# Ultrahigh-fidelity composite quantum phase gates

Hayk L. Gevorgyan†,‡,§ Nikolay V. Vitanov† †Faculty of Physics, St Kliment
Ohridski University of Sofia, 5 James Bourchier blvd, 1164 Sofia, Bulgaria
‡A. Alikhanyan National Science Laboratory (Yerevan Physics Institute), 2
Alikhanian Brothers st, 0036 Yerevan, Armenia
§Institute for Physical Research, Armenian National Academy of Sciences,
Ashtarak-2, 0203, Armenia

###### Abstract

A number of composite pulse (CP) sequences for four basic quantum phase gates
— the Z, S, T and general phase gates — are presented. The CP sequences
contain up to 18 pulses and can compensate up to eight orders of experimental
errors in the pulse amplitude and duration. The short CP sequences (up to 8
pulses) are calculated analytically and the longer ones numerically. The
results demonstrate the remarkable flexibility of CPs accompanied by extreme
accuracy and robustness to errors — three features that cannot be
simultaneously achieved by any other coherent control technique. These CP
sequences, in particular the Z, S and T gates, can be very useful quantum
control tools in quantum information applications, because they provide a
variety of options to find the optimal balance between ultrahigh fidelity,
error range and speed, which may be different in different physical
applications.

##  I Introduction

Phase coherence is of paramount importance in modern quantum information
technologies and it is one of the most significant differences between
classical and quantum computing [1, 3, 4, 2]. Phase coherence is controlled by
quantum phase gates, such as the Z, S and T gates, which are key elements in
any quantum circuit. Because of the vast number of such gates involved even in
moderate quantum circuits their fidelity is of crucial significance for the
success of any quantum algorithm.

Among the existing quantum control techniques capable of efficient
manipulation of quantum systems, composite pulse (CP) sequences [5, 6] stand
out as a very powerful tool which offers a unique combination of accuracy of
operations, robustness to experimental errors, flexibility and versatility as
it can be adopted and applied to essentially any quantum control task — a set
of features that can only be found in composite pulses. A composite pulse is
actually a sequence of pulses with well defined relative phases which are used
as control parameters in order to shape the excitation profile, and generally,
the propagator, in essentially any desired manner.

The vast majority of composite pulses are designed to produce complete and
partial rotations on the Bloch sphere [5, 7, 8, 9, 6, 10, 11, 12]. Among
these, a clear distinction exists between the so-called variable and constant
rotations. Variable rotations start on one of the poles of the Bloch sphere
and move the Bloch vector at a particular latitude, i.e. on a particular
parallel, without controlling the longitude. Constant rotations do not require
a specific initial condition and produce the desired rotation starting at any
point on the Bloch sphere. In quantum control language, the variable rotations
are characterized by well-defined absolute values (i.e. populations) of the
propagator elements but not well-defined phases. Constant rotations (or phase-
distortionless rotations) are characterized by both well-defined populations
and phases of the propagator, i.e. the quantum gate. Obviously, constant
rotations are much more demanding to generate, but they are exactly what is
required for reliable and scalable quantum computing circuits. Over the years,
variable and constant composite rotations have been demonstrated on multiple
occasions in NMR [14, 15, 16, 17, 5, 6, 7, 8, 9], trapped ions [18, 19, 20,
21, 25, 22, 23, 24], neutral atoms [26, 28, 29, 30, 27], quantum dots [31, 33,
34, 35, 36, 32], doped solids [38, 40, 41, 39], superconducting qubits [42,
43], etc., featuring remarkable accuracy and robustness. A variation of the
composite idea, with the detuning rather than the phase of each constituent
pulse in the composite sequence used as the control parameter, has also been
proposed and experimentally demonstrated [44].

Compared to rotations, very few proposals exist for composite phase gates
[45]. An arbitrary phase shift at an angle $\phi$, being rotation around the
$z$ axis, can be implemented by two resonant $\pi$ pulses with an appropriate
relative phase between them. However, resonant driving is prone to errors in
the experimental parameters, e.g. the pulse amplitude, duration, and detuning.
Here the phase gates are implemented as sequences of $\pi$ pulses with
specific phases. Application of composite pulses to quantum phase gates has
been discussed in Ref. [45], where composite sequences of $2(2n+1)$
($n=1,2,\ldots$) pulses have been presented, with the 6-pulse and 10-pulse
sequences being the shortest ones. In the present paper, we make a step toward
filling this gap: we supplement the library of composite pulses with composite
sequences of $2n$ ($n=2,3,\ldots$) pulses, which produce arbitrary quantum
phase gates, with a focus at the most important ones for quantum information
processing: the Z, S, and T gates. We use analytic approaches and brute-force
numerics to derive composite sequences, which achieve error compensation of up
to 8th order.

This paper is organized as follows. In Sec. II we explain the derivation
method. The design and performance of the composite phase gates are presented
in Sec. III. Finally, Sec. IV presents the conclusions.

##  II SU(2) Approach

###  II.1 Method of derivation

Our objective in this article is to construct the quantum phase-shift gate
$\mathbf{F}(\phi)=e^{-i(\phi/2)\hat{\sigma}_{z}}$, or in matrix form,

$\mathbf{F}(\phi)=\mathbf{R}_{z}(\phi)=\left[\begin{array}[]{cc}e^{-i\phi/2}&0\\\
0&e^{i\phi/2}\end{array}\right]:=\left[\begin{array}[]{cc}1&0\\\
0&e^{i\phi}\end{array}\right],$ (1)

which is equal to the standard one up to the unimportant global phase factor
$e^{-i\phi/2}$. For the Z gate we have $\phi=\pi$, for the S gate
$\phi=\pi/2$, and for the T gate $\phi=\pi/4$.

The derivation of the robust ultrahigh-fidelity quantum phase gates via
composite pulses is similar to the rotation gates in our previous work [12].
Basically, starting from the time-dependent Schrödinger equation for a two-
state system, one can write down the evolution operator for a single-qubit,
which is referred to as Rabi rotation gate in experimental quantum computing
[2, 1], or theta pulse in nuclear magnetic resonance [9],

$\mathbf{U}_{\phi}(\mathcal{A})=\left[\begin{array}[]{cc}\cos(\mathcal{A}/2)&-ie^{i\phi}\sin(\mathcal{A}/2)\\\
-ie^{-i\phi}\sin(\mathcal{A}/2)&\cos(\mathcal{A}/2)\end{array}\right].$ (2)

Here $\mathcal{A}=\int_{t_{\rm{i}}}^{t_{\rm{f}}}\Omega(t)\text{d}t$ is the
temporal pulse area and $\phi$ stands for the phase of the coupling. A train
of $N$ resonant theta pulses, each with a specific pulse area
$\mathcal{A}_{k}$ and a specific phase $\phi_{k}$ ($k=1,2,\ldots,N$),

$(\mathcal{A}_{1})_{\phi_{1}}(\mathcal{A}_{2})_{\phi_{2}}(\mathcal{A}_{3})_{\phi_{3}}\cdots(\mathcal{A}_{N})_{\phi_{N}},$
(3)

produces the propagator

$\bm{\mathcal{U}}=\mathbf{U}_{\phi_{N}}(\mathcal{A}_{N})\cdots\mathbf{U}_{\phi_{3}}(\mathcal{A}_{3})\mathbf{U}_{\phi_{2}}(\mathcal{A}_{2})\mathbf{U}_{\phi_{1}}(\mathcal{A}_{1}).$
(4)

In Eq. (4) the evolution matrices $\mathbf{U}_{\phi_{k}}(\mathcal{A}_{k})$ act
chronologically, from right to left, while in Eq. (3) the pulses
$(\mathcal{A}_{k})_{\phi_{k}}$ are applied from left to right.

Under the assumption of a single systematic pulse area error $\epsilon$, i.e.
when each pulse is replaced by the errant one,
$\mathcal{A}_{k}\rightarrow\mathcal{A}_{k}(1+\epsilon)$, we can expand the
errant composite propagator

$\bm{\mathcal{U}}(\epsilon)=\left[\begin{array}[]{cc}\mathcal{U}_{11}(\epsilon)&\mathcal{U}_{12}(\epsilon)\\\
-\mathcal{U}_{12}^{\ast}(\epsilon)&\mathcal{U}_{11}^{\ast}(\epsilon)\end{array}\right].$
(5)

in a Taylor series versus $\epsilon$. Because of the SU(2) symmetry of the
errant overall propagator, it suffices to expand only two of its elements, say
$\mathcal{U}_{11}(\epsilon)$ and $\mathcal{U}_{12}(\epsilon)$. We set their
zero-error (nominal) values to the target values,

$\mathcal{U}_{11}(0)=e^{-i\phi/2},\quad\mathcal{U}_{12}(0)=0,$ (6)

and we set to zero as many of their derivatives with respect to $\epsilon$, in
the increasing order, as possible,

$\mathcal{U}^{(m)}_{11}(0)=0,\quad\mathcal{U}^{(m)}_{12}(0)=0,\quad(m=1,2,\ldots,n),$
(7)

where $\mathcal{U}^{(m)}_{jl}=\partial_{\epsilon}^{m}\mathcal{U}_{jl}$ denotes
the $m$th derivative of $\mathcal{U}_{jl}$ with respect to $\epsilon$. The
largest derivative order $n$ satisfying Eqs. (7) gives the order of the error
compensation $O(\epsilon^{n})$.

Equations (6) and (7) generate a system of $2(n+1)$ algebraic equations for
the nominal pulse areas $A_{k}$ and the composite phases $\phi_{k}$
($k=1,2,\ldots,N$). The equations are complex-valued and generally we have to
solve $4(n+1)$ equations with the $2N$ free parameters (nominal pulse areas
and phases). Equation (6) alone can be satisfied at least by two $\pi$ pulses,

$\pi_{\nu}\pi_{\nu+\pi-\phi/2},\\\ $ (8)

with the propagator

$\mathbf{F}(\phi)=\mathbf{U}_{\nu+\pi-\phi/2}(\pi)\mathbf{U}_{\nu}(\pi).$ (9)

For the Z, S and T gates we have the sequences (setting $\nu=0$)

$\displaystyle{Z2}$ $\displaystyle=\pi_{0}\pi_{\frac{1}{2}\pi},$ (10a)
$\displaystyle{S2}$ $\displaystyle=\pi_{0}\pi_{\frac{3}{4}\pi},$ (10b)
$\displaystyle{T2}$ $\displaystyle=\pi_{0}\pi_{\frac{7}{8}\pi},$ (10c)

Taking into account this fact, and because of the normalization condition
$|\mathcal{U}_{11}|^{2}+|\mathcal{U}_{12}|^{2}=1$ ($\mathcal{U}_{11}$ and
$\mathcal{U}_{12}$ are the complex-valued Cayley-Klein parameters), an error
compensation of order $n$ requires a CP sequence of $N=2(n+1)$ $\pi$ pulses.

As stated above, the derivation of the CP sequences requires the solution of
Eqs. (6) and (7). For composite sequences of a small number of pulses (up to
eight $\pi$ pulses), Eqs. (6) and the first, second and third pairs of
equations ($n=3$) of Eqs. (7) can be solved analytically. For longer pulse
sequences, Eqs. (7) are solved numerically. We do this by using standard
routines in Mathematica©.

###  II.2 Quantum gate fidelity

If Eqs. (6) and (7) are satisfied, then the overall propagator can be written
as

$\bm{\mathcal{U}}(\epsilon)=\mathbf{F}(\phi)+O(\epsilon^{n+1}),$ (11)

with $\mathbf{F}(\phi)=\bm{\mathcal{U}}(0)$. Then the _Frobenius distance
fidelity_ ,

$\mathcal{F}=1-\|\bm{\mathcal{U}}(\epsilon)-\mathbf{F}(\phi)\|=1-\sqrt{\tfrac{1}{4}\sum\nolimits_{j,k=1}^{2}\left|\mathcal{U}_{jk}-F_{jk}\right|^{2}},$
(12)

is of the same error order $O(\epsilon^{n})$ as the propagator,
$\mathcal{F}=1-O(\epsilon^{n+1})$. Another possibility is to use the _trace
distance fidelity_ ,

$\mathcal{F}_{\text{T}}=\tfrac{1}{2}\text{Tr}\,[\bm{\mathcal{U}}(\epsilon)\mathbf{F}(\phi)^{\dagger}].$
(13)

As in our previous work [12], we will use the Frobenius distance fidelity,
because it explicitly includes information about both major and minor diagonal
elements, as required for phase-distortionless phase gates.

###  II.3 Structure of composite phase gates

Based on numerical evidence, we consider CP sequences, which consist of
$2(n+1)$ nominal $\pi$ pulses, with asymmetrically ordered phases,

$\displaystyle R_{n+1}(\nu)R_{n+1}(\nu+\pi-\phi/2),$ (14a) $\displaystyle
R_{n+1}(\nu)=\pi_{\nu}\pi_{\nu+\phi_{1}}\pi_{\nu+\phi_{2}}\cdots\pi_{\nu+\phi_{n}},$
(14b)

which is equivalent to

$R_{n+1}(\nu+\pi+\phi/2)R_{n+1}(\nu).$ (15)

These sequences generalize the initial two-pulse sequence (9) and have similar
design. Due to this specific structure of composite phases, Eqs. (6) are
satisfied, all odd-order derivatives $\mathcal{U}^{(2k+1)}_{11}(0)$ of the
major-diagonal elements in Eq. (7) vanish, and so do all even-order
derivatives $\mathcal{U}^{(2k)}_{12}(0)$ of the minor-diagonal elements. This
facilitates reaching the compensation order $n$, which is the maximum number
for which the derivatives of all major-diagonal and minor-diagonal elements
vanish. This can be obtained with a suitable choice of the available composite
phases in Eq. (14).

From the infinite number of solutions, we choose those of the type (14) and
with a free parameter $\nu=0$, since only the choice of the relative phases
$\phi_{1},\phi_{2},\ldots,\phi_{n}$ is of importance. Henceforth, we use the
design

$\displaystyle R_{n+1}R_{n+1}(\pi-\phi/2),$ (16a) $\displaystyle
R_{n+1}=\pi_{0}\pi_{\phi_{1}}\pi_{\phi_{2}}\cdots\pi_{\phi_{n}},$ (16b)
$\displaystyle
R_{n+1}(\pi-\phi/2)=\pi_{\pi-\frac{1}{2}\phi}\pi_{\phi_{1}+\pi-\frac{1}{2}\phi}\cdots\pi_{\phi_{n}+\pi-\frac{1}{2}\phi}.$
(16c)

Other possible but equivalent solutions can be obtained by choosing an
arbitrary parameter $\nu$ in (14) or/and by using the type (15).

##  III Composite phase gates

As it is well known, such a gate can be produced by two resonant $\pi$ pulses,
see Eq. (9) with $\nu=0$. The propagator elements of the overall propagator
read

$\displaystyle\mathcal{U}_{11}(\epsilon)$
$\displaystyle=e^{-i\phi/2}\cos^{2}(\pi\epsilon/2)+\sin^{2}(\pi\epsilon/2),$
(17a) $\displaystyle\mathcal{U}_{12}(\epsilon)$
$\displaystyle=\tfrac{1}{2}i(1-e^{-i\phi/2})\sin(\pi\epsilon),$ (17b)

where $\epsilon$ is the pulse area error. The Frobenius distance fidelity (12)
reads

$\mathcal{F}=1-\sqrt{2}\left|\sin\frac{\pi\epsilon}{2}\right|\left|\sin\frac{\phi}{4}\right|,$
(18)

and it has 0th order error compensation, $\mathcal{F}=1+O(\epsilon)$. For
comparison, the trace fidelity is

$\mathcal{F}_{T}=1-2\sin^{2}\frac{\pi\epsilon}{2}\sin^{2}\frac{\phi}{4},$ (19)

which has 1st order error compensation, $\mathcal{F}=1+O(\epsilon^{2})$
Obviously the error of the Frobenius distance fidelity (18) is far greater
than the value of the error of the trace fidelity (19). Below we consider
longer sequences, in the increasing order of error compensation.

---
Figure 1:  Frobenius distance fidelity $\mathcal{F}$ (top) and infidelity
$1-\mathcal{F}$ (bottom) of composite Z gates. The infidelity is in
logarithmic scale in order to better visualize the high-fidelity (low-
infidelity) range. The numbers $N$ on the curves refer to CP sequences Z$N$
listed in the Table 1.
---
Figure 2:  Frobenius distance fidelity $\mathcal{F}$ (top) and infidelity
$1-\mathcal{F}$ (bottom) of composite S gates. The infidelity is in
logarithmic scale in order to better visualize the high-fidelity (low-
infidelity) range. The numbers $N$ on the curves refer to CP sequences S$N$
listed in the Table 2.
---
Figure 3:  Frobenius distance fidelity $\mathcal{F}$ (top) and infidelity
$1-\mathcal{F}$ (bottom) of composite T gates. The infidelity is in
logarithmic scale in order to better visualize the high-fidelity (low-
infidelity) range. The numbers $N$ on the curves refer to CP sequences T$N$
listed in the Table 3.

###  III.1 First-order error compensation

The careful analysis of Eqs. (6) and (7) shows that the shortest possible CP
which can compensate first-order errors (both in major and minor diagonal
elements) consists of four pulses, each with a nominal pulse area of $\pi$,
and asymmetric phases, with the structure similar to the two-pulse case,

$\pi_{0}\pi_{\phi_{1}}\pi_{\pi-\frac{1}{2}\phi}\pi_{\phi_{1}+\pi-\frac{1}{2}\phi}.$
(20)

Solving Eqs. (6) along with Eqs. (7) for the first derivatives gives four
solutions for the phases,

$\displaystyle\pi_{0}\pi_{-\frac{1}{4}\phi}\pi_{\pi-\frac{1}{2}\phi}\pi_{\pi-\frac{3}{4}\phi},$
(21a)
$\displaystyle\pi_{0}\pi_{\pi-\frac{1}{4}\phi}\pi_{\pi-\frac{1}{2}\phi}\pi_{-\frac{3}{4}\phi},$
(21b)
$\displaystyle\pi_{\frac{1}{4}\phi}\pi_{0}\pi_{\pi-\frac{1}{4}\phi}\pi_{\pi-\frac{1}{2}\phi},$
(21c)
$\displaystyle\pi_{\pi+\frac{1}{4}\phi}\pi_{0}\pi_{-\frac{1}{4}\phi}\pi_{\pi-\frac{1}{2}\phi}.$
(21d)

These four sequences generate the same propagator and hence the same fidelity.
The Frobenius distance and trace distance fidelities read

$\displaystyle\mathcal{F}=1-\sqrt{2}\sin^{2}\frac{\pi\epsilon}{2}\left|\sin\frac{\phi}{4}\right|,$
(22a)
$\displaystyle\mathcal{F}_{T}=1-2\sin^{4}\frac{\pi\epsilon}{2}\sin^{2}\frac{\phi}{4}.$
(22b)

Obviously, the Frobenius distance infidelity for four sequences is of order
$O(\epsilon^{2})$ and it is much larger than the trace distance infidelity,
which is of order $O(\epsilon^{4})$. The trace distance fidelity is much
higher than the Frobenius distance fidelity, similar to rotation gates [12].

For the Z, S and T gates of the form (21a) we have

$\displaystyle{Z4}$
$\displaystyle=\pi_{0}\pi_{-\frac{1}{4}\pi}\pi_{\frac{1}{2}\pi}\pi_{\frac{1}{4}\pi},$
(23a) $\displaystyle{S4}$
$\displaystyle=\pi_{0}\pi_{-\frac{1}{8}\pi}\pi_{\frac{3}{4}\pi}\pi_{\frac{5}{8}\pi},$
(23b) $\displaystyle{T4}$
$\displaystyle=\pi_{0}\pi_{-\frac{1}{16{}}\pi}\pi_{\frac{7}{8}\pi}\pi_{\frac{13}{16}\pi}.$
(23c)

The Frobenius distance fidelity $\mathcal{F}$ (top) and infidelity
$1-\mathcal{F}$ (bottom) for the composite Z4, S4 and T4 gates is shown in
Figs. 1, 2, and 3, respectively. With respect to the ubiquitous quantum
computation benchmark fidelity value of $1-10^{-4}$, the Frobenius distance
fidelity (22a) for the four-pulse composite Z4 gates of Eqs. (21) remains
above this value in the pulse area interval $(0.9936\pi,1.0064\pi)$, i.e. for
relative errors up to $|\epsilon|<0.0064$. For comparison, the trace distance
fidelity (22b) remains above this value in the pulse area interval
$(0.936\pi,1.064\pi)$, i.e. for relative errors up to $|\epsilon|<0.064$, a
factor of 10 larger. Again we stress that the Frobenius distance fidelity is a
much more stringent measure.

For the four-pulse composite S4 gate, the Frobenius infidelity remains below
$10^{-4}$ for $|\epsilon|<0.0087$, and the trace infidelity requires
$|\epsilon|<0.087$, a factor of 10 larger. For the four-pulse composite T4
gate, the Frobenius infidelity requires $|\epsilon|<0.0121$, and the trace
infidelity demands $|\epsilon|<0.122$, a factor of 10 larger.

Obviously, both the Frobenius and trace distance fidelities depend on the
phase flip angle $\phi$. The pulse area intervals for the four-pulse composite
phase gates are broadest for T4 and narrowest for Z4, with S4 in the middle.
This is clearly visible in Figs. 1, 2, and 3. This monotonic pattern persists
for longer sequences as well.

###  III.2 Second-order error compensation

For sequences of six $\pi$ pulses, it becomes possible to annul also the
second-order derivatives in Eq. (7). It is still possible to derive analytic
solutions of the form

$\pi_{\phi_{0}}\pi_{\phi_{1}}\pi_{\phi_{2}}\pi_{\phi_{0}+\pi-\frac{1}{2}\phi}\pi_{\phi_{1}+\pi-\frac{1}{2}\phi}\pi_{\phi_{2}+\pi-\frac{1}{2}\phi},$
(24)

The solutions can be written as

$\displaystyle\pi_{\chi}(2\pi)_{0}\pi_{\chi+\pi-\frac{1}{2}\phi}(2\pi)_{\pi-\frac{1}{2}\phi},$
(25a)
$\displaystyle\pi_{\pi+\frac{1}{2}\phi-\chi}(2\pi)_{0}\pi_{-\chi}(2\pi)_{\pi-\frac{1}{2}\phi},$
(25b)
$\displaystyle(2\pi)_{0}\pi_{\pi-\frac{1}{2}\phi+\chi}(2\pi)_{\pi-\frac{1}{2}\phi}\pi_{-\phi+\chi},$
(25c)
$\displaystyle(2\pi)_{0}\pi_{-\chi}(2\pi)_{\pi-\frac{1}{2}\phi}\pi_{-\chi+\pi-\frac{1}{2}\phi},$
(25d)

where

$\chi=\frac{\phi}{4}+\arcsin\left(\frac{1}{2}\sin\frac{\phi}{4}\right).$ (26)

For the Z6 ($\phi=\pi$), S6 ($\phi=\pi/2$), and T6 ($\phi=\pi/4$) gates, we
have $\chi=0.3650\pi$, $0.1863\pi$, and $0.0936\pi$, respectively. Explicitly,
for the Z, S and T gates of the form (25a) we have

$\displaystyle{Z6}$
$\displaystyle=\pi_{0.3650\pi}(2\pi)_{0}\pi_{0.8650\pi}(2\pi)_{\frac{1}{2}\pi},$
(27a) $\displaystyle{S6}$
$\displaystyle=\pi_{0.1863\pi}(2\pi)_{0}\pi_{0.9363\pi}(2\pi)_{\frac{3}{4}\pi},$
(27b) $\displaystyle{T6}$
$\displaystyle=\pi_{0.0936\pi}(2\pi)_{0}\pi_{0.9686\pi}(2\pi)_{\frac{7}{8}\pi}.$
(27c)

The Frobenius distance and trace distance fidelities for these second-order
sequences read

$\displaystyle\mathcal{F}=1-\sqrt{2}\left|\sin^{3}\frac{\pi\epsilon}{2}\right|\left|\sin\frac{\phi}{4}\right|,$
(28a)
$\displaystyle\mathcal{F}_{T}=1-2\sin^{6}\frac{\pi\epsilon}{2}\sin^{2}\frac{\phi}{4}.$
(28b)

The Frobenius infidelity is of order $O(\epsilon^{3})$, and the trace
infidelity is of order $O(\epsilon^{6})$. The Frobenius distance fidelity
$\mathcal{F}$ (top) and infidelity $1-\mathcal{F}$ (bottom) for the composite
Z6, S6 and T6 gates is shown in Figs. 1, 2, and 3, respectively.

###  III.3 Third-order error compensation

Nullification of up to the third-order derivatives in Eq. (7) requires eight
$\pi$ pulses,

$\pi_{\phi_{0}}\pi_{\phi_{1}}\pi_{\phi_{2}}\pi_{\phi_{3}}\pi_{\phi_{0}+\pi-\frac{1}{2}\phi}\pi_{\phi_{1}+\pi-\frac{1}{2}\phi}\pi_{\phi_{2}+\pi-\frac{1}{2}\phi}\pi_{\phi_{3}+\pi-\frac{1}{2}\phi}.$
(29)

The explicit solutions we have found are

$\displaystyle\pi_{\chi}(2\pi)_{0}\pi_{\chi+\pi-\frac{1}{4}\phi}\pi_{\chi+\pi-\frac{1}{2}\phi}(2\pi)_{\pi-\frac{1}{2}\phi}\pi_{\chi-\frac{3}{4}\phi},$
(30a)
$\displaystyle\pi_{-\chi+\pi+\frac{1}{4}\phi}(2\pi)_{0}\pi_{-\chi}\pi_{-\chi-\frac{1}{4}\phi}(2\pi)_{\pi-\frac{1}{2}\phi}\pi_{-\chi+\pi-\frac{1}{2}\phi},$
(30b)
$\displaystyle(2\pi)_{0}\pi_{\chi+\pi-\frac{1}{4}\phi}\pi_{\chi+\pi-\frac{1}{2}\phi}(2\pi)_{\pi-\frac{1}{2}\phi}\pi_{\chi-\frac{3}{4}\phi}\pi_{\chi-\phi},$
(30c)
$\displaystyle(2\pi)_{0}\pi_{-\chi}\pi_{-\chi-\frac{1}{4}\phi}(2\pi)_{\pi-\frac{1}{2}\phi}\pi_{-\chi+\pi-\frac{1}{2}\phi}\pi_{-\chi+\pi-\frac{3}{4}\phi},$
(30d)
$\displaystyle\pi_{\chi+\frac{1}{4}\phi}\pi_{\chi}(2\pi)_{0}\pi_{\chi+\pi-\frac{1}{4}\phi}\pi_{\chi+\pi-\frac{1}{2}\phi}(2\pi)_{\pi-\frac{1}{2}\phi},$
(30e)
$\displaystyle\pi_{-\chi+\pi+\frac{1}{2}\phi}\pi_{-\chi+\pi+\frac{1}{4}\phi}(2\pi)_{0}\pi_{-\chi}\pi_{-\chi-\frac{1}{4}\phi}(2\pi)_{\pi-\frac{1}{2}\phi},$
(30f)

where

$\chi=\frac{\phi}{8}+\arcsin\left(\frac{1}{2}\sin\frac{\phi}{8}\right).$ (31)

For the Z8 ($\phi=\pi$), S8 ($\phi=\pi/2$), and T8 ($\phi=\pi/4$) gates, we
have $\chi=0.1863\pi$, $0.0936\pi$, and $0.0469\pi$, respectively. Explicitly,
for the Z, S and T gates of the form (30a) we have

$\displaystyle{Z8}$
$\displaystyle=\pi_{0.1863\pi}(2\pi)_{0}\pi_{0.9363\pi}\pi_{0.6863\pi}(2\pi)_{\frac{1}{2}\pi}\pi_{1.4363\pi},$
(32a) $\displaystyle{S8}$
$\displaystyle=\pi_{0.0936\pi}(2\pi)_{0}\pi_{0.9686\pi}\pi_{0.8436\pi}(2\pi)_{\frac{3}{4}\pi}\pi_{1.7186\pi},$
(32b) $\displaystyle{T8}$
$\displaystyle=\pi_{0.0469\pi}(2\pi)_{0}\pi_{0.9844\pi}\pi_{0.9219\pi}(2\pi)_{\frac{7}{8}\pi}\pi_{1.8594\pi}.$
(32c)

The Frobenius distance and trace distance fidelities for these third-order
sequences read

$\displaystyle\mathcal{F}=1-\sqrt{2}\sin^{4}\frac{\pi\epsilon}{2}\left|\sin\frac{\phi}{4}\right|,$
(33a)
$\displaystyle\mathcal{F}_{T}=1-2\sin^{8}\frac{\pi\epsilon}{2}\sin^{2}\frac{\phi}{4}.$
(33b)

The Frobenius infidelity is of order $O(\epsilon^{4})$, and the distance
infidelity is of order $O(\epsilon^{8})$. The Frobenius distance fidelity
$\mathcal{F}$ (top) and infidelity $1-\mathcal{F}$ (bottom) for the composite
Z8, S8 and T8 gates is shown in Figs. 1, 2, and 3, respectively.

###  III.4 Higher-order error compensation

For CP sequences of more than eight $\pi$ pulses, the equations for the
composite phases quickly get very bulky and hard to derive analytically. The
general form for these sequences is given by Eq. (16). They reiterate the
pattern of the sequences of four, six and eight pulses above: the CP sequences
of $2(n+1)$ $\pi$ pulses produce error compensation of the order
$O(\epsilon^{n})$ and fidelity profiles

$\displaystyle\mathcal{F}=1-\sqrt{2}\left|\sin^{n+1}\frac{\pi\epsilon}{2}\right|\left|\sin\frac{\phi}{4}\right|,$
(34a)
$\displaystyle\mathcal{F}_{T}=1-2\sin^{2n+2}\frac{\pi\epsilon}{2}\sin^{2}\frac{\phi}{4}.$
(34b)

The fidelities are sensitive to the choice of the gate phase $\phi$, as
before. These sequences are shown below.

We have derived numerically composite sequences consisting of only $\pi$ and
$2\pi$ pulses. Other possible sequences are listed in Appendix B. All
composite sequences have the familiar structure of Eq. (16), viz.

$R_{n+1}R_{n+1}(\pi-\phi/2),$ (35)

with

$\displaystyle R_{5}$
$\displaystyle=\pi_{0}(2\pi)_{\phi_{1}}\pi_{\phi_{3}}\pi_{\phi_{4}},$ (36a)
$\displaystyle R_{6}$
$\displaystyle=(2\pi)_{0}(2\pi)_{\phi_{2}}\pi_{\phi_{4}}\pi_{\phi_{4}-\frac{1}{4}\phi},$
(36b) $\displaystyle R_{7}$
$\displaystyle=\pi_{0}(2\pi)_{\phi_{1}}(2\pi)_{\phi_{3}}\pi_{\phi_{5}}\pi_{\phi_{6}},$
(36c) $\displaystyle R_{8}$
$\displaystyle=(2\pi)_{0}(2\pi)_{\phi_{2}}(2\pi)_{\phi_{4}}\pi_{\phi_{6}}\pi_{\phi_{6}-\frac{1}{4}\phi},$
(36d) $\displaystyle R_{9}$
$\displaystyle=\pi_{0}(2\pi)_{\phi_{1}}(2\pi)_{\phi_{3}}(2\pi)_{\phi_{5}}\pi_{\phi_{7}}\pi_{\phi_{8}}.$
(36e)

They compensate errors of orders ranging from $O(\epsilon^{4})$ for $R_{5}$ to
$O(\epsilon^{8})$ for $R_{9}$. Other equivalent (in terms of total pulse area
and fidelity) configurations can be obtained by interchanging pulses in the
corresponding sequence similarly to Eqs. (21), (25) and (30).

We have derived numerically the composite phases of this type of sequences of
an even number of pulses. They are presented in Tables 1, 2 and 3 for Z, S and
T gates correspondingly. The fidelities of these composite Z, S and T gates
are plotted in Figures 1, 2, and 3 respectively.

###  III.5 Discussion

It can be seen from the tables and the figures that the two-pulse sequences
Z2, S2 and T2 have very little room for errors, since high-fidelity Z, S and T
gates allow pulse area errors of less than 0.01%, about 0.01%, about 0.02%,
respectively. The four-pulse composite phase gates Z4, S4 and T4 offer some
leeway, with the admissible error of 0.6%, 0.9% and 1.2% for Z, S and T cases.
The significant pulse area error correction effect is achieved with the CP
sequences of 6 to 10 pulses, for which the high-fidelity range of admissible
errors increases from 3% to 10.1% for Z, from 3.6% to 11.5% for S, and from
4.5% to 13.1% for T. Quite remarkably, errors of up to 23.4%, 25.1% and 27.1%
can be eliminated for Z, S and T, and ultrahigh fidelity maintained, with the
18-pulse composite phase gates Z18, S18 and T18. Note that these error ranges
are calculated by using the rather tough Frobenius distance fidelity (12). Had
we used the much more relaxed trace distance fidelity (13), these ranges would
be much broader.

It is obvious from the discussion that by using longer composite sequences one
can compensate increasingly large errors. However, very long sequences are
barely practical because the gate is much slower. Moreover, the quantum
computer is not supposed to operate with a pulse area error of 10% or more.
Clearly, there is some “sweet spot” of speed and error tolerance. To this end,
the CP sequences of 4 to 8 pulses, for which the phases are given by analytic
formulas, seem to offer the best fidelity-to-speed ratio.

Appendix B presents composite pulse sequences for general phase gates with
different phase angles.

##  IV Comments and conclusions

In this paper we presented a number of CP sequences for four basic quantum
gates — the Z gate, the S gate, the T gate and general phase gates. The CP
sequences contain up to 18 pulses and can compensate up to eight orders of
experimental errors in the pulse amplitude and duration. The short CP
sequences (up to 8 pulses) are calculated analytically and the longer ones
numerically. Although longer composite phase gates are derived numerically,
their fidelity profiles have an analytic dependence on the pulse area error,
Eqs. (34a) and (34b), and show trigonometric dependence on the phase-shift
angle.

A similar class of CP sequences for phase gates is derived in [45], where they
are build from the $\theta$ rotation gates, containing $2n+1$ ($n=1,2,\ldots$)
pulses, in a similar scenario as prescribed by Eq. (16). Hence the composite
sequences contain $2(2n+1)$ pulses, i.e. 6, 10, 14, … pulses. Here we fill the
missing numbers of 4, 8, 12, … pulses, of which particularly important appear
to be the 4-pulse sequences, as they are the fastest ones. The sequences with
the same number of pulses here and in Ref. [45], although different in
construction, have performance equal to the earlier composite gates.

The results presented in this article demonstrate the remarkable flexibility
of CPs accompanied by extreme accuracy and robustness to errors — three
features that cannot be achieved together by any other coherent control
technique. We expect these CP sequences, in particular the Z, the S and the T
gates, to be very useful quantum control tools in quantum computing
applications, because they provide a variety of options to find the optimal
balance between ultrahigh fidelity, error range and speed, which may be
different in different physical applications.

We note that in addition to quantum computing, the results presented in this
paper can be applied in polarization optics to obtain broadband polarization
rotators using stacked single polarization half-wave plates with the optical
axes rotated by precisely chosen rotation angles (composite phases). This is
possible due to quantum-classical analogy of composite rotations on the Bloch
and the Poincaré spheres [46]. Hereby, we demonstrate the possibility to
design the broadband polarization rotators with $\pi/2$, $\pi/4$, $\pi/8$ and
arbitrary phase shift angles, by up to 18 CP sequences.

###### Acknowledgements.

HLG acknowledges support from the EU Horizon-2020 ITN project LIMQUET
(Contract No. 765075), and also from the RA Science Committee in the frames of
the research project 20TTAT-QTc004. NVV acknowledges support from the
Bulgarian national plan for recovery and resilience, contract BG-
RRP-2.004-0008-C01 (SUMMIT), project number 3.1.4.

##  Appendix A Composite phases for Z, S and T phase gates

Here we present the complete sets of phases of the composite pulse sequences
generating phase gates with various orders of error compensation.

Name | Pulses | $O(\epsilon^{n})$ |  Phases $\phi_{0},\phi_{1},\phi_{2},\ldots,\phi_{n},\phi_{n+1},\ldots,\phi_{2n+1}$ (in units $\pi$) | High-fidelity
---|---|---|---|---
|  |  | (according to (14)) | error correction range
two | 2 | $O(\epsilon^{0})$ | $0,\frac{1}{2}$ | $[0.99994\pi,1.00006\pi]$
Z4 | 4 | $O(\epsilon)$ | $0,\frac{7}{4},\frac{1}{2},\frac{1}{4}$ | $[0.994\pi,1.006\pi]$
Z6 | 6 | $O(\epsilon^{2})$ | $0,0,1.6350,\frac{1}{2},\frac{1}{2},0.1350$ | $[0.970\pi,1.030\pi]$
Z8 | 8 | $O(\epsilon^{3})$ | $0,0,1.8137,1.5637,\frac{1}{2},\frac{1}{2},0.3137,0.0637$ | $[0.936\pi,1.064\pi]$
Z10 | 10 | $O(\epsilon^{4})$ | $0,1.0992,1.0992,1.8315,0.0203,\frac{1}{2},1.5992,1.5992,0.3315,0.5203$ | $[0.899\pi,1.101\pi]$
Z12 | 12 | $O(\epsilon^{5})$ | $0,0,0.4492,0.4492,1.4099,1.1599,\frac{1}{2},\frac{1}{2},0.9492,0.9492,1.9099,1.6599$ | $[0.862\pi,1.138\pi]$
Z14 | 14 | $O(\epsilon^{6})$ | $0,0.7815,0.7815,1.9963,1.9963,0.8915,0.3245,\frac{1}{2},1.2815,1.2815,0.4963,$ |
|  |  |  $0.4963,1.3915,0.8245$ | $[0.823\pi,1.177\pi]$
Z16 | 16 | $O(\epsilon^{7})$ | $0,0,1.8969,1.8969,1.0586,1.0586,0.0214,1.7714,\frac{1}{2},\frac{1}{2},0.3969,0.3969,1.5586,$ |
|  |  |  $1.5586,0.5214,0.2714$ | $[0.795\pi,1.205\pi]$
Z18 | 18 | $O(\epsilon^{8})$ | $0,0.1421,0.1421,1.0834,1.0834,0.5572,0.5572,1.4991,1.0352,\frac{1}{2},0.6421,0.6421,$ |
|  |  |  $1.5834,1.5834,1.0572,1.0572,1.9991,1.5352$ | $[0.766\pi,1.234\pi]$
Table 1:  Phases of asymmetric composite sequences of $N=2(n+1)$ nominal $\pi$ pulses, which produce the Z gate with a pulse area error compensation up to order $O(\epsilon^{n})$. The last column gives the high-fidelity range $[\pi(1-\epsilon_{0}),\pi(1+\epsilon_{0})]$ of pulse area error compensation wherein the Frobenius distance fidelity is above the value $0.9999$, i.e. the fidelity error is below $10^{-4}$. Name | Pulses | $O(\epsilon^{n})$ |  Phases $\phi_{0},\phi_{1},\phi_{2},\ldots,\phi_{n},\phi_{n+1},\ldots,\phi_{2n+1}$ (in units $\pi$) | High-fidelity
---|---|---|---|---
|  |  | (according to (14)) | error correction range
two | 2 | $O(\epsilon^{0})$ | $0,\frac{3}{4}$ | $[0.99988\pi,1.00012\pi]$
S4 | 4 | $O(\epsilon)$ | $0,\frac{15}{8},\frac{3}{4},\frac{5}{8}$ | $[0.991\pi,1.009\pi]$
S6 | 6 | $O(\epsilon^{2})$ | $0,0,1.8137,\frac{3}{4},\frac{3}{4},0.5637$ | $[0.964\pi,1.036\pi]$
S8 | 8 | $O(\epsilon^{3})$ | $0,0,1.9064,1.7814,\frac{3}{4},\frac{3}{4},0.6564,0.5314$ | $[0.926\pi,1.074\pi]$
S10 | 10 | $O(\epsilon^{4})$ | $0,0.8226,0.8226,1.9152,0.4416,\frac{3}{4},1.5726,1.5726,0.6652,1.1916$ | $[0.885\pi,1.115\pi]$
S12 | 12 | $O(\epsilon^{5})$ | $0,0,1.3587,1.3587,0.3367,0.2117,\frac{3}{4},\frac{3}{4},0.1087,0.1087,1.0867,0.9617$ | $[0.847\pi,1.153\pi]$
S14 | 14 | $O(\epsilon^{6})$ | $0,0.8197,0.8197,1.6756,1.6756,0.7586,1.1000,\frac{3}{4},1.5697,1.5697,0.4255,$ |
|  |  |  $0.4255,1.5086,1.8500$ | $[0.811\pi,1.189\pi]$
S16 | 16 | $O(\epsilon^{7})$ | $0,0,1.9466,1.9466,1.1420,1.1420,0.1251,0.0001,\frac{3}{4},\frac{3}{4},0.6966,0.6966,1.8920,$ |
|  |  |  $1.8920,0.8751,0.7501$ | $[0.778\pi,1.222\pi]$
S18 | 18 | $O(\epsilon^{8})$ | $0,0.3453,0.3453,1.4636,1.4636,0.2616,0.2616,1.3543,0.0643,\frac{3}{4},1.0953,1.0953,$ |
|  |  |  $0.2136,0.2136,1.0116,1.0116,0.1043,0.8143$ | $[0.749\pi,1.251\pi]$
Table 2:  Phases of asymmetric composite sequences of $N=2n+2$ nominal $\pi$ pulses, which produce the S gate with a pulse area error compensation up to order $O(\epsilon^{n})$. The last column gives the high-fidelity range $[\pi(1-\epsilon_{0}),\pi(1+\epsilon_{0})]$ of pulse area error compensation wherein the Frobenius distance fidelity is above the value $0.9999$, i.e. the fidelity error is below $10^{-4}$. Name | Pulses | $O(\epsilon^{n})$ |  Phases $\phi_{0},\phi_{1},\phi_{2},\ldots,\phi_{n},\phi_{n+1},\ldots,\phi_{2n+1}$ (in units $\pi$) | High-fidelity
---|---|---|---|---
|  |  | (according to (14)) | error correction range
two | 2 | $O(\epsilon^{0})$ | $0,\frac{7}{8}$ | $[0.99977\pi,1.00023\pi]$
T4 | 4 | $O(\epsilon)$ | $0,\frac{31}{16},\frac{7}{8},\frac{13}{16}$ | $[0.988\pi,1.012\pi]$
T6 | 6 | $O(\epsilon^{2})$ | $0,0,1.9064,\frac{7}{8},\frac{7}{8},0.7814$ | $[0.955\pi,1.045\pi]$
T8 | 8 | $O(\epsilon^{3})$ | $0,0,1.9531,1.8906,\frac{7}{8},\frac{7}{8},0.8281,0.7656$ | $[0.912\pi,1.088\pi]$
T10 | 10 | $O(\epsilon^{4})$ | $0,1.1086,1.1086,0.0218,0.2593,\frac{7}{8},1.9836,1.9836,0.8968,1.1343$ | $[0.869\pi,1.131\pi]$
T12 | 12 | $O(\epsilon^{5})$ | $0,0,0.5488,0.5488,1.5386,1.4761,\frac{7}{8},\frac{7}{8},1.4238,1.4238,0.4136,0.3511$ | $[0.828\pi,1.172\pi]$
T14 | 14 | $O(\epsilon^{6})$ | $0,0.9406,0.9406,0.2214,0.2214,1.1532,1.3379,\frac{7}{8},1.8156,1.8156,1.0964,$ |
|  |  |  $1.0964,0.0282,0.2129$ | $[0.791\pi,1.209\pi]$
T16 | 16 | $O(\epsilon^{7})$ | $0,0,1.9724,1.9724,0.7247,0.7247,1.7171,1.6546,\frac{7}{8},\frac{7}{8},0.8474,0.8474,1.5997,$ |
|  |  |  $1.5997,0.5921,0.5296$ | $[0.758\pi,1.242\pi]$
T18 | 18 | $O(\epsilon^{8})$ | $0,0.9424,0.9424,0.5711,0.5711,1.3429,1.3429,0.3645,0.6381,\frac{7}{8},1.8174,1.8174,$ |
|  |  |  $1.4461,1.4461,0.2179,0.2179,1.2395,1.5131$ | $[0.728\pi,1.272\pi]$
Table 3:  Phases of asymmetric composite sequences of $N=2n+2$ nominal $\pi$
pulses, which produce the T gate with a pulse area error compensation up to
order $O(\epsilon^{n})$. The last column gives the high-fidelity range
$[\pi(1-\epsilon_{0}),\pi(1+\epsilon_{0})]$ of pulse area error compensation
wherein the Frobenius distance fidelity is above the value $0.9999$, i.e. the
fidelity error is below $10^{-4}$.

##  Appendix B Arbitrary phase gates

The fourth-order compensating sequences of ten $\pi$ pulses can be written in
the compact form

$(3\pi)_{0}\pi_{\phi_{3}}\pi_{\phi_{4}}(3\pi)_{\pi-\frac{1}{2}\phi}\pi_{\phi_{3}+\pi-\frac{1}{2}\phi}\pi_{\phi_{4}+\pi-\frac{1}{2}\phi}.$
(37)

For brevity, we do not show other configurations consisting of $3\pi$ pulse
and $\pi$ pulses, because all these designs have equal total pulse area, i.e.
operation run-time, and equal fidelity. Such solutions can be obtained by
interchanging pulses in the sequence similar to (21), (25) and (30).

The fifth-order compensating sequence consists of twelve $\pi$ pulses and it
can be written in the compact form

$(3\pi)_{0}\pi_{\phi_{3}}\pi_{\phi_{4}}\pi_{\phi_{5}}(3\pi)_{\pi-\frac{1}{2}\phi}\pi_{\phi_{3}+\pi-\frac{1}{2}\phi}\pi_{\phi_{4}+\pi-\frac{1}{2}\phi}\pi_{\phi_{5}+\pi-\frac{1}{2}\phi},$
(38)

with $\phi_{5}=\phi_{4}-\phi_{3}-\frac{1}{4}\phi$. The sixth-order
compensating sequence contains fourteen $\pi$ pulses,

$(4\pi)_{0}\pi_{\phi_{4}}\pi_{\phi_{5}}\pi_{\phi_{6}}(4\pi)_{\pi-\frac{1}{2}\phi}\pi_{\phi_{3}+\pi-\frac{1}{2}\phi}\pi_{\phi_{4}+\pi-\frac{1}{2}\phi}\pi_{\phi_{5}+\pi-\frac{3}{4}\phi}.$
(39)

The composite phases for this type of composite phase gates for arbitrary
phase flip angles are presented in Table 4. The structure of these sequences
corresponds to Eq. (14) with $\nu=0$ and zero first phases, i.e. with
accordance to Eqs. (21a), (25d), (30d), (37), (38) and (39).

| 4 pulses, $O(\epsilon)$ | 6 pulses, $O(\epsilon^{2})$ | 8 pulses, $O(\epsilon^{3})$ | 10 pulses, $O(\epsilon^{4})$ | 12 pulses, $O(\epsilon^{5})$ | 14 pulses, $O(\epsilon^{6})$
---|---|---|---|---|---|---
$\phi$ | $\phi_{1}$ | $\phi_{2}$ | $\phi_{2},\phi_{3}$ | $\phi_{3},\phi_{4}$ | $\phi_{3},\phi_{4},\phi_{5}$ | $\phi_{4},\phi_{5},\phi_{6}$
| (see (21a)) | (see (25d)) | (see (30d)) | (see (37)) | (see (38)) | (see (39))
$\frac{1}{16}$ | $\frac{127}{64}=1.984375$ | 1.9766 | 1.9883, 1.9727 | 0.9980, 0.9883 | 1.0316, 1.7227, 0.6755 | 0.9995, 0.9960, 0.9857
$\frac{1}{12}$ | $\frac{95}{48}=1.9791(6)$ | 1.9688 | 1.9844, 1.9635 | 0.9974, 0.9844 | 1.0342, 1.6996, 0.6446 | 0.9993, 0.9947, 0.9809
$\frac{1}{8}$ | $\frac{63}{32}=1.96875$ | 1.9531 | 1.9766, 1.9453 | 0.9961, 0.9765 | 1.0379, 1.6646, 0.5955 | 0.9990, 0.9922, 0.9716
$\frac{1}{6}$ | $\frac{47}{24}=1.958(3)$ | 1.9375 | 1.9688, 1.9271 | 0.9948, 0.9687 | 1.0405, 1.6378, 0.5556 | 0.9987, 0.9895, 0.9620
$\frac{1}{4}$ | $\frac{31}{16}=1.9375$ | 1.9064 | 1.9531, 1.8906 | 0.9922, 0.9530 | 1.0439, 1.5966, 0.4901 | 0.9980, 0.9843, 0.9431
$\frac{1}{3}$ | $\frac{23}{12}=1.91(6)$ | 1.8754 | 1.9375, 1.8542 | 0.9895, 0.9371 | 1.0459, 1.5642, 0.4349 | 0.9974, 0.9790, 0.9240
$\frac{1}{2}$ | $\frac{15}{8}=1.875$ | 1.8137 | 1.9064, 1.7814 | 0.9842, 0.9050 | 1.0477, 1.5126, 0.3399 | 0.9961, 0.9684, 0.8855
$\frac{2}{3}$ | $\frac{11}{6}=1.8(3)$ | 1.7529 | 1.8754, 1.7087 | 0.9787, 0.8721 | 1.0479, 1.4703, 0.2558 | 0.9947, 0.9575, 0.8460
$\frac{3}{4}$ | $\frac{29}{16}=1.8125$ | 1.7229 | 1.8599, 1.6724 | 0.9759, 0.8552 | 1.0475, 1.4512, 0.2162 | 0.9941, 0.9520, 0.8259
$\frac{5}{6}$ | $\frac{43}{24}=1.791(6)$ | 1.6932 | 1.8444, 1.6361 | 0.9731, 0.8381 | 1.0470, 1.4332, 0.1779 | 0.9934, 0.9464, 0.8056
$\frac{7}{8}$ | $\frac{57}{32}=1.78125$ | 1.6785 | 1.8368, 1.6180 | 0.9717, 0.8294 | 1.0467, 1.4245, 0.1591 | 0.9930, 0.9436, 0.7953
$\frac{11}{12}$ | $\frac{85}{48}=1.7708(3)$ | 1.6639 | 1.8291, 1.5999 | 0.9702, 0.8206 | 1.0463, 1.4161, 0.1405 | 0.9927, 0.9407, 0.7849
$\frac{15}{16}$ | $\frac{113}{64}=1.765625$ | 1.6566 | 1.8252, 1.5908 | 0.9695, 0.8161 | 1.0462, 1.4119, 0.1314 | 0.9925, 0.9393, 0.7797
$1$ | $\frac{7}{4}=1.75$ | 1.6350 | 1.8137, 1.5637 | 0.9673, 0.8027 | 1.0456, 1.4000, 0.1041 | 0.9920, 0.9350, 0.7638
Table 4:  Phases of composite pulse sequences which produce phase gates of
angle $\phi$ (according to (14)). The all phases are given in units $\pi$. The
cases of $\phi=\pi$, $\phi=\frac{1}{2}\pi$ and $\phi=\frac{1}{4}\pi$ repeat
the asymmetric Z, S and T gates respectively, already presented partly in Sec.
III, and partly in Appendix B; they are given here for the sake of comparison
and completeness.

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#  Mobile Manipulator for Autonomous Localization, Grasping and Precise
Placement of Construction Material in a Semi-structured Environment

Petr Štibinger${}^{1\href https://orcid.org/0000-0002-7662-9230}$, George
Broughton${}^{2\href https://orcid.org/0000-0003-0071-5834}$, Filip
Majer${}^{2\href https://orcid.org/0000-0002-4921-3360}$, Zdeněk Rozsypálek1,
Anthony Wang3, Kshitij Jindal3, Alex Zhou4, Dinesh Thakur${}^{4\href
https://orcid.org/0000-0001-5046-8160}$, Giuseppe Loianno${}^{3\href
https://orcid.org/0000-0002-3263-5401}$, Tomáš Krajník${}^{2\href
https://orcid.org/0000-0002-4408-7916}$, Martin Saska${}^{1\href
https://orcid.org/0000-0001-7106-3816}$ 1The authors are with the Czech
Technical University in Prague, Faculty of Electrical Engineering, Department
of Cybernetics, Karlovo Náměstí 13, 121 35 Prague, Czech Republic. email:
{stibipet, rozsyzde<EMAIL_ADDRESS>authors are with the Czech
Technical University in Prague, Faculty of Electrical Engineering, Department
of Computer Science, Karlovo Náměstí 13, 121 35 Prague, Czech Republic. email:
{brouggeo, majerfil<EMAIL_ADDRESS>authors are with the New York
University, Tandon School of Engineering, 5 MetroTech Center, 11201 Brooklyn
NY, USA. email: {kshitij.jindal, aw3645<EMAIL_ADDRESS>authors are
with the University of Pennsylvania, GRASP Lab, 3330 Walnut Street, 19104,
Philadelphia, PA, USA. email: {tdinesh<EMAIL_ADDRESS>
###### Abstract

Mobile manipulators have the potential to revolutionize modern agriculture,
logistics and manufacturing. In this work, we present the design of a ground-
based mobile manipulator for automated structure assembly. The proposed system
is capable of autonomous localization, grasping, transportation and deployment
of construction material in a semi-structured environment. Special effort was
put into making the system invariant to lighting changes, and not reliant on
external positioning systems. Therefore, the presented system is self-
contained and capable of operating in outdoor and indoor conditions alike.
Finally, we present means to extend the perceptive radius of the vehicle by
using it in cooperation with an autonomous drone, which provides aerial
reconnaissance. Performance of the proposed system has been evaluated in a
series of experiments conducted in real-world conditions.

##  I Introduction

Robotic assembly has become a staple of the manufacturing process over the
past decades. For many years, industrial robots have been purpose-built to
perform a singular task with nearly zero versatility or possibility to
reconfigure the assembly process. The status quo is slowly changing with the
increasing availability of compact but powerful computational units and new
lightweight actuators. The available processing power allows for extensive use
of advanced planning and perception methods, as well as deep learning
techniques supported by a widespread availability of datasets [1].

The growth in e-commerce also led into the development of robots in
warehouses, where they are used for automated storage or product retrieval
[2]. Moreover, robots also see increased prevalence outside of the structured
environments, e.g., in households and offices [3, 4]. The construction task
addressed in this paper takes the logistic aspects of warehouse automation and
employs them into a less organized semi-structured environment, as shown in
Fig. 1.

Figure 1: The described system loading and placing building material during
the MBZIRC 2020 contest.

Despite the undeniable progress made in recent years, common household robots
like intelligent vacuum cleaners are still a long way from versatile robotic
assistants. The challenges of designing such system include navigation in a
cluttered dynamic environment and interaction with a wide variety of household
objects. Most importantly, the operation has to be reliable and safe,
especially since the robot is expected to share its operational space with
humans.

Unfortunately, the design requirements often contradict each other. Excellent
dexterity can only be satisfied by a large robotic arm with many actuators,
which in turn results in a need for a large and stable mobile base. However,
safe and precise navigation in a cluttered environment is often difficult for
bulky robots. Therefore, present day mobile manipulators appear as clumsy and
require human supervision while operating, which somewhat negates the purpose
of an automated system. This becomes even more pronounced when the operation
spans over larger time periods [3, 5]. The need for human supervision also
makes the mobile manipulator economically unviable.

The economical aspects may be offset by considering environments, which are
potentially hazardous for humans. These include search and rescue operations
in underground environments [6, 7], clearing areas of toxic or radioactive
debris [8, 9], high-rise building maintenance [10] or power line inspection
[11, 12].

###  I-A Related work

The deployment of complex cyber-physical systems and machine learning in
industry marks a historic shift in design paradigms, which has been dubbed the
fourth industrial revolution, or Industry 4.0 [13, 14].

Commercially available mobile manipulators include the KUKA KMR
iiwa111https://www.kuka.com/en-de/products/mobility/mobile-robots/kmr-iiwa,
Robotnik RB-1222https://robotnik.eu/products/mobile-manipulators/rb-1 or the
Fetch Robotics Mobile Manipulator [15]. Despite availability, mobile
manipulators remain predominantly a subject of academic research.

A 5 DOF manipulator mounted to an omni-directional mobile platform proposed in
[16] demonstrated the ability to localize and pick up colored objects in a
semi-structured environment. A dexterity analysis in [17, 18] considers the
task of picking up an object from a shelf by a mobile manipulator. In [19],
manipulation and transportation of a heavy payload is addressed. Finally in
[20], a robot is shown to fetch various warehouse objects, while also avoiding
a human worker, who occupies the same area.

Mobile manipulators will play a crucial role in the transition to Industry
4.0, as they offer an easy way to reconfigure the assembly line by self-
reorganization, and provide more options for cooperation with human workers.
In the near future, mobile manipulators may become an integral component of
smart factories, workshops and highly modular assembly lines [21, 22, 23, 24].

The field of precision agriculture may also greatly benefit from mobile
manipulators. However, robots working outdoors have to overcome changing
weather and lighting conditions [25]. In [26], more than 50 agricultural
mobile manipulation systems were reviewed. The results show a significant lack
of reliability, if the robots are deployed outside of laboratory conditions.
However, the challenges of reliable robotic perception in the real-world
outdoor environment are slowly being overcome, as new sensors and processing
methods are developed [27, 28, 29]. Novel approaches also allow for long-term
deployment of agricultural robotic systems [30]. A study in [31] shows, that
specializing in one crop type and adjusting the environment for robotic
operation significantly improves reliability in real-world conditions.

Aiming to push the boundaries of research, the Mohamed Bin Zayed International
Robotics Challenge333http://mbzirc.com (MBZIRC) held in 2017 and in 2020 also
focused on mobile manipulation tasks. The competition attracted attention and
participants from renowned research institutions from all over the world. In
2017, a ground robot was tasked to autonomously locate a valve, pick up a tool
of an appropriate size and turn the valve [32]. It also featured the “Treasure
hunt” challenge, where a team of up to 3 aerial robots was tasked to locate,
grasp and transport magnetic discs to a designated drop-off zone [33, 34, 35].
In 2020, a successor to the Treasure hunt was introduced in the “Brick
building” challenge. The difficulty was increased significantly by adding
objects of varying sizes, replacing the drop-off by precision placement, and
requiring both ground and aerial vehicles to be deployed simultaneously [36].

###  I-B Contribution

We present a self-contained robotic system for autonomous localization,
grasping, transportation and precise placement of magnetic blocks. The system
design is provided to the community together with the necessary onboard
software as open-source [37]. In contrast to the commercially available mobile
manipulators, the proposed system is highly modular and dexterous, while
maintaining a compact form-factor. As was mentioned earlier, these two design
requirements often contradict each other. The proposed system greatly benefits
of a unique combination of a compact, yet powerful mobile base, and a very
lightweight ($8.2$ kg) manipulator arm with 7 DOF, which is able to carry
objects weighing up to $3$ kg. Another major advantage is the ability to
traverse uneven terrain, which is essential for outdoor operation, but is not
present with the aforementioned industrial solutions.

We propose a modified variant of the fast segmentation algorithm introduced in
[38] to efficiently extract objects of interest from both depth and RGB image
streams. The system is designed to be invariant to external lighting, and was
proven to work in darkness as well as broad daylight. We also show, how to
enhance the limited perception range of a ground-based robot by employing a
cooperating aerial vehicle equipped with an onboard camera, to quickly detect
visually distinct features of the area.

Finally, we share the experience gained by our participation in the Brick
challenge of the MBZIRC 2020. Out of 19 teams participating in this challenge,
only 2 teams managed to complete the task with their ground robots in
autonomous mode. As the winners of the competition, we believe that presenting
the complete system to the community will make a valuable contribution.

##  II Problem description

This work tackles a construction task in an outdoor semi-structured
environment. The area of operation is rectangular, approximately $50$ m
$\times$ $60$ m in size. It is largely devoid of features and obstacles. One
unmanned ground vehicle (UGV) and three unmanned aerial vehicles (UAV) are
deployed in the area simultaneously. The following objects of interest are
located in the area: stacked bricks for the UGV to pick up and a colored
pattern to place the bricks to. The arena also contains stacked bricks and a
placement area for the UAVs. These objects are not of interest to the UGV and
therefore will be treated as obstacles. The objects may be located at an
arbitrary position in the arena. A precise position and orientation of these
objects is not known in advance.

The stacked bricks are arranged in a predefined pattern, as shown in Fig. 2.
There are four classes of bricks, which differ in size, weight and color. Each
brick features a white ferrous plate attached to the top side. The placement
area consists of two flat segments forming an L-shape. The segments are
covered in a high contrast yellow-magenta checkered pattern, see Fig. 2, and
each segment is $4$ m long, and $0.4$ m wide. The pattern is laid directly on
the ground.

The structures related to UAVs are clearly distinguishable, with the pickup
area only consisting of one layer of bricks, and the placement area being
placed on an elevated platform with a height of $1.7$ m.

The goal is to use the colored bricks to construct a wall in the area marked
by the checkered pattern. Each layer of the wall consists of exactly 2 orange,
1 blue, 2 green and 4 red bricks. The ordering of the bricks is specified in
advance, but changes with each trial. Each correctly placed brick is awarded
by a score gain, with larger bricks being worth more points. The time limit to
complete the challenge is 30 minutes.

Figure 2: Objects of interest, with which the UGV will interact. The brick
pickup area (left) consists of colored bricks stacked in multiple layers. The
checkered pattern (right) indicates the deployment zone.

##  III Robotic platform

(a) Proposed mobile manipulator

(b) Cargo bay attached

Figure 3: An overview of the proposed robotic system. A Gen3 Kinova
manipulator arm is attached to a Clearpath Husky wheeled base. A custom-
designed cargo bay is attached to the robot to increase payload capacity and
minimize the time spent by traversing the construction site.

The proposed system is built using mostly off-the-shelf components, which
corresponds well with our modular and open design philosophy. A Clearpath
Robotics Husky A200 serves as the mobile base for the system. An Intel NUCi7,
with the operating system Ubuntu 18.04, is used as the main onboard computer.
The Robot Operating System (ROS) serves as a middleware interconnecting the
hardware and software parts [39]. For object manipulation, a 7 DOF Kinova
Robotics Gen3 arm is employed. The arm is mounted to the mobile base as shown
in Fig. 3.

The arm is equipped with a custom-designed, force compliant end-effector with
a magnetic gripper as shown in Fig. 4. Its design is based on our previous
experience with magnetic object grasping by aerial vehicles [33, 35]. The
gripper features two YJ-40/20 solenoid electromagnets, mounted at a $130$ mm
center-to-center pitch. The magnets are rated to operate at $12$ V, with each
magnet providing up to $25$ kg of holding force. However, the force is
strongly affected by the relative distance and orientation of contact the
surfaces. In our setup, the magnets are overcharged with a $24$ V input, which
effectively doubles the available grasping force. On the other hand, it makes
the gripper susceptible to overheating when powered on for extended periods of
time. Each magnet is fitted with a Hall-effect sensor, which serves as a
proximity detector for ferromagnetic objects. The sensoric readout and power
management of the magnets is handled by an Arduino Nano board, which is
connected via USB to the main computer.

(a) CAD design

(b) 3D printed and mounted

Figure 4: Detail of the custom-designed magnetic gripper for the end-effector.
The gripper is fitted with two electromagnets and Hall-effect sensors. The
design also incorporates a mounting point for the Realsense D435 camera, which
is used for visual servoing.

To minimize the time spent by transitions between the material loading area
and the deployment area, a custom-designed cargo bay is attached to the rear
of the robot. The cargo bay allows the robot to store up to 7 bricks (4 red, 2
green and 1 blue), which are sufficient for construction of an entire section
of the structure. Loading a brick into the cargo bay takes around 30 seconds.
During this time period, the gripper temperature is kept to reasonable levels.

For environment sensing, the robot is fitted with a Velodyne Puck VLP-16
LiDAR444Light Detection and Ranging, consisting of a rotating infrared laser
range-finder. The LiDAR provides a $360^{\circ}$ horizontal and a $30^{\circ}$
vertical field of view, with an effective range of around $50$ m. The sensor
is connected to the main computer via Ethernet, and provides measurements at a
rate of $10$ Hz in the form of a 3D point cloud.

The manipulator is already fitted with an Intel Realsense D410 RGBD camera
integrated into the wrist link. This camera was intended to provide visual
feedback during the brick grasping process. However, we have experienced a
significant delay in the image output, which rendered it impossible to use in
a feedback loop. Therefore, an additional Intel Realsense D435 RGBD camera was
mounted to the gripper and connected directly to the onboard computer. The
camera provides an RGB resolution of up to $1920\times 1080$ pixels, and a
depth image resolution of up to $1280\times 720$ pixels.

##  IV Control architecture

The following sections summarize the key software components used to
successfully complete the task in autonomous mode. As mentioned before, the
environment layout is unknown prior to the deployment. The robot therefore has
to explore the area and localize the objects of interest. Despite using an
array of powerful onboard sensors, the object detection range of the robot is
still quite limited due to the finite resolution of the sensors and the size
of the objects. For this reason, we concluded that a full exploration of the
area would not be feasible given the limited time budget. On the other hand,
specifying navigational waypoints in an unknown environment is not possible.

We approach the problem using a “guided autonomy” concept [40, 41, 42], which
aims to strike the balance between exhaustive exploration and going directly
to a well-defined target. The system is able to investigate coarsely defined
areas of interest, which may be provided on-the-fly by a human operator or by
other cooperating robots. Limiting the search space allows the robot to
conserve some of the limited time budget, and if no objects of interest are
found within the prioritized areas, the full exploration is resumed.

##  V Arm manipulation

The arm manipulation is controlled by the Kinova Kortex API and a ROS driver
provided by the manufacturer555https://github.com/Kinovarobotics/ros_kortex.
We have designed a custom wrapper for the API called Kortex Control Manager
(KCM), which integrates the API into the control pipeline.

The KCM includes a library of short pre-programmed actions, which are chained
to form more complex tasks, such as the brick grasping and loading. In
addition to self-collision avoidance, which is integrated into the API, the
KCM also expands this by enforcing an extended no-go zone over the UGV’s
frame. The no-go zone is further expanded by half a brick size to prevent
collisions with the grasped brick. These safety features are necessary, but
significantly reduce the operational space of the arm, as shown in Fig. 5. The
limited reach of the arm therefore needs to be compensated by the mobile base.

The KCM offers position control of the manipulator in joint space, and
velocity control of the end-effector in Cartesian space. In the position
control mode we utilize the MoveIt! planning interface [43] to interpolate the
trajectory between current and desired position. The trajectory is sampled
with a time step of $1$ ms and the corresponding joint commands are sent to
the actuators via the Kortex API.

The KCM also monitors the data returned by the integrated sensors, which
measure position, velocity and torque in each of the joints at a rate of $100$
Hz. The data is analyzed, and the motion is terminated if excessive torque or
no-go zone violation is detected.

Figure 5: Grasping reach of the proposed mobile manipulator with the 7 DOF
Kinova Gen3 manipulator. The arm is equipped with the custom end-effector and
a magnetic gripper. If a brick center is located within the range highlighted
in red, it can be reliably grasped. The dimensions are shown in millimeters
unless stated otherwise.

##  VI Navigation and localization

The UGV uses a default ROS navigation stack, as implemented by the STRANDS
project [3]. The main source of positional data for navigation is the onboard
LiDAR. During the development, we have discovered that the dynamically
changing area of a construction site and a non-flat ground plane introduce a
lot of false-positive measurements to the localization stack.

By experimental evaluation, it was found that the localization performance can
be significantly improved by focusing on the upper layers of the point cloud.
Therefore, the LiDAR scan is separated into two horizontal slices, with the
upper slice being used for robot localization, and the lower slice for
obstacle avoidance. For this particular application, the slicing threshold was
empirically set to $1.5$ m above the sensor.

Prior to the deployment, a high-resolution scan of the surrounding environment
is taken by a Leica BLK360 3D scanner. The point cloud obtained by the 3D
scanner is used to augment the measurements taken by the onboard LiDAR. The
augmentation contributes significantly to the quality of localization,
especially when the UGV moves near the area center, where the feature density
is the lowest.

The pose of the UGV is estimated using the Adaptive Monte Carlo Localization
(AMCL)666http://wiki.ros.org/amcl. The AMCL essentially attempts to correct
the drift in measured wheel odometry by matching onboard laser scans to the
area map [44]. However, the odometry estimator of the Husky gets distorted by
shifting the center of mass of the vehicle, which occurs with every
manipulator movement and with every brick added into the cargo bay. We have
managed to solve this issue by exhaustively tuning the parameters of the
odometry estimator for various vehicle and payload configurations.

The ROS move_base777http://wiki.ros.org/move_base package is employed for path
planning. Several adaptations of the code were made to allow the UGV to
dynamically change the precision of path following. During the exploration
phase, the precision is relaxed, as the focus is on covering large area in the
shortest amount of time. On the contrary, while approaching the stacked
bricks, high precision is strongly required, since the area reachable by the
manipulator is limited. In the high precision mode, the UGV performs a series
of “parallel parking” manoeuvres to reach the desired position with very tight
margins.

Without any additional information, the UGV conducts a full exploration of the
area by generating a grid of waypoints to achieve a full sensoric coverage of
the area, see Fig. 6. An early termination of the procedure is triggered, if
all objects of interest are located.

Figure 6: Visualization of the full coverage exploration strategy. The high
resolution map obtained by the 3D scanner is outlined in black. The yellow
frame determines the operational area, the navigational waypoints are shown in
purple. The green circles centered around each waypoint represent the maximal
perception range of the system. The UGV then follows a zig-zag pattern in
order to visit all the waypoints.

##  VII Automated construction

###  VII-A Inventory management

The proposed design features a cargo bay, which can hold up to 7 bricks.
However, in order to preserve a compact form factor, the bricks in the cargo
bay are stacked into three layers. This renders some bricks unreachable, if
other bricks are loaded on top of them.

We have developed an inventory management system, which monitors the
reachability of individual bricks. The inventory manager also processes the
blueprint for the desired wall assembly, and determines which bricks to pick
up, and where to place them. This module also allows the UGV to dynamically
change the building strategy. This feature is especially useful when the UGV
encounters an unexpected issue, such as an inability to grasp some of the
bricks. In such case, a new strategy is to immediately navigate to the
building pattern and unload all the remaining cargo.

###  VII-B Brick stack detection

During the exploration phase, the LiDAR is used for detection of the stacked
bricks. The Velodyne VLP-16 has a $30^{\circ}$ vertical field of view divided
into 16 evenly spaced scan layers. To reliably distinguish a brick in the
point cloud, at least two scan layers have to hit it. The number of rays
hitting a brick can be determined as

$\phantom{,.}N=\frac{\mathrm{arccos}\left(1-\frac{a^{2}}{2b^{2}}\right)}{\alpha}\phantom{.},$
(1)

where $a$ is the brick height, $b$ is the distance between the sensor and the
point (ray length) and $\alpha$ is the angular pitch of two neighboring scan
layers. The maximal detection distance $d$ is then estimated as

$\phantom{..}d=\frac{\frac{a}{4}}{\mathrm{tan}\left(\frac{\alpha}{2}\right)}\phantom{.}.$
(2)

For a single brick with a height of $0.2$ m, the estimated detection range is
approximately $3.055$ m. The formula assumes an idealized scenario, in which
the rays and one brick side form an isosceles triangle. In practice, the range
would be even shorter. Fortunately, the apriori knowledge of the stack layout
can be leveraged to increase the range, since all brick types are arranged in
at least two layers.

The point cloud is processed using the Iterative End Point Fit (IEPF)
algorithm [45], which extracts line segments from the measurement. All
segments of length matching one of the brick classes are considered brick
candidates. In the real-world, this approach alone is not sufficient, as the
structure of the environment causes generation of false-positive candidates.

To filter out the erroneous data, an Expectation Maximization (EM) algorithm
is employed. The algorithm uses the following probabilistic model:

$\phantom{,.}P(\vec{x}_{m})\sim\mathcal{N}(\vec{x}_{m},\,\vec{\mu}+k_{m}\vec{v},\,\bm{\Sigma_{m}})\phantom{.},$
(3)

where $m$ is the brick class index, $\mathcal{N}$ is the Gaussian
distribution, $\vec{\mu}$ is the mean of the model, $\bm{\Sigma_{m}}$ is the
covariance matrix of a brick class, $k_{m}$ is the scalar multiplier unique
for each class and $\vec{v}=(\cos\phi,\sin\phi)^{T}$ is a direction vector
representing orientation. The probability density of individual brick classes
is visualized in Fig. 7. The model provides the most likely estimate of the
parameters $\vec{\mu}$ and $\phi$. The parameters correspond with the position
of the brick stack center, and the orientation of the stack. The results are
then used to generate a navigational waypoint near the red bricks, so that the
bricks are on the right hand side of the robot.

Figure 7: Probability density of the brick class position in the pickup area.
This model is used by the brick classifier to estimate the position and
orientation of the pickup area.

After approaching the red bricks, a line-fitting algorithm based on the Random
Sampling Consensus (RANSAC [46]) is applied to the point cloud. The UGV
performs a parallel parking action to align its heading with the major axis of
the brick stack, which is estimated by the RANSAC algorithm. After aligning,
the manipulator is extended into a predefined position to the right hand side
of the Husky. In this position, the wrist-mounted camera and the gripper point
directly towards the ground.

###  VII-C Brick detection, grasping and loading

The brick detection pipeline is built on the fast image segmentation proposed
in [38]. The algorithm has been successfully deployed in the Treasure hunt
challenge of MBZIRC 2017 [47, 33, 35] for detection of circular objects. The
original algorithm works with data provided by an RGB camera. For the brick
detection, we opted to use a depth camera, which proved to be more resilient
to illumination changes. Therefore, the algorithm was adapted for depth image
processing, which enabled the robot to operate in direct sunlight and also at
night.

The algorithm factors in the position of the arm to estimate the height of the
camera above the ground plane. The depth information contained within each
pixel is transformed into height above the estimated ground level. The
algorithm begins by discarding all data below the height of $0.1$ m. Then, a
flood-fill algorithm described in [38] is initiated to extract brick candidate
segments. In addition to the segments, the algorithm also obtains their
centers, sizes, eccentricities and distances from the camera. Finally, the
method determines, whether all edges of the segment are visible by the camera.

The processing pipeline then attempts to assign a brick class to the fully
visible segments. It starts by locating a pixel with the highest distance from
the center, which we denote as $c_{0}$. Then, a pixel $c_{1}$ is found by
searching for the maximal distance from $c_{0}$. The third corner $c_{2}$ is
found by searching for a maximal distance from both $c_{0},c_{1}$. The final
corner $c_{3}$ is located by maximizing the sum of distances from the other
three corners. Using the coordinates of the corner pixels and the depth
information, the segment is transformed from camera coordinates into 3D and
the length and width of the object are calculated. Since the brick classes are
clearly distinguishable by dimensions only, RGB processing may be omitted.
Therefore, the system does not require color calibration to the current
lighting conditions.

(a) RGB image

(b) Raw depth image

(c) Processed depth

Figure 8: Visual servoing based on the wrist-mounted RGBD camera. The RGB
image is only shown for comparison. Due to the tight packing of the bricks,
multiple segments are visible at the same time. The segmented depth image is
shown on the right, with detected corners marked in red, and the current
alignment target in yellow. Note the robot wheels at the bottom of the image.

The processed depth image (see Fig. 8) is used for visual servoing of both the
mobile base and the manipulator. If the image contains the brick class
requested by the inventory manager, the segment position is transformed into
the coordinate system of the gripper, and the arm attempts to align with its
center. If multiple segments fit the request, the system prioritizes the
segment closest to the lower right corner of the image. If the segment center
is unreachable by the arm, the mobile base starts to perform small turns to
adjust its position. The motion is stopped, once the brick center is located
directly below the gripper with a $\pm 8$ cm tolerance. Afterwards, the
gripper is aligned with the brick center (tolerance $\pm 2$ cm) and the arm
begins to descend.

During the descent, lateral adjustments to the gripper position are made as
long as the brick is fully visible. Afterwards, the magnets are powered on,
and the arm descends straight down until the Hall-effect sensors report a
successful grasp. Grasping failure is detected by either reaching lower than
expected (brick missed completely), or by detecting an excessive torque on the
arm joints (ferromagnetic plate missed). In case of failure, the robot will
attempt the grasping process again. After two failed attempts, the brick is
marked invalid, and the UGV moves on to the next one.

Once the brick is grasped, the inventory manager assigns it a position in the
cargo bay. A series of predefined actions ensures that the arm stores the
brick in the assigned inventory slot. Based on the brick class and the current
payload status, additional actions are added into the loading sequence to
avoid potential collisions.

###  VII-D Build pattern detection

The checkered building pattern is impossible to detect by the LiDAR, as it
blends in with the ground plane. Therefore we use the RGB portion of the data
provided by the arm-mounted RGBD camera. For the pattern detection, the fast
segmentation algorithm based on [38, 47] is used. The algorithm uses a 3D RGB
lookup grid, to classify individual pixels as background, object or unknown.
The method enables pixel-wise classification in constant time, which is
achieved by pre-computing the lookup grid during a calibration process.

In the case of the high-contrast pattern, the grid is initialized using a
Gaussian mixture to model the desired color in the HSV color space. The RGB
image is then searched for the desired color. Once a satisfactory pixel is
found, the flood-fill algorithm is initiated to extract the segment to which
the pixel belongs. In the default state, the algorithm would only find
individual squares in the checkered pattern, which would require costly post-
processing to merge the objects back together. Instead, we propose a modified
flood-fill algorithm capable of bypassing small discontinuities in the
segments.

The algorithm, as presented in [47], naturally obtains the number of colors
surrounding each pixel in the segment, allowing for an easy corner detection.
For each corner, the searched neighborhood is increased to 5 pixels. If a new
pixel of the desired color is found within this neighborhood, it is added to
the original segment. This way, half of the entire checkered pattern may be
extracted as a single segment. Comparing the dimensions of the segment with
the apriori knowledge of the build area allows for effective filtration of
false positive candidates.

During the pattern search, the UGV fully extends the arm, placing the camera
into an elevated position, see Fig. 9. The camera is then rotated to fully
scan the surroundings of the robot. This process requires the UGV to stop
moving, as the center of mass is greatly offset by the arm. Due to the camera
resolution and density of the pattern, detection range is limited to
approximately $10$ m. Achieving a full area coverage would require performing
the scan at 20 different locations, see Fig. 6, which is not feasible within
the given time limit.

(a) Long range pattern search

(b) Precise pattern alignment

Figure 9: While searching for the pattern, the UGV extends the arm upwards to
provide the camera an elevated view point (left). Doing so requires the
vehicle to stop moving due to a large shift in the center of gravity. Once the
building area is located, the UGV uses visual servoing to precisely align the
mobile base with the pattern (right). Green highlights detected segment edges.
The proposed segmentation algorithm enables extraction of all the magenta
squares as one object in a single pass, without the need for post-processing.

Since the UGV shares the construction area with a team of aerial vehicles, we
opted to use a cooperative approach to detect the building pattern. The UAVs
are equipped with a downward facing RGBD camera, and are able to run the
pattern detection pipeline on board. Due to the orientation of the camera, the
UAV does not have to stop moving, and the detection may run continuously while
in motion. The detections are reported to the UGV over a WiFi network. We
employ the NimbroNetwork [48], which allows ROS message sharing between
multiple independent robotic systems.

Once the approximate position of the build pattern is reached, the arm is
folded so that the camera aims forward, as shown in Fig. 3(a). The UGV then
drives in an outward spiral until the pattern appears in the image. Finally,
the arm is extended to the right hand side, and the UGV uses a similar visual
servoing approach to align itself with the pattern, as it does with the brick
stack, see Fig. 9. The bricks are then deployed in an order determined by the
inventory manager.

During the MBZIRC 2020 finals, the UAV assistance was not used due to the fear
of overloading the communication channel and disrupting the mutual
coordination of the UAVs. Instead, the pattern position was estimated from
images taken during the previous contest rounds, and the cooperation was only
emulated.

##  VIII Experiments

Early experimental evaluation was performed at the tennis courts of the New
York University campus in Abu Dhabi. In here, the system was repeatably able
to locate, load and deploy the bricks in a specified order. The success rate
of brick loading was around $75\%$, with the grasping accuracy reaching $\pm
3$ cm. In the remaining $25\%$ of cases, the system was not able to fill the
cargo bay with all seven bricks. However, in case of a brick loading failure,
the system was still able to autonomously interrupt the loading procedure,
navigate to the checkered pattern, and unload the contents of the cargo bay.

During the experiments, we measured that navigating to the stacked bricks
takes up to 2 minutes, assuming the stack of four orange bricks is visible
from almost any starting position. The precise alignment and loading all 7
bricks into the cargo bay takes nearly 10 minutes. Without the UAV assistance,
the building pattern search is by far the most time consuming operation. To
acquire the image from an elevated position, the robot has to stop, extend the
arm upward, look around with the camera and then fold the arm back. This
procedure alone requires 15 minutes on average, which only leaves 3 minutes
for the brick deployment.

Increasing the manipulator speed was not possible, as higher accelerations
might cause the brick to be dropped. With the time budget in mind, we also
prepared the emergency protocol, which focused on reliably placing a single
brick on the first run. This would reduce the loading and deployment times
significantly, leaving enough margin for difficult arena configurations
requiring prolonged exploration.

The grasping procedure was extensively tested under various lighting
conditions to ensure robust performance at any time of day, as shown in Fig.
10. This feature greatly enhances the economic feasibility of the proposed
system as a robotic construction worker, since night-time construction
operations fall into the “potentially hazardous for humans” category.

(a)

(b)

(c)

Figure 10: Testing of the grasping procedure in various kinds of lighting
conditions. By using the depth image rather than RGB, the grasping can be
performed in a repeatable manner at night (a), in direct sunlight (b) or in
strong lateral illumination at sunset (c).

The performance of the brick extraction and classification in the LiDAR data
was evaluated offline using a dataset obtained during the first competition
rehearsal. In Table I, we show a success rate of brick classification by the
IEPF algorithm without any additional filtration or odometry fusion. The blue
bricks are the least populous in the pickup area, and therefore are the most
susceptible to erroneous detections. The accuracy is greatly improved by
filtering the false positive measurements and by employing the EM algorithm,
which incorporates the a priori knowledge of the brick pile layout. Snapshots
of the point cloud processing are shown in Fig. 11.

Brick color | Correct | Incorrect | Success rate [%]
---|---|---|---
Red | 174 | 70 | 71
Green | 57 | 19 | 75
Blue | 24 | 63 | 28
Orange | 32 | 11 | 62
TABLE I: Accuracy of brick candidates generation by the IEPF algorithm. The
evaluation was performed offline using data collected during the first
rehearsal in the competition arena.

(a) Extracted line segments

(b) Red pile center estimation

Figure 11: Detection of stacked brick in the LiDAR data. The left image
showcases the line segments as extracted and classified by the IEPF algorithm.
The right image shows the five initial iterations of the EM algorithm. The red
points represent the estimated position of the red pile center.

The final evaluation took place during the MBZIRC 2020 Brick challenge finals.
During the contest, the brick pickup area was placed on the inclined ramp,
which posed additional challenges for the participants. With no time to
reconfigure the system to this new environment, we opted to engage the
emergency protocol, which was optimized for reliability and ensured task
completion with at least one brick. The successful brick placement during the
contest finals is shown in Fig. 1.

##  IX Conclusion

We have presented an autonomous robotic system for localization, grasping,
transportation and precise deployment of construction blocks. The system is
capable of autonomous operation in challenging lighting conditions on uneven
terrain. The positioning and navigation pipeline does not rely on external
systems, such as the GPS. The system may therefore be deployed in an indoor
environment without any adjustments, and even perform the complicated
transition from an outdoor to an indoor environment. The presented computer
vision algorithms, which are application independent, offer fast and reliable
segmentation of both depth and RGB images.

We provide all the software to the community as open-source [37]. The
presented system was successfully deployed during the MBZIRC 2020 Brick
challenge, which simulated a possible application of mobile manipulators in a
construction task. As the winners of the contest, we hope this work will
inspire further development and facilitate the mobile manipulation endeavors
of other research groups.

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