prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are the benefits of using binary variables in the SLAS formulation?

Read the following text and answer briefly.

Context 1: Paper Info

Title: SLAS: Speed and Lane Advisory System for Highway Navigation
Publish Date: Unkown
Author List: Faizan Tariq, David Isele, John Baras, Sangjae Bae

Figure

Fig  1 Motivational Example With a slow moving vehicle ahead, the ego vehicle (in blue) may decide to either change lane to the fast moving lane (left) to minimize travel time or adjust its speed without changing lanes to preserve safety but it would be unwise for it to switch to the slow moving lane (right) as that would not benefit travel time or safety 
Fig  3  Simulation Setup Scenario Runner sets up the scenario for the CARLA Simulator, which then communicates with the SLAS and the Planning and Control ROS (Robot Operating System) nodes through the ROS bridge node 
Fig  4  Testing scenario with three lanes: lane 0 (left), lane 1 (center) and lane 2 (right) The expected motion of the ego vehicle, over the course of the simulation, is shown with numbered frames The right most lane (lane 3) is reserved for merging traffic so it is not utilized in our simulation 
Fig  5  Left: Travel time comparison Center: Lane choice (lateral position) comparison The center lines of lanes 0 (left), 1 (center) and 2 (right) have fixed lateral displacements of 0m, 3 5m and 7m respectively Right: Headway comparison With no leading vehicle, the headway is restricted by the visibility range of 50m 

abstract

This paper proposes a hierarchical autonomous vehicle navigation architecture, composed of a high-level speed and lane advisory system (SLAS) coupled with low-level trajectory generation and trajectory following modules  Specifically, we target a multi-lane highway driving scenario where an autonomous ego vehicle navigates in traffic 
We propose a novel receding horizon mixed-integer optimization based method for SLAS with the objective to minimize travel time while accounting for passenger comfort  We further incorporate various modifications in the proposed approach to improve the overall computational efficiency and achieve real-time performance 
We demonstrate the efficacy of the proposed approach in contrast to the existing methods, when applied in conjunction with state-of-the-art trajectory generation and trajectory following frameworks, in a CARLA simulation environment 

INTRODUCTION

Lane changing is considered to be one of the most risky driving behaviors since it is highly contingent upon multimodal trajectory predictions of neighboring vehicles and requires timely decision making   It is further influenced by a number of uncertainty factors such as road conditions, measurement accuracy, and a long tail of behavioral uncertainty of on-road agents 
However, if executed efficiently, lane changing coupled with speed adjustment can yield significant improvement in minimizing overall travel time while ensuring passenger comfort   To elaborate further, consider the scenario presented in Fig    Based on the predicted motion (shown in a lighter shade) of the neighboring vehicles (shown in orange), the ego vehicle (shown in blue) may decide to either change lane left in an attempt to minimize its travel time or slow down in the current lane to maintain safety 
However, it would be imprudent for the ego vehicle to risk changing lane right and consequently get stuck behind a slow moving vehicle even though there is presently a greater headway  This simple scenario highlights the importance of foresight and long planning-horizon in strategic decision making for autonomous vehicles 
Existing methods like MOBIL give us the ability to change lanes but behave greedily (prioritizing immediate rewards) oftentimes, which can lead to sub-optimal performance  It was shown in that the lane changing performance can be improved with an A inspired approach, but the formulation was limited to constant speed 
Such an approach is unable to assess the benefits of speed adjustment 1 University of Maryland, College Park, MD, USA  Email: {mftariq,baras}@umd edu  2 Honda Research Institute, San Jose, CA, USA  Email: {disele,sbae}@honda-ri com  Research supported by Honda Research Institute, USA   in minimizing overall travel time 
As will become apparent in Section IV, it may be necessary at times to sacrifice on shortterm benefits to gain long-term performance improvements  In such a scenario, an approach with speed adjustment coupled with long planning horizon has the foresight to deliver significantly better results  Moreover, the inclusion of speed adjustment in the decision making process inhibits the risk of incurring trajectory infeasibility as the environment conditions may prevent the ego vehicle from traveling at a constant reference speed and the low-level planner may be unable to handle such a discrepancy 
Therefore, in this work, we propose a low complexity receding horizon optimization based approach that outputs the lane change maneuvers coupled with speed adjustments for long planning horizons (> 15s) while guaranteeing safety  The long horizon strategic decision making gives ego vehicle the ability to proactively anticipate and handle challenging driving situations 
Literature review: In the literature, speed and lane changing decisions are generally considered from a motion planner's perspective , which allows for a simultaneous determination of target lanes and waypoints to perform the maneuver  The motion planning methods present in the literature can broadly be categorized into sampling-based, learning-based and optimization-based approaches 
In regards to the sampling-based approaches, single-query methods, in particular the different variants of RRT, are preferred over multi-query methods, like roadmap-based methods, due to the faster execution time and their ability to incorporate non-holonomic constraints   Even though these methods are able to incorporate safety guarantees by sampling feasible trajectories from a reachable safe set , the overall driving experience is often rather uncomfortable due to the concatenation of individual trajectories 
Moreover, the asymptotic optimality guarantees availed by arXiv:2303 00861v1 [cs RO] 1 Mar 2023 these methods do not help with real-world implementation in complex driving scenarios since they tend to have high sample complexity   In terms of the learning-based methods, the preferred approach seems to be the variations of Reinforcement Learning techniques applied in a simulated environment , , , ,  
These approaches, although seeming to work well in simulation, have concerns regarding real-world implementation due to the large amount of training data that they require, the exploration of unsafe behaviors during training, and a general inability to handle edge cases  They mainly utilize neural networks as function approximators which yields low computational complexity but also results in a lack of explainability and safety guarantees 
Lastly, the optimization-based approaches, especially the derivatives of optimal control methods, are abundant in the literature  In contrast to the potential-field based approaches that yield decent collision avoidance performance but are unable to accommodate vehicle dynamics, the optimal control methods , especially the derivatives of Model Predictive Control (MPC) approach , , , yield excellent collision avoidance performance while accommodating vehicle dynamics 
However, this performance comes at a cost of high computational complexity, arising mainly from the non-convex collision avoidance and the non-linear dynamics constraints  This, in turn, restricts the planning horizon to merely a few seconds  The key requirements for the algorithmic design of an autonomous vehicle include real-time operation, safety guarantees, optimality with respect to some metric(s), and accounting for the behavior variability of on-road agents 
Considering these requirements, we propose an optimization-based behavioral planning framework that enables autonomous vehicle maneuvering on multilane highways  While having the benefits of optimizationbased approaches, our method achieves a low computational complexity by employing a binary representation of the decoupled lane indicator dynamics in lieu of lateral dynamics, and utilizing algorithmic modifications to aid numerical computations 
Specifically, our method provides: • optimality with respect to travel time and comfort; • safety and feasibility guarantees; • real-time applicability for a long planning horizon; and • modularity in design, which enables the integration of external trajectory prediction modules  The proposed method fills in the research gap by meeting all the key algorithmic requirements while simultaneously gaining the foresight to make strategic decisions that yield long-term performance benefits, as verified in Section IV 
In this section, we present the algorithmic pipeline and formalize the road, observation and vehicle dynamics models that will be utilized in the subsequent sections 

Algorithmic Pipeline

Fig  illustrates the algorithmic pipeline of the proposed navigation architecture, in reference to the various existing   Algorithmic pipeline of the proposed navigation architecture  The raw sensory input data is processed by the Perception, and Simultaneous Localization and Mapping (SLAM) modules to place the autonomous vehicle relative to the various environmental entities in a unified frame of reference 
This information is then passed on to the navigation stack, composed of the behavioral planning, motion planning, and vehicle control modules  The output of the navigation module is passed down further in terms of actuation commands (brake, throttle and steering) to the actuators  algorithmic modules deployed on an autonomous vehicle 
The taxonomy of the various components of the navigation stack (highlighted by the dotted rectangle) is borrowed from   This pipeline essentially improves the pipeline introduced in by adding a speed advisory system  Our main focus is the development of the behavior planning module, highlighted as SLAS in Fig    SLAS outputs the target lane and reference speed which are utilized by the motion planning module to generate a reference trajectory for the ego vehicle 
The vehicle controllers compute the throttle and steering commands to track the trajectory accordingly  For the motion planning module, we adopt the Neural Networks integrated Model Predictive Control (NNMPC) due to its ability to accommodate the behaviors of neighboring vehicles in the trajectory generation process 
In our approach, we assume that the perception (of other vehicles) and the localization (of ego vehicle) are known without any uncertainty, for simplicity, but the modular architecture avails us the ability to integrate any perception or SLAM module in the overall framework  Throughout the manuscript, Z will denote the set of integers and R the set of real numbers 
For some a, c ∈ Z and a < c, we will write For some e, g ∈ R and e < g, we will write

Road Model

The physical road structure is modeled as a continuous multi-lane highway with negligible curvature and unidirectional traffic flow  The lanes on the highway are clearly demarcated and at any given time k, the number of available lanes for the vehicles to travel on is denoted by N l (k) while the road speed limit is denoted by V l  
Therefore, the set of lanes available for traveling at a given time instant k is denoted by   We work with the Frenet coordinate system where the distance along the road is denoted by the longitudinal displacement (s) and the distance perpendicular to the road is defined by the lateral displacement (d) 
Each lane is assigned a lane indicator variable l  The leftmost lane, with respect to the direction of traffic flow, is assigned a value of l = 0 while each subsequent lane is assigned an increasing integer value for l, as depicted in Fig   

Vehicle Model

Since we aim to have real-time computations for a long planning horizon (> 15s), we model the vehicle dynamics with a linearized decoupled dynamical system  For the highway driving scenario, where the road curvature is typically small, it is reasonable to assume a decoupling between the lateral and the longitudinal dynamics , especially for the behavior planning layer 
Therefore, we utilize a linear constant acceleration model for the longitudinal dynamics and abstract out the lateral dynamics with a lane indicator variable  For the lane change dynamics, we use a moving average filter coupled with a rounding function to model the time required by the ego vehicle to change lanes 
This is compactly represented as: where s 0 (k), v 0 (k), l 0 (k) and L(k) denote the ego vehicle's longitudinal displacement, speed, lane indicator and target lane, respectively, at time instant k; the subscript i indexes the vehicles on the road with 0 being reserved for the ego vehicle; T s denotes the discretization time step; and N corresponds to the number of time steps required to change lane 
The state (x 0 (k)) and control input (u 0 (k)) to the system at time instant k are defined as: where V m denotes the maximum speed of the ego vehicle 

Observation Model

For practical considerations, we restrict the ego vehicle's visibility range to the sensory perception limit, denoted by R v   Then, the set of vehicles in ego vehicle's visibility range at time instant k, represented by O(k), is defined as: where s i (k) corresponds to the longitudinal displacement of the observed vehicle 
Remark 1: For the multi-lane highway driving scenario, occlusion does not play a prominent role so we do not account for it in the existing formulation  However, the proposed framework can easily accommodate occlusion and measurement uncertainties since the receding horizon approach bases its decision on the most up-to-date information available at any given time, as demonstrated in  
In this section, we describe the prediction model to generate the predicted future trajectories of observed vehicles and present a discussion on the proposed receding horizon optimization-based behavioral planning module 

Trajectory Prediction

Reliable behavior and trajectory prediction of other traffic participants is crucial for safe maneuvering of autonomous vehicles  The algorithm proposed in Section III-B is able to incorporate any generic prediction module available in the literature as long as it can provide a deterministic predicted future trajectory for a given vehicle 
In this work, we formulate a low-complexity prediction model that highlights the flexibility and efficiency of our proposed approach  For an observed vehicle i ∈ O(k), the future speed profile is predicted using a piece-wise linear function while the lane profile is assumed to stay constant for the duration of the prediction horizon 
At a given time step k, the estimated acceleration (ā k i ) and the estimated speed (v k i ) parameters are obtained through linear regression with mean-squared error on the past o k i > 1 speed observations  Based on the estimated parameters, we predict the future speed and longitudinal displacement as follows:
Here, H a corresponds to the acceleration horizon while vk i (j) and ŝk i (j) respectively represent the predicted speed and longitudinal displacement for vehicle i, j time steps into the future starting from the current time instant k  Remark 2: Due to the modular nature of the proposed framework, the behavior planning module detailed in Section III-B can work with advanced maneuver-based (e g 
Markov Chain ) and interaction-based (e g  Social Generative Adversarial Networks ) trajectory prediction modules, allowing for interactive maneuvering behaviors 

Speed and Lane Advisory System

The goal of our behavior planning module, Speed and Lane Advisory System or in short, SLAS, is to determine a sequence of speed and lane change commands that would enable the ego vehicle to maximize its speed, thus minimizing the travel time, while accounting for driver comfort and abiding by its dynamical, actuator, and safety limits 
The output of this module is a relatively smooth speed and lane change profile which is then passed on to a motion planner  It is necessary to incorporate the dynamical and actuator limits in the behavioral planning module so as not to provide the motion planner with goals that are not reachable, and jeopardize the safety of the overall system as a result 
In the subsequent discussion, we provide a formulation of the optimization problem for SLAS; highlight the modifications necessary to improve the computational complexity; and, present safety and feasibility analysis  1) Optimization Problem with Integer Constraints: SLAS is posed as an optimization problem, with the objective to maximize speed while minimizing frequent lane changes and abrupt changes in speed 
The output of SLAS, at time instant k, is the control input u 0 (k + 1), as defined in   The optimization problem is formulated as follows: Objective Function: In the formulation above, the optimization variables are the ego vehicle's speed (v k (j)) and target lane (L k (j)), j step into the future, starting from time instant k 
Here, H corresponds to the planning horizon  The scalarization parameters γ 1 , γ 2 and γ 3 in the objective function account for a relative tradeoff between maximizing speed, minimizing lane changes and minimizing abrupt changes in speed respectively  Increasing γ 1 yields a more aggressive behavior with the priority placed on maximizing speed while γ 2 and γ 3 combine to place an emphasis on maximizing passenger comfort by reducing lane and speed changes respectively 
Dynamical Constraints: These constraints are put in place to ensure the dynamical feasibility of the solution  The constraints ( ), and serve to initialize the longitudinal displacement, speed and target lane respectively for the optimizer, based on the values observed at time instant k  The constraints and ( ) bound the ego vehicle's speed by the speed limit and the acceleration limits of the vehicle respectively 
The ego vehicle's speed is then used to calculate the projected longitudinal displacement in   The target lane values at any planning step (j) are restricted to the set of reachable values by ( ), ( ) and   Here, restricts the target lane to the set of available lanes (L(k)), ensures that the lane change, if needed, is made to the adjacent lane only and (17) models the time steps (N ) required for a lane change 
The flooring function can easily be transformed into a couple of linear constraints by the introduction of an auxiliary integer variable, as shown in the Appendix  Finally, l k (j) is merely the internal representation of the lane the ego vehicle is projected to travel on at planning step j  Safety Constraint: The safety constraint ensures that the ego vehicle maintains a minimum safe distance (L s i (j)) to the nearest vehicle i, in its projected lane of travel (l k (j)), at planning instant j 
We borrow the definition of this safe distance from , where the authors provide a formalization, based on the clause from Vienna Convention on Road Traffic that states that "A vehicle [   ] shall keep at a sufficient distance [   ] to avoid collision if the vehicle in front should suddenly slow down or stop "
Furthermore, the absolute value constraint can be decomposed into linear constraints by the application of big-M method and the introduction of an auxiliary variable, as shown in the Appendix  Remark 3: The proposed formulation can accommodate arbitrary number of lanes at any given time instant k  This means that if at any given time, the number of available lanes for traveling either increases or decreases, the proposed formulation will still continue to hold 
This is an important consideration since many a times on highways, some lanes are blocked due to various unanticipated situations such as road accidents, roadwork, narrowing of road etc  2) Computational Complexity Reduction: This section details the optimization problem reformulation with binary variables, optimization warm start technique and lazy constraint implementation, all of which combine to improve the computational complexity of our SLAS module 
Binary Variables: The proposed formulation in Section III-B has relatively high computation complexity (computation time of ∼ 2s in the worst case scenario -slow moving traffic blocking all the lanes) due to the integer decision variables yielding a mixed-integer optimization problem   To circumvent the computational overload, we reformulate the problem with binary variables that replace the integer variables, as follows:
where the Lk (i, j) represents the modified target lane variable, indexed by the lane (i) as well as the planning step (j) and Lk (a, b) = 1 represents the choice of lane a ∈ L as the target lane at planning step b ∈ Z   Then, some of the constraints from the SLAS formulation in Section III-B are modified as follows:
Here, initializes the target lane, (21) restricts the target lane at any planning step to the set of available lanes, restricts the lane change between consecutive planning steps to the adjacent lanes, and ( ) represents the augmented safety constraint  The implication ( =⇒ ) in ( ) can easily be transformed into a linear constraint (see Appendix) 
The augmented minimum safety distance ( Ls i (j)) incorporates the time required to execute the lane change maneuver (N ) from ( ) into the following unified safety constraint: where L l is the width of the lanes (see Fig  ), δ(k) is the signed lateral deviation of the ego vehicle from the previous target lane's boundary at time step k, and γ d (δ k (j)) is the dynamic cost of deviation from the previous target lane (L(k − 1)) 
Moreover, in the cost function , we take L k (0) = L(k − 1)  These costs are introduced to prevent the swerving (canceling of lane switch before completion) behavior, unless absolutely necessary (for safety purposes)  Remark 4: Since the ego vehicle is considered to have changed lane once it crosses a lane boundary, the deviation δ k (j) is considered from the lane boundary instead of the center of the target lane to maintain the continuity of γ d (δ k (j)) with respect to the lateral displacement of the ego vehicle 
Specifically, δ k (j) > 0 if the ego vehicle has crossed the previous target lane boundary and 0 otherwise  This is an important consideration since a discontinuity in γ d (δ k (j)), upon completion of lane change, may lead to infeasibility  Remark 5: The swerving behavior is suppressed but not completely eliminated with a hard constraint since such a behavior is necessary at times to react to the environment's unpredictability 
This reactive strategy, which is a distinctive feature of our approach, avails the algorithm the ability to proactively 'change its mind' in case something unanticipated happens in the environment that can jeopardize safety  Optimization Warm Start: To aid the optimizer in finding an initially feasible solution, we provide the solution from the previous time step as a reference 
Formally, This doesn't imply that the solution from time step k − 1 will hold exactly at time step k, owing to the unmodeled disturbances, but providing this reference aids the optimizer in finding an initially feasible solution in the vicinity of the reference solution  This observation is rooted in the premise that the solution for the long planning horizon is not expected to change significantly between time steps, given the sampling time is not too large, and the predicted behavior of on-road agents does not alter significantly 
It is also worth pointing out that the priority here is quickly finding a feasible solution that obeys the safety constraints and actuator limits, and recursively improving it rather than excessively iterating to reach at a global optimum  In our experiments, it was observed that a suboptimal solution was qualitatively not significantly different from the optimal one 
Therefore, we utilize the cutting planes method for optimization , which first looks for a feasible solution, using our provided reference, and then recursively updates it until either the globally minimal solution is found or the time limit is reached  Lazy Constraints: To further enhance the computational efficiency, we introduce a lazy implementation of the lane changing constraints  
It was observed in our experiments that a feasible solution without the lane changing constraints ( ) can be found several order of magnitude (∼ 10×) quicker than if we include these constraints so we decided to have a lazy implementation for them  With a lazy implementation , the solver finds a set of feasible solutions without the inclusion of these constraints and then determines the feasibility of those solutions from the reduced problem with respect to the lazy constraints 
3) Feasibility: By an argument similar to the one presented in , it is a relatively straightforward proof for recursive feasibility of the problem, i e  the optimization problem will continue to stay feasible, if initially feasible, with the trivial solution being matching the speed of the leading vehicle and not changing lanes 
In this section, we detail our experimental setup, demonstrate the performance of SLAS, and report a qualitative as well as a quantitative comparative analysis  The baselines in our comparative analysis are set to: Extended-Astar (EA ) , MOBIL , and no lane-change model (No-Change) 

Experimental Setup

The implementation setup, depicted in Fig  , is composed of the CARLA simulator (Version 0 9 11) , SLAS module (Section III-B), and the planner and controller module   To solve the optimization problem for SLAS, we use Gurobi Optimizer (Version 9 1 1)   The simulations are performed on a computer equipped with an Intel Xeon(R) CPU E5-2643 v4 @ 3 40GHz × 12 and NVIDIA Titan XP, running Ubuntu 20 04 LTS 
On average, the time required for each optimization step is ∼ 0 096s, while the maximum time limit for the optimizer is set to 0 2s, indicating the strong potential for real-time applicability 

Case Study

Figure illustrates the test case scenario for our comparative analysis  The scenario is composed of a highway segment with four lanes and the rightmost lane reserved for merging vehicles  The ego vehicle is initialized to follow a slow moving vehicle in lane 1 and has even slower moving traffic to its right in lane 2  Thus, the only option for it, in order to minimize travel time, is to switch left to lane 0 with faster moving traffic and greater headway 
Once it moves to lane 0, and overtakes the slow moving vehicle in lane 1, it has two options: either to keep traveling in lane 0 without making any lane change decisions until getting close to the lead vehicle or proactively exploiting the gap in lane 2 to switch to lane 3 in anticipation of traffic buildup in lanes 1 and 2  A strategic decision maker with foresight will choose to take the later option and make the decision proactively for a greater overall benefit 
The evaluation metrics for the comparative analysis include: travel time, lateral displacement, headway and distance to the closest vehicle  As for the simulation parameters, the simulation step size is set to 0 05s (simulation frequency of 20Hz); the velocities of vehicles in lanes 0, 1 and 2 are set to 8, 5 and 2 m/s respectively while the speed limit V l is set to 15m/s; the length of the highway patch is set to 350m while the width between the lanes is set to 3 5m; and the sensor visibility range is set to R v = 50m 
The parameters for SLAS are set as follows: T s = 0 4s, H = 40, N = 3, A min = −5m/s 2 , A max = 3 5m/s 2 , γ 1 = 1, γ 2 = 0 1 and γ 3 = 0 01  The values of these parameters can be tuned to yield an aggressive or defensive behavior of the algorithm  1) Travel Time: The left plot in Fig  depicts the travel time as a function of longitudinal displacement for the four algorithms 
As seen in the plot, our method (SLAS) maintains a lower overall travel time as compared to the other methods  Quantitatively speaking, SLAS outperforms EA , MOBIL and No-change methods by 12 72%, 23 52% and 54 34% respectively in terms of the time required to complete the simulation scenario  This shows that our method's foresight compensates for its apparent conservativeness arising from the need to preserve passenger comfort 
2) Lateral Displacement: To identify the differences in lane changing behaviors between the four approaches, the relationship between lateral and longitudinal displacements over the course of the simulation is highlighted in the center plot of Fig    In the plot, the lateral displacement of 0 corresponds to the center of lane 0 while the center of each following lane is 3 5m away 
Comparing the performance of the four algorithms, we see SLAS and EA showing relatively similar performances, resulting from proactive decision making  In contrast, since MOBIL only assesses the advantage of switching to the adjacent lanes, it is unable to see the benefit of proactively switching to lane 2  This explains why EA and SLAS start outperforming MOBIL in terms of travel time (left plot) at around the 130 [m] mark for longitudinal displacement 
As for a direct comparison between SLAS and EA , the benefits of having speed advisory system become apparent in this center plot  Due to speed control, SLAS is able to constantly maintain a greater headway (right plot) without having to brake significantly upon getting too close to the lead vehicle  This results in a smooth lateral displacement profile which allows the vehicle to change lanes with minimal jerk (quantitative analysis to follow in Section IV-C) and deliver better overall timing performance (left plot) 
3) Headway: The right plot in Fig  shows the headway maintained by the ego vehicle over the course of the simulation  In accordance with our prior discussion, MOBIL cruises behind the front vehicle, maintaining a relatively low headway until a sufficient space in the adjacent lane is found to perform the lane-change maneuver 
On the other hand, EA and SLAS show a comparable headway trajectory, however, SLAS maintains a greater headway throughout and achieves the maximum headway prior to EA   Quantitatively, SLAS maintains on average 9 43%, 36 57% and 113 17% more headway than the EA , MOBIL and No-change approaches respectively 
This strong performance by SLAS can be attributed to its incorporation of safety guarantees coupled with its consideration for passenger comfort  4) Distance to closest vehicle: Finally, we compare the distance that ego vehicle maintains from the closest vehicle throughout the simulation  On average, SLAS maintains 9 28%, 32 01%, and 22 84% more distance in comparison to EA , MOBIL and No-change approaches respectively 
These numbers are a testament to the strength of our approach resulting from consideration of long planning horizon coupled with speed control 

Monte Carlo Simulations

To demonstrate the long-term performance of the three approaches (SLAS, EA and MOBIL), we run a series of Monte Carlo simulations on scenarios with randomized initial positions (within a range of 8m) and velocities (within    ranges of 8, 5 and 2 m/s assigned to each of the three lanes randomly) of traffic participants 
The result from 50 simulations is presented in Table   In this table, the columns represent the different evaluation metrics, the rows identify the three algorithms, and the values highlighted in green represent the best result with respect to each evaluation metric  The evaluation metrics, going from left to right in the table, are completion time (s), brake (R [−1,0] ), brake jerk (R [−1,0] ), throttle (R [0,1] ), throttle jerk (R [0,1] ), angular acceleration ( • /s 2 ) and angular jerk ( • /s 3 ) 
Apart from completion time, the remaining metrics, based on the commands passed to the vehicular actuators (Fig  ), are used to model passenger comfort  In terms of average performance, SLAS greatly outperforms the other methods when it comes to passenger comfort since it explicitly accounts for comfort in the formulation 
However, it does so at a cost of slightly reduced performance in regards to travel time, when compared to EA , since SLAS tries to strike a balance between minimizing travel time and maximizing passenger comfort  SLAS also secures the lowest standard deviation, for each of the evaluation metrics, when compared to the other methods, which points to the consistency in its long-term performance 

CONCLUSION

We propose a novel behavior planning module for the multi-lane highway maneuvering scenario, that outputs strategic target lane and reference speed commands, and incorporate it with a state-of-the-art motion planning and control framework  We formulate the approach as a receding horizon mixed integer optimization with the goal to minimize travel time while accounting for passenger comfort for a long planning horizon 
In order to reduce the computational overload, we reformulate the problem by replacing integer variables with binary ones and further incorporate various modifications to aid numerical computations  We also carry out a detailed comparative analysis to demonstrate the performance of our approach on the CARLA simulator 
Our future work includes incorporating various delays and uncertainty measures in the perception, localization and prediction modules to evaluate the robustness properties of our approach 

Flooring Constraint

For y ∈ Z and x ∈ R, the constraint y = x can be represented by the following linear constraints: y ≤ x, y + 1 ≥ x + where > 0 accounts for the feasibility tolerance  where M 0 (big-M) and > 0 accounts for numerical errors (chosen to be 0 1 in our implementations) 

Absolute Value Constraint

For ∆s, L s ∈ R, the constraint |∆s| − L s ≥ 0 can be represented as: This can further be generalized, as done in our implementation, to have different forward and rear safety margins as: ∆s ≥ L f s ∨ ∆s ≤ −L r s where L f s and L r s are forward and rear safety margins respectively  This can be represented with the following linear constraints:
s ≥ 0 where M 0 (big-M) and c ∈ {0, 1} is responsible for making a choice between the two constraints 
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新品快览 FS312A：PD3 0 诱骗- FS312A支持PD2 0/PD3 0最高诱骗电压：20V - FS312AE支持PD2 0/PD3 0 最高诱骗电压：20V支持Emarker模拟功能 - 封装：SOT23-5 VBUS CC1 CC2 DM DP 用电电路 4 7K 0 47uF R C C 1 V D D F U N C C C 2F S 3 1 2 B D M D P EP GND 应用图 FS8628：A+C快充协议CC2 CC1 VBUS CC2 CC1 FS312A FUNC GND VDD 4 7K GND R 用电电路 1uF GND 应用图 多口极简方案 FS8611SP*2+CCM-8611SP-A+7533B-T 双C智能降功率方案 FS8611S USB-C AC-DC 双变压器 7533B-T CCM-8611SP-A FS8611S USB-C 采用2颗FS8611SP搭配CCM-8611SP-A （MCU），7533B-T配合工作 - 支持多种协议 - 支持I2C控制 - 任意单 C 的为 35W - 双 插 降 功 率 ， 三 档 功 率 智 能 配 置：27 4W+7 4W；17 4W+17 4W； 27 4W - BOM极简，成本低 FS312B：PD3 1 诱骗FS8611K*2+CCM-8611K-A+7550B-T 双C方案 - FS312BL支持PD2 0/PD3 0/PD3 1/第三方协议最高诱骗电压：20V - FS312BLE支持PD2 0/PD3 0/PD3 1/第三方协议最高诱骗电压：20V支持Emarker模拟功能 - FS312BH支持PD2 0/PD3 0/PD3 1/第三方协议最高诱骗电压：48V - FS312BHE支持PD2 0/PD3 0/PD3 1/第三方协议最高诱骗电压：48V 支持Emarker模拟功能 - 封装：DFN2x2-6L - 兼容兼容BC1 2、Apple2 4A、 QC2 0 Class A、QC3 0 Class A/B、 FCP、SCP、AFC、低压直充等 - 兼容Type-C PD2 0、Type-C PD3 0、 Type-C PD3 0 PPS、QC4 0协议 - 支持两路DP/DM - 支持CV/CC（分段CC）功能 - 支持定制PDO - 支持A+C双口工作，电压自动回5V - 支持FB/OPTO反馈 - 封装：QFN3x3-20L VPWR FB PowerSystem 100K GND R1 GND 19 VIN 17 FB FUNC1 FUNC2 20 15 18 13 PLUGIND VFB FS8628 QFN3x3-20L AGATE 47K 7 5K 47K 7 5K 1 16 8 7 3 4 5 6 10 9 11 CGATE CVBUS CC2 CC1 CDP CDM AVBUS DM DP ISP ISN 12 应用图 2 V3P3 100Ω 1u EP GND GND CVBUS TYPE- C CC2 CC1 CDP CDM CGND TYPE-A AVBUS DM DP 10n 200 AGND 5mΩ GND FS8611K USB-C AC-DC DC-DC 7550B-T CCM-8611K-A FS8611K USB-C 采用2颗FS8611K搭配CCM-8611K-A （MCU）工作，7550B-T配合工作 - 支持PD2 0/PD3 0/QC2 0/AFC/FCP - 支持PDO定制 - 任意单 C 的为 35W(可定制) - 双插18W（可定制15W/20W） - BOM极简，成本低 FS212C+ACM-212C-A+7550B-T 双C方案 FS212C USB-C AC-DC DC-DC 7550B-T ACM-212C-A FS8623B-A+C方案 AC-DC DC-DC FS8623B USB-A USB-C USB-A 采 用 1 颗 F S 2 1 2 C 搭 配 ACM-212C-A 工 作，7550B-T配合工作 - 支持PD2 0/PD3 0 - 支持PDO定制 - 任意单 C 的为20W - 双插7 5W回5V - BOM极简，成本低 采用一颗FS8623B实现A+C方案 - 兼容兼容Apple2 4A/低压直充 QC2 0 Class A/QC3 0 Class A/B/ FCP/SCP等 - 兼 容Type -C PD2 0 / PD3 0 / PD3 0PPS/QC4 0协议 - 支持PDO定制 - 双插回5V 
多口方案选型 产品选型 受电端芯片选型 速芯微现有多种多口的方案选择：A+C，C+C，C+C+A，C+C+C，C+C+A+A等方案。对于 A+C的方案，可使用1颗芯片实现，也可用多颗芯片来实现。 速芯微现有多种受电端诱骗芯片，客户可根据应用需求进行选择。 受电端诱骗芯片应用领域 筋膜枪 无线充 线材 无人机 产品型号 PD2 0 PD3 0 PD3 1 第三方协议 诱骗电压(V) 控制方式 内置Emarker 定制 封装 FS312A √ √ 5/9/12/15/20 电阻阻值 可变电压策略 SOT23-5 FS312AE √ √ 5/9/12/15/20 电阻阻值 √ (公头专用) 可变电压策略 SOT23-5 FS312BL √ √ √ √ 5/9/12/15/20 电阻阻值 可变电压策略 DFN2x2-6 FS312BLE √ √ √ √ 5/9/12/15/20 电阻阻值 √ (公头专用) 可变电压策略 DFN2x2-6 FS312BH √ √ √ √ 5/20/28/36/48 电阻阻值 可变电压策略 DFN2x2-6 FS312BHE √ √ √ √ 5/20/28/36/48 电阻阻值 √ (公头专用) 可变电压策略 DFN2x2-6 FS312LC √ √ √ 5/9/12 电阻阻值 可变第三方 协议 SSOP10 FS312HC √ √ √ 5/9/12/15/20 电阻阻值 可变第三方 协议 SSOP10 FS2711Q √ √ √ 任意设置 I2C √ QFN3x3-16 FS2711P √ √ √ 任意设置 I2C √ QFN3x3-16 FS2711PA √ √ 全协议 任意设置 I2C √ SSOP10 FS2711SW √ √ 全协议 SSOP10 FS512 √ √ 全协议 任意设置 I2C √ SSOP10 方案 类型 产品型号 单C 单A 双插 A+C方案 FS8623 20W（PPS）（可定制） A口全协议18w 5V共享3A FS8623B 20W（PPS）(可定制) A口全协议18w 5V共享3A FS8628 20W（PPS）(可定制) A口全协议18w 5V共享3A FS8611RPC+FS116DB 65W（PPS）(可定制) A口全协议18w A口：5V/2 4A C口：45W FS8628RC+FS116DB 35W(可定制) A口全协议18w A口：5V（BC1 2，Apple 2 4） C口：20W 方案类型 产品型号 单C1 单C2 C1/C2 C+C方案 FS8611RPB*2 30W(可定制) 30W(可定制) C1/C2：5V/3A(或5V/2 4A) FS8611GH*2 35W（可定制） 35W（可定制） C1/C2:18W（可定制） FS8628P*2 35W（可定制） 35W（可定制） C1/C2：17 4W可定制） FS8611KL*2 20W（可定制） 20W（可定制） C1/C2：5V/1 5 A FS8611PC*2 35W 35W C1/C2：18W FS8611BH*2 65W（可定制） 65W（可定制） C1：45W（可定制）C2：20W（可定制） FS8628RPC+FS8611RB 45W（可定制）） 36W （可定制）） C1：30W（可定制）C2：5V/1 5A（可定制） 方案类型 产品型号 单C1 单C2 单A C1+C2 C1/C2+A C1+C2+A C+C+A FS8611S*2+FS116DB 65W（可定制） 65W（ 可定制）） A口全协议18w 智能分配功率 45W+18W C1/C2：智能分配功率 A：18W（或5V1 5A） FS8612C+FS8628P 100W（可定制） 35W （可定制）） 20W C1：65W C2：20W C1+A：65W+20W C2+A：7 5W+7 5W C1：65W C2：7 5W A：7 5W 其他 
Source-TYPE C协议芯片选型 Source-TYPE A协议芯片选型 速芯微现有多种TYPE-C的快充协议芯片，支持多种协议，支持客户定制，多样化，满 足客户对TYPE C的各种快充需求。 速芯微现有多种TYPE A快充协议芯片，支持全协议，支持定制，满足客户对A口协议的各种需 求。速芯微的TYPE-A快充协议芯片的协议丰富，FS112系列拥有多种的型号；FS116D 系列带插入指示，可搭配TYPE-C快充协议芯片，实现A+C，A+C+C，A+A+C+C等多口方 案，协议丰富，其中FS116A一般用于插入指示使用 Source-TYPE A协议芯片引脚封装图 D+ VSS FB 1 2 3 FS112 6 5 4 D- VDD FUNC GATE VIN FUNC FB LED/PLUG_IN 1 2 3 4 5 FS116D 10 DM 9 8 7 6 DP CSP CSN VSS速芯微的各TYPE-C快充协议芯片之间可搭配使用，实现多口方案，更多详情请咨 询我司工作人员。 多口降功率专用快充协议芯片：FS8611RB，FS8611RC，FS8611RPB，FS8611RPC， FS8612CP。 带I2C快充协议芯片：FS8611S，FS8611SP 产品型号 BC1 2 Apple 2 4 QC2 0 QC3 0 AFC FCP SCP HISCP 大电流直充 封装 FS112 √ √ √ √ √ √ √ SOT23-6 FS112H √ √ √ √ √ √ √ √ √ SOT23-6 FS113 √ v √ √ √ √ √ √ √ SOT23-6 FS116DP √ √ √ √ √ √ √ √ SSOP10 FS116DB √ √ √ √ √ √ √ √ SSOP10 FS116E √ √ √ √ √ √ √ √ √ SSOP10 FS116A √ √ SSOP10 其他 可定制 PD2 0 PD3 0 PD3 0 PPS 第三方协议 反馈方式 MOS CV/CC 定制 封装 FS212C √ √ FB √ SOT23-6 FS212CM √ √ FB PMOS(可省) √ SOT23-6 FS212D √ √ √ FB √ SOT23-6 FS212DH √ √ √ FB √ SOT23-6 FS212DP √ √ √ FB PMOS √ SOT23-6 FS212DG √ √ √ FB PMOS √ SOT23-6 FS8611G √ √ FB PMOS(可省) √ SOP-8 FS8611K √ √ QC2 0/AFC/FCP FB PMOS(可省) √ SOP8 FS8611J √ √ √ 全协议 FB PMOS(可省) √ SOP8 FS8611B √ √ √ 全协议 FB PMOS(可省) √ SSOP10 FS8611RB √ √ 全协议 FB PMOS √ SSOP10 FS8611RC √ √ 全协议 FB PMOS √ SSOP10 FS8611S √ √ √ 全协议 FB PMOS √ SSOP10 FS8611PP √ √ √ 全协议 FB PMOS √ SSOP10 FS8611BP √ √ √ 全协议 FB PMOS(可省) √ SSOP10 FS8611RPB √ √ √ 全协议 FB PMOS √ SSOP10 FS8611RPC √ √ √ 全协议 FB PMOS √ SSOP10 FS8611SP √ √ √ 全协议 FB PMOS(可省) SSOP10 FS8612 √ √ √ 全协议 OPTO PMOS √ √ SSOP16 FS8612B √ √ √ 全协议 FB PMOS √ √ SSOP16 FS8612BP √ √ √ 全协议 FB PMOS √ √ SSOP16 FS8612C √ √ √ 全协议 FB/OPTO PMOS √ √ QFN4x4-16 FS8612CP √ √ √ 全协议 FB/OPTO PMOS √ √ QFN4x4-16 
'
Context 3: 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



ground_truth: Reduced computational complexity.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: 对于PD3.0协议，FS312BH支持的最高诱骗电压是多少？

Read the following text and answer briefly.

Context 3:  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



ground_truth: 48V.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Can the denoiser be applied to circuits with non-Clifford noise?

Read the following text and answer briefly.

Context 3: 
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 



ground_truth: Yes, the denoiser works for non-Clifford local noise channels.