# No-go theorem and optimization of dynamical decoupling against noise with
soft cutoff

Zhen-Yu Wang  Ren-Bao Liu<EMAIL_ADDRESS>Department of Physics and
Center for Quantum Coherence, The Chinese University of Hong Kong, Shatin, N.
T., Hong Kong, China

###### Abstract

We study the performance of dynamical decoupling in suppressing decoherence
caused by soft-cutoff Gaussian noise, using short-time expansion of the noise
correlations and numerical optimization. For the noise with soft cutoff at
high frequencies, there exists no dynamical decoupling scheme to eliminate the
decoherence to arbitrary orders of the short time, regardless of the timing or
pulse shaping of the control under the population conserving condition. We
formulate the equations for optimizing pulse sequences that minimizes
decoherence up to the highest possible order of the short time for the noise
correlations with odd power terms in the short-time expansion. In particular,
we show that the Carr-Purcell-Meiboom-Gill sequence is optimal in short-time
limit for the noise correlations with a linear order term in the time
expansion.

###### pacs:

03.67.Pp, 03.65.Yz, 03.67.Lx, 82.56.Jn

##  I Introduction

Quantum information processing Nielsen and Chuang (2000) relies on the
coherence of quantum systems. Unavoidable interactions between a quantum
system and its environment (bath) introduce noise on the system and lead to
error evolutions (decoherence) of the quantum system. Various methods have
been proposed to combat the decoherence, including decoherence-free subspaces
Duan and Guo (1997); Zanardi (1998); Lidar et al. (1998), error-correction
codes Shor (1995); Steane (1996), and dynamical decoupling (DD) Viola and
Lloyd (1998); Ban (1998); Zanardi (1999); Viola et al. (1999); Yang et al.
(2011). In particular, the DD scheme uses rapid unitary control pulses acting
only on the systems to suppress the effects of the noise from the
environments. DD has the advantages of suppressing decoherence without
measurement, feedback, or redundant encoding Viola et al. (1999). DD
originated from the seminal spin echo experiment Hahn (1950), in which the
effect of a static random magnetic field (inhomogeneous broadening) is
canceled. And more complex DD pulse sequences, such as the Carr-Purcell-
Meiboom-Gill (CPMG) sequence Carr and Purcell (1954); Meiboom and Gill (1958),
were designed to prolong the spin coherence time Mehring (1983).

The early DD schemes only eliminate low-order errors, i.e., the errors of
quantum evolutions up to some low order in the Magnus expansion. By unitary
symmetrization procedure Zanardi (1999); Viola et al. (1999), DD cancels the
first order (i.e., leading order) errors. To eliminate errors to the second
order in short time, mirror-symmetric arrangement of two DD sequences can be
used Viola et al. (1999). The first explicit arbitrary $M$th order DD scheme,
which suppresses errors to $\mathcal{O}(T^{M+1})$ for short evolution time
$T$, is the concatenated DD (CDD) Khodjasteh and Lidar (2005, 2007) proposed
by Khodjasteh and Lidar. CDD sequences against pure dephasing were
investigated for electron spin qubits in realistic solid-state systems with
nuclear spins as baths Yao et al. (2007); Witzel and Das Sarma (2007); Zhang
et al. (2007). Experiments Peng et al. (2011); Álvarez et al. (2010);
Tyryshkin et al. ; Wang et al. (2012); Barthel et al. (2010) have tested the
performance of CDD. CDD works for generic quantum systems coupled to a finite
bath Santos and Viola (2008); Wang and Liu (2011a). However, since CDD uses
recursively constructed pulse sequences to suppress decoherence, the number of
pulses increases exponentially with the decoupling order. As pulse errors are
inevitably introduced in each control pulse in experiments, finding efficient
DD schemes with fewer control pulses is desirable. A remarkable advance is the
Uhrig DD (UDD) Uhrig (2007); Lee et al. (2008); Uhrig (2008); Yang and Liu
(2008). UDD is optimal in the short-time limit in the sense that it suppresses
the pure dephasing of a qubit coupled to a finite bath to the $M$th order
using only $M$ qubit flips. The performance bounds for UDD against pure
dephasing were established Uhrig and Lidar (2010). Shaped pulses Pasini et al.
(2008); Fauseweh et al. (2012) of finite amplitude can be incorporated into
UDD Pasini et al. (2008). Many recent experimental studies Biercuk et al.
(2009a); Du et al. (2009); Biercuk et al. (2009b); Uys et al. (2009); Jenista
et al. (2009); Álvarez et al. (2010); Barthel et al. (2010) demonstrated the
performance of UDD.

It is important to find efficient schemes to suppress general decoherence
(including pure dephasing and population relaxation). Yang and Liu extended
UDD to the suppression of population relaxation Yang and Liu (2008). This
inspired efficient ways to suppress the general decoherence of single qubits,
including concatenation of UDD sequences (CUDD) Uhrig (2009) and a much more
efficient one called quadratic DD (QDD) West et al. (2010); Wang and Liu
(2011a); Quiroz and Lidar (2011); Kuo and Lidar (2011); Jiang and Imambekov
(2011) discovered by West _et al_ West et al. (2010). Based on the proof in
Yang and Liu (2008), Mukhtar _et al_ generalized UDD to protect arbitrary
multilevel systems with full prior knowledge of the initial states Mukhtar et
al. (2010). One can actually preserve the coherence of arbitrary multi-qubit
systems by protecting a mutually orthogonal operation set (MOOS) Wang and Liu
(2011a). By nesting UDD sequences for protecting the elements in the MOOS, the
nested UDD (NUDD) Wang and Liu (2011a); Mukhtar et al. (2010); Jiang and
Imambekov (2011) requires only a polynomially increasing number of pulses in
the decoupling order. These universal DD schemes also work for analytically
time-dependent baths Wang and Liu (2011b).

The above-mentioned variations of UDD, however, rely on the finiteness of the
baths, i.e., the existence of hard high-frequency cutoff in the noise spectra.
Legitimate questions are: For quantum systems coupled to an infinite quantum
bath or affected by soft-cutoff noise, can any DD be designed to eliminate the
decoherence to arbitrary orders of precision in the short-time limit? And if
yes, how can such DD be designed? Such questions have been previously
addressed in some specific noise models. Comparing the efficiency of various
DD sequences in suppressing pure dephasing of a qubit due to classical noise,
Cywiński _et al_ observed that if the noise spectrum cutoff is not reached,
CPMG sequences Carr and Purcell (1954); Meiboom and Gill (1958) actually
performs better than CDD and UDD sequences Cywiński et al. (2008). With the
consideration of minimum pulse separations in physical systems, Viola _et al_
observed that low-order DD sequences provide better performance than high-
order DD when the rate of pulses is not faster than the correlation time of
the noise Hodgson et al. (2010); Khodjasteh et al. (2011). It was confirmed by
experiments that for ${}^{13}\text{C}$ spin qubits in a ${}^{1}\text{H}$ spin
bath of which the high frequency cutoff was not reached by the DD sequences,
CPMG outperforms UDD Ajoy et al. (2011). Also, Pasini and Uhrig derived the
equations for minimizing decoherence for power-law spectra, and found that the
numerically optimized sequences resemble CPMG Pasini and Uhrig (2010). Chen
and Liu proved that for telegraphlike noise the CPMG sequences are the most
efficient scheme in protecting the qubit coherence in the short-time limit and
the decoherence can be suppressed at most to the third order of short
evolution time by DD Chen and Liu (2010). These results suggest that for noise
with soft cutoff in the spectrum, there are certain constraints on the optimal
order and decoupling scaling of DD. Ref. Gordon et al. (2008) presented
numerical optimization of bounded-strength DD for specific noise spectra.
However, no conclusion has been drawn on the performance of DD with arbitrary
timing and shaping for the general cases of soft-cutoff noise.

In this paper, we address the general question of the performance of DD
against soft-cutoff noise based on the general modulation function induced by
arbitrary DD with bounded-strength or pulsed control. We show that for the
noise spectrum with a power-law asymptote at high frequencies, there exists no
modulation function to eliminate the decoherence to an arbitrary order of the
short time, regardless of the timing and shaping of the DD control under the
population conserving condition. Although the decohernce can be suppressed to
be arbitrarily small by DD with a sufficiently large number of pulses, the
existence of the largest achievable decoupling order shows that DD against
soft-cutoff noise does not have the order-by-order decoupling efficiency,
which is possible for hard-cutoff noise. Since for soft-cutoff noise the
decoherence cannot be eliminated at a certain order of short time, we derive a
set of equations to minimize the leading-order term in the short-time
expansion while eliminating the lower orders. These equations are numerically
solved for optimal solutions. In particular, for noise correlations with a
linear order term in time, we prove that the CPMG sequences are optimal. For
other noise correlations with odd-order terms, the minimum pulse interval of
the optimized sequences is larger than in UDD sequences. This feature is
important in realistic experiments when there is a minimum pulse switching
time Khodjasteh et al. (2011).

This paper is organized as follows: In Sec. II, we analyze the performance of
DD against soft-cutoff noise, and we give the condition under which
decoherence suppression to an arbitrary order of short-time scaling is
impossible. The relation between the high-frequency cutoff and the short-time
expansion of correlations is also discussed. In Sec. III, we derive the
equations for sequence optimization and obtain optimal DD for noise
correlations with odd-power expansion terms. Finally, the conclusions are
drawn in Sec. IV.

##  II No-go theorem on dynamical decoupling against noise with soft cutoff

We consider the pure-dephasing Hamiltonian for a single spin (qubit)

$H=\frac{1}{2}\sigma_{z}[\omega_{a}+\beta(t)],$ (1)

where $\sigma_{z}=|+\rangle\langle+|-|-\rangle\langle-|$ is the Pauli operator
of the qubit, $\omega_{a}$ is the energy splitting of the qubit, and
$\beta(t)$ describes random noise with average $\overline{\beta(t)}=0$. Here
the over bar denotes averaging over the noise realizations. We assume that the
statistics of the noise fluctuations are Gaussian.

After a duration of free evolution time $T$, the noise induces between the two
states $|\pm\rangle$ a random phase shift $\int_{0}^{T}\beta(t)$ that destroys
the quantum coherence. We can suppress the decoherence by DD control on the
qubit. There is only one noise source $\beta(t)$ in the model Eq. (1), and to
suppress the decoherence we need to protect a MOOS which consists of a Pauli
operator $\sigma_{x}$ (more generally $\sigma_{x}\cos\phi+\sigma_{y}\sin\phi$
with $\phi$ being real) Wang and Liu (2011a). We will prove later that DD can
suppress the decoherence (i.e., the protection of the MOOS $\\{\sigma_{x}\\}$)
only to a certain order of short evolution time for noise correlations that
have odd-power expansion terms in time. We expect that the proof also applies
to other quantum systems (e.g., multi-qubit systems) when the noise
correlations have odd-power terms, since in those systems there are more noise
sources and more system operators (e.g., a MOOS consisting of $L>1$ Pauli
operators) should be protected.

When we apply a sequence of instantaneous unitary operations $\sigma_{x}$ at
the moments $T_{1}$, $T_{2}$, $\ldots$, $T_{N}$, the controlled evolution
operator reads

$\displaystyle U(T)$ $\displaystyle=$
$\displaystyle(\sigma_{x})^{N}U(T_{N+1},T_{N})\sigma_{x}U(T_{N},T_{N-1})\cdots$
(2) $\displaystyle\times\sigma_{x}U(T_{2},T_{1})\sigma_{x}U(T_{1},T_{0}),$

where $T_{0}=0$, $T_{N+1}=T$, and the free evolution operator

$U(T_{j+1},T_{j})=e^{-i\frac{\sigma_{z}}{2}\int_{T_{j}}^{T_{j+1}}\left[\omega_{a}+\beta(t)\right]dt}.$
(3)

Note that when $N$ is odd, we may apply an additional $\sigma_{x}$ pulse at
the end of the sequence for the identity evolution. Using

$\sigma_{x}U(T_{j+1},T_{j})\sigma_{x}=e^{-i\frac{\sigma_{z}}{2}\int_{T_{j}}^{T_{j+1}}\left[-\omega_{a}-\beta(t)\right]dt},$
(4)

we write the evolution operator as

$U(T)=e^{-i\frac{\sigma_{z}}{2}\int_{0}^{T}\omega_{a}F_{\pi}(t/T)dt}e^{-i\frac{\sigma_{z}}{2}\int_{0}^{T}\beta(t)F_{\pi}(t/T)dt},$
(5)

where we have defined the modulation function for instantaneous $\pi$-pulse
sequences Uhrig (2007); Cywiński et al. (2008)

$F_{\pi}(t/T)=\begin{cases}(-1)^{j}&\text{for }t\in(T_{j},T_{j+1}]\\\
0&\text{for }t>T,\mbox{\text{ or }}t\leq 0\end{cases}.$ (6)

The DD control is parametrized by the relative pulse locations $T_{j}/T$. At
the moment $T$, the off-diagonal density matrix element of an ensemble is

$\rho_{\uparrow\downarrow}(T)=\rho_{\uparrow\downarrow}(0)e^{-i\int_{0}^{T}\omega_{a}F_{\pi}(t/T)dt}\overline{e^{-i\int_{0}^{T}\beta(t)F_{\pi}(t/T)dt}}.$
(7)

The coherence is characterized by the ensemble-averaged phase factor

$W(T)\equiv\overline{e^{-i\int_{0}^{T}\beta(t)F_{\pi}(t/T)dt}}.$ (8)

For Gaussian noise, the ensemble-averaged phase factor $W(T)$ is determined by
the two-point correlation function $\overline{\beta(t_{1})\beta(t_{2})}$ and
$W(T)$ becomes Anderson (1954); Kubo (1954); Cywiński et al. (2008)

$W(T)=e^{-\chi_{\pi}(T)},$ (9)

where the phase correlation for instantaneous pulses

$\chi_{\pi}(T)=\frac{1}{2}\int_{0}^{T}dt_{1}\int_{0}^{T}dt_{2}\overline{\beta(t_{1})\beta(t_{2})}F_{\pi}\left(\frac{t_{1}}{T}\right)F_{\pi}\left(\frac{t_{2}}{T}\right).$
(10)

can be written as the overlap between the noise power spectrum and a filter
function determined by the Fourier transform of the modulation function
Cywiński et al. (2008).

Under DD control, the qubit is flipped at different moments, and the random
field $\beta(t)$ is modulated by the modulation function $F_{\pi}(t/T)$. For
multilevel systems, the modulation functions resulting from instantaneous
$\pi$-pulse sequences may have values not restricted to $\\{\pm 1\\}$ for
$t\in(0,T]$ (see Appendix A). In Ref. Uhrig and Pasini (2010), it is shown
that for DD composed of specially engineered finite-duration pulses, the
effective modulation functions can take values from $\\{+1,-1,0\\}$
alternatively. We may also encounter effective modulation functions which are
triangle wave functions during the time of system evolution Wang and Liu . For
a more general analysis, we assume that the control conserves the populations
and the phase modulation function $F_{\pi}(t/T)$ has a general form as

$\displaystyle F\left(\frac{t}{T}\right)$ $\displaystyle=\begin{cases}\text{a
bounded function}&\text{for }t\in(0,T]\\\ 0&\text{otherwise}\end{cases},$ (11)

which has a finite number of discontinuities. The more general phase
correlation considered in this paper reads

$\displaystyle\chi(T)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\int_{0}^{T}dt_{1}\int_{0}^{T}dt_{2}\overline{\beta(t_{1})\beta(t_{2})}F^{*}\left(\frac{t_{1}}{T}\right)F\left(\frac{t_{2}}{T}\right)$
(12a) $\displaystyle=$
$\displaystyle\Re\int_{0}^{T}dt_{1}\int_{0}^{t_{1}}dt_{2}\overline{\beta(t_{1})\beta(t_{2})}F^{*}\left(\frac{t_{1}}{T}\right)F\left(\frac{t_{2}}{T}\right),$
(12b)

where we have used
$\overline{\beta(t_{1})\beta(t_{2})}=\overline{\beta(t_{2})\beta(t_{1})}$ to
derive Eq. (12b) [For quantum noise, this may not be true. But in Eq. (12a),
$t_{1}$ and $t_{2}$ can be exchanged without changing the integration. So the
noise correlation can always be symmetrized]. It was shown that under suitable
approximation, Eq. (12) with a complex $F(t/T)=e^{-i\int_{0}^{t/T}V(s)ds}$
describes the average dephasing of a qubit under bounded-strength control with
the amplitude $V(s)$ Gordon et al. (2008, 2006); Gordon and Kurizki (2007);
Gordon et al. (2007). In Ref. Gordon et al. (2008) based on minimization of
$\chi(T)$, some optimal control fields $V(s)$ were obtained for some specific
noise spectra. Therefore we analyse and minimize $\chi(T)$ given by Eq. (12)
for DD design. For the special case of instantaneous $\pi$-pulse sequences for
the dephasing of a qubit, $F(t/T)=F_{\pi}(t/T)$ and $\chi(T)=\chi_{\pi}(T)$.

We assume the noise is stationary, i.e., of time translation symmetry,
$\overline{\beta(t_{1})\beta(t_{2})}=\overline{\beta(t_{1}-t_{2})\beta(0)}$.
Another symmetry is $\overline{\beta(t)\beta(0)}=\overline{\beta(0)\beta(t)}$.
These symmetries indicate that the noise correlation is an even function of
time, i.e.,

$\overline{\beta(t)\beta(0)}\equiv C_{\text{corr}}(t)=C_{\text{corr}}(|t|).$
(13)

The noise correlation can be transformed from the noise power spectrum
$S(\omega)$ as

$\overline{\beta(t)\beta(0)}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}S(\omega)e^{-i\omega
t}.$ (14)

Note that both $\overline{\beta(t)\beta(0)}$ and $S(\omega)$ are real even
functions.

The general filter function is defined as the Fourier transform of the general
modulation function,

$\displaystyle\tilde{f}(\omega T)$
$\displaystyle\equiv\frac{1}{T}\int_{-\infty}^{\infty}F\left(\frac{t}{T}\right)e^{i\omega
t}dt$ (15a) $\displaystyle=\int_{0}^{1}F(s)e^{i\omega Ts}ds,$ (15b)

which has the power expansion

$\displaystyle\tilde{f}(\omega T)=\sum_{m=0}^{\infty}\frac{(i\omega
T)^{m}}{m!}\lambda_{m},$ (16a)
$\displaystyle\lambda_{m}\equiv\int_{0}^{1}F(s)s^{m}ds.$ (16b)

Eq. (15b) shows that $\tilde{f}(\omega T)$ is bounded by $|\tilde{f}(\omega
T)|\leq\int_{0}^{1}|F(s)|ds.$

As $F(s)$ has a finite number of discontinuities, we use integration by parts
and get

$\tilde{f}(u)=\frac{1}{iu}\sum_{j}\left[\left.F(s)e^{ius}\right|_{s_{j}}^{s_{j+1}}-I_{j}(u)\right],$
(17)

where $s_{j}$ are the discontinuous points and
$|I_{j}(u)|=|\int_{s_{j}}^{s_{j+1}}F^{\prime}(s)e^{ius}ds|\leq\int_{s_{j}}^{s_{j+1}}|F^{\prime}(s)|ds\equiv\tilde{I}_{j}$
is finite. Therefore we have

$|\tilde{f}(u)|\leq a_{\tilde{f}}/u,$ (18)

where the coefficient
$a_{\tilde{f}}=\sum_{j}\left[|F(s_{j})|+|F(s_{j+1})|+\tilde{I}_{j}\right]$ is
bounded.

For a sequence of $N$ instantaneous $\pi$ pulses, $\lambda_{m}$ reads

$\lambda_{m}^{(\pi)}=\sum_{j=0}^{N}\frac{(-1)^{j}}{(m+1)}\left[\left(\frac{T_{j+1}}{T}\right)^{m+1}-\left(\frac{T_{j}}{T}\right)^{m+1}\right].$
(19)

Using Eqs. (14) and (15b), Eq. (12a) can be written as the overlap of the
noise spectrum and filter function

$\displaystyle\chi(T)$
$\displaystyle=T^{2}\int_{0}^{\infty}\frac{d\omega}{\pi}S(\omega)|\tilde{f}(\omega
T)|^{2}.$ (20)

It should be stressed that in Eq. (20) the filter function $\tilde{f}(\omega
T)$ is general and not limited to the case of instantaneous pulse sequences.

###  II.1 Scaling of decoupling orders

We separate the noise spectrum into two parts by a frequency $\Omega$. As the
noise spectrum $S(\omega)$ in Eq. (20) induces decoherence linearly,

$\chi(T)=\chi_{[0,\Omega]}(T)+\chi_{[\Omega,\infty]}(T),$ (21)

where

$\displaystyle\chi_{[0,\Omega]}(T)=T^{2}\int_{0}^{\Omega}\frac{d\omega}{\pi}S(\omega)|\tilde{f}(\omega
T)|^{2},$ (22a)
$\displaystyle\chi_{[\Omega,\infty]}(T)=T^{2}\int_{\Omega}^{\infty}\frac{d\omega}{\pi}S(\omega)|\tilde{f}(\omega
T)|^{2}.$ (22b)

$\chi_{[0,\Omega]}(T)$ and $\chi_{[\Omega,\infty]}(T)$ account for the effects
of the low- and high-frequency noise, respectively. Both
$\chi_{[0,\Omega]}(T)$ and $\chi_{[\Omega,\infty]}(T)$ cause decoherence as
$S(\omega)\geq 0$.

####  II.1.1 Effects of low-frequency noise

For the spectrum $S(\omega)=\mathcal{O}(1/\omega^{P})$ with $P<1$ when
$\omega\rightarrow 0$, Eq. (14) gives the noise correlation of the noise with
the frequencies $\omega<\Omega$,

$\displaystyle\overline{\beta(t)\beta(0)}_{[0,\Omega]}$
$\displaystyle=\int_{0}^{\Omega}\frac{d\omega}{\pi}S(\omega)\cos\left(\omega
t\right)$ (23a) $\displaystyle=\sum_{m=0}^{\infty}C_{2m}\Omega^{2m+1}t^{2m}.$
(23b)

Here the coefficients

$C_{2m}=(-1)^{2m}\int_{0}^{1}\frac{du}{\pi}S(u\Omega)\frac{u^{2m}}{(2m)!},$
(24)

which depend on the noise spectrum $S(\omega)$ and $\Omega$, converge at low
frequencies $\omega\rightarrow 0$.

Eq. (12b) gives

$\chi_{[0,\Omega]}(T)=\sum_{m=0}^{\infty}C_{2m}\phi_{2m}\Omega^{2m+1}T^{2m+2},$
(25)

where the decoherence functions

$\phi_{k}\equiv\Re\int_{0}^{1}ds_{1}\int_{0}^{s_{1}}ds_{2}(s_{1}-s_{2})^{k}F^{*}\left(s_{1}\right)F\left(s_{2}\right),$
(26)

are modified by the modulation function of DD. The even-order decoherence
functions $\phi_{2k}$ control the effects of the low-frequency noise.

Therefore if the modulation function $F(t/T)$ is designed to make

$\\{\phi_{2m}=0\\}_{m=0}^{M-1}\equiv\\{\phi_{0}=\phi_{2}=\cdots=\phi_{2M-2}=0\\},$
(27)

the decoherence from low-frequency noise is eliminated to
$\chi_{[0,\Omega]}(T)=\mathcal{O}(T^{2M+2})$ (the prefactor of the scaling
depends on the noise spectrum and $\Omega$). Note that
$e^{-i\int_{0}^{T}\omega_{a}F(t/T)dt}=1$ in Eq. (7) when $\phi_{0}=0$.

In Appendix B, we simplify the even-order $\phi_{2m}$ as

$\displaystyle\phi_{2m}$
$\displaystyle=\frac{(2m)!}{2}\Re\sum_{r=0}^{2m}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2m-r}}{(2m-r)!}.$
(28)

From Eq. (28), we find that the following two sets of equations are equivalent

$\\{\phi_{2m}=0\\}_{m=0}^{M-1}\Leftrightarrow\\{\lambda_{m}=0\\}_{m=0}^{M-1}.$
(29)

For instantaneous $\pi$-pulse sequences, the optimal solution of the equation
set $\\{\lambda_{m}=\lambda_{m}^{(\pi)}=0\\}_{m=0}^{N-1}$ is

$T_{j}^{\text{UDD}}=T\sin^{2}\left[\frac{\pi
j}{2N+2}\right],\>(j=1,2,\ldots,N),$ (30)

which is the timing of UDD sequences Uhrig (2007). The conditions Eqs. (29)
and (16b) are more general than the one that leads to UDD and may lead to more
general designs of optimal DD.

For the power-law spectrum $S(\omega)\approx\alpha/\omega^{P}$ with $P\geq 1$
at low frequencies,
$\chi_{[0,\Omega]}(T)=T^{2}\int_{0}^{\Omega}\frac{d\omega}{\pi}\frac{\alpha}{\omega^{P}}|\tilde{f}(\omega
T)|^{2}$ in general diverges. For the modulation function $F_{\pi}(t/T)$, it
was shown that the divergence of the integral can be eliminated by high-order
DD sequences Pasini and Uhrig (2010). For the general modulation function
$F(t/T)$ under the conditions $\\{\lambda_{m}=0\\}_{m=0}^{M-1}$, using Eq.
(16a) we have $|\tilde{f}(u)|^{2}\sim\mathcal{O}(u^{2M})$ and
$\chi_{[0,\Omega]}(T)=\mathcal{O}(\Omega^{2M+1-P}T^{2M+2})$ when $M>(P-1)/2$.

When the noise with the frequency $\omega>\Omega$ is negligible (i.e., a hard
cutoff frequency $\omega_{c}=\Omega$ in the noise spectrum and
$\chi_{[\Omega,\infty]}\approx 0$), the decoherence can be suppressed order by
order, and $\chi(T)\approx\chi_{[0,\Omega]}(T)$ has the scaling
$\chi(T)\approx\mathcal{O}(T^{2M+2})$ in short-time limit.

####  II.1.2 Effects of high-frequency noise

Consider the noise spectrum with a power-law asymptote at high frequencies
(i.e., a soft high-frequency cutoff),

$S(\omega)\approx\frac{\alpha}{\omega^{P}},\text{ for
}\Omega\leq\omega\leq\Omega_{\infty},$ (31)

where $\Omega_{\infty}\gg 1/T$ and $\Omega\lesssim 1/T$. We assume that the
decay of the noise at the frequencies $\omega>\Omega_{\infty}$ is not slower
than $\alpha/\omega^{P}$ and the contribution is negligible. We set
$\Omega_{\infty}=\infty$. The high-frequency contribution Eq. (22b) reads

$\chi_{[\Omega,\infty]}(T)\approx\chi_{P}(T)\equiv T^{P+1}\int_{\Omega
T}^{\infty}\frac{du}{\pi}\frac{\alpha}{u^{P}}|\tilde{f}(u)|^{2}.$ (32)

Since $\tilde{f}(u)=\mathcal{O}\left(1/u\right)$ when $u\rightarrow\infty$
[Eq. (18)], $\chi_{[\Omega,\infty]}(T)$ converges when $P>-1$.

Using $|\tilde{f}(0)|^{2}=|\lambda_{0}|^{2}$, we get $\lim_{T\rightarrow
0}\chi_{[\Omega,\infty]}(T)=0$ even though the integral $\lim_{T\rightarrow
0}\int_{\Omega
T}^{\infty}\frac{du}{\pi}\frac{\alpha}{u^{P}}|\tilde{f}(u)|^{2}$ may diverge
when $T\rightarrow 0$. We have shown in Sec. II.1.1 that under the conditions
$\\{\lambda_{m}=0\\}_{m=0}^{M-1}$ with $2M+1-P>0$, $\int_{0}^{\Omega
T}\frac{du}{\pi}\frac{\alpha}{u^{P}}|\tilde{f}(u)|^{2}=\mathcal{O}(\Omega^{2M+1-P}T^{2M+1-P})$.
Therefore when $M>(P-1)/2$, in Eq. (32) we extend the bounds of integration to
$(0,\infty)$,

$\displaystyle\chi_{[\Omega,\infty]}(T)$
$\displaystyle=C_{P}^{\text{soft}}T^{P+1}-\mathcal{O}(T^{2M+1-P})=\mathcal{O}(T^{P+1}),$
(33)

where
$C_{P}^{\text{soft}}=\int_{0}^{\infty}\frac{du}{\pi}\frac{\alpha}{u^{P}}|\tilde{f}(u)|^{2}$
is bounded when $\\{\lambda_{m}=0\\}_{m=0}^{M-1}$ with $2M+1-P>0$ and $P>-1$.
The scaling $\chi_{[\Omega,\infty]}(T)=\mathcal{O}(T^{P+1})$ is the largest
order of decoupling that can be achieved for the noise with soft cutoff. We
have the following theorem.

######  Theorem 1.

For the noise spectrum with a power-law asymptote $\alpha/\omega^{P}$ with
$P>-1$ at high frequencies, the largest achievable decoupling order of DD with
a general modulation function given by Eq. (11) is
$\chi(T)=\mathcal{O}(T^{P+1})$ in short-time limit $T\rightarrow 0$.

This theorem holds for arbitrary non-zero modulation function $F(t/T)$, and it
shows that one cannot suppress the decoherence to arbitrary order when the
noise has a soft cutoff in the spectrum.

For example, the $1/f$ noise and the Lorentz spectrum
$\alpha/(\Omega^{2}+\omega^{2})$ correspond to the cases of $P=1$ and $P=2$,
respectively. After eliminating the effect of low-frequency noise by high-
order DD which satisfy $\lambda_{0}=0$, we achieve the largest decoupling
order $\chi_{P=1}(T)=\mathcal{O}\left(T^{2}\right)$ and
$\chi_{P=2}(T)=\mathcal{O}\left(T^{3}\right)$.

Note that Theorem 1 applies to the order of short-time scaling and it does not
mean that DD can not protect the coherence to arbitrarily high precision. The
decoherence can be suppressed further by reducing the prefactor
$C_{P}^{\text{soft}}$, i.e., the the overlap of the filter function
$|\tilde{f}(\omega T)|^{2}$ and noise spectrum $S(\omega)$. We will discuss
the optimization of pulse sequences based by minimizing the overlap in Sec.
III.

The result in Ref. Chen and Liu (2010) that DD can suppress decoherence at
most to the third order of short evolution time for telegraphlike noise is
general for noise with arbitrary statistics. In deriving Theorem 1, we have
made the assumption that the statistics of the noise are Gaussian and the
decoherence is determined by the two-point noise correlation. It would be
interesting to generalize the theorem to non-Gaussian noise.

The perturbative regime $T\lesssim 1/\Omega$ is limited by the technology of
experiments. For power-law noise (e.g., $1/f$ noise), $\Omega\rightarrow 0$
and the conclusion applies to arbitrary duration of $T$ when $M>(P-1)/2$.

###  II.2 Noise correlation expansion and high-frequency cutoff

The correlation in the short-time limit, which is due to high-frequency noise,
can be written as

$C_{\Omega,P}(t)=\Re\int_{\Omega}^{\infty}\frac{d\omega}{\pi}\frac{\alpha}{\omega^{P}}e^{-i\omega
t}.$ (34)

As $C_{\Omega,P}(t)$ is an even function, we just calculate the integral for
the case of $t>0$. For $P>1$ and $t>0$, we have

$\displaystyle C_{\Omega,P}(t)$
$\displaystyle=\frac{\Re}{\pi}\left[\int_{c_{\Omega}}+\int_{c_{i}}+\int_{c_{\infty}}\right]\frac{\alpha}{z^{P}}e^{izt}dz,$

where the paths $c_{\Omega}$, $c_{i}$, and $c_{\infty}$ are shown in Fig. (1).

Figure 1:  The paths for the integral of $\alpha/z^{P}$.

Since the maximum of $\alpha/z^{P}\rightarrow 0$ as $|z|\rightarrow\infty$ in
the upper half-plane, the contribution
$\int_{c_{\infty}}\frac{\alpha}{z^{P}}e^{izt}dz=0$. The contribution

$\frac{\Re}{\pi}\int_{c_{i}}\frac{\alpha}{z^{P}}e^{izt}dz=\frac{\alpha}{\pi}\Re\int_{\Omega}^{\infty}i^{1-P}y^{-P}e^{-yt}dy$
(35)

vanishes for even $P$. And

$\displaystyle\frac{\Re}{\pi}\int_{c_{\Omega}}\frac{\alpha}{z^{P}}e^{izt}dz$
$\displaystyle=\frac{\Re}{\pi}\int_{0}^{\frac{\pi}{2}}\frac{\alpha}{\Omega^{P}e^{iP\theta}}e^{i\Omega
te^{i\theta}}(i\Omega)e^{i\theta}d\theta$
$\displaystyle=I_{c_{\Omega}}^{(1)}+I_{c_{\Omega}}^{(2)},$ (36)

where

$\displaystyle I_{c_{\Omega}}^{(1)}$
$\displaystyle=\frac{\Re}{\pi}\sum_{r=0,r\neq
P-1}^{\infty}\frac{\alpha}{r!}\frac{\Omega^{1-P}(i\Omega
t)^{r}}{(r+1-P)}\left(i^{r+1-P}-1\right),$ (37a) $\displaystyle
I_{c_{\Omega}}^{(2)}$ $\displaystyle=\begin{cases}\frac{1}{2}\alpha\Re
i^{P}t^{P-1}/(P-1)!,&\text{if }r\text{ is an integer,}\\\
0,&\text{otherwise.}\end{cases}$ (37b)

For even $P$, $I_{c_{\Omega}}^{(1)}$ is an expansion of even powers of $t$

$I_{c_{\Omega}}^{(1)}=\sum_{k=0}^{\infty}C_{2k}t^{2k},C_{2k}=\frac{\alpha(-1)^{k+1}\Omega^{1-P+2k}}{\pi(2k)!(2k+1-P)}\neq
0,$ (38)

and there is only one odd-order expansion term, which is

$I_{c_{\Omega}}^{(2)}=\frac{(-1)^{P/2}\alpha}{2(P-1)!}|t|^{P-1}.$ (39)

The existence of an odd-order term means that the correlation function is non-
analytical, which indicates that the noise source cannot be a finite quantum
bath. The noise with non-analytical correlation functions must come from the
fluctuations of an infinite bath, since otherwise the unitary evolution of a
finite quantum system will always lead to analytical correlation functions.
For example, the noise correlation
$\overline{\beta(t)\beta(0)}=e^{-|t|/t_{c}}$ has odd-order terms in the time
expansion, and the noise has a Lorentz spectrum, which has a power-law
decrease at high frequencies. This kind of noise can be caused by Markovian
(or instantaneous) collisions in the bath Berman and Brewer (1985).

As an example, we consider the following noise spectrum,

$S_{2K}^{\prime}(\omega)\equiv\frac{\alpha}{\Omega_{c}^{2K}+\omega^{2K}},$
(40)

for $K\in\\{1,2,\ldots\\}$. $S_{2K}^{\prime}(\omega)\approx\alpha/\omega^{2K}$
when $\omega\gg\Omega_{c}$. For example, the measured ambient noise for ions
in a Penning trap has an approximate $1/\omega^{4}$ spectrum at high
frequencies and a flat dependence at low frequencies Biercuk et al. (2009a),
which approximately corresponds to the noise spectrum Eq. (40) with $K=2$. The
corresponding correlation function of Eq. (40) is obtained by the inverse
transform

$\overline{\beta(t)\beta(0)}_{2K}=\int_{0}^{\infty}\frac{d\omega}{\pi}\frac{\alpha
e^{-i\omega t}}{\Omega_{c}^{2K}+\omega^{2K}}.$ (41)

Using the residue theorem, we have

$\overline{\beta(t)\beta(0)}_{2K}=\frac{i\alpha}{2K}\Omega_{c}^{1-2K}\sum_{n=0}^{K-1}\frac{\exp[-ie^{i\frac{\pi}{2K}(2n+1)}|\Omega_{c}t|]}{\exp[i\frac{\pi}{2K}(2n+1)(2K-1)]}.$
(42)

Expanding the right-hand side in powers of $t$, we get

$\displaystyle\overline{\beta(t)\beta(0)}_{2K}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{(-i|\Omega_{c}t|)^{m}}{m!}\frac{e^{i\frac{(m-2K+1)\pi}{2K}}[(-1)^{m}+1]}{e^{i(m+1)\pi/K}-1}$
$\displaystyle\times\frac{-i\alpha}{2K}\Omega_{c}^{1-2K}.$ (43)

In Eq. (43), the leading odd-order term of the time expansion is
$\frac{(-1)^{K}\alpha}{2(2K-1)!}|t|^{2K-1}$, as predicted by Eq. (39).

For simplicity, let us consider the noise correlations that have the power
expansion

$\overline{\beta(t)\beta(0)}\equiv
C_{\text{corr}}(t)=\sum_{k=0}^{\infty}C_{k}\left|t\right|^{k},$ (44)

where the expansion coefficients

$C_{k}\equiv\left.\frac{1}{k!}\frac{d^{k}C_{\text{corr}}(t)}{dt^{k}}\right|_{t\rightarrow
0^{+}}$ (45)

are finite real numbers with

$\displaystyle C_{2k-1}$ $\displaystyle=0,\text{ for }k<K,$ (46a)
$\displaystyle C_{2K-1}$ $\displaystyle\neq 0.$ (46b)

The leading odd-order term in the short-time expansion of $C_{\text{corr}}(t)$
is $C_{2K-1}|t|^{2K-1}.$ An example is $C_{\text{corr}}(t)=e^{-|t|^{3}}$ with
$C_{1}=0$ and $C_{3}=-1$. We assume that the noise correlation decreases
smoothly at long correlation times, that is,

$\lim_{t\rightarrow\infty}\frac{d^{k}}{dt^{k}}C_{\text{corr}}(t)=0,\text{ for
}k=0,1,\ldots,$ (47)

and

$\displaystyle I_{L}(\omega)$ $\displaystyle\equiv\int_{0}^{\infty}e^{i\omega
t}\frac{d^{L}}{dt^{L}}C_{\text{corr}}(t)dt,$ (48)

vanishes at large $\omega$ for $L=0,1,\ldots$ Sun and Liu (1996). For the
noise correlations that decay in the correlation time smoothly without fast
oscillation, $I_{L}(\omega)$ vanishes at large frequency $\omega$. For
example, the noise correlation $e^{-|t|}$ has
$I_{L}(\omega)=i(-1)^{L}/(\omega+i)\rightarrow 0$ when
$\omega\rightarrow\infty$.

We consider the high frequency behavior of the noise spectrum,
$S_{2K}(\omega)=\int_{-\infty}^{\infty}C_{\text{corr}}(t)e^{i\omega t}dt.$
Integration by parts $L\geq(2K+1)$ times gives

$S_{2K}(\omega)=2\Re\left[\sum_{r=1}^{L}\frac{(r-1)!}{(-i\omega)^{r}}C_{r-1}+\frac{I_{L}(\omega)}{(-i\omega)^{L}}\right],$
(49)

where we have used Eqs. (45) and (47).

Using Eqs. (46) and (48), we obtain for large $\omega$,

$S_{2K}(\omega)\approx
2\frac{(2K-1)!}{(i\omega)^{2K}}C_{2K-1}+\mathcal{O}\left(\frac{1}{\omega^{2K+1}}\right),$
(50)

which is a power-law decrease at high frequencies.

When the noise correlation expansion contains only even-order terms, from Eq.
(49) we have the noise spectrum $S_{\text{even}}(\omega)=2\Re
I_{L}(\omega)/(-i\omega)^{L}$ for an arbitrarily large $L$. From the
assumption $\lim_{\omega\rightarrow\infty}I_{L}(\omega)=0$, we have
$\lim_{\omega\rightarrow\infty}S_{\text{even}}(\omega)\omega^{L}=0$ for an
arbitrarily large $L$ and therefore the noise spectrum has a hard high-
frequency cutoff. One example is the correlation function $e^{-t^{2}}$, which
has the noise spectrum of exponential form $\sim\exp(-\frac{\omega^{2}}{4})$,
and obviously the UDD sequence can suppress the noise effect order by order.
The large $\omega$ behavior of other correlation functions of the form
$\exp(-\sum_{j=1}^{p}\alpha_{2j}t^{2j})$ can be calculated by the saddle point
integration method, which gives a result of an exponential decrease at high
frequencies (i.e., hard cutoff). For example, when $\omega$ is very large,
$\int_{-\infty}^{\infty}e^{-t^{4}}e^{i\omega
t}dt\simeq\frac{1}{2}\Im\sqrt{\frac{2\pi}{a(\omega)}}e^{g(\omega)}$, where
$g(\omega)=3(\frac{\omega}{4})^{\frac{4}{3}}e^{-i2\pi/3}$ and
$a(\omega)=12(\frac{\omega}{4})^{2/3}e^{i2\pi/3}$.

##  III SEQUENCE OPTIMIZATION IN SHORT-TIME LIMIT

In this section, we optimize DD for the noise correlations that have the power
expansion given by Eq. (44),
$\overline{\beta(t)\beta(0)}\equiv\sum_{k=0}^{\infty}C_{k}\left|t\right|^{k}$.

The performance of DD in the short-time limit is directly derived from the
time-domain expansion. The time-domain expansion of noise correlations has the
advantage to avoid the divergence of the decoherence $\chi(T)$. Using the
expansion Eq. (44), we write

$\chi(T)=\sum_{k=0}^{\infty}C_{k}\phi_{k}T^{k+2},$ (51)

where the decoherence functions $\phi_{k}$ is given by Eq. (26). It seems that
we can find DD schemes to suppress the decoherence to an arbitrary order
$\chi(T)=\mathcal{O}(T^{M+2})$ by solving the equations $\phi_{k}=0$ with
$C_{k}\neq 0$ for $k<M$. However, we have shown in Sec. II.1.2 by Theorem 1
that for soft-cutoff noise, there is a largest decoupling order.

The even-order functions $\phi_{2m}$ are given by Eq. (28). Using Eq. (82), we
have the odd-order functions $\phi_{2M-1}$,

$\displaystyle\phi_{2M-1}=-\frac{(2M-1)!}{2}\sum_{r=0}^{2M-1}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2M-1-r}}{(2M-1-r)!}$
$\displaystyle+(2M-1)!\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{2M}}\right.$
$\displaystyle\left.-\sum_{r=0}^{2M-1}\sum_{n=0}^{2M-1-r}\frac{(-1)^{r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right].$
(52)

The condition Eq. (29) gives

$\displaystyle\phi_{2M-1}=(2M-1)!\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{2M}}\right.$
$\displaystyle\left.-\sum_{r=M}^{2M-1}\sum_{n=M}^{2M-1-r}\frac{(-1)^{r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right].$
(53)

Notice in the summation $2M-1-r<M$. We obtain

$\phi_{2M-1}=(-1)^{M}(2M-1)!\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{|\tilde{f}(\omega)|^{2}}{\omega^{2M}}.$
(54)

In Eq. (54), the integrand $|\tilde{f}(\omega)|^{2}/\omega^{2M}\geq 0$ and it
cannot vanish for all $\omega$ from $-\infty$ to $\infty$. Thus we have the
following theorem.

######  Theorem 2.

There is no non-zero modulation function $F(t/T)$ to eliminate the errors so
that the equations $\\{\phi_{2m}=0\\}_{m=0}^{M-1}$ and $\phi_{2M-1}=0$ satisfy
simultaneously.

For example, for the noise with the correlation function $e^{-|t/t_{c}|}$, one
cannot simultaneously eliminate the two leading decoherence terms
$C_{0}\phi_{0}$ and $C_{1}\phi_{1}$, and the error induced by the noise is at
least $\mathcal{O}(T^{3})$. The result is consistent with Theorem 1, since the
noise correlation $e^{-|t/t_{c}|}$ implies a spectrum with a soft cutoff at
high frequencies $S(\omega)=\frac{t_{c}}{1+(\omega t_{c})}$.

###  III.1 Sequence optimization

In this paper, we optimize the DD performance in the short-time limit. As
indicated in Eq. (54), a smaller $\tilde{f}(\omega T)$ at low frequencies will
give a smaller $\phi_{2M-1}$. Here we focus on DD with ideal instantaneous
$\pi$ pulses. We use more pulses to construct a more efficient modulation
function $F(t/T)=F_{\pi}(t/T)$ to minimize $\phi_{2M-1}$, with the conditions
$\\{\phi_{2m}=0\\}_{m=0}^{M-1}$ [Eq. (29)]. Using the method of Lagrange
multipliers, we need to solve a set of nonlinear equations as

$\displaystyle\nabla_{\\{y,T\\}}G_{M}=0,$ (55a) $\displaystyle
G_{M}\equiv\sum_{j=0}^{M-1}y_{j}\lambda_{j}^{(\pi)}+\phi_{2M-1},$ (55b)
$\displaystyle\nabla_{\\{y,T\\}}\equiv(\frac{\partial}{\partial
T_{1}},...,\frac{\partial}{\partial T_{N}},\frac{\partial}{\partial
y_{0}},...,\frac{\partial}{\partial y_{M-1}}).$ (55c)

The introduced variables $\\{y_{j}\\}$ are the Lagrange multipliers. Note that
the sequence optimization in Ref. Pasini and Uhrig (2010) also used the
constraints $\\{\lambda_{m}^{(\pi)}=0\\}$, but the constraints were used there
to guarantee the convergence of the calculation of $\chi(T)$. Here, the
constraints eliminate the lowest orders of errors
($\\{\phi_{2m}=0\\}_{m=0}^{M-1}$) [see Eq. (29)] in short-time limit. In
particular, the decoherence from inhomogeneous broadening is eliminated when
$\phi_{0}=0$.

We calculate Eq. (26) by separating the domain of integration according to the
value of $F_{\pi}(t_{1}/T)F_{\pi}(t_{2}/T)$. For $k\geq 0$, we obtain

$\displaystyle\phi_{k}$
$\displaystyle=\frac{-1}{T^{k+2}(k+1)(k+2)}\left[4\sum_{j=2}^{N}\sum_{i=1}^{j-1}(T_{j}-T_{i})^{k+2}(-1)^{i+j}\right.$
$\displaystyle+(T_{N+1}-T_{0})^{k+2}(-1)^{N+1}+2\sum_{j=1}^{N}(T_{j}-T_{0})^{k+2}(-1)^{j}$
$\displaystyle\left.+2\sum_{i=1}^{N}(T_{N+1}-T_{i})^{k+2}(-1)^{N+1+i}\right].$
(56)

Then we have

$\displaystyle\frac{\partial\phi_{2M-1}}{\partial(T_{k}/T)}=\frac{(-1)^{k}}{T^{2M}M}\left[2\sum_{j=k+1}^{N}(T_{j}-T_{k})^{2M}(-1)^{j}-(T_{k}-T_{0})^{2M}\right.$
$\displaystyle\left.+(T_{N+1}-T_{k})^{2M}(-1)^{N+1}-2\sum_{j=1}^{k-1}(T_{k}-T_{j})^{2M}(-1)^{j}\right].$
(57)

For the special case of $M=1$, $G_{1}=y_{0}\lambda_{0}^{(\pi)}+\phi_{1}$, we
find that the CPMG sequences are solutions to Eq. (55). The timing of an
$N$-pulse CPMG sequence reads

$T_{j}^{\text{CPMG}}=\frac{2j-1}{2N}T,\text{ for }j=1,...,N.$ (58)

The CPMG sequences obviously satisfy the constraint $\lambda_{0}^{(\pi)}=0$
[see Eq. (19)], which is the so-called echo condition that eliminates the
effect of static inhomogeneous broadening. Eqs. (57) and (19) give

$\displaystyle\left.\frac{\partial\phi_{1}}{\partial(T_{k}/T)}\right|_{\text{CPMG}}=(-1)^{k+1}\frac{1}{4N^{2}}\left[1+(-1)^{N}\right],$
(59a)
$\displaystyle\left.y_{0}\frac{\partial\lambda_{0}^{(\pi)}}{\partial(T_{k}/T)}\right|_{\text{CPMG}}=2(-1)^{k+1}y_{0}.$
(59b)

Thus the CPMG sequences also satisfy Eq. (55) with
$y_{0}=-\frac{1}{8N^{2}}\left[1+(-1)^{N}\right]$, so they are at least the
locally optimal pulse sequences. It has been proved that CPMG sequences are
the most efficient pulse sequences in protecting the qubit coherence against
telegraph-like noise in the short-time limit Chen and Liu (2010). With
numerical evidence, we conjecture that it is also true that the CPMG sequences
are globally optimal in the short-time limit when the time expansion of the
noise correlation function has the two leading terms $C_{0}$ and $C_{1}|t|$.

For other cases of minimizing $\phi_{2M-1}$ with the condition
$\\{\lambda_{m}^{(\pi)}=0\\}_{m=0}^{M-1}$, one can see that the short-time
optimized DD (ODD) coincides with UDD for pulse number $N\leq M$. And as $N$
increases, the ODD sequences gradually approach the CPMG sequences. For
example, ODD for the noise correlation

$\overline{\beta(t)\beta(0)}=C_{0}+C_{2}t^{2}+C_{3}|t|^{3}+\mathcal{O}(t^{4}),$
(60)

is shown in Fig. 2 in comparison with UDD and CPMG. The ODD sequences resemble
the CPMG sequences when $N$ is large.

Figure 2:  (Color online). Comparison of the ODD, UDD and CPMG sequences for
different pulse number $N$. Squares (blue), triangles (green), and circles
(red) correspond to UDD, CPMG, and ODD. The ODD sequences are optimized to
minimize $\phi_{3}$ under the constraints $\phi_{0}=\phi_{2}=0$.

We show in Fig. 3(a) the performance of three DD schemes against the noise
described by Eq. (60). A comparison is also shown in Fig. 3(b) by considering
a hard high-frequency cutoff $\omega_{c}=40$. In Fig. 3(a), we can see that
ODD sequences give better performance than UDD and CPMG sequences. These ODD
sequences are optimal for a wide range of noise which has the noise
correlation given by Eq. (60). When we introduce a high-frequency cutoff in
the noise spectrum, as the case in Fig. 3(b), the ODD is the best initially
when the number of pulses $N\lesssim\omega_{c}T/2\approx 10$, and the UDD
sequences become better and suppress the decoherence order by order when $N$
is large and the hard cutoff is reached. In Fig. 3(b), for large $N$ UDD is
better than ODD, since the ODD sequences are optimized for soft-cutoff noise
rather than hard-cutoff noise. In Fig. 3(a) the decreasing of $\chi(T)$ is a
linear decrease in the double-logarithm plot, but in Fig. 3(b) it is much
faster. This confirms that DD is not so efficient against soft-cutoff noise.

Figure 3: (Color online). The decoherence function $\chi(T)$ as a function of
pulse number $N$ (a) without hard cutoff and (b) with a hard high-frequency
cutoff $\omega_{c}=40$. Here the noise spectrum
$S(\omega)=\frac{10^{5}}{1+\omega^{4}}$ and $T=0.5$. Squares (blue), triangles
(green), and circles (red) correspond to UDD, CPMG, and ODD. The ODD sequences
are the same as those shown in Fig. 2.

##  IV Summary and conclusions

We have studied the dynamical decoupling control of decoherence caused by
Gaussian noise with soft cutoff in a general modulation formalism. We have
proved Theorem 1 which shows that, for the soft cutoff with the power-law
asymptote $~{}\alpha/\omega^{P}$ at high frequencies, DD can only suppress
decoherence to $\mathcal{O}(T^{P+1})$, where $P$ does not need to be an
integer. When the short-time expansion of noise correlation has the $(2K-1)$th
odd expansion term, DD can only suppress decoherence to
$\mathcal{O}(T^{2K+1})$ (Theorem 2). The existence of odd-order terms in the
short-time expansion corresponds to a soft high-frequency cutoff
($\sim\alpha/\omega^{2K}$) in the noise spectrum. For these noise spectra, we
have derived the equations for pulse sequence optimization, which minimizes
the leading odd-order decoherence function and eliminates even-order
decoherence functions of lower orders. The ODD sequences obtained by this
method coincide with the UDD sequences when the pulse number $N$ is small, and
they resemble CPMG sequences when $N$ is large. For the special case that the
short-time correlation function expansion has a linear term in time (i.e., a
soft cutoff $\sim\alpha/\omega^{2}$), the ODD sequences are exactly the CPMG
sequences.

Although we derived the theorems from a pure dephasing model, we expect that
the results of the existence of the largest decoupling order in short-time
limit can be extended to the general decoherence model (including both
dephasing and relaxation) of quantum systems. It is desirable to study the DD
in suppressing the general decoherence of quantum systems in the soft-cutoff
noise in the future.

###### Acknowledgements.

We thank Yi-Fan Luo and Bobo Wei for discussions. This work was supported by
the Hong Kong GRF CUHK402209, the CUHK Focused Investments Scheme, and the
National Natural Science Foundation of China Project No. 11028510.

##  Appendix A Modulation Functions in Multiqubit Systems

Consider an $L$-qubit (or $2^{L}$-level) system suffering pure dephasing
described by the Hamiltonian

$H=\sum_{m=0}^{2^{L}-1}|m\rangle\langle m|[\omega_{m}+\beta_{m}(t)],$ (61)

where $\omega_{m}$ is the energy and $\beta_{m}(t)$ is the fluctuation on the
state $|m\rangle$. Here $m=(m_{L}\cdots m_{2}m_{1})$ is a binary code with
$m_{l}=0$ or $1$ for the $l$th qubit.

The Pauli operator for the $l$th qubit is

$\sigma_{x}^{(l)}=\sum_{m_{l}=0}\left(|m+2^{l-1}\rangle\langle
m|+\text{H.c.}\right),$ (62)

which exchanges two basis states $|m\rangle$ and $|m^{\prime}\rangle$ if $m$
and $m^{\prime}$ differ at and only at the $l$th bit.

After a sequence of $\sigma_{x}^{(l)}$ pulses and a final pulse
$\sigma_{\text{add}}=\sigma_{x}^{(l_{1})}\cdots\sigma_{x}^{(l_{N})}$, the
evolution operator is

$\displaystyle U(T)$
$\displaystyle=\sigma_{\text{add}}U(T_{N+1},T_{N})\sigma_{x}^{(l_{N})}U(T_{N},T_{N-1})\cdots$
$\displaystyle\times\sigma_{x}^{(l_{2})}U(T_{2},T_{1})\sigma_{x}^{(l_{1})}U(T_{1},T_{0})\sigma_{x}^{(l_{0})},$
(63)

where $T_{0}=0$, $T_{N+1}\equiv T$, and $\sigma_{x}^{(l_{0})}\equiv I$. The
free evolution operator

$U(T_{j+1},T_{j})=\exp\left[-i\int_{T_{j}}^{T_{j+1}}\sum_{m=0}^{2^{L}-1}|m\rangle\langle
m|[\omega_{m}+\beta_{m}(t)]dt\right].$ (64)

We write the evolution operator as

$U(T)=e^{-i\sum_{j=0}^{N}\int_{T_{j}}^{T_{j+1}}\sum_{m=0}^{2^{L}-1}\Xi_{m,j}[\omega_{m}+\beta_{m}(t)]dt},$
(65)

with
$\Xi_{m,j}\equiv\sigma_{x}^{(l_{0})}\cdots\sigma_{x}^{(l_{j})}|m\rangle\langle
m|\sigma_{x}^{(l_{j})}\cdots\sigma_{x}^{(l_{0})}$. The phase factor between
the states $|p\rangle$ and $|q\rangle$ changed during the evolution time $T$
is

$\varphi_{pq}(T)=\langle p|U(T)|p\rangle\langle q|U^{\dagger}(T)|q\rangle,$
(66)

where

$\langle p|U(T)|p\rangle=e^{-i\sum_{j=0}^{N}\int_{T_{j}}^{T_{j+1}}\langle
p|\sum_{m=0}^{2^{L}-1}\Xi_{m,j}|p\rangle[\omega_{m}+\beta_{m}(t)]dt}.$ (67)

Note that
$\sigma_{x}^{(l_{j})}\cdots\sigma_{x}^{(l_{0})}|p\rangle=|p\oplus[l_{0}\cdots
l_{j}]\rangle$ with

$p\oplus[l_{0}\cdots l_{j}]\equiv p\oplus 2^{l_{1}}\oplus
2^{l_{2}}\cdots\oplus 2^{l_{j}},$ (68)

for $j>0$ and $p\oplus[l_{0}]\equiv p$. Here $\oplus$ denotes addition on
binary digits, that is, $p\oplus[l_{0}\cdots l_{j}]$ is obtained by flipping
the $l_{1}$th,$\ldots$,$l_{j}$th bits of $p=(p_{L}\cdots p_{2}p_{1})$ in
binary code. We obtain

$\langle
p|U(T)|p\rangle=e^{-i\sum_{j=0}^{N}\int_{T_{j}}^{T_{j+1}}[\omega_{p\oplus[l_{0}\cdots
l_{j}]}+\beta_{p\oplus[l_{0}\cdots l_{j}]}(t)]dt}.$ (69)

Therefore the coherence between the states $|p\rangle$ and $|q\rangle$
decreases by the average of the random phase

$\overline{e^{-i\sum_{j=0}^{N}\int_{T_{j}}^{T_{j+1}}[\beta_{p\oplus[l_{0}\cdots
l_{j}]}(t)-\beta_{q\oplus[l_{0}\cdots l_{j}]}(t)]dt}}.$ (70)

When each of the qubits feels the same noise,
$\beta_{m}(t)=(\sum_{k=1}^{L}m_{k})\beta(t)$ for the $m$th level, and

$\overline{\varphi_{pq}(T)}=e^{-i\int_{0}^{T}F_{pq}(t)\omega
dt}\overline{e^{-i\int_{0}^{T}F_{pq}(t)\beta(t)dt}},$ (71)

where the the modulation function is defined as

$F_{pq}(t)=(\sum_{k=1}^{L}\tilde{p}_{k})-(\sum_{k=1}^{L}\tilde{q}_{k}),$ (72)

with $\tilde{p}=p\oplus[l_{0}\cdots l_{j}]$ and $\tilde{q}=q\oplus[l_{0}\cdots
l_{j}]$ for $t\in(T_{j},T_{j+1}]$. For example, when $L=2$,
$F_{pq}(t)\in\\{0,\pm 1,\pm 2\\}$.

##  Appendix B Decoherence Functions $\phi_{k}$

As $F(t/T)=0$ for $t\notin(0,T]$, we extend the bounds of integration for $t$
to infinity and transform Eq. (26) to

$\displaystyle\phi_{k}$ $\displaystyle=$
$\displaystyle\Re\frac{\partial^{k}}{\partial(i\eta)^{k}}\int_{-\infty}^{\infty}\frac{d\omega_{1}}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega_{2}}{2\pi}\int_{-\infty}^{\infty}dt_{1}\int_{0}^{t_{1}}dt_{2}$
(73)
$\displaystyle\times\tilde{f}^{*}(\omega_{1})\tilde{f}(\omega_{2})e^{i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}e^{i\eta(t_{1}-t_{2})},$

where we set $\eta\rightarrow 0$ after differentiation. Integrations over
$t_{2}$, $t_{1}$ and $\omega_{1}$ give

$\phi_{k}=\Re\frac{\partial^{k}}{\partial(i\eta)^{k}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{\tilde{f}(\omega)}{i(\omega+\eta)}[\tilde{f}^{*}(-\eta)-\tilde{f}^{*}(\omega)].$
(74)

Using the formulas

$\frac{\partial^{k}}{\partial(i\eta)^{k}}\left[u(\eta)v(\eta)\right]=\sum_{r=0}^{k}\frac{k!}{r!(k-r)!}\left[\frac{\partial^{k-r}u(\eta)}{\partial(i\eta)^{k-r}}\right]\left[\frac{\partial^{r}v(\eta)}{\partial(i\eta)^{r}}\right],$
(75)
$\left.\frac{\partial^{k-r}}{\partial(i\eta)^{k-r}}\frac{1}{i(\omega+\eta)}\right|_{\eta=0}=\frac{-(k-r)!}{(-i\omega)^{k-r+1}},\text{
for }r\geq 0,$ (76)

and

$\left.\frac{\partial^{r}}{\partial(i\eta)^{r}}\tilde{f}^{*}(-\eta)\right|_{\eta=0}=\lambda_{r}^{*},\text{
for }r\geq 0,$ (77)

we have

$\phi_{k}=k!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{k+1}}-\sum_{r=0}^{k}\frac{\tilde{f}(\omega)}{(-i\omega)^{k-r+1}}\frac{\lambda_{r}^{*}}{r!}].$
(78)

Changing the summation index, we obtain

$\phi_{k}=k!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{k+1}}-\sum_{r=0}^{k}\frac{\tilde{f}(\omega)}{(-i\omega)^{k-r+1}}\frac{\lambda_{r}^{*}}{r!}\right].$
(79)

Note that the summation over $r$ and the integration over frequency cannot be
exchanged when the integration does not converge for each individual term.
Using Eq. (16a), we have

$\displaystyle\phi_{k}$ $\displaystyle=$ $\displaystyle
k!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{k+1}}+\sum_{r=0}^{k}\sum_{n=0}^{k-r}\frac{(-1)^{k-r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right.$
(80) $\displaystyle+$
$\displaystyle\left.\sum_{r=0}^{k}\sum_{n=k-r+1}^{\infty}\frac{(-1)^{k-r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right].$

We simplify the last line by using Eq. (16b) and the equality

$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\sum_{n=r}^{\infty}\frac{(\pm
i\omega)^{n-r}}{n!}t^{n}=\frac{1}{2}\frac{t^{r-1}}{(r-1)!},\text{ for }r\geq
1,t\geq 0,$ (81)

which is proved in Appendix C. We obtain

$\displaystyle\phi_{k}=k!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{k+1}}+\sum_{r=0}^{k}\sum_{n=0}^{k-r}\frac{(-1)^{k-r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right]$
$\displaystyle+\frac{k!}{2}\Re\sum_{r=0}^{k}(-1)^{k-r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{k-r}}{(k-r)!}.$
(82)

For even number $2m$, $\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{2m+1}}$ is
an odd function and its integral vanishes. Eq. (82) gives

$\displaystyle\phi_{2m}=(2m)!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\sum_{r=0}^{2m}\sum_{n=0}^{2m-r}\frac{(-1)^{r}}{(i\omega)^{2m-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right]$
$\displaystyle+\frac{(2m)!}{2}\Re\sum_{r=0}^{2m}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2m-r}}{(2m-r)!},$
(83)

which is decomposed as (with the changes of summation order and indices)

$\displaystyle\phi_{2m}$ $\displaystyle=$
$\displaystyle(2m)!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\sum_{r=0}^{2m}\sum_{n=0}^{2m-r}\frac{(-1)^{r}/2}{(i\omega)^{2m-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right.$
(84)
$\displaystyle+\left.\sum_{r=0}^{2m}\sum_{n=0}^{2m-r}\frac{(-1)^{n}/2}{(i\omega)^{2m-r-n+1}}\frac{\lambda_{r}}{r!}\frac{\lambda_{n}^{*}}{n!}\right]$
$\displaystyle+\frac{(2m)!}{2}\Re\sum_{r=0}^{2m}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2m-r}}{(2m-r)!},.$

The integrals of the terms with odd-power of $\omega$ vanish. And for even
functions of $\omega$, the sum $n+r$ is an odd number, so
$(-1)^{r}+(-1)^{n}=0$. Thusthe real part of the integral in Eq. (84) vanishes.
From the last line of Eq. (84), we obtain Eq. (28), i.e.,

$\displaystyle\phi_{2m}$
$\displaystyle=\frac{(2m)!}{2}\sum_{r=0}^{2m}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2m-r}}{(2m-r)!}.$
(85)

##  Appendix C Proof of Equation (81)

To prove Eq. (81), we just need to prove

$\lim_{R\rightarrow\infty}\int_{-R}^{R}dx\sum_{n=r}^{\infty}\frac{(\pm
ix)^{n-r}}{n!}=\frac{\pi}{(r-1)!},\text{ for }r\geq 1,$ (86)

where the bounds in the integral guarantee that the modulation function $F(t)$
is a real function. Using

$\displaystyle\int_{-R}^{R}dx\sum_{n=r}^{\infty}\frac{(\pm
ix)^{n-r}}{n!}=\sum_{n=1}^{\infty}\frac{R^{n}}{(n+r-1)!n}(i^{n-1}+\text{c.c.}),$
(87a) $\displaystyle\frac{1}{(r-1)!}\int_{-R}^{R}\frac{\sin
x}{x}dx=\sum_{n=1}^{\infty}\frac{R^{n}}{n!n}\frac{1}{(r-1)!}(i^{n-1}+\text{c.c.}),$
(87b)

and $\lim_{R\rightarrow\infty}\int_{-R}^{R}\frac{\sin x}{x}dx=\pi$, we just
need to prove

$\lim_{R\rightarrow\infty}\sum_{n=1}^{\infty}\left[\frac{R^{n}}{(n+r-1)!n}-\frac{R^{n}}{n!n(r-1)!}\right](i^{n-1}+\text{c.c.})=0.$
(88)

For $r=1$, it obviously holds. For $r\geq 2$, we can show the difference

$\Delta\equiv\sum_{n=1}^{\infty}\frac{R^{n}(i^{n-1}+\text{c.c.})}{(n+r-1)!n}\left[(r-1)!-\prod_{j=1}^{r-1}(n+j)\right]=\mathcal{O}\left(\frac{1}{R}\right),$
(89)

so $\text{lim}_{R\rightarrow\infty}\Delta=0$. By expanding the terms in the
square brackets of Eq. (89), we have

$\Delta=\sum_{n=1}^{\infty}\frac{R^{n}(i^{n-1}+\text{c.c.})}{(n+r-1)!}\left(\sum_{k=0}^{r-2}a_{k}n^{k}\right),$
(90)

where $a_{k}$ is a number independent of $n$. We arrange the terms in the
square brackets and get

$\Delta=\sum_{n=1}^{\infty}\frac{R^{n}(i^{n-1}+\text{c.c.})}{(n+r-1)!}\left[\sum_{k=1}^{r-2}b_{k}\prod_{j=1}^{k}[(n+r-j)]+b_{0}\right],$
(91)

with $b_{j}$ independent of $n$. After some simplification it becomes for
$r\geq 2$

$\sum_{k=1}^{r-1}\frac{b_{r-k-1}}{i^{k+1}R^{k}}\left(e^{iR}-\sum_{n=0}^{k-1}\frac{R^{n}}{n!}i^{n}\right)+\text{c.c.}=\mathcal{O}\left(\frac{1}{R}\right).$
(92)

Hence $\Delta=\mathcal{O}(\frac{1}{R})$, and Eq. (81) is proved.

## References

  * Nielsen and Chuang (2000) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information_ (Cambridge University, Cambridge, 2000).
  * Duan and Guo (1997) L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997).
  * Zanardi (1998) P. Zanardi, Phys. Rev. A 57, 3276 (1998).
  * Lidar et al. (1998) D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998).
  * Shor (1995) P. W. Shor, Phys. Rev. A 52, 2493 (1995).
  * Steane (1996) A. M. Steane, Proc. R. Soc. Lond. A 452, 2551 (1996).
  * Viola and Lloyd (1998) L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 (1998).
  * Ban (1998) M. Ban, J. Mod. Opt. 45, 2315 (1998).
  * Zanardi (1999) P. Zanardi, Phys. Lett. A 258, 77 (1999).
  * Viola et al. (1999) L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).
  * Yang et al. (2011) W. Yang, Z.-Y. Wang, and R.-B. Liu, Front. Phys. 6, 2 (2011).
  * Hahn (1950) E. L. Hahn, Phys. Rev. 80, 580 (1950).
  * Carr and Purcell (1954) H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954).
  * Meiboom and Gill (1958) S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958).
  * Mehring (1983) M. Mehring, _Principles of High Resolution NMR in Solids_ (Spinger-Verleg, Berlin, 1983), 2nd ed.
  * Khodjasteh and Lidar (2005) K. Khodjasteh and D. A. Lidar, Phys. Rev. Lett. 95, 180501 (2005).
  * Khodjasteh and Lidar (2007) K. Khodjasteh and D. A. Lidar, Phys. Rev. A 75, 062310 (2007).
  * Yao et al. (2007) W. Yao, R.-B. Liu, and L. J. Sham, Phys. Rev. Lett. 98, 077602 (2007).
  * Witzel and Das Sarma (2007) W. M. Witzel and S. Das Sarma, Phys. Rev. B 76, 241303(R) (2007).
  * Zhang et al. (2007) W. X. Zhang, V. V. Dobrovitski, L. F. Santos, L. Viola, and B. N. Harmon, Phys. Rev. B 75, 201302(R) (2007).
  * Peng et al. (2011) X. Peng, D. Suter, and D. A. Lidar, J. Phys. B: At. Mol. Opt. Phys. 44, 154003 (2011).
  * Álvarez et al. (2010) G. A. Álvarez, A. Ajoy, X. Peng, and D. Suter, Phys. Rev. A 82, 042306 (2010).
  * (23) A. M. Tyryshkin, Z.-H. Wang, W. Zhang, E. E. Haller, J. W. Ager, V. V. Dobrovitski, and S. A. Lyon, arXiv:1011.1903v2.
  * Wang et al. (2012) Z.-H. Wang, W. Zhang, A. M. Tyryshkin, S. A. Lyon, J. W. Ager, E. E. Haller, and V. V. Dobrovitski, Phys. Rev. B 85, 085206 (2012).
  * Barthel et al. (2010) C. Barthel, J. Medford, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. 105, 266808 (2010).
  * Santos and Viola (2008) L. F. Santos and L. Viola, New J. Phys. 10, 083009 (2008).
  * Wang and Liu (2011a) Z.-Y. Wang and R.-B. Liu, Phys. Rev. A 83, 022306 (2011a).
  * Uhrig (2007) G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007), ibid, 106, 129901 (2011).
  * Lee et al. (2008) B. Lee, W. M. Witzel, and S. Das Sarma, Phys. Rev. Lett. 100, 160505 (2008).
  * Uhrig (2008) G. S. Uhrig, New J. Phys. 10, 083024 (2008).
  * Yang and Liu (2008) W. Yang and R.-B. Liu, Phys. Rev. Lett. 101, 180403 (2008).
  * Uhrig and Lidar (2010) G. S. Uhrig and D. A. Lidar, Phys. Rev. A 82, 012301 (2010).
  * Pasini et al. (2008) S. Pasini, T. Fischer, P. Karbach, and G. S. Uhrig, Phys. Rev. A 77, 032315 (2008).
  * Fauseweh et al. (2012) B. Fauseweh, S. Pasini, and G. S. Uhrig, Phys. Rev. A 85, 022310 (2012).
  * Biercuk et al. (2009a) M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Nature 458, 996 (2009a).
  * Du et al. (2009) J. Du, X. Rong, N. Zhao, Y. Wang, J. Yang, and R.-B. Liu, Nature 461, 1265 (2009).
  * Biercuk et al. (2009b) M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Phys. Rev. A 79, 062324 (2009b).
  * Uys et al. (2009) H. Uys, M. J. Biercuk, and J. J. Bollinger, Phys. Rev. Lett. 103, 040501 (2009).
  * Jenista et al. (2009) E. R. Jenista, A. M. Stokes, R. T. Branca, and W. S. Warren, J. Chem. Phys. 131, 204510 (2009).
  * Uhrig (2009) G. S. Uhrig, Phys. Rev. Lett. 102, 120502 (2009).
  * West et al. (2010) J. R. West, B. H. Fong, and D. A. Lidar, Phys. Rev. Lett. 104, 130501 (2010).
  * Quiroz and Lidar (2011) G. Quiroz and D. A. Lidar, Phys. Rev. A 84, 042328 (2011).
  * Kuo and Lidar (2011) W.-J. Kuo and D. A. Lidar, Phys. Rev. A 84, 042329 (2011).
  * Jiang and Imambekov (2011) L. Jiang and A. Imambekov, Phys. Rev. A 84, 060302 (2011).
  * Mukhtar et al. (2010) M. Mukhtar, T. B. Saw, W. T. Soh, and J. Gong, Phys. Rev. A 81, 012331 (2010).
  * Wang and Liu (2011b) Z.-Y. Wang and R.-B. Liu, Phys. Rev. A 83, 062313 (2011b).
  * Cywiński et al. (2008) Ł. Cywiński, R. M. Lutchyn, C. P. Nave, and S. Das Sarma, Phys. Rev. B 77, 174509 (2008).
  * Hodgson et al. (2010) T. E. Hodgson, L. Viola, and I. D’Amico, Phys. Rev. A 81, 062321 (2010).
  * Khodjasteh et al. (2011) K. Khodjasteh, T. Erdélyi, and L. Viola, Phys. Rev. A 83, 020305 (2011).
  * Ajoy et al. (2011) A. Ajoy, G. A. Álvarez, and D. Suter, Phys. Rev. A 83, 032303 (2011).
  * Pasini and Uhrig (2010) S. Pasini and G. S. Uhrig, Phys. Rev. A 81, 012309 (2010).
  * Chen and Liu (2010) K. Chen and R.-B. Liu, Phys. Rev. A 82, 052324 (2010).
  * Gordon et al. (2008) G. Gordon, G. Kurizki, and D. A. Lidar, Phys. Rev. Lett. 101, 010403 (2008).
  * Anderson (1954) P. W. Anderson, J. Phys. Soc. Jpn. 9, 316 (1954).
  * Kubo (1954) R. Kubo, J. Phys. Soc. Jpn. 9, 935 (1954).
  * Uhrig and Pasini (2010) G. S. Uhrig and S. Pasini, New J. Phys. 12, 045001 (2010).
  * (57) Z.-Y. Wang and R.-B. Liu, unpublished.
  * Gordon et al. (2006) G. Gordon, G. Kurizki, and A. G. Kofman, Opt. Commun. 264, 398 (2006).
  * Gordon and Kurizki (2007) G. Gordon and G. Kurizki, Phys. Rev. A 76, 042310 (2007).
  * Gordon et al. (2007) G. Gordon, N. Erez, and G. Kurizki, J. Phys. B 40, S75 (2007).
  * Berman and Brewer (1985) P. R. Berman and R. G. Brewer, Phys. Rev. A 32, 2784 (1985).
  * Sun and Liu (1996) C. P. Sun and X. J. Liu, Acta Phys. Sin. 5, 343 (1996).


Creighton Rosentallogo_ttlM. Antonia Huertas, João Marcos, María Manzano,
Sophie Pinchinat,
François Schwarzentruber54th International Conference on Tools for Teaching
Logic11191TTL2015

# Teaching Logics through Their Philosophical Commitments: “Logical
Worldviews”

Creighton Rosental  Associate Professor of Philosophy
Mercer University
<EMAIL_ADDRESS>
###### Abstract

I have developed a pedagogy and textbook for teaching logic centered on what I
call “logical worldviews”. A logical worldview examines the close connection
between philosophical commitments and the logical principles and method for a
particular historical logical system. The class examines multiple historical
logical worldviews to show how philosophical positions and logical systems
have been closely intertwined, and how changes in philosophical positions have
over time yielded corresponding changes in the development of logic. Such an
approach has great benefits for teaching logic to undergraduates.

###### keywords:

history of logic, logic education

##  1 A “New” Approach to Teaching Logic to Undergraduates: Logical Worldviews

Since 2006, I have developed an approach to teaching logic to undergraduate
philosophy majors that bases logic instruction in the philosophical background
underpinning historic logical systems. Starting in 2013, I expanded this
pedagogical approach to a general education introductory logic class, which is
open to students of any major and at any stage of their college career. This
curricular development has culminated in a textbook designed for an
introductory logic course that can be used by teachers with minimal or no
experience in the history of logic.

In the textbook I present a re-contextualization of classical formal logic as
just one of many approaches to logic, each having its own philosophical
commitments and unique value. I suggest that there is more to logic than is
construed by only considering its contemporary formal aspects; after all,
formalism in mathematics and logic is a relatively recent innovation, adopted
in part because of other, non-mathematical considerations and commitments.

I hold that the history of logical thought includes a series of logics, or
logical systems, each of which has non-formal philosophical commitments. I
further hold that these commitments are not separate and independent add-ons
to the logical system, but that the logical method and fundamental principles
of the system are shaped and determined by these commitments. Though a
historical logic can be considered formally and in the abstract, such a
perspective yields an inaccurate account of what the logical system was
intended to do by its author, of the true power of its methods of reasoning,
and of the precise meaning of the logical principles employed. When the
logical system is considered in conjunction with the intertwined philosophical
commitments, I characterize this broader conceptualization as a “logical
worldview.”

##  2 Using Philosophy to Better Understand a Logical System

It will be useful at this point to consider a particular example, and
Aristotle is perhaps the best place to start: after all, as the first to fully
develop a logical worldview, many later logics were (as I present in the
textbook) a response to rejecting one or more Aristotelian philosophical
commitments. What follows is a very brief (and incomplete) characterization of
some of the philosophical commitments in Aristotle’s logic.111Here and
throughout this paper, I present the philosophical views in an introductory
and casual manner, to match the form in which they are presented in the
textbook. Though some of these Aristotelian interpretations are contested by
scholars of Aristotle, a close and well-defended history of Aristotle’s
logical and philosophical positions would not be to the point. What matters
for the sake of this pedagogical approach is that philosophically and
historically inexperienced students see that Aristotle’s logical system is
amenable to being better understood by framing it in terms of philosophical
positions that Aristotle (plausibly) held.

  * •

Theory of truth: Correspondence. Propositions are true if and only if they
correspond to how the world actually is; false otherwise. Somewhat
anachronistically, we might refer to propositions as being true if and only if
they correspond to facts about the world. This view also includes bivalence –
propositions are either true or false.222See Aristotle _Categories_ chapter
14.

  * •

Access to truth: Direct Realism. Human beings can directly access the world,
and the properties and entities they perceive are (generally, in principle)
the same as the actual properties and entities that exist in the world.333See
Aristotle _de Anima_ , particularly Book II chapter 12 through Book III
chapter 4.

  * •

Theory of mind: Simple Apprehension and Judgment. The human mind has a
capacity to discern the true properties of things. Further, we are capable of
abstracting from particular properties to universal forms shared by more than
one individual entity, and recombining such forms in the imagination and in
memory. Finally, we can combine previously unseen combinations of forms and
judge whether or not such propositions correspond to reality.444See Aristotle
_On Interpretation_ chapter 1 and Posterior Analytics Book II chapter 19.

  * •

Metaphysics: Substances, Forms and a Hierarchical Ontology. In Aristotle’s
account, the world is constituted by individual entities, or primary
substances, each of which has a number of properties, or forms. Substances can
be grouped together into secondary substances, or species, for which some
forms are essential, others not. Species can be grouped together into a
higher-order grouping, or genus, which have their own essential forms.
Collectively the interconnection of species and genus through essential forms
constitutes a hierarchical ontology, which is rationally discernible.555See
Aristotle _Categories_ ; also _Metaphysics._

In my textbook I explain how these (and other) features of Aristotle’s
philosophical worldview determine the specifics of his formal logic. I’ll
sketch this dependence briefly here.

First, individual substances and their forms are the fundamental perceivable
elements of reality. Collectively, these are referred to as “terms”, which are
the atoms of Aristotle’s logical system.

Second, propositions are assertions about the actual facts of the world. Any
fact will fundamentally involve two components: an identified subject and one
of its forms. Since propositions correspond to these fundamental facts, every
proposition will have only two terms: a subject term (what the proposition is
about), and a predicate term (what is being said of that subject).

Third, through experience, induction, abstraction, and natural talent, humans
are capable of discerning some of the essential qualities of substances. This
allows us to generalize beyond particular substances to be able to quantify
the scope of the assertion (all, some, one) and whether the subject and
predicate are linked (affirmation) or separated (denial).

Fourth, through our capacity to discern the ontological structure of reality,
we are able (in principle) to choose any two propositions and connect them by
means of something in the hierarchy that links them (a middle term). The
fundamental form of reasoning from one proposition (with two terms) and
another (sharing one of those terms) is through a bridge proposition, sharing
a “middle” term with our premise and conclusion. This is the minimum size of
an argument productive of new knowledge, and thus the syllogistic form must
have three propositions sharing exactly three distinct terms.

This is a rough and quick sketch of how Aristotle’s philosophical commitments
determine the form of his logical method, which culminates in the syllogism.
This perspective on Aristotle’s logical worldview also helps to explain many
other curiosities about Aristotle’s logic that have perplexed many trying to
understand his logic. I can’t get into them in detail here, but here are a few
positions in Aristotle’s logic that are explained by considering his
philosophical commitments.

  * •

Reduction to the first figure. Rather than be satisfied with identifying valid
syllogistic moods, he takes great pains to reduce them to four figure one
moods. He does so because he believes that these four moods are the
fundamental natural and self-evident reasoning patterns for human
beings.666See Aristotle _Prior Analytics_ Book II chapters 1-15.

  * •

The missing fourth figure and two moods. In inventorying valid moods,
Aristotle neglects to identify two valid moods in figure one and figure four
altogether. These moods do not fit the “natural” reasoning patterns determined
by Aristotle’s view and are thus neglected as inappropriate forms of human
reason.777For a good discussion of this issue see Lynn E. Rose, “Aristotle’s
Syllogistic and the Fourth Figure,”  _Mind_ vol. 74, no. 295 (July 1965):
382-389.

  * •

Existential import. Many have found that Aristotle’s inference that “All S are
P” implies “Some S is P” to be logically invalid. However, given his
philosophical commitments, no universal claim could ever be made without first
establishing the truth of the particular, so the truth of the universal will
always imply the particular.

Though this exposition of this logical worldview is seriously abridged, it
should demonstrate that the form, method and logical principles of Aristotle’s
logic are heavily dependent on Aristotle’s philosophical commitments. Those
who wish to teach Aristotle’s logic to students can use the logical worldview
in order to have Aristotle’s logic make much more sense to students, and show
how his logic is closely connected to many of Aristotle’s other philosophical
positions.

##  3 Using Philosophical Commitments to Explain Transitions from One Logical
System to Another

I’ll briefly discuss how later logical systems can be understood as having
rejected or modified some part or other of Aristotle’s logical worldview. I’m
not suggesting that these later logics were the result of simply abandoning
some philosophical commitment or other of Aristotle’s; rather, by rejecting
one or more commitments, new opportunities for logical method are opened up,
which in combination with some innovations led to new ways to do and conceive
logic. I’ll briefly consider here two logical worldviews post-Aristotle.

_Bacon’s rejection of Aristotle’s theory of mind: a motivation for natural
history and modern natural sciences._ In Bacon’s _New Organon_ , he takes
Aristotelian philosophers to task for relying too heavily on our natural
capacity to understand the forms present in reality and argues that nature is
far too subtle for our unaided sensory and cognitive capacities. He argues
that we must develop new instruments to enhance our senses (microscopes,
thermometers, etc.) and to enhance our cognitive discernment of what we are
seeing by means of experiments.888He makes this case over the course of Book
I. Since our capacity to naturally discern the forms in reality is flawed, we
must develop a new logical technique, which is basically induction from a very
large number of particular instances towards generalizations of increasing
universality and confidence.999This technique is laid out in Book II.
Therefore, the syllogism, which relies on gathering information about forms
and then deducing new truths is for Bacon the wrong logical method. Bacon does
not reject Aristotle’s other philosophical commitments mentioned above, and to
varying degrees and with some modification, the natural sciences still endorse
a correspondence theory of truth, a capacity to discern forms of substances,
and a hierarchical ontology.

_Boole’s rejection of Aristotle’s Direct Realism and metaphysics: an effort to
mathematize logic._ To a great extent Newton’s mathematical approach to
physics follows similar rejections as Boole, but for the sake of what will
soon become clear, Boole’s contribution to logic is easier to present to
students. Starting with the Early Modern philosophers in the 17th century and
culminating in the modern approach to science in the 19th, the rejection of
Direct Realism became fairly common. Philosophers were no longer inclined to
believe that we had direct access to the substances and forms through our
senses exactly as they are in reality. Because we lacked the capacity to see
the world as it actually was, Boole changed the focus of logic from terms to
classes in his seminal work The Laws of Thought. Instead of the logical atom
being a term, which required discerning the true forms of individuals, he made
the fundamental element the class, a collection of individuals gathered in
some (perhaps unknowable, perhaps arbitrary) way. One could identify a class
without having to know exactly what was required for membership. His
particular brilliance was to show that the logical method of Aristotle could
be encompassed within an algebra for classes. By shifting from term to class,
Boole needed a new logical method, which he found in algebra. Though Boole
mathematized logic, one should note that he did not abandon Aristotle’s
philosophical commitments completely: he still largely conceived of reality as
being metaphysically ordered in a hierarchy, and also that truth was
established by correspondence. Further, he did believe that he had discovered
the “laws of thought”: the actual laws by which human minds reasoned,
maintaining a close link between logic and theory of mind.101010See Boole _The
Laws of Thought_ chapter 1. As an interesting philosophical divergence into
the history of computing and artificial intelligence, I often have students
learn and discuss the development of logic machines in the 1870s following
Boole’s logical system. These machines prompted a brief but lively debate
amongst American logicians such as C. S. Peirce as to whether machines that
followed the laws of thought actually were thinking. (see C. S. Peirce,
“Logical Machines,”  _The American Journal of Psychology_ , vol. 1, no. 1
(Nov. 1887): 165-170) The view seemed generally to be in the affirmative.
Eventually, however, those doing algebra of logic dropped their commitment to
linking logic to laws of thinking. (see, for example, chapter 1 in Louis
Couturat, _The Algebra of Logic_ (Chicago: Open Court Publishing, 1914))

I hope that these two brief sketches outline how disparate logical systems in
history may appear to be radically different, but that their differences are
in part due to rejecting or modifying some basic philosophical (and non-
logical) positions. These transitions can show students that historical logics
are not useless or obsolete; instead, they are evolutionary predecessors to
later logics (and ultimately contemporary classical logic).111111By the way,
in my textbook I use logical worldviews to show how contemporary propositional
logic results from abandoning a commitment to a correspondence theory of truth
and the resulting modification of the algebraic method of Boole’s logic. This
philosophical position is partly inspired by formalism – often identified with
Giuseppe Peano and many of his Italian colleagues. See Paolo Mancosu, Richard
Zach, and Calixto Badesa, “The Development of Mathematical Logic from Russell
to Tarski, 1900-1935” in _The Development of Modern Logic_ , edited by Leila
Haaparanta, p. 318-470. Oxford: Oxford University Press, 2009. These
historical logics can still be valuable and useful to the contemporary
student, should one embrace the philosophical commitments underlying the
logical worldview.

Over the years, I have taught and analyzed a number of different historical
logical systems and perspectives by means of a close examination of their
philosophical commitments. In addition to those systems mentioned above, I
have found that at least the following logical approaches can be analyzed in
this way: Medieval faith and reason, propositional and predicate logic,
mathematical physics (a la Newton), Ancient/Medieval problem of universals,
nominalism, Leibniz’s logical innovations, logical fatalism, logical atomism,
logicism, and the theory of relations. I anticipate many more, if not most,
historical logics and logical issues would also be amenable to this
pedagogical approach.

##  4 Pedagogical Benefits of Teaching Logical Worldviews

In teaching introductory formal logic to undergraduates one may have
encountered the following scenario: those students who have an affinity to
formal reasoning (e.g. math, computer science, and science students) take
rather well to logic, but other students (e.g. humanities and social science
students) struggle. This should not be too surprising, given the mathematical
basis for contemporary formal logic. But what to do with those students left
behind? And though the power and flexibility of formal logic is well
demonstrated, especially relative to earlier approaches to logic (such as
Aristotelian logic), students that are not mathematically inclined find formal
logic both confusing and limiting. Such students may be capable of reasoning
well, and may have a great deal of experience and success in reasoning, but
often find that their previously successful reasoning practices are not well
captured by the methods found in classical formal logic. One solution to this
mis-fit is addressed by courses in informal logic (sometimes called “critical
thinking”). And though such an approach to reasoning more closely fits the
more natural, intuitive, and practical forms of reasoning of interest to many
college students, it typically achieves this result by minimizing or avoiding
formal reasoning altogether.

The contemporary setting of formal logic in introductory undergraduate
education is a difficult, if not paradoxical one. Students not already open to
formal reasoning often find formal logic not very “useful”, and many struggle
to see the value or purpose of learning it. This is further compounded when
formal logic is taught by philosophy departments, a discipline that otherwise
frequently engages in applied reasoning in a variety of fields of contemporary
relevance (science, religion, art, ethics, etc.), and in “big” questions that
are quite meaningful to students in their lives (“What is the meaning of
life?” “What is the best life one can live?” “What is the right thing to
do?”). It is rare for a logic class to turn from the study of formal reasoning
methods to connecting those to issues of current interest (“informal” or
philosophical reasoning), and so students who do ultimately learn formal logic
have great difficulty applying their new skills to questions of particular
interest to them, unless they continue on with more advanced philosophy
courses.

By understanding logical systems and methods in the context of their
philosophical commitments, students are exposed to the close connection
between logic and philosophy. They learn why logic has the method it does,
which helps them engage and understand the material more deeply than simply
challenging them to master the mathematical methods of contemporary formal
logic. Further, students learning logical worldviews are more capable of
knitting together formal logic and “informal” or philosophical reasoning, as
these approaches are explicitly linked. Finally, logical worldviews
demonstrate how logic is closely connected to philosophy, and it is rather
easy to introduce philosophical questions of interest to students in the midst
of learning a logical system.121212For example, students can explore logical
fatalism (and their more philosophically interesting versions concerning free
will and determinism or divine foreknowledge) by exploring the consequences of
Aristotle’s commitment to a correspondence theory of truth and bivalence.
Also, students can better understand Anselm’s ontological argument and
Guanilo’s reply by seeing how Anselm’s position is grounded in Aristotle’s own
reductio proofs, and Guanilo’s on Aristotle’s theory of mind and direct
realism.

##  5 Benefits of Teaching Logical Worldviews to History of Philosophy

Most logic textbooks, and by implication, most undergraduate logic classes
teach a system of formal logic developed in the 20th century (except, in some
cases, a cursory examination of syllogistic reasoning). This approach to logic
has some disturbing implications for the 2,500 years of philosophical
reasoning that preceded the 20th century. The considerate student (as well as
many professors) may draw any or all of the following conclusions about
pre-20th century logical reasoning:

  1. 1.

Logic from these time periods was either wrong, incomplete, or unacceptably
limited in scope and methods. After all, if such logics were suitable, why
develop contemporary formal logic that encompasses and surpasses such logics?

  2. 2.

By extension from this first point, philosophical reasoning based on earlier
logics could similarly be considered to be flawed, obsolete, or otherwise
useless. Some schools of philosophical thought have explicitly embraced this
perspective (e.g. the Logical Positivists); today, this perspective is often
implicit in attitudes about historical philosophy from a wide variety of
fields (every so often one will make such an assertion explicitly, as Stephen
Hawking recently did).131313See Stephen Hawking and Leonard Mlodinow, _The
Grand Design_ (New York: Bantam Books, 2010) chapter 1: “Traditionally these
are questions for philosophy, but philosophy is dead. Philosophy has not kept
up with modern developments in science, particularly physics. Scientists have
become the bearers of the torch of discovery in our quest for knowledge.”

  3. 3.

Many contemporary philosophers (students and professors) educated in
contemporary formal logic enough to use it to parse examples of natural
reasoning will use it to parse examples of philosophical reasoning that pre-
dates mathematical logic. This is anachronistic to say the least, and results
in a misleading conception of the actual reasoning made by these historic
figures. Further, it is likely to construe historical philosophy in a bad
light, since it makes it appear that these historical figures engaged
regularly in invalid reasoning for most of the intellectual history of Western
thought.

For some, these considerations are not of much concern: many do, in fact,
believe that history is rife with examples of bad reasoning, just as up to the
scientific revolution, history was plagued with examples of bad science. This
perspective is reinforced by the (mostly legitimate) observation that some of
history’s most highly regarded philosophical reasoners seem to have made grave
errors in their knowledge claims (Aristotle is a significant target here).

In math and science an ahistorical perspective may be appropriate: after all,
most science and math textbooks teach the completed current state of
knowledge, not the long and difficult (and in many cases wrong) path that led
us here. Outside of these fields, however, the value of accurately
understanding past perspectives is more apparent: many areas of study in the
humanities and social sciences to this day engage well-reasoned theories
dating back to Ancient Greece.

Setting concerns about history of philosophy aside, a large number of people
presently endorse Aristotle’s philosophical commitments enumerated above and
teachers would likely find that most of those people would have little problem
with syllogistic reasoning. In contrast, I expect that most teachers of logic
find that many students find much of contemporary formal logic counter-
intuitive and alien to how they reason. In contrast, logical worldviews can
help the logic teacher show that in certain philosophical contexts, particular
historic logics may be the best approach; but that should those philosophical
commitments be rejected or disproved in other reasoning contexts, different
logics are called for. A student’s intuitive reasoning can be embraced rather
than replaced, which would certainly seem to be a desirable outcome for a
logic class.

##  6 Benefits for the Teacher of Logic

I’d like to finish up by discussing a few benefits for the logic teacher who
adopts this approach. I’ve designed the textbook to contain a number of
different modules, each examining a different logical worldview. As I complete
more modules, the textbook will fill up with a fair number of unique instances
of logical systems spanning the last 2,500 years. Each module contains several
key components: (1) an inventory of the philosophical commitments of the
logical system; (2) a trace of the connection between the philosophical
commitments and the development and form of the logical method and principles
employed; (3) an introduction and discussion of some interesting philosophical
issues that arise under this worldview.

Each module is designed to be teachable by those with only a minimal expertise
in the history of philosophy.141414As mentioned earlier, this involves
presenting a simplistic interpretation which historians of philosophy might
find unsupported by sufficient research and scholarship. However, most
textbooks are likely subject to the same complaint, and thus this publication
genre permits, if not requires, such oversimplification. The characterization
of a logical worldview does depend on a historical interpretation of the
philosophy and logic under study, but a background in historical scholarship
is not required in order to understand or teach the main components of each
module. Instead, anyone with a good background in philosophy should be able to
comprehend and easily teach the first and third components (inventory of
philosophical commitments, and linkage to philosophical issues). The second
component of each module (tracing the connection between the philosophy and
the logic) is more subtle and complex, but should be within reach for anyone
experienced in teaching logic.

Modules are designed to be able to be taught independently, although many are
closely related to others. This allows the teacher to pick and choose a path
through the history of logic, and thereby emphasize the impact of different
philosophical views on the development of logic and/or our understanding of
certain logical concepts and methods. For example, in a logic class that
culminates in teaching propositional and predicate logic, one might start with
Aristotle and then pick modules that progressively show how logic became more
mathematical and abstract (by for instance, selecting logical worldviews of
Leibniz, Boole and Russell as transitional modules). On the other hand, a
course interested in the logic of the scientific method might feature the
logical worldviews of Aristotle, Medieval faith and reason, Bacon, and Newton.

In conclusion, I have found this approach to be of great benefit to students
and to myself by expanding teaching of logic beyond the boundaries of formal
methods and systems. Students seem to derive a deeper understanding of logic
by seeing how it connects with other philosophical positions. Further, this
approach helps make logic more comprehensible to the broad pool of students
who are either disinclined or not well suited towards the strictly formal,
mathematical approaches to teaching logic so commonly taught today.