# Online Action Representation using Change Detection and Symbolic Programming

Vishnu S Nair  Department of Electrical Engineering
Indian Institute of Technology Madras
<EMAIL_ADDRESS>Sneha Sree  Department of Electrical Engineering
Indian Institute of Technology Madras
<EMAIL_ADDRESS>Jayaraj Joseph  Mohanasankar Sivaprakasam
Department of Electrical Engineering
Indian Institute of Technology Madras
<EMAIL_ADDRESS>Department of Electrical Engineering
Indian Institute of Technology Madras
<EMAIL_ADDRESS>
###### Abstract

This paper addresses the critical need for online action representation, which
is essential for various applications like rehabilitation, surveillance, etc.
The task can be defined as representation of actions as soon as they happen in
a streaming video without access to video frames in the future. Most of the
existing methods use predefined window sizes for video segments, which is a
restrictive assumption on the dynamics. The proposed method employs a change
detection algorithm to automatically segment action sequences, which form
meaningful sub-actions and subsequently fit symbolic generative motion
programs to the clipped segments. We determine the start time and end time of
segments using change detection followed by a piece-wise linear fit algorithm
on joint angle and bone length sequences. Domain-specific symbolic primitives
are fit to pose keypoint trajectories of those extracted segments in order to
obtain a higher level semantic representation. Since this representation is
part-based, it is complementary to the compositional nature of human actions,
i.e., a complex activity can be broken down into elementary sub-actions. We
show the effectiveness of this representation in the downstream task of class
agnostic repetition detection. We propose a repetition counting algorithm
based on consecutive similarity matching of primitives, which can do online
repetition counting. We also compare the results with a similar but offline
repetition counting algorithm. The results of the experiments demonstrate
that, despite operating online, the proposed method performs better or on par
with the existing method.

###### Index Terms:

online action representation, symbolic programming, change detection

##  I Introduction

Action representation is important for many tasks like surveillance, human-
computer interface, rehabilitation monitoring, sports analysis, behavior
analysis, autonomous navigation, etc. In most of the existing methods,
processing is done on fixed-sized segments of an incoming video, and an output
will be produced for each segment. Some methods do segmentation to get
meaningful windows as a pre-processing step, but they are mostly offline
methods, i.e., they require the whole video to find the segments. But for
tasks like rehabilitation feedback, surveillance, etc., it’s crucial that the
methods work online. Setting a fixed window size is not optimal, as different
motions have different dynamics. Certain approaches use image-by-image
methods, but they don’t capture the temporal dynamics of actions. We identify
this gap in the existing works and propose an online action representation
method, where we use change detection to segment out action sequences and then
use primitive fitting to represent the motion segments at a higher level. The
segmentation works online as it uses an online change detection algorithm.
Thus, we get meaningful segments of actions as soon as they happen in the
streaming video without needing future video frames. Also, since
representation is derived over time periods, they capture temporal
information.

Figure 1: (a) Joint angle sequences over one instance of jumping jack
(b) Illustration of primitives fit to each keypoint

As an introduction, we intend to provide an intuition to the method. The
motivation for this representation comes from the idea that an activity(e.g.,
walking) is composed of elementary sub-actions like moving left leg forward,
moving right leg forward, etc. Also, these sub-actions can be detected from
the increasing and decreasing changes in the joint angles that occur along
with the action. So, we can extract the regions of such sub-actions and then
make higher-level representations of them. Such a representation will be both
discriminative and generalizable because of its compositionality. In order to
segment the sub-actions, we first identify the regions of change. Next, we fit
each segment to symbolic primitives to create more semantic representations.
Symbolic programming uses domain knowledge to predefine generative symbols and
fit the data optimally to those symbols. Here, we fit the trajectories to
simple shapes like lines, circles, etc. These symbols are generative as they
can be used for synthesis as well. So, with such fits, we will be able to
state something like: “the arm is rotated in this time segment.” This method
can work sequentially on an incoming video by the choice of an online change
detection algorithm. Fig. 2 shows the complete pipeline of our proposed
method.

We explain each step in the method in more detail. Later on, we use repetition
detection as an example downstream task to show the effectiveness of the
representation. We also compare the results of our method with the existing
offline counterpart.

##  II Related Works

Existing works that attempt to do rehabilitation and exercise monitoring
include [1, 2, 3], but they are not online. In [4], an image-based approach is
used that gives feedback on each frame. As it is image-based, it fails to
capture the temporal information. Being data-driven makes these methods less
feasible in specific domains where data is sparse, such as rehabilitation.

In [5], a motion programming idea of fitting primitive motions to action
segments is introduced. That method uses a dynamic programming approach to
automatically segment the videos based on an optimal error of fitting. But
this method only works offline. We improve on this by replacing the
segmentation part with a change detection algorithm and then fitting a single
primitive to the segmented part.

Change detection is the process of identifying meaningful changes in a time
series over time. They can be classified into online and offline algorithms.
Though we use a classical change detection algorithm, it can be replaced with
any online change detection method.

Loop/repetition detection is the task of counting the number of repetitions of
approximately the same action in an input video sequence. Some of the existing
works on loop detection are [6, 7]. In [6], online loop detection is done, but
it uses windows of a predetermined fixed size.

##  III Methods

Figure 2: Action representation pipeline

Here we provide a comprehensive overview of the entire pipeline, followed by
detailed explanations of relevant parts. The primary objective of our work is
to make the task of action recognition online. The input is an incoming
sequence of frames.

First, we employ an off-the-shelf pose estimator to extract the keypoints from
the frames. Here we use the mediapipe pose estimator[8]. We calculate the
angles at several joints and also the length of bones using vectors
constructed from the keypoints. Joint angles refer to angles subtended at body
joints, such as the shoulder angle and the elbow angle, while bone lengths
refer to the lengths of body parts between two joints, such as the calf length
and the torso length. Consequently, we have several sequences of joint angles
and bone lengths as illustrated in Fig.1(a). These values are computed frame
by frame, and hence they are online incoming time series signals. We use the
CuSum change detection algorithm[9] to detect change in each of these signals.
Once a change is triggered in any one of the signals, that particular signal
is utilized to determine the starting and ending times for temporal
segmentation. While CuSum provides the time of change and the starting time of
change, it does not provide the ending time. Therefore, a piece-wise line-
fitting method is devised to estimate the ending time after waiting for a few
more frames from the point of change. The number of extra frames to take is
decided based on a heuristic. We observe that the clip between the starting
and the ending time of the detected change forms a meaningful action segment.
The keypoint sequences, the joint angle sequences, and the bone length
sequences are trimmed according to those time points. In order to obtain a
higher-level representation of the action, we use a motion primitive fitting
method. This method adopts a symbolic programming approach, wherein the data
is matched to a set of predefined symbols. We fit the trajectories of each
keypoint in the trimmed time span to primitive shapes(circle, line,
stationary)(Fig. 1(b)). Note that the change detection is done on angles and
lengths signals, whereas the primitives are fit on 2D coordinates of each
keypoint. This symbolic representation brings us one step closer to the
temporal semantics of the motion. Thus, they are better features for
downstream tasks. Fig. 2 provides a block diagrammatic depiction of the
process.

Figure 3: The signal represents the variation of left thigh bone
length(indicated in red) as a knee raise is performed. First, a meaningful
segment is trimmed out using change detection, followed by fitting a primitive
to each trajectory of the keypoints within that segment. In the figure, a
representative fit to one of the keypoint(left knee) is indicated.

###  III-A Temporal Segmentation using Change Detection

Given an incoming time series, we want to find out the window of time around
in which a meaningful change in body part motion has happened. We define the
interval from the starting time of the change to the ending time of the change
as the segment interval. In the proposed method, we perform change detection
on joint angles and bone lengths, as they are indicative of sub-actions. It
was demonstrated in [4] that the feature constituting joints and angles gives
the best performance in a pose-based classification task.

####  III-A1 CuSum Change Detection

Change detection algorithms work by identifying a shift in certain statistical
characteristics of the input signal. Change detection proves to be
particularly effective in this application because of the fact that human
motions are constrained and hence always vary within particular ranges. Here,
we use the CuSum(Cumulative Sum) change detection algorithm. CuSum is a
classical change detection method developed by E.S. Page[9]. It detects a
shift in the mean of the signal. It is a sequential method and hence can be
executed on an online time series.

We explain the method using Fig. 4. The time series in the figure is the
variation of left shoulder angle of a subject performing a jumping jack. As it
can be observed, the sub-action of raising the hand is signified by the
ascending section of the signal. Such similar variations in joint angles and
bone lengths serve as indicators for human sub-actions. We leverage this
property of human motion to segment out relevant sections to be further
represented. To do this, we need to detect the starting time and ending time
of the change. In the above example, we are interested in determining the
starting point and ending point of the $S$-shaped segment. We do this using
change detection. Let $t_{\text{start}},t_{\text{change}},t_{\text{end}}$
denote the start time of the change, the time at which change is detected, and
the end time of the change, respectively.

In CuSum change detection, given input time series, $x[t]$, the cumulative sum
of positive and negative changes ($g_{t}^{+}$ and $g_{t}^{-}$) are calculated
and compared to a threshold. When the threshold is crossed, it is flagged that
a change is detected, and the cumulative sums are reset to 0. Since both
positive and negative changes are tracked, it can detect both increasing and
decreasing changes. We make use of the CuSum implementation of the detecta
python package [10]. The detailed steps are given in Algorithm 1.

Algorithm 1 CuSum Change Detection

1:for each $t$ do

2: $s[t]=x[t]-x[t-1]$

3: $g^{+}[t]=\max(g^{+}[t-1]+s[t],0)$

4: $g^{-}[t]=\max(g^{-}[t-1]+s[t],0)$

5: if $g^{+}[t]<0$ then

6: $g^{+}[t]=0$

7: $t^{+}=t$

8: end if

9: if $g^{-}[t]<0$ then

10: $g^{-}[t]=0$

11: $t^{-}=t$

12: end if

13: if $g^{+}[t]>\text{threshold}$ or $g^{-}[t]>\text{threshold}$ then

14: $t_{\text{change}}=t$

15: if $g^{+}[t]>\text{threshold}$ then

16: $t_{\text{start}}=t^{+}$

17: else

18: $t_{\text{start}}=t^{-}$

19: end if

20: $g^{+}[t]=0$

21: $g^{-}[t]=0$

22: end if

23:end for

####  III-A2 End-time of Change

After using CuSum, we have the start point of change and the point of change,
but we do not have the end point of change. We estimate the end of the change
by fitting two piece-wise lines, where the second line is a constant. By
fitting such a function, we are assuming that the change continues to happen
for some more time, and then it reaches a constant value. We are interested in
the time when the transition happens, and we define that as the end time of
the change.

The following piece-wise linear function is fit to the time series between
$t_{\text{change}}$ and $t_{\text{change}}$ \+ ($t_{\text{change}}$ \-
$t_{\text{start}}$). The choice of this interval for fitting is under the
assumption that the rate of the range would remain approximately constant.

$y=\begin{cases}ax+b&\text{when }x<c\\\ d&\text{when }x\geq c\end{cases}$

$c$ is the transition point, and is taken as $t_{\text{end}}$.

In Fig. 4, the red dot indicate $t_{\text{change}}$ and the green arrows
indicate $t_{\text{start}}$ and $t_{\text{end}}$. The piece-wise linear fit
for estimation of $t_{\text{end}}$ is also indicated in the figure. As it can
be observed, the transition point of the two lines is $t_{\text{end}}$.

Now we have start and end times. Action is segmented from $t_{\text{start}}$
to $t_{\text{end}}$. Segments obtained this way capture meaningful movements
like arm swings, knee raises, etc.

Figure 4: In a jumping jack instance, a change was detected in the left
shoulder joint angle in the upward motion.

###  III-B Fitting Motion Primitives

Over the window of frames that we segmented out, each keypoint has a sequence
of 2D coordinates. We fit a primitive shape like a line or circle to each
keypoint sequence. This approach falls under symbolic programming, and its
application to represent motion was introduced in [5]. In this manner, we
obtain a higher-level representation of the sub-action.

We use least-squares to fit a given set of points to line or
circle($(x_{i}-x_{c})^{2}+(y_{i}-y_{c})^{2}=r^{2}$). Further, polynomial
fitting with respect to time is done in order to get a smoother parametric
variation with time. x and y coordinates of a line are fit as :
$\text{line}_{x}=a_{x}t^{2}+b_{x}t+c_{x}$ and
$\text{line}_{y}=a_{y}t^{2}+b_{y}t+c_{y}$, where $t$ is the parametric time
and $a_{x},b_{x},c_{x},a_{y},b_{y},c_{y}$ are fitted parameters. Similarly,
for circle, a polynomial fit for angle($\theta$) is done. First $\theta$ value
for each time value is found using $x=x_{c}+r\cos{\theta}$ and
$y=y_{c}+r\sin{\theta}$. Then those $(t,\theta)$ value pairs are fitted as a
polynomial: $\theta=a_{c}t^{2}+b_{c}t+c_{c}$.

If the deviation of a set of coordinates from their mean is smaller than a
threshold, those are fitted as a stationary primitive and represented using
the mean point($\overline{x},\overline{y}$).

The sequence of coordinates for each keypoint is fitted to all three
primitives using the aforementioned methods, and the one with the minimum
error is chosen as the fit.

###  III-C Action Representation

Next, we compute the start, mid, and end coordinates of each keypoint of each
primitive by using the fitted parameters. fit(t=0), fit(t=T/2) and fit(t=T)
gives start, mid and end points respectively. Our final motion primitive
representation is $[\text{PRIM TYPE},[(Sx,Sy),(Mx,My),(Ex,Ey)]]$, where “PRIM
TYPE” is the fitted symbol. In addition, we determine the start, mid, and end
values of each joint angle and each bone length signal. i.e. we have
$\text{joint}_{i}$: start, mid, end and $\text{bone}_{i}$ : start, mid, end.

These higher-level representations are very effective and can be used for a
wide range of downstream tasks. It encapsulates the starting, middle, and
ending points of an action sequence and also the approximate trajectory shape
of each keypoint. We demonstrate the applicability of this representation in
repetition detection.

Using start, mid, and end values, we can figure out whether an angle is
increasing, or a part is moving left or right, etc. In this way, they capture
relevant information about the dynamics of that sub-action. Therefore, these
serve as features of good discriminative power for recognition tasks as well.

##  IV Repetition Detection

We use this action representation for class agnostic repetition detection,
which is one example of a downstream task.

We propose a consecutive match search algorithm for repetition detection.
Match is defined using an error metric between the start, mid, and end points
of two primitives. The error is given by
$|\overline{s}_{i}-\overline{s}_{j}|+|\overline{m}_{i}-\overline{m}_{j}|+|\overline{e}_{i}-\overline{e}_{j}|$,
where
$\overline{s}_{i},\overline{m}_{i},\overline{e}_{i},\overline{s}_{j},\overline{m}_{j},\overline{e}_{j}$
are start, mid and end coordinates of two primitives $i$ and $j$ respectively
of one keypoint, and the norm is L2 norm. This error is defined for all
keypoints, and the total error is taken as the sum of all. Two motion
primitives are said to match if the error is less than a threshold.

A buffer of the recent motion primitives is maintained. When a new motion
primitive is encountered, it is matched with the existing primitives in the
buffer. If a match is found, then consecutive matches are searched for by
drawing more primitives and checking matches with adjacent primitives. This
process is continued until a mismatch. A group of consecutive primitives that
are matched belong to the same loop. The buffer is updated by removing the
primitives which are matched. Counts get updated as the instances occur, as
opposed to giving a total count of a given video offline.

Figure 5: Once a match is found in the buffer, consecutive matches are
searched for by drawing more primitives, and this continues until the match
chain is broken. Groups of consecutive matches are loops.

##  V Experiments & Results

We demonstrate change detection and primitive fitting in 4 different exercise
examples(Fig. 6, Fig. 7, Fig. 8, Fig. 9). The segmentation window found using
change detection is indicated by the red arrows in the angle/length sequences.
Corresponding primitive fit(for each keypoint) is overlaid on the skeletons
below. The trajectories that appear curvy are those which were fitted as
circles, those which appear straight were fitted as lines and those which are
single points were fitted as stationary. It can be observed that majority of
the extracted primitives are meaningful. Fig. 9 shows an example of false
positives, where primitives are detected though not meaningful. From the
figure, it can be seen that the middle two primitives are segmented based on
change in elbow-angle signal, which is irrelevant in crunches. Ideally, we
want two primitives to be detected(the up motion and the down motion), but two
additional noisy primitives were added.

Figure 6: Change detection and primitive fitting on two instances of push-up
Figure 7: Change detection and primitive fitting on jumping jack Figure 8:
Change detection and primitive fitting on high knee raise Figure 9: Change
detection and primitive fitting on one instance of crunches (showing false
positives)

In [5], a loop detection method that uses similar start, mid, and end points
of extracted primitives is proposed. However, those primitives are extracted
offline from videos. We compare our method with that method in terms of per-
class percentage accuracy on repetition counting. Percentage accuracy is
defined as $(1-\frac{\left|\text{Predicted count}-\text{True
count}\right|}{\text{True count}}*100)$. The validation is done on moticon
dataset from [11] after manually labeling the number of counts of actions in
the videos. We validate the methods only on videos where the count is more
than 1, as the algorithms are based on finding matches and hence require at
least two instances. Table I shows the per-class percentage accuracy of our
online method and the aforementioned method. We managed to get either
comparable or better results in each class. Note that this was achieved by
operating online, which is the key contribution. The offline method failed to
do prediction on bulk of arm-cross-swings instances, and hence it was left
out. Meanwhile, the proposed method gives consistent results for all classes.

TABLE I: loop counting on moticon dataset Class | # vid | Online % acc. | Offline % acc.
---|---|---|---
Jumping jack | 89 | 73.9% | 78.4%
High knee march | 15 | 83.7% | 74.7%
Arm cross swings | 39 | 72.0% | -
All classes | 136 | 74.0% | 78.1%

Further, we collected more exercise videos and evaluated the method using
those. The data collection involved 3 subjects, each performing 4
exercises(push-ups, crunches, arm raises, arm rotations). Each participant
performed 5 instances of varying counts for each exercise, totaling 60 videos
of different counts for different exercises. We annotated each video with the
actual count and then calculated the percentage accuracy of the proposed
method on this data. The exercises were performed in different view
orientations w.r.t the camera in order to check the performance of the method
in such cases. Table II shows per class percentage accuracy on repetition
counting using the proposed method. It can be observed that the accuracy is
high for simple motion exercises like arm raises and push-ups, where the
movements remain mostly regular. There is a higher chance of irregularity in
the repeated motion for exercises like crunches and arm rotation. This factor
has an effect on the performance because the method relies on similarity
matching.

TABLE II: Loop counting on custom data Class | # videos | Percentage accuracy
---|---|---
Push ups | 15 | 81.5%
Crunches | 15 | 58.3%
Arm raises | 15 | 100%
Arm rotations | 15 | 61.8%
All classes | 60 | 76.0%

##  VI Discussion

The main advantage of the proposed method is that it operates online.
Furthermore, the choice of symbolic representation makes it compositional,
generalizable, and interpretable. The compositionality here is with respect to
body parts. A human activity can be composed of elementary actions of each
part. It can be argued that these elementary primitives are finite in number
as human motion is constrained. Then, any activity can be represented as a
composition of those finite primitives. This makes it generalizable. A part-
based representation also proves to be advantageous in scenarios where the
view is incomplete, such as when the camera is positioned such that just a
portion of the body is visible. Since we explicitly know which symbol a part
is fit to, the representation is interpretable. Also, this method doesn’t
require any data for training and instead relies on domain knowledge for good
choices for primitives.

Noisy involvement of other body parts when performing an exercise will
compromise the effectiveness of the method. For e.g., if a person is randomly
moving his arms while performing a knee raise, those hand motions might get
triggered as changes. In that case, the segmentation done is not meaningful
(Fig. 9). One way to address this drawback is to use a consensus of change
detections over multiple time series instead of considering only a single
series. Usage of adaptive thresholds for different signals may also be a
promising approach.

We use bone lengths along with joint angles because we are working with 2D
coordinates. We expect bone lengths to capture surrogates of some of the 3D
motions. If we use a 3D pose estimation method directly, then we need only use
angle sequences for change detection, and further we can directly fit 3D
curves as motion primitives. However, most of the 3D pose methods are noisy
along the depth dimension.

##  VII Conclusion

We address the problem of online action representation. We use a change
detection algorithm, which has been shown to give meaningful segments for
further symbolic fitting. Using a symbolic programming approach allows us to
use this method in sparse data scenarios. Additionally, the nature of
segmentation and primitive fitting captures the compositional aspect of
actions as a consequence. This representation can effectively work for many
downstream tasks, and more importantly, it can be operated online, which is
the primary gap that is addressed.

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# Scheme for parity-controlled multi-qubit gates with superconducting qubits

Kasper Sangild Christensen  Institute for physics and astronomy, Aarhus
University, 8000-DK Aarhus C, Denmark  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, 2100-DK Copenhagen, Denmark  Nikolaj
Thomas Zinner  Institute for physics and astronomy, Aarhus University, 8000-DK
Aarhus C, Denmark  Kvantify Aps, DK-2300 Copenhagen S, Denmark  Morten
Kjaergaard  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, 2100-DK Copenhagen, Denmark

###### Abstract

Multi-qubit parity measurements are at the core of many quantum error
correction schemes. Extracting multi-qubit parity information typically
involves using a sequence of multiple two-qubit gates. In this paper, we
propose a superconducting circuit device with native support for multi-qubit
_parity-controlled gates_ (PCG). These are gates that perform rotations on a
parity ancilla based on the multi-qubit parity operator of adjacent qubits,
and can be directly used to perform multi-qubit parity measurements. The
circuit consists of a set of concatenated Josephson ring modulators and
effectively realizes a set of transmon-like qubits with strong longitudinal
nearest-neighbor couplings. PCGs are implemented by applying microwave drives
to the parity ancilla at specific frequencies. We investigate the scheme’s
performance with numerical simulation using realistic parameter choices and
decoherence rates, and find that the device can perform four-qubit PCGs in
30$\mathrm{ns}$ with process fidelity surpassing $99\%$. Furthermore, we study
the effects of parameter disorder and spurious coupling between next-nearest
neighboring qubits. Our results indicate that this approach to realizing PCGs
constitute an interesting candidate for near-term quantum error correction
experiments.

##  I Introduction

The capabilities of quantum computers and simulators have been consistently
improving over the last decades, and experiments controlling a vast number of
quantum degrees of freedom have been performed on a variety of platforms [1,
2, 3, 4, 5, 6]. Despite this impressive progress, significant challenges
remain to be addressed in demonstrating large-scale error-corrected quantum
computing. Quantum error correction (QEC) uses many physical qubits to encode
a single logical qubit so that errors can be identified and corrected before
they spread and corrupt the computation [7]. A central requirement for many
QEC schemes is that operations on the physical qubits that make up the logical
qubit can be implemented with sufficiently low errors, the so-called
_threshold_ for a given QEC code [8, 9]. Depending on the specific code, the
threshold value varies [7, 10, 11, 12], and each code comes with its own set
of requirements on connectivity, gates required and strategies for ensuring
fault tolerance. Several experiments have recently demonstrated quantum error
correction using physical qubits beyond (or close to) the relevant thresholds
[13, 14, 15, 16].

One of the most promising protocols for quantum error correction is the
_surface codes_ , due to their lenient demand on threshold error rate and
nearest-neighbor connectivity [17, 18, 19, 20, 7, 21]. Here logical qubits are
encoded on a two-dimensional lattice of qubits, and errors can be detected by
measuring the parity of four adjacent qubits, i.e., extracting expectation
values of operators of the form $Z_{1}Z_{2}Z_{3}Z_{4}$ and
$X_{1}X_{2}X_{3}X_{4}$, where $X_{i}$ and $Z_{i}$ are Pauli operators acting
on qubit $i$. These operators are the _stabilizers_ of the quantum error
correcting code and are equivalent to finding the parity (in an appropriate
basis) of a collection of qubits [22]. Reliable measurement of multi-qubit
parity operators is thus a central feature of implementing fault-tolerant
quantum computing with qubits [23, 24]. For several quantum computing
platforms, the error rates of state-of-the-art operations are (or close to)
being limited by the decoherence of the qubits [25, 26, 27]. Thus, to further
reduce error rates towards implementing logical qubits, there are (at least)
two issues to address: increasing the coherence of the qubits and improving
the speed/efficiency of the operations required to extract the parity.

In the context of superconducting qubits, there are several avenues for
improving the coherence of the physical qubits. This includes advancements in
the materials and fabrication strategies used for superconducting qubits, as
showcased by recent experiments using, e.g., tantalum-based transmons [28,
29], by optimizing the physical layout of the qubits on-chip, or by novel
qubit circuit designs that come with error-suppressing properties out-of-the-
box [30], e.g. the 0-$\pi$ qubit [31, 32], the parity-protected qubit [33] or
the bifluxon [34]. On the other hand, there have also been significant
improvements in increasing the speed and reducing control errors on operations
with superconducting qubits [35]. This work includes implementing and
optimizing various types of two-qubit gates [36, 37, 38], new types of
coupling mechanisms including tunable couplers [39, 40, 41, 42], multi-qubit
gates [43], and new control schemes [25, 44, 45]. In the context of quantum
error correction, two-qubit gates are concatenated to implement the desired
parity operations, i.e., a four-qubit parity check (as in the surface code)
requires concatenating four two-qubit gates [14, 13].

In this paper, we propose an alternative scheme for efficiently measuring
multi-qubit parity operators with superconducting circuits by a new coupling
mechanism that enables multi-qubit _parity-controlled gates_ (PCG). The system
we propose consists of $N$ qubits and a _parity ancilla_ , which is assumed to
be initialized in its ground state $\ket{0_{P}}$. The key feature of our
proposal is the ability to perform fast operations on the parity ancilla
conditioned on the $N$-qubit parity operator $\bigotimes_{i}^{N}Z_{i}$, which
allows for direct measurements of the multi-qubit parity. The notion of PCGs
also has applications beyond efficient stabilizer readout for QEC, and in
particular, PCGs naturally arise in performing specific quantum simulation
tasks. For instance, multi-qubit parity operators arise in the Kogut-Susskind
Hamiltonian for $\mathbb{Z}_{2}$ lattice gauge theories [46, 47]. Another
application of PCGs is for implementing the quantum approximate optimization
algorithm [48], which can be used to find approximate solutions to
combinatorial optimization problems that are intractable on classical
hardware. Specifically, using parity compilation [49, 50], multi-qubit parity
plays a central role. This technique has the advantage of requiring only
nearest-neighbor connectivity with the added cost of encoding additional
constraints in multi-qubit parity operators. Parity-controlled gates may prove
useful for implementing precisely this type of dynamics.

The paper is structured as follows: In Section II, we review how PCGs can be
used to perform operations that are highly relevant in the context of quantum
error correction, simulation, and optimization. In Section III, we propose a
superconducting circuit device that implements PCGs efficiently. The circuit
consists of multiple concatenated Josephson ring modulators (JRM) [51, 52] and
effectively realizes a system of qubits with strong longitudinal couplings to
a parity ancilla. Specifically, we consider the two- and four-qubit cases and
use numerical simulation to gauge performance with realistic parameter
choices. In Section IV, we investigate the impact of parameter disorder on
device performance and the effect of unwanted next-nearest-neighbor couplings
between qubits.

##  II Parity controlled gates

In this section, we introduce the notion of _parity controlled gates_ , that
is, gates conditioned on whether a set of qubits have an even or odd number of
$\ket{1}$’s. We will denote the Hilbert space of the qubits as
$\mathcal{H}_{Q}=\bigotimes\mathcal{H}_{i}$, where $\mathcal{H}_{i}$ is the
Hilbert space of qubit $i$, and $\mathcal{H}_{P}$ is the Hilbert space of the
parity ancilla. The total subspace is then the tensor product
$\mathcal{H}_{Q}\otimes\mathcal{H}_{P}$. Our basic building block for multi-
qubit control is the _parity controlled flip_ (PCF) gate, which is
parameterized by a phase $\varphi$ and represented by the unitary

$U_{\text{PCF}}(\varphi)=P_{+}\otimes\mathds{1}_{P}+iP_{-}\otimes\left(\cos(\varphi)X_{P}+\sin(\varphi)Y_{P}\right).$
(1)

Here $\mathds{1}_{P},X_{P},Y_{P},Z_{P}$ are parity ancilla identity and Pauli
operators and

$P_{\pm}=\frac{1\pm\bigotimes_{i=1}^{N}Z_{i}}{2},$ (2)

are the projectors onto subspaces of different qubit parity. For the remainder
of this paper, we will omit explicit tensor products. As the name indicates,
the purpose of the PCF-gate is to flip the parity ancilla conditioned on the
parity of the adjacent qubits. The phase $\varphi$ dictates whether the flip
occurs by rotating the parity ancilla around the $X$\- or $Y$-axis on the
Bloch sphere.

A clear use case of the PCF-gate is a parity meter, which can be realized by
applying the gate followed by a subsequent measurement of the parity ancilla.
The complete parity measurement process can be described using the Kraus
operators

$\displaystyle K_{0}=P_{+}\ket{0_{P}}\bra{0_{P}},$ (3) $\displaystyle
K_{1}=P_{-}\ket{1_{P}}\bra{0_{P}}.$ (4)

Through this measurement, we obtain information on the parity of the adjacent
qubits. The PCF-gate can also implement a _parity controlled phase_ (PCP)
gate. This is done using two subsequent PCF gates, which ensures that the
parity ancilla ends its evolution in its ground state and, thus, that the
system state remains in the computational subspace. Projected on the
computational subspace, the total unitary reads

$\begin{split}U_{\text{PCP}}(\varphi)=&\bra{0_{P}}U_{\text{PCF}}(\pi)U_{\text{PCF}}(\varphi)\ket{0_{P}}\\\
=&P_{+}+e^{i\varphi}P_{-}=e^{i\frac{\varphi}{2}}e^{-i\frac{\varphi}{2}\prod_{i}Z_{i}},\end{split}$
(5)

which simulates a Hamiltonian on form $H\propto\prod_{i}Z_{i}$. It also
provides an alternative method for measuring the multi-qubit parity (see
Appendix A). As a demonstration of its utility, consider the case where the
gate is applied to the product state $\ket{+}^{\otimes
N}=2^{-1/2}\left(\ket{0}+\ket{1}\right)^{\otimes N}$ with $\varphi=\pi/2$. A
simple calculation shows that the final state becomes (ignoring an unimportant
overall phase factor)

$U_{\text{PCP}}\left(\frac{\pi}{2}\right)\ket{+}^{\otimes
N}=\frac{1}{\sqrt{2}}\left(\ket{+}^{\otimes N}-i\ket{-}^{\otimes N}\right),$
(6)

which is equivalent to a $N$-qubit GHZ-state up to single qubit gates. This
simple example elucidates the PCP-gates ability to generate a large amount of
entanglement with only a few physical gates.

In the following sections, we propose a superconducting circuit device capable
of performing the PCF-gate using flux tuning and microwave control pulses.
Furthermore, we demonstrate its performance through numerical simulations of
the two- and four-qubit cases.

##  III Physical implementation

Figure 1:  (a) (Top) Diagram of the proposed circuit. The system consists of
two capacitively shunted Josephson ring modulators that effectively realize
two transmon-like qubits with a strong Cross-Kerr type coupling to a third
ancillary parity qubit. (b) Qubit frequencies and their dependence on flux.
(c) Flux dependence of the couplings. Multi-qubit operations are done with
$g_{1}=g_{2}$.

We propose a realization using superconducting circuits. The circuit consists
of concatenated JRMs, a circuit element previously used in the context of
quantum amplifiers [51, 42], interleaved with large shunt capacitors. Our
scheme uses the JRM modes as the physical qubits and parity ancilla. For
example, the proposed circuit with two qubits can be seen in Fig. 1a.
Effectively, this realizes a system of qubits longitudinally connected to the
central parity ancilla (see Appendix B for a complete circuit analysis). The
coupling strengths and qubit frequencies depend on the flux applied through
the JRM loops, and by tuning this flux, the system can be brought to a point
where the coupling strengths are equal, i.e., $g_{i}=g$ for all
$i\in\\{1,...,N\\}$, as demonstrated for the two-qubit case in Fig. 1b and c.
By projecting the qubit and ancilla modes onto their two lowest states, we
obtain the effective hamiltonian

$H_{0}=-\frac{1}{2}\left(\omega_{P}+\Delta(Z_{1},...,Z_{N})\right)(Z_{P}-1)-\frac{1}{2}\sum_{i=1}^{N}\omega_{i}Z_{i},$
(7)

where $\omega_{i}$ is the frequency of the $i$’th qubit, $\omega_{P}$ is the
frequency of the parity ancilla and

$\Delta(Z_{1},...,Z_{N})=-\frac{g}{2}\sum_{i}^{N}Z_{i},$ (8)

is the detuning of the parity ancilla arising from the coupling to the
adjacent qubits.

Before we continue, it is informative to consider how this coupling scheme
compares to other examples commonplace in superconducting circuits. The modes
of the JRMs are akin to transmons [53], so it is natural to ask how our
proposal differs from the highly successful transmon-based architectures
commonly used in state-of-the-art quantum computing experiments. Typically,
transmons couple transversely [54, 55], i.e. through terms on the form
$(b_{1}b_{2}^{\dagger}+b_{1}^{\dagger}b_{2})$, where $b(b^{\dagger})$ is the
annihilation(creation) operator. This type of coupling can be achieved by
capacitively coupling the qubits or by coupling the qubits through a coupler,
e.g., a resonator [56, 57]. Furthermore, the coupling can be made flux tunable
and can be entirely switched off [39, 58]. The main difference between
traditional coupling schemes and ours is that our coupling is purely
longitudinal, i.e., on the form $b_{1}^{\dagger}b_{1}b_{2}^{\dagger}b_{2}$.
The longitudinal coupling axis allows us to have strong coupling without
worrying about leakage to higher levels since
$\bra{11}b_{1}^{\dagger}b_{1}b_{2}^{\dagger}b_{2}\ket{20}=0$. The coupling
arises because of the non-linearity of the Josephson inductance, the same
mechanism that gives rise to the anharmonicity of transmons. As such, it is
not surprising that the coupling strength is of the same order as the
anharmonicity, typically a few hundred $h$\mathrm{MHz}$$. This is
significantly larger than the typical coupling strength between transmons,
which is on the order of tens of $h$\mathrm{MHz}$$. The drawback of this large
longitudinal coupling is that it cannot be switched off, even though the
coupler is tunable. Our proposed workaround is to couple the qubits to an
ancillary mode, in this case, the parity ancilla.

In the rest of the analysis, we will work in the interaction picture given by
$\ket{\psi_{I}}=\exp(iH_{0}t)\ket{\psi_{S}}$. The coupling strength in our
scheme is large, comparable to the anharmonicity, which makes the control of
individual qubits more difficult. Thus, our first order of business is to
investigate how single qubit control can be implemented in the rotating frame.
Consider a microwave drive applied to qubit $i$ at frequency $\omega_{i}$. The
drive couples to the qubits charge degree of freedom $Q_{i}\propto Y_{i}$, and
in the Schrödinger picture, the driving Hamiltonian reads

$H_{d,S}(t)=\Omega(t)\sin\left(\omega_{i}t-\varphi\right)Y_{i},$ (9)

where $\Omega(t)$ is the pulse envelope and $\varphi$ is the phase shift of
the drive. In the interaction picture, this becomes

$H_{d,I}(t)=-\frac{\Omega(t)}{2}\left(e^{-i\left(\varphi-g(Z_{P}-1)\right)}\ket{0_{i}}\bra{1_{i}}+h.c.\right),$
(10)

after employing rotating wave approximation (RWA). In general, this operator
depends on the state of the parity ancilla, which complicates single qubit
control. This is the price of having strongly coupled qubits. In our scheme,
the problem is circumvented by only applying single qubit gates when the
parity ancilla is known to be in its ground state. In this case Eq. 10 reduces
to

$\bra{0_{P}}H_{d,I}(t)\ket{0_{P}}=-\frac{\Omega(t)}{2}\left(\cos(\varphi)X_{i}+\sin(\varphi)Y_{i}\right),$
(11)

which can perform arbitrary single-qubit rotations. In summary, the system
effectively decouples when the parity ancilla is in its ground state, allowing
for individual qubit control and readout.

In the rest of this section, we investigate how parity-controlled gates can be
performed by applying a microwave drive to the parity ancilla.

###  III.1 Two qubit case

Figure 2:  (a) Drive applied to the parity ancilla qubit. The drive is
resonant with the $\ket{010}\leftrightarrow\ket{011}$ and
$\ket{100}\leftrightarrow\ket{101}$ transitions, as indicated on the level
diagram. (b) Process fidelity error as a function of gate time between the
simulated process and the ideal PCF gate. The solid lines are simulations
including noise in the form of single qubit relaxation and dephasing with
rates $T_{\phi}=T_{1}=40$\mathrm{\SIUnitSymbolMicro s}$$. The advanced pulse
scheme includes corrections due to AC-Stark-induced energy shifts. (c) The
shape of the applied $\pi$-pulse applied to the parity ancilla qubit. (d)
Excited state probability of the parity ancilla during a single PCF gate for
different qubit states.

We now focus on the $N=2$ case shown in Fig. 1a. The coupling strength and, to
a lesser extent, the qubit frequencies depend on the externally applied flux,
as shown in Fig. 1b and c, and the system is operated at a bias flux so that
$g_{1}=g_{2}=g$. We apply a drive to the parity ancilla at the frequency
$\omega_{P}$, which is resonant with the parity ancilla when
$\Delta_{P}(Z_{1},Z_{2})=0$ corresponding to an odd parity configuration of
the qubits as shown on Fig. 2a. In the Schrödinger picture, the driving
Hamiltonian reads

$H_{d,S}(t)=\Omega(t)\sin\left(\omega_{P}t+\varphi\right)Y_{P}.$ (12)

Again we will switch to the interaction frame and eliminate counter-rotating
terms using RWA. The resulting Hamiltonian is

$H_{d,I}(t)=-\frac{\Omega(t)}{2}\left(e^{-i\left(\varphi+g\left(Z_{1}+Z_{2}\right)t/2\right)}\ket{0_{P}}\bra{1_{P}}+h.c.\right).$
(13)

Note that in the case of odd parity, we have
$(Z_{1}+Z_{2})\ket{\psi_{\text{odd}}}=0$. This time we will also assume that
$|\Omega/2|\ll|g|$ such that terms rotate with frequencies of $|g|$ or higher
can be eliminated. The driving Hamiltonian then becomes

$H_{d,I}(t)=-P_{-}\frac{\Omega(t)}{2}\left(\cos(\varphi)X_{P}+\sin(\varphi)Y_{P}\right),$
(14)

with $P_{\pm}$ as defined in Eq. 2. For a fixed phase $\varphi$ the time
evolution operator at time $T$ becomes

$\begin{split}U&=e^{-i\int_{0}^{T}H_{d,I}(t)dt}\\\
&=P_{+}+P_{-}e^{i\left(\cos(\varphi)X_{P}+\sin(\varphi)\right)\int_{0}^{T}\frac{\Omega(t)}{2}dt}.\end{split}$
(15)

After a $\pi$-pulse, where $\int_{0}^{T}\Omega(t)dt=\pi$, we recover the PCF
unitary from Eq. 1. Until now, we have assumed that off-resonant terms have no
effect and can be eliminated using RWA. A more realistic model can be built by
accounting for the AC-Stark shifts caused by the driving by using time-
averaging techniques [59]. The AC-stark-induced errors can be corrected by
slightly modifying the pulse (see Appendix C). Undoubtedly, more sophisticated
pulse engineering techniques can provide better results. Still, this
relatively simple correction is sufficient to reach fidelities above 99% in
our case with reasonable circuit parameters and decoherence times.

To gauge the feasibility of the proposed protocol, we have carried out
numerical simulations of the driven system by numerical integrating the
Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation [60]

$\dot{\rho}=-i\left[H_{d,I},\rho\right]+\sum_{i}\gamma_{i}\left(L_{i}^{\dagger}\rho
L_{i}-\frac{1}{2}\\{L_{i}^{\dagger}L_{i},\rho\\}\right),$ (16)

where $\rho$ is the system density matrix, and $L_{i}$ are the systems
Lindblad operators with rates $\gamma_{i}$. We note in passing that we use
SciPy [61] and NumPy [62] for all numerical experiments. Transmon-like modes
such as the ones used in this scheme are weakly anharmonic. As such, we model
the parity ancilla as a three-level system with anharmonicity $\alpha_{P}$ for
simulation. We have used parameters $g=-250h$\mathrm{MHz}$$,
$\alpha_{P}=-100h$\mathrm{MHz}$$ consistent with realistic hardware parameters
(see Appendix B). The system relaxes to the ground state for low temperatures,
so we model relaxation through the annihilation operator
$L_{\text{rel},i}=b_{i}$, and dephasing using the number operators of each
mode $L_{\phi,i}=b_{i}^{\dagger}b_{i}$. We use relaxation time
$T_{1}=40$\mathrm{\SIUnitSymbolMicro s}$$ and pure dephasing time
$T_{\phi}=40$\mathrm{\SIUnitSymbolMicro s}$$. We use a Tukey window function
for the pulse shape and set $\varphi=0$ as shown in Fig. 2b. As expected, the
parity ancilla is only excited when the qubits are in state $\ket{01}$ or
$\ket{10}$ as shown in Fig. 2d. By subsequently measuring the parity ancilla,
we realize the parity measurement operation given in Eq. 4. As a measure of
process quality, we use the process fidelity [63] defined as

$F_{\text{pro}}\left(\mathcal{E}_{\text{sim}},\mathcal{E}_{\text{ideal}}\right)=F\left(\rho_{\mathcal{E},\text{sim}},\rho_{\mathcal{E},\text{ideal}}\right),$
(17)

here $\mathcal{E}_{\text{sim}}$($\mathcal{E}_{\text{ideal}}$) is the simulated
(ideal) process, $\rho_{\mathcal{E}}$ is the Choi density matrix
representation [64, 65] of the process $\mathcal{E}$ and
$F(\rho,\sigma)=\left(\text{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^{2}$
is the fidelity [66]. The simulations show process fidelities above 99% can be
achieved in roughly 25$\mathrm{ns}$ (see Fig. 2c). For low gate times, the
error is dominated by coherent errors, such as leakage and dynamical phase
error, while relaxation and incoherent dephasing limit fidelity for longer
gate times.

###  III.2 Four qubit case

Figure 3:  (a) Level diagram of a four-qubit plaquette. A two-tone drive is
applied to the parity ancilla, as shown. The drive is resonant if the qubits
have an odd number of excitations. (b) Plaquette configuration with four
qubits and a parity ancilla. The four qubits couple longitudinally to the
central parity ancilla with equal strength. (c) Process fidelity as a function
of gate time for the four qubit PCF gate. The inset shows the pulse shape used
for the (basic) simulation. The advanced simulation includes corrections to
AC-Stark-induced errors. The solid lines are simulated with decoherence in the
form of single qubit dephasing and relaxation. The rates used are
$T_{1}=T_{\phi}=40$\mathrm{\SIUnitSymbolMicro s}$$. (d) Simulation of the GHZ-
factory circuit. The inset shows the ideal circuit and the pulse sequence used
to realize it. The plot shows the state fidelity of the output state with the
target GHZ state.

An important feature of our scheme is its extension to two dimensions, which
we do by connecting two additional qubits to the parity ancilla, as shown in
Fig. 3b. As in the two-qubit case, multi-qubit operations are performed by
applying a drive to the parity ancilla. In this case, we use a two-tone drive
at frequencies $\omega_{\pm}=\omega_{P}\pm g$, as shown on Fig. 3a. In the
Schrödinger picture, the driving Hamiltonian becomes

$H_{d,S}(t)=\Omega(t)\left(\sin\left(\omega_{+}t-\varphi\right)+\sin\left(\omega_{-}t-\varphi\right)\right)Y_{P}.$
(18)

Note that one of the tones is resonant in the case where
$\Delta_{P}(Z_{1},...,Z_{4})=-g$ and the other is when
$\Delta_{P}(Z_{1},...,Z_{4})=g$. Eq. 8 and Fig. 2a shows that these two cases
correspond to cases where there are either 1 or 3 excitations amongst the
qubits, i.e., the states with odd parity. Switching to the interaction picture
and employing RWA, we obtain

$\begin{split}H_{d,I}(t)=&-\frac{\Omega(t)}{2}e^{-i\left(\varphi+gt\left(1+\sum_{i=1}^{4}Z_{i}/2\right)\right)}\ket{0_{P}}\bra{1_{P}}\\\
&-\frac{\Omega(t)}{2}e^{-i\left(\varphi-
gt\left(1-\sum_{i=1}^{4}Z_{i}/2\right)\right)}\ket{0_{P}}\bra{1_{P}}\\\
&+h.c.\\\
\approx&-P_{-}\frac{\Omega(t)}{2}\left(\cos(\varphi)X_{P}+\sin(\varphi)Y_{P},\right)\end{split}$
(19)

where we have employed RWA a second time in the last line to eliminate terms
rotating with frequencies $\sim g$. Eq. 19 is the four-qubit version of Eq.
14, and the PCF unitary can be realized using a single $\pi$-pulse following
the same arguments as before.

As with the two-qubit case, we have conducted numerical simulations of the
PCF-gate. We use a coupling strength of $g=-200h$\mathrm{GHz}$$ consistent
with realistic hardware parameters (see Appendix B), and an anharmonicity of
$-100h$\mathrm{GHz}$$. Fig. 3c shows the process fidelity of the simulated
PCF-gate with and without AC-stark compensation. We achieve process fidelities
above 99% in roughly 30$\mathrm{ns}$. Furthermore, we have carried out
simulations of the PCP-gate. Specifically, we simulate its performance as a
GHZ-state factory, as shown on Fig. 3d, where we apply the simulated PCP gate
to create the GHZ-like state described in Eq. 6. Here we can initialize the
GHZ state with 99% fidelity in roughly $75$\mathrm{ns}$$.

##  IV Practical considerations

Figure 4:  (a) Sketch illustrating the proposed physical layout of the
circuit. Using cross-over junctions, as shown on the inset, we eliminate the
need for air bridges. (b) A plot of the coupling strengths illustrating the
effects of parameter disorder. The mean and standard deviation of 200
realizations of the circuit with randomized Josephson energies have been
calculated. Each Josephson energy is picked from a normal distribution with
mean at the ideal energy and a standard deviation of 10%. (c) Cancellation of
the parasitic coupling between qubits through off-resonant driving of the
parity ancilla.

We propose a physical layout using crossover junctions as indicated on Fig.
4a. This compact design eliminates the need for air bridges and may prove
advantageous for scalability. Furthermore, the design is easily adapted to two
dimensions.

So far, our analysis has assumed that junctions belonging to the same JRM have
identical $E_{J}$’s. In practice, fabrication imperfections will almost always
allow for some degree of parameter disorder among the junctions. The disorder
results in transverse couplings between qubits and parity ancilla. The
coupling is relatively small (on average $\sim 50h$\mathrm{MHz}$$) compared to
the qubit-parity ancilla detuning and can therefore be treated as a
perturbation. Ultimately, the parameter disorder amounts to a shift in the
coupling strength between qubits and parity ancilla as shown on Fig. 4b.
Importantly, the shift is small enough that the system can be tuned such that
$g_{1}=g_{2}$, even in the presence of disorder.

Another issue common to qubit couplers in superconducting circuits is the
unwanted coupling between non-neighboring qubits. This issue is also present
in our scheme, although its origin differs from typical superconducting
couplers. The coupling arises from higher order correction to the Hamiltonian
and is on the form $g_{12}Z_{1}Z_{2}$ where $g_{12}\propto
g_{1}g_{2}/(2\omega_{P})$. Although the parasitic coupling is suppressed by
$|g/\omega_{P}|\ll 1$, it is still significant due to the large value of $g$.
This coupling can be compensated for by applying an off-resonant drive to the
parity ancilla, as demonstrated on Fig. 4c where a drive at frequency
$\omega_{P}-g/2$ has been applied. By tuning the amplitude, the parasitic
coupling is canceled by the driving-induced AC-Stark shift. Similar schemes
have been proposed to compensate for unwanted couplings in other coupler
designs [67]. Additionally, we note that the parasitic coupling scales as
$g^{2}$, whereas the gate time scale as $g^{-1}$. Thus, by reducing the
coupling strength between qubits and parity ancilla, the parasitic coupling is
suppressed while only increasing gate times moderately.

##  V Conclusion

In this paper, we have presented a novel scheme for implementing parity-
controlled gates in superconducting circuits. The scheme relies on a strong
longitudinal coupling between qubits and an ancillary parity qubit. We have
performed numerical simulations of the scheme with realistic circuit and
decoherence parameters for the two- and four-qubit cases. In both cases, we
achieve process fidelities above 99% in the order of tens of nanoseconds.
Furthermore, we examine the effects of parameter disorder in the junction
energies and find that this only slightly modifies the qubit-ancilla coupling.
Our simulations indicate that the tunability of the proposed device can
compensate for these effects. We have also demonstrated how unwanted parasitic
coupling between qubits may be canceled by applying an off-resonant drive to
the parity ancilla.

All in all, the proposed scheme natively implements features that are highly
desirable in many areas of quantum computing and presents an exciting test-bed
for surface codes and specific quantum simulation and optimization schemes.

##  VI Acknowledgements

The authors thank András Gyenis for enlightening discussions on the proposed
device. KSC and NTZ acknowledge support from the Independent Research Fund
Denmark. MK gratefully acknowledges support from the Danish National Research
Foundation, the Danish Council for Independent Research — Natural Sciences,
Villum Foundation (grant 37467) through a Villum Young Investigator grant.

##  Appendix A Parity readout using the PCP-gate

In the main text, we proposed performing a PCF gate followed by measurement of
the parity ancilla to measure the multi-qubit parity operator
$\prod_{i}Z_{i}$. In this section, we propose an alternate scheme that instead
involves measuring one of the qubits, say qubit 1, for concreteness. This may
prove advantageous since the system effectively decouples while the parity
ancilla is in its ground state, and it is straightforward to perform qubit
measurements in this case. This alternate readout protocol involves a single
PCP gate sandwiched between single-qubit gates on qubit 1. Specifically:

$U_{1}=H_{1}S_{1}^{\dagger}U_{PCP}\left(\frac{\pi}{4}\right)H_{1},$ (20)

where $H_{1}=(X_{1}+Z_{1})/\sqrt{2}$ is the Hadamard gate and
$S_{1}=\text{diag}(1,i)$ the phase gate. It can be seen that this unitary
allows for parity measurements by considering
$U_{1}^{\dagger}Z_{1}U_{1}=\prod_{i}Z_{i}$. Thus, after a single $U_{1}$ gate,
the parity information is encoded in qubit one and can be measured. The
measurement may change the state of qubit 1; however, this can be undone by
subsequently applying $U_{1}^{\dagger}$.

##  Appendix B Circuit

Figure 5: (a) Frequencies of qubits and parity ancilla computed using naive
RWA, second order perturbation theory using Schrieffer wolf transformation
(SW) and exact diagonalization. (b) Comparison of coupling strengths between
RWA, perturbative and exact diagonalization approaches.

The circuit we propose is shown in Fig. 1a. The circuit consists of three
large capacitors coupled through four Josephson junctions in a ring
configuration. This section shows how our system reduces to a chain of three
modes with strong, independently tunable longitudinal couplings between
neighboring qubits. For convenience, we will work in units where $2e=\hbar=1$.

The first step is to define the coordinates in terms of the node fluxes of the
circuit. We write this transformation as

$\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\ \phi_{P}\\\ \theta_{1}\\\
\theta_{2}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}1&-1&0&0&0&0\\\
0&0&1&-1&0&0\\\ 0&0&0&0&1&-1\\\ 1&1&0&0&-1&-1\\\
0&0&1&1&-1&-1\end{pmatrix}\begin{pmatrix}\varphi_{1}\\\ \varphi_{2}\\\
\varphi_{3}\\\ \varphi_{4}\\\ \varphi_{5}\\\ \varphi_{6}\end{pmatrix},$ (21)

where the $\varphi_{i}$’s are the node fluxes, and $\phi_{i}$ and $\theta_{i}$
constitute the chosen coordinates. In these coordinates, the potential energy
takes the form

$\begin{split}V=\sum_{i=1}^{2}-4E_{Ji}&\left(\cos{\phi_{i}}\cos{\phi_{P}}\cos{\theta_{i}}\cos{\frac{\phi_{xi}}{4}}\right.\\\
&-\left.\sin{\phi_{i}}\sin{\phi_{P}}\sin{\theta_{i}}\sin{\frac{\phi_{xi}}{4}}\right),\end{split}$
(22)

where we have defined the normalized external flux
$\phi_{x1}=2\pi\Phi_{x}/\Phi_{0}$. The kinetic energy is

$K=E_{CP}q_{P}^{2}+\sum_{i=1}^{2}\left(E_{Ci}q_{i}^{2}+\tilde{E}_{Ci}Q_{i}^{2}\right),$
(23)

where we have defined the canonical conjugate momenta
$\\{\phi_{i},q_{i}\\}=\\{\theta_{i},Q_{i}\\}=\\{\phi_{P},q_{P}\\}=1$, where
$\\{\cdot,\cdot\\}$ denotes the Poisson bracket. The charging energies are
given by

$\displaystyle E_{Ci}=$ $\displaystyle\frac{1}{8(C_{Qi}+C_{Ji})},$ (24)
$\displaystyle E_{CP}=$ $\displaystyle\frac{1}{8(C_{P}+C_{J1}+C_{J2})},$ (25)
$\displaystyle\tilde{E}_{Ci}=$ $\displaystyle\frac{1}{8C_{Ji}},$ (26)

where $i\in\\{1,2\\}$. The device is operated near $\phi_{xi}=0$, meaning that
the potential has a minimum at $\phi_{i}=\theta_{i}=\phi_{P}=0$. Therefore we
assume that the wavefunction is reasonably localized around this point, and we
can approximate the potential by its Taylor expansion. The system then reduces
to a system of coupled anharmonic oscillators and is quantized by introducing
the ladder operators $b_{\pm}=\left(\zeta^{-1/2}\phi\mp
i\zeta^{1/2}q\right)/\sqrt{2}$, where $\zeta=\sqrt{E_{C}/(2E_{J})}$ is the
modes characteristic impedance. After expanding, the Hamiltonian becomes

$\begin{split}H=&E_{CP}q_{P}^{2}+4\sum_{i=1}^{2}c_{i}E_{Ji}\left(\frac{\phi_{P}^{2}}{2}-\frac{\phi_{P}^{4}}{24}\right)\\\
&+\sum_{i=1}^{2}\left(E_{Ci}q_{i}^{2}+4c_{i}E_{Ji}\left(\frac{\phi_{i}^{2}}{2}-\frac{\phi_{i}^{4}}{24}\right)\right)\\\
&+\sum_{i=1}^{2}\left(\tilde{E}_{Ci}Q_{i}^{2}+4c_{i}E_{Ji}\left(\frac{\theta_{i}^{2}}{2}-\frac{\theta_{i}^{4}}{24}\right)\right)\\\
&-\sum_{i=1}^{2}c_{i}E_{Ji}\left(\phi_{i}^{2}\phi_{P}^{2}+\phi_{i}^{2}\theta_{i}^{2}+\phi_{P}^{2}\theta_{i}^{2}\right)\\\
&+4\sum_{i=1}^{2}s_{i}E_{Ji}\phi_{i}\theta_{i}\phi_{P},\end{split}$ (28)

where we have introduced $c_{i}=\cos\left(\phi_{xi}/4\right)$ and
$s_{i}=\sin\left(\phi_{xi}/4\right)$ to ease notation. In the following, we
will work with a basis of eigenstates for the quadratic Hamiltonian and treat
off-diagonal, which consists only of cubic and quartic terms, as a
perturbation. We denote the diagonal part of the Hamiltonian as
$H_{D}=\sum_{\alpha}E_{\alpha}\ket{E_{\alpha}}\bra{E_{\alpha}}$ and the off-
diagonal as $H_{X}$. Projecting onto the relevant subspace, where the parity
ancilla and qubits are restricted to their two lowest eigenstates and the
$\theta$-modes are in their ground states, yields a Hamiltonian on the form

$\begin{split}H_{D}=-\frac{1}{2}\left(\omega_{P}-\frac{g_{1}}{2}Z_{1}-\frac{g_{2}}{2}Z_{2}\right)(Z_{P}-1)-\frac{1}{2}\sum_{i=1}^{2}\omega_{i}Z_{i},\end{split}$
(29)

with the unperturbed frequencies $\omega_{P}$ and $\omega_{i}$, and
unperturbed coupling strengths $g_{i}$. The primary mechanism behind the
strong longitudinal coupling is the quartic interaction since the diagonal
part of $\phi_{P}^{2}\phi_{i}^{2}\propto(b_{P}^{\dagger}b_{P}+1)(b_{\phi
i}^{\dagger}b_{\phi i}+1)$. However, the off-resonant interactions of $H_{X}$
introduce energy shifts which can be approximated using standard perturbation
techniques. To second order in $H_{X}$, the corrections are

$E^{(2)}_{\alpha}=\sum_{\beta\neq\alpha}\frac{\left|\bra{E_{\alpha}}H_{X}\ket{E_{\beta}}\right|^{2}}{E_{\alpha}-E_{\beta}},$
(30)

where the $\beta$ runs over all states, not only those in the relevant
subspace. These energy shifts modify the coupling strengths and mode
frequencies. This is shown on Fig. 5, which demonstrates a significant
discrepancy between the naive RWA approach, where the off-diagonal terms are
eliminated, and the exact diagonalization of the system. The second-order
perturbative approach is much closer to the exact diagonalization and captures
the coupling strength’s flux dependence. Furthermore, the quartic interaction
terms introduce a small parasitic coupling between the qubits, which can be
canceled by applying an off-resonant driving, as described in the main text.

In our numerical simulations, we are only interested in the lowest $N$ levels
of each mode, where $N=3$ for the parity ancilla, $N=2$ for the qubits, and
$N=1$ for the $\theta$ modes. We compute the energy shifts of each level
according to Eq. 30 using the lowest $N+2$ levels of each mode, thus including
all shifts due to interactions with higher levels. We wish to choose the
external flux biases so that $g_{i}=g$ for all $i$ by numerically minimizing
$\text{Var}(g_{i})$. Although it is possible to diagonalize the system for two
qubits numerically, this becomes unfeasible in the four-qubit case for the
hardware used for this work since the total dimension of the Hilbert space
becomes $5\cdot 4^{4}\cdot 3^{4}\sim 10^{5}$ dimensional. As such, we rely on
the perturbative approach for diagonalization.

For the simulations of the two-qubit system, we have used the capacitances
$C_{1}=C_{2}=22$\mathrm{fF}$$, $C_{P}=19$\mathrm{fF}$$ and
$C_{J1}=C_{J2}=4$\mathrm{fF}$$, together with junction energies
$E_{J1}=10h$\mathrm{GHz}$$ and $E_{J2}=11h$\mathrm{GHz}$$. This give an
optimal flux bias of $\Phi_{x1}/\Phi_{0}=\phi_{x}/(2\pi)\approx 0.06$ and
$\Phi_{x2}=0$ with a coupling strength of roughly $280h$\mathrm{MHz}$$, and
frequencies ranging from $7.0-7.4h$\mathrm{GHz}$$ for the qubit and
$9.8h$\mathrm{GHz}$$ for the parity ancilla, as seen on Fig. 1c. For the four-
qubit case, we use Josephson energies $E_{J1}=E_{J2}=13h$\mathrm{GHz}$$ and
$E_{J3}=E_{J4}=12h$\mathrm{GHz}$$ with capacitances
$C_{1}=C_{3}=C_{P}=13$\mathrm{fF}$$, $C_{2}=C_{4}=15$\mathrm{fF}$$ and
$C_{Ji}=4$\mathrm{fF}$$ for $i=1,...,4$. In this case, the qubit frequencies
are all between $8.8h$\mathrm{GHz}$$ and $9.6h$\mathrm{GHz}$$, while the
parity ancillas frequency is $15h$\mathrm{GHz}$$. The coupling strength
between qubits and parity ancilla is $214h$\mathrm{MHz}$$. In both the two-
and four-qubit cases, the theta modes have frequencies between
$19.4h$\mathrm{GHz}$$ and $20.6h$\mathrm{GHz}$$.

##  Appendix C Accounting for AC-stark shifts

The rotating frame Hamiltonian given by Eq. 14 only holds in the limit where
$|\Omega/g|\rightarrow 0$. Furthermore, when driving the
$\ket{0}\xrightarrow{}\ket{1}$ transition of a weakly anharmonic qubit, such
as the transmon-like qubits of our system, one also drives the
$\ket{1}\xrightarrow{}\ket{2}$ transition off-resonantly. The off-resonant
driving of specific transitions leads to AC-stark shifts of the levels,
ultimately leading to phase errors if not accounted for.

In this section, we demonstrate a simple technique for suppressing this type
of error using time-averaging techniques. Intuitively, this can be considered
a more sophisticated version of RWA. To gauge the effect of leakage, we model
the parity ancilla as an anharmonic oscillator with anharmonicity $\alpha$.
This means modifying Eq. 7:

$\begin{split}H_{0}=&-\sum_{i}\omega_{i}Z_{i}/2\\\
&+\left(\omega_{P}+\Delta_{P}\right)b^{\dagger}_{P}b_{P}+\frac{\alpha}{2}b^{\dagger}_{P}b_{P}(b^{\dagger}_{P}b_{P}-1),\end{split}$
(31)

where $b^{\dagger}_{P}$ and $b_{P}$ are the parity ancilla creation and
annihilation operators. Since there are only longitudinal couplings and no
driving on the qubits, it is not necessary to include the higher energy levels
of the qubit degrees of freedom. We show that the AC-stark-induced errors can
be accounted for by changing the phase of the driving pulse. It is convenient
to define the projection onto the relevant subspace

$P=P_{+}\ket{0_{P}}\bra{0_{P}}+P_{-}\left(\ket{0_{P}}\bra{0_{P}}_{P}+\ket{1_{P}}\bra{1_{P}}\right).$
(32)

If the qubits are in an even parity state, this projects onto the ground state
of the parity ancilla, and if the qubits are in an odd parity state, it
projects onto the two lowest states instead.

###  C.1 Two qubit case

We apply a drive to the parity ancilla coupled to two qubits at frequency
$\omega_{d}$. In the Schrödinger picture, the Hamiltonian reads

$H_{S}=H_{0}+H_{d},$ (33)

is the free Hamiltonian of the system and

$H_{d}=i\Omega(t)\sin(\omega_{d}t-\phi(t))(b^{\dagger}-b).$ (34)

We now switch to a frame defined by

$\ket{\psi(t)}_{R}=V(t)\ket{\psi(t)}_{S},$ (35)

with $V(t)=\exp\left(i\int_{0}^{t}H_{R}(t)dt\right)$ and

$H_{R}(t)=H_{0}+H_{ac}(t),$ (36)

where

$H_{ac}=\delta_{P}(t)b^{\dagger}_{P}b_{P}-\frac{\delta_{Q}(t)}{2}\sum_{i}Z_{i},$
(37)

is introduced to cancel AC-stark shifts induced by the driving term.
Crucially, the rotating frame Hamiltonian depends only on diagonal single
qubit operators, and we can therefore switch between this frame and the lab
frame using only virtual rotation. It is also worth noting that in the
relevant subspace, we have

$PH_{ac}P=\delta_{Q}\left(P_{2}-P_{0}\right)\ket{0_{P}}\bra{0_{P}}+\delta_{P}P_{-}\ket{1_{P}}\bra{1_{P}},$
(38)

where $P_{n}$ is the projector onto the qubit subspace of Hamming weight $n$.
We now choose the driving frequency as $\omega_{d}=\omega_{P}$ and the phase
as $\phi(t)=\varphi-\int_{0}^{t}\delta_{P}(t^{\prime})dt^{\prime}$. Switching
to the rotating frame and eliminating terms rotating with frequencies of
$2|\omega_{d}|$, we find

$\begin{split}H_{d,I}=&VH_{d}V^{\dagger}-H_{ac}\\\
=&\frac{\Omega(t)}{2}\left(b^{\dagger}(t)+b(t)\right)-H_{ac},\end{split}$ (39)

where for compactness, we have defined

$\displaystyle
b(t)=e^{-i\left(\Delta_{P}t+\varphi\right)}\left(\ket{0_{P}}\bra{1_{P}}+\sqrt{2}e^{-i\alpha
t}\ket{1_{P}}\bra{2_{P}}\right).$ (40)

Computing the effective time-averaged Hamiltonian [59] and projecting onto the
relevant subspace yields

$\begin{split}PH_{eff}P=&H_{XY}(t,\varphi)\\\
&+\left(\delta_{Q}(t)+\frac{\Omega(t)^{2}}{4g}\right)\left(P_{0}-P_{2}\right)\ket{0_{P}}\bra{0_{P}}\\\
&-\left(\frac{\Omega(t)^{2}}{2\alpha}+\delta_{P}(t)\right)P_{-}\ket{1_{P}}\bra{1_{P}}.\end{split}$
(41)

By setting

$\displaystyle\delta_{Q}(t)=$ $\displaystyle-\frac{\Omega(t)^{2}}{4g},$ (42)
$\displaystyle\delta_{P}=$ $\displaystyle-\frac{\Omega(t)^{2}}{2\alpha}.$ (43)

we thus recover Eq. 14.

###  C.2 Four qubit case

In the four-qubit case, we apply a two-tone drive to the parity ancilla on the
form

$H_{d}=i\Omega(t)\left(\sin(\omega_{3}t-\phi_{3}(t))+\sin(\omega_{1}t-\phi_{1}(t))\right)(b^{\dagger}-b),$
(44)

where we choose $\omega_{d}=\omega_{m}+\delta_{m}$ and $\omega_{s}=g+g_{ac}$,
where $|g_{ac}|\ll|g|$ is a small correction to the effective coupling
strengths that arise due to AC-stark shifts of off-resonantly driven
transitions. Switching to the rotating frame defined as before but with

$H_{ac}(t)=\left(\delta_{P,1}(t)P_{1}+\delta_{P,3}(t)P_{3}\right)b^{\dagger}_{P}b_{P}-\frac{\delta_{Q}(t)}{2}\sum_{i}Z_{i}.$
(45)

Again we switch to a frame rotating with $H_{R}(t)=H_{0}+H_{ac}(t)$, this time
choosing
$\phi_{i}(t)=\varphi_{i}-\int_{0}^{t}\delta_{P,i}(t^{\prime})dt^{\prime}$. For
the moment, $\varphi_{i}$ is an undetermined constant phase. In the
interaction picture, the Hamiltonian becomes identical to Eq. 39 with

$\displaystyle b(t)=2e^{-i\left(\Delta
t+\varphi\right)}\cos(gt)\left(\ket{0}\bra{1}_{P}+\sqrt{2}e^{-i\alpha
t}\ket{1}\bra{2}_{P}\right).$ (46)

Here we have implicitly used that $\exp\left(i(\Delta\pm
g)t+\int_{0}^{t}H_{ac}\right)\sim\exp\left(i(\Delta\pm g)t\right)$ for the
purposes of time-averaging, when acting on the subspace where $\Delta\pm g\neq
0$. In the following, we will drop the explicit time dependence for
compactness. The effective Hamiltonian projected onto the relevant subspace
becomes

$\begin{split}PH_{eff}P=&-P_{1}\frac{\Omega}{2}\left(\cos(\varphi_{-})X_{P}+\sin(\varphi_{-})Y_{P}\right)\\\
&-P_{3}\frac{\Omega}{2}\left(\cos(\varphi_{+})X_{P}+\sin(\varphi_{+})Y_{P}\right)\\\
&+\sum_{n=0}^{4}\delta_{n,0}P_{n}\ket{0_{P}}\bra{0_{P}}\\\
&+\sum_{m\in\\{1,3\\}}\delta_{m,1}P_{m}\ket{1_{P}}\bra{1_{P}}\end{split}$ (47)

with

$\displaystyle\delta_{0,0}=$
$\displaystyle-\delta_{4,0}=2\delta_{Q}+\frac{\Omega^{2}}{3g}$ (48)
$\displaystyle\delta_{1,0}=$
$\displaystyle-\delta_{3,0}=\delta_{Q}+\frac{\Omega^{2}}{8g}$ (49)
$\displaystyle\delta_{1,1}=$
$\displaystyle\delta_{Q}-\delta_{P,1}-\frac{\Omega^{2}}{8g}-\frac{\Omega^{2}}{2\alpha}-\frac{\Omega^{2}}{2\alpha-4g}$
(50) $\displaystyle\delta_{3,1}=$
$\displaystyle-\delta_{Q}-\delta_{P,3}+\frac{\Omega^{2}}{8g}-\frac{\Omega^{2}}{2\alpha}-\frac{\Omega^{2}}{2\alpha+4g}.$
(51)

By setting

$\displaystyle\delta_{Q}=$ $\displaystyle-\frac{\Omega^{2}}{6g}$ (52)
$\displaystyle\delta_{P,1}=$
$\displaystyle-\frac{\Omega^{2}}{4g}-\frac{\Omega^{2}}{2\alpha}-\frac{\Omega^{2}}{2\alpha-4g}$
(53) $\displaystyle\delta_{P,3}=$
$\displaystyle\delta_{Q}+\frac{\Omega^{2}}{8g}-\frac{\Omega^{2}}{2\alpha}$
(54)

the effective Hamiltonian reads

$\begin{split}PH_{eff}P=&-P_{1}\frac{\Omega}{2}\left(\cos(\varphi_{-})X_{P}+\sin(\varphi_{-})Y_{P}+\frac{\Omega}{12g}\right)\\\
&-P_{3}\frac{\Omega}{2}\left(\cos(\varphi_{+})X_{P}+\sin(\varphi_{+})Y_{P}-\frac{\Omega}{12g}\right)\end{split}$
(55)

After a $\pi$-pulse, we achieve a slight variation on the PCF gate:

$\begin{split}U_{PCF}(\varphi)=&P_{+}+ie^{i\chi}P_{1}\left(\cos(\varphi_{1})X_{P}+\sin(\varphi_{1})Y_{P}\right)\\\
&+ie^{-i\chi}P_{3}\left(\cos(\varphi_{3})X_{P}+\sin(\varphi_{3})Y_{P}\right),\end{split}$
(56)

where $\chi=\int_{0}^{T}\Omega(\tau)^{2}/(24g)d\tau$. For parity measurements,
this unitary serves precisely the same function as that of Eq. 1 since the
phases become irrelevant once the parity ancilla is measured. The PCP may be
realized using a two-pulse sequence, the first with
$\varphi_{1}=\varphi_{3}=\pi$, and the second with $\varphi_{1}=\varphi-2\chi$
and $\varphi_{3}=\varphi+2\chi$.

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