Q, whose modulus is the fidelity, followed by verification that the URn phase prescription satisfies them for even n, thereby proving 1-F = O(ε^n). We appreciate the emphasis on the analytical grounding this provides for the linear-pulse-overhead construction.

Circularity Check

0 steps flagged

Derivation of error scaling via independent coefficient cancellation conditions is self-contained

full rationale

The paper expands a quantity Q (with |Q| equal to the fidelity F) in powers of the error parameter ε, derives necessary and sufficient conditions for the coefficients of Q through order n-1 to vanish, and verifies that the URn phase prescription satisfies these conditions exactly, yielding Q = O(ε^n) and therefore 1-F = O(ε^n). This chain relies on direct algebraic verification against the given phase choices rather than any fitted parameter, self-definition of the target scaling, or load-bearing self-citation whose validity is assumed. The conditions are shown to hold independently of the specific arbitrary pulse-error values, consistent with the universal robustness claim. No step reduces the claimed scaling to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the approach relies on standard perturbative expansion in quantum control theory with no free parameters or new entities introduced in the abstract.

axioms (1)

domain assumption Fidelity can be analyzed via series expansion of a complex quantity whose modulus equals F.
Central technical step stated in the abstract.

pith-pipeline@v0.9.0 · 5452 in / 1178 out tokens · 82842 ms · 2026-05-07T16:34:15.084752+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

Hence1−F=O(ϵ n)

The normalized HS overlap satisfies G(ϵ)=1+O(ϵ n)(17) for arbitraryαandβ. Hence1−F=O(ϵ n). Equiva- lently, all coefficients ofG(ϵ)−1of orders1, . . . , n−1 vanish; since only even powers occur, this is cancella- tion through ordern−2

work page
[2]

Hence, for genericα, the fidelity error is of exact orderϵ n

For this UR phase family, the coefficient of orderϵ n in 1−Fis not identically zero as a function ofα. Hence, for genericα, the fidelity error is of exact orderϵ n

work page
[3]

Our goal is to provide a rigorous mathematical proof of this theorem

This order is optimal at fixed sequence lengthn: no α-independent choice of phases{φ k}n k=1 can make all non-constant coefficients ofG(ϵ)−1through orderϵ n vanish identically as functions ofα. Our goal is to provide a rigorous mathematical proof of this theorem. For that purpose, it is convenient to write the phase differences∆ k defined in Eq. (9) in te...

work page
[4]

internal

Moreover, n ∑ j=1 cj =S(r), n ∑ j=1 cjj=L r(XN).(63) Hence Lr(XD)=D 1 + n ∑ j=2 cjDj =D 1 −(φ 2 +nΦ) n ∑ j=2 cj +Φ n ∑ j=2 cjj =D 1 −(φ 2 +nΦ)(S(r)−1)+Φ(L r(XN)−1). (64) Substituting the value ofD 1 and simplifying gives Eq. (59). The second formula follows immediately by substituting Eq. (58) into Eq. (59). The preceding lemmas now combine to give the fi...

work page
[5]

Casen=4m Chooseγ=c=− 1 2 in Eq. (124). Then βj ={ j−1, jodd, −j, jeven. (134) Write P(t)=T 1(t)⋯T 2m(t)T2m+1(t)⋯T 4m(t),(135) where Tj(t)∶=I+tK j =( 1tω βj tω−βj 1 ).(136) Sinceω ±2m=−1, Eq. (134) gives ωβj+2m =−ω βj ,1≤j≤2m.(137) Thus Tj+2m(t)=ET j(t)E, E∶=( 1 0 0−1 ).(138) If Q(t)∶=T 1(t)⋯T 2m(t),(139) then P(t)=Q(t)EQ(t)E.(140) For1≤j≤m, direct verific...

work page
[6]

Casen=4m+2,modd Chooseγ=0andc=N/2in Eq. (124). Then βj =(−1) j+1j+ N 2 .(149) Write again P(t)=T 1(t)⋯T n(t), T j(t)=( 1tω βj tω−βj 1 ).(150) For everyj=1, . . . , N, βj+N =−β j +(1+(−1) j)N.(151) Sincemis odd,ω N =−1, and hence ωβj+N =ω −βj .(152) Therefore, with J∶=( 0 1 1 0 ),(153) 11 we have Tj+N(t)=J T j(t)J,1≤j≤N.(154) Defining Q(t)∶=T 1(t)⋯T N(t),(...

work page
[7]

Casen=4m+2,meven Again set N∶= n 2 =2m+1.(166) Choose γ= N 2m , c=0(167) in Eq. (124). Once again write P(t)=T 1(t)⋯T n(t), T j(t)=( 1tω βj tω−βj 1 ).(168) For everyj=1, . . . , N, direct calculation gives βj+N =−β j ±N.(169) Sincemis even,ω N =1, and hence ωβj+N =ω −βj .(170) Therefore Tj+N(t)=J T j(t)J,1≤j≤N,(171) and, withQ(t)∶=T 1(t)⋯T N(t), P(t)=Q(t)...

work page
[8]

E. L. Hahn, Spin echoes, Phys. Rev.80, 580 (1950)

work page 1950
[9]

H. Y . Carr and E. M. Purcell, Effects of diffusion on free pre- cession in nuclear magnetic resonance experiments, Phys. Rev. 94, 630 (1954)

work page 1954
[10]

Meiboom and D

S. Meiboom and D. Gill, Modified spin-echo method for mea- suring nuclear relaxation times, Rev. Sci. Instrum.29, 688 (1958)

work page 1958
[11]

Viola and S

L. Viola and S. Lloyd, Dynamical suppression of decoherence in two-state quantum systems, Phys. Rev. A58, 2733 (1998)

work page 1998
[12]

Viola, E

L. Viola, E. Knill, and S. Lloyd, Dynamical decoupling of open quantum systems, Phys. Rev. Lett.82, 2417 (1999)

work page 1999
[13]

Khodjasteh and D

K. Khodjasteh and D. A. Lidar, Fault-tolerant quantum dynam- ical decoupling, Phys. Rev. Lett.95, 180501 (2005)

work page 2005
[14]

L. F. Santos and L. Viola, Enhanced convergence and robust performance of randomized dynamical decoupling, Phys. Rev. Lett.97, 150501 (2006)

work page 2006
[15]

G. S. Uhrig, Keeping a quantum bit alive by optimizedπ-pulse sequences, Phys. Rev. Lett.98, 100504 (2007)

work page 2007
[16]

J. R. West, B. H. Fong, and D. A. Lidar, Near-optimal dynami- cal decoupling of a qubit, Phys. Rev. Lett.104, 130501 (2010)

work page 2010
[17]

Wang and R.-B

Z.-Y . Wang and R.-B. Liu, Protection of quantum systems by nested dynamical decoupling, Phys. Rev. A83, 022306 (2011)

work page 2011
[18]

Y . Xia, G. S. Uhrig, and D. A. Lidar, Rigorous performance bounds for quadratic and nested dynamical decoupling, Phys. Rev. A84, 062332 (2011)

work page 2011
[19]

A. D. Bookatz, M. Roetteler, and P. Wocjan, Improved bounded-strength decoupling schemes for local hamiltonians, IEEE Transactions on Information Theory62, 2881 (2016)

work page 2016
[20]

A. F. Brown and D. A. Lidar, Efficient chromatic-number-based multiqubit decoherence and crosstalk suppression, PRX Quan- tum6, 020354 (2025)

work page 2025
[21]

C. Yi, L. Kim, and M. Marvian, Faster randomized dynamical decoupling, Physical Review Letters136, 010601 (2026)

work page 2026
[22]

Viola, S

L. Viola, S. Lloyd, and E. Knill, Universal control of decoupled quantum systems, Phys. Rev. Lett.83, 4888 (1999)

work page 1999
[23]

Zanardi, Symmetrizing evolutions, Physics Letters A258, 77 (1999)

P. Zanardi, Symmetrizing evolutions, Physics Letters A258, 77 (1999)

work page 1999
[24]

M. S. Byrd and D. A. Lidar, Bang–bang operations from a geometric perspective, Quantum Information Processing1, 19 (2002)

work page 2002
[25]

A. M. Souza, G. A. ´Alvarez, and D. Suter, Robust dynami- cal decoupling for quantum computing and quantum memory, Phys. Rev. Lett.106, 240501 (2011)

work page 2011
[26]

Souza, G

A. Souza, G. A. Alvarez, and D. Suter, Robust dynamical de- coupling, Phil. Trans. R. Soc. A370, 4748 (2012)

work page 2012
[27]

Quiroz and D

G. Quiroz and D. A. Lidar, Optimized dynamical decoupling via genetic algorithms, Phys. Rev. A88, 052306 (2013)

work page 2013
[28]

Suter and G

D. Suter and G. A. Alvarez, Colloquium: Protecting quantum information against environmental noise, Rev. Mod. Phys.88, 041001 (2016)

work page 2016
[29]

M. H. Levitt, Composite pulses, Progress in Nuclear Magnetic Resonance Spectroscopy18, 61 (1986)

work page 1986
[30]

K. R. Brown, A. W. Harrow, and I. L. Chuang, Arbitrarily accu- rate composite pulse sequences, Physical Review A70, 052318 (2004)

work page 2004
[31]

G. T. Genov, D. Schraft, N. V . Vitanov, and T. Halfmann, Ar- bitrarily accurate pulse sequences for robust dynamical decou- pling, Phys. Rev. Lett.118, 133202 (2017)

work page 2017
[32]

G. T. Genov, M. Hain, N. V . Vitanov, and T. Halfmann, Univer- sal composite pulses for efficient population inversion with an arbitrary excitation profile, Phys. Rev. A101, 013827 (2020)

work page 2020

[1] [1]

Hence1−F=O(ϵ n)

The normalized HS overlap satisfies G(ϵ)=1+O(ϵ n)(17) for arbitraryαandβ. Hence1−F=O(ϵ n). Equiva- lently, all coefficients ofG(ϵ)−1of orders1, . . . , n−1 vanish; since only even powers occur, this is cancella- tion through ordern−2

work page

[2] [2]

Hence, for genericα, the fidelity error is of exact orderϵ n

For this UR phase family, the coefficient of orderϵ n in 1−Fis not identically zero as a function ofα. Hence, for genericα, the fidelity error is of exact orderϵ n

work page

[3] [3]

Our goal is to provide a rigorous mathematical proof of this theorem

This order is optimal at fixed sequence lengthn: no α-independent choice of phases{φ k}n k=1 can make all non-constant coefficients ofG(ϵ)−1through orderϵ n vanish identically as functions ofα. Our goal is to provide a rigorous mathematical proof of this theorem. For that purpose, it is convenient to write the phase differences∆ k defined in Eq. (9) in te...

work page

[4] [4]

internal

Moreover, n ∑ j=1 cj =S(r), n ∑ j=1 cjj=L r(XN).(63) Hence Lr(XD)=D 1 + n ∑ j=2 cjDj =D 1 −(φ 2 +nΦ) n ∑ j=2 cj +Φ n ∑ j=2 cjj =D 1 −(φ 2 +nΦ)(S(r)−1)+Φ(L r(XN)−1). (64) Substituting the value ofD 1 and simplifying gives Eq. (59). The second formula follows immediately by substituting Eq. (58) into Eq. (59). The preceding lemmas now combine to give the fi...

work page

[5] [5]

Casen=4m Chooseγ=c=− 1 2 in Eq. (124). Then βj ={ j−1, jodd, −j, jeven. (134) Write P(t)=T 1(t)⋯T 2m(t)T2m+1(t)⋯T 4m(t),(135) where Tj(t)∶=I+tK j =( 1tω βj tω−βj 1 ).(136) Sinceω ±2m=−1, Eq. (134) gives ωβj+2m =−ω βj ,1≤j≤2m.(137) Thus Tj+2m(t)=ET j(t)E, E∶=( 1 0 0−1 ).(138) If Q(t)∶=T 1(t)⋯T 2m(t),(139) then P(t)=Q(t)EQ(t)E.(140) For1≤j≤m, direct verific...

work page

[6] [6]

Casen=4m+2,modd Chooseγ=0andc=N/2in Eq. (124). Then βj =(−1) j+1j+ N 2 .(149) Write again P(t)=T 1(t)⋯T n(t), T j(t)=( 1tω βj tω−βj 1 ).(150) For everyj=1, . . . , N, βj+N =−β j +(1+(−1) j)N.(151) Sincemis odd,ω N =−1, and hence ωβj+N =ω −βj .(152) Therefore, with J∶=( 0 1 1 0 ),(153) 11 we have Tj+N(t)=J T j(t)J,1≤j≤N.(154) Defining Q(t)∶=T 1(t)⋯T N(t),(...

work page

[7] [7]

Casen=4m+2,meven Again set N∶= n 2 =2m+1.(166) Choose γ= N 2m , c=0(167) in Eq. (124). Once again write P(t)=T 1(t)⋯T n(t), T j(t)=( 1tω βj tω−βj 1 ).(168) For everyj=1, . . . , N, direct calculation gives βj+N =−β j ±N.(169) Sincemis even,ω N =1, and hence ωβj+N =ω −βj .(170) Therefore Tj+N(t)=J T j(t)J,1≤j≤N,(171) and, withQ(t)∶=T 1(t)⋯T N(t), P(t)=Q(t)...

work page

[8] [8]

E. L. Hahn, Spin echoes, Phys. Rev.80, 580 (1950)

work page 1950

[9] [9]

H. Y . Carr and E. M. Purcell, Effects of diffusion on free pre- cession in nuclear magnetic resonance experiments, Phys. Rev. 94, 630 (1954)

work page 1954

[10] [10]

Meiboom and D

S. Meiboom and D. Gill, Modified spin-echo method for mea- suring nuclear relaxation times, Rev. Sci. Instrum.29, 688 (1958)

work page 1958

[11] [11]

Viola and S

L. Viola and S. Lloyd, Dynamical suppression of decoherence in two-state quantum systems, Phys. Rev. A58, 2733 (1998)

work page 1998

[12] [12]

Viola, E

L. Viola, E. Knill, and S. Lloyd, Dynamical decoupling of open quantum systems, Phys. Rev. Lett.82, 2417 (1999)

work page 1999

[13] [13]

Khodjasteh and D

K. Khodjasteh and D. A. Lidar, Fault-tolerant quantum dynam- ical decoupling, Phys. Rev. Lett.95, 180501 (2005)

work page 2005

[14] [14]

L. F. Santos and L. Viola, Enhanced convergence and robust performance of randomized dynamical decoupling, Phys. Rev. Lett.97, 150501 (2006)

work page 2006

[15] [15]

G. S. Uhrig, Keeping a quantum bit alive by optimizedπ-pulse sequences, Phys. Rev. Lett.98, 100504 (2007)

work page 2007

[16] [16]

J. R. West, B. H. Fong, and D. A. Lidar, Near-optimal dynami- cal decoupling of a qubit, Phys. Rev. Lett.104, 130501 (2010)

work page 2010

[17] [17]

Wang and R.-B

Z.-Y . Wang and R.-B. Liu, Protection of quantum systems by nested dynamical decoupling, Phys. Rev. A83, 022306 (2011)

work page 2011

[18] [18]

Y . Xia, G. S. Uhrig, and D. A. Lidar, Rigorous performance bounds for quadratic and nested dynamical decoupling, Phys. Rev. A84, 062332 (2011)

work page 2011

[19] [19]

A. D. Bookatz, M. Roetteler, and P. Wocjan, Improved bounded-strength decoupling schemes for local hamiltonians, IEEE Transactions on Information Theory62, 2881 (2016)

work page 2016

[20] [20]

A. F. Brown and D. A. Lidar, Efficient chromatic-number-based multiqubit decoherence and crosstalk suppression, PRX Quan- tum6, 020354 (2025)

work page 2025

[21] [21]

C. Yi, L. Kim, and M. Marvian, Faster randomized dynamical decoupling, Physical Review Letters136, 010601 (2026)

work page 2026

[22] [22]

Viola, S

L. Viola, S. Lloyd, and E. Knill, Universal control of decoupled quantum systems, Phys. Rev. Lett.83, 4888 (1999)

work page 1999

[23] [23]

Zanardi, Symmetrizing evolutions, Physics Letters A258, 77 (1999)

P. Zanardi, Symmetrizing evolutions, Physics Letters A258, 77 (1999)

work page 1999

[24] [24]

M. S. Byrd and D. A. Lidar, Bang–bang operations from a geometric perspective, Quantum Information Processing1, 19 (2002)

work page 2002

[25] [25]

A. M. Souza, G. A. ´Alvarez, and D. Suter, Robust dynami- cal decoupling for quantum computing and quantum memory, Phys. Rev. Lett.106, 240501 (2011)

work page 2011

[26] [26]

Souza, G

A. Souza, G. A. Alvarez, and D. Suter, Robust dynamical de- coupling, Phil. Trans. R. Soc. A370, 4748 (2012)

work page 2012

[27] [27]

Quiroz and D

G. Quiroz and D. A. Lidar, Optimized dynamical decoupling via genetic algorithms, Phys. Rev. A88, 052306 (2013)

work page 2013

[28] [28]

Suter and G

D. Suter and G. A. Alvarez, Colloquium: Protecting quantum information against environmental noise, Rev. Mod. Phys.88, 041001 (2016)

work page 2016

[29] [29]

M. H. Levitt, Composite pulses, Progress in Nuclear Magnetic Resonance Spectroscopy18, 61 (1986)

work page 1986

[30] [30]

K. R. Brown, A. W. Harrow, and I. L. Chuang, Arbitrarily accu- rate composite pulse sequences, Physical Review A70, 052318 (2004)

work page 2004

[31] [31]

G. T. Genov, D. Schraft, N. V . Vitanov, and T. Halfmann, Ar- bitrarily accurate pulse sequences for robust dynamical decou- pling, Phys. Rev. Lett.118, 133202 (2017)

work page 2017

[32] [32]

G. T. Genov, M. Hain, N. V . Vitanov, and T. Halfmann, Univer- sal composite pulses for efficient population inversion with an arbitrary excitation profile, Phys. Rev. A101, 013827 (2020)

work page 2020