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 was the best performing model for the Spanish language in Track-1?

Read the following text and answer briefly.

Context 1: Paper Info

Title: Two-stage Pipeline for Multilingual Dialect Detection
Publish Date: Unkown
Author List: Ankit Vaidya (from Pune Institute of Computer Technology), Aditya Kane (from Pune Institute of Computer Technology)

Figure

Figure 1: Class distribution of dialects
Figure 2: System diagram for dialect classification The LID classifies the input into one of 3 languages The sample is then further classified into dialects by language specific models 
Figure 3: Confusion matrix of 9-way classification Note that rows are normalized according to the number of samples is that class 
Our complete results for Track-1 using the two-stage dialect detection pipeline Model-* denotes the language of the models used for the experiments 
Performance on Track-1 validation dataset of individual models used in the two-stage pipeline "Lg" stands for language of the model used 
Comparative results of two-way classification using the finetuned (F T ) predictions and predictions adapted from three-way classification models 

abstract

Dialect Identification is a crucial task for localizing various Large Language Models  This paper outlines our approach to the VarDial 2023 DSL-TL shared task  Here we have to identify three or two dialects from three languages each which results in a 9-way classification for Track-1 and 6-way classification for Track-2 respectively 
Our proposed approach consists of a two-stage system and outperforms other participants' systems and previous works in this domain  We achieve a score of 58 54% for Track-1 and 85 61% for Track-2  Our codebase is available publicly 1  

Introduction

Language has been the primary mode of communication for humans since the pre-historic ages  Studies have explored the evolution of language and outlined mathematical models that govern the intricacies of natural language   Inevitably, as humans established civilization in various parts of the world, this language was modified by, and for the group of people occupied by that particular geographical region 
This gave rise to multiple national dialects of the same language  The VarDial workshop (colocated with EACL 2023) explores various dialects and variations of the same language  We participated in the Discriminating Between Similar Languages -True Labels (DSL-TL) shared task  In this task, the participants were provided with data from three languages, with each language having three varieties 
This shared task consisted of two tracks -Track-1 featuring nine-way classification and Track-2 featuring six-way classification  The second track included two particular national dialects of each language (eg  American English and British English), and the first track had one general  We ranked 1 st in both of the tracks 
Moreover, we beat the next best submission by a margin of 4 5% in the first task and 5 6% in the second task We were the only team to surpass the organizer baseline scores  We present our winning solution in this paper  We used an end-to-end deep learning pipeline which consisted of a language identification model and three language-specific models, one for each language 
We converged upon the best combination by doing an elaborate analysis of various models available  Furthermore, in this work we also analyze the performance of the pipeline as a whole and also provide an ablation study  Lastly, we provide some future directions in this area of research 

Related Work

The present literature encompasses various aspects of dialect identification  We study this from three perspectives: large language models, language identification and dialect classification problems 

Large Language Models

The success of transformers and BERT based models was inevitable since the initial boom of the transformer  2017) model  In recent years, many other architectures like RoBERTa and ELECTRA have further pushed the state-of-the-art in this domain  Moreover, autoregressive models like GPT and GPT-2 have also shown their prowess 
Multilingual versions of RoBERTA, namely XLM-RoBERTa are also available  Lastly, language specific models like Spanish BERT (la Rosa y Eduardo G  Ponferrada y Manu Romero y Paulo Villegas y Pablo González de Prado Salas y María Grandury, 2022) and Portuguese BERT are available as well  Our winning solution makes use of these large language models trained on specific languages 

Language Identification Models

Many multilingual language identification models have been developed in order to classify the language of the input sentence beforehand  Even though the initial works used n-gram models and generative mixture models or even conditional random fields and other classical machine learning methods like naive bayes , modern methods have shifted to the use of deep learning for language identification  
Recent works have mainly focused on deep learning based language identification, where handling codemixed data is a big challenge in the domain  For our experiments, we use a version of XLM-RoBERTa finetuned on a language identification dataset 2   This model has near-perfect test accuracy of 99 6% 

Dialect Classification

Dialect classification has been previously solved using statistical methods like Gaussian Mixture Models and Frame Selection Decoding or Support Vector Machines (SVM)   It has been explored relatively sparsely, mostly in the case for local languages   Deep learning approaches have been explored in previous editions of the VarDial workshop shared tasks and otherwise  
Dialect classification was also explored previously as a part of other shared tasks   We want to stress that given the multilingual nature of the dataset, using the present methods directly was not an option  In our work, although we take inspiration from the previous works, we propose a novel system that surpasses the performance of the previous systems by a large margin 

Data

The dataset  We observed that the class PT-BR had the most number of samples (2,724) and the class EN had the least number of samples (349), and thus the imbalance ratio was almost 1:8  We have illustrated the data distribution in Figure   We tried to mitigate this imbalance using over-sampling and weighted sampling methods 
However, the improved data sampling method did not affect the performance 

System Description

This was a problem of multi-class classification having 9 classes for Track-1 and 6 classes for Track-2  The samples were belonging to 3 languages having 3 varieties each, so the classification pipeline was made in 2 stages  The Language Identification (LID) model which is the first stage classifies the sentence into 3 languages: English (EN), Spanish (ES) and Portuguese (PT) 
The LID is a pretrained XLM-RoBERTa that is fine-tuned for the task of language identification  It is able to classify the input sentence into 20 languages  We classify and separate the samples according to their language  The samples corresponding to the specific languages are then fed into the language specific models for dialect identification 
For dialect identification we have used models like BERT and RoBERTa with a linear layer connected to the pooler output of the models  Then fine-tuning is done on the models for dialect identification   using the samples corresponding to the specific languages  For the task of dialect identification we experimented with several pretrained models like XLM-RoBERTa, BERT, ELECTRA, GPT-2 and RoBERTa 
All models were fine-tuned for 20 epochs with a learning rate of 1e-6 and weight decay 1e-6 with a batch size of 8  The best performing model checkpoint was chosen according to the epoch-wise validation macro-F1 score  5 Experiments and Results

Experiments using Large Language Models

For the task of Dialect Identification we have tried various language specific models like XLM-RoBERTa, BERT, ELECTRA, RoBERTa and GPT- 2  The base variant of all these models were used and all the models were used through the Hugging-Face library  The pooler output of these models was passed through a linear layer and the models were fine-tuned 
First, we experimented with different models for Track-1  All the models were trained for 20 epochs with learning rate 1e-6, weight decay 1e-6 and a batch size of 8  We used XLM-RoBERTa as the baseline for all 3 languages  The best performing models for the English language were RoBERTa and BERT whereas GPT-2 was the worst performing 
Similarly the language specific versions of RoBERTa and BERT performed well for the Spanish and Portuguese respectively  Overall the worst performing model was GPT-2 across all 3 languages  The validation F1 scores are present in Table   The two best-performing models for every language were chosen for Track-2 
The same procedure as specified above was used and the F1 scores are present in Table   The train and validation F1 scores for 2-class classification are higher for all models as compared to the F1 score of the same models for 3-class classification  This was mainly due to the poor representation and accuracy of classification of the third class 
We observed symptoms of overfitting in all models after 12-15 epochs and the best validation F1 score was obtained in the range of 4-8 epochs 

LID experiments

The pipeline for dialect identification is divided into two parts as the sentences in the dataset belong to different languages  The stages are described in Section 4  The XLM-RoBERTa we have used for language classification has a test accuracy of 99 6% meaning it correctly classifies all input sentences and hence, can be considered as a perfect classifier 
For the final pipeline we experimented using the two best performing models for each language in Track-1 and Track-2  For both the tracks we experimented with all 8 (2 3 ) possible combinations of models and calculated the validation F1 score for the combined validation dataset which had sentences belonging to all languages 
The validation scores for Track-1 and Track-2 are shown in Table and Table respectively  For both the tracks, the three pipelines with the best validation F1 scores were chosen for submission 

Using 3-way classifier as a 2-way classifier

In Track-1, participants are expected to train a classifier which classifies amongst 9 classes, and in Track-2, participants are expected to train a classifier which classifies amongst 6 classes  These 6 classes are a proper subset of the 9 classes from Track-1  Thus, an intuitive baseline for Track-2 is to use the model finetuned for Track-1, whilst considering only the relevant classes for the latter task 
The classes EN , ES and P T , i e  the classes without any national dialect associated with them are not included in Track-2 as compared to Track-1  Thus, we calculate the predictions for the Track-2 validation dataset using the models for Track-1 and exclude the metrics for Track-1 specific classes to get the metrics for this "adapted" 2-way classification 
We show the results of this experiment in Table and observe that, as expected, the adapted 2-way classification performs worse compared to the explicitly finetuned variant 

Results for Track-1 and Track-2

We now present our experiments and their performance for both tracks  Our experiments for Track-1 are described in Table and our experiments for Track-2 are described in Table   The participants were allowed three submissions for evaluation on the test set, so we submitted predictions using the three systems which performed the best on the validation set 
As mentioned in Section 5 2, we performed 2 3 , i e  a total of 8 experiments using the two best models for each language  We observed that RoBERTa base on English, Spanish BERT base on Spanish and Portuguese BERT base performed the best on the testing set for Track-1  The same combination, with RoBERTa base for English, worked best for Track-2 
All of our submissions were the top submissions for each track, which surpassed the next best competitors by a margin of 4 5% and 5 6% for Track-1 and Track-2 respectively 

Ablation of best submissions

We hereby make some observations of our submissions and other experiments  To assist this, we plot the confusion matrices of our best submissions for Track-1 and Track-2 in Figures respectively  Note that these confusion matrices have their rows (i e  true labels axes) normalized according to the number of samples in the class 
Here are observations from our experiments: 1  BERT-based models outperform other models across all languages: We observe that BERT-based models outperform ELECTRA-based and GPT-2-based models, as shown in Table   We speculate this is because of the inherent architecture of BERT, which combines semantic learning with knowledge retention 
This combination of traits is particularly useful for this task  2  Common labels perform the worst across all languages: We observe that the common labels EN , ES and P T perform the worst, both in the individual as well as the two-stage setup  We hypothesize this is because of the absence of dialect specific words, or words that are specific to the geographical origin of the national dialect (for example, "Yankees" for EN-US and "Oxford" for EN-GB) 
3  English models work better than models of other languages: It can be noted from Figures 4 and 3 that the English models have the best performance across all classes  This can be attributed to two reasons: absence of national dialect specific words and lesser pretraining data in the case of Portuguese 
4  British English is most correctly classified class: We can observe that the Spanish or Portuguese models make equal number of mistakes in the case of either national dialect, in the case of Track-2 (see Figure )  However, in the case of English, the label EN − GB is correctly classified for more than 95% of the cases 
We speculate this is because British English involves slightly distinctive grammar and semantics, which help the model separate it from other classes  5  The proposed 2-step method is scalable for multiple language dialect classification: We can strongly assert that the novel 2-step deep learning method for multilingual dialect classification is a scalable method for the task due to two specific reasons: firstly, the multilingual models (like XLM-RoBERTa) might not have the vocabulary as well as the learning capabilities to learn the minute differences between individual dialects 
Secondly, this system can be quickly expanded for a new language by simply adding a language specific dialect classifier, provided the language identification model supports that particular language 

Conclusion

In this paper we propose a two-stage classification pipeline for dialect identification for multilingual corpora  We conduct thorough ablations on this setup and provide valuable insights  We foresee multiple future directions for this work  The first is to expand this work to many languages and dialects 
Secondly, it is a worthwhile research direction to distill this multi-model setup into a single model with multiple prediction heads  The obvious limitation of this system is the excessive memory consumption due to the usage of language specific models  For low resource languages this system is difficult to train and scale 
We hope that these problems will be addressed by researchers in future works 
Context 2: The major actions taken by the board of directors and council during the national meeting in Chicago were reported in C&EN, April 30 (page 32) 
The Society Committee on Budget & Finance met on Saturday, March 24, to review the society's 2006 financial performance  The society ended 2006 with a net contribution from operations of $12 2 million, on revenues of $424 0 million and expenses of $411 8 million  This was $7 8 million favorable to the approved budget 
After including the results of the Member Insurance Program and new ventures, the society's overall net contribution for 2006 was $11 5 million, which was $7 4 million favorable to the approved budget  The favorable variance was primarily attributable to higher than budgeted electronic services revenue and investment income, as well as expense savings from lower than budgeted health care costs and reduced IT spending  In addition, the society ended the year in compliance with the board-established financial guidelines 
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The committee ratified two interim actions taken since the Dec  3, 2006, meeting: to remove the financial restriction of the ACS Petroleum Research Fund (ACS PRF) Supplement for Underrepresented Minority Research Programs (SUMR) and to contact nominators whose nominations for the Volunteer Service Award had expired and to invite them to reactivate their nomination packet for the 2008 Volunteer Service Award 
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The committee also reviewed the final report from the Special Board Task Force on the Review of the ACS National Awards Program, chaired by Ronald Breslow; established a Canvassing & Selection Subcommittee; and reviewed a list of external awards for which ACS may want to nominate candidates  The committee agreed to include the list of significant external awards in the awards locator database that is being developed 
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ACS President Hunt discussed her 2007-08 Presidential Task Force on Enhancing Science & Technology, which is charged with developing advocacy best practices that can enhance ACS's attainment of its public policy priorities  The task force is composed of a diverse set of ACS members as well as former U S  Representative and chairman of the House Science Committee, Sherwood Boehlert, who will cochair the task force 
• Results of the 2007 Public Policy Priorities Survey, which resulted in a four-tiered ranking of ACS's 2007 public policies  The ranking will help focus staff resources in conducting outreach and advocacy on behalf of ACS members 
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The committee's Task Force on Laboratory Environment, Health & Safety is working on a new edition of "Laboratory Waste Management " Task force members also commented on the recent Environmental Protection Agency Proposed Rule for Hazardous Waste in Academic Laboratories  Our Video Safety Resources Task Force is developing video resources to be distributed over the Web 
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The CWD Ambassador Program that was announced at CWD's 25th anniversary celebration at the Washington, D C , meeting was discussed  Zelda Wasserman reported on the status of the letter from CWD to the ACS Board regarding captioning of ACS video materials  Janelle Kasper-Wolf, of ACS staff, discussed adding new questions to the ACS annual employment salary survey to obtain information for the committee 
At the Chicago national meeting, the Committee on Community Activities (CCA) partnered with the ACS Education Division and the Office of the President to host "Chemistry In Action—It's Easy Being Green" at the Peggy Notebaert Nature Museum on Saturday, March 24  More than 250 children participated in the hands-on activities focused on recycling  ACS President Hunt presented a Salutes to Excellence plaque to the museum for its dedication to community outreach 
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The subcommittee also requested that ACS staff provide CCA with an official annual statement of Corporation Associates' financial reserves as of Jan  1 of each year 
The Programs Subcommittee reported on planned programming activities in 2007 and beyond between CCA and SCI  The subcommittee gave an update on a Boston symposium cosponsored by Corporation Associates and the Medicinal Chemistry Division featuring past ACS Heroes of Chemistry from the pharmaceutical industry 
By request of the subcommittee, representatives from Chemical Abstracts Service gave an overview on AnaVist—a tool with potential applications for CCA's efforts to provide member companies with industry-relevant information and reports  The subcommittee also requested that CCA earmark approximately $20,000 of Corporation Associates funds in 2008 for its activities 
The Educational Outreach Subcommittee reported on its decision to collaborate with the Graduate Student Symposium Programming Committee of the Chemical Education Division on a graduate student industry roundtable program in Boston 
The subcommittee requested $5,000 in support of this effort  The subcommittee also discussed a request for corporate executive support of an American Association of Physics Teachers initiative to promote undergraduate research 
The Committee on Environmental Improvement (CEI) continues to be focused on the sustainability of the chemical enterprise  In Chicago, the committee introduced a multiyear effort to make ACS meetings more sustainable  This effort is designed to take advantage of the large size and diverse programs of the society to lead in the sustainability arena by "walking the talk "
The committee held a dialogue with representatives of the U S  Environmental Protection Agency who are trying to "green" federal conferences and work with the travel and tourism industry to change practices and shrink the environmental footprint of meetings  The committee also used advertising and student volunteers to engage individual meeting participants in a campaign to increase recycling by asking "Are you sustainable?" Moving forward, CEI looks forward to working closely with the Committee on Meetings & Expositions to advance this agenda 
CEI was also pleased to participate in the meeting theme on sustainability through the ACS presidential programming  CEI cohosted the Monday presidential luncheon to discuss sustainability issues with the Committee on Science and is leading the follow-up to that luncheon, which will include recommendations on advancing sustainability in the three focal areas of the meeting—energy, food, and water 
The committee also continued its dialogue with the Committee on Corporation Associates about a collaborative workshop  This activity, tentatively slated for the New Orleans meeting, will seek additional insights from chemical and allied products companies about public policy barriers that limit adoption of more sustainable products and practices as well as policy incentives that would lead to increased sustainability in the chemical enterprise 
At its Chicago meeting, the committee welcomed the president of the Jordanian Chemical Society and the past-president of the Arab Union of Chemists 
The committee was briefed on Pittcon 2007, where, with financial support from the Society of Analytical Chemists of Pittsburgh, ACS coordinated participation of a scientific delegation from Adriatic nations 
The committee heard reports on the 2007 Frontiers of Chemical Science III: Research & Education in the Middle East meeting; the 2007 Transatlantic Frontiers of Chemistry meeting, which was jointly sponsored by ACS, the German Chemical Society, and the Royal Society of Chemistry; planned workshops to engage U S  and Chinese early-career scientists in chemical biology, supramolecular, and new materials chemistry; and ACS Discovery Corps U S /Brazil Research Collaboration Project in Biomass Conversion to Biofuels, Biomaterials & Chemicals 
The committee discussed Latin American engagement opportunities created through Puerto Rico's involvement in three key chemical science events there: the 2009 ACS Southeast Regional Meeting, the 2008 Federation of Latin American Chemical Associations (FLAQ) meeting, and the proposed IUPAC 2011 Congress & General Assembly 
The committee heard reports on letter-writing efforts by the ACS president to government officials in Libya and Mexico expressing concerns about challenges to the scientific freedom and human rights of scientists there 
The Committee on Minority Affairs (CMA) approved new vision, mission, and values statements at the Chicago national meeting  The mission of CMA is to increase the participation of minority chemical scientists and influence policy on behalf of minorities in ACS and the chemical enterprise 
An aggressive new strategic plan was approved by CMA to guide its activities over the next three years  By the end of 2009, CMA will increase the number of ACS Scholars that graduate to 100 per year, add 100 new minorities to leadership positions in ACS, engage in several collaborations, and increase the number of minority members of ACS by 5,000  CMA will focus initially on increasing minorities in ACS leadership  In working toward this goal, CMA began work on two new leadership-development programs for minority chemists 
CMA continues to support the work of the Joint Subcommittee on Diversity (JSD) in developing programs, products, and services to ensure full participation of all members in ACS  In Chicago, JSD premiered a diversity booth at the meeting exposition hall and cosponsored symposia 
The Committee on Patents & Related Matters (CPRM) discussed proposed legislative and regulatory changes to the U S  patent system as well as open-access legislation and the potential effects such matters might have on industry and academia as well as on ACS 
CPRM also continued its work on several new educational tools to assist and inform members on patent issues and other intellectual property matters important to a successful career in the chemical enterprise  Many of these tools are now available on the committee's expanded website, membership acs org/C/CPRM/ 
At the March 2007 meeting, the Committee on Professional Training (CPT) reviewed 42 new and additional information reports from ACS-approved chemistry programs  CPT held conferences with four schools seeking approval, discussed three updates and five site visit reports, and approved three new schools  The total number of ACS-approved chemistry programs is now 642 
The committee released the full draft of the ACS guidelines for review and comment  Copies of the draft were distributed to the department chairs at all ACS-approved schools, the chairs of all ACS committees, and the chairs of all ACS technical divisions 
Several CPT members met with the ACS technical divisions during the Chicago meeting to present an overview of the draft and obtain feedback  The draft guidelines document is available on the CPT website, and the committee invites any comments to be sent to cpt@acs org 
In other business, the committee continued development of the two workshops with minority-serving institutions that will be held in 2007  The committee reviewed the editorial policies for the 2007 edition of the ACS Directory of Graduate Research, which is using a new protocol for retrieving research publication titles in an effort to improve the accuracy of the directory 
C&EN finished 2006 with an exceptionally strong editorial package  The first months of 2007 are proving to be equally successful in fulfilling the magazine's mission of keeping its readers informed  On the advertising side, revenues in 2006 increased for the second year in a row, and the early months of 2007 show continuing positive signs  The most significant editorial achievement was the successful launch of the redesign of the print edition of C&EN with the Oct  16, 2006, issue 
The Subcommittee on Copyright has successfully updated the Copyright Module on the ACS Publications website  The subcommittee is looking into the possibility of conducting copyright programs at future ACS national and regional meetings 
The final monitoring reports for Chemistry of Materials, Journal of Agricultural & Food Chemistry, and Molecular Pharmaceutics were presented and accepted by the committee  Journal of Chemical Information & Modeling, Organic Letters, Accounts of Chemical Research, and the Journal of Chemical Theory & Computation will be monitored next 
3  Examining the scientific basis of public policies related to the chemical sciences and making recommendations to the appropriate ACS units
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: The best performing model for the Spanish language in Track-1 was Spanish BERT.

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: How many people attend the 233rd ACS national meeting?

Read the following text and answer briefly.

Context 2: 
In the first of these areas, ComSci partnered with President Hunt and the Committee on Environmental Improvement in planning and hosting a sustainability luncheon that featured roundtable discussions centering on a key sustainability question  At the Boston national meeting, ComSci will deliver a full-day program on the subject of "Partnerships in Innovation & Competitiveness "
Regarding the second thrust, ComSci will present two programs in Boston: a box lunch that will feature two speakers taking opposing sides on the subject of "Genetic Screening & Diagnostic Testing: Do You Really Want to Know?" and a symposium titled "Creating & Sustaining International Research Collaborations "
In support of the last thrust, ComSci is planning two events for 2008: "Balancing Security & Openness" will gather data to determine if the recent emphasis on security is hindering scientific progress and "Transitioning Chemical Science to Commercially Successful Products "
The Women Chemists Committee (WCC) hosted more than 70 attendees at its open meeting recently in Chicago, where representatives from Iota Sigma Pi, Women in Science & Engineering, the Association of Women in Science, and the Chicago local section helped WCC celebrate the committee's 80th anniversary 
The Women in Industry Breakfast was also highly successful with a new format of speed networking  More than 100 participants had the opportunity to practice their elevator speeches and make several professional connections  A related workshop will be offered by WCC in Boston 
In Chicago, WCC sponsored two symposia, "Women Achieving Success: The ACS as a Platform in Leadership Development" in honor of Madeleine Joullié's 80th birthday and the ACS Award for Encouraging Women into Careers in the Chemical Sciences: Symposium in Honor of Bojan H  Jennings 
More than 225 ACS meeting attendees were present for the biannual WCC Luncheon and heard the keynote speaker Laura Kiessling, 2007 Francis P  Garvan-John Olin Medal Recipient  Twelve women presented their research at this meeting with funding by the WCC/Eli Lilly Travel Grant Award  WCC members also spent time educating expo attendees on programs offered by the ACS Office of Diversity Programs at its new booth 
In Chicago, the Younger Chemists Committee (YCC) welcomed its new committee members with an information session centered on YCC's charter as well as on its strategic plan: to make ACS relevant to younger chemists, to involve younger chemists in all levels of the society, and to integrate younger chemists into the profession 
In January, YCC again hosted a Leadership Development Workshop during the ACS Leaders Conference  There were more than 80 applications for the 15 awards, which covered travel and registration for the conference  YCC plans to again fund the travel awards and provide leadership training for young chemists in 2008  YCC also solicited applications and selected a new graduate student representative on the Graduate Education Advisory Board 
During the Chicago meeting, YCC programs included "Starting a Successful Research Program at a Predominantly Undergraduate Institution," "Career Experiences at the Interface of Chemistry & Biology," and "Chemistry Pedagogy 101 "
In addition to these programs, YCC cosponsored five programs with various committees and divisions  YCC continues to reach out to ACS committees and divisions and has initiated liaisonships with 11 technical divisions to encourage technical programming that highlights the contributions of younger chemists  Looking forward to Boston, YCC is planning symposia including "The Many Faces of Chemistry: International Opportunities for Chemists"; "Being a Responsible Chemist: Ethics, Politics & Policy"; and "Changing Landscapes of the Bio-Pharma Industry "
The Committee on Committees (ConC) conducted its annual training session for new national committee chairs at the ACS Leaders Conference in January 2007  ConC's interactive session for committee chairs in Chicago served as an opportune follow-on and a forum for informative interchange among seasoned and new chairs 
ConC began developing its recommendations for the 2008 committee chair appointments for consideration by the president-elect and chair of the board  ConC continues to focus efforts to identify members with the skills and expertise specified by the committee chairs using the councilor preference form 
The form will be sent to councilors in May  ConC also seeks the names of noncouncilor members for consideration for service on council-related committees, especially those with no prior appointment 
As part of ongoing activities with the joint CPC-Board Governance Review Task Force, ConC has collected data on committee liaisons to other committees  This information will be distributed to committee chairs  The number of liaisons indicates that unofficial but strong communication channels exist within the ACS committee structure 
On Sunday evening, the Committee on Nominations & Elections (N&E) sponsored its fifth successful Town Hall Meeting for President-Elect Nominees  An estimated 200 people attended this session  This forum facilitated communication among the 2008 president-elect nominees, councilors, and other members  N&E will hold another Town Hall Meeting featuring the candidates for director-at-large at the fall meeting in Boston 
Now that voting over the Internet has become an accepted procedure for ACS national elections, the ACS technical divisions and local sections have expressed strong interest in using this method for their elections  N&E has developed protocols for elections for local sections and divisions  This document will be forwarded to the appropriate committees for their review and distribution 
N&E is responsible for reviewing annually the distribution of member populations within the six electoral districts to ensure that the districts have equitable representation  According to bylaw V, section 4(a), the member population of each electoral district must be within 10% of the result of dividing by six the number of members whose addresses lie within these districts  The committee is happy to report that the six electoral districts are in compliance 
The committee has developed a petition on election procedures for president-elect and district director  The proposed election mechanism provides for a preferential (ranked) ballot and an "instant runoff " N&E continues to address the areas of campaigning and the timing of our national election process  Between the Chicago and Boston meetings, the committee plans to sponsor online forums for input from councilors and other interested members on these issues 
In response to member concerns regarding the collection of signatures for petition candidates, N&E reviewed the society's bylaws  The bylaws state that an endorsement is required, but does not stipulate the method of endorsement  N&E has determined that original or electronic signatures are acceptable and will establish appropriate procedures for receipt of electronic signatures 
The Committee on Constitution & Bylaws (C&B), acting for the council, issued new certified bylaws to the Corning Section, the Portland Section, the Division of Colloid & Surface Chemistry, and the Division of Chemical Education  The committee reviewed new proposed amendments for the Division of Medicinal Chemistry, the Columbus Section, the Detroit Section, and the Southern Arizona Section 
Three petitions were presented to council for action at this meeting  Regarding the "Petition on Election Procedures 2006," a motion to separate the petition was approved, and the petition was divided  Provisions affecting bylaw V, sec  2d, bylaw V, sec  3c, and bylaw V, sec  4f, which deal with election procedures and the timing of run-off elections, were approved by council and will become effective following confirmation by the board of directors 
The second part of the petition regarding bylaw V, sec  2c, and bylaw V, sec  3b, which deal with signature requirements for petition candidates for president-elect and director-at-large respectively, was recommitted to the Committee on Nominations & Elections, which has primary substantive responsibility for the petition 
The Committee on Nominations & Elections was asked to reconsider the signature requirements, procedures for acceptance of electronic signatures, and recommendations from the Governance Review Task Force on election procedures 
The second petition presented to council for action was the "Petition on Rules for Nominating Members of N&E for National Offices " This petition was not approved by council  The third petition, the "Petition on Multiyear Dues," was amended by incidental motion on the council floor, calling for the petition to become effective when technical components are instituted to track payments, but no later than Jan  1, 2010  Council approved the incidental motion and then approved the petition 
The committee reviewed one petition for consideration, the "Petition on Local Section Affiliations," which will be submitted to council for action at the fall 2007 meeting in Boston 
The committee met with representatives of the Committee on Membership Affairs and the Governance Review Task Force to continue discussions on proposals currently being formulated on membership requirements and student membership  In addition, the committee discussed election issues of concern to the Southern California Section 
We hope you enjoyed the presidential and division thematic program, "Sustainability of Energy, Food & Water" in Chicago  A small, dedicated group of volunteers and staff labored tirelessly to create and coordinate this programming; to them the Committee on Divisional Activities (DAC) offers sincere thanks 
DAC has committed to transfer the process of choosing and organizing future national meeting themes to a body that represents all divisions  We made substantial progress in Chicago, where division, secretariat, and committee representatives convened to discuss national meeting program concepts  They proposed themes for the 2008 Philadelphia national meeting as well as a framework for a future national programming group 
Divisions have successfully served their members fortunate enough to attend national meetings  To maximize benefits to division members, DAC encourages divisions to consider extending the reach of the content they deliver at national meetings through Internet-based distribution channels and will support worthy efforts in this direction via Innovative Program Grants 
The committee voted in Chicago to propose modifications to the division funding formula that will more greatly reward interdisciplinary programming  The new formula will also be simpler and more transparent to divisions  DAC will present the revised plan to council for action in Boston 
The Committee on Economic & Professional Affairs (CEPA), working with ACS staff in the Departments of Career Management & Development and Member Research & Technology, continues to update and implement its strategic plan to address the career needs of society members 
Specifically, the committee reviewed and revised existing workshops and materials to help ACS members get jobs  CEPA is developing new programs to address the needs of mid- and late-career chemists to ensure their continued competitiveness in the workplace and to ease their career transitions  New initiatives in these areas include the development of workshops, online training, surveys to assess member needs, suggested changes to public policies, and updates to professional and ethical workplace guidelines  As a result of discussions at the Public Policy Roundtable, which was held in San Francisco, a background paper is being developed on trends in health care issues 
The newly revised "Chemical Professional's Code of Conduct" was presented to council, which approved it  The Standards & Ethics Subcommittee is preparing a revision of the "Academic Professional Guidelines" to be presented to council for consideration in Boston 
CEPA reviewed the Globalization Task Force Report  As our science diffuses around the globe, we want to make sure that our members are aware of the economic and professional challenges they will face and that they have the tools they need to succeed  Therefore, CEPA made a commitment to work with other committees, divisions, and ACS staff to develop programs and policies that position our membership to compete in the global workforce 
CEPA heard and discussed a presentation on the proposal from the Membership Affairs Committee on broadening the requirements of membership  CEPA supports the spirit of this proposal and encourages further detailed studies to assess financial impacts on local sections and student affiliates chapters 
The Local Section Activities Committee (LSAC) recognized local sections celebrating significant anniversaries in 2007, including Savannah River (50 years), Northeast Tennessee (75 years), and the St  Louis and Syracuse local sections (both celebrating 100 years) 
LSAC hosted the local section leaders track in conjunction with the ACS Leaders Conference in Baltimore on Jan  26–28  A total of 135 delegates from 124 local sections participated in the weekend leadership conference 
LSAC also hosted a Local Section Summit on March 2–4 in Arlington, Va  The summit focused on practical operational issues that will support local sections' long-term success  Specific areas that were discussed include the development of a multiyear plan to expand or develop programming for local sections, opportunities to encourage innovation and experimentation within and among local sections, and capitalizing on existing opportunities to facilitate partnerships between local sections and other ACS groups 
Following the San Francisco national meeting, LSAC launched a local section Science Café minigrant program  Fifty-five local sections accepted LSAC invitation to host Science Cafés in 2007 
A DVD entitled "ACS Close to Home: Local Sections Connecting Chemistry & the Community" was released earlier this year  The video provides a seven-minute overview of the many outreach and educational programs sponsored by local sections and the critical role they play in positively influencing the public's perception of chemistry and its practitioners  Copies of the DVD were sent to all local section officers 
The Committee on Meetings & Expositions (M&E) reported that the 233rd ACS national meeting hosted 14,520 attendees  This included 7,152 chemical scientists, 5,059 students, 1,283 exhibitors, 119 precollege teachers, 573 exposition visitors, and 453 guests  The exposition had 424 booths with 268 companies 
The 10 2006 regional meetings set a new standard for excellence with attendance exceeding 8,000, a 30% increase in average meeting attendance compared to the 2005 meetings  A total of 4,717 abstracts were submitted  A region summit was held in February at which the final report of the ReACT study group was reviewed 
The practice of tracking the number of presenter no-shows continues  M&E will collaborate with the Committee on Divisional Activities to study options for addressing this problem  Suggestions will be presented at the Boston meeting for implementation in 2008 
It is the intent of M&E to pursue the goal of making our meetings "greener " We will communicate with staff and governance units to identify actions for both the short and long term 
The American Institute of Chemical Engineers (AIChE) and ACS will hold their 2008 spring meetings simultaneously in New Orleans  An ad hoc working group consisting of members from M&E, DAC, and AIChE are actively exploring joint programming opportunities for this meeting 
The Committee on Membership Affairs (MAC) met in executive session on Saturday and Sunday in Chicago and reported that the ACS closed 2006 with 160,491 members, our highest year-end membership count since 2002  Of the 17,857 applications processed in 2006, more than 1,000 came from the Member-Get-a-Member campaign in which many councilors participated  The society's retention rate in 2006 remained strong at 92%  The committee also reported that recruitment for the first two months of 2007 netted 2,844 new applications—729 more than for the same time period last year 
MAC continues to work with deliberate speed on the proposed new bylaw language for members, student members, and society affiliates-the three ways to connect to the society  The committee received input from the Governance Review Task Force and its action teams, the Council Policy Committee, the board of directors, the Committee on Constitution & Bylaws, and several other committees between the San Francisco and Chicago meetings  These interactions have resulted in the current bylaw change recommendations 
In Chicago, representatives from MAC attended several committee meetings and all seven councilor caucuses to summarize the current proposal for membership changes, answer questions, and seek input  In addition, all committee chairs were invited to have their respective committees review these bylaw changes and respond to MAC—if possible—before council met on Wednesday  MAC received 11 responses: eight supported the proposed changes as is, and three supported the proposed language with specified changes or considerations 
The comprehensive petition will likely represent the most significant and voluminous change in the ACS bylaws that has occurred in decades, and MAC is proud to be among the leaders in its development and in efforts to get it right the first time  Hundreds of individuals have contributed to this major effort, since MAC began such discussions at the spring 2004 national meeting 
The Committee on Ethics met in Chicago and discussed the possibility of organizing and scheduling a committee retreat in the near future to enable the committee to move from the current stage of exploring the needs and interests of ACS members to setting priorities for the next few years 
The Project SEED program offers summer research opportunities for high school students from economically disadvantaged families  Since its inception in 1968, the program has had a significant impact on the lives of more than 8,400 students  At the selection meeting in March, the committee approved research projects for 340 SEED I students and 98 SEED II students for this summer in more than 100 institutions 
The 2006 annual assessment surveys from 300 students indicate that 78% of the Project SEED participants are planning to major in a chemistry-related science, and 66% aspire to continue to graduate education  This program is made possible by contributions from industry, academia, local sections, ACS friends and members, the ACS Petroleum Research Fund, and the Project SEED Endowment 
The committee formally submitted a request to ConC to amend the Project SEED acronym and the committee duties described in the Supplementary Information of the "ACS Charter, Constitution, Bylaws & Regulations "
In Chicago, the committee's agenda focused on the ACS Strategic Plan and how Project SEED fits into it, the Program Review Advisory Group (PRAG) review of the Project SEED program, the committee's review of an online application form, and planning of the 40th anniversary celebration to be held at the Philadelphia meeting in the fall of 2008  The committee selected a task force to review the criteria for selection of the Project SEED ChemLuminary Award 
3  Making ACS relevant to technicians 
Last year, CTA, along with the Division of Chemical Technicians, the Committee on Economic & Professional Affairs, and ChemTechLinks, started the Equipping the 2015 Chemical Technology Workforce initiative  This year, the initiative awarded six $500 minigrants to activities and programs that support the educational and professional development of chemical technicians 
We are pleased to announce that the winners of the minigrants are the ACS Division of Environmental Chemistry; the Chemical Technician Program Chair for the 39th ACS Central Regional Meeting in Covington, Ky ; Delta College, University Center, Mich ; Grand Rapids Community College, in Michigan; Mount San Antonio College, Walnut, Calif ; and Southwestern College in Chula Vista, Calif 
The winners are collaborating with industry, academia, and ACS local sections on such activities as chemical technology career fairs for high school students, discussion panels on employability skills for technicians, and technical programming at regional and national meetings on the vital role technicians have in the chemical enterprise 
Because of the enthusiastic response to the minigrants, Equipping the 2015 Chemical Technology Workforce will be supporting another round of minigrants to be distributed in the fall  Details will be available on the website  For more information, go to www ChemTechLinks org and click on "Equipping the 2015 Chemical Technology Workforce "
CTA has also joined with the Joint Subcommittee on Diversity, formerly known as the Collaboration of Committees Working Group  Because this group is focused on increasing diversity in ACS and the chemical enterprise, we believe that this is an opportunity to raise awareness of the value of technicians  CTA looks forward to collaborating on the promotion of traditionally underrepresented chemical professionals 
In 2007, CTA will be placing renewed focus on distribution of the ACS Chemical Technology Student Recognition Award  The award recognizes academic excellence in students preparing for careers as chemical technicians  For more information on the award, please visit the CTA website at chemistry org/committees/cta 
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: There are 14,520 attendees, including 7,152 chemical scientists, 5,059 students, 1,283 exhibitors, 119 precollege teachers, 573 exposition visitors, and 453 guests.

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 size chains were used in the benchmarking?

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: L = 8 and L = 14.