Input: How many experiments were demonstrated to test the capabilities of the controller?

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Paper Info

Title: Force Feedback Control For Dexterous Robotic Hands Using Conditional Postural Synergies
Publish Date: Unkown
Author List: Dimitrios Dimou, José Santos-Victor, Plinio Moreno

Figure

Fig. 1.Example of modeling the contacts and friction during manipulation.
Fig. 2. Schematic representation of the proposed force controller.The input is the state (GRASP or RELEASE) and the force readings.Based on that the grasp size is adjusted by a value C and is given to the posture mapping function along with the desired grasp type.A finger configuration is then generated and commanded to the robot.
Fig. 3. Our control algorithm in Python-like pseudocode.
Fig. 4. Our first experiment.The robot picks up a bottle, transports it, and places down on the desk.In the bottom part of the figure, you can see the control signals during this task.
Fig. 5.The household objects used in our experiments.
Under the pictures of the execution you can see the signals recorded by the controller: the average normal force applied by all fingers (blue line), the thresholds f threshold high n .(purple dashed line) and f threshold low n.(yellow dashed line), the average tangential force (green), and the grasp size used in each time-step (red).The task is divided four stages: 1) (red part) the initial grasp of the object, in this stage the force controller closes the grasp until the applied normal
Fig.6.In the upper row of images, you can see our second experiment.The robot picks up the chips can, rotates it 90 degrees, and places back down.In the middle row, for our third experiment, the robot picks up the chips can, rotates it 90 degrees, and hands it over to a person.In the bottom row, for our forth experiment, the robot picks up a foam brick, rotates it 180 degrees, and hands it over to a person, using a pinch grasp.

abstract

We present a force feedback controller for a dexterous robotic hand equipped with force sensors on its fingertips. Our controller uses the conditional postural synergies framework to generate the grasp postures, i.e. the finger configuration of the robot, at each time step based on forces measured on the robot's fingertips.
Using this framework we are able to control the hand during different grasp types using only one variable, the grasp size, which we define as the distance between the tip of the thumb and the index finger. Instead of controlling the finger limbs independently, our controller generates control signals for all the hand joints in a (lowdimensional) shared space (i.e.
synergy space). In addition, our approach is modular, which allows to execute various types of precision grips, by changing the synergy space according to the type of grasp. We show that our controller is able to lift objects of various weights and materials, adjust the grasp configuration during changes in the object's weight, and perform object placements and object handovers.

INTRODUCTION

To perform complex manipulation tasks in unstructured environments, humans use tactile feedback from their fingers. This feedback is provided by tactile afferents located in the skin of the hand. Particularly, for handling small objects with precise movements, the afferents located in the fingertips are used, which have high density and adapt fast to pressure changes .
These afferents provide information about the characteristics of the exerted contact forces, such as the magnitude and the direction. For anthropomorphic robots to be able to perform dexterous tasks similar force feedback signals must be used to alleviate problems arising from uncertainty in measurements, and handle external perturbations.
For example, using open-loop position control to lift a heavy object may fail due to slip without any feedback mechanism to provide tactile information. Previous works have used tactile sensors to design force controllers that use slip prediction to update the desired normal forces applied by the fingertips.
The slip predictors are based on machine learning models such as neural networks and random forests to classify multi-modal signals from a tactile sensor. In all previous works, each finger was separately controlled by an independent force controller. In addition, they required labeled data to train the slip predictors and because each finger is controlled independently is not obvious how to implement different anthropomorphic grasp types.
In this work we develop a force controller that takes as input the force readings of the fingertips and computes the grasp size which is then used along with a grasp type label to generate a grasp posture with the desired characteristics. To avoid slippage the desired normal contact force is calculated to be proportional to the tangential contact forces.
The applied normal force is then controlled using the size of the grasp as a control variable. Larger grasp sizes mean less force is applied to the object. So the grasp size is calculated from the error between the desired normal force and the actual measured normal force. The grasp size is then given to the posture sampler that generates a grasp posture, i.e. the finger joint angles.
The posture sampler is modeled with a conditional Variational Auto-Encoder (cVAE) based on the framework proposed in . With this framework we abstract away the low-level control of the fingers and generate hand postures based on high-level properties such as the type and the size of the grasp. So it works as a mapping function that takes as input a low-dimensional vector and the grasp type and size as conditional variables and maps them to a set of joint angles.
We show that with our controller we can control a dexterous robotic hand to lift objects of different weights using three precision grasps. Our controller is also able to compensate and retain a stable grasp during changes in the objects' weight, for example when filling up a cup or emptying it. In addition we show how with the addition of the hand pose information we can use the controller to calculate if the tangential force is due to gravity or due to a support surface and use this information to perform handovers and place down objects on surfaces.
We perform several real-world experiments with a dexterous robotic hand to showcase the capabilities of our controller and support our design choices. To sum up our main contributions are • We develop a controller for a dexterous robotic hand that uses force feedback and the conditional synergies framework to perform dexterous manipulation tasks.
• We show that with our controller we can easily use different precision grasp types, by changing only the grasp type variable which is given to the grasp posture mapping function. • We demonstrate by incorporating information about the world pose of the hand we can use our controller to perform additional tasks such as placing down and handing over objects.
Roboticists have looked for inspiration in humans for developing methods for complex object manipulation . Neuroscientists have studied for a long time the processes that allow humans to use tactile feedback to perform complex manipulation tasks. Humans tend to adjust the grip force according to the object's weight, its friction and they use a safety margin to account for uncertainties .
To gather information about the tactile states they use multiple afferents that are located in the skin of the fingers . There are different afferents in different parts of the hand depending on their usage, e.g. fast adapting afferents in the fingertips for precise manipulation. Based on signals from these afferents, humans encode simple contact events into action phases, such as grasping, lifting or releasing, which they combine in order to perform more complex and long-horizon manipulation tasks .
In robotics tactile sensors have been used for object stabilization and slip prediction in a variety of settings. For example, in , a compliant anthropomorphic prosthetic hand was controlled using force sensing to maintain object stability and avoid slip. In , they develop a control approach that uses integrated force and spatial tactile signals to avoid slip with unknown objects in real world settings.
In , , grasp quality metrics are computed based on the tactile feedback from the robots fingertips. In these works, simple two or three fingered grippers were considered for simple grasping tasks. Force control with anthropomorphic robotic hands has also been explored in more recent works. In , they employ three slip prediction methods to estimate when slip starts and based on the force signals at that moment they calculate the friction coefficient value.
Based on the calculated friction coefficient, they design a force controller that independently controls each finger to achieve a desired normal force. The desired normal contact force is set to be proportional to the tangential contact force and a safety margin based on the evidence found in . In , they train a random forest to classify the contact states into the classes: no contact, contact, slip.
Based on this classification signal, when slip is detected they increase the desired normal contact force to avoid it. In they train a recurrent neural network to estimate slip and the object material from the readings of a Biotac sensor. The force controller is increasing the desired normal contact force when slip is detected.
All these works , , use tactile feedback sensors to predict slip. They collect labeled data, on which they train their models. This approach is based on complex and expensive tactile sensors, and the process of collecting data is cumbersome. In addition, the data do not cover all possible hand poses, which would be impractical.
In contrast, in our work we do not rely on slip prediction, we avoid slip by defining a tangential force gain and a safety margin that work for a large number of objects. Furthermore, instead of independently controlling each finger we use a synergistic framework to generate grasp postures, that is conditioned on two variables: the grasp type and the grasp size.
This way, instead of controlling the values of each joint of each finger, we control only the two conditional variables greatly simplifying the control pipeline. This also, gives us the ability to use different grasp types in our manipulation tasks by changing only the grasp type variable. In also a synergistic framework was used to prevent an object from slipping from a humanoid hand, but they modeled only one synergy for a tripod grasp and they used the forces on the robotic arm as feedback, while we use force feedback from the fingertips.
Our control algorithm could also be applied to different hands as it does not depend on the hands configuration. Finally, in previous approaches only lifting tasks had been considered. In our work we demonstrate that our approach can be used to perform more complex tasks, such as placing objects on surfaces and performing handovers, which was not done in previous works.
Our goal in this work is to design a control algorithm for an anthropomorphic robotic hand to perform dexterous manipulation skills such as lifting and placing down objects. Our control algorithm will use tactile feedback from the force sensors on the fingertips of the hand to decide the forces that need to be applied to the object in each step of the task.
Given the desired forces to be applied, the size of the grasp will be computed. Given the grasp size and a desired grasp type, the posture generator will generate a grasp posture, i.e. the hand configuration, such that the force constraints are satisfied. To model the contacts and friction we use Coulombs' law, which states that in order to avoid slip, the normal contact force f n to the contact surface of an object, times the fiction coefficient µ, has to be larger than the tangential force f t :
µf n ≥ f t You can see an example in Figure , where an object is pressed against a wall by an applied normal force f n , and we have the tangential force f t = mg due to gravity. In order for the object to remain stable we need to apply a normal force: where µ is the friction coefficient between the object and the wall.
In the case of a dexterous hand manipulating an object, we want the normal forces applied by all fingers to be greater than the tangential force divided by the friction coefficient of the materials of the object and the fingertip. Since it is hard to accurately compute the friction coefficient between all possible object materials previous works have used multi-modal tactile sensors like the BioTac sensor, which provides information about the pressure, skin deformation, and temperature, to predict slip and based on that signal to increase the applied normal force.
In our work we use the FTS3 sensors which is a low-cost sensor that measures the 3D force applied in each fingertip. In addition, previous works gathered labeled datasets in order to train their slip prediction models which is time-consuming and limits the possible orientations of the hand, because gathering labeled data for all possible orientations is impractical.
To overcome this we experimentally selected the parameters that determine the value of the applied normal force such that we avoid slip for all objects in our dataset, from the lightest to the heaviest. In order to guarantee contact between the fingertip and the object, in the beginning of the grasping phase, we use an offset f of f set n as the minimum normal force applied by each finger.
In they also suggest that humans use an additional safety margin which is proportional to the tangential force, f margin n ∝ f t . So the final desired normal contact force becomes: where G is the gain that includes the friction coefficient and the additional safety margin. To alleviate the effects of noise in the sensors, the running average of the measured normal force f n and tangential force f t is used, as a low pass filter.
So for each force measurement we have the following relation: where α ∈ (0, 1) is a parameter that determines how much new measurements affect the value, and is experimentally selected. Given the measured normal force f n from the fingertip sensors we can compute the error f err n = f des n − f n . We use this error signal to control the grasp size variable g size , that we use as a conditional variable in our posture mapping function.
The grasp size represents the distance between the thumb and the index finger in a grasp posture. So a smaller grasp size will result in a tighter grasp and greater normal force applied to the surface of the object. We use a linear controller for the grasp size variable that is implemented as follows: where K is a parameter that controls the rate of decrease of the grasp size, and is experimentally selected.
So when the error between the desired normal force and the actual normal force is large the grasp size decreases so tighter grasp postures are generated in order to apply more normal force. In practice, in order to avoid oscillations in the grasp size we use the desired normal force as a high threshold that we want the measured normal force to be below:
If the normal force is below that threshold the grasp size does not change even if there are small oscillations in the measured tangential and normal forces. Also, in order to avoid the hand applying too much force that damages the hardware or the object we use a low threshold, that is: where w threshold is the width of the threshold in mN .
If the measured normal force is below the grasp size increases in order to apply less force. So the final grasp size variable for grasping is calculated as follows: where This is similar to the deadband control method , where instead of having a fixed reference point, an operating range is set. If the response is in this range, the controller does not exert any correction.
In our case, the operating range changes according to the force signals from the robot's fingertips. The grasp posture mapping function is based on the conditional postural synergies model presented in . It uses a conditional Variational Auto-Encoder model to generate grasps postures conditioned on additional variables such as the grasp size.
In this work we augment this model to also generate grasp postures conditioned on the grasp type. The model is trained on a set of labeled grasp samples acquired by teleoperating a robotic hand using a data-glove. Using this model we are able to abstract away the low-level control of each joint of each finger and generate grasps based on more general characteristics such as the type and the size of the grasp.
In this way we can control all the fingers jointly by a single value, the grasp size, thus greatly reducing the control parameters. In addition we are able to use the same control algorithm for different precision grasp types, by changing the grasp type conditional variable. Finally, we can modify our controller to release objects instead of grasping them.
Given the pose of the hand in the world coordinate frame, which we can acquire from the robotic arm that is attached to, we can use the forward kinematics of the hand to compute the poses of each fingertip. Then using the force readings of each fingertip we can calculate the global direction of the net tangential force.
If the angle between the direction of the net tangential force and the direction of gravity is less than 90 degrees, i.e. the net tangential force's direction is towards the ground, we assume that the tangential force is due to gravity pulling the object, so the force controller tries to grasp it. If the angle is more than 90 degrees, i.e. the net tangential force's direction is upward, it means that something is pushing (or pulling) the object upward, in which case we assume that the object is touching on a support surface or someone is pulling the object so the controller increases the grasp size given to the posture mapping function proportionally to the normal force measured thus slowly releasing the object.
Opening the grasp is done by controlling the grasp size variable as follows: That way we can place objects on surfaces but also perform robot to human handovers, where the robot holds the object and the human grasps the object and slightly pushes or pulls it up, signaling to the robot that there is a support surface.
The robot then slowly releases the object by opening its grasp. We showcase these scenarios in the experiments' section. Based on these observations, we present our force controller in Figure . The hand starts in an open pre-grasp position, a latent point is sampled from the prior distribution of the posture mapping function, and given the desired grasp type and the grasp size a grasp posture, i.e. the joint angles of the fingers, is sampled.
The initial grasp size is set to the maximum value, and when the force controller comes into effect and depending on the state of the system and the forces on the fingertips grasp size changes by some value C, according to equations 1,2, until the desired normal force is achieved. To choose between grasping or releasing an object we use a finite state machine formulation.
When the hand reaches the desired grasp pose, which we assume is provided, the GRASP state is activated, in which the controller tries to grasp the object. When the controller detects that the tangential force applied to the object is coming from a support surface the state changes to the RELEASE state, in which the controller releases the object by opening the grasp.
You can see the full algorithm in Python-like pseudocode in Figure . To summarize, the advantages of our controller compared with previous approaches are threefold: 1) instead of controlling each joint of each finger of the hand we use only two variables, the grasp size and the grasp type, which allows us to perform multiple grasp types by changing only one variable while the grasp size variable is common among all grasp types, that greatly reduces the complexity of the control process compared to independently controlling a 21 DoF hand to perform different grasp types, 2) we do not rely on slip prediction for controlling the desired normal force, which involves gathering labeled data and works only for the hand poses in the training dataset, and 3) we can use our controller to also release objects instead of only grasping them.

Experimental Set-up.

For our experiments we used the Seed Robotics RH8D Hand , which is a robotic hand with 7 DoFs. The hand is equipped with the FTS-3 force sensors in each fingertip, which are high resolution tactile sensors that provide the 3D force applied in each fingertip. The sensor provides data at a rate of 50Hz. For the experiments the hand was mounted on a Kinova Gen3 7DoF robot.
To train the posture mapping function we used the CyberGlove to teleoperate the hand and collect 468 grasps belonging to three precision grasp  types: tripod, pinch, lateral tripod. The architecture of the cVAE model was the same as in , with the addition of the grasp type as a conditional variable, which was one-hot encoded.
We used 10 household objects shown in Figure . With the heaviest object weighing 380g and the lightest 1g. During the experiments the trajectories of the arm were prerecorded, while the hand was controlled online by our control algorithm.

Parameter tuning.

To select the values of the parameters in our controllers we conducted preliminary experiments where we tested lifting and releasing several objects, with different physical properties. To select the value of the normal offset force f of f set n , we used an empty plastic cup as our test object, and we choose a value such that the fingers do not deform the cup.
The final value of the parameter was set to -50 mN. To select the values of the gain G and the rate of decrease K, of the grasp size, we experimented with the heaviest object in our dataset, which is the mustard bottle and weighs 380g. The gain G was set to 2.0 such that the desired normal force would be enough to hold the object.
The rate of change of the grasp size was set to 100.0, based on the operating frequency of the force sensor and the range of values of the tangential force. For the tangential force averaging process we used a parameter value of α t = 0.7, because we want the controller to be sensitive to fast changes in its value, that can arise for example during lifting an object.
For the normal force averaging process we used a parameter value of α n = 0.5, as we do not want it to be affected by noise that could make the controller overconfident.

Experiments.

To explore the capabilities of our controller, we demonstrate five experiments of increasing complexity: 1) we picked and placed a bottle using a tripod grasp, 2) we picked, rotated and placed a chips can on a box using a tripod grasp, 3) we picked, rotated and handed over the chips can to a person using a tripod grasp, 4) we picked, rotated and handed over a brown foam brick to a person using a pinch grasp, 5) a person handed over a plastic cup to the robot, filled it with coins to increase its weight, and the robot then handed it back to the person using a tripod grasp.
You can see the execution of the first experiment in  In the middle row, for our third experiment, the robot picks up the chips can, rotates it 90 degrees, and hands it over to a person. In the bottom row, for our forth experiment, the robot picks up a foam brick, rotates it 180 degrees, and hands it over to a person, using a pinch grasp.
Fig. . In our fifth experiment, a person hands over an empty plastic cup to the robot, throws coins in it to increase its weight while the robot adjusts its grip to stabilize the object, and then hand overs the cup back to the person. force is below the offset f of f set n , 2) (green part) the robot lifts the object, as it tries to lift the tangential force increases, increasing the threshold, so the grasp size decreases to apply more normal force, 3) (orange part) the robot transports the object, you can see, in point A in the Figure, a perturbation in the tangential force when the robot begins to move, the controller responds by decreasing the grasp thus stabilizing the object, and 4) (blue part) the robot enters the releasing phase, where it lowers the arm until it detects that the tangential force is due to a support surface, then it stops lowering the arm and increases the grasp size slowly releasing the object.
In point B in the Figure, you can see that there is noise in the tangential force, due to the arm moving to place the object on the table, that is also reflected in the desired normal force. Because we use the desired normal force as a threshold and not as a reference signal this noise is not manifested in the control of the grasp size.
You can see the execution of the second experiment in the upper part of Figure . This experiment demonstrates the ability of the controller to handle arbitrary hand poses. The experiment is divided in four parts: 1) the robot enters the GRASP phase and the force controller generates grasps to achieve a normal contact force below the f of f set n threshold, 2) the robot lifts the object and adjusts the grasp size to avoid the object falling, 3) the hand rotates to place the chips can on the horizontal position, and 4) the robot enters the RELEASE phase, and the arm lowers until the object touches the box, when the hand detects the supporting surface, it starts to slowly release the object.
You can see the execution of the third experiment in the middle part of Figure . This experiment demonstrates the ability of the controller to perform robot to human handovers. The experiment is divided in four parts: 1) the robot enters the GRASP phase and the force controller generates grasps to achieve a normal contact force below the f of f set n threshold, 2) the robot lifts the object and adjusts the grasp size to avoid the object falling, 3) the hand rotates to place the chips can on the vertical position, and 4) the robot enters the RELEASE phase, the arm stays still, the human grasps the object from the bottom and slightly pushes it up, the hand then detects that there is a supporting surface and starts to slowly release the object.
You can see the execution of the fourth experiment in the bottom part of Figure . This experiment is similar to previous one, but the grasp type that the robot uses is a pinch grasp, that involves only the thumb and the index finger. To perform this we only had to alter the grasp type conditional variable that was given to the posture mapping function.
You can see the execution of the fifth experiment in the bottom part of Figure . In the first part (blue) of the experiment the robot closes its grasp, by reducing the grasp size, until the normal force is below the force offset. In the next three parts (pink, green, red) the person throws coins in the cup to increase its weight.
You can see in the signal plots that each time coins are added the tangential force decreases so the normal force threshold decreases too. The grasp sizes then decreases as well in order to apply more normal force. This experiment demonstrates the ability of the controller to handle perturbations in the weight of the object during grasping.

CONCLUSION

In summary, we presented a controller that uses force feedback integrated with conditional synergies to control a dexterous robotic hand to grasp and release objects. We demonstrated that our controller can lift objects of different weights and materials while avoiding slip, react online when the weight of the object changes, place them down on surfaces, and hand them over to humans.
In addition, the control architecture is modular, so the synergy grasp mapping component can be easily changed in order to control several precision grasp types. However, our experiments also revealed various limitations of our controller. For example our method fails to stabilize the object when rotational slip occurs.
In addition hardware limitations such as, slow update rates and noise in the force measurements can create problems that result in the object falling. In future work we plan to incorporate additional sensing modalities, such as vision to alleviate some of these issues.

Answer: 5.


Input: What size chains were used in the benchmarking?

Context:
Paper Info

Title: Compressed quantum error mitigation
Publish Date: 10 May 2023
Author List: Maurits Tepaske (from Physikalisches Institut, Universität Bonn), David Luitz (from Physikalisches Institut, Universität Bonn)

Figure

FIG.3.The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and time t, for the infinite temperature initial state, for a denoised second-order Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2.We consider evolution times t = 0.5, 1, ..., 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise with p = 0.01.
FIG. 4. The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right).The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0.0046.The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars.It is evident that the denoiser recovers all the noiseless eigenvalues from the noisy circuit.
FIG. 2. The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right).The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0.036.The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars.It is clear that the denoiser recovers with high accuracy the noiseless eigenvalues from the noisy circuit.
FIG. 3. The half-chain channel entanglement entropy S at different two-qubit depolarizing noise strengths p, for a secondorder Trotter supercircuit with Mtrot = 16 and t = 2, for a M = 4 denoiser.The Trotter circuit is for a Heisenberg model with PBC of size L = 6.The different curves correspond to the different supercircuits, i.e. the noisy supercircuit, the denoiser, the corresponding denoised supercircuit, and the noiseless variant.
FIG. 4. The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and stacked time t, for the infinite temperature initial state, for a denoised secondorder Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2.It is optimized at t = 2 and stacked up to ten times.The calculations are for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarization with p = 0.01.The denoiser is affected by the same noise.
FIG.6.The distribution of the ZZ angle α of M = 2 denoisers (top panels) and M = 8 denoisers (bottom panels), with the lightest color corresponding to the denoiser for the Trotter supercircuit with t = 0.5, and the darkest color with t = 5.As usual, we consider the Heisenberg model on a periodic chain, and second-order Trotter supercircuits with depths Mtrot = 8, 16, 32, 64, which together with the denoiser is affected by a two-qubit depolarizing noise with p = 0.01.The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively.
FIG. 7. The sampling overhead γ of the optimized denoisers from Fig. 2 of the main text, with denoiser depths M = 1, 2, 4, 6, 8 and Trotter depths Mtrot = 8, 16, 32, 64 at times t = 0.5, 1, ..., 5, for the Heisenberg model on a chain with PBC affected by two-qubit depolarizing noise with p = 0.01.The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively.
FIG.8.The domain wall magnetization Z dw after evolving a periodic density wall |dw |dw * with the denoised second-order Trotter supercircuits D C from Fig.2of the main text.These supercircuits have various Trotter depths Mtrot = 8, 16, 32, 64, denoiser depths M = 1, 2, 4, 6, 8, and evolution times t = 0.5, 1, ..., 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise of strength p = 0.01.The denoiser is affected by the same noise.The non-denoised results are labelled with M = 0 and the noiseless results with p = 0.The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively.We see that the denoiser allows us to recover the noiseless behavior.

abstract

We introduce a quantum error mitigation technique based on probabilistic error cancellation to eliminate errors which have accumulated during the application of a quantum circuit. Our approach is based on applying an optimal "denoiser" after the action of a noisy circuit and can be performed with an arbitrary number of extra gates.
The denoiser is given by an ensemble of circuits distributed with a quasiprobability distribution. For a simple noise model, we show that efficient, local denoisers can be found, and we demonstrate their effectiveness for the digital quantum simulation of the time evolution of simple spin chains. Introduction.
-Quantum information processing has been theoretically shown to hold great promises, and quantum algorithms were developed which can in principle achieve an exponential speed-up over their classical counterparts, both for general purpose computing and quantum simulation . However, present day quantum computing prototypes still suffer from significant noise processes which hinder the execution of many potentially groundbreaking quantum algorithms .
Nontrivial quantum algorithms typically require large sequences of quantum gates, each of which introduces dissipation and hence an overall loss of coherence, eventually rendering the results useless. Until quantum error correction becomes practical, quantum error mitigation seems to be more feasible to increase the accuracy of expectation values.
Here the goal is to induce the (partial) cancellation of errors that stem from noisy quantum gates by extending the circuit corresponding to the desired algorithm with an ensemble of gates , sampled from a quasiprobability distribution. The traditional way to accomplish this is with the gatewise method from , where noise is mitigated by inverting the noise channel of each gate separately, i.e. the cancellation of errors is performed for each gate on its own.
Here the local noise channel is approximated in a way such that it can be easily inverted analytically, e.g. using Pauli twirling . Gates are then sampled from the inverted noise channel by interpreting it as a quasiprobability distribution. Because in this gate-wise approach every noisy gate has to be modified separately, the sign problem is exponentially large in the number of gates, limiting the practicality of the mitigation.
The success of the gate-wise approach resulted in a large body of work concerning these methods , including extensions for simultaneous mitigation of multiple gates by Pauli-twirling entire layers or variationally constructing a mitigating matrix product operator . In principle, errors during the execution of a circuit can propagate and accumulate.
These propagated errors * david.luitz@uni-bonn.de ≈ C

C

FIG. 1. An example of the quantum error mitigation procedure used in this work for the time evolution of the wave function of a spin chain. The ideal second-order Trotter supercircuit C of depth Mtrot = 1 (light blue) is approximated by applying a denoiser D of depth M = 1 (red) to the noisy Trotter supercircuit C (dark blue).
Because the denoiser is applied after fully executing the noisy Trotter supercircuit, it represents an approximate inverse of the global noise channel with a precision tunable by the depth of the denoiser. can potentially blow up and lead to large errors for the circuit as a whole . Here we introduce a mitigation technique that takes into account the propagation of errors, can be performed with a tunable number of extra gates, and works for non-Clifford local noise channels since the inversion of the accumulated global noise channel is implicit.
We first execute the targeted noisy circuit completely, letting the noise propagate and accumulate, and only afterwards we apply an extra random circuit sampled from a quasiprobability distribution. We call the corresponding ensemble of random circuits a denoiser, and we construct it such that upon averaging the accumulated errors cancel.
Essentially, the denoiser inverts a global noise channel. Since we will construct it as a local brickwall circuit, following the classical preprocessing approach from , we call this compressed quantum error mitigation. Method. -Due to the inevitable coupling of a quantum processor to its environment, every qubit operation is affected by noise.
Therefore, the simplest technique to minimize the impact of the resulting noise is to minimize the number of operations when performing a quantum algorithm. In we showed that many-body time evolution operators can be efficiently compressed into brick-wall circuits with high fidelity per gate. In this Letter, we consider the noise explicitly by treating quantum operations as (generally non-unitary) quantum channels, corresponding to completely positive and trace preserving (CPTP) maps .
For example, instead of a noiseless two-qubit gate G, which acts on a quantum state |ρ in superoperator form as G|ρ = G⊗G * |ρ , we get the noisy channel G = N G, where the noise channel N implements the two-qubit noise . These channels are used to construct a "supercircuit" C = N G i=1 Gi , consisting of N G channels, which is affected by multi-qubit accumulated noise.
This supercircuit encodes an ensemble of circuits . For simplicity, we assume that the noisy channels Gi in each half brickwall layer are lattice inversion and translation invariant, such that we can construct a denoiser with these properties, limiting the number of variational parameters. The purpose of quantum error mitigation is to modify the ensemble of circuits described by C in a way that we can use it to obtain the noiseless expectation values.
In superoperator language, we do this by following the supercircuit C with a denoiser supercircuit D, such that D C is as close to the noiseless supercircuit C = C ⊗ C * as possible. Here C is the target unitary circuit. Because the noise channel N is non-unitary, hence making the supercircuit C non-unitary, we need to use a non-unitary denoiser to retrieve the unitary C.
We illustrate the mitigation procedure in Fig. , where a denoiser with one layer is used to mitigate errors for a second-order Trotter supercircuit with one layer. This circuit architecture is commonly used to simulate the time evolution of a quantum many-body system, until some time t, with controllable precision , and we will use it to benchmark the denoiser.
In practice, we cannot directly implement a supercircuit, and so we have to utilize its interpretation as an ensemble of circuits. Essentially, after executing a shot of the noisy circuit we sample the denoiser and apply it. The goal is to construct the denoiser in a way that averaging over many of its samples cancels the accumulated errors and gives us a good approximation of the noiseless expectation values.
It should be noted that our approach requires more gate applications on the quantum processor than with the gate-wise scheme, since there each sample from the mitigation quasiprobability distribution can be absorbed into the original circuit, whereas our approach increases the circuit depth. We take this into account by imposing the same noise on the denoiser.
Furthermore, within our scheme, the dimensionality of the quasiprobabilistic mitigating ensemble can be controlled, in contrast to the gate-wise approach where it is equal to the gate count. To facilitate the stochastic interpretation we parameterize each two-qubit denoiser channel G i as a sum of CPTP maps, such that we can sample the terms in this sum and execute the sampled gate on the quantum processor.
Concretely, we use a trace preserv-ing sum of a unitary and a non-unitary channel. For the unitary part we take a two-qubit unitary channel U( φ i ) = U ( φ i ) ⊗ U * ( φ i ), with U ( φ i ) a two-qubit unitary gate parameterized by φ i . For this we take the two-qubit ZZ rotation exp(−iα(σ z ⊗ σ z )) with angle α, which can be obtained from native gates on current hardware , and dress it with four general one-qubit unitaries, only two of which are independent if we want a circuit that is space inversion symmetric around every bond.
The resulting gate has 7 real parameters φ i . For the non-unitary part, which is essential because D has to cancel the non-unitary accumulated noise to obtain the noiseless unitary circuit, we use a general onequbit measurement followed by conditional preparation channel M( , with V a general one-qubit unitary and each κ i a 3-dimensional vector, resulting in a real 9-dimensional ζ i .
This yields the two-qubit correlated measurement M( With these parts we construct the parameterization with coefficients η i ∈ R that satisfy η 0 + η 1 = 1 because G i is trace preserving. Note that here the tensor product symbol corresponds to combining two one-qubit channels to make a two-qubit channel, whereas in most of the paper it is used to link the column and row indices of a density matrix.
We construct the denoiser from the noisy channels Gi = N G i . With this parameterization one denoiser channel has 17 independent real parameters, such that a denoiser of depth M , i.e. consisting of M brickwall layers, has 34M real parameters (we use one unique channel per half brickwall layer). For reference, a general channel has 544M parameters.
To determine the mitigated expectation values we use the full expression where |ρ 0 is the initial state and |1 is the vectorized identity operator on the full Hilbert space. To evaluate this on a quantum processor, we use the stochastic interpretation of (1) to resample . In particular, from each channel (1) we get a unitary with probability p 0 = |η 0 |/γ and a measurement followed by conditional preparation with probability p 1 = |η 1 |/γ.
Here γ = |η 0 | + |η 1 | is the sampling overhead, which characterizes the magnitude of the sign problem from negative η i . For quasiprobability distributions, i.e. with γ > 1, every denoiser sample has an extra sign sgn(η) = N G g=1 sgn(η g ), 2. The normalized distance between the denoised Trotter supercircuit D C and the noiseless Trotter supercircuit C (top panels), at evolution times t = 0.5, 1, ..., 5, and the twopoint z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t (bottom panels), for the infinite temperature initial state.
We consider denoisers with depths M = 1, 2, 4, 6, 8 and second-order Trotter circuits with depths Mtrot = 16, 32, 64. In the top panels we use a Heisenberg chain with L = 8, and in the bottom panels with L = 14, both with periodic boundary conditions. All gates are affected by two-qubit depolarizing noise with p = 0.01.
The non-denoised results are labelled with M = 0, and the noiseless values with p = 0. where sgn(η g ) is the sign of the sampled coefficient of the gth channel. γ = 1 means that all signs are positive. Observables Ô p=0 for the noiseless circuit are then approximated by resampling the observables from the denoiser ensemble
where γ = N G g=1 γ g is the overall sampling overhead, with γ g the overhead of the gth gate. Clearly, a large γ implies a large variance of Ô p=0 for a given number of samples, with accurate estimation requiring the cancellation of large signed terms. The number of samples required to resolve this cancellation of signs is bounded by Hoeffding's inequality, which states that a sufficient number of samples to estimate Ô p=0 with error δ at probability 1 − ω is bounded by (2γ 2 /δ 2 ) ln(2/ω) .
Since γ scales exponentially in γ g , it is clear that a denoiser with large M and γ 1 will require many samples. We observed that decompositions with γ > 1 are crucial for an accurate denoiser. Restricting to γ = 1 leads to large infidelity and no improvement upon increasing the number of terms in or the depth M of the denoiser.
Simply put, probabilistic error cancellation of gate noise introduces a sign problem and it is crucial to find optimal parameterizations (1) which minimize γ to make the approach scalable. This issue arises in all high performance error mitigation schemes , because the inverse of a physical noise channel is unphysical and cannot be represented as a positive sum over CPTP maps.
This is clearly visible in the spectra of the denoiser, which lies outside the unit circle (cf. Fig. ). This makes the tunability of the number of gates in each denoiser sample a crucial ingredient, which allows control over the sign problem, because we can freely choose the η i in . For the parametrization (1) of denoiser channels, we try to find a set of parameters for error mitigation by minimizing the normalized Frobenius distance between the noiseless and denoised supercircuits
which bounds the distance of output density matrices and becomes zero for perfect denoising. We carry out the minimization of on a classical processor, using gradient descent with the differential programming algorithm from . Instead of explicitly calculating the accumulated global noise channel and subsequently inverting it, we approximate the noiseless supercircuit C with the denoised supercircuit D C, effectively yielding a circuit representation D of the inverse noise channel.
Results. -To benchmark the denoiser we apply it to the second-order Trotter circuits of the spin-1/2 Heisenberg chain with periodic boundary conditions (PBC) where is the Pauli algebra acting on the local Hilbert space of site i. A second-order Trotter circuit for evolution time t with depth M trot consists of M trot − 1 half brickwall layers with time step t/M trot and two layers with half time step .
We consider circuits that are affected by uniform depolarizing noise with probability p for simplicity, but our approach can be used for any non-Clifford noise. The two-qubit noise channel is which acts on neighboring qubits i and i + 1 and is applied to each Trotter and denoiser gate, and p = 0.01 unless stated otherwise.
We study circuits with depths M trot = 16, 32, 64 for evolution times t = 0.5, 1, ..., 5, and denoisers D with depths M = 1, 2, 4, 6, 8. In the top panels of Fig. we show (4) for a chain of size L = 8 as a function of time t. Here it can be seen that even for M trot = 32 a denoiser with M = 1 already improves by roughly an order of magnitude at all considered t.
Depending on M trot and t, further increasing M lowers , with the biggest improvements occurring for high precision Trotter circuits with large depth M trot = 64 and short time t = 0.5, where the Trotter gates are closer to the identity than in the other cases. At the other extreme, for M trot = 16 the improvements are relatively small upon increasing M > 2. In all cases the denoiser works better at early times than at late times, again indicating that it is easier to denoise Trotter gates that are relatively close to the identity.
To probe the accuracy of the denoiser on quantities that do not enter the optimization, as a first test we consider the two-point correlator between spins at different times where we have chosen the infinite temperature initial state, and C(t) is the Trotter supercircuit for time t. In the bottom panels of Fig. we show C zz i=L/2,j=L/2 (t) for the supercircuits from the upper panels, now for a L = 14 chain.
Here we see that at M trot = 16 we can retrieve the noiseless values already with M = 1, but that increasing M trot makes this more difficult. At M trot = 64 we see larger deviations, and improvement upon increasing M is less stable, but nonetheless we are able to mitigate errors to a large extent. As a further test, we compute the out-of-time-ordered correlator (OTOC) ]
In Fig. we show the results for i = L/2, for a Trotter circuit with depth M trot = 32 and a denoiser with depth M = 2. Here we see that a denoiser with M M trot is able to recover the light-cone of correlations, which are otherwise buried by the noise. In the Supplementary Material we consider how the denoiser performs at different noise levels p, and how the denoised supercircuits perform under stacking.
There we also calculate domain wall magnetization dynamics, and show the distribution of the optimized denoiser parameters and the sampling overhead associated to the denoiser as a whole. In Fig. we show the eigenvalues of the noisy supercircuits for a noisy second-order Trotter supercircuit with M trot = 16 at t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised supercircuit (right).
The eigenvalues λ of a unitary supercircuit lie on the unit circle, and in the presence of dissipation they are pushed to the center. We see that the spectrum of the denoiser lies outside the unit circle, making it an unphysical channel which cures the effect of the noise on the circuit, such that the spectrum of the denoised circuit is pushed back to the unit circle.
The noiseless eigenvalues are shown as blue bars, making it clear that the denoiser is able to recover the noiseless eigenvalues from the noisy circuit. In the Supplementary Material we show the spectra for a p = 0.036 denoiser, where we observe a clustering of eigenvalues reminiscent of Refs. . There we also investigate the channel entropy of the various supercircuits .
Conclusion. -We have introduced a probabilistic error cancellation scheme, where a classically determined denoiser mitigates the accumulated noise of a (generally non-Clifford) local noise channel. The required number of mitigation gates, i.e. the dimensionality of the corresponding quasiprobability distribution, is tunable and the parameterization of the corresponding channels provides control over the sign problem that is inherent to probabilistic error cancellation.
We have shown that a denoiser with one layer can already significantly mitigate errors for second-order Trotter circuits with up to 64 layers. This effectiveness of low-depth compressed circuits for denoising, in contrast with the noiseless time evolution operator compression from , can be understood from the non-unitarity of the denoiser channels.
In particu-lar, measurements can have non-local effects, since the measurement of a single qubit can reduce some highly entangled state (e.g. a GHZ state) to a product state, whereas in unitary circuits the spreading of correlations forms a light-cone. To optimize a denoiser with convenience at L > 8, the optimization can be formulated in terms of matrix product operators or channels , which is convenient because the circuit calculations leading to the normalized distance and its gradient are easily formulated in terms of tensor contractions and singular value decompositions .
This provides one route to a practical denoiser, which is relevant because the targeted noiseless circuit and the accompanying noisy variant in (4) need to be simulated classically, confining the optimization procedure to limited system sizes with an exact treatment or limited entanglement with tensor networks.
Nonetheless, we can use e.g. matrix product operators to calculate (4) for some relatively small t, such that the noiseless and denoised supercircuits in (4) have relatively small entanglement, and then stack the final denoised supercircuit on a quantum processor to generate classically intractable states.
Analogously, we can optimize the channels exactly at some classically tractable size and then execute them on a quantum processor with larger size. Both approaches are limited by the light-cone of many-body correlations, as visualized in Fig. , because finite-size effects appear when the light-cone width becomes comparable with system size.
1. The normalized distance (left) and z spin correlator C zz i=L/2,j=L/2 (right), for a second-order Trotter supercircuit of depth Mtrot = 16 for time t = 1, affected by various twoqubit depolarizing errors p. We compare the values obtained with and without a denoiser, i.e. M > 0 and M = 0, to the noiseless values (p = 0).
The denoiser is affected by the same noise as the Trotter circuit. We consider denoisers with depths M = 1, 2, 4, 6, 8, and we use a L = 8 Heisenberg chain with PBC for the normalized distance, while for the correlator we use L = 14. * david.luitz@uni-bonn.de to observe that even for larger noise strength p, the local observable C zz improves significantly even with denoisers of depth M = 1.
For large noise strengths, we generally see that the optimization of the denoiser becomes difficult, leading to nonmonotonic behavior as a function of p, presumably because we do not find the global optimum of the denoiser. It is interesting to analyze the spectra of the supercircuits considered in this work.
As mentioned in the main text, the spectrum of the ideal, unitary supercircuit C lies on the unit circle. The comparison to this case is therefore instructive. In the main text, we showed an example of the spectra in Fig. for moderate noise strength. Here, we show additional data for stronger noise p = 0.036 in Fig. for a denoiser with M = 4 layers, optimized to mitigate errors for a second-order Trotter supercircuit with M trot = 16 layers at time t = 1.
The eigenvalues λ of the noisy supercircuit C are clustered close to zero, far away from the unit circle (except for λ = 1), showing that the circuit is strongly affected by the noise. To mitigate the impact of the noise, the denoiser consequently has to renormalize the spectrum strongly. If it accurately represents the inverse of the global noise channel, its spectrum has to lie far outside the unit circle, which is the case.
Interestingly, we observe a clustering of eigenvalues which is reminiscent to the spectra found in . By comparison to these works, we suspect that this is due to the local nature of the denoiser, and warrants further investigation. The right panel of Fig. shows the result of the denoiser, pushing the eigenvalues back to the unit circle, nearly with the exact same distribution along the circle as the noiseless eigenvalues (blue bars).
Due to the strong noise, this is not achieved perfectly, and it is clear that this cannot work in principle if the global noise channel has a zero eigenvalue. The complexity of an operator can be quantified by its operator entanglement entropy . Here we calculate the half-chain channel entanglement entropy S of the noiseless C, noisy C, denoiser D, and denoised D C supercircuits.
We define S as the entanglement entropy of the state that is related to a supercircuit C via the Choi-Jamio lkowski isomorphism, i.e. ψ C = χ C /N , where the process matrix χ ab,cd C = C ac,bd is simply a reshaped supercircuit and N ensures normalization. Then we have S = −Tr [ψ C ln ψ C ]. This entropy measure is a particular instance of the "exchange entropy", which characterizes the information exchange between a quantum system and its environment .
In Fig. we plot the various S for a second-order Trotter circuit with M trot = 16 at t = 2, for a denoiser with M = 4, both affected by two-qubit depolarizing noise with p ∈ [10 −3 , 10 −1 ]. The Trotter circuit is for a Heisenberg model with L = 6 and PBC. We see that at large p, the noise destroys entanglement in the noisy supercircuit, and that the denoiser S increases to correct for this, such that the denoised supercircuit recovers the noiseless S.
Here we investigate how denoised supercircuits perform upon repeated application. We optimize the denoiser for a Trotter supercircuit for a fixed evolution time t. Then, to reach later times, we stack the denoised supercircuit n times to approximate the evolution up to time nt: In Fig. we stack a denoised t = 1 supercircuit up to n = 20 times and calculate the correlation function, defined in the main text, for the middle site.
We consider Trotter depths M trot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8, for a L = 14 Heisenberg chain with p = 0.01 depolarizing two-qubit noise. The noisy results correspond to M = 0 and the noiseless results to p = 0. In Fig. we calculate the OTOC, defined in the main text, with stacked time evolution for a denoised t = 2 supercircuit with M trot = 32 and M = 2, stacked up to ten times.
We see that the stacked supercircuit performs very well, and the additional precision obtained by using deep denoisers (M = 8) pays off for long evolution times, where we see convergence to the exact result (black dashed lines in Fig. ) as a function of M . FIG. . The two-point z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t, for the infinite temperature initial state, for denoised second-order Trotter supercircuits that are optimized at evolution time t = 1 and then stacked up to twenty times.
We use Trotter depths Mtrot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8. The calculations were performed for a periodic Heisenberg model with L = 14 and PBC, affected by two-qubit depolarizing noise with strength p = 0.01, which also affects the denoiser. The non-denoised results are labelled with M = 0, and the noiseless results with p = 0.
The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively. The costliest and most noise-susceptible operation is the two-qubit ZZ rotation with angle α, which is the foundation of the unitary piece in our channel parameterization, defined in the main text.
For completeness, we here present the α angles of the optimized denoisers. The results are shown in Fig. , which contains histograms for the channel count N G versus α. The histograms are stacked, with the lightest color corresponding to the angles of the denoiser at t = 0.5 and the darkest at t = 5. The top four panels are for a denoiser with M = 2 and the bottom four with M = 8.
We consider M trot = 8, 16, 32, 64. We see that in both cases the distribution widens upon increasing M trot , indicating that the unitary channels start deviating more from the identity. Moreover, while the M = 2 denoisers in all cases except M trot = 64 have ZZ contributions close to the identity, this is clearly not the case for M = 8.
For simplicity, we did not focus on obtaining denoisers with the smallest sampling overhead γ, which is required to minimize the sign problem and hence ease the sampling of mitigated quantities. Instead, we let the optimization freely choose the η i in the denoiser parameterization, as defined in the main text.
In Fig. we show the sampling overhead of the denoisers from Fig. of the main text. We see that for M = 1 and M = 2 the sampling overhead is relatively small and uniform across the different t, whereas for M > 2 the optimization sometimes yields a denoiser with large γ and other times with small γ. This could be related to the difference in α distributions from Fig. .
The large fluctuations of γ appears to stem from the difficulty in finding optimal deep denoisers, and our optimization procedure likely only finds a local minimum in these cases. Here C(t) is the Trotter supercircuit for time t. In Fig. we show Z dw for the circuits from Fig.

Answer: L = 8 and L = 14.


Input: What experimental techniques were used to study the quantum dot structures in this research?

Context:
\section{Introduction}

Despite the rise of graphene and other 2D materials, semiconducting single-walled carbon nanotubes (SWNT) are still regarded as strong candidates for the next generation of high-performance ultrascaled transistors~\cite{Cao_IBM_2015,IBM_2017,3D_CNT_FET} as well as for opto-electronic devices~\cite{Review_Avouris,CNT_photonics} such as chip-scale electronic-photonic platforms~\cite{Pernice_2016} or low-threshold near-infrared tunable micro-lasers~\cite{Graf_2017}.
Engineering a quantum dot (QD) along a (suspended) semiconducting SWNT foreshadows promising opportunities in the field of quantum information processing and sensing through recently proposed schemes such as detection and manipulation of single spins via coupling to vibrational motion~\cite{Palyi_2012}, optomechanical cooling~\cite{Wilson_Rae_2012} as well as all optical manipulation of electron spins~\cite{Galland_all_optical_2008}. Furthermore, the quasi one-dimensional geometry of SWNTs allows for defining tunable p-n junctions induced by electrostatic doping through local gates~\cite{Buchs_JAP,tunable_pn_2011}. Combining a well-defined QD within such a p-n junction structure could constitute a crucial building-block for the realization of highly desirable electrically driven, on-demand single photon emitters operating at telecom wavelength, based $e.g.$ on a turnstile device architecture~\cite{turnstile_1994,turnstile_1999}.
In practice, QDs in carbon nanotubes have been reported predominantly for two different confinement structures: i) Engineered tunneling barriers at metal-nanotube contacts~\cite{Pablo04nat} and/or by gate electrodes, used \emph{e.g.} to manipulate single electron spins~\cite{Laird:2015}, ii) Unintentional localization potentials stemming from environmental disorder~\cite{Hofmann_2016}, allowing for single-photon emission mediated by localization of band-edge excitons to QD states~\cite{CNT_photonics,Hoegele_2008,Walden_Newman_2012,Hofmann_2013,Pernice_2016_2}. Both types of structures are usually operated at cryogenic temperature due to small energy scales ranging from a few to few tens of millielectronvolts.
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\indent Another technique for achieving confinement in SWNTs makes use of artificial defects such as covalently bound oxygen or aryl functionalization groups on the side walls of semiconducting SWNTs, inducing deep exciton trap states allowing for single-photon emission at room temperature~\cite{Htoon_2015,tunable_QD_defects}. Also, carrier confinement between defect pairs acting as strong scattering centers has been reported for mechanically induced defects~\cite{Postma_SET} as well as for ion-induced defects with reported level spacings up to 200 meV in metallic SWNTs~\cite{Buchs_PRL}. The latter technique, combined with recent progress in controlling defects structure and localization~\cite{Robertson_2012,Yoon_2016,Laser_writing_2017} offers a high potential for engineering a broad set of SWNT-based quantum devices operating at room temperature.
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\indent Here, we demonstrate confinement of electrons and holes in sub-10 nm QD structures defined by ion-induced defect pairs along the axis of semiconducting SWNTs. Using low temperature scanning tunneling microscopy and spectroscopy (STM/STS), bound states with level spacings of the order of 100 meV and larger are resolved in energy and space. By solving the one-dimensional Schr\"odinger equation over a piecewise constant potential model, the effects of asymmetric defect scattering strength as well as the influence of the Au(111) substrate such as terrace edges on the bound states structure are remarkably well reproduced. By means of ab-initio calculations based on density functional theory and Green's functions, we find that single (SV) and double vacancies (DV) as well as chemisorbed nitrogen ad-atoms are good candidates to produce QDs with the experimentally observed features. These simulations also allow to study the scattering profile as a function of energy for different defect combinations.

\section{Experimental section}

The experiments have been performed in a commercial (Omicron) low temperature STM setup operating at $\sim5$~K in ultra high vacuum. Topography images have been recorded in constant current mode with a grounded sample, using mechanically cut Pt/Ir tips. Differential conductance $dI/dV$ spectra, proportional in first approximation to the local density of states (LDOS)~\cite{Tersoff85} have been recorded using a lock-in amplifier technique. The LDOS spatial evolution along a nanotube axis is obtained by $dI/dV(x,V)$ maps built by a series of equidistant $dI/dV$ spectra. Spatial extent mismatches between topography images and consecutive $dI/dV(x,V)$ maps have been systematically corrected~\cite{Buchs_Ar}, and the metallic nature of the tip has been systematically checked on the gold substrate to prevent any tip artefacts before recording STM or/and STS data sets.
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\indent Nanotube samples were made of extremely pure high-pressure CO conversion (HiPCo) SWNTs~\cite{Smalley01} with a diameter distribution centered around 1 nm, FWHM $\sim$ 0.3 nm~\cite{Buchs_conf}. The measured intrinsic defect density was below one defect every 200 nm. SWNTs were deposited on atomically flat Au(111) surfaces from a 1,2-dichloroethane suspension, followed by an in-situ annealing process~\cite{Buchs_APL_07,Buchs_Ar}.
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\indent Local defects in SWNTs have been created in-situ by exposure to: (i) Medium energy $\sim$ 200 eV argon ions (Ar$^{+}$) produced by an ion gun \cite{Buchs_Ar,Buchs_PRL}, (ii) Low energy (few eV's) nitrogen ions (N$^{+}$) produced by a 2.45 GHz ECR plasma source~\cite{Buchs_APL_07,Buchs_NJP_07}. In both cases, the exposure parameters have been calibrated to reach an average defect separation along the SWNTs of about 10 nm~\cite{Buchs_Ar,Buchs_APL_07}.

\section{Results and discussion}
\subsection{Experimental LDOS patterns}
\begin{figure}
  \includegraphics[width=8cm]{Figure_1.pdf}
  \caption{\label{exp_data_1} (a)-(b) 3D topography images (processed with WSXM~\cite{WSXM}) of SWNT I with Ar$^{+}$ ions-induced defects, with sample-tip bias voltage ($V_\mathrm{S}$) 1 V and tunneling current $I_\mathrm{S}$ 0.1 nA. (c) Corresponding $dI/dV(x,V)$ map recorded along the horizontal dashed lines in (b), with $V_\mathrm{S}=1$ V, $I_\mathrm{S}=0.2$ nA. Spatial resolution $\sim$ 0.3 nm. (d) 3D topography image of SWNT II with N$^{+}$ ions-induced defects, with $V_\mathrm{S}=1$ V, $I_\mathrm{S}=128$ pA. (e) Corresponding $dI/dV(x,V)$ map recorded along the horizontal dashed lines in (d), with $V_\mathrm{S}=1.5$ V, $I_\mathrm{S}=0.3$ nA. Spatial resolution $\sim$ 0.2 nm.}
\end{figure}
In Fig.~\ref{exp_data_1} (a) and (b), we show 3D STM images of the same semiconducting SWNT (referred as SWNT I in the following) with Ar$^{+}$ ions-induced defect sites labeled $d1-d5$ . Panel (d) shows a 3D STM image of a second semiconducting SWNT (referred as SWNT II) with N$^{+}$ ions-induced defect sites labeled $d6-d7$. In both cases, defect sites typically appear as hillock-like protrusions with an apparent height ranging from 0.5~{\AA} to 4~{\AA} and an apparent lateral extension varying between 5~{\AA} and 30~{\AA}~\cite{Buchs_NJP_07,Buchs_Ar,Thesis_Buchs}.
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\indent The resulting $dI/dV(x,V)$ maps recorded along the horizontal dashed line drawn in the STM images (b) and (d) are displayed in panels (c) and (e) in Fig.~\ref{exp_data_1}, respectively. Defect signatures in the LDOS in both cases are characterized by deep in-gap states at the defects positions. This is consistent with the expected defect structures, $i.e.$ mainly SVs, DVs and combinations thereof for collisions with Ar$^{+}$ ions~\cite{Buchs_Ar} and bridgelike N ad-atom for collisions with N$^{+}$ ions~\cite{Thesis_Buchs,Nitrogen_prb_07}. Note that gap states at energy levels $\sim$~0.2 eV and $\sim$~0.05 eV in panels (c) and (e), respectively, are shifted to the right from $d3$ by about 1 nm and to the right from $d6$ by about 2 nm. This indicates the presence of intrinsic or ion-induced defects on the lateral or bottom side wall of the SWNTs~\cite{Kra01prb}, not visible in the topographic images. These defects are labelled $d3'$ and $d6'$, respectively.
\\
\begin{figure}
  \includegraphics[width=12cm]{Figure_2.pdf}
  \caption{\label{exp_data_Ar} (a)-(b) QD I detailed $dI/dV(x,V)$ maps in conduction and valence bands. Lower subpanels contain QD states linecut profiles and stationary wave-like fits in left and right QD parts. Right subpanels contain experimental energy dispersion relation data sets $k_\mathrm{n}(E_\mathrm{n})$ and tight-binding calculations. (c)-(d) Resulting LDOS calculated from a one-dimensional piecewise constant potential model featuring potential barriers and a potential step (gray area), with position of the potential step: 5.09 nm from the right barrier's center, potential step heigth: $U_\mathrm{C}=V_\mathrm{L}-V_\mathrm{R}=60$ meV, barrier heights: $V_\mathrm{d3'}=1$ eV, $V_\mathrm{d4}=0.85$ eV, barrier widths: $a_\mathrm{d3'}=a_\mathrm{d4}=3.4$ nm.  Valence band: $V_\mathrm{d3'}=-0.4$ eV, $a_\mathrm{d3'}=a_\mathrm{d4}=2.5$ nm, $V_\mathrm{d4}=-0.4$ eV. $E_\mathrm{g}$ stands for bandgap energy.}
\end{figure}
\begin{figure}
  \includegraphics[width=12cm]{Figure_3.pdf}
  \caption{\label{exp_data_N} (a) QD II detailed $dI/dV(x,V)$ map. Lower subpanels contain QD states linecut profiles and stationary wave-like fits in the left and right QD parts. Right subpanel contains experimental energy dispersion relation data sets $k_\mathrm{n}(E_\mathrm{n})$ and tight-binding calculations. (b) Resulting LDOS calculated from a one-dimensional piecewise constant potential model featuring potential barriers and a potential step (gray area) with position of the potential step: 4.7 nm from the right barrier's center, potential step heigth: $U_\mathrm{C}=V_\mathrm{L}-V_\mathrm{R}=60$ meV, barrier heights: $V_\mathrm{d6'}=0.6$ eV, $V_\mathrm{d7}=0.6$ eV, barrier widths: $a_\mathrm{d6'}=1.5$ nm, $a_\mathrm{d7}=2.6$ nm.}
\end{figure}
\indent Remarkably, the $dI/dV(x,V)$ maps in Fig.~\ref{exp_data_1} exhibit several broad discrete states in the conduction bands of SWNT I, II (white dashed boxes in panel (c) and (e), respectively) and in the valence band of SWNT I (white dashed box in panel (c)), characterized by a modulation of the $dI/dV$ signals in the spatial direction between pairs of consecutive defect sites $d3'-d4$ and $d6'-d7$. Enlarged plots of these boxed regions are displayed in Fig.~\ref{exp_data_Ar}(a)-(b) and Fig.~\ref{exp_data_N}(a) for SWNTs I and II, respectively. In the conduction bands, cross-sectional curves recorded along the black horizontal dashed lines labelled m1--m3 in Fig.~\ref{exp_data_Ar}(a) and m1--m4 in Fig.~\ref{exp_data_N}(a) are plotted below the LDOS panels. These clearly reveal one to three and respectively one to four spatially equidistant maxima. The number of maxima increases for increasing $\left|V_\mathrm{bias}\right|$ and the measured level spacings between consecutive discrete states is of the order of 100 meV and larger for both cases. This indicates that defect sites $d3'-d4$ and $d6'-d7$, respectively separated by 12.1 nm and 11.2 nm, act as strong scattering centers able to confine carriers in semiconducting SWNTs~\cite{Buchs_PRL,Bercioux_prb_2011}. Such intrananotube QD structures will be referred as QD I (in SWNT I) and QD II (in SWNT II) in the following. We estimated the level spacings in the conduction band of QD I to 98 meV (m1-m2) and 116 meV (m2-m3). For QD II, we measured 122 meV (m1-m2), 185 meV (m2-m3) and 210 meV (m3-m4).
\\
\indent In the valence band of SWNT I, discrete states with level spacings of the order of 80-90 meV, with one clear maximum at the level m-1, can also be distinguished between defect sites $d3'-d4$ in Fig.~\ref{exp_data_Ar}(b). The discretization of the states indicates that this QD structure also confines holes. Discrete states starting from m-2 and lower show less well defined structures compared to the conduction band states. In the case of SWNT II, no clear discrete states are observed in the valence band (see supplementary information). These observations are most probably the result of an energy dependent scattering strength of the defects, respectively $d3'$-$d4$ and $d6'$-$d7$, leading here to a weaker confinement in the valence band. Such energy dependence is well known for metallic SWNTs~\cite{Chico96,vac_2007,mayrhofer:2011,Bockrath_Science01} and is corroborated by our ab-initio calculations. Note that mixing effects with defect states and substrate-induced effects~\cite{substrate_effects} cannot be ruled out.
\\
\indent Another remarkable feature in the LDOS is the strong spatial asymmetry of the lowest energy states m1 and m-1 in QD I and m1 in QD II. In QD I, m1 is shifted to the right side of the dot while m-1 is shifted to the left side. Higher states m2 and m3 show more symmetry in terms of position of the maxima relative to the center of the QD. In QD II, m1 is shifted to the right side of the QD. We attribute the observed lowest energy states asymmetry (for electrons as well as for holes) in part to their strong sensitivity to weak potential modulations within the QD structure (as we will show in section \ref{1D}). For QD I, this assertion is supported by the observation of a 0.25 nm high Au(111) terrace edge located around the center of the QD, leading to a supported-suspended interface (see white dashed lines in Fig.~\ref{exp_data_1}(b) and more topographic details in Fig.~S2(a)-(d) in supplementary information). Such configurations have been reported to induce a rigid shift in the SWNT bands~\cite{Clair_2011}, for instance here a down-shift in the right side of QD I corresponding to the "suspended" portion between two terraces. In QD II, we attribute the spatial shift of m1 to a potential modulation induced by a layer of disordered impurities, most probably residua from the 1,2-dichloroethane suspension, lying between the gold substrate and the SWNT (see Fig.~\ref{exp_data_1}(d) and Fig.~S2(e)-(h) in supplementary information).
\\
\indent Also, the LDOS in QD I and II (Fig.~\ref{exp_data_Ar}(a) and Fig.~\ref{exp_data_N}(a), respectively) reveals asymmetric patterns with curved stripes oriented from top left to bottom right for QD I and from bottom left to top right for QD II. These are characteristic signatures for defect pairs with different scattering strengths~\cite{Bercioux_prb_2011,Buchs_PRL}. For instance here, the left defect in QD I ($d3'$) has a larger scattering strength than the right one ($d4$), while the right defect in QD II ($d7$) has a larger scattering strength than the left one ($d6'$).
\\
\indent The exact atomic structure of the defects could in principle be determined from a comparison of $dI/dV$ spectra with simulated first-principle LDOS signatures of expected defect types. In reality, this is hampered by the large number of possible geometries to simulate, including complex multiple defect structures~\cite{Buchs_Ar}, together with the large unit cells of the semiconducting chiral SWNTs studied here.
\\
\subsection{1D piecewise constant potential model}
\label{1D}
To better understand the physical origins of the non-trivial signatures of the quantized states, we model the experimental $dI/dV$ maps by solving the time independent one-dimensional Schr\"odinger equation over a piecewise constant potential model of QD I and QD II. The scattering centers are approximated by semi-transparent rectangular tunneling barriers leading to a square confinement potential~\cite{Laird:2015}. This is supported by previous results on defect-induced confinement in metallic SWNTs using the same experimental conditions~\cite{Buchs_PRL} and is consistent with ab-initio simulations presented later in this work. The potential modulation within the QD is approximated by a potential step. The resulting potential geometries are illustrated with gray shaded areas in Fig.~\ref{exp_data_Ar} (c) and (d) and Fig.~\ref{exp_data_N}(b). Dispersion relations $E(k)$ can be extracted experimentally from the quantized states wavefunctions by measuring the energy and corresponding momenta in the left and right sides of the QDs. The wavevectors $k$ are determined using stationary wave-like fitting functions~\cite{Buchs_PRL} displayed with dashed red curves in Figs.~\ref{exp_data_Ar}(a)-(b) and ~\ref{exp_data_N}(a)). From this procedure, the potential step height and position can be estimated (see supplementary information). The experimental data sets $E(k)$ are plotted in the right panels of Figs.~\ref{exp_data_Ar}(a) and \ref{exp_data_N}(a) together with dispersion relations from a third-nearest neighbor tight-binding calculation closely approximating ab-initio results~\cite{Reich_TB_2002}. These chirality-dependent tight-binding dispersion relations, calculated within an extended Brillouin zone resulting from the defect-induced breaking of the translation invariance~\cite{Bercioux_prb_2011}, are used in the Hamiltonian of our one-dimensional model. Taking into account the measured chiral angle, diameter distribution~\cite{Buchs_conf} and measured bandgaps, we find the best match with chiralities $(7,6)$ for QD I and $(11,1)$ for QD II (see supplementary information).
\\
\indent Once chiralities together with potential step heights and positions are optimized, one can fit the height and width of the rectangular tunneling barriers in order to reproduce the experimental level spacings and general LDOS patterns. On a qualitative ground, a symmetric double barrier system results in the formation of spatially symmetric discrete bound states. Increasing both barrier heights simultaneously shifts the bound state energy levels and level spacings up. This leads to sharper bound states as the confinement in the QD is made stronger thus increasing the lifetime of the confined electrons. Increasing the barrier thickness with constant inner edge separation does not affect much the level spacings but further sharpens the bound states. Any asymmetry introduced by a change in the width or height of one single barrier leads to broader bound states. The presence of a potential step modifies the LDOS in lifting the levels of the bound states, with a more pronounced effect on the lower states. In QD I and II, the center of each barrier is aligned with the center of the gap states ($d3'$-$d4$ for QD I and $d6'$-$d7$ in QD II) and the width ratio is kept proportional to the ratio of the spatial extent of the gap states. Thus, by increasing the width of the barriers, we decrease the length of the QD leading to higher level spacings, and vice versa. The experimental level spacings can then be approximated by tuning both barrier widths in the same ratio and the heights individually, knowing that the scattering strength of $d3'$ ($d7$) is larger than $d4$ ($d6'$) according to the observed asymmetry in the LDOS described above \footnote{The transmission probability through a rectangular tunneling barrier is given by $T=\left( 1+\frac{V^{2}\sinh^{2}\left( a \cdot \sqrt{2m^{*}(V-E)}/\hbar \right)}{4E(V-E)} \right)^{-1}$, where $V$ and $a$ are respectively the barrier height and width. For the argument in the $\sinh$ sufficiently small such that $\sinh(x)\simeq x$, it can be shown that $a$ and $V$ can be coupled such that the transmission probability becomes a function of the area under the barrier $A=a\cdot V$, with $T=\left( 1+ \frac{m^{*}A^{2}}{2\hbar^{2}E} \right)^{-1}$. In our case, this condition is not satisfied and thus the barrier geometries are tuned empirically to fit the experimental level spacings.}.
\\
\indent For QD I, we find a good match in the conduction band for the barrier heights $V_\mathrm{d3'}=1$ eV and $V_\mathrm{d4}=0.85$ eV, widths $a_\mathrm{d3'}=a_\mathrm{d4}=$ 3.4 nm,  and potential step $V_\mathrm{L}-V_\mathrm{R}=60$ meV. With these parameters, the spatial profile of the obtained quantized states (see lower subpanels in Fig.~\ref{exp_data_Ar}(a) and (c)) reproduces the experimental modulation features remarkably well. Also, the simulated LDOS displays a pattern with curved stripes oriented from top left to bottom right, as observed experimentally, due to a left barrier with a larger scattering strength. In the valence band, although modes m-2 and lower do not show a well defined structure in the spatial direction, thinner barriers with dimensions $a_\mathrm{d3'/d4}=2.5$ nm, $V_\mathrm{d3'/d4}=-0.4$ eV, leading to a slightly longer QD length (9.6 nm compared to 8.7 nm in the conduction band) can reproduce the measured level spacings very well.
\\
\indent For QD II, we observed that the measured energy levels are overestimated by a factor $\alpha\sim1.29$, presumably due to a voltage division effect induced by the impurity layer mentioned above (see details in supplementary information). We find a good agreement with the experimental LDOS with the parameters: $V_{d3'}=V_{d4}\simeq$ 0.47 eV, $a_\mathrm{d6'}=1.5$ nm, $a_\mathrm{d7}=2.6$ nm and $U_\mathrm{C}=V_\mathrm{L}-V_\mathrm{R}\simeq 47$ meV. Note that in Fig.~\ref{exp_data_N}(b) the barrier and potential heights are multiplied by $\alpha$ to allow a direct comparison with the experimental LDOS. The simulated LDOS shows a pattern with curved stripes oriented from bottom left to top right, as observed experimentally, due to a right barrier exhibiting a larger scattering strength. Also, the spatial profile of the obtained bound states (see lower subpanels in Fig.~\ref{exp_data_N}(a) and (b)) reproduces the experimental features quite well. Note also that one can distinguish an isolated state in the experimental LDOS at an energy level between m1 and m2, about in the middle of the QD. This state that prevented an accurate fit of the state m2 in the right QD part is attributed to a spatial feature visible in the STM topography image in Fig.~\ref{exp_data_Ar}(d) (see also supplementary information, Fig.S2(f)), probably a physisorbed impurity which does not affect the LDOS significantly.
\\
\subsection{Ab-initio calculations}
\begin{figure}
  \includegraphics[width=16cm]{Figure_4.pdf}
  \caption{\label{num_data} (a)-(c) LDOS ab-initio simulations of a semiconducting $(16,0)$ SWNT with combinations of vacancies defects separated by 11.1 nm. Subpanels display QD state linecut profiles. (d) Tight-binding (black curve) and ab-initio dispersion relations (green circles) for a pristine $(16,0)$ SWNT with $E_\mathrm{n}(k_\mathrm{n})$ data sets extracted from (a)-(c). (e)-(g) LDOS ab-initio simulations of a semiconducting $(17,0)$ SWNT with combinations of N ad-atoms and vacancies defects separated by 10.7 nm. (h) Tight-binding (black curve) and ab-initio dispersion relations (green circles) for a pristine $(17,0)$ SWNT with $E_\mathrm{n}(k_\mathrm{n})$ data sets extracted from (e)-(g).}
\end{figure}
In order to elucidate the physical nature of the electron/hole confining scattering centers, we performed ab-initio simulations based on a combination of density functional theory~\cite{pbe,paw,vasp_paw,VASP2}, maximally localized Wannier orbitals~\cite{transportwannier90} and Green's functions (see supplementary information). Without loss of generality, we have simulated short unit cell semiconducting zigzag SWNTs with different combinations of the most probable defect structures. Results for vacancy defects likely being induced by 200 eV Ar$^{+}$ ions, separated by about 11 nm in a $(16,0)$ SWNT are shown in Fig.~\ref{num_data}(a)-(c) with DV-DV, DV-SV and SV-SV pairs, respectively. The LDOS displays midgap states at the defect positions as expected as well as defect states in the valence band~\cite{Buchs_Ar}. Most importantly, clear quantized states with a number of maxima increasing with energy are observed between the defects in the conduction band, emphasizing the ability of SVs and DVs to confine carriers. For the asymmetric configuration DV-SV, one can distinguish faint curved stripe patterns oriented from top left to bottom right, indicating a larger scattering strength for DVs compared to SVs. This is consistent with observations in transport experiments~\cite{Gomez05nm}. On the other hand, the patterns in the valence band strongly depend on the defect types. Discrete states can be distinguished for the DV-DV case, with m-2 being mixed with defect states. For the DV-SV case, clear curved stripe patterns oriented from bottom left to top right indicate again a stronger scattering strength for DV. Also, broader states are observed, indicating that the scattering strength of DVs and SVs is weaker in the valence band compared to the conduction band.
\\
\indent More insight on the energy dependent scattering strength for each defect pair configuration can be obtained by extracting the wavevector $k_\mathrm{n}(E_\mathrm{n})$ for each resonant state. This data set is plotted in Fig.~\ref{num_data}(d) for the conduction and valence bands together with the $(16,0)$ dispersion relations calculated from the third-nearest neighbor TB model and from the ab-initio calculation for the pristine nanotube. A first observation is the excellent agreement between TB and ab-initio results, further validating the method used in Figs.~\ref{exp_data_Ar}(a)-(b) and ~\ref{exp_data_N}(a). The vertical dashed lines indicate the limiting $k_\mathrm{n,\infty}=\frac{\pi \cdot n}{L}$ values corresponding to the closed system (infinite hard walls potential) with $L=11.1$ nm being the defect-defect distance. In the conduction band, we find that $k_\mathrm{n}(E_\mathrm{n})=\frac{\pi \cdot n}{L_\mathrm{eff}(n)} < k_\mathrm{n,\infty}$, indicating that the effective lengths $L_\mathrm{eff}(n)$ of the QD are larger than $L$ ($i.e.$ the resonant states wavefunctions are characterized by penetrating evanescent modes inside the defect scattering potential), as expected for an open system. The shortest $L_\mathrm{eff}(n)$ are obtained for the DV-DV configuration with 12.1 nm (m1), 13.1 nm (m2) and 12.9 nm (m3), which we attribute to wider scattering potential profiles for DVs compared to SVs. In the valence band, we find that $k_\mathrm{n}(E_\mathrm{n})=\frac{\pi \cdot n}{L_\mathrm{eff}(n)} > k_\mathrm{n,\infty}$, with $L_\mathrm{eff}(n)$ values between 7.9 nm (DV-DV, m-1) and 9.66 nm (DV-SV, m-2). We attribute this pronounced QD shortening to wider scattering potential profiles of both DVs and SVs in the valence band, probably due to mixing with wide spread defect states in the valence band.
\\
\indent Ab-initio calculations for different defect pairs combinations containing at least one N ad-atom, $i.e.$ N-DV, N-SV and N-N, are presented in Fig.~\ref{num_data}(e)-(h) for a $(17,0)$ SWNT, along with details on the defects geometries. Remarkably, clear QD states are generated for all three configurations, underlining the potential of N ad-atoms to confine carriers in semiconducting SWNTs and thus to generate intrananotube QDs.
\\
\indent In order to demonstrate the scattering strengths of the different defects, we calculated the energy dependent conductance in addition to the LDOS for the different combinations of the QD defining scattering defects on the $(16,0)$ and $(17,0)$ SWNTs, see supplementary information. Generally we can observe strong conductance modulation of the order of 30-40\% with regard to the pristine CNT for all three tested defects (double vacancies DV, single vacancies SV and chemisorbed C-N) with the DVs having the largest scattering strength in the CB and VB.
\\
\indent Note that the choice of the zigzag SWNT chiralities in the two different ab-initio scenarios is motivated by the different effective masses of both chiralities ($m^{*}_{(17,0)}>m^{*}_{(16,0)}$) which is typical for chirality families $(3n-1,0)$ and $(3n-2,0)$~\cite{ZZ_families}. Taking advantage of recent reports on SWNT chirality control~\cite{chirality_control_EMPA,chirality_control_chinese,chirality_chemistry}, this property could be used in practice to design QDs with different level spacings for the same QD length. From an application point of view, however, QDs generated by DVs will have far superior stability at room temperature due to their high migration barrier above 5 eV ($\sim$~1 eV for single vacancy)~\cite{Kra06vm}. This value drops down by at least 2 eV for N ad-atoms depending on their chemisorption configuration~\cite{Nitrogen_prb_07,Yma05nitr}.
\\
\indent Our ab-initio simulations do not take into account any substrate effect. In the experimental case, the carriers can decay through the substrate, thus limiting their lifetime. This leads to state broadening, measured between about 60 meV up to 120 meV in QD I and II, while the quantized states widths in ab-initio simulations vary between about 5 meV and 45 meV. This suggests that a better contrast of the experimental quantized states, especially in the valence band, could be achieved by lowering the nanotubes-substrate interaction through $e.g.$ the insertion of atomically thin insulating NaCl films~\cite{Ruffieux_Nature_2016}. This would allow to gain more insight on the electronic structure of the QDs as well as in the associated scattering physics at the confining defects~\cite{Buchs_PRL}.

\section{Conclusions and outlook}
In summary, using low-temperature STM/STS measurements supported by an analytical model and ab-initio simulations, we have demonstrated that intrananotube quantum dots with confined electron and hole states characterized by energy level spacings well above thermal broadening at room temperature can be generated in semiconducting SWNTs by structural defects such as vacancies and di-vacancies, as well as nitrogen ad-atoms. These results, combined with recent progresses in type and spatial control in the formation of defects~\cite{Robertson_2012,Yoon_2016,Laser_writing_2017} as well as chirality control~\cite{tunable_QD_defects}, hold a high potential for applications in the design of SWNT based quantum devices. These include $e.g.$ electrically driven single-photon emitters operating at room temperature and telecom wavelength. In this context, the observation of quantum confinement effects in the emitted light of cut, sub-10 nm, semiconducting SWNTs~\cite{Dai_2008} shall be seen as an additional motivation for investigating the optical properties of our "QD with leads" building-blocks. These would include $e.g.$ studying optical transitions selection rules for different types and configurations of defect pairs~\cite{sel_rules_2006} associated with experimental studies such as photoluminescence~\cite{Lefebvre06} combined to $g^{(2)}$ correlation measurements~\cite{Hofmann_2013} in suspended SWNT devices as well as photocurrent imaging~\cite{Buchs_Nat_comm} and spectroscopy~\cite{Gabor_2009}.

\section*{Acknowledgements}
The authors thank Ethan Minot, Lee Aspitarte, Jhon Gonzalez, Andres Ayuela, Omjoti Dutta and Arkady Krasheninnikov for fruitful discussions.
The work of DB is supported by Spanish Ministerio de Econom\'ia y Competitividad (MINECO) through the project  FIS2014-55987-P and by the (LTC) QuantumChemPhys. LM acknowledges support from the BMBF-project WireControl (FKZ16ES0294) and computing time for the supercomputers JUROPA and JURECA at the J\"ulich Supercomputer Centre (JSC).

\clearpage

\section*{References}

Answer: Low temperature scanning tunneling microscopy and spectroscopy (STM/STS).