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 does the new Iraqi Body Count organization do?

Read the following text and answer briefly.

Context 1: Ann's Mega Dub: 12/19/10 - 12/26/10
Got o have a penis to be an expert
Thursday on NPR's Fresh Air, Terry Gross wanted to talk film and music  Since women don't know a thing about either and aren't interested in either, Terry had to find men who were 'experts 'This is C I 's " Iraq snapshot Friday, December 24, 2010  Chaos and violence continue, Nouri's incomplete Cabinet continues to receive criticism, a father offers an 'excuse' for killing his own daughter, and more Marci Stone (US Headlines Examiner) reports, "Friday afternoon, Santa is currently in Baghdad, Iraq and on his next stop is Moscow, Russia, according to the 2010 NORAD Santa Tracker  The North American Aerospace Defense Command (NORAD) has been tracking Santa as he makes his annual journey throughout the world " Gerald Skoning (Palm Beach Post) quotes Santa saying, "We send our special wishes for peace and goodwill to all  That includes the people of Iraq, Afghanistan, Iran and North Korea " Please note that this is Santa's seventh trip to Iraq since the start of the Iraq War and, as usual, his journey was known in advance  No waiting until he hit the ground to announce he was going to Iraq -- the way George The Bully Boy Bush had to and the way US President Barack Obama still has to  In the lead up to Santa's yearly visit, many 'authorities' in Iraq began insisting that Christmas couldn't be celebrated publicly, that even Santa was banned  Gabriel Gatehouse (BBC News) quotes Shemmi Hanna stating, "I wasn't hurt but I wish that I had been killed  I wish I had become a martyr for this church, but God kept me alive for my daughters " Shemmi Hanna was in Our Lady of Salvation Church in Baghdad when it was assaulted October 31st and she lost her husband, her son, her daughter-in-law and her infant grandson in the attack  The October 31st attack marks the latest wave of violence targeting Iraqi Christians  The violence has led many to flee to northern Iraq (KRG) or to other countries  Zvi Bar'el (Haaretz) notes, "This week the Iraqi legislature discussed the Christians' situation and passed a resolution in principle to help families who fled  However, the parliament does not know where the Christians are, how many are still in Iraq, in their homes, and how many have found asylum in Iraqi Kurdistan " John Leland (New York Times) reports:The congregants on Friday night were fewer than 100, in a sanctuary built for four or five times as many  But they were determined  This year, even more than in the past, Iraqi's dwindling Christian minority had reasons to stay home for Christmas  "Yes, we are threatened, but we will not stop praying," the Rev  Meyassr al-Qaspotros told the Christmas Eve crowd at the Sacred Church of Jesus, a Chaldean Catholic church  "We do not want to leave the country because we will leave an empty space " Raheem Salman (Los Angeles Times) reports, "Rimon Metti's family will go to Christian services on Christmas Day, but his relatives will be praying for their own survival and wondering whether this is their last holiday season in Baghdad  If they had any grounds for optimism about the future of their faith in Iraq, it vanished this year amid repeated attacks on fellow believers " Shahsank Bengali (McClatchy Newspapers) adds, "Nearly two months after a shocking assault by Islamist militants, Our Lady of Salvation Catholic Church will commemorate Christmas quietly, with daytime mass and prayers for the dead, under security fit more for a prison than a house of worship  It is the same at Christian churches across Baghdad and northern Iraq, where what's left of one of the world's oldest Christian communities prepares to mark perhaps the most somber Christmas since the start of the Iraq war "Meanwhile Taylor Luck (Jordan Times) reports on Iraqi refugees in Jordan:Although the calendar will say December 25, for Theresa, Saturday will not be Christmas  There will be no cinnamon klecha cooling on the dining room table, no outdoor ceramic nativity scene, no readings of hymns with relatives  The 63-year-old Iraqi woman has even refused to put up Christmas lights in the crowded two-room Amman hotel apartment she has called home since fleeing Baghdad last month "There is no holiday spirit  All we have is fear," she said This holiday will instead mark another year without news from her 46-year-old son, who was kidnapped outside Baghdad in late 2006 From Turkey, Sebnem Arsu (New York Times -- link has text and video) notes the increase in Iraq refugees to the country since October 31st and quotes Father Emlek stating, "I've never seen as many people coming here as I have in the last few weeks  They also go to Lebanon, Jordan and Syria but it seems that Turkey is the most popular despite the fact that they do not speak the language " Jeff Karoub (AP) reports on the small number of Iraqi refugees who have made it to the US and how some of them "struggle with insomnia, depression and anxiety "One group in Iraq who can openly celebrate Christmas are US service members who elect to  Barbara Surk (AP) reports that tomorrow Chief Warrant Officer Archie Morgan will celebrate his fourth Christmas in Iraq and Captain Diana Crane is celebrating her second Christmas in Iraq: "Crane was among several dozen troops attending a Christmas Eve mass in a chapel in Camp Victory, an American military base just outside Baghdad " Marc Hansen (Des Moines Reigster) speaks with six service members from Iowa who are stationed in Iraq  Sgt 1st Class Dennis Crosser tells Hansen, "I certainly understand from reading the paper what's going on in Afghanistan and the attention definitely needs to be on the troops there  But everyone serving here in Operation New Dawn appreciates a little bit of attention as we finish this up "Today Jiang Yu, China's Foreign Minister, issued the following statement, "We welcome and congratulate Iraq on forming a new government  We hope that the Iraqi Government unite all its people, stabilize the security situation, accelerate economic reconstruction and make new progress in building its country " James Cogan (WSWS) reports:US State Department official Philip Crowley declared on Wednesday that Washington had not "dictated the terms of the government"  In reality, constant American pressure was applied to Maliki, Allawi, Kurdish leaders and other prominent Iraqi politicians throughout the entire nine-month process to form a cabinet  The US intervention included numerous personal phone calls and visits to Baghdad by both President Barack Obama and Vice President Joe Biden The key objective of the Obama administration has been to ensure that the next Iraqi government will "request" a long-term military partnership with the US when the current Status of Forces Agreement (SOFA) expires at the end of 2011  The SOFA is the legal basis upon which some 50,000 American troops remain in Iraq, operating from large strategic air bases such as Balad and Tallil and Al Asad  US imperialism spent billions of dollars establishing these advanced bases as part of its wider strategic plans and has no intention of abandoning them Cogan's only the second person to include the SOFA in his report  Some are impressed with the 'feat' of taking nearly ten months to form a government, stringing the country along for ten months while no decisions could go through  The editorial board of the Washington Post, for example, was full of praise yesterday  Today they're joined by Iran's Ambassador to Iraq, Hassan Danaiifar  The Tehran Times reports that Danaiifar was full of praise today hailing the "positive and final step which ended the 10-month political limbo in Iraq " However, Danaiifar was less pie-in-the-sky than the Post editorial board because he can foresee future problems as evidenced by his statement, "We may witness the emergence of some problems after one and half of a year -- for example, some ministers may be impeached " Of course, there are already many clouds on the horizon, even if Iranian diplomats and Post editorial boards can't suss them out  For example, Ben Bendig (Epoch Times) noted the objection of Iraq's female politicians to Nouri al-Maliki's decision to nominate only one woman (so far) to his Cabinet: "Some 50 female lawmakers went to the country's top leadership, the United Nations and the Arab League to voice their concern and desire for increased representation " BNO notes that protest and also that a group of Iraqi MPs are alleging that Iraqiya bought seats in the Cabinet via money exchanged in Jordan  UPI adds, "Maliki, a Shiite who has a long history of working with Tehran, has named himself acting minister of defense, interior and national security, three most powerful and sensitive posts in the government he is stitching together  Although Maliki appears to be bending over backward to accommodate rivals among Iraq's Shiite majority as well as minority Sunnis and Kurds in his administration in a spirit of reconciliation, he is unlikely to relinquish those ministries that dominate the security sector " DPA reports, "Sheikh Abdel-Mahdi al-Karbalaei, a confident of influential Shiite spiritual leader Ayatollah Ali al-Sistani, said that the new cabinet is 'below the standards' Iraqi citizens had hoped for and suggested it could prove to be weaker than the previous government " Ranj Alaaldin (Guardian) also spots clouds on the horizon:Lasting peace and stability depends on resolving outstanding disputes with the Kurds on oil, revenue-sharing, security and the disputed territories (Kirkuk in particular)  The Kurds, rather than exploiting their kingmaker position to take a stronger proportion of ministries in Baghdad (they are taking just one major portfolio – the foreign ministry), are instead banking on guarantees from Maliki to implement their list of 19 demands that includes resolving the above disputes in their favour They may have been naive, though  With their historical and federalist partners, the Islamic supreme council of Iraq in decline, the Kurds may be isolated in the new government – a government dominated by the nationalistic and centrist characteristics of the INM, the Sadrists and indeed State of Law Maliki may, therefore, turn out to be unable to grant concessions even if he wanted to and could use Osama Nujayfi, the new ultra-nationalist speaker of parliament and Kurdish foe, to absorb the Kurdish criticism and insulate himself from any attacks AP reports that Iraqi police sought out a 19-year-old woman because of rumors that she was working with al Qaida in Mesopotamia only to be greeted with the news that her father allegedly killed her and the father showed the police where he buried the woman       last month  The story begs for more than it offers  The most obvious observation is: what does it say that a woman's allegedly killed by her father and no one says a word for over a month? After that, it should probably be noted that there are many men in Iraq killing women who, no doubt, would love to also be able to pin the blame on al Qaida  In other violence, Reuters notes a house bombing in Haswa which claimed the life of Mohammed al-Karrafi, "his wife, two sons and a nephew" -- as well as injuring four more people, and a Samarra roadside bombing which claimed the lives of 2 police officers  DPA notes it was two homes bombed in Haswa and that the Samarra roadside bombing also injured four Iraqi soldiers  Jomana Karadsheh (CNN) reports, "Another policeman was wounded in Baghdad Friday night when a roadside bomb detonated by a police patrol, an Interior Ministry official told CNN "And we'll close with this from Peace Mom Cindy Sheehan's latest Al Jazeera column:The recent repeal of the US military policy of "Don't ask, don't tell" is far from being the human rights advancement some are touting it to be  I find it intellectually dishonest, in fact, illogical on any level to associate human rights with any military, let alone one that is currently dehumanising two populations as well as numerous other victims of it's clandestine "security" policies Placing this major contention aside, the enactment of the bill might be an institutional step forward in the fight for "equality"; however institutions rarely reflect reality Do we really think that the US congress vote to repeal the act and Obama signing the bill is going to stop the current systemic harassment of gays in the military?While I am a staunch advocate for equality of marriage and same-sex partnership, I cannot - as a peace activist - rejoice in the fact that now homosexuals can openly serve next to heterosexuals in one of the least socially responsible organisations that currently exists on earth: The US military It is an organisation tainted with a history of intolerance towards anyone who isn't a Caucasian male from the Mid-West  Even then I'm sure plenty fitting that description have faced the terror and torment enshrined into an institution that transforms the pride and enthusiasm of youth into a narrow zeal for dominating power relations And we'll close with this from Francis A  Boyle's "2011: Prospects for Humanity?" (Global Research):Historically, this latest eruption of American militarism at the start of the 21st Century is akin to that of America opening the 20th Century by means of the U S -instigated Spanish-American War in 1898  Then the Republican administration of President William McKinley stole their colonial empire from Spain in Cuba, Puerto Rico, Guam, and the Philippines; inflicted a near genocidal war against the Filipino people; while at the same time illegally annexing the Kingdom of Hawaii and subjecting the Native Hawaiian people (who call themselves the Kanaka Maoli) to near genocidal conditions  Additionally, McKinley's military and colonial expansion into the Pacific was also designed to secure America's economic exploitation of China pursuant to the euphemistic rubric of the "open door" policy  But over the next four decades America's aggressive presence, policies, and practices in the "Pacific" would ineluctably pave the way for Japan's attack at Pearl Harbor on Dec  7, 194l, and thus America's precipitation into the ongoing Second World War  Today a century later the serial imperial aggressions launched and menaced by the Republican Bush Jr  administration and now the Democratic Obama administration are threatening to set off World War III  By shamelessly exploiting the terrible tragedy of 11 September 2001, the Bush Jr  administration set forth to steal a hydrocarbon empire from the Muslim states and peoples living in Central Asia and the Persian Gulf under the bogus pretexts of (1) fighting a war against international terrorism; and/or (2) eliminating weapons of mass destruction; and/or (3) the promotion of democracy; and/or (4) self-styled "humanitarian intervention " Only this time the geopolitical stakes are infinitely greater than they were a century ago: control and domination of two-thirds of the world's hydrocarbon resources and thus the very fundament and energizer of the global economic system – oil and gas  The Bush Jr / Obama administrations have already targeted the remaining hydrocarbon reserves of Africa, Latin America, and Southeast Asia for further conquest or domination, together with the strategic choke-points at sea and on land required for their transportation  In this regard, the Bush Jr  administration announced the establishment of the U S  Pentagon's Africa Command (AFRICOM) in order to better control, dominate, and exploit both the natural resources and the variegated peoples of the continent of Africa, the very cradle of our human species  This current bout of U S  imperialism is what Hans Morgenthau denominated "unlimited imperialism" in his seminal work Politics Among Nations (4th ed  1968, at 52-53): The outstanding historic examples of unlimited imperialism are the expansionist policies of Alexander the Great, Rome, the Arabs in the seventh and eighth centuries, Napoleon I, and Hitler  They all have in common an urge toward expansion which knows no rational limits, feeds on its own successes and, if not stopped by a superior force, will go on to the confines of the political world  This urge will not be satisfied so long as there remains anywhere a possible object of domination--a politically organized group of men which by its very independence challenges the conqueror's lust for power  It is, as we shall see, exactly the lack of moderation, the aspiration to conquer all that lends itself to conquest, characteristic of unlimited imperialism, which in the past has been the undoing of the imperialistic policies of this kind…  On 10 November 1979 I visited with Hans Morgenthau at his home in Manhattan  It proved to be our last conversation before he died on 19 July 1980  Given his weakened physical but not mental condition and his serious heart problem, at the end of our necessarily abbreviated one-hour meeting I purposefully asked him what he thought about the future of international relations  iraqbbc newsgabriel gatehousethe new york timesjohn lelandhaaretzzvi bar'elthe jordan timestaylor luckthe associated pressjeff karoubthe los angeles timesraheem salmancnnjomana karadsheh
Terry thinks she's a man
Yesterday on NPR's Fresh Air the hour went to a male TV critic  It's always a man with Terry  Always  And somebody tell her that a snotty, snooty TV critic really doesn't make for good programming This is C I 's "Iraq snapshot:" Thursday, December 23, 2010  Chaos and violence continue, Iraqi women make clear their displeasure over the Cabinet make up, Daniel Ellsberg and Veterans for Peace get some recognition, and more  Last Thursday a protest held outside the White House  One of the organizers was Veterans for Peace and Pentagon Papers whistle blower Daniel Ellsberg participated and spoke  Juana Bordas (Washington Post) advocates for both of them to be named persons of the year: Veterans for Peace and Daniel Ellsberg should be this year's person of the year because of their courage and bravery to stand up for all of us who believe that "war is not the answer " Moreover in a time of economic recession, the war machine is bankrupting our country  As John Amidon, a Marine Corps veteran from Albany asked at the White House protest, "How is the war economy working for you?"While unemployment rates hover near 10 percent, there is no doubt that the U S  economy and quality of life is faltering  Worldwide we are 14th in education, 37th in the World Health Organization's ranking on medical systems, and 23rd in the U N  Environmental Sustainability Index on being most livable and greenest benefits  There is one place we take the undeniable world lead  The US military spending accounts for a whopping 46 5 percent of world military spending--the next ten countries combined come in at only 20 7 percent  Linda Pershing (Truthout) reports, "Responding to a call from the leaders of Stop These Wars(1) - a new coalition of Veterans for Peace and other activists - participants came together in a large-scale performance of civil resistance  A group of veterans under the leadership of Veterans for Peace members Tarak Kauff, Will Covert and Elaine Brower, mother of a Marine who has served three tours of duty in Iraq, sponsored the event with the explicit purpose of putting their bodies on the line  Many participants were Vietnam War veterans; others ranged from Iraq and Afghanistan war veterans in their 20s and 30s to World War II vets in their 80s and older  They were predominately white; men outnumbered women by at least three to one  After a short rally in Lafayette Park, they formed a single-file procession, walking across Pennsylvania Avenue to the solemn beat of a drum  As they reached the police barricade (erected to prevent them from chaining themselves to the gate, a plan they announced on their web site), the activists stood shoulder to shoulder, their bodies forming a human link across the 'picture postcard' tableau in front of the White House " Maria Chutchian (Arlington Advocate) quotes, participant Nate Goldshlag (Vietnam veteran) stating, ""There was a silent, single file march around Lafayette Park to a drum beat  Then we went in front of the White House,  There were barricades set up in front of white house fence  So when we got there, we jumped over barricades and were able to get right next to the White House fence " Participant Linda LeTendre (Daily Gazette) reports: At the end of the rally, before the silent, solemn procession to the White House fence, in honor of those killed in Iraq and Afghan wars of lies and deceptions, the VFP played taps and folded an American flag that had been left behind at a recent funeral for the veteran of one of those wars  Two attendees in full dress uniform held and folded the flag  I had the image of all of the people who stood along the roads and bridges when the bodies of the two local men, Benjamin Osborn and David Miller, were returned to the Capital District  I thought if all of those people were here now or spoke out against war these two fine young men might still be with us I was blessed enough to be held in custody with one of those in uniform; a wonderful young man who had to move from his hometown in Georgia because no one understood why as a veteran he was against these wars  Even his family did not understand  (He remains in my prayers )Our plan was to attach ourselves to the White House fence until President Obama came out and talked to us or until we were arrested and dragged away  I don't have to tell you how it ended Mr  Ellsberg was one of 139 people arrested at that action  We've noted the protest in pretty much every snapshot since last Thursday  If something else comes out that's worth noting on the protest, we'll include it  We will not include people who don't have their facts and it's really sad when they link to, for example, Guardian articles and the links don't even back them up  It's real sad, for example, when they're trashing Hillary (big strong men that they are) and ripping her apart and yet Barack? "Obama's inaccurate statements"??? What the hell is that? You're inferring he lied, say so  Don't be such a little chicken s**t  It's especially embarrasing when you're grandstanding on 'truth ' Especially when you're the little s**t that clogged up the public e-mail account here in the summer of 2008 whining that you were holding Barack to a standard, then admitting that you weren't, then whining that if you did people would be mean to you  Oh, that's sooooooo sad  Someone might say something bad about you  The horror  You must suffer more than all the people in Iraq and Afghanistan combined  While the action took place in DC, actions also took place in other cities  We've already noted NYC's action this week, Doug Kaufmann (Party for Socialism & Liberation) reports on the Los Angeles action: Despite heavy rain, over 100 people gathered in Los Angeles on the corner of Hollywood and Highland to demand an end to the U S  wars on Afghanistan and Iraq  People came from as far as Riverside to protest, braving what Southern California media outlets have dubbed the "storm of the decade " The demonstration, initiated and led by the ANSWER Coalition, broke the routine of holiday shopping and garnered support from activists and even passers by, who joined in chanting "Money for jobs and education -- not for war and occupation!" and "Occupation is a crime -- Iraq, Afghanistan, Palestine!" Protesters held banners reading, "U S /NATO Out of Afghanistan!" and "Yes to jobs, housing and education -- no to war, racism and occupation!"Speakers at the demonstration included representatives of Korean Americans for Peace, ANSWER Coalition, KmB Pro-People Youth, Veterans for Peace, Party for Socialism and Liberation and National Lawyers Guild  Tuesday, Nouri al-Maliki managed to put away the political stalemate thanks to a lot of Scotch -- tape to hold the deal together and booze to keep your eyes so crossed you don't question how someone can claim to have formed a Cabinet when they've left over ten positions to be filled at a later date  One group speaking out is women  Bushra Juhi and Qassmi Abdul-Zahra (AP) report, "Iraq's female lawmakers are furious that only one member of the country's new Cabinet is a woman and are demanding better representation in a government that otherwise has been praised by the international community for bringing together the country's religious sects and political parties " As noted Tuesday, though represenation in Parliament is addressed in Iraq's Constitution, there is nothing to address women serving in the Cabinet  Aseel Kami (Reuters) notes one of the most damning aspects of Nouri's chosen men -- a man is heaing the Ministry of Women's Affairs  Iraqiya's spokesperson Maysoon Damluji states, "There are really good women who could do wel       they cannot be neglected and marginalized " Al-Amal's Hanaa Edwar states, "They call it a national (power) sharing government  So where is the sharing? Do they want to take us back to the era of the harem? Do they want to take us back to the dark ages, when women were used only for pleasure " Deborah Amos (NPR's All Things Considered) reports that a struggle is going on between secular impulses and fundamentalist ones  Gallery owner Qasim Sabti states, "We know it's fighting between the religious foolish man and the civilization man  We know we are fighting like Gandhi, and this is a new language in Iraqi life  We have no guns  We do not believe in this kind of fighting " Deborah Amos is the author of Eclipse of the Sunnis: Power, Exile, and Upheaval in the Middle East  Meanwhile Nizar Latif (The National) reports that distrust is a common reaction to the new government in Baghdad and quotes high school teacher Hussein Abed Mohammad stating, "Promises were made that trustworthy, competent people would be ministers this time around, but it looks as if everything has just been divided out according to sectarian itnerests  No attention has been paid to forming a functioning government, it is just a political settlement of vested interests  I'm sure al Maliki will have the same problems in his next four years as he had in the last four years " Days away from the ten months mark, Nouri managed to finally end the stalemate  Some try to make sense of it and that must have been some office party that the editorial board of the Washington Post is still coming down from judging by "A good year in Iraq " First up, meet the new Iraqi Body Count -- an organization that provides cover for the war and allows supporters of the illegal war to point to it and insist/slur "Things aren't so bad!" Sure enough, the editorial board of the Post does just that noting the laughable "civilian deaths" count at iCasualities  As we noted -- long, long before we walked away from that crap ass website, they're not doing a civilian count 
Context 2: Paper Info

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

Figure

FIG 3 The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and time t, for the infinite temperature initial state, for a denoised second-order Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2 We consider evolution times t = 0 5, 1,    , 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise with p = 0 01 
FIG  4  The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right) The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0 0046 The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars It is evident that the denoiser recovers all the noiseless eigenvalues from the noisy circuit 
FIG  2  The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right) The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0 036 The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars It is clear that the denoiser recovers with high accuracy the noiseless eigenvalues from the noisy circuit 
FIG  3  The half-chain channel entanglement entropy S at different two-qubit depolarizing noise strengths p, for a secondorder Trotter supercircuit with Mtrot = 16 and t = 2, for a M = 4 denoiser The Trotter circuit is for a Heisenberg model with PBC of size L = 6 The different curves correspond to the different supercircuits, i e  the noisy supercircuit, the denoiser, the corresponding denoised supercircuit, and the noiseless variant 
FIG  4  The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and stacked time t, for the infinite temperature initial state, for a denoised secondorder Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2 It is optimized at t = 2 and stacked up to ten times The calculations are for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarization with p = 0 01 The denoiser is affected by the same noise 
FIG 6 The distribution of the ZZ angle α of M = 2 denoisers (top panels) and M = 8 denoisers (bottom panels), with the lightest color corresponding to the denoiser for the Trotter supercircuit with t = 0 5, and the darkest color with t = 5 As usual, we consider the Heisenberg model on a periodic chain, and second-order Trotter supercircuits with depths Mtrot = 8, 16, 32, 64, which together with the denoiser is affected by a two-qubit depolarizing noise with p = 0 01 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively 
FIG  7  The sampling overhead γ of the optimized denoisers from Fig  2 of the main text, with denoiser depths M = 1, 2, 4, 6, 8 and Trotter depths Mtrot = 8, 16, 32, 64 at times t = 0 5, 1,    , 5, for the Heisenberg model on a chain with PBC affected by two-qubit depolarizing noise with p = 0 01 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively 
FIG 8 The domain wall magnetization Z dw after evolving a periodic density wall |dw |dw * with the denoised second-order Trotter supercircuits D C from Fig 2of the main text These supercircuits have various Trotter depths Mtrot = 8, 16, 32, 64, denoiser depths M = 1, 2, 4, 6, 8, and evolution times t = 0 5, 1,    , 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise of strength p = 0 01 The denoiser is affected by the same noise The non-denoised results are labelled with M = 0 and the noiseless results with p = 0 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively We see that the denoiser allows us to recover the noiseless behavior 

abstract

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

C

FIG  1  An example of the quantum error mitigation procedure used in this work for the time evolution of the wave function of a spin chain  The ideal second-order Trotter supercircuit C of depth Mtrot = 1 (light blue) is approximated by applying a denoiser D of depth M = 1 (red) to the noisy Trotter supercircuit C (dark blue) 
Because the denoiser is applied after fully executing the noisy Trotter supercircuit, it represents an approximate inverse of the global noise channel with a precision tunable by the depth of the denoiser  can potentially blow up and lead to large errors for the circuit as a whole   Here we introduce a mitigation technique that takes into account the propagation of errors, can be performed with a tunable number of extra gates, and works for non-Clifford local noise channels since the inversion of the accumulated global noise channel is implicit 
We first execute the targeted noisy circuit completely, letting the noise propagate and accumulate, and only afterwards we apply an extra random circuit sampled from a quasiprobability distribution  We call the corresponding ensemble of random circuits a denoiser, and we construct it such that upon averaging the accumulated errors cancel 
Essentially, the denoiser inverts a global noise channel  Since we will construct it as a local brickwall circuit, following the classical preprocessing approach from , we call this compressed quantum error mitigation  Method  -Due to the inevitable coupling of a quantum processor to its environment, every qubit operation is affected by noise 
Therefore, the simplest technique to minimize the impact of the resulting noise is to minimize the number of operations when performing a quantum algorithm  In we showed that many-body time evolution operators can be efficiently compressed into brick-wall circuits with high fidelity per gate  In this Letter, we consider the noise explicitly by treating quantum operations as (generally non-unitary) quantum channels, corresponding to completely positive and trace preserving (CPTP) maps  
For example, instead of a noiseless two-qubit gate G, which acts on a quantum state |ρ in superoperator form as G|ρ = G⊗G * |ρ , we get the noisy channel G = N G, where the noise channel N implements the two-qubit noise   These channels are used to construct a "supercircuit" C = N G i=1 Gi , consisting of N G channels, which is affected by multi-qubit accumulated noise 
This supercircuit encodes an ensemble of circuits   For simplicity, we assume that the noisy channels Gi in each half brickwall layer are lattice inversion and translation invariant, such that we can construct a denoiser with these properties, limiting the number of variational parameters  The purpose of quantum error mitigation is to modify the ensemble of circuits described by C in a way that we can use it to obtain the noiseless expectation values 
In superoperator language, we do this by following the supercircuit C with a denoiser supercircuit D, such that D C is as close to the noiseless supercircuit C = C ⊗ C * as possible  Here C is the target unitary circuit  Because the noise channel N is non-unitary, hence making the supercircuit C non-unitary, we need to use a non-unitary denoiser to retrieve the unitary C 
We illustrate the mitigation procedure in Fig  , where a denoiser with one layer is used to mitigate errors for a second-order Trotter supercircuit with one layer  This circuit architecture is commonly used to simulate the time evolution of a quantum many-body system, until some time t, with controllable precision , and we will use it to benchmark the denoiser 
In practice, we cannot directly implement a supercircuit, and so we have to utilize its interpretation as an ensemble of circuits  Essentially, after executing a shot of the noisy circuit we sample the denoiser and apply it  The goal is to construct the denoiser in a way that averaging over many of its samples cancels the accumulated errors and gives us a good approximation of the noiseless expectation values 
It should be noted that our approach requires more gate applications on the quantum processor than with the gate-wise scheme, since there each sample from the mitigation quasiprobability distribution can be absorbed into the original circuit, whereas our approach increases the circuit depth  We take this into account by imposing the same noise on the denoiser 
Furthermore, within our scheme, the dimensionality of the quasiprobabilistic mitigating ensemble can be controlled, in contrast to the gate-wise approach where it is equal to the gate count  To facilitate the stochastic interpretation we parameterize each two-qubit denoiser channel G i as a sum of CPTP maps, such that we can sample the terms in this sum and execute the sampled gate on the quantum processor 
Concretely, we use a trace preserv-ing sum of a unitary and a non-unitary channel  For the unitary part we take a two-qubit unitary channel U( φ i ) = U ( φ i ) ⊗ U * ( φ i ), with U ( φ i ) a two-qubit unitary gate parameterized by φ i   For this we take the two-qubit ZZ rotation exp(−iα(σ z ⊗ σ z )) with angle α, which can be obtained from native gates on current hardware , and dress it with four general one-qubit unitaries, only two of which are independent if we want a circuit that is space inversion symmetric around every bond 
The resulting gate has 7 real parameters φ i   For the non-unitary part, which is essential because D has to cancel the non-unitary accumulated noise to obtain the noiseless unitary circuit, we use a general onequbit measurement followed by conditional preparation channel M( , with V a general one-qubit unitary and each κ i a 3-dimensional vector, resulting in a real 9-dimensional ζ i  
This yields the two-qubit correlated measurement M( With these parts we construct the parameterization with coefficients η i ∈ R that satisfy η 0 + η 1 = 1 because G i is trace preserving  Note that here the tensor product symbol corresponds to combining two one-qubit channels to make a two-qubit channel, whereas in most of the paper it is used to link the column and row indices of a density matrix 
We construct the denoiser from the noisy channels Gi = N G i   With this parameterization one denoiser channel has 17 independent real parameters, such that a denoiser of depth M , i e  consisting of M brickwall layers, has 34M real parameters (we use one unique channel per half brickwall layer)  For reference, a general channel has 544M parameters 
To determine the mitigated expectation values we use the full expression where |ρ 0 is the initial state and |1 is the vectorized identity operator on the full Hilbert space  To evaluate this on a quantum processor, we use the stochastic interpretation of (1) to resample   In particular, from each channel (1) we get a unitary with probability p 0 = |η 0 |/γ and a measurement followed by conditional preparation with probability p 1 = |η 1 |/γ 
Here γ = |η 0 | + |η 1 | is the sampling overhead, which characterizes the magnitude of the sign problem from negative η i   For quasiprobability distributions, i e  with γ > 1, every denoiser sample has an extra sign sgn(η) = N G g=1 sgn(η g ), 2  The normalized distance between the denoised Trotter supercircuit D C and the noiseless Trotter supercircuit C (top panels), at evolution times t = 0 5, 1,    , 5, and the twopoint z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t (bottom panels), for the infinite temperature initial state 
We consider denoisers with depths M = 1, 2, 4, 6, 8 and second-order Trotter circuits with depths Mtrot = 16, 32, 64  In the top panels we use a Heisenberg chain with L = 8, and in the bottom panels with L = 14, both with periodic boundary conditions  All gates are affected by two-qubit depolarizing noise with p = 0 01
Context 3: Paper Info

Title: Interpretable reduced-order modeling with time-scale separation Interpretable reduced-order modeling with time-scale separation
Publish Date: 7 March 2023
Author List: Sebastian Kaltenbach (from CSE-Lab, ETH Zurich, Harvard SEAS), Phaedon-Stelios Koutsourelakis (from CSE-Lab, ETH Zurich, Harvard SEAS), Petros Koumoutsakos (from CSE-Lab, ETH Zurich, Harvard SEAS), Harvard Seas (from CSE-Lab, ETH Zurich, Harvard SEAS)

Figure

FIG  5  Comparison between the phase-space of the reference solution (left) and the phase-space of the predictions
FIG  7  Comparison between predictions and reference solutions for a new initial condition fort = 1 25, 3 75, 7 5, 12 5, 20, 30  (from left to right and top to down) We note that with longer prediction time the uncertainty bounds increases Despite the chaotic nature of the KS equation, the predictive posterior mean is close to the reference solution for t ≤ 12 5

abstract

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering  However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial  We propose such a data-driven scheme that automates the identification of the time-scales involved and, can produce stable predictions forward in time as well as under different initial conditions not included in the training data 
To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space  It readily allows for the inclusion of sparse and irregularly sampled training data  The learned, latent dynamics are interpretable and reveal the different temporal scales involved 
We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix  Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation 
Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework 

INTRODUCTION

High-fidelity simulations of critical phenomena such as ocean dynamics and epidemics have become essential for decision-making  They are based on physically-motivated PDEs expressing system dynamics that span multiple spatiotemporal scales and which necessitate cumbersome computations   In recent years there is increased attention to the development of data-driven models that can accelerate the solution of these PDEs as well as reveal salient, lower-dimensional features that control the long-term evolution 
In most cases, data-driven reduced-order models are not interpretable  In particular, models based on neural networks despite good predictive capabilities , they offer a black-box description of the system dynamics  A possible remedy is applying a symbolic regression to the learned neural network representation , but this adds additional computational cost due to the two-step procedure 
A number of frameworks such as SINDy allows to learn interpretable dynamics but it relies on the a-priori availability of lower-dimensional descriptors and of time-derivatives which can be very noisy for both simulation and experimental data  Other frameworks are tailored to specific problems such as molecular dynamics  
Here, we present a framework that only needs the value of the observables, and not their derivatives, as training data and is capable of identifying interpretable latent dynamics  The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model  
The present method is related to approaches based on the Koopman-operator extended Dynamic Mode Decomposition (eDMD) but uses continuous complex-valued latent space dynamics and only requires one scalar variable per latent dimension to describe the latent space dynamics  Therefore we do not have to enforce any parametrizations on the Koopman matrix  
The time-continuous formulation moreover allows to incorporate sparse and irregularly sampled training data and fast generation of predictions after the training phase  By using a complex-valued latent space we can also incorporate harmonic effects and reduce the number of latent variables needed  Linear and non-linear autoencoders are used to map the observed, high-dimensional time-series to the lower-dimensional, latent representation and we identify simultaneously the autoencoder as well as the latent dynamics by optimizing a combined loss function 
Hence the to tasks of dimensionality reduction and discovery of the reduced dynamics are unified while other frameworks treat the two parts separately   Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed   We are also proposing a probabilistic version of our algorithm ) that makes use of probabilistic Slow Feature Analysis  
This allows for a latent representation that arart from being time-continuous, can quantify the predictive uncertainty and hierarchically decompose the dynamics into their pertinent scales while promoting the discovery of slow processes that control the system's evolution over long time horizons  The rest of the paper is structured as follows: We introduce the methodological framework as well as algorithmic details in section II 
Particular focus is paid on the interpretability of the inferred lower-dimensional dynamics  In section III we present three numerical illustrations, i e  a system of linear ODEs, a hidden Markov Model and the discretized KS-equation  We then present in section IV the probabilistic extension of the framework and apply it to the KS-equation 
We conclude with a summary and a short discussion about possible next steps  We introduce the autoencoders deployed in this work, followed by the interpretable latent space dynamic and discuss the training process  We consider data from high-dimensional time series x n ∈ R f with n = 1,    , T   We remark that the intervals between the different states do not need to be uniformly spaced 

Autoencoder

A core assumption of the method is that each high-dimensional state x n can be compressed to a lower-dimensional representation z n ∈ C c with c << f   We identify this lower-dimensional representation by an autoencoder consisiting of a parameterized encoder and decoder  The encoder maps the high-dimensional representation to the latent space as:
The latent space is complex-valued  The decoder reconstructs the high-dimensional representation based on the latent variables as: We denote the parameters of the encoder as well as the decoder by θ  As discussed later in Section II C, both set of parameters are optimized simultaneously during training and therefore there is no need for differentiating them 

Interpretable Latent Space Dynamics

We employ a propagator in the latent space to capture the reduced-order dynamics of the system  In contrast to other time-extended variational autoencoder frameworks, our representation uses complex valued latent variables  In addition the latent variables are treated independently  The latter feature enables us to have an interpretable latent dynamics as well as a model that is especially suitable for being trained in the Small Data regime due to the small number of required parameters 
This is in contrast to temporal propagators such as LSTMs   For each dimension i of the latent variable z we are using the following continuous ODE in the complex plane: By solving this ODE, we can define the operator: Interpretable reduced-order modeling with time-scale separation Here, λ is a vector containing all the individual λ's and ∆t n indicates the time-step between the latent states 
The symbol is used to indicate a component-wise multiplication  We remark that the latent variables and the parameter governing the temporal evolution are complex numbers and their role in describing the system dynamics is similar to that of an eigenvalue  The real part is associated with growth and decay whereas the imaginary part is representing the periodic component 
This approach has similarities with the Koopman-operator based methods and the extended dynamic mode decomposition   In contrast to the methods mentioned before we are using a continuous formulation in the latent space that allows us to incorporate scarce and irregularly sampled training data and directly rely on complex numbers in the latent space 

Training and Predictions

We optimize a loss function that combines both a reconstruction loss as well as a loss associated with the error of our learned propagator in the latent space: (5) We note that we could directly incorporate mini-batch training by only taking the summation over a subset of the N available training data 
For new predictions of unseen states, we use the encoder to generate a latent representations which is then advanced in time by the learned propagator  At a designated time step we are using the decoder to reconstruct the high-dimensional solution  We applied our algorithm to three systems  First, we show that the algorithm is capable of exactly reproducing the solution of a linear ODE and to identify its eigenvalues 
Afterwards we are applying the framework to a high-dimensional process generated by a complex latent dynamics, which is correctly identified  As a final test case, we are applying the algorithm to a Kuramoto Shivashinski (KS) equation  Interpretable reduced-order modeling with time-scale separation

Linear ODE

We are considering a two-dimensional ODE system for x = y 1 y 2 : Based on the obtained training data we run our algorithm using a linear encoder and decoder structure as well as two latent variables z



ground_truth: It provides cover for the war and allows supporters of the illegal war to point to it.

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 2: 
The non-denoised results are labelled with M = 0, and the noiseless values with p = 0  where sgn(η g ) is the sign of the sampled coefficient of the gth channel  γ = 1 means that all signs are positive  Observables Ô p=0 for the noiseless circuit are then approximated by resampling the observables from the denoiser ensemble
where γ = N G g=1 γ g is the overall sampling overhead, with γ g the overhead of the gth gate  Clearly, a large γ implies a large variance of Ô p=0 for a given number of samples, with accurate estimation requiring the cancellation of large signed terms  The number of samples required to resolve this cancellation of signs is bounded by Hoeffding's inequality, which states that a sufficient number of samples to estimate Ô p=0 with error δ at probability 1 − ω is bounded by (2γ 2 /δ 2 ) ln(2/ω)  
Since γ scales exponentially in γ g , it is clear that a denoiser with large M and γ 1 will require many samples  We observed that decompositions with γ > 1 are crucial for an accurate denoiser  Restricting to γ = 1 leads to large infidelity and no improvement upon increasing the number of terms in or the depth M of the denoiser 
Simply put, probabilistic error cancellation of gate noise introduces a sign problem and it is crucial to find optimal parameterizations (1) which minimize γ to make the approach scalable  This issue arises in all high performance error mitigation schemes , because the inverse of a physical noise channel is unphysical and cannot be represented as a positive sum over CPTP maps 
This is clearly visible in the spectra of the denoiser, which lies outside the unit circle (cf  Fig  )  This makes the tunability of the number of gates in each denoiser sample a crucial ingredient, which allows control over the sign problem, because we can freely choose the η i in   For the parametrization (1) of denoiser channels, we try to find a set of parameters for error mitigation by minimizing the normalized Frobenius distance between the noiseless and denoised supercircuits
which bounds the distance of output density matrices and becomes zero for perfect denoising  We carry out the minimization of on a classical processor, using gradient descent with the differential programming algorithm from   Instead of explicitly calculating the accumulated global noise channel and subsequently inverting it, we approximate the noiseless supercircuit C with the denoised supercircuit D C, effectively yielding a circuit representation D of the inverse noise channel 
Results  -To benchmark the denoiser we apply it to the second-order Trotter circuits of the spin-1/2 Heisenberg chain with periodic boundary conditions (PBC) where is the Pauli algebra acting on the local Hilbert space of site i  A second-order Trotter circuit for evolution time t with depth M trot consists of M trot − 1 half brickwall layers with time step t/M trot and two layers with half time step  
We consider circuits that are affected by uniform depolarizing noise with probability p for simplicity, but our approach can be used for any non-Clifford noise  The two-qubit noise channel is which acts on neighboring qubits i and i + 1 and is applied to each Trotter and denoiser gate, and p = 0 01 unless stated otherwise 
We study circuits with depths M trot = 16, 32, 64 for evolution times t = 0 5, 1,    , 5, and denoisers D with depths M = 1, 2, 4, 6, 8  In the top panels of Fig  we show (4) for a chain of size L = 8 as a function of time t  Here it can be seen that even for M trot = 32 a denoiser with M = 1 already improves by roughly an order of magnitude at all considered t 
Depending on M trot and t, further increasing M lowers , with the biggest improvements occurring for high precision Trotter circuits with large depth M trot = 64 and short time t = 0 5, where the Trotter gates are closer to the identity than in the other cases  At the other extreme, for M trot = 16 the improvements are relatively small upon increasing M > 2  In all cases the denoiser works better at early times than at late times, again indicating that it is easier to denoise Trotter gates that are relatively close to the identity 
To probe the accuracy of the denoiser on quantities that do not enter the optimization, as a first test we consider the two-point correlator between spins at different times where we have chosen the infinite temperature initial state, and C(t) is the Trotter supercircuit for time t  In the bottom panels of Fig  we show C zz i=L/2,j=L/2 (t) for the supercircuits from the upper panels, now for a L = 14 chain 
Here we see that at M trot = 16 we can retrieve the noiseless values already with M = 1, but that increasing M trot makes this more difficult  At M trot = 64 we see larger deviations, and improvement upon increasing M is less stable, but nonetheless we are able to mitigate errors to a large extent  As a further test, we compute the out-of-time-ordered correlator (OTOC) ]
In Fig  we show the results for i = L/2, for a Trotter circuit with depth M trot = 32 and a denoiser with depth M = 2  Here we see that a denoiser with M M trot is able to recover the light-cone of correlations, which are otherwise buried by the noise  In the Supplementary Material we consider how the denoiser performs at different noise levels p, and how the denoised supercircuits perform under stacking 
There we also calculate domain wall magnetization dynamics, and show the distribution of the optimized denoiser parameters and the sampling overhead associated to the denoiser as a whole  In Fig  we show the eigenvalues of the noisy supercircuits for a noisy second-order Trotter supercircuit with M trot = 16 at t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised supercircuit (right) 
The eigenvalues λ of a unitary supercircuit lie on the unit circle, and in the presence of dissipation they are pushed to the center  We see that the spectrum of the denoiser lies outside the unit circle, making it an unphysical channel which cures the effect of the noise on the circuit, such that the spectrum of the denoised circuit is pushed back to the unit circle 
The noiseless eigenvalues are shown as blue bars, making it clear that the denoiser is able to recover the noiseless eigenvalues from the noisy circuit  In the Supplementary Material we show the spectra for a p = 0 036 denoiser, where we observe a clustering of eigenvalues reminiscent of Refs    There we also investigate the channel entropy of the various supercircuits  
Conclusion  -We have introduced a probabilistic error cancellation scheme, where a classically determined denoiser mitigates the accumulated noise of a (generally non-Clifford) local noise channel  The required number of mitigation gates, i e  the dimensionality of the corresponding quasiprobability distribution, is tunable and the parameterization of the corresponding channels provides control over the sign problem that is inherent to probabilistic error cancellation 
We have shown that a denoiser with one layer can already significantly mitigate errors for second-order Trotter circuits with up to 64 layers  This effectiveness of low-depth compressed circuits for denoising, in contrast with the noiseless time evolution operator compression from , can be understood from the non-unitarity of the denoiser channels 
In particu-lar, measurements can have non-local effects, since the measurement of a single qubit can reduce some highly entangled state (e g  a GHZ state) to a product state, whereas in unitary circuits the spreading of correlations forms a light-cone  To optimize a denoiser with convenience at L > 8, the optimization can be formulated in terms of matrix product operators or channels , which is convenient because the circuit calculations leading to the normalized distance and its gradient are easily formulated in terms of tensor contractions and singular value decompositions  
This provides one route to a practical denoiser, which is relevant because the targeted noiseless circuit and the accompanying noisy variant in (4) need to be simulated classically, confining the optimization procedure to limited system sizes with an exact treatment or limited entanglement with tensor networks 
Nonetheless, we can use e g  matrix product operators to calculate (4) for some relatively small t, such that the noiseless and denoised supercircuits in (4) have relatively small entanglement, and then stack the final denoised supercircuit on a quantum processor to generate classically intractable states 
Analogously, we can optimize the channels exactly at some classically tractable size and then execute them on a quantum processor with larger size  Both approaches are limited by the light-cone of many-body correlations, as visualized in Fig  , because finite-size effects appear when the light-cone width becomes comparable with system size 
1  The normalized distance (left) and z spin correlator C zz i=L/2,j=L/2 (right), for a second-order Trotter supercircuit of depth Mtrot = 16 for time t = 1, affected by various twoqubit depolarizing errors p  We compare the values obtained with and without a denoiser, i e  M > 0 and M = 0, to the noiseless values (p = 0) 
The denoiser is affected by the same noise as the Trotter circuit  We consider denoisers with depths M = 1, 2, 4, 6, 8, and we use a L = 8 Heisenberg chain with PBC for the normalized distance, while for the correlator we use L = 14  * david luitz@uni-bonn de to observe that even for larger noise strength p, the local observable C zz improves significantly even with denoisers of depth M = 1 
For large noise strengths, we generally see that the optimization of the denoiser becomes difficult, leading to nonmonotonic behavior as a function of p, presumably because we do not find the global optimum of the denoiser  It is interesting to analyze the spectra of the supercircuits considered in this work 
As mentioned in the main text, the spectrum of the ideal, unitary supercircuit C lies on the unit circle  The comparison to this case is therefore instructive  In the main text, we showed an example of the spectra in Fig  for moderate noise strength  Here, we show additional data for stronger noise p = 0 036 in Fig  for a denoiser with M = 4 layers, optimized to mitigate errors for a second-order Trotter supercircuit with M trot = 16 layers at time t = 1 
The eigenvalues λ of the noisy supercircuit C are clustered close to zero, far away from the unit circle (except for λ = 1), showing that the circuit is strongly affected by the noise  To mitigate the impact of the noise, the denoiser consequently has to renormalize the spectrum strongly  If it accurately represents the inverse of the global noise channel, its spectrum has to lie far outside the unit circle, which is the case 
Interestingly, we observe a clustering of eigenvalues which is reminiscent to the spectra found in   By comparison to these works, we suspect that this is due to the local nature of the denoiser, and warrants further investigation  The right panel of Fig  shows the result of the denoiser, pushing the eigenvalues back to the unit circle, nearly with the exact same distribution along the circle as the noiseless eigenvalues (blue bars) 
Due to the strong noise, this is not achieved perfectly, and it is clear that this cannot work in principle if the global noise channel has a zero eigenvalue  The complexity of an operator can be quantified by its operator entanglement entropy   Here we calculate the half-chain channel entanglement entropy S of the noiseless C, noisy C, denoiser D, and denoised D C supercircuits 
We define S as the entanglement entropy of the state that is related to a supercircuit C via the Choi-Jamio lkowski isomorphism, i e  ψ C = χ C /N , where the process matrix χ ab,cd C = C ac,bd is simply a reshaped supercircuit and N ensures normalization  Then we have S = −Tr [ψ C ln ψ C ]  This entropy measure is a particular instance of the "exchange entropy", which characterizes the information exchange between a quantum system and its environment  
In Fig  we plot the various S for a second-order Trotter circuit with M trot = 16 at t = 2, for a denoiser with M = 4, both affected by two-qubit depolarizing noise with p ∈ [10 −3 , 10 −1 ]  The Trotter circuit is for a Heisenberg model with L = 6 and PBC  We see that at large p, the noise destroys entanglement in the noisy supercircuit, and that the denoiser S increases to correct for this, such that the denoised supercircuit recovers the noiseless S 
Here we investigate how denoised supercircuits perform upon repeated application  We optimize the denoiser for a Trotter supercircuit for a fixed evolution time t  Then, to reach later times, we stack the denoised supercircuit n times to approximate the evolution up to time nt: In Fig  we stack a denoised t = 1 supercircuit up to n = 20 times and calculate the correlation function, defined in the main text, for the middle site 
We consider Trotter depths M trot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8, for a L = 14 Heisenberg chain with p = 0 01 depolarizing two-qubit noise  The noisy results correspond to M = 0 and the noiseless results to p = 0  In Fig  we calculate the OTOC, defined in the main text, with stacked time evolution for a denoised t = 2 supercircuit with M trot = 32 and M = 2, stacked up to ten times 
We see that the stacked supercircuit performs very well, and the additional precision obtained by using deep denoisers (M = 8) pays off for long evolution times, where we see convergence to the exact result (black dashed lines in Fig  ) as a function of M   FIG    The two-point z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t, for the infinite temperature initial state, for denoised second-order Trotter supercircuits that are optimized at evolution time t = 1 and then stacked up to twenty times 
We use Trotter depths Mtrot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8  The calculations were performed for a periodic Heisenberg model with L = 14 and PBC, affected by two-qubit depolarizing noise with strength p = 0 01, which also affects the denoiser  The non-denoised results are labelled with M = 0, and the noiseless results with p = 0 
The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively  The costliest and most noise-susceptible operation is the two-qubit ZZ rotation with angle α, which is the foundation of the unitary piece in our channel parameterization, defined in the main text 
For completeness, we here present the α angles of the optimized denoisers  The results are shown in Fig  , which contains histograms for the channel count N G versus α  The histograms are stacked, with the lightest color corresponding to the angles of the denoiser at t = 0 5 and the darkest at t = 5  The top four panels are for a denoiser with M = 2 and the bottom four with M = 8 
We consider M trot = 8, 16, 32, 64  We see that in both cases the distribution widens upon increasing M trot , indicating that the unitary channels start deviating more from the identity  Moreover, while the M = 2 denoisers in all cases except M trot = 64 have ZZ contributions close to the identity, this is clearly not the case for M = 8 
For simplicity, we did not focus on obtaining denoisers with the smallest sampling overhead γ, which is required to minimize the sign problem and hence ease the sampling of mitigated quantities  Instead, we let the optimization freely choose the η i in the denoiser parameterization, as defined in the main text 
In Fig  we show the sampling overhead of the denoisers from Fig  of the main text  We see that for M = 1 and M = 2 the sampling overhead is relatively small and uniform across the different t, whereas for M > 2 the optimization sometimes yields a denoiser with large γ and other times with small γ  This could be related to the difference in α distributions from Fig   
The large fluctuations of γ appears to stem from the difficulty in finding optimal deep denoisers, and our optimization procedure likely only finds a local minimum in these cases  Here C(t) is the Trotter supercircuit for time t  In Fig  we show Z dw for the circuits from Fig 
Context 3:  The loss function was optimized using the Adam algorithm   As we consider a linear ODE we can analytically compute the eigenvalues involved and compare it with the parameters λ identified by our algorithm 
We observe in Figure that the algorithm was able to recover the correct values, i e  the eigenvalues 7 and 3 of the given linear ODE  The system does not have a periodic component and the two imaginary parts correctly go to zero, whereas the real parts converge to the reference value  Moreover we are also able to identify for the linear mapping between our latent variables z and the training data a matrix consisting of a multiple of the eigenvectors (1,1) and (1,-1) and thus the correct solution 
This example was chosen to show that the algorithm is able to quickly identify the exact solution of a linear ODE in terms of its linearly independent components 

Hidden multiscale dynamics

We consider eight-dimensional synthetic time series data produced by an underlying twodimensional complex valued process  In particular, the data points x are generated by first solving for the temporal evolution for the two complex-valued processes p 1 and p 2 and than mapping to the eight-dimensional space by using a randomly sampled linear mapping W  
One of the two processes used to generate the data is chosen to be much slower than the other one and both processes have a periodic component  dp 2 dt = (−0 9 + 1 5i)p 2 (8) As training data we consider 40 time series with 150 data points each, obtained by simulating the described processes for a maximum of t = 15 s and then sampling from the obtained data points 
Hence the training data consists of: • 40 time-series • with each consisting 150 observations of the x at a uniform time-step ∆t = 0 0025 The autoencoder obtained consists of one linear layer for both the decoder as well as the encoder  The model is trained for 5000 iterations using the Adam optimizer and a learning rate of 10 −3  
The results for the convergence of the parameters λ 1 and λ 2 can be found in Figure   We note that the process which is slower decaying and thus more responsible for the long-term evolution of the system has a higher convergence rate than the faster process  With the obtained parameters λ as well as the trained autoencoder, we compute predictions based on the last time step used for training, i e  we apply the encoder to obtain our latent representation and than use the latent dynamics to advance the latent representation in time 
Afterwards, we employ the decoder to reconstruct the full high-dimensional system  The results can be found in Figure and show very good agreement between predictions and reference data  This example shows that our model is successfully able to carry out dimensionality reduction and moreover indicates that the convergence rate between latent processes can be different 
The latter is relevant when training models as for accurate predictions all latent processes and their dynamics should be converged 

Kuramoto-Sivashinsky

Finally, we applied our algorithm to the KS equation and aim to identify a reduced-order model for the solution u(y, t): We employed periodic boundary conditions, µ = 1 and a domain size y ∈ [0, 22]  For this domain-size, the KS-equation exhibits a structurally stable chaotic attractor as discussed in The black lines divides the area for which training data was given from the area without raining data 
;   The equation is discretized in space using a discretization step of 22 64 resulting in a state vector x of dimension 64 and a nonlinear system of coupled ODEs  This is solved using a stiff fourth-order solver  We employed a non-linear encoder and decoder with four fully-connected layers each and ReLU-activation functions as well as Dropout Layers between the fully-connected layers 
We trained the model for 200000 iterations using Adam and a learning rate of 5 • 10 4 and assuming a five-dimensional latent space  We obtained the λ's in Figure   Four latent variables have λ's close to zero and thus a slow temporal dynamic that is responsible for the long-term evolution whereas one latent variable is quickly decaying 
Based on the obtained parameters, we do predictions based on an unseen initial condition not contained in the training data  We are able to reconstruct the correct phase space based on our predictions despite only using a very limited amount of training data  The results for the phase space can be seen in Figure  
Although the small-scale fluctuations in the temporal dynamics are not well captured, the model identifies the correct manifold which has a good accuracy compared to the reference solution  All phase-spaces were obtained by using a finite-difference operator on the data or predictions  These results are in accordance Interpretable reduced-order modeling with time-scale separation with whose LSTM-based temporal dynamic model was also able to find the correct phase space but not to track the actual dynamics for long-term predictions 
Our model is not able to account for noise in the temporal evolution and thus dealing with chaotic, small-scale fluctuations is challenging  We believe that a probabilistic version of our algorithm could be advantageous here  This section contains a fully probabilistic formulation for the deterministic model discussed before 
We replace the Autoencoder with a Variational Autoencoder and the ODE in the latent space with a SDE  The loss function which we optimize is the Evidence-Lower Bound (ELBO) 

Model Structure

We postulate the following relations for our probabilistic model using an Ornstein-Uhlenbeck (OU) for each dimension i of the latent space and a Wiener process W t in the latent space: We again assume that the latent variables z t are complex-valued and a priori independent  Complex variables were chosen as their evolution includes a harmonic components which are observed in many physical systems 
We assume an initial conditions z 0,i ∼ CN (0, σ 2 0,i )  The total parameters associated with the latent space dynamics of our model are thus {σ 2 0,i , σ 2 i , λ i } c i=1 and will be denoted by θ together with all parameters responsible for the decoder mapping G (see next section)  These parameters along with the state variables z t have to be inferred from the data x t  
Based on probabilistic Slow Feature Analysis (SFA) , we set σ 2 i = 2; (λ j ) and σ 2 0,i = 1  As a consequence, a priori, the latent dynamics are stationary  A derivation and reasoning for this choice can be found in Appendix A  Hence the only independent parameters are the λ i , the imaginary part of which can account for periodic effects in the latent dynamics 

Variational Autoencoder

We employ a variational autoencoder to account for a probabilistic mappings from the lower-dimensional representation z n to the high-dimensional system x n   In particular we are employing a probabilistic decoder The encoder is used to infer the state variables z based on the given data and thus defined in the inference and learning section 

Inference and Learning

Given the probabilistic relations , our goal is to infer the latent variables z 0:T as well as all model parameters θ  We follow a hybrid Bayesian approach in which the posterior of the state variables is approximated using amortized Variational Inference and Maximum-A-Posteriori (MAP) point-estimates for θ are computed 
The application of Bayes' rule for each data sequence x 0:T leads to the following posterior: where p(θ) denotes the prior on the model parameters  In the context of variational inference, we use the following factorization of the approximate posterior i e  we infer only the mean µ and variance σ for each state variable based on the given data points 
This conditional density used for inference is the encoder-counterpart to the probabilistic decoder defined in the section before  It can be readily shown that the optimal parameter values are found by maximizing the Evidence Lower Bound (ELBO) F(q φ (z 0:T ), θ) which is derived in Appendix B  We compute Monte Carlo estimates of the gradient of the ELBO with respect to φ and θ with the help of the reparametrization trick and carry out stochastic optimization with the ADAM algorithm  

Results for the probabilistic extension

We applied our probabilistic version to the KS-equation  We used the same settings as for the deterministic approach but considered up to 10 complex latent variables  The obtained λ's are in Figure   The probabilistic model allows us to quantify the uncertainty in predictions  In Figure predictions for various time-steps and the respective uncertainty bounds are shown for an unseen initial condition 
Due to the chaotic nature of the KS-equation and the small amount of training data, the underlying linear dynamic of our model is only able to capture the full dynamics for a limited time horizon  Fortunately, due to the probabilistic approach the model is capable of capturing chaotic fluctuations with increasingly wide uncertainty bounds 
We also computed the phase space representation for the KS-equation based on the predictions obtained by our model and compare it with the reference solution  The probabilistic model identifies the correct manifold with a better accuracy than the deterministic model  As some of the small-scale fluctuations are accounted as noise, the resulting manifold is more concentrated at the origin and the obtained values are slightly smaller than the reference manifold although their shape is very similar 



ground_truth: L = 8 and L = 14.

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 does the framework capture the reduced-order dynamics?

Read the following text and answer briefly.

Context 4:  Voltage pulses ranging between 4-6 V are applied by the tip of a combined STM/AFM system, which result in cleavage of one C-H bond at each of the pentagonal apices of 6, thereby leading to the generation of 5 (Fig  ) 
In the main text, we focus on the generation and characterization of 5 on insulating surfaces  Generation and characterization of 5 on coinage metal surfaces is shown in Supplementary Fig    )  Blue and orange colors represent spin up and spin down densities, respectively  c, Probability density of the SOMOs of 5OS (isovalue: 0 001 e -Å -3 ) 
d, DFT-calculated bond lengths of 5OS  e, Constant-height I(V) spectra acquired on a species of 5 assigned as 5OS, along with the corresponding dI/dV(V) spectra  Open feedback parameters: V = -2 V, I = 0 17 pA (negative bias side) and V = 2 V, I = 0 17 pA (positive bias side)  Acquisition position of the spectra is shown in Supplementary Fig    f, Scheme of many-body transitions associated to the measured ionic resonances of 5OS 
Also shown are STM images of assigned 5OS at biases where the corresponding transitions become accessible  Scanning parameters: I = 0 3 pA (V = -1 2 V and -1 5 V) and 0 2 pA (V = 1 3 V and 1 6 V)  g, Laplace-filtered AFM image of assigned 5OS  STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3
Å  The tip-height offset Δz for each panel is provided with respect to the STM setpoint, and positive (negative) values of Δz denote tip approach (retraction) from the STM setpoint  f and g show the same molecule at the same adsorption site, which is next to a trilayer NaCl island  The bright and dark features in the trilayer NaCl island in g correspond to Cl -and Na + ions, respectively 
Scale bars: 10 Å (f) and 5 Å (g)  To experimentally explore the electronic structure of 5, we used bilayer NaCl films on coinage metal surfaces to electronically decouple the molecule from the metal surfaces  Before presenting the experimental findings, we summarize the results of our theoretical calculations performed on 5 in the neutral charge state (denoted as 5 0 ) 
We start by performing DFT calculations on 5 0 in the gas phase  Geometry optimization performed at the spin-unrestricted UB3LYP/6-31G level of theory leads to one local minimum, 5OS, the geometry of which corresponds to the open-shell resonance structure of 5 (Fig  , the label OS denotes open-shell) 
The triplet electronic configuration of 5OS is the lowest-energy state, with the openshell singlet configuration 90 meV higher in energy  Geometry optimization performed at the restricted closed-shell RB3LYP/6-31G level reveals two local minima, 5para and 5ortho, the geometries of which (Fig  ) exhibit bond length alternations in line with the presence of a para-or an ortho-QDM moiety, respectively, in the as-indacene core of the closed-shell resonance structures of 5 (Fig  )  
Relative to 5OS in the triplet configuration, 5para and 5ortho are 0 40 and 0 43 eV higher in energy, respectively  Additional DFT results are shown in Supplementary Fig    To gain more accurate insights into the theoretical electronic structure of 5, we performed multireference perturbation theory calculations (Supplementary Fig  ) based on quasi-degenerate second-order n-electron valence state perturbation theory (QD-NEVPT2) 
In so far as the order of the ground and excited states are concerned, the results of QD-NEVPT2 calculations qualitatively match with DFT calculations  For 5OS, the triplet configuration remains the lowest-energy state, with the open-shell singlet configuration 60 meV higher in energy  The energy differences between the open-and closed-shell states are substantially reduced in QD-NEVPT2 calculations, with 5para and 5ortho only 0 11 and 0 21 eV higher in energy, respectively, compared to 5OS in the triplet configuration 
We also performed nucleus-independent chemical shift calculations to probe local aromaticity of 5 in the openand closed-shell states  While 5OS in the triplet configuration exhibits local aromaticity at the terminal benzenoid rings, 5OS in the open-shell singlet configuration, 5para and 5ortho all display antiaromaticity (Supplementary Fig  ) 
The choice of the insulating surface determines the charge state of 5: while 5 adopts neutral charge state on the high work function bilayer NaCl/Au(111) surface (irrespective of its openor closed-shell state, Supplementary Fig  ), 5 exhibits charge bistability between 5 0 and the anionic state 5 -1 on the lower work function bilayer NaCl/Ag(111) and Cu(111) surfaces (Supplementary Figs  ) 
In the main text, we focus on the characterization of 5 on bilayer NaCl/Au(111)  Characterization of charge bistable 5 is reported in Supplementary Figs    We first describe experiments on 5 on bilayer NaCl/Au(111), where 5 exhibits a geometry corresponding to the calculated 5OS geometry, and an open-shell electronic configuration 
We compare the experimental data on this species to calculations on 5OS with a triplet configuration, as theory predicts a triplet ground state for 5OS  For 5OS, the calculated frontier orbitals correspond to the SOMOs ψ1 and ψ2 (Fig  ), whose spin up levels are occupied and the spin down levels are empty 
Figure shows the DFT-calculated bond lengths of 5OS, where the two salient features, namely, the small difference in the bond lengths within each ring and the notably longer bond lengths in the pentagonal rings, agree with the open-shell resonance structure of 5 (Fig  )  Figure shows an AFM image of 5 adsorbed on bilayer NaCl/Au(111) that we assign as 5OS, where the bond-order differences qualitatively correspond to the calculated 5OS geometry (discussed and compared to the closed-shell state below) 
Differential conductance spectra (dI/dV(V), where I and V denote the tunneling current and bias voltage, respectively) acquired on assigned 5OS exhibit two peaks centered at -1 5 V and 1 6 V (Fig  ), which we assign to the positive and negative ion resonances (PIR and NIR), respectively  Figure shows the corresponding STM images acquired at the onset (V = -1 2
V/1 3 V) and the peak (V = -1 5 V/1 6 V) of the ionic resonances  To draw a correspondence between the STM images and the molecular orbital densities, we consider tunneling events as many-body electronic transitions between different charge states of 5OS (Fig  )  Within this framework, the PIR corresponds to transitions between 5 0 and the cationic state 5  
At the onset of the PIR at -1 2 V, an electron can only be detached from the SOMO ψ1 and the corresponding STM image at -1 2 V shows the orbital density of ψ1  Increasing the bias to the peak of the PIR at -1 5 V, it becomes possible to also empty the SOMO ψ2, such that the corresponding STM image shows the superposition of ψ1 and ψ2, that is, |ψ1| 2 + |ψ2| 2 (ref 
)  Similarly, the NIR corresponds to transitions between 5 0 and 5 -1   At the NIR onset of 1 3 V, only electron attachment to ψ2 is energetically possible  At 1 6 V, electron attachment to ψ1 also becomes possible, and the corresponding STM image shows the superposition of ψ1 and ψ2  The observation of the orbital densities of SOMOs, and not the hybridized HOMO and LUMO, proves the open-shell ground state of assigned 5OS 
Measurements of the monoradical species with a doublet ground state are shown in Supplementary Fig    Unexpectedly, another species of 5 was also experimentally observed that exhibited a closedshell ground state  In contrast to 5OS, where the frontier orbitals correspond to the SOMOs ψ1 and ψ2, DFT calculations predict orbitals of different shapes and symmetries for 5para and 5ortho, denoted as α and β and shown in Fig   
For 5ortho, α and β correspond to HOMO and LUMO, respectively  The orbitals are inverted in energy and occupation for 5para, where β is the HOMO and α is the LUMO  Fig  shows an AFM image of 5 that we assign as 5para  We experimentally infer its closed-shell state first by using qualitative bond order discrimination by AFM 
In high-resolution AFM imaging, chemical bonds with higher bond order are imaged brighter (that is, with higher frequency shift Δf) due to stronger repulsive forces, and they appear shorter   In Fig  , we label seven bonds whose bond orders show significant qualitative differences in the calculated 5ortho, 5para (Fig  ) and 5OS (Fig  ) geometries 
In 5para, the bonds b and d exhibit a higher bond order than a and c, respectively  This pattern is reversed for 5ortho, while the bond orders of the bonds a-d are all similar and small for 5OS  Furthermore, in 5para bond f exhibits a higher bond order than e, while in 5ortho and 5OS bonds e and f exhibit similar bond order (because they belong to Clar sextets) 
Finally, the bond labeled g shows a higher bond order in 5para than in 5ortho and 5OS  The AFM image of assigned 5para shown in Fig  indicates higher bond orders of the bonds b, d and f compared to a, c and e, respectively  In addition, the bond g appears almost point-like and with enhanced Δf contrast compared to its neighboring bonds, indicative of a high bond order (see Supplementary Fig  for height-dependent measurements) 
These observations concur with the calculated 5para geometry (Fig  )  Importantly, all these distinguishing bond-order differences are distinctly different in the AFM image of 5OS shown in Fig  , which is consistent with the calculated 5OS geometry (Fig  )  In the AFM images of 5OS (Fig  and Supplementary Fig  ), the bonds a-d at the pentagon apices appear with similar contrast and apparent bond length 
The bonds e and f at one of the terminal benzenoid rings also exhibit similar contrast and apparent bond length, while the central bond g appears longer compared to assigned 5para  Further compelling evidence for the closed-shell state of assigned 5para is obtained by STM and STS  dI/dV(V) spectra acquired on an assigned 5para species exhibit two peaks centered at -1 4 V (PIR) and 1 6 V (NIR) (Fig  ) 
STM images acquired at these biases (Fig  ) show the orbital densities of β (-1 4 V) and α (1 6 V)  First, the observation of α and β as the frontier orbitals of this species, and not the SOMOs, strongly indicates its closed-shell state  Second, consistent with AFM measurements that indicate good correspondence to the calculated 5para geometry, we observe β as the HOMO and α as the LUMO 
For 5ortho, α should be observed as the HOMO and β as the LUMO  We did not observe molecules with the signatures of 5ortho in our experiments  We observed molecules in open-(5OS, Fig  ) and closed-shell (5para, Fig  ) states in similar occurrence after their generation from 6 on the surface  We could also switch individual molecules between open-and closed-shell states as shown in Fig  and Supplementary Fig   
To this end, a change in the adsorption site of a molecule was induced by STM imaging at ionic resonances, which often resulted in movement of the molecule  The example presented in Fig  shows a molecule that was switched from 5para to 5OS and back to 5para  The switching is not directed, that is, we cannot choose which of the two species will be formed when changing the adsorption site, and we observed 5OS and 5para in approximately equal yields upon changing the adsorption site 
The molecule in Fig  is adsorbed on top of a defect that stabilizes its adsorption geometry on bilayer NaCl  At defect-free adsorption sites on bilayer NaCl, that is, without a third layer NaCl island or atomic defects in the vicinity of the molecule, 5 could be stably imaged neither by AFM nor by STM at ionic resonances (Supplementary Fig  ) 
Without changing the adsorption site, the state of 5 (open-or closedshell) never changed, including the experiments on bilayer NaCl/Ag(111) and Cu(111), on which the charge state of 5 could be switched (Supplementary Figs  )  Also on these lower work function surfaces, both open-and closed-shell species were observed for 5 0 and both showed charge bistability between 5 0 (5OS or 5para) and 5 -1 (Supplementary Figs  ) 
The geometrical structure of 5 -1 probed by AFM, and its electronic structure probed by STM imaging at the NIR (corresponding to transitions between 5 -1 and the dianionic state 5 -2 ), are identical within the measurement accuracy for the charged species of both 5OS and 5para  When cycling the charge state of 5 between 5 0 and 5 -1 several times, we always observed the same state (5OS or 5para) when returning to 5 0 , provided the molecule did not move during the charging/discharging process 
Based on our experimental observations we conclude that indeno[1,2-a]fluorene (5), the last unknown indenofluorene isomer, can be stabilized in and switched between an open-shell (5OS) and a closed-shell (5para) state on NaCl  For the former, both DFT and QD-NEVPT2 calculations predict a triplet electronic configuration 
Therefore, 5 can be considered to exhibit the spin-crossover effect, involving magnetic switching between high-spin (5OS) and low-spin (5para) states, coupled with a reversible structural transformation  So far, the spin-crossover effect has mainly only been observed in transition-metal-based coordination compounds with a near-octahedral geometry  
The observation that the switching between open-and closedshell states is related to changes in the adsorption site but is not achieved by charge-state cycling alone, indicates that the NaCl surface and local defects facilitate different electronic configurations of 5 depending on the adsorption site 
Gas-phase QD-NEVPT2 calculations predict that 5OS is the ground state, and the closed-shell 5para and 5ortho states are 0 11 and 0 21 eV higher in energy  The experiments, showing bidirectional switching between 5OS and 5para, indicate that a change in the adsorption site can induce sufficient change in the geometry of 5 (leading to a corresponding change in the ground state electronic configuration) and thus induce switching 
Switching between open-and closed-shell states in 5 does not require the breaking or formation of covalent bonds , but a change of adsorption site on NaCl where the molecule is physisorbed  Our results should have implications for single-molecule devices, capitalizing on the altered electronic and chemical properties of a system in π-diradical open-shell and closed-shell states such as frontier orbital and singlet-triplet gaps, and chemical reactivity 
For possible future applications as a single-molecule switch, it might be possible to also switch between open-and closed-shell states by changing the local electric field, such as by using chargeable adsorbates   Scanning probe microscopy measurements and sample preparation  STM and AFM measurements were performed in a home-built system operating at base pressures below 1×10 -10 mbar and a base temperature of 5 K  Bias voltages are provided with respect to the sample 
All STM, AFM and spectroscopy measurements were performed with carbon monoxide (CO) functionalized tips  AFM measurements were performed in non-contact mode with a qPlus sensor   The sensor was operated in frequency modulation mode with a constant oscillation amplitude of 0 5 Å  STM measurements were performed in constantcurrent mode, AFM measurements were performed in constant-height mode with V = 0 V, and I(V) and Δf(V) spectra were acquired in constant-height mode 
Positive (negative) values of the tip-height offset Δz represent tip approach (retraction) from the STM setpoint  All dI/dV(V) spectra are obtained by numerical differentiation of the corresponding I(V) spectra  STM and AFM images, and spectroscopy curves, were post-processed using Gaussian low-pass filters 
Au(111), Ag(111) and Cu(111) surfaces were cleaned by iterative cycles of sputtering with Ne + ions and annealing up to 800 K  NaCl was thermally evaporated on Au(111), Ag(111) and Cu(111) surfaces held at 323 K, 303 K and 283 K, respectively  This protocol results in the growth of predominantly bilayer (100)-terminated islands, with a minority of trilayer islands 
Sub-monolayer coverage of 6 on surfaces was obtained by flashing an oxidized silicon wafer containing the precursor molecules in front of the cold sample in the microscope  CO molecules for tip functionalization were dosed from the gas phase on the cold sample  Density functional theory calculations  DFT was employed using the PSI4 program package  
All molecules with different charge (neutral and anionic) and electronic (open-and closed-shell) states were independently investigated in the gas phase  The B3LYP exchangecorrelation functional with 6-31G basis set was employed for structural relaxation and singlepoint energy calculations  The convergence criteria were set to 10 −4 eV Å −1 for the total forces and 10 −6 eV for the total energies 
Multireference calculations  Multireference calculations were performed on the DFToptimized geometries using the QD-NEVPT2 level of theory , with three singlet roots and one triplet root included in the state-averaged calculation  A (10,10) active space (that is, 10 electrons in 10 orbitals) was used along with the def2-TZVP basis set  
Increasing either the active space size or expanding the basis set resulted in changes of about 50 meV for relative energies of the singlet and triplet states  These calculations were performed using the ORCA program package   Nucleus-independent chemical shift (NICS) calculations  Isotropic nucleus-independent chemical shift values were evaluated at the centre of each ring using the B3LYP exchangecorrelation functional with def2-TZVP basis set using the Gaussian 16 software package  
Starting materials (reagent grade) were purchased from TCI and Sigma-Aldrich and used without further purification  Reactions were carried out in flame-dried glassware and under an inert atmosphere of purified Ar using Schlenk techniques  Thin-layer chromatography (TLC) was performed on Silica Gel 60 F-254 plates (Merck) 
Column chromatography was performed on silica gel (40-60 µm)  Nuclear magnetic resonance (NMR) spectra were recorded on a Bruker Varian Mercury 300 or Bruker Varian Inova 500 spectrometers  Mass spectrometry (MS) data were recorded in a Bruker Micro-TOF spectrometer  The synthesis of compound 6 was developed following the two-step synthetic route shown in Supplementary Fig  , which is based on the preparation of methylene-bridge polyarenes by means of Pd-catalyzed activation of benzylic C-H bonds  
Supplementary Figure | Synthetic route to obtain compound 6  The complex Pd2(dba)3 (20 mg, 0 02 mmol) was added over a deoxygenated mixture of 1,3-dibromo-2,4-dimethylbenzene (9, 100 mg, 0 38 mmol), boronic acid 10 (178 mg, 1 14 mmol), K2CO3 (314 mg, 2 28 mmol) and XPhos (35 mg, 0 08 mmol) in toluene (1:1, 10 mL), and the resulting mixture was heated at 90 °C for 2 h 
After cooling to room temperature, the solvents were evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording 11 (94 mg, 76%) as a colorless oil  The complex Pd(OAc)2 (7 mg, 0 03 mmol) was added over a deoxygenated mixture of terphenyl 11 (90 mg, 0 27 mmol), K2CO3 (114 mg, 0 83 mmol) and ligand L (26 mg, 0 06 mmol) in NMP (2 mL) 
The resulting mixture was heated at 160 °C for 4 h  After cooling to room temperature, H2O (30 mL) was added, and the mixture was extracted with EtOAc (3x15 mL)  The combined organic extracts were dried over anhydrous Na2SO4, filtered, and evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording compound 6 (8 mg, 11%) as a white solid  in AFM imaging due to their reduced adsorption height compared to the rest of the carbon atoms 
We attribute this observation to the significantly different lattice parameter of Cu(111) (2 57 Å) compared to Au(111) and Ag(111) (2 95 Å and 2 94 Å, respectively) , such that the apical carbon atoms of the pentagonal rings of 5 adsorb on the on-top atomic sites on Au(111) and Ag(111), but not on Cu(111) 
Our speculation is based on a previous study of polymers of 1 on Au(111) by Di Giovannantonio et al  , where both tilted and planar individual units of 1 were observed depending on whether the apical carbon atoms of the pentagonal rings in 1 adsorbed on the on-top or hollow sites of the surface, respectively 
Given the strong molecule-metal interaction, we found no electronic state signatures of 5 on all three metal surfaces  STM set point for AFM images: V = 0  e, Frontier orbital spectrum of 5 -1   In the anionic state, ψ2 becomes doubly occupied and ψ1 is the SOMO  Filled and empty circles denote occupied and empty orbitals, respectively 
For each panel, zero of the energy axis has been aligned to the respective highest-energy occupied orbital 



ground_truth: By using a propagator in the latent space.