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: An CUSUM Test with Observation-Adjusted Control Limits in Change Detection
Publish Date: March 9, 2023
Author List: Fuquan Tang (from Department of Statistics, Shanghai Jiao Tong University), Dong Han (from Department of Statistics, Shanghai Jiao Tong University)

Figure

exp{−cg(µ)(θ − x Hv (θ) + o(1))} for 1 ≤ k ≤ ac − 1, bc ≤ n ≤ m, where Zi = −g ′ (µ)(Z i − µ)/a and Hv (θ) = ln hv (θ) + ( ac k − 1) ln ĥv (θ), ĥv (θ) = E v (e θ Zi ) 
i < cg(µ)(1 + o(1))) exp{−cg(µ)θ * v (1 + o(1))} (A  5) for ac ≤ k ≤ bc − 1, bc ≤ n ≤ m,andP v (
i + g ′ (µ)a −1 Tc(g)−1 i=Tc(g)−ac (Z i − µ)] −→ µas c → ∞ By the uniform integrability of {T c (g)/c} and using Theorem A 1 1 in Gut's book(1988), we haveE v (T c (g)) = (1 + o(1)) cg(µ) µfor a large c This completes the proof of Theorem 2 Proof of Theorem 4  Since g(x) < 0 for x > a * , a * ≤ µ * and µ * ≥ 0, it follows thatP v m Ẑm < cg( Ẑm ), Ẑm > a * ≤ P v ( Ẑm < µ * )andP v (T c (g) > m) = P v n i=n−k+1 Z i < cg( Ẑn ), 1 ≤ k ≤ n, 1 ≤ n ≤ m ≤ P v m Ẑm < cg( Ẑm ) = P v m Ẑm < cg( Ẑm ), Ẑm ≤ a * + P v m Ẑm < cg( Ẑm ), Ẑm > a * ≤ 2P v ( Ẑm < µ * ) Furthermore,P v ( Ẑm < µ * ) = P v ( m i −Z i > −mµ * ) = P v ( m i (µ − Z i ) > m(µ − µ * )) = P v (e θ m i (µ−Z i ) > e θm(µ−µ * ) ) ≤ e −m[θ(µ−µ * )−ln M (θ)] ,whereM(θ) = E v (e θ(µ−Z 1 )) and the last inequality follows from Chebychev's inequality Note thath(θ) = θ(µ − µ * ) − ln M(θ) attains its maximum value h(θ * ) = θ * (µ − µ * ) − ln M(θ * ) > 0 at θ = θ * > 0, where h ′ (θ * ) = 0  So, E v (T c (g)) = 1 + ∞ m=1 P v (T c (g) > m) ≤ 1 + m=1 −m[θ * (µ−µ * )−ln M (θ * )] = e θ * (µ−µ * )−ln M (θ * ) + 1 e θ * (µ−µ * )−ln M (θ * ) − Let k > 1 It follows that E vk (T c (g) − k + 1) + = ∞ m=1 P vk (T c (g) > m + k − 1, T c (g) > k − 1) ≤ (a 0 + 1)(k − 1)P 0 (T c (g) > k − 1) + ∞ m≥(a 0 +1)(k−1) P vk (T c (g) > m + k − 1) Similarly, we haveP vk (T c (g) > m + k − 1) = P vk n i=n−k+1 Z i < cg( Ẑn ), 1 ≤ k ≤ n, 1 ≤ n ≤ m + k − 1 ≤ 2P vk ( Ẑm+k−1 < µ * ) − Z i ) > m(µ − µ * ) + (k − 1)(µ 0 − µ * ) ≤ 2 exp{−m θ * (µ − µ * ) − ln M(θ * ) + k − 1 m [µ 0 − µ * − ln M 0 (θ * )] } ≤ e −mb for m ≥ (a 0 + 1)(k − 1), since θ * (µ − µ * ) − ln M(θ * ) + k − 1 m [µ 0 − µ * − ln M 0 (θ * )] ≥ b for m ≥ (a 0 + 1)(k−1) Thus, E vk (T c (g) − k + 1) + ≤ (a 0 + 1)(k − 1)P 0 (T c (g) ≥ k) + m≥(a 0 +1)(k−1) e −mb ≤ (a 0 + 1)(k − 1)P 0 (T c (g) >≥ k) + 2e −(a 0 +1)(k−1)b 1 − e −b  
Simulation of E τ i ,v and J ACE for detecting two mean shifts v = 0 1, v = 1 The parameters for T * M are k1=1, k2=150, r 1 = 5 2 * 10 −5 , r 2 = 1 1 * 10 −5 , and the expectation and standard deviation in both cases are 1717 06with 13459 80 and 3918 33 with 16893 25,respectively 

abstract

In this paper, we not only propose an new optimal sequential test of sum of logarithmic likelihood ratio (SLR) but also present the CUSUM sequential test (control chart, stopping time) with the observation-adjusted control limits (CUSUM-OAL) for monitoring quickly and adaptively the change in distribution of a sequential observations 
Two limiting relationships between the optimal test and a series of the CUSUM-OAL tests are established  Moreover, we give the estimation of the in-control and the out-of-control average run lengths (ARLs) of the CUSUM-OAL test  The theoretical results are illustrated by numerical simulations in detecting mean shifts of the observations sequence 

INTRODUCTION

In order to quickly detect a change in distribution of observations sequence without exceeding a certain false alarm rate, a great variety of sequential tests have been proposed, developed and applied to various fields since proposed a control chart method, see, for example, , , One of popular used sequential tests is the following upper-sided CUSUM test which was proposed by  
where c > 0 is a constant control limit, Z i = log[p v 1 (X i )/p v 0 (X i )], p v 0 (x) and p v 1 (x) are prechange and post-change probability density functions respectively for a sequence of mutually independent observations {X i , i ≥ 1}, that is, there is a unknown change-point τ ≥ 1 such that X 1 ,    , X τ −1 have the probability density function p v 0 , whereas, X τ , X τ +1 ,     have the probability density function p v 1  
By the renewal property of the CUSUM test T C we have , where E 1 (T C ) is the out-of-control average run length (ARL 1 ), P k and E k denote the probability and expectation respectively when the change from p v 0 to p v 1 occurs at the change-point τ = k for k ≥ 1  Though we know that the CUSUM test is optimal under Lorden's measure (see Moustakides 1986 and Ritov 1990), the out-of-control ARL 1 of the CUSUM test is not small, especially in detecting small mean shifts ( see Table in Section 4) 
In other words, the CUSUM test is insensitive in detecting small mean shifts  Then, how to increase the sensitivity of the CUSUM test ? Note that the control limit in the CUSUM test is a constant c which does not depend on the observation samples  Intuitively, if the control limit of the CUSUM test can become low as the samples mean of the observation sequence increases, then the alarm time of detecting the increasing mean shifts will be greatly shortened 
Based on this idea, by selecting a decreasing function g(x) we may define the ( upper-sided ) CUSUM chart T C (cg) with the observation-adjusted control limits cg( Ẑn ) ( abbreviated to the CUSUM-OAL chart ) in the following where c > 0 is a constant and Ẑn = n i=1 Z i /n  In other words, the control limits cg( Ẑn ) of the CUSUM-OAL test can be adjusted adaptively according to the observation information { Ẑn } 
Note that the control limits cg( Ẑn ) may be negative  In the special case, the CUSUM-OAL chart T C (cg) becomes into the conventional CUSUM chart T C (c) in (1) when g ≡ 1  Similarly, we can define a down-sided CUSUM-OAL test  In this paper, we consider only the upper-sided CUSUM-OAL test since the properties of the down-sided CUSUM-OAL test can be obtained by the similar method 
The main purpose of the present paper is to show the good detection performance of the CUSUM-OAL test and to give the estimation of its the in-control and out-of-control ARLs  The paper is organized as follows  In Section 2, we first present an optimal SLR sequential test, then define two sequences of the CUSUM-OAL tests and prove that one of the two sequences of CUSUM-OAL tests converges to the optimal test, another sequences of CUSUM-OAL tests converges to a combination of the optimal test and the CUSUM test 
The estimation of the in-control and out-of-control ARLs of the CUSUM-OAL tests and their comparison are given in Section 3  The detection performances of the three CUSUM-OAL tests and the conventional CUSUM test are illustrated in Section 4 by comparing their numerical out-ofcontrol ARLs  Section 5 provides some concluding remarks 
Proofs of the theorems are given in the Appendix 

AN OPTIMAL SLR TEST, TWO CUSUM-OAL TESTS AND THEIR LIMITING RELATIONSHIPS

Let P 0 and E 0 denote the probability and the expectation respectively with the probability density p v 0 when there is no change for all the time  It is known that It follows from Proposition 2 38 in and (5 8)-(5 9) in Chow et al, P 108) that the following sequence test of sum of logarithmic likelihood ratio (SLR)
for B > 1, is optimal in the following sense min for P 0 (T SLR < ∞) = α, where c = log B and 0 < α < 1  In particular, if P 0 is the standard normal distribution with mean shift µ > 0 after changepoint, we have Z j − µ 0 = µX j , where µ 0 = −µ 2 /2  It follows from proposition 4 in that the SLR test T SLR in (4) is also optimal (minimal ARL 1 ) with the same false alarm probability P 0 (T < τ ) 
It can be seen that the in-control average run length of T SLR is infinite, that is, ARL 0 = E 0 (T SLR ) = ∞  However, the minimal ARL 1 with finite ARL 0 is a widely used optimality criterion in statistical quality control (see ) and detection of abrupt changes (see   In order to get finite ARL 0 for T SLR , we replace the constant control limit c of T SLR in (3) or (4) with the dynamic control limit n(µ 0 − r) and obtain a modified SLR test T SLR (r) in the following
for r ≥ 0  For comparison, the in-control ARL 0 of all candidate sequential tests are constrained to be equal to the same desired level of type I error, the test with the lowest out-of-control ARL v has the highest power or the fastest monitoring (detection) speed  In the following example 1, the numerical simulations of the out-of-control ARLs of the CUSUM-OAL tests T C (cg u,0 ) in detecting the mean shifts of observations with normal distribution will be compared with that of the SLR tests T * (r) and T * (0), and that of the CUSUM-SLR test T C (c) ∧ T * (0) := min{T C (c), T * (0)} in the following Table  
These comparisons lead us to guess that there are some limiting relationships between T C (cg u,r ) and T * (r), and T C (c g u ) and T C (c) ∧ T * (0), respectively  Example 1  Let X 1 , X 2 ,      be mutually independent following the normal distribution N(0, 1) if there is no change  After the change-point τ = 1, the mean E µ (X k ) ( k ≥ 1 ) will change from v 0 = 0 to v = 0 1, 0 25, 0 5, 0 75, 1, 1 25, 1 5, 3  Here, we let
, where v 1 = 1 is a given reference value which for the CUSUM test is the magnitude of a shift in the process mean to be detected quickly  We conducted the numerical simulation based on 1,000,000 repetitions  The following Table lists the simulation results of the ARLs of the tests T C (c), T C (c g u ) for u = 1, 10, 10 2 , 10 3 , 10 4 , T * (0 0007), T C (c) ∧ T * (0) and T * (0) for detecting the mean shifts, where the mean shift 0 0 means that there is no change which corresponds to the in-control ARL 0 and all tests have the common ARL 0 ≈ 1000 except the test T * (0) which has ARL 0 = ∞ 
The values in the parameters are the standard deviations of the tests  From the last row in Table , it's a little surprising that though the ARL 0 of T * (0) is infinite, that is, E 0 (T * (0)) = ∞, the detection speed of T * (0) is faster than that of the CUSUM chart T C for all mean shifts, in particular, for detecting the small mean shift 0 1, the speed of T * (0) is only 7 47 which is very faster than the speed, 439, of the CUSUM test 
Moreover, both control charts T * (0 0007) and T C (11 9271) ∧ T * (0) not only have the nearly same detection performance as T * (0) but also can have the finite in-control ARL 0   Note particularly that when the number u in g u is taken from 0 to 1, 10, 10 2 , 10 3 , 10 4 , the detection speed of T C (c g u ) is getting faster and faster, approaching to that of T C (c) ∧ T * (0) 
This inspires us to prove the following theoretic results  Let τ = 1 and {X k , k ≥ 1} be an i i d  observations sequence with Theorem 2 shows that when the constant control limit c of the CUSUM test T C (c) is replaced with the observation-adjusted control limits {cg u,r ( Ẑn )} and {c g u ( Ẑn )} respectively, the corresponding two CUSUM-OAL tests {T C (cg u,r )} and {T C (c g u )} will converge to the optimal SLR test T * (r) and the CUSUM-SLR test T C (c) ∧ T * (0) as u → ∞, respectively 
In other words, the fastest alarm times that {T C (cg u,r )} and {T C (c g u )} can be reached are T * (r) and T C (c) ∧ T * (0), respectively  u ≥ 0} can be seen as two "long bridges" connecting T C (c) and T * (r), and T C (c) and T C (c) ∧ T * (0), respectively 

ESTIMATION AND COMPARISON OF ARL OF THE CUSUM-OAL TEST

In this section we will give an estimation of the ARLs of the following CUSUM-OAL test that can be written as where g( ) is a decreasing function, Ẑn (ac x] denotes the smallest integer greater than or equal to x  Here Ẑn (ac) is a sliding average of the statistics, Next we discuss on the the post-change probability distribution in order to estimate the ARLs of T C (cg) 
Usually we rarely know the post-change probability distribution P v of the observation process before it is detected  But the possible change domain and its boundary (including the size and form of the boundary) about v may be determined by engineering knowledge, practical experience or statistical data 
So we may assume that the region of parameter space V and a probability distribution Q on V are known  If we have no prior knowledge of the possible value of v after the change time τ , we may assume that v occurs equally on V , that is, the probability distribution Q is an equal probability distribution (or uniform distribution ) on V  
For example, let P v be the normal distribution and v = (µ, σ), where µ and σ denote the mean and standard deviation respectively, we can take the set V = {(µ, σ) : and Q is subject to the uniform distribution U(V ) on V if v occurs equally on V , where the numbers µ 1 , µ 2 , σ 1 and σ 2 are known  It means that we know the domain of the possible post-change distributions, P v , v ∈ V , i e , the boundary ∂V of the parameter space V is known 
Next we shall divide the parameter space V into three subsets V + , V 0 and V − by the Kullback-Leibler information distance  Let and are two Kullblak-Leibler information distances between P v , P v 0 and P v , P v 1   Since I(p|q) = 0 if and only if p = q, where p and q are two probability measures, it follows that
, it means that P v is closer to P v 0 than to P v 1 according to the Kullblak-Leibler information distance  There is a similar explanation for v ∈ V + or ∈ V 0   Suppose the post-change distribution P v and the function g(x) satisfy the following conditions: (I) The probability P v is not a point mass at E v (Z 1 ) and P v (Z 1 > 0) > 0 
(II) The moment-generating function h v (θ) = E v (e θZ 1 ) satisfies h v (θ) < ∞ for some θ > 0  (III) The function g(x) is decreasing, its second order derivative function g ′′ (x) is continuous and bounded, and there is a positive number x * such that g(x * ) = 0  ) and and therefore, Θ ′ (θ(u)) = −H(θ(u)) = −H(θ * v ) = 0, Θ ′ (θ(1/x)) > 0 for x > 1/u and Θ ′ (θ(1/x)) < 0 for x > 1/u 
Hence, there exists a positive number b defined in (??)  It can be seen, the main part of ARL v (T c (g)) will be an exponential function, square function, and linear function of c when the process {Z k : k ≥ 0} has no change or a "small change", a "medium change" and a "large change" from P v 0 to P v , respectively 
Here, the "small change" (v ∈ V − ) means that P v is closer to P v 0 than to P v 1 , i e , I(P v |P v 0 ) < I(P v |P v 1 ), and the "large change" is just the opposite  The "medium change" (v ∈ V 0 ) corresponds to In this paper, we will use another method to prove Theorem 3 since Wald's identity and the martingale method do not hold or can not work for showing the ARLs estimation of the test T c (g) when g is not constant 
Next we compare the detection performance of the CUSUM-OAL test (ARL v (T c ′ (g))) with that of the CUSUM test (ARL v (T C (c))) by using (??) in Theorem 4 1  ) when µ 0 < µ < 0 and for θ * v 0 > g(µ)/g(µ 0 ) when µ ≥ 0  This means that ARL v (T c (g)) can be smaller than ARL v (T C (c)) as long as g(µ)/g(µ 0 ) is small for all µ > µ 0  

NUMERICAL SIMULATION AND A REAL EX-AMPLE ILLUSTRATION

4 1 Numerical Simulation of ARLs for τ ≥ 1 By the simulation results of ARLs in Table , we see that the detection performance of T * (r), T C (c)∧T * (0), T * (0) and T C (c g u ) for large u is much better than that of the conventional CUSUM test T C for τ = 1  The following Table illustrates the simulation values of E τ i ,v and J ACE of nine tests in detecting two mean shifts v = 0 1 and v = 1 after six change-points, τ i , 1 ≤ i ≤ 6 with ARL 0 (T ) = E 0 (T ) ≈ 500 
Note that H v (θ) is a convex function and H ′ v (0) = µ < 0  This means that there is a unique positive number   It follows from (A 9) that for a large c  Taking θ ց θ * v and u ′ ց u, we have for a large c  Thus, by (A 11) we have as c → ∞  By the properties of exponential distribution, we have for a large c 
To prove the downward inequality of (A 10), let where b is defined in (??) and without loss of generality, we assume that b > a  Obviously, Let k = xcg(µ)  By Chebyshev's inequality, we have Since Hv (θ) and H v (θ) are two convex functions and Let m = tcg(µ)θ * v /bc for t > 0  By (A 13), (A 14), (A 15) and Theorem 5 1 in Esary, Proschan and Walkup (1967) we have
Finally, as c → +∞, where θ 0 > 0 satisfies h v (θ 0 ) = 1  Thus as c → ∞  This implies that for a large c  This completes the proof of (A 10)  Let v ∈ V 0   Let m 1 = (cg(0)) 2 /σ 2   It follows that Note that for a large c, where A = |g ′ (0)|/a, and , where Φ( ) is the standard normal distribution  Let m 2 = (cg(0)) 2 /(8σ 2 ln c) 
Note that as c → ∞, where the third inequality comes from Theorem 5 1 in Esary, Proschan and Walkup (1967)  Thus, we have Let v ∈ V + and let The uniform integrability of {T c (g)/c} for c ≥ 1, follows from the well-known uniform integrability of {T 0 /c} (see Gut (1988)) 
Context 3: Paper Info

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

Figure

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

abstract

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

C

FIG  1  An example of the quantum error mitigation procedure used in this work for the time evolution of the wave function of a spin chain  The ideal second-order Trotter supercircuit C of depth Mtrot = 1 (light blue) is approximated by applying a denoiser D of depth M = 1 (red) to the noisy Trotter supercircuit C (dark blue) 
Because the denoiser is applied after fully executing the noisy Trotter supercircuit, it represents an approximate inverse of the global noise channel with a precision tunable by the depth of the denoiser  can potentially blow up and lead to large errors for the circuit as a whole   Here we introduce a mitigation technique that takes into account the propagation of errors, can be performed with a tunable number of extra gates, and works for non-Clifford local noise channels since the inversion of the accumulated global noise channel is implicit 
We first execute the targeted noisy circuit completely, letting the noise propagate and accumulate, and only afterwards we apply an extra random circuit sampled from a quasiprobability distribution  We call the corresponding ensemble of random circuits a denoiser, and we construct it such that upon averaging the accumulated errors cancel 
Essentially, the denoiser inverts a global noise channel  Since we will construct it as a local brickwall circuit, following the classical preprocessing approach from , we call this compressed quantum error mitigation



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

Question: What are the three subsets into which the parameter space V is divided?

Read the following text and answer briefly.

Context 3:  Method  -Due to the inevitable coupling of a quantum processor to its environment, every qubit operation is affected by noise 
Therefore, the simplest technique to minimize the impact of the resulting noise is to minimize the number of operations when performing a quantum algorithm  In we showed that many-body time evolution operators can be efficiently compressed into brick-wall circuits with high fidelity per gate  In this Letter, we consider the noise explicitly by treating quantum operations as (generally non-unitary) quantum channels, corresponding to completely positive and trace preserving (CPTP) maps  
For example, instead of a noiseless two-qubit gate G, which acts on a quantum state |ρ in superoperator form as G|ρ = G⊗G * |ρ , we get the noisy channel G = N G, where the noise channel N implements the two-qubit noise   These channels are used to construct a "supercircuit" C = N G i=1 Gi , consisting of N G channels, which is affected by multi-qubit accumulated noise 
This supercircuit encodes an ensemble of circuits   For simplicity, we assume that the noisy channels Gi in each half brickwall layer are lattice inversion and translation invariant, such that we can construct a denoiser with these properties, limiting the number of variational parameters  The purpose of quantum error mitigation is to modify the ensemble of circuits described by C in a way that we can use it to obtain the noiseless expectation values 
In superoperator language, we do this by following the supercircuit C with a denoiser supercircuit D, such that D C is as close to the noiseless supercircuit C = C ⊗ C * as possible  Here C is the target unitary circuit  Because the noise channel N is non-unitary, hence making the supercircuit C non-unitary, we need to use a non-unitary denoiser to retrieve the unitary C 
We illustrate the mitigation procedure in Fig  , where a denoiser with one layer is used to mitigate errors for a second-order Trotter supercircuit with one layer  This circuit architecture is commonly used to simulate the time evolution of a quantum many-body system, until some time t, with controllable precision , and we will use it to benchmark the denoiser 
In practice, we cannot directly implement a supercircuit, and so we have to utilize its interpretation as an ensemble of circuits  Essentially, after executing a shot of the noisy circuit we sample the denoiser and apply it  The goal is to construct the denoiser in a way that averaging over many of its samples cancels the accumulated errors and gives us a good approximation of the noiseless expectation values 
It should be noted that our approach requires more gate applications on the quantum processor than with the gate-wise scheme, since there each sample from the mitigation quasiprobability distribution can be absorbed into the original circuit, whereas our approach increases the circuit depth  We take this into account by imposing the same noise on the denoiser 
Furthermore, within our scheme, the dimensionality of the quasiprobabilistic mitigating ensemble can be controlled, in contrast to the gate-wise approach where it is equal to the gate count  To facilitate the stochastic interpretation we parameterize each two-qubit denoiser channel G i as a sum of CPTP maps, such that we can sample the terms in this sum and execute the sampled gate on the quantum processor 
Concretely, we use a trace preserv-ing sum of a unitary and a non-unitary channel  For the unitary part we take a two-qubit unitary channel U( φ i ) = U ( φ i ) ⊗ U * ( φ i ), with U ( φ i ) a two-qubit unitary gate parameterized by φ i   For this we take the two-qubit ZZ rotation exp(−iα(σ z ⊗ σ z )) with angle α, which can be obtained from native gates on current hardware , and dress it with four general one-qubit unitaries, only two of which are independent if we want a circuit that is space inversion symmetric around every bond 
The resulting gate has 7 real parameters φ i   For the non-unitary part, which is essential because D has to cancel the non-unitary accumulated noise to obtain the noiseless unitary circuit, we use a general onequbit measurement followed by conditional preparation channel M( , with V a general one-qubit unitary and each κ i a 3-dimensional vector, resulting in a real 9-dimensional ζ i  
This yields the two-qubit correlated measurement M( With these parts we construct the parameterization with coefficients η i ∈ R that satisfy η 0 + η 1 = 1 because G i is trace preserving  Note that here the tensor product symbol corresponds to combining two one-qubit channels to make a two-qubit channel, whereas in most of the paper it is used to link the column and row indices of a density matrix 
We construct the denoiser from the noisy channels Gi = N G i   With this parameterization one denoiser channel has 17 independent real parameters, such that a denoiser of depth M , i e  consisting of M brickwall layers, has 34M real parameters (we use one unique channel per half brickwall layer)  For reference, a general channel has 544M parameters 
To determine the mitigated expectation values we use the full expression where |ρ 0 is the initial state and |1 is the vectorized identity operator on the full Hilbert space  To evaluate this on a quantum processor, we use the stochastic interpretation of (1) to resample   In particular, from each channel (1) we get a unitary with probability p 0 = |η 0 |/γ and a measurement followed by conditional preparation with probability p 1 = |η 1 |/γ 
Here γ = |η 0 | + |η 1 | is the sampling overhead, which characterizes the magnitude of the sign problem from negative η i   For quasiprobability distributions, i e  with γ > 1, every denoiser sample has an extra sign sgn(η) = N G g=1 sgn(η g ), 2  The normalized distance between the denoised Trotter supercircuit D C and the noiseless Trotter supercircuit C (top panels), at evolution times t = 0 5, 1,    , 5, and the twopoint z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t (bottom panels), for the infinite temperature initial state 
We consider denoisers with depths M = 1, 2, 4, 6, 8 and second-order Trotter circuits with depths Mtrot = 16, 32, 64  In the top panels we use a Heisenberg chain with L = 8, and in the bottom panels with L = 14, both with periodic boundary conditions  All gates are affected by two-qubit depolarizing noise with p = 0 01 
The non-denoised results are labelled with M = 0, and the noiseless values with p = 0  where sgn(η g ) is the sign of the sampled coefficient of the gth channel  γ = 1 means that all signs are positive  Observables Ô p=0 for the noiseless circuit are then approximated by resampling the observables from the denoiser ensemble
where γ = N G g=1 γ g is the overall sampling overhead, with γ g the overhead of the gth gate  Clearly, a large γ implies a large variance of Ô p=0 for a given number of samples, with accurate estimation requiring the cancellation of large signed terms  The number of samples required to resolve this cancellation of signs is bounded by Hoeffding's inequality, which states that a sufficient number of samples to estimate Ô p=0 with error δ at probability 1 − ω is bounded by (2γ 2 /δ 2 ) ln(2/ω)  
Since γ scales exponentially in γ g , it is clear that a denoiser with large M and γ 1 will require many samples  We observed that decompositions with γ > 1 are crucial for an accurate denoiser  Restricting to γ = 1 leads to large infidelity and no improvement upon increasing the number of terms in or the depth M of the denoiser 
Simply put, probabilistic error cancellation of gate noise introduces a sign problem and it is crucial to find optimal parameterizations (1) which minimize γ to make the approach scalable  This issue arises in all high performance error mitigation schemes , because the inverse of a physical noise channel is unphysical and cannot be represented as a positive sum over CPTP maps 
This is clearly visible in the spectra of the denoiser, which lies outside the unit circle (cf  Fig  )  This makes the tunability of the number of gates in each denoiser sample a crucial ingredient, which allows control over the sign problem, because we can freely choose the η i in   For the parametrization (1) of denoiser channels, we try to find a set of parameters for error mitigation by minimizing the normalized Frobenius distance between the noiseless and denoised supercircuits
which bounds the distance of output density matrices and becomes zero for perfect denoising  We carry out the minimization of on a classical processor, using gradient descent with the differential programming algorithm from   Instead of explicitly calculating the accumulated global noise channel and subsequently inverting it, we approximate the noiseless supercircuit C with the denoised supercircuit D C, effectively yielding a circuit representation D of the inverse noise channel 
Results  -To benchmark the denoiser we apply it to the second-order Trotter circuits of the spin-1/2 Heisenberg chain with periodic boundary conditions (PBC) where is the Pauli algebra acting on the local Hilbert space of site i  A second-order Trotter circuit for evolution time t with depth M trot consists of M trot − 1 half brickwall layers with time step t/M trot and two layers with half time step  
We consider circuits that are affected by uniform depolarizing noise with probability p for simplicity, but our approach can be used for any non-Clifford noise  The two-qubit noise channel is which acts on neighboring qubits i and i + 1 and is applied to each Trotter and denoiser gate, and p = 0 01 unless stated otherwise 
We study circuits with depths M trot = 16, 32, 64 for evolution times t = 0 5, 1,    , 5, and denoisers D with depths M = 1, 2, 4, 6, 8  In the top panels of Fig  we show (4) for a chain of size L = 8 as a function of time t  Here it can be seen that even for M trot = 32 a denoiser with M = 1 already improves by roughly an order of magnitude at all considered t 
Depending on M trot and t, further increasing M lowers , with the biggest improvements occurring for high precision Trotter circuits with large depth M trot = 64 and short time t = 0 5, where the Trotter gates are closer to the identity than in the other cases  At the other extreme, for M trot = 16 the improvements are relatively small upon increasing M > 2  In all cases the denoiser works better at early times than at late times, again indicating that it is easier to denoise Trotter gates that are relatively close to the identity 
To probe the accuracy of the denoiser on quantities that do not enter the optimization, as a first test we consider the two-point correlator between spins at different times where we have chosen the infinite temperature initial state, and C(t) is the Trotter supercircuit for time t  In the bottom panels of Fig  we show C zz i=L/2,j=L/2 (t) for the supercircuits from the upper panels, now for a L = 14 chain 
Here we see that at M trot = 16 we can retrieve the noiseless values already with M = 1, but that increasing M trot makes this more difficult  At M trot = 64 we see larger deviations, and improvement upon increasing M is less stable, but nonetheless we are able to mitigate errors to a large extent  As a further test, we compute the out-of-time-ordered correlator (OTOC) ]
In Fig  we show the results for i = L/2, for a Trotter circuit with depth M trot = 32 and a denoiser with depth M = 2  Here we see that a denoiser with M M trot is able to recover the light-cone of correlations, which are otherwise buried by the noise  In the Supplementary Material we consider how the denoiser performs at different noise levels p, and how the denoised supercircuits perform under stacking 
There we also calculate domain wall magnetization dynamics, and show the distribution of the optimized denoiser parameters and the sampling overhead associated to the denoiser as a whole  In Fig  we show the eigenvalues of the noisy supercircuits for a noisy second-order Trotter supercircuit with M trot = 16 at t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised supercircuit (right) 
The eigenvalues λ of a unitary supercircuit lie on the unit circle, and in the presence of dissipation they are pushed to the center  We see that the spectrum of the denoiser lies outside the unit circle, making it an unphysical channel which cures the effect of the noise on the circuit, such that the spectrum of the denoised circuit is pushed back to the unit circle



ground_truth: The three subsets are V+, V0, and V-, determined by the Kullback-Leibler information distance.

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

Question: What size chains were used in the benchmarking?

Read the following text and answer briefly.

Context 3: 
The noiseless eigenvalues are shown as blue bars, making it clear that the denoiser is able to recover the noiseless eigenvalues from the noisy circuit  In the Supplementary Material we show the spectra for a p = 0 036 denoiser, where we observe a clustering of eigenvalues reminiscent of Refs    There we also investigate the channel entropy of the various supercircuits  
Conclusion  -We have introduced a probabilistic error cancellation scheme, where a classically determined denoiser mitigates the accumulated noise of a (generally non-Clifford) local noise channel  The required number of mitigation gates, i e  the dimensionality of the corresponding quasiprobability distribution, is tunable and the parameterization of the corresponding channels provides control over the sign problem that is inherent to probabilistic error cancellation 
We have shown that a denoiser with one layer can already significantly mitigate errors for second-order Trotter circuits with up to 64 layers  This effectiveness of low-depth compressed circuits for denoising, in contrast with the noiseless time evolution operator compression from , can be understood from the non-unitarity of the denoiser channels 
In particu-lar, measurements can have non-local effects, since the measurement of a single qubit can reduce some highly entangled state (e g  a GHZ state) to a product state, whereas in unitary circuits the spreading of correlations forms a light-cone  To optimize a denoiser with convenience at L > 8, the optimization can be formulated in terms of matrix product operators or channels , which is convenient because the circuit calculations leading to the normalized distance and its gradient are easily formulated in terms of tensor contractions and singular value decompositions  
This provides one route to a practical denoiser, which is relevant because the targeted noiseless circuit and the accompanying noisy variant in (4) need to be simulated classically, confining the optimization procedure to limited system sizes with an exact treatment or limited entanglement with tensor networks 
Nonetheless, we can use e g  matrix product operators to calculate (4) for some relatively small t, such that the noiseless and denoised supercircuits in (4) have relatively small entanglement, and then stack the final denoised supercircuit on a quantum processor to generate classically intractable states 
Analogously, we can optimize the channels exactly at some classically tractable size and then execute them on a quantum processor with larger size  Both approaches are limited by the light-cone of many-body correlations, as visualized in Fig  , because finite-size effects appear when the light-cone width becomes comparable with system size 
1  The normalized distance (left) and z spin correlator C zz i=L/2,j=L/2 (right), for a second-order Trotter supercircuit of depth Mtrot = 16 for time t = 1, affected by various twoqubit depolarizing errors p  We compare the values obtained with and without a denoiser, i e  M > 0 and M = 0, to the noiseless values (p = 0) 
The denoiser is affected by the same noise as the Trotter circuit  We consider denoisers with depths M = 1, 2, 4, 6, 8, and we use a L = 8 Heisenberg chain with PBC for the normalized distance, while for the correlator we use L = 14  * david luitz@uni-bonn de to observe that even for larger noise strength p, the local observable C zz improves significantly even with denoisers of depth M = 1 
For large noise strengths, we generally see that the optimization of the denoiser becomes difficult, leading to nonmonotonic behavior as a function of p, presumably because we do not find the global optimum of the denoiser  It is interesting to analyze the spectra of the supercircuits considered in this work 
As mentioned in the main text, the spectrum of the ideal, unitary supercircuit C lies on the unit circle  The comparison to this case is therefore instructive  In the main text, we showed an example of the spectra in Fig  for moderate noise strength  Here, we show additional data for stronger noise p = 0 036 in Fig  for a denoiser with M = 4 layers, optimized to mitigate errors for a second-order Trotter supercircuit with M trot = 16 layers at time t = 1 
The eigenvalues λ of the noisy supercircuit C are clustered close to zero, far away from the unit circle (except for λ = 1), showing that the circuit is strongly affected by the noise  To mitigate the impact of the noise, the denoiser consequently has to renormalize the spectrum strongly  If it accurately represents the inverse of the global noise channel, its spectrum has to lie far outside the unit circle, which is the case 
Interestingly, we observe a clustering of eigenvalues which is reminiscent to the spectra found in   By comparison to these works, we suspect that this is due to the local nature of the denoiser, and warrants further investigation  The right panel of Fig  shows the result of the denoiser, pushing the eigenvalues back to the unit circle, nearly with the exact same distribution along the circle as the noiseless eigenvalues (blue bars) 
Due to the strong noise, this is not achieved perfectly, and it is clear that this cannot work in principle if the global noise channel has a zero eigenvalue  The complexity of an operator can be quantified by its operator entanglement entropy   Here we calculate the half-chain channel entanglement entropy S of the noiseless C, noisy C, denoiser D, and denoised D C supercircuits 
We define S as the entanglement entropy of the state that is related to a supercircuit C via the Choi-Jamio lkowski isomorphism, i e  ψ C = χ C /N , where the process matrix χ ab,cd C = C ac,bd is simply a reshaped supercircuit and N ensures normalization  Then we have S = −Tr [ψ C ln ψ C ]  This entropy measure is a particular instance of the "exchange entropy", which characterizes the information exchange between a quantum system and its environment  
In Fig  we plot the various S for a second-order Trotter circuit with M trot = 16 at t = 2, for a denoiser with M = 4, both affected by two-qubit depolarizing noise with p ∈ [10 −3 , 10 −1 ]  The Trotter circuit is for a Heisenberg model with L = 6 and PBC  We see that at large p, the noise destroys entanglement in the noisy supercircuit, and that the denoiser S increases to correct for this, such that the denoised supercircuit recovers the noiseless S 
Here we investigate how denoised supercircuits perform upon repeated application  We optimize the denoiser for a Trotter supercircuit for a fixed evolution time t  Then, to reach later times, we stack the denoised supercircuit n times to approximate the evolution up to time nt: In Fig  we stack a denoised t = 1 supercircuit up to n = 20 times and calculate the correlation function, defined in the main text, for the middle site 
We consider Trotter depths M trot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8, for a L = 14 Heisenberg chain with p = 0 01 depolarizing two-qubit noise  The noisy results correspond to M = 0 and the noiseless results to p = 0  In Fig  we calculate the OTOC, defined in the main text, with stacked time evolution for a denoised t = 2 supercircuit with M trot = 32 and M = 2, stacked up to ten times 
We see that the stacked supercircuit performs very well, and the additional precision obtained by using deep denoisers (M = 8) pays off for long evolution times, where we see convergence to the exact result (black dashed lines in Fig  ) as a function of M   FIG    The two-point z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t, for the infinite temperature initial state, for denoised second-order Trotter supercircuits that are optimized at evolution time t = 1 and then stacked up to twenty times 
We use Trotter depths Mtrot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8  The calculations were performed for a periodic Heisenberg model with L = 14 and PBC, affected by two-qubit depolarizing noise with strength p = 0 01, which also affects the denoiser  The non-denoised results are labelled with M = 0, and the noiseless results with p = 0 
The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively  The costliest and most noise-susceptible operation is the two-qubit ZZ rotation with angle α, which is the foundation of the unitary piece in our channel parameterization, defined in the main text 
For completeness, we here present the α angles of the optimized denoisers  The results are shown in Fig  , which contains histograms for the channel count N G versus α  The histograms are stacked, with the lightest color corresponding to the angles of the denoiser at t = 0 5 and the darkest at t = 5  The top four panels are for a denoiser with M = 2 and the bottom four with M = 8 
We consider M trot = 8, 16, 32, 64  We see that in both cases the distribution widens upon increasing M trot , indicating that the unitary channels start deviating more from the identity  Moreover, while the M = 2 denoisers in all cases except M trot = 64 have ZZ contributions close to the identity, this is clearly not the case for M = 8 
For simplicity, we did not focus on obtaining denoisers with the smallest sampling overhead γ, which is required to minimize the sign problem and hence ease the sampling of mitigated quantities  Instead, we let the optimization freely choose the η i in the denoiser parameterization, as defined in the main text 
In Fig  we show the sampling overhead of the denoisers from Fig  of the main text  We see that for M = 1 and M = 2 the sampling overhead is relatively small and uniform across the different t, whereas for M > 2 the optimization sometimes yields a denoiser with large γ and other times with small γ  This could be related to the difference in α distributions from Fig   
The large fluctuations of γ appears to stem from the difficulty in finding optimal deep denoisers, and our optimization procedure likely only finds a local minimum in these cases  Here C(t) is the Trotter supercircuit for time t  In Fig  we show Z dw for the circuits from Fig 



ground_truth: L = 8 and L = 14.