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 hedge fund's collapse in 1998 highlighted the need for regulation of derivatives?

Read the following text and answer briefly.

Context 1: Brooksley Elizabeth Born (born August 27, 1940) is an American attorney and former public official who, from August 26, 1996, to June 1, 1999, was chair of the Commodity Futures Trading Commission (CFTC), the federal agency which oversees the U S  futures and commodity options markets  During her tenure on the CFTC, Born lobbied Congress and the President to give the CFTC oversight of off-exchange markets for derivatives, in addition to its role with respect to exchange-traded derivatives, but her warnings were ignored or dismissed, and her calls for reform resisted by other regulators <ref name="nytimes">Goodman, Peter S  The Reckoning - Taking Hard New Look at a Greenspan Legacy, The New York Times, October 9, 2008 </ref> Born resigned as chairperson on June 1, 1999, shortly after Congress passed legislation prohibiting her agency from regulating derivatives 

In 2009, Born received the John F  Kennedy Profiles in Courage Award, along with Sheila Bair of the Federal Deposit Insurance Corporation, in recognition of the "political courage she demonstrated in sounding early warnings about conditions that contributed" to the 2007-08 financial crisis 

Early life and education
Born graduated from Abraham Lincoln High School (San Francisco, California) at the age of 16  She then attended Stanford University, where she majored in English and was graduated with the class of 1961   She initially wanted to become a doctor, but a guidance counsellor at Stanford advised her against medicine, so she majored in English literature instead 

She then attended Stanford Law School, one of only seven women in her class   She was the first female student ever to be named president of the Stanford Law Review  She received the "Outstanding Senior" award and graduated as valedictorian of the class of 1964 

Legal career
Immediately after law school Born was selected as a law clerk to judge Henry Edgerton of the U S  Court of Appeals for the District of Columbia Circuit  It was during this time that she met her first husband, Jacob C  Landau, who was a journalist covering the Federal courts at the time  Following her clerkship, she became an associate at the Washington, D C -based international law firm of Arnold & Porter  Born was attracted to Arnold & Porter because it was one of the few major law firms to have a woman partner at that time, Carolyn Agger, who was the head of the tax practice  Born took a two-year leave of absence from Arnold & Porter to accompany her first husband to Boston, where he had received a fellowship  During that time she worked as a research assistant to law professor Alan Dershowitz 

Born's early career at Arnold & Porter focused on international trade law, in which she represented a number of Swiss industries and the government of Switzerland  She developed a practice representing clients in numerous complex litigation and arbitration cases involving financial market transactions  Among her high-profile cases was the matter of the Hunt Brothers attempt to corner the market in silver in the 1970s  She made partner at Arnold & Porter, after moving to a three-day schedule to help raise her second child, and eventually rose to be the head of the firm's derivatives practice 

Born was among the first female attorneys to systematically address inequities regarding how the laws treated women  Born and another female lawyer, Marna Tucker, taught what is considered to have been the first "Women and the Law" course at Catholic University’s Columbus School of Law  The class exclusively concerned prejudicial treatment of women under the laws of the United States, past and present   Born and Tucker were surprised to discover that there was no textbook on the issue at the time   Born is also one of the co-founders of the National Women's Law Center  Born also helped rewrite the American Bar Association rules to make it possible for more women and minorities to sit on federal bench 

During her long legal career, and into her retirement, Born did much pro bono and other types of volunteer work  She was active in the American Bar Association, the largest professional organization of lawyers in the United States   Initially Born was named a member of the governing council of the ABA's Individual Rights Section, eventually becoming chairperson   Born and Tucker founded the ABA Women's Caucus, the first organization of female lawyers in the ABA   She held several other senior positions in the ABA, including being named the first woman member of the ABA's Standing Committee on the Federal Judiciary   As a member of the Judiciary Committee, Born provided testimony and opinion on persons nominated for federal judgeships  In 1980 she was named chair of the committee   As chair of the committee, Born was invited to address the U S  Congress regarding the nomination of Judge Sandra Day O'Connor to the U S  Supreme Court 

In 1993, Born's name was floated as a possible candidate for Attorney General of the United States, but Janet Reno was nominated 

In July 2009, Nancy Pelosi appointed Brooksley Born as a commissioner to the Financial Crisis Inquiry Commission (FCIC) 

Born and the OTC derivatives market
Born was appointed to the CFTC on April 15, 1994, by President Bill Clinton  Due to litigation against Bankers Trust Company by Procter and Gamble and other corporate clients, Born and her team at the CFTC sought comments on the regulation of over-the-counter derivatives, a first step in the process of writing CFTC regulations to supplement the existing regulations of the Federal Reserve System,  the Options Clearing Corporation, and the National Association of Insurance Commissioners  Born was particularly concerned about swaps, financial instruments that are traded over the counter between banks, insurance companies or other funds or companies, and thus have no transparency except to the two counterparties and the counterparties' regulators, if any   CFTC regulation was strenuously opposed by Federal Reserve chairman Alan Greenspan, and by Treasury Secretaries Robert Rubin and Lawrence Summers  On May 7, 1998, former SEC Chairman Arthur Levitt joined Rubin and Greenspan in objecting to the issuance of the CFTC's concept release  Their response dismissed Born's analysis and focused on the hypothetical possibility that CFTC regulation of swaps and other OTC derivative instruments could create a "legal uncertainty" regarding such financial instruments,  hypothetically reducing the value of the instruments  They argued that the imposition of regulatory costs would "stifle financial innovation" and encourage financial capital to transfer its  transactions offshore  The disagreement between Born and the Executive Office's top economic policy advisors has been described not only as a classic Washington turf war, but also a war of ideologies,  insofar as it is possible to argue that Born's actions were consistent with Keynesian and neoclassical economics while Greenspan, Rubin, Levitt, and Summers consistently espoused neoliberal, and neoconservative policies 

In 1998, a trillion-dollar hedge fund called Long Term Capital Management (LTCM) was near collapse   Using mathematical models to calculate debt risk, LTCM used derivatives to leverage $5 billion into more than $1 trillion, doing business with fifteen of Wall Street's largest financial institutions   The derivative transactions were not regulated, nor were investors able to evaluate LTCM's exposures   Born stated, "I thought that LTCM was exactly what I had been worried about"   In the last weekend of September 1998, the President's working group was told that the entire American economy hung in the balance   After intervention by the Federal Reserve, the crisis was averted   In congressional hearings into the crisis, Greenspan acknowledged that language had been introduced into an agriculture bill that would prevent CFTC from regulating the derivatives which were at the center of the crisis that threatened the US economy   U S  Representative Maurice Hinchey (D-NY) asked "How many more failures do you think we'd have to have before some regulation in this area might be appropriate?"  In response, Greenspan brushed aside the substance of Born's warnings with the simple assertion that "the degree of supervision of regulation of the over-the-counter derivatives market is quite adequate to maintain a degree of stability in the system"  Born's warning was that there wasn't any regulation of them   Born's chief of staff, Michael Greenberger summed up Greenspan's position this way: "Greenspan didn't believe that fraud was something that needed to be enforced, and he assumed she probably did  And of course, she did "  Under heavy pressure from the financial lobby, legislation prohibiting regulation of derivatives by Born's agency was passed by the Congress   Born resigned on June 1, 1999 

The derivatives market continued to grow yearly throughout both terms of George W  Bush's administration  On September 15, 2008, the bankruptcy of Lehman Brothers forced a broad recognition of a financial crisis in both the US and world capital markets   As Lehman Brothers' failure temporarily reduced financial capital's confidence, a number of newspaper articles and television programs suggested that the failure's possible causes included the conflict between the CFTC and the other regulators Faiola, Anthony, Nakashima, Ellen and Drew, Jill  The Crash: Risk and Regulation - What Went Wrong, The Washington Post, October 15, 2008 

Born declined to publicly comment on the unfolding 2008 crisis until March 2009, when she said: "The market grew so enormously, with so little oversight and regulation, that it made the financial crisis much deeper and more pervasive than it otherwise would have been " She also lamented the influence of Wall Street lobbyists on the process and the refusal of regulators to discuss even modest reforms 

An October 2009 Frontline documentary titled "The Warning"  described Born's thwarted efforts to regulate and bring transparency to the derivatives market, and the continuing opposition thereto  The program concluded with an excerpted interview with Born sounding another warning: "I think we will have continuing danger from these markets and that we will have repeats of the financial crisis -- may differ in details but there will be significant financial downturns and disasters attributed to this regulatory gap, over and over, until we learn from experience "

In 2009 Born, along with Sheila Bair of the FDIC, was awarded the John F  Kennedy Profiles in Courage Award in recognition of the "political courage she demonstrated in sounding early warnings about conditions that contributed" to the 2007-08 financial crisis   According to Caroline Kennedy, "Brooksley Born recognized that the financial security of all Americans was being put at risk by the greed, negligence and opposition of  powerful and well connected interests     The catastrophic financial events of recent months have  proved them [Born and Sheila Bair] right "  One member of the President's working group had a change of heart about Brooksley Born   SEC Chairman Arthur Levitt stated "I've come to know her as one of the most capable, dedicated, intelligent and committed public servants that I have ever come to know", adding that "I could have done much better  I could have made a difference" in response to her warnings 

In 2010, a documentary film Inside Job further alleged that derivatives regulation was ineffective from the Clinton administration on  Along with fellow whistleblower, former IMF Chief Economist Raghuram Rajan, who was also scorned by the economic establishment, Brooksley Born was cited as one of the authorities arguing that financial derivatives increase economic risk 

 Personal life 
Born is married to Alexander E  Bennett (also retired from Arnold & Porter)   She has five adult children - two from a previous marriage to Jacob Landau and three stepchildren  Notably, Born was named a partner at Arnold & Porter while working part-time so she could raise her two young children   When both of her children were school-age, Born returned to practice full-time 

References

External links
Attorney profile at Arnold & Porter
Brooksley Born (2009 Winner) of the Profiles in Courage Award, with acceptance speech transcript and NECN video

Profile at MarketsWiki
Speeches and statements
"Testimony Of Brooksley Born Chairperson of the CFTC Concerning The Over-The-Counter Derivatives Market", before the House Committee On Banking And Financial Services, July 24, 1998 
"The Lessons of Long Term Capital Management L P ", Remarks of Brooksley Born, Chairperson of the CFTC, Chicago-Kent-IIT Commodities Law Institute, Chicago, Illinois, October 15, 1998 
 Interview: Brooksley Born for "PBS Frontline: The Warning", PBS, (streaming VIDEO 1 hour), October 20, 2009 
Articles
Manuel Roig-Franzia  "Credit Crisis Cassandra:Brooksley Born's Unheeded Warning Is a Rueful Echo 10 Years On", The Washington Post, May 26, 2009
 Taibbi, Matt  "The Great American Bubble Machine", Rolling Stone'', July 9–23, 2009

1940 births
American women lawyers
Arnold & Porter people
Clinton administration personnel
Columbus School of Law faculty
Commodity Futures Trading Commission personnel
Heads of United States federal agencies
Lawyers from San Francisco
Living people
Stanford Law School alumni
21st-century American women

Context 3: How Oxycontin, Florida and the Sackler Family Created the Opioid Crisis In America
Why are the Sacklers worth $13 billion today? Answer: “The Oxy Express Explained”
(MASS TORT NEXUS MEDIA)
A COMPARISON OF OXYCODONE PRESCRIBING
In the first six months of 2010, Ohio doctors and health care practitioners bought the second-largest number of oxycodone doses in the country at just under 1 million pills 
Florida doctors bought 40 8 million in the same period, the comparison is astounding, yet it flew under the DEA, Opioid Big Pharma and everyone elses radar for years and years 
Of the country’s top 50 oxycodone-dispensing clinics, 49 were in Florida  From August 2008 to November 2009, a new pain clinic opened in Broward and Palm Beach counties on average of every three days 
Pharmacies and distributors are at fault as well, pharmacies ordered jaw-dropping numbers of pills from opioid drug distributors, the middlemen between manufacturers and pharmacies 
90 of 100 of the nation’s top 100 oxy-buying doctors in 2010, were in Florida  49 of 50 of the country’s top oxy-dispensing clinics were in Florida  For some reason this didn’t raise an alarm or cause anyone to look further at the time 
Purdue Pharma New What Was Happening In Florida
Purdue and the Sacklers chose to ignore Florida, because apparently nobody there sued them or complained  In 2007, in other states, the infamous drug maker and three of its executives pled guilty in federal court and paid out $634 5 million in fines for purposefully misleading regulators, doctors, and patients about the addictiveness of their opioid painkiller  Around the same time, Purdue was also sued by several states, including Washington, over similar allegations  Purdue agreed to a $19 5 million multi-state settlement  And in 2015, Purdue settled a case with Kentucky, agreeing to pay $24 million 
As part of the state settlements, Purdue was supposed to set up monitoring programs to make sure that its opioid drug didn’t wind up in the wrong hands  It was supposed to watch out for shady pharmacies, unusually large orders, or suspiciously frequent orders  But on this front, Everett alleges that Purdue once again put profits over people 
Obviously, this was ignored as the Florida based “Oxy Expres”; rolled on for years and years with np input, comment or oversight by Purdue Pharma and the Sackler family other than “show me the money” and enjoying a life of luxury on the misery created and managed in the Purdue Pharma boardroom  But, the Purdue boardroom isn’t the only guilty “Opioid Big Pharma” industry player who designed and supported the opioid prescribing crisis 
For the current status of efforts to make Opioid Big Pharma accept responsibility in litigation filed in federal and state courts across the country, see: https://www masstortnexus com/Briefcases/254/OPIOID-CRISIS-BRIEFCASE-INCLUDING-MDL-2804-OPIATE-PRESCRIPTION-LITIGATION
Why Distributors Are Liable
Cardinal Health, one of the nation’s biggest distributors, sold two CVS pharmacies in Sanford a combined 3 million doses of oxycodone, flooding the town of 54,000 with an average of 250,000 oxycodone pills every month 
West of Jupiter, a Walgreens drug distribution center sold 2 2 million tablets to a single Walgreens’ pharmacy in tiny Hudson, a roughly six-month supply for each of its 12,000 residents  It shipped more than 1 1 million pills to each of two Fort Pierce Walgreens pharmacies 
For 40 days starting in late 2010, the distribution center shipped 3,271 bottles of oxycodone — 327,100 doses of the drug — to a Port Richey Walgreens pharmacy, prompting a distribution manager to ask: “How can they even house this many bottles?”
There were 53 million oxycodone prescriptions filled in 2013 by US pharmacies, according to NIDA  This translates to approximately one bottle of this addictive drug for every 6 people in the country  How was this not noticed by those responsible for monitoring narcotics prescribing in the United States?
Charts and Data On Florida’s Oxycontin Gold Mine
https://www documentcloud org/documents/3936665-Purdue-Pharma-1-in-48-Study html
https://www documentcloud org/documents/3534759-uS-Atty-on-Purdue-Settle html#document/p2/a384323
A Boardroom Contrived Opioid Epidemic
This is the pain chart created by the “Opioid Big Pharma Industry” to support massive over-prescribing of opioids across the country to everyone who walked in to a medical treatment facility, this was an effort to increase narcotic prescribing practices in mainstream medical care–and it worked very very well! This chart became a standard treatment assessment protocol tool across the country 
https://www documentcloud org/documents/3936646-DEA-NATL-DRUG-ASSESSMENT-2010 html#document/p51/a383739
HOW WEST VIRGINIA WAS TARGETED
It-Was-Raining-Opiates-How-drug-companies-submerged-West-Virginia-in-opioids-for-years
Reliably red on the political map, Huntington is a West Virginia town with a 182-year-old university, a storied football team and more than 100 churches 
It’s where Will Lockwood graduated from high school  It’s where he enrolled at Marshall University  It’s where he first tried OxyContin  By the time Lockwood entered Marshall, Detroit dealers were trickling into Huntington, selling OxyContin and pills with OxyContin’s active ingredient, oxycodone 
Even though Lockwood could step out his front door and get the drug, Detroit street dealers weren’t the preferred supplier, they were in Florida 
It may have been 1,000 miles away, but to Lockwood, getting OxyContin and oxycodone from Florida’s loosely regulated pain clinics “was legal, in a sense ”
Twice a month, different “crews” from Huntington crowded into vans and headed south to Palm Beach and Broward counties, home to more than 200 pill mills, the pain clinics where anyone with a fake ache and hard cash could walk out with pills and prescriptions 
After hitting a string of clinics, the Huntington crews drove back with “around 500 to 600 pills per person,” said Lockwood 
But it wasn’t just a few hundred pills  It was tens of thousands 
And it wasn’t just Huntington, The West Virginia vans were part of a nationwide caravan heading to South Florida  Cars bearing tags from Kentucky, Tennessee, the Carolinas, Virginia and Ohio crowded into one clinic parking lot after another, loading up on pills and prescriptions 
News stories and law enforcement focused on those “parking lot” states in Appalachia, where dealers and addicts with a tank of gas or a cheap plane ticket traveled the “Oxy Express” to Palm Beach and Broward 
But Florida’s pill pipeline reached far beyond those roadways 
By 2010, Florida was the oxycodone drug dealer of choice for drug users and dealers in the Great Lakes, Northeast and Mid-Atlantic regions as well as the Southeast, DEA records show, an area spanning virtually every state east of the Mississippi  It wasn’t just that Florida guaranteed a flow of cheap oxycodone  For 10 years, key lawmakers and agency heads repeatedly looked the other way as crooked doctors and bogus clinics flooded almost half the nation with the highly addictive drug 
In failing to crack down, Florida extended by years the amount of time highly addictive oxycodone would be available to both first-time experimenters and addicts  It gave criminals the raw materials for trafficking  It gave Will Lockwood the OxyContin needed to feed his growing habit, It paved the way for his eventual jump to heroin 
Jumping state lines
Teenage high-school wrestling buddies in New Port Richey ran oxycodone into Tennessee; they were paid with cash hidden in teddy bears  A Hillsborough County man mailed 17,000 pills to Glen Fork, W Va , a month’s supply for every man woman and child in the tiny town 
A Boston Chinatown crime boss trafficked pills from Sunrise into Massachusetts, New York, Rhode Island and South Carolina  Wellington twins and pill mill kingpins Paul and Phil George, brothers who oversaw one of the largest operations in the country from their five Palm Beach and Broward clinics, pushing oxycodone into Kentucky, Tennessee, Ohio and South Carolina 
A husband and wife team operating out of a Forest Hill Boulevard clinic funneled pills to Delaware  At Palm Beach International Airport, two federal security agents accepted $500 a pop each time they waved through thousands of pillsbound for Connecticut and New York 
A Palm Bay man’s Puerto Rican family bought local pills destined for the working class town of Holyoke, Mass  In Rhode Island, police pulled over a Lauderhill man caught speeding through Providence  They found 903 oxycodone tablets and 56 morphine pills in the car 
Senior citizen and Tulane business graduate Joel Shumrak funneled more than 1 million pills into eastern Kentucky from his South Florida and Georgia clinics, much of it headed for street sales — an estimated 20 percent of the illicit oxycodone in the entire state 
Van loads of pill-seekers organized by “VIP buyers” traveled from Columbus, Ohio, to three Jacksonville clinics, where armed guards handled crowd control (federal indictment) and doctors generated prescriptions totaling 3 2 million pills in six months  In Miami, Vinny Colangelo created 1,500 internet website names to entice drug users throughout the nation to one of his six South Florida pain clinics or pharmacies 
Even the Mafia got in on the Florida oxy express action: A Bonanno crime family associate oversaw a local crew stocking up on Palm Beach and Broward pain clinic oxycodone, upstreaming profits to the New York family 
At times, it seemed almost no section of the country was free of Florida-supplied pills: When Olubenga Badamosi was arrested driving his Bentley Continental in Miami in 2011, the Oregon man was one of two traffickers overseeing a crew smuggling South Florida oxycodone to sell in Salt Lake City, Seattle and Denver as well as Oregon, Nevada, Texas and even Alaska 
Pharmacy delivers oxy ‘pot of gold’
It would be hard to overstate Florida’s role in feeding the country’s voracious appetite for oxycodone  Oxycodone 30-milligram tablets were favored by addicts  And in 2009 and 2010, roughly four of every 10 of those pills were sold in Florida  Small wonder: Of the nation’s top 100 oxycodone-buying doctors, 90 were in Florida 
Pharmacies, too, ordered jaw-dropping numbers of pills from drug distributors, the middlemen between manufacturers and pharmacies 
West of Jupiter, a Walgreens drug distribution center sold 2 2 million tablets to a single Walgreens’ pharmacy in tiny Hudson, a roughly six-month supply for each of its 12,000 residents  It shipped more than 1 1 million pills to each of two Fort Pierce Walgreens pharmacies  By contrast, a single Walgreens pharmacy in the Central Florida townOviedo bought 169,700 doses of oxycodone in 30 days 
People on both sides of the counter knew what was going on: In a letter to the chief executive of Walgreens, Oviedo’s police chief warned that people were walking out of the town’s two Walgreens stores and selling their drugs on the spot, crushing and snorting them, or — still in the pharmacy’s parking lot — injecting them 
Why Pharmacies are LIABLE
In Fort Pierce, a Walgreens pharmacist accidentally provided an extra 120 oxycodone pills to a customer  When the druggist called to ask that the man return the pills, the customer’s girlfriend bluntly responded that he was an addict, that he sold oxycodone and the 120 pills were “a pot of gold,” DEA records show 
That was in September  The same man came back to the same Walgreens in December and January with a prescription in hand, and the pharmacy filled his prescriptions every time 
‘Wild West of Oxycodone Prescribing’
Cincinnati-based Masters Pharmaceuticals Inc  was a middling-sized drug distributor selling oxycodone to Florida pharmacies 
It sold to other customers in other states  But mostly, it sold to Florida: Oxycodone made up more than 60 percent of its drug sales in 2009 and 2010, according to federal records  Of its top 55 oxycodone customers, 44 were in Florida 
Company CEO Dennis Smith worried that the Florida-bound oxycodone was getting in the wrong hands  A trip to Broward did nothing to ease his mind  “It was,” he later testified, “the Wild West of oxycodone prescribing ”
Bus and park benches touted pain clinics  When Smith picked up and thumbed through City Beat, a free magazine, he found pages of ads for pain clinics  “It would show young people sitting around a pool and it named the pain clinic and say (sic) ‘we dispense on site,’ and that really hit home hard ”
Smith stopped selling to pain clinics  But the company continued to shovel millions of oxycodone pills to Florida pharmacies  Masters executives figured the pharmacies would keep an eye out for excessive prescriptions written by pill mill doctors  But not all pharmacies were worrying about doctors at pain clinics, many pharmacies were courting the pill mills prescribers 
A Lake Worth Family Pharmacy
In 2009, the small pharmacy off Lucerne Avenue in Lake Worth had a history  It had been in business for 43 years  The owner and head pharmacist had been there for 32  It had shaded parking and a downtown location, a stone’s throw from the City Hall Annex 
When a Masters inspector visited, he was alarmed to find Tru-Valu Drugs bustling with a long line of young, thin, tattooed customers arriving in groups of 10 to pick up pills  There were signs in the pharmacy warning of limits on the number of oxycodone pills handed out  Even Mallinckrodt Pharmaceuticals, an oxycodone manufacturer, was worried about the volume of its pill sales there 
Of the 300,000 doses of all drugs the small pharmacy dispensed in December 2008, 192,000 were for oxycodone 30 mg, the dosage preferred by traffickers and users alike 
The huge oxycodone volume was no accident  The owner and head pharmacist, unidentified in DEA records, told a Masters inspector that the pharmacy “has pushed for this (narcotic) business with many of the area pain doctors ”
And, despite the torrent of oxycodone going out the door, the pharmacy owner expressed frustration that drug distributors were limiting the amount of narcotics they would sell to his now-closed pharmacy 
Ohio to Florida and Back
Pharmacy after pharmacy benefited from the combination of Masters’ Ohio oxycodone business and Florida’s unregulated pill mills 
In Englewood, north of Fort Myers, the pharmacy owner filled prescriptions for six pain clinics — including clinics an hour’s drive away  A Masters inspector found cars from Tennessee and Kentucky in the parking lot and young men leaving the pharmacy carrying large trash bags 
Superior Pharmacy not only filled oxycodone prescriptions for pain clinics, it shared waiting room space with a pain clinic in a Temple Terrace strip mall outside Tampa  Neither Masters nor Superior had so much as Googled the background of pain clinic doctors writing those prescriptions, the DEA later said 
Had they done so, the DEA dryly noted, they “would likely have come across a press release” announcing one of the doctors had been arrested and charged with trafficking in prescription drugs 
Hundreds of thousands of oxycodone pills were sent from Ohio distributors to Florida pharmacies  Unknown thousands of pills headed right back up to Ohio 
When Ohio police burst into Christopher Thompson’s home outside Columbus, they found an assault rifle, $80,000 in cash and oxycodone from his Florida deals  A construction worker whose own pill habit started at age 14, Thompson oversaw a ring of 15 Ohio buyers who traveled to Florida to pick up oxycodone to resell in Central Ohio 
Two hours to the west in Martin’s Ferry, David L  Kidd orchestrated a ring of buyers traveling to West Palm Beach and Central Florida to pick up oxycodone for resale on the streets of eastern Ohio and West Virginia 
Doctors and pharmacies from Florida were complicit with Kidd’s ring in fueling Ohio’s opioid epidemic, wrote the U S  attorney for West Virginia after Kidd’s 2011 arrest: “The steady flow of pain pills into the Ohio Valley from Florida must stop ”
Driving To Pick Up Death By Rx
With more drugs came more deaths, in January 2010, say police, Fort Lauderdale pathologist Dr  Lynn Averill started a seven-month oxycodone shopping spree, buying 437,880 oxycodone pills from drug distributors 
The same month, Matthew Koutouzis drove from Toms River, N J , to see Averill in her Broward County pain clinic  The 26-year-old collected prescriptions for 390 pills and overdosed two days later  Brian Moore traveled 13 hours from his Laurel County, Ky , home to see Averill  He left with prescriptions for 600 pills and also overdosed within 48 hours 
Kenneth Hammond didn’t make it back to his Knoxville, Tenn , home  He had a seizure after picking up prescriptions for 540 pills and died in an Ocala gas station parking lot 
Keith Konkol didn’t make it back to Tennessee, either  His body was dumped on the side of a remote South Carolina road after he overdosed in the back seat of a car the same day of his clinic visit  He had collected eight prescriptions totaling 720 doses of oxycodone, methadone, Soma and Xanax  Konkol had every reason to believe he would get those prescriptions: In three previous visits to the Plantation clinic, he had picked up prescriptions for 1,890 pills 
An estimated 60 percent of her patients were from out of state, a former medical assistant told the DEA  In 2015, Averill pleaded not guilty to eight manslaughter charges  She is awaiting trial in Broward County  Averill was just one doctor at just one clinic  In 2010, the year Averill’s patients overdosed, Florida received applications to open 1,026 more pain clinics 
An online message board advising drug users summed it up: “Just go anywhere in South Florida and look for a ‘pain management clinic ’ It shouldn’t be too hard; you can’t swing a dead cat without hitting one ” Complain about anything from a back injury to a hangnail, it advised, “and they’ll set you right up ”
By this time, Kentucky had reined in its pill mills  It didn’t matter, Ohio, Delaware, North Carolina, Connecticut acted as well, but other state’s efforts didn’t matter either, Florida continued ignoring the pill mills and rogue doctors feeding the nation’s oxycodone habit, the pills flowed 
“There were folks down there, where if I had an opportunity to, get my hands around their throat, I would have wrung their neck,” said Huntington Mayor Steve Williams  On Florida’s inaction he stated, “There was total evidence as to what was happening  It lays at the foot, in my opinion, of the public officials there that allowed it to continue on ”
Governor Jeb Bush Backed A Solution
One of the first dinners Florida Gov  Jeb Bush hosted after moving into the governor’s mansion in 1999 was a small one  Among those sitting at the table with Bush were Lt  Gov  Toni Jennings, state Sen  Locke Burt and James McDonough, who would become the state’s hard-nosed drug czar  There was an urgent topic on the agenda that night: the explosion of prescription painkillers  For the state’s first family, it may have been personal  Bush had talked publicly about one of his children’s struggle with addiction 
By the time the meal ended, all had agreed on the need for establishing a prescription drug monitoring program that would collect information and track prescriptions written for controlled substances, such as oxycodone 
Absent a prescription drug monitoring database, there was no way to know whether someone was “doctor shopping,” going from doctor to doctor, getting more and more prescriptions to feed their habit 
And there was no way to know whether a doctor was overprescribing, key to pinpointing whether a pill mill was operating, and where  Similar databases had been adopted by more than a dozen states  It was being described as a “silver bullet” to curb overprescribing  Soon enough, $2 million to get the database up and running would be on the table — but it came with a catch 
Florida Attorney General Misfires Against Purdue
In 2001, OxyContin-maker Purdue Pharma was fending off early criticism of its blockbuster painkiller  At issue was whether Purdue’s aggressive marketing campaign had misled doctors and patients alike  Purdue and three top executives later pleaded guilty to federal charges of illegally marketing the drug



ground_truth: Long Term Capital Management (LTCM).

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

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

Read the following text and answer briefly.

Context 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 
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:  Far from being safe and non-addictive, OxyContin carried the same addiction risk as morphine, and was every bit as potent 
But that was six years away  In 2001, towns in Maine reported an alarming uptick in crime tied to OxyContin  The first of several congressional hearings was ramping up  Critics and parents who lost children were piling on  Reporters were starting to write stories 
In November, Florida Attorney General Bob Butterworth appeared poised to take on the company  Calling OxyContin street sales “a major threat to public health,” Butterworth told a state Board of Medicine committee that Purdue should consider temporarily taking the drug off the market  It wasn’t only traffickers concerning Butterworth  It was the sales pitch 
In late 2001, Butterworth called a young assistant attorney general into his office and gave him a magazine article on OxyContin and an assignment: Look into Purdue marketing  Former Florida Attorney General Bob Butterworth and Palm Beach County State Attorney Dave Aronberg  The young lawyer, now-Palm Beach County State Attorney Dave Aronberg, said he knew nothing about OxyContin  But he didn’t like what he read 
During the yearlong inquiry, 589 Floridians died after taking oxycodone  Nothing criminal was found, Aronberg later said  Instead, Butterworth and Purdue struck a settlement  As part of a $2 million deal, Purdue would pay to establish a prescription monitoring database, the same silver bullet sought by Bush  After Florida’s computerized system was up and running, the same system would be free to any other state  The entire country, not just Florida, would benefit 
It could have been a groundbreaking deal  There was one catch  State lawmakers had to vote to create the prescription monitoring program by 2004, or Purdue would keep its money 
Marco Rubio Kills The Anti-Oxy Rx Bill
A political gight killed the program  “And there was one person who was responsible,” said former state Sen  Burt, now an Ormond Beach insurance executive  “And it was Marco Rubio ”
A rising state lawmaker in 2002, now-U S  Sen  Marco Rubio had the clout to make or break the legislation  He had been one of two state House majority whips and was on the fast track to becoming House speaker 
Rubio didn’t kill the 2002 bill out of opposition to prescription monitoring—it was politics “as usual” yet nobody blamed Rubio for the resulting opioid crisis that seems to have started in his political backyard and flourished beyond belief  
U S  Sen  Marco Rubio, R-Fla , was a leader in the Florida House in 2002 when he blocked a vote on prescription monitoring  That year, Rubio favored a bill changing the Miami-Dade County charter, which failed to pass because of a single “no” vote in the Senate  Burt cast the vote 
Angered by what he saw as Burt’s betrayal, Rubio killed the prescription drug monitoring bill  “When I found out he broke his word, it made the choice easy,” Rubio told The Miami Herald 
It’s not certain that the full Legislature would have passed the bill had it made it to a floor vote  Rubio was the first, not the last, in a line of state legislative leaders over years who would refuse to seriously consider the bill  Most cited privacy concerns 
But prescription monitoring databases in Florida and other states free to use Florida’s model would have pinpointed rogue doctors, would-be pill mills and doctor-shoppers across the country, just as all three were beginning to converge  In doing so, it could have curbed a national opioid epidemic when it was just an emerging problem, not the monster it would become 
Only weeks after the 2002 bill was killed, Bush suppressed a sob as he discussed his daughter’s arrest for forging a prescription  Court-ordered to drug treatment and then briefly to jail, Noelle Bush survived her pill addiction  The 2004 deadline for greenlighting a monitoring system passed  So did Purdue’s million-dollar obligation to pay for it 
Between 2002, the year Rubio killed the database that could have identified doctor-shoppers, and late 2011, when the database finally came online, more than 20,800 Floridians died after taking prescription opioids, including OxyContin, annual Florida Medical Examiners’ reports show  “Not getting that bill through the Legislature resulted in Florida becoming the pill mill capital of the United States,” said Burt 
“There was heartache for thousands of families beyond measure and it didn’t have to happen ”
Florida Officials Were Told Of The Oxy Express
The East Kentucky hills and valleys of Greenup County suit Keith Cooper, a long-haired undercover cop-turned-sheriff: “It’s a backwater  I tell people all the time I am a hick sheriff from a hick location, and by 2011, the rural county and its sheriff had big city problems 
Greenup is near the stretch of interstate highways that provided drug traffickers and users with a straight shot to Palm Beach and Broward pill mills  It’s less than an hour’s ride to Huntington Tri-State Airport, where a $27 flight to Fort Lauderdale was a popular draw for dealers hoping to stock up 
Arrests for Florida pills soon eclipsed local arrests for pot 
“When we locked ’em up, we take all their pill bottles and all their paperwork, and we found maps to the doctors offices and everything,” recalled Cooper 
“I called the (Florida) medical board and gave them a big list of doctors,” Cooper said  He called the state pharmacy board, too  He got no response 
“So then I called the Attorney General’s Office and the Governor’s Office  I was calling them all, the whole state  Of course, I was talking to the state police the entire time  “I told them, all of the profits were down there  And all of the pain’s up here ” Nothing happened  Florida’s oxycodone pipeline continued to flow 
On the other side of the law in Greenup, Mikey Frazier was banking on it 
The Oxy Express
Frazier was on a scholarship to play baseball at his junior college in Chicago when he suffered a torn rotator cuff  Doctors prescribed Percocet, a pill containing oxycodone, in 2002  When doctors cut him off, he bought it on the street  In 2006, he moved to OxyContin, nearly pure oxycodone  In 2007, he gave his friends money to go to Florida and bring him back pills 
“My buddy had a minivan and he would actually go down one week and take two to three people with him, and then the following week I’d go,” said Frazier  He still remembers the route: “I’d take 64 East to 77 South to 95 South  And it’s just a straight shot ”
Others followed suit  “What got everyone started was because the doctors around here won’t write a strong enough prescription,” he recalled  OxyContin and generic oxycodone still could be had — just not in Kentucky, which had a prescription drug monitoring database 
In Florida, “there was none of that … stuff that they check and find out what doctor you’ve been to,” said Frazier 
“And one person does it, and then they tell a friend, and then they go do it, and that’s how it all really got started here ”
MEDICAID-MEDICAIRE PAID MILLIONS FOR OXY
Tallahassee wasn’t just ignoring the epidemic, It was financing it 
Before her office was raided by law enforcement in December 2001, Asuncion M  Luyao’s patients would wait in a line in the rain to get prescriptions from the Port St  Lucie internist and acupuncturist  She was one of the most prolific prescribers of OxyContin in the state 
And hundreds of thousands of those pills were being paid for by Medicaid, Florida’s taxpayer-financed health program for the state’s poorest and sickest citizens  Between 1999 and 2001, Medicaid shelled out $935,634 for OxyContin prescriptions written by Luyao  That was just OxyContin  Luyao was prescribing an array of addictive drugs  In the 12 months leading up to the clinic raid, Medicaid paid roughly $1 million for 7,000 prescriptions, only about 17 percent of them for OxyContin 
Nor did the raid slow her down  Between the raid and her arrest on trafficking charges four months later, Luyao wrote another 282 OxyContin prescriptions billed to Medicaid  She was not an outlier  In 24 months, taxpayers footed the bill for more than 49 million doses of pills containing oxycodone, even though there were only 1 36 million Medicaid patients  Half were children 
The sheer volume of pills might have been a tipoff that the drugs were not all intended for legitimate use  So were arrest reports dating to 2001  One man had used his 7-year-old son’s Medicaid number to doctor-shop for OxyContin  A Miramar pharmacist who billed Medicaid $3 7 million for OxyContin pills was charged with paying Medicaid patients $150 each to use their IDs 
Medicaid paid for more than $300,000 to fill Dr  James Graves’ OxyContin prescriptions  The Florida Panhandle physician was the first doctor in the nation convicted of killing patients by overprescribing OxyContin 
Addiction risk for people taking high doses of oxycodone begins climbing after just three days, a recent study concluded  And most people on Florida Medicaid getting oxycodone prescriptions in 2011 were getting much more than a few days worth  They were getting an average of nine months worth of pills, state officials said 
Pill mill doctors prescribed 1 million of those pills:
Doctors working for the George twins’ trafficking empire prescribed at least 102,081 oxycodone pills billed to Medicaid before the ring collapsed in 2010 
Working out of a Delray Beach pain clinic founded by a convicted drug smuggler, Zvi Harry Perper, son of the Broward County medical examiner, was arrested on trafficking charges, but not before he wrote prescriptions to Medicaid patients for 115,977 doses of oxycodone in 90 days 
In Lake Worth, Cesar Deleon was arrestedas part of a DEA pill mill sweep and charged with 55 counts of illegally distributing drugs  Deleon wrote orders for 20,302 oxycodone pills for Medicaid patients 
Miami internist Dr  Selwyn Carrington authorized 32,411 doses of oxycodone for Medicaid patients in just two years  He was busted for signing his name to hundreds of prescriptions 
Further, Florida wasn’t in any hurry to stop doctors linked to pill mills 
Carrington was arrested for overprescribing in March 2011  The state’s emergency order to suspend his license was signed months after he had pleaded guilty in 2012 
Perper was busted at a Delray Beach pill mill operated by a former felon in 2011  The state did not act against his license until 2014 
Joseph M  Hernandez was writing prescriptions from his car, a veritable pill mill on wheels, when he was busted in February 2010 on one count of trafficking in oxycodone 
 Florida’s Department of Health didn’t file paperwork to restrict his license for almost 18 months 
During that time, Hernandez wrote oxycodone prescriptions for Medicaid patients totaling 258,940 doses representing a taxpayer-footed bill of $130,165 
Purdue Pharma’s Profits Before Patients Creed
Kelly Skidmore is exactly the type of person Purdue Pharma’s OxyContin marketing was intended to reach: Diagnosed with juvenile arthritis, the former state legislator’s struggle with chronic pain began at age 4 
Skidmore was wary of opioid painkillers, though, one reason her willingness in 2009 to work with Purdue was surprising  But she did it to get Florida’s dormant drug monitoring database up and running 
Then a state representative in a district straddling Palm Beach and Broward counties, Skidmore recalled that, “They came to me and said, ‘Could you help get it across the finish line?’ ”
OxyContin and prescription opioids, a serious problem in 2002, had evolved into a full-blown crisis in the ensuing seven years  Broward alone had more pain clinics than it had McDonald’s  Deaths tied to oxycodone had exploded, up by 263 percent since the prescription monitoring database had first been proposed and killed  Overdoses from prescription opioids were claiming more than seven lives a day 
“By God, if we had had seven dolphins a day dying and washing up on Florida beaches, we would have been appropriating money and solving it,” Skidmore said 
Skidmore believed a database wasn’t going to resolve the underlying addiction crisis  Still, it was a start  Not a silver bullet, but “maybe silver buckshot,” she said  The database law passed with gaping loopholes  No health care professional would have to report opioid prescriptions or check the database before prescribing more, and the state refused to pay for it 
“Just to get that one little piece … took nine years of filing bills and then it had no teeth,” Skidmore said  “And it should have been the easiest piece ”
Where Was The DEA and Everyone Else?
The DEA all but wrung its hands over Florida’s lethal inaction  The agency ticked off a devil’s brew of regulatory loopholes: Florida’s Health Department regulated health care professionals but not pain clinics  The state’s Agency for Health Care Administration regulated pain clinics that accepted insurance, but pill mills were most often on a cash-only basis  And the prescription monitoring database, mired in a vendor dispute, remained stalled 
In early 2011, when Gov  Rick Scott took office, just one drug — oxycodone — was tied to six fatal overdoses a day  Deaths tied to all drugs claimed 25 a day  In the handful of Appalachian states where traffickers were bringing back South Florida pills, it was worse 
Ohio’s death rate for oxycodone and similar opioids had doubled in 24 months, federal records show  Kentucky’s was up by more than 50 percent  And in West Virginia, home to hard-hit Huntington, death rates tied to pill mill drugs such as oxycodone and Opana had climbed by 341 percent 
The DEA formally pinpointed Palm Beach, Broward and Miami-Dade counties as the nation’s single biggest hub for trafficking pills across state lines  Within weeks of being sworn in, Scott abolished Florida’s Office of Drug Control, eliminating the state drug czar position, announced plans to drive a final stake in the heart of the database and rebuffed Purdue Pharma’s renewed offer to help pay for it 
Scott, a tea party conservative, cited privacy concerns, expressed skepticism the monitoring program would work and raised the possibility taxpayers would be left with a $500,000-a-year bill to operate it 
Attorney General Pam Bondi had also ridden the tea party wave to her position  She shared many of Scott’s conservative convictions  Unlike Scott, the former prosecutor relentlessly lobbied to keep the database alive  Florida’s failure to adopt the drug monitoring database was so out of step with the rest of the country that it began spawning conspiracy theories on both sides of the law 
Everyone knew prescription monitoring was going to kill the pill smuggling business, said a corrupt Florida Highway Patrol trooper as he drove a load of pills out of Florida, according to a federal lawsuit  Talking to the confidential informant in the seat next to him, the trooper speculated someone in Tallahassee must have a piece of the action, “because (Scott) was so adamant about not putting that system in place  Right?”
In Greenup, an infuriated Cooper told a reporter, “In my opinion, (Scott’s) getting money from somewhere  He has to be ” A few days later, recalled Cooper, “A lieutenant with the state police I’d been talking to down there called me, said, ‘Man, just a head’s up: I wouldn’t come to Florida ’” In states on the receiving end of the Florida pill pipeline and among federal officials, Scott’s resistance triggered outrage 
In Kentucky, where as much as 60 percent of the illicit oxycodone in that state flowed from Florida, Lt  Gov  Daniel Mongiardo proposed erecting billboards at the Florida line: “Welcome to the Oxy Tourism Capital of the World ”
U S  House Appropriations Chairman Hal Rogers, also from Kentucky, twice wrote Scott  “Canceling Florida’s prescription drug monitoring program is equal to firing firefighters while your house is ablaze,” he wrote 
Gil Kerlikowske, director of the White House Office of National Drug Control Policy, asked to meet with Scott  So did DEA Administrator Michele Leonhart 
Three U S  senators — New York’s Chuck Schumer, West Virginia’s Joe Manchin and Rhode Island’s Sheldon Whitehouse — joined Florida’s Bill Nelson in pointing out that the pills weren’t just a Florida problem: There were “serious ramifications for the rest of the country,” wrote Nelson of Scott’s reluctance to crack down  This is a perfect example of how political rhetoric, in-fighting and contrived agendas prevented an early stop to the emerging opioid crisis many years ago 
WHY DIDN’T THE DEA, DRUG DISTRIBUTORS AND PHARMACIES TAKE NOTICE BEFORE THE OPIOID CRISIS SPREAD ACROSS THE COUNTRY LIKE WILDFIRE? WAS IT BECAUSE OF THE BILLIONS IN PROFITS, QUARTERLY BONUSES AND DIVIDENDS? STOCK OPTIONS CASHED IN BY BOARDROOMS AT EVERY OPIOID BIG PHARMA COMPANY? STAY TUNED FOR HOW “PROFITS BEFORE PATIENTS” BECAME THE NORM
(article excerpts and quotes have been taken from publicly available media sources and court records)



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

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

Question: What was the reason given by Governor Rick Scott for not implementing a prescription drug monitoring database in Florida?

Read the following text and answer briefly.

Context 4:  We compare our algorithm with standard RLS and three other LMS-based algorithms: LMS, NLMS \cite{sayed2008adaptive}, VSS-LMS \cite{shin2004variable} \footnote{The used parameters for each algorithm are: for RLS $\lambda=1$, $\epsilon^{-1}=0 01$; for LMS $\mu=0 01$; for NLMS $\mu=0 5$; and for VSS-LMS $\mu_{max}=1$, $\alpha=0 95$, $C=1e-4$ } The probabilistic LMS algorithm in \cite{park2014probabilistic} is not simulated because it is not suitable for stationary environments 

In stationary environments, the proposed algorithm has only one parameter, $\sigma^2_n$  We simulate both the scenario where we have perfectly knowledge of the amount of noise (probLMS1) and the case where the value $\sigma^2_n$ is $100$ times smaller than the actual value (probLMS2)  The Mean-Square Deviation (${\sf MSD} = {\mathbb E} \| {\bf w}_0 - {\bf w}_k \|^2$), averaged out over $50$ independent simulations, is presented in Fig  \ref{fig:msd_statationary} 



\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{results_stationary_MSD}}
\end{minipage}
\caption{Performance in terms of MSD of probabilistic LMS with both optimal (probLMS1) and suboptimal (probLMS2) compared to LMS, NLMS, VS-LMS, and RLS }
\label{fig:msd_statationary}
\end{figure}

The performance of probabilistic LMS is close to RLS (obviously at a much lower computational cost) and largely outperforms previous variable step-size LMS algorithms proposed in the literature  Note that, when the model is stationary, i e  $\sigma^2_d=0$ in \eqref{eq:trans_eq},  both the uncertainty $\hat{\sigma}^2_k$, and the adaptive step size $\eta_k$, vanish over time  This implies that the error tends to zero when $k$ goes to infinity  Fig  \ref{fig:msd_statationary} also shows that the proposed approach is not very sensitive to a bad choice of its only parameter, as demonstrated by the good results of probLMS2, which uses a $\sigma^2_n$ that is $100$ times smaller than the optimal value  


\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{fig2_final}}
\end{minipage}
\caption{Real part of one coefficient of the measured and estimated channel in experiment two  The shaded area represents two standard deviations from the prediction {(the mean of the posterior distribution)} }
\label{fig_2}
\end{figure}


\begin{table}[ht]
\begin{footnotesize}
\setlength{\tabcolsep}{2pt}
\def1 5mm{1 5mm}
\begin{center}
\begin{tabular}{|l@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|}
\hline
Method &  LMS &  NLMS & LMS-2013 & VSSNLMS & probLMS & RLS \\
\hline
\hline
MSD (dB) &-28 45 &-21 07 &-14 36 &-26 90 &-28 36 &-25 97\\
\hline                                                                     
\end{tabular}
\end{center}
\caption{Steady-state MSD of the different algorithms for the tracking of a real MISO channel }
\label{tab:table_MSD}
\end{footnotesize}

\end{table}
\newpage
In a second experiment, we test the tracking capabilities of the proposed algorithm with {real} data of a wireless MISO channel acquired in a realistic indoor scenario  More details on the setup can be found in \cite{gutierrez2011frequency}  Fig  \ref{fig_2} shows the real part of one of the channels, and the estimate of the proposed algorithm  The shaded area represents the estimated uncertainty for each prediction, i e  $\hat{\mu}_k\pm2\hat{\sigma}_k$  Since the experimental setup does not allow us to obtain the optimal values for the parameters, we fix these parameters to their values that optimize the steady-state mean square deviation (MSD)  \hbox{Table \ref{tab:table_MSD}} shows this steady-state MSD of the estimate of the MISO channel with different methods  As can be seen, the best tracking performance is obtained by standard LMS and the proposed method  





\section{Conclusions and Opened Extensions}
\label{sec:conclusions}

{We have presented a probabilistic interpretation of the least-mean-square filter  The resulting algorithm is an adaptable step-size LMS that performs well both in stationary and tracking scenarios  Moreover, it has fewer free parameters than previous approaches and these parameters have a clear physical meaning  Finally, as stated in the introduction, one of the advantages of having a probabilistic model is that it is easily extensible:}

\begin{itemize}
\item If, instead of using an isotropic Gaussian distribution in the approximation, we used a Gaussian with diagonal covariance matrix, we would obtain a similar algorithm with different step sizes and measures of uncertainty, for each component of ${\bf w}_k$  Although this model can be more descriptive, it needs more parameters to be tuned, and the parallelism with LMS vanishes 
\item Similarly, if we substitute the transition model of \eqref{eq:trans_eq} by an Ornstein-Uhlenbeck process, 

\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;\lambda {\bf w}_{k-1}, \sigma_d^2), \nonumber
\label{eq:trans_eq_lambda}
\end{equation}
a similar algorithm is obtained but with a forgetting factor $\lambda$ multiplying ${\bf w}_{k-1}^{(LMS)}$ in \eqref{eq:lms}  This algorithm may have improved performance under such a kind of autoregresive dynamics of ${\bf w}_{k}$, though, again, the connection with standard LMS becomes dimmer 

\item As in \cite{park2014probabilistic}, the measurement model \eqref{eq:mess_eq} can be changed to obtain similar adaptive algorithms for classification, ordinal regression, and Dirichlet regression for compositional data  

\item A similar approximation technique could be applied to more complex dynamical models, i e  switching dynamical models \cite{barber2010graphical}  The derivation of efficient adaptive algorithms that explicitly take into account a switch in the dynamics of the parameters of interest is a non-trivial and open problem, though the proposed approach could be useful 

\item Finally, like standard LMS, this algorithm can be kernelized for its application in estimation under non-linear scenarios 

\end{itemize}


\begin{appendices}

\section{KL divergence between a general gaussian distribution and an isotropic gaussian}
\label{sec:kl}

 We want to approximate  $p_{{\bf x}_1}(x) = \mathcal{N}({\bf x}; \boldsymbol\mu_1,\boldsymbol\Sigma_1)$ by $p_{{\bf x}_2}({\bf x}) = \mathcal{N}({\bf x}; \boldsymbol\mu_2,\sigma_2^2 {\bf I})$  In order to do so, we have to compute the parameters of $p_{{\bf x}_2}({\bf x})$, $\boldsymbol\mu_2$ and $\sigma_2^2$, that minimize the following Kullback-Leibler divergence,

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) &=&\int_{-\infty}^{\infty} p_{{\bf x}_1}({\bf x}) \ln{\frac{p_{{\bf x}_1}({\bf x})}{p_{{\bf x}_2}({\bf x})}}d{\bf x} \nonumber  \\
&= &  \frac{1}{2} \{ -M + {\sf Tr}(\sigma_2^{-2} {\bf I}\cdot \boldsymbol\Sigma_1^{-1})  \nonumber \\
  & &  + (\boldsymbol\mu_2 - \boldsymbol\mu_1 )^T \sigma^{-2}_2{\bf I} (\boldsymbol\mu_2 - \boldsymbol\mu_1 )  \nonumber \\
 & &   +  \ln \frac{{\sigma_2^2}^M}{\det\boldsymbol\Sigma_1} \}   
\label{eq:divergence}
\end{eqnarray}
Using symmetry arguments, we obtain 
\begin{equation}
\boldsymbol\mu_2^{*} =\arg \displaystyle{  \min_{\boldsymbol\mu_2}} \{ D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) \} = \boldsymbol\mu_1 
\end{equation}
Then, \eqref{eq:divergence} gets simplified into 

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) = \frac{1}{2}\lbrace { -M + {\sf Tr}(\frac{\boldsymbol\Sigma_1}{\sigma_2^{2}}) + \ln \frac{\sigma_2^{2M}}{\det\boldsymbol\Sigma_1}}\rbrace 
\end{eqnarray}
The variance $\sigma_2^2$ is computed in order to minimize this Kullback-Leibler divergence as

\begin{eqnarray}
\sigma_2^{2*} &=& \arg\min_{\sigma_2^2} D_{KL}(P_{x_1}\| P_{x_2}) \nonumber \\
 &=& \arg\min_{\sigma_2^2}\{ \sigma_2^{-2}{\sf Tr}\{\boldsymbol\Sigma_1\} + M\ln \sigma_2^{2} \}  
\end{eqnarray}
Deriving and making it equal zero leads to

\begin{equation}
\frac{\partial}{\partial \sigma_2^2} \left[ \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{\sigma_2^{2}} + M \ln \sigma_2^{2} \right] = \left  {\frac{M}{\sigma_2^{2}}-\frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{(\sigma_2^{2})^2}}\right|_{\sigma_2^{2}=\sigma_2^{2*}}\left  =0 \right   
\nonumber
\end{equation}
Finally, since the divergence has a single extremum in $R_+$,
\begin{equation}
\sigma_2^{2*} = \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{M} 
\end{equation}




\end{appendices}

\vfill
\clearpage

\bibliographystyle{IEEEbib}




ground_truth: Privacy concerns and skepticism about its effectiveness.