Trapped-ion entangling gates robust against qubit frequency errors

Abstract

Entangling operations are a necessary tool for large-scale quantum information processing, but experimental imperfections can prevent current schemes from reaching sufficient fidelities as the number of qubits is increased. Here it is shown numerically how multitoned generalizations of standard trapped-ion entangling gates can simultaneously be made robust against noise and mis-sets of the frequencies of the individual qubits. This relaxes the degree of homogeneity required in the trapping field, making physically larger systems more practical.

Full text

PHYSICAL REVIEW RESEARCH 2, 033117 (2020)
Trapped-ion entangling gates robust against qubit frequency errors
Jake Lishman and Florian Mintert
Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
(Received 15 April 2020; accepted 22 June 2020; published 22 July 2020)
Entangling operations are a necessary tool for large-scale quantum information processing, but experimental
imperfections can prevent current schemes from reaching sufficient fidelities as the number of qubits is increased.
Here it is shown numerically how multitoned generalizations of standard trapped-ion entangling gates can
simultaneously be made robust against noise and mis-sets of the frequencies of the individual qubits. This relaxes
the degree of homogeneity required in the trapping field, making physically larger systems more practical.
DOI: 10.1103/PhysRevResearch.2.033117
I. INTRODUCTION
A major goal in quantum information processing is to
reach the level of a fast, highly scalable universal quantum
computer. A device at this level is proven to have compu-
tational capabilities for certain classes of problems which
exceed any possible classical computer [1,2] and would have
major applications in a broad range of fields spanning all
the physical and computational sciences [3–5], making an
inherently quantum world accessible to simulation and inves-
tigation. Several physical technologies are being developed
in parallel in search of this target [6–8], of which trapped
ions are commonly recognized as one of the two leading
platforms, along with superconducting qubits [9,10]. To reach
universality for a constant number of qubits, only a small
set of operations is absolutely required: a small number of
single-qubit operations, and a single two-qubit entangling
operation.
Throughout their development, quantum gate implemen-
tations have always contended with noise reduction, with
varying estimates placing the maximum allowable probability
of failure per gate at between 10−2and 10−4[11]. Single-qubit
gates have been achieved in ion traps at fidelities over 99% for
over a decade [12], with more recent works taking the average
gate infidelity to 10−6[13]. The current state-of-the-art fideli-
ties for two-qubit gates are performed in ion traps, achieving
infidelities of less than 10−3with laser-induced gates [14,15]
and 3 ×10−3with microwave-controlled schemes [16], re-
quiring very low tolerances in homogeneity and stability of
control and trapping fields, with scalability remaining a large
problem. Proposals to enlarge ion-trap computers typically
focus on producing modular systems, either on physically
shuttling ions [17] or introducing probabilistic photonic inter-
connects between separated traps [18]. Both of these methods
exacerbate existing sources of noise by increasing either the
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physical distance or the number of external supplies and bulk
optics that field coherence must be maintained across. It is still
imperative that entangling operations can be achieved that are
robust against degraded conditions and controls.
These two primary qubit technologies in ion traps typically
suffer from differing dominant degradation effects, though
their methods of action are similar. In both the optical and
microwave regimes, the qubits are separated by too great
distances due to their Coulomb repulsion to interact directly,
but this same force can be used to engineer an interaction
using the shared motion as a temporary bus mode [19]. The
reliance on the motion creates another potential source of infi-
delity, alongside the necessity of keeping the qubit frequencies
entirely coherent with each other and the driving fields. In
ion-trap gates, these noise sources are typically macroscopic
components; laboratory temperature and electrode voltage
drifts decohere the motional mode, while long-term laser-
and microwave-field frequency and amplitude fluctuations
primarily affect qubit frequency splittings.
For ion-trap qubits, there has been significant interest in
making gates resilient against unwanted heating and fre-
quency errors of the bus mode using multitoned driving fields
[20,21] or by amplitude or phase modulation [22,23]. Early
microwave-controlled gates necessitated dynamical decou-
pling methods to protect against overall fluctuations in the
magnetic field [16,24], with more recent proposals for hyper-
fine qubits considering gate speedups by inserting coupling to
more motional modes [25], or experimental simplifications by
decoupling from global qubit frequency mis-sets and oscilla-
tions without additional fields [26].
The scheme illustrated here extends the previous litera-
ture by using a multitone extension to the Mølmer-Sørensen
scheme to produce a gate resilient against all frequency errors
on one or both of the qubits individually. These sources of
error have previously not found as wide interest as those
in the motional frequency in existing literature outside of
microwave-controlled qubits, although the physical scaling
of apparatus will only exacerbate driving amplitude and fre-
quency drift considerations for all technologies [10]. This
scheme is applicable to all ion-trap qubit encodings, including
magnetic-field-sensitive optical qubits, and produces an im-
provement in infidelity around the current threshold of error
2643-1564/2020/2(3)/033117(7) 033117-1 Published by the American Physical Society

JAKE LISHMAN AND FLORIAN MINTERT PHYSICAL REVIEW RESEARCH 2, 033117 (2020)
correction of over two orders of magnitude, without being
specifically generated for any particular offset magnitude.
The same numerical optimization methods can be applied
to produce a driving scheme that minimizes the average
infidelity for any error model, as the errors are considered
nonperturbatively. It may also be implemented simply in
experiments, requiring no fields to be added; an arbitrary
waveform generator is the sole necessity over the original
Mølmer-Sørensen implementation [27].
II. MODEL
A. System
The system Hamiltonian for two harmonically trapped ions
considering only a single motional mode is
ˆ
HS/¯h=1
2¯ω(1)
eg ˆσ(1)
z+¯ω(2)
eg ˆσ(2)
z+¯ωmˆa†ˆa,(1)
where ¯ω(n)
eg is the qubit frequency separation of the nth ion
and ¯ωmis the frequency of a phonon of motion that has
ˆaand ˆa†as annihilation and creation operators. For ideal
gate operation, the two separate qubit frequencies should
be equal and all frequencies should be exactly known. In
reality, however, several noise sources conspire to modify
these values over the course of a complete experiment, and
the true frequency ¯ωis formed of a known component ωwith
the addition of some deviation δas ¯ω=ω+δ. Modifications
to the motional frequency δmoccur primarily due to endcap
voltage drifts, causing apparent dephasing effects when av-
eraged over several gate realizations. The dominant sources
of error on the nth qubit frequency δ(n)
eg depend strongly on
the encoding of the qubits; magnetic-field-sensitive qubits will
generally suffer most from local variations in the field, while
the frequency separation of optical qubits is more commonly
mis-set due to slow drift of the spectroscopy laser. The errors
on the two qubit frequencies can be parametrized as individual
differences from the estimated frequency, as in Eq. (1), but
it is more convenient for the analysis to consider an error in
the estimation of the average frequency δavg =(δ(1)
eg +δ(2)
eg )/2
and the distance of each individual value from this average
δspl =(δ(1)
eg −δ(2)
eg )/2. These frequencies are diagramed in the
context of the energy-level scheme in Fig. 1for the standard
Mølmer-Sørensen gate detuned from the sidebands by an
amount .
Moving to a rotating frame defined by ˆ
U=exp(iˆ
HSt/¯h),
the driving-ion interaction Hamiltonian is
ˆ
HI/¯h=˜
f(t)ei(ωeg+δavg )t(eiδspltˆσ(1)
++e−iδspltˆσ(2 )
+)
×(1 +iηei(ωm+δm)tˆa†+iηe−i(ωm+δm)tˆa)
+H.c.,(2)
where the Lamb-Dicke condition that η√2n+11 has
been assumed, and terms oscillating at the same or-
der as ωeg are omitted. The complete driving term ˜
f(t)
is ˜
f(t)=jfj(t)exp[−i(ω(j)
s+δ(j)
s)t] and comprises two
terms: the latter sideband-selection term and a slowly oscil-
lating driving term fj, such that each term in Eq. (2)isof
a comparable frequency to the acoustic trap frequency. The
selection frequency ωsis set to ωeg +nωmto pick out the nth
FIG. 1. The energy levels of the standard Mølmer-Sørensen gate
operation in the presence of frequency errors. Thick black lines
denote the expected energy levels, whereas thin ones show the
modified structure, and the driving is marked in red and blue for
the appropriate sideband. An error δmin the motional frequency ωm
causes phonon levels to shift but maintains resonance of all transition
paths. Any error in qubit frequencies causes the two-photon process
to be off resonant for some starting states; a shift in the average of
the carrier frequencies δavg changes the energy gap of |gg↔|eeby
2δavg while leaving |ge↔|eg, and an energy splitting between the
two qubits 2δspl has the opposite effect.
sideband, where n=0 is the carrier, n=1 is the blue side-
band, and n=−1 is the red sideband. The sideband-driving
term fj(t) has frequency components that are small compared
to the sideband separation frequency ωm, so that only the
targeted sideband is excited. For the Mølmer-Sørensen gate,
the ions are globally illuminated by a blue field with a selec-
tion frequency ω(b)
s=ωeg +ωmusing a constant-amplitude,
slightly off-resonant drive fb(t)=eit, simultaneously with
a red field at ω(r)
s=ωeg −ωmand fr(t)=f∗
b(t), leading to a
Hamiltonian
ˆ
HMS =−ηfb(t)eiδmtˆa†·⎛
⎜
⎜
⎝
cos((δavg +δspl )t)ˆσ(1)
y
+sin((δavg +δspl )t)ˆσ(1)
x
+cos((δavg −δspl )t)ˆσ(2)
y
+sin((δavg −δspl )t)ˆσ(2)
x
⎞
⎟
⎟
⎠+H.c.
(3)
In the absence of errors, this degrades to the standard Hamil-
tonian ˆ
HMS =−η[fb(t)ˆa†+f∗
b(t)ˆa]( ˆσ(1)
y+ˆσ(2)
y). In Eq. (3),
the two selection error terms δ(r)
sand δ(b)
shave been
reparametrized to average and splitting terms with the same
treatment as the qubit error terms. In this form, the splitting
term appears only as an addition to the motional detuning δm,
while the average similarly modifies the average qubit detun-
ing δavg, allowing these two preexisting terms to completely
represent static mis-sets in the selection frequencies.
Equation 3is analytically solvable only when no qubit
frequency errors are present, resulting in a time-evolution
operator
ˆ
UMS (t)=ˆ
Dˆσ(1)
y+ˆσ(2)
yt
0fb(t1)dt1
×exp2iˆσ(1 )
yˆσ(2)
yIm t
0dt1t1
0dt2fb(t1)f∗
b(t2),
(4)
where ˆ
Dis the motional phase-space displacement operator
ˆ
D(α)=exp(αˆa†−α∗ˆa). As such, the first term defines the
coupling of the qubits individually to the excitation of the mo-
tional mode, and the second term represents a true entangling
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interaction between the two qubits. Together, these two terms
form two conditions that must be satisfied simultaneously at
the gate time: the motional phase-space displacement must
return to zero, and the qubit entanglement phase accumulation
must reach the desired level.
The driving function fb(t) cannot be designed to eliminate
error terms from the complete Hamiltonian, and with their
effects active, an exact time-evolution operator like Eq. (4)
cannot be found. Series-expansion methods neither truncate
nor converge in a computable number of steps; the noncom-
mutation of the Pauli operators σxand σyalong with depen-
dence on increasingly large motional excitations at higher
orders frustrate the Magnus and similar expansions, while
the aperiodicity of the system prevents a reasonable Floquet
approach. Instead, numerical techniques are used here to
access and minimize the gate’s response to errors.
B. Optimization
It is first important to quantify how performant a quantum
gate is so that some form of improvement can be found and
optimized. Any meaningful measure of the success of an
operation must take into account all possible states that the
system may exist in. One such measure is the gate infidelity,
defined by I=1−k|ψk|ˆ
UMS (
δ)|χk|2/K, in terms of all
types of detunings 
δand Kpairs of start and ideal target
states {|χk,|ψk}, respectively, where the start states span the
Hilbert space concerned. To ensure that the gate is resilient to
detunings with a wide range of magnitudes, an appropriate fig-
ure of merit is an expectation of the total infidelity E[I(
δ)] =
I(
δ)dw(
δ) for a suitable weight function w; typically this
can be taken as an adequately dimensioned normal distri-
bution as a reasonable proxy for experimental uncertainties.
This modified target causes the optimizer to prefer parameters
which provide good fidelities over a range of errors, with a
hyperparameter 
σδbeing the standard deviations of the error
distributions, affecting how heavily larger errors are weighted.
Notably, the use of an expectation does not require the optimal
schemes to have perfect infidelity at zero error, but a suitable
choice of weight function may ensure that any remnant error
will be negligible.
The optimizations presented here will consider a shaped
driving field f(t)—the subscript bis dropped for simplicity—
with multiple frequency components (“tones”) simultaneously
to minimize the effects of the error terms on the final gate
operation, taking f(t)=n
k=1cn,keikt, where the cn,kare
complex variables with dimensions of frequency. The targeted
figure of merit is the expectation of gate infidelity, averaged
over all possible electronic starting states weighted equally
and over all possible detunings weighted as a normally dis-
tributed error model. An understanding of the precise details
of the numerical methodology is not necessary to appreciate
the subsequent results, so further discussion is deferred to
Sec. IV.
III. RESULTS
We perform optimizations using the multitone parametriza-
tion of the driving field to produce gates resilient against all
forms of static errors on the qubit frequencies and an average
FIG. 2. Main figure: gate fidelity for optimized driving schemes
compared to the standard Mølmer-Sørensen scheme. Inset: total drive
amplitude during the gate operation. The detuning error considered
is in the ratio δavg =2δspl. An error which causes the single-tone gate
to leave the error-correction threshold of 99.9% causes an infidelity
of only 2.5×10−5when four or more tones are used. The two-
and three-tone gates are minor modifications of the standard driving
yet produce a three- to four-times improvement over the range of
meaningful infidelities.
offset in the sideband-selection frequencies. The maximum
peak power usage of the interrogation source is fixed across
all numbers of tones so as to form a fair comparison with the
base gate, while the gate time is allowed to vary to facilitate
this by making the base detuning a control parameter. Aside
from this detuning of the closest tone to the sideband, the other
optimization variables are the relative strengths and phases of
the tones in the driving field.
In Fig. 2, the best driving schemes obtained are compared
to the performance of the base gate at varying qubit detunings.
Due to the nature of any numerical optimization, and as
the optimization landscape is infinite and nonperiodic, it is
impossible to ascertain if a true global maximum has been
found. However, sampling the initial parameter space increas-
ingly finely can arbitrarily reduce the possibility of having
missed a better result. The results presented here then are most
correctly lower bounds on the maximum achievable fidelities.
The hyperparameter 
σδwas chosen to prioritize the minimiza-
tion of infidelity for detunings of such a magnitude that the
base gate is close to, but not quite in, the error-correcting
region. This prioritizes cases where qubit frequency errors
would prevent current gates from being computationally vi-
able and is largely unconcerned with situations where such
errors would not be the dominant terms. At lower detuning
magnitudes, the monotone gate is able to outperform these
numerical schemes, but only in regions where the error is
insignificant.
The driving fields resulting from these optimizations are
specified in Table I, and their time-dependent amplitudes are
shown in the inset of Fig. 2. For two- or three-tone gates,
the optimized drivings are minor perturbations of a standard
gate performed with two loops in phase space, with maximum
relative amplitude variation of 0.132 and 0.033, respectively,
but the largest improvements are seen once four tones are
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JAKE LISHMAN AND FLORIAN MINTERT PHYSICAL REVIEW RESEARCH 2, 033117 (2020)
TABLE I. Tabulated values of coefficients for the multitone driving of the Mølmer-Sørensen gate optimized to reduce the effects of static
qubit frequency errors. The driving field with ntones takes the form fn(t)=n
k=1|cn,k|eiφn,keiknt, where the units are scaled such that 1=4
and c1,1=1 in the base case, and all driving fields have the same peak power usage. The gate time τnis given in terms of the standard gate,
which has constant power, while the multitone gates have a maximum variation in the power of δfn. The last phase is chosen as zero for all
pulses; driving fields are equivalent up to a global phase.
Tones τn/τ1δfnncn,1cn,2cn,3cn,4cn,5cn,6
|c|0.066 0.934
2 3.368 0.132 1.188 φ/π −0.032 0
|c|0.103 0.979 0.090
3 3.185 0.033 1.256 φ/π −0.005 −0.003 0
|c|0.051 0.405 0.539 0.359
4 4.836 0.555 0.827 φ/π −0.609 −0.817 0.108 0
|c|0.048 0.450 0.516 0.414 0.183
5 4.542 0.622 0.881 φ/π −0.899 −0.930 0.045 −0.242 0
|c|0.055 0.098 0.413 0.733 0.215 0.128
6 6.529 0.482 0.613 φ/π −0.616 −0.785 −0.954 0.007 −0.043 0
included. For errors which cause the base gate to have fi-
delities on the thresholds of the error-correcting regions, 99%
and 99.9%, depending on the particular definition, a four-tone
gate using the same amount of peak power has infidelities of
1.0×10−3and 2.5×10−5, respectively—10 and 250 times
smaller. This does, however, come at a cost in gate time; this
gate requires slightly under five times the amount of time to
complete, largely because of the increase of the number of
loops completed in phase space.
The causes of the infidelity in the presence of these errors
are best understood in terms of the qubit-motion intermediary
entanglement and the qubit-qubit phase accumulation condi-
tions represented by Eq. (4) for ideal operation, and additional
ion-asymmetric terms which arise from frequency errors. As
the motional frequency remains well known, the phase-space
trajectories depicted in Fig. 3close well even as qubit fre-
quency errors are increased, and the residual qubit-motion
entanglement is low. Instead, shifts in qubit frequencies that
FIG. 3. Motional phase-space trajectories of the different multi-
tone gates also plotted in Fig. 2, with the same peak power usage
and different gate times. Structural changes to the trajectories only
occur on even numbers of tones. The relative time through the gate
is represented by the line color, moving from purple (dark) to orange
(light). Valid qubit-phase advancements are (4n+1)π/4 for integer
n; the single-tone gate has n=0, while two to five tones have n=1
and six tones has n=2.
cause the two-photon process to become off resonant, such
as an average qubit shift when considering the |gg↔|ee
transition, result in phase decoherence between the starting
and double-spin-flipped state, degrading the fidelity of the
targeted Bell state. For configurations where the gate process
is on resonant but has a path asymmetry, such as a qubit
splitting on the same |gg↔|eetransition, the dominant
infidelity source is due to residual population left in states with
only a single spin-flip.
The infidelity of the base gate varies predominantly
quadratically with a change in the magnitude of the qubit
errors. We have found numerically that a minimum of four
tones are required to improve this scaling behavior over any
sizable region of interest; Fig. 2shows this improvement
in the scaling through steeper gradients for four and higher
numbers of tones. This new scaling is quartic for realistic
errors, although a new constant offset is introduced at lower
magnitudes. The minimum infidelity is then nonzero for the
optimized gates; however, it can be made negligibly small
with the addition of further tones. As this method also allows
the easy selection of the region of interest, this does not pose
any hard limit of fidelity from this multitone driving.
It is notable that only even numbers of tones make sig-
nificant changes to the fidelity response of the gate; Fig. 3
shows that structural changes to the phase-space trajectories
only occur at these points, despite the amplitude modulation
being rather different between members of each even-odd pair.
IV. OPTIMIZATION METHODS
A. Evaluation of figure of merit
A successful optimization over several parameters typi-
cally requires hundreds of evaluations of the figure of merit,
and the number of repetitions needed to adequately sample the
initial parameter possibilities can easily push this to millions.
As the number of free parameters increases, so too does
the average number of evaluations needed for convergence.
This can easily place restrictions on the driving fields that
can be considered or severely limit the exploration of the
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TRAPPED-ION ENTANGLING GATES ROBUST AGAINST … PHYSICAL REVIEW RESEARCH 2, 033117 (2020)
optimization landscape if the calculation complexity is not
carefully attended to.
The inclusion of an integral in the figure of merit poses a
particular speed concern; numerical integration must always
evaluate the integrand several times, and each evaluation re-
quires a complete numerical solution to the Schrödinger equa-
tion. The number of operations required to achieve a certain
precision generally scales exponentially with the dimensional-
ity of the integral, mandating that the integrand should be sam-
pled in minimal locations. In one dimension, integrals over a
weight function can be evaluated to a high degree of accuracy
with few abscissae using Gaussian quadrature. The integrand
is considered in terms of a polynomial set orthogonal under
the real inner product f,g=f(x)g(x)dw(x), allowing the
integration of Iaccurate to degree 2n−1tobeexpressedasa
sum n
i=1wiI(xi), where the xiare the roots of the nth-order
polynomial and the weights wiare precalculated [28]. For a
weight function of the form w(x)=exp(−x2), as is the case
for normally distributed errors, the relevant polynomial set
is the Hermite polynomials. This method does not require
recursive subdivision of integration regions to reach a desired
accuracy, unlike simpler Newton-Cotes schemes, reducing
the total number of evaluations, and can also handle infinite
regions without truncation. Similar methods allow the exten-
sion to ddimensions with better performance than the naïve
ndachieved by nesting, although the lack of well-defined
orthogonal polynomials does not permit generic constructions
to arbitrary degree [29].
The evaluation of the figure of merit is also accelerated by
considering the symmetry of the integrand under transforma-
tions of the detunings and spanning basis. After the absorption
of the selection errors into other terms, the three remaining
errors specified in Eq. (3) manifest differently throughout the
action. Shifts in the motional frequency leave all two-photon
processes on resonance but modify the true gate time. The
qubit errors cause certain transitions to become energetically
mismatched; a shift in the average carrier transition causes
the |gg↔|eeflopping to have an energy difference of 2δavg
from the sum of the red and blue photons required, but without
lifting the energy degeneracy of the |egand |gelevels, the
blue-blue and red-red processes which mediate entanglement
between this manifold remain favorable, albeit with a mod-
ified detuning from the virtual levels. If instead the average
is well known but there are different carrier frequencies, the
|eg↔|getransition cannot be on resonance but the blue-red
process to promote |ggto |eeis.
This similarity can be quantified by considering how the
evolution of the system changes when its initial state is mod-
ified by a time-independent unitary operator ˆ
V, such as ˆσ(1)
y
that maps |ggto −i|ge. When a Hamiltonian ˆ
Hsatisfying the
Schrödinger equation i∂tˆ
U=ˆ
Hˆ
Uis modified by ˆ
Vto ˆ
H=
ˆ
V†ˆ
Hˆ
V, the resultant time-evolution operator is ˆ
U=ˆ
V†ˆ
Uˆ
V.
With the qubit error terms in Eq. (3) as explicit arguments,
taking ˆ
V=ˆσ(1)
yleads to ˆ
H
MS (δavg,δ
spl )=ˆ
HMS (−δspl,−δavg ),
while ˆ
V=ˆσ(2)
ymakes ˆ
H
MS (δavg,δ
spl )=ˆ
HMS (δspl,δ
avg ). We
therefore find that δavg and δspl have equivalent effects on
different starting states and cause equal infidelities when
totaled over the complete basis of gate operation. Any shaped
driving function f(t) which minimizes a total gate error for an
offset in the average qubit frequency will consequently also
minimize the error due to a splitting between the two. Further,
the oscillations |gg↔|eeand |eg↔|geare symmetric
with respect to exchange of starting state if the signs of both
errors simultaneously flip, i.e., the dynamics of the transition
|gg→(|gg−i|ee)/√2 exhibit the same infidelity behavior
for δavg and δspl as |eedoes for −δavg and −δspl . In tandem,
these two symmetries allow complete information of the
average fidelity to be obtained by considering only half the
possible initial states, thus taking half the time.
B. Power-usage constraints
Unlike the standard Mølmer-Sørensen scheme, the n-tone
driving fn(t)=n
k=1cn,keikntconsidered here has variable
power usage ∝|fn|2throughout the gate. The supremum lo-
cation for an arbitrary number of tones with given control
parameters is calculated by reformulating the natural maxi-
mization problem into one of polynomial root finding, which
can be solved by eigenvalue methods on a companion matrix
[28]. All extrema of the power constraint are located at the
zeros of the derivative ∂t|fn(t)|2, which can be recast via
multiplication by the nonzero term exp[i(n−1)nt] into the
complex polynomial in z=eint:
2n−2

k=0
k=n−1

j
cjc∗
j−k+n−1(k−n+1)zk=0,(5)
where jruns from 1 to k+1fork<n−1 and from k−n+
2tonfor k>n−1. The roots zare related to the tempo-
ral locations of extrema t,mby nt,m=arg(z)−iln |z|+
2πm, where the integer mdenotes the period of the driving,
and the only roots of interest are in the first period and real,
where |z|=1 and m=0. The peak power usage follows
simply by testing the 2n−2 or fewer abscissae to find the
global maximum.
The optimizations presented in the paper are performed
using a standard unconstrained Broyden-Fletcher-Goldfarb-
Shanno (BFGS) algorithm [28] over the free ratios |cn,k/cn,1|,
the relative phases φn,k, and the principle detuning n, which
can vary entirely freely. The constraint that the peak power
usage is equal under both schemes is then achieved by
fixing the absolute value of cn,1such that maxt|fn(t)|2=
maxt|f1(t)|2, inside the figure-of-merit calculation. The final
free parameter—the coupling strength of the base gate—is
chosen to be c1,1=1/4 to coincide with the shortest possible
single-tone gate.
C. Agreement with prior results
In order to gauge the reliability of the numerical method, a
comparison can be made with analytically constructed control
solutions, such as the shaped pulses rendering gates robust
against errors solely in the motional frequency. The optimal
pulse shapes found with this method reproduce those previ-
ously reported [20,21], which are significantly different from
those presented in Table Iand illustrated in Figs. 2and 3.In
particular, the average absolute phase-space displacement is
kept as close as possible to zero to lessen the effects of thermal
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JAKE LISHMAN AND FLORIAN MINTERT PHYSICAL REVIEW RESEARCH 2, 033117 (2020)
fluctuations and trap-frequency offsets [30], whereas this is
not the case for qubit frequency errors.
These methods have reproduced the analytically known
optimum solutions in situations where they are known to
exist but have also been numerically shown to produce highly
robust gates in qualitatively different situations where a pen-
and-paper construction is not possible, highlighting the versa-
tility of the methodology.
V. CONCLUSIONS
Simply synthesized multitone drivings can massively re-
duce errors due to qubit frequency shifts on one or both
qubits simultaneously in the standard Mølmer-Sørensen gate,
without increasing the amount of peak power required. With
four or more tones, the quadratic scaling of the infidelity with
respect to the qubit error size can be improved to fourth order,
with a constant maximum fidelity ceiling which is raised by
the addition of further tones. This method is not unique to
any method of driving nor qubit encoding and can be applied
universally across all standard trapped-ion processors with
little-to-no additional hardware required. It is most useful,
though, in the field of optical qubit systems where previous
microwave techniques do not readily apply. The techniques
used to quickly numerically optimize pulse sequences with
a minimum number of simulations and to apply the strictly
nonlinear power-usage constraints are general, applicable to
all numerical infidelity optimizations.
ACKNOWLEDGMENTS
We are grateful for stimulating discussions with Oliver
Corfield, Jacopo Mosca Toba, Mahdi Sameti, Frédéric
Sauvage, Richard Thompson, and Simon Webster. Financial
support by EPSRC through the Training and Skills Hub in
Quantum Systems Engineering (Grant No. EP/P510257/1) is
gratefully acknowledged.
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