title: Real Irish Republican Army

text: On 15 August 1998 the RIRA left a car containing 500 lb of home-made explosives in the centre of Omagh, County Tyrone. The bombers could not find a parking space near the intended target of the courthouse, and the car was left 400 metres away. As a result, three inaccurate telephone warnings were issued, and the Royal Ulster Constabulary (RUC) believed the bomb was located outside the courthouse. They attempted to establish a security cordon to keep civilians clear of the area, which inadvertently pushed people closer to the location of the bomb. Shortly after, the bomb exploded killing 29 people and injuring 220 others, in what became the single deadliest strike of the Troubles.


title: Real Irish Republican Army

text: The bombing caused a major outcry throughout the world, and the Irish and British governments introduced new legislation in an attempt to destroy the organisation. The RIRA also came under pressure from the Provisional IRA, when Provisional IRA members visited the homes of 60 people connected with the RIRA and ordered them to disband and stop interfering with Provisional IRA arms dumps. With the organisation under intense pressure, which included McKevitt and Sands-McKevitt being forced from their home after the media named McKevitt in connection with the bombing, the RIRA called a ceasefire on 8 September.


title: Real Irish Republican Army

text: Following the declaration of the ceasefire the RIRA began to regroup, and by the end of October had elected a new leadership and were planning their future direction. In late December Irish government representative Martin Mansergh held a meeting with McKevitt in Dundalk, in an attempt to convince McKevitt to disband the RIRA. McKevitt refused, stating that members would be left defenceless against attacks by the Provisional IRA. In 1999 the RIRA began preparations for a renewed campaign, and in May three members travelled to Split in Croatia to purchase arms, which were smuggled back to Ireland. On 20 October, ten people were arrested when Gardaí raided a RIRA training camp near Stamullen, County Meath.


title: Real Irish Republican Army

text: Officers found a firing range inside a disused wine cellar being used as an underground bunker, and seized weapons including an assault rifle, a submachine gun, a semi-automatic pistol and an RPG-18 rocket launcher. An earlier version of the rocket launcher, the RPG-7, had been in the possession of the Provisional IRA from as early as 1972, but this was the first time the RPG-18 had been found in the possession of a paramilitary organisation in Ireland.


title: Real Irish Republican Army

text: On 20 January 2000 the RIRA issued a call-to-arms in a statement to the "Irish News". The statement condemned the Northern Ireland Executive, and stated: "Once again, Óglaigh na hÉireann declares the right of the Irish people to the ownership of Ireland. We call on all volunteers loyal to the Irish Republic to unite to uphold the Republic and establish a permanent national parliament representative of all the people." The RIRA launched its new campaign on 25 February with an attempted bombing of Shackleton Army Barracks in Ballykelly. The bombers were disturbed as they were assembling the device, which would have caused mass murder if detonated, according to soldiers.


title: Real Irish Republican Army

text: On 29 February a rocket launcher similar to one seized in the 1999 raid was found near an army base in Dungannon, County Tyrone, and on 15 March three men were arrested following the discovery of 500 lb of home-made explosives when the RUC searched two cars in Hillsborough, County Down. On 6 April a bomb attack took place at Ebrington Barracks in Derry. RIRA members lowered a device consisting of 5 lb of homemade explosives over the perimeter fence using ropes, and the bomb subsequently exploded damaging the fence and an unmanned guardhouse.


title: Real Irish Republican Army

text: After the Omagh bombing, the RIRA leadership were unwilling to launch a full-scale campaign in Northern Ireland due to the possibility of civilians being killed. Instead they decided to launch a series of attacks in England, in particular London, which they hoped would attract disenchanted Provisional IRA members to join the RIRA. On 1 June 2000 a bomb damaged Hammersmith Bridge, a symbolic target for Irish republican paramilitary groups. The bridge had been targeted by the Irish Republican Army on 29 March 1939 as part of its Sabotage Campaign, and by the Provisional IRA on 24 April 1996.


title: Real Irish Republican Army

text: On 19 July, security forces carried out a controlled explosion on a bomb left at Ealing Broadway station and public transport was disrupted when the Metropolitan Police closed Victoria and Paddington train stations and halted services on the London Underground. On 21 September a rocket-propelled grenade was fired at the MI6 headquarters using an RPG-22 rocket launcher, which generated headlines around the world. In November 2000, security forces foiled a plot to drive 500 lb of homemade explosives to central London that month, a bomb twice as powerful as the one in Omagh. At the time police were warning for weeks that a terrorist attack in London could be imminent.


title: Real Irish Republican Army

text: On 21 February 2001 a bomb disguised as a torch left outside a Territorial Army base in Shepherd's Bush seriously injured a 14-year-old cadet, who was blinded and had his hand blown off. A second attack in Shepherd's Bush, the 4 March BBC bombing, injured a civilian outside the BBC Television Centre. The explosion was captured by a BBC cameraman, and the footage was broadcast on TV stations worldwide, and gained mass publicity for the group. On 14 April a bomb exploded at a postal sorting office in Hendon, causing minor damage but no injuries. Three weeks later on 6 May a second bomb exploded at the same building, causing slight injuries to a passer-by. The 3 August 2001 Ealing bombing injured seven people, and on 3 November a car bomb containing 60 lb of home-made explosives was planted in the centre of Birmingham. The bomb did not fully detonate and no one was injured.


title: Real Irish Republican Army

text: The successful attack on Hammersmith Bridge encouraged the RIRA leadership to launch further attacks in Northern Ireland. On 19 June 2000 a bomb was found in the grounds of Hillsborough Castle, home of Secretary of State for Northern Ireland Peter Mandelson. On 30 June a bomb exploded on the Dublin-to-Belfast railway line near the village of Meigh in County Armagh. The explosion damaged the tracks, and caused disruption to train services. On 9 July a car bomb damaged buildings in Stewartstown, County Tyrone including an RUC station, and on 10 August an attack in Derry was thwarted by the RUC after a van containing a 500 lb bomb failed to stop at a police checkpoint. Following a car chase the bombers escaped across the Irish border, and the Irish Army carried out a controlled explosion on the bomb after the van was found abandoned in County Donegal.


title: Real Irish Republican Army

text: On 13 September 2000, two 80 lb bombs were planted at the Magilligan army camp in County Londonderry, one of which was planted in a wooden hut and partially exploded when a soldier opened the door to the hut. The second bomb was found during a follow-up search and made safe by bomb disposal experts. On 11 November the RUC and British Army prevented a mortar attack after stopping a van near Derrylin, County Fermanagh, and the RUC prevented a further attack on 13 January 2001 when an 1100 lb bomb was found in Armagh – the largest bomb found in several years according to the RUC.


title: Real Irish Republican Army

text: On 23 January the RIRA attacked Ebrington Army Barracks in Derry for a second time, firing a mortar over a perimeter fence. A mortar similar to the one used in the attack was found by Gardaí near Newtowncunningham on 13 February, and British army bomb disposal experts made safe another mortar found between Dungannon and Carrickmore on 12 April. On 1 August a 40 lb bomb was discovered in a car at the long-stay car park of Belfast International Airport following a telephone warning, and was made safe with two controlled explosions by bomb disposal experts. In December a six-day security operation ended when a 70 lb bomb found under railway tracks at Killeen Bridge near Newry was defused. The operation began following telephone warnings, and the road and railway line connecting Newry to Dundalk were closed due to security alerts.


title: Real Irish Republican Army

text: A pipe bomb was discovered at a police officer's home in Annalong, County Down on 3 January 2002, and two teenage boys were injured in County Armagh on 2 March when a bomb hidden in a traffic cone exploded. On 29 March 2002 the RIRA targeted a former member of the Royal Irish Regiment from Sion Mills, County Tyrone, with a bomb attached to his car that failed to explode. On 1 August 2002 a civilian worker was killed by an explosion at a Territorial Army base in Derry. The man, a 51-year-old former member of the Ulster Defence Regiment, was the thirtieth person killed by the RIRA.


title: Real Irish Republican Army

text: Despite the RIRA's renewed activity, the organisation was weakened by the arrest of key members and continued infiltration by informers. McKevitt was arrested on 29 March 2001 and charged with membership of an illegal organisation and directing terrorism, and remanded into custody. In July 2001, following the arrests of McKevitt and other RIRA members, British and Irish government sources hinted that the organisation was now in disarray. Other key figures were jailed, including the RIRA's Director of Operations, Liam Campbell, who was convicted of membership of an illegal organisation, and Colm Murphy who was convicted of conspiring to cause the Omagh bombing, although this conviction was overturned on appeal.


title: Real Irish Republican Army

text: On 10 April 2002 Ruairi Convey, from Donaghmede, Dublin, was jailed for three years for membership of the RIRA. During a search of his home a list of names and home addresses of members of the Gardaí's Emergency Response Unit was found. Five RIRA members were also convicted in connection with the 2001 bombing campaign in England, and received sentences varying from 16 years to 22 years' imprisonment. In October 2002, McKevitt and other RIRA members imprisoned in Portlaoise Prison issued a statement calling for the organisation to stand down. After a two-month trial, McKevitt was sentenced to twenty years' imprisonment in August 2003 after being convicted of directing terrorism.


title: Real Irish Republican Army

text: After McKevitt's imprisonment, the RIRA regrouped and claimed responsibility for a series of firebomb attacks against premises in Belfast in November 2004, and an attack on a Police Service of Northern Ireland (PSNI) patrol in Ballymena during March 2006 was attributed to the RIRA by the Independent Monitoring Commission (IMC). On 9 August 2006, fire bomb attacks by the RIRA hit businesses in Newry, County Down. Buildings belonging to JJB Sports and Carpetright were destroyed, and ones belonging to MFI and TK Maxx were badly damaged. On 27 October 2006, a large amount of explosives was found in Kilbranish, Mount Leinster, County Carlow by police, who believe the RIRA were trying to derail the peace process with a bomb attack. The IMC believe the RIRA were also responsible for a failed mortar attack on Craigavon PSNI Station on 4 December 2006. The IMC's October 2006 report stated that the RIRA remains "active and dangerous" and that it seeks to "sustain its position as a terrorist organisation". The RIRA has stated it has no intention of calling a ceasefire unless a declaration of intent to withdraw from Northern Ireland is made by the British Government.


title: Real Irish Republican Army

text: In a lengthy interview with the newspaper "An Phoblacht" in 2003, the leadership of the Provisional IRA said that the RIRA had "no coherent strategy".


title: Real Irish Republican Army

text: On 8 November 2007 two RIRA members shot an off-duty PSNI officer as he sat in his car on Bishop Street in Derry, causing injuries to his face and arm. On 12 November another PSNI member was shot by RIRA members in Dungannon, County Tyrone. On 7 February 2008, the RIRA stated that, after experiencing a three-year period of reorganisation, it intended to "go back to war" by launching a new offensive against "legitimate targets". It also, despite having apologised for the Omagh bombing, denied any large scale involvement with the attack and said that their part had only gone as far as their codeword being used. On 12 May 2008 the RIRA seriously injured a member of the PSNI when a booby trap bomb exploded underneath his car near Spamount, County Tyrone. On 25 September 2008 the RIRA shot a man in the neck in St Johnston, near the County Londonderry border. The same man was targeted in a pipe bomb attack on his home on 25 October, the RIRA did not claim responsibility for the attack, but security forces believe they were responsible for it.


title: Real Irish Republican Army

text: On 7 March 2009 the RIRA claimed responsibility for the 2009 Massereene Barracks shooting. This shooting occurred outside the Massereene Barracks as four soldiers were receiving a pizza delivery. Two soldiers were killed, and the other two soldiers and two deliverymen were injured. On 3 April 2009 the RIRA in Derry claimed responsibility for carrying out a punishment shooting of a man who was awaiting sentencing for raping a 15-year-old girl. The RIRA were also blamed for orchestrating rioting in the Ardoyne area of Belfast on 13 July 2009 as an Apprentice Boys parade was passing. Several PSNI officers were injured in the rioting and at least one shot was fired at police. In early November, the Independent Monitoring Commission released a report stating that the threat from the RIRA and other dissident republicans was at its most serious level since the 1998 Good Friday Agreement.


title: Real Irish Republican Army

text: When drug dealer Sean Winters was shot dead in Portmarnock, north Dublin, in September 2010, the Real IRA "emerged as the chief suspects". They were also suspected of shooting dead drugs gang leader Michael Kelly in Coolock in September 2011.


title: Real Irish Republican Army

text: On 5 October 2010, a car bomb exploded outside a branch of the Ulster Bank on Culmore Road in Derry. Two police officers were slightly injured in the blast, which also damaged a hotel and other businesses. Several telephone warnings were received an hour prior to the blast allowing police to cordon off the area. The RIRA later claimed responsibility in a telephone call to the "Derry Journal".


title: Real Irish Republican Army

text: A large Real IRA explosives dump and arms cache were discovered in Dunleer, County Louth by Gardaí in October 2010, following a weekend of searches and arrests in the east of the country. In addition, two Real IRA men were charged in Dublin's non-jury Special Criminal Court of membership of an illegal organisation. The Real IRA claimed responsibility for kidnapping and shooting dead of one of their members, Kieran Doherty, for alleged drug dealing. Further seizures of the group's arms and explosives by the Gardaí in 2012 and 2013 led to over a dozen more arrests. In 2011 Michael Campbell, brother of Liam, was found guilty in Vilnius, Lithuania, of trying to purchase arms and explosives and was sentenced to twelve years in prison. In October 2013 Campbell was freed on appeal, only to have the Supreme Court of Lithuania order a retrial in June 2014. Campbell has maintained his innocence, accusing British intelligence of attempting to frame him.


title: Real Irish Republican Army

text: On 26 July 2012, it was reported that Republican Action Against Drugs (RAAD) and other small republican militant groups were merging with the Real IRA. As before, the group would continue to refer to itself as "the Irish Republican Army", though some media began to refer to the group as a "new IRA".


title: Real Irish Republican Army

text: The RIRA has a command structure similar to the Provisional IRA, with a seven-member Army Council consisting of a chief of staff, quartermaster general, director of training, director of operations, director of finance, director of publicity, and adjutant general. The rank-and-file members operate in active service units of covert cells to prevent the organisation from being compromised by informers. In June 2005, the organisation was believed to have a maximum of about 150 members, according to a statement by the Irish Minister for Justice, Equality and Law Reform, Michael McDowell.


title: Real Irish Republican Army

text: The RIRA also has political wings: the 32 County Sovereignty Movement (formerly the 32 County Sovereignty Committee), led by Francis Mackey, and unregistered political party Saoradh, led by Brian Kenna.


title: Real Irish Republican Army

text: The RIRA is distinct from the Continuity IRA, another Provisional IRA splinter group founded in 1986, although the two groups have been known to co-operate at a local level. The Provisional IRA has been hostile to the RIRA and issued threats to RIRA members, and in October 2000 was alleged to be responsible for the fatal shooting of Belfast RIRA member Joe O'Connor according to O'Connor's family and 32 County Sovereignty Movement member Marian Price.


title: Real Irish Republican Army

text: Organisations called "Irish Republican Army" are illegal in both UK law and Irish law; both proscriptions have been held to apply to the RIRA as to other groups of the name. Membership in the organisation is punishable by a sentence of up to ten years' imprisonment under UK law. In 2001 the United States government designated the RIRA (and its aliases) as a "Foreign Terrorist Organization" (FTO). This makes it illegal for Americans to provide material support to the RIRA, requires American financial institutions to freeze the group's assets, and denies suspected RIRA members visas into the United States.


title: Real Irish Republican Army

text: In 2014, "Forbes" magazine estimated the group's annual turnover at US$50 million. According to the police in Northern Ireland, the main sources of the Real IRA's funding are illegal fuel operations and various smuggling activities. Illicit cigarettes were also said to be a significant source of income for the group. There are also other significant sources of funding from the group, including funding from sympathisers based in the US and other countries.


title: Real Irish Republican Army

text: The RIRA initially took small amounts of materiel from Provisional IRA arms dumps under the control of McKevitt and other former Provisional IRA members, including the plastic explosive Semtex, Uzi submachine guns, AK-47 assault rifles, handguns, detonators and timing devices. The defection of senior Provisional IRA members also gave the RIRA the ability to manufacture home-made explosives and improvised mortars, including the Mark 15 mortar capable of firing a shell.


title: Real Irish Republican Army

text: In 1999 the organisation supplemented its equipment by importing arms from Croatia, including military explosive TM500, CZ Model 25 submachine guns, modified AK-47 assault rifles with a folding stock, and RPG-18 and RPG-22 rocket launchers but a July 2000 attempt to smuggle a second consignment of arms was foiled by Croatian police, who seized seven RPG-18s, AK-47 assault rifles, detonators, ammunition, and twenty packs of TM500.


title: Real Irish Republican Army

text: In 2001 RIRA members travelled to Slovakia to procure arms, and were caught in a sting operation by the British security agency MI5. The men attempted to purchase five tonnes of plastic explosives, 2,000 detonators, 500 handguns, 200 rocket-propelled-grenades, and also wire-guided missiles and sniper rifles. Three men from County Louth were arrested and extradited to the UK and subsequently imprisoned for 30 years each after pleading guilty to conspiring to cause explosions and other charges.


title: Real Irish Republican Army

text: In June 2006, the PSNI made arrests following an MI5 sting operation targeting a dissident republican gun smuggling plot. The RIRA had attempted to procure arms from France including Semtex and C-4 plastic explosives, SA-7 surface-to-air missiles, AK-47s, rocket launchers, heavy machine guns, sniper rifles, pistols with silencers, anti-tank weapons and detonators. On 30 June 2010, two of those arrested were found guilty following a trial by judge in Belfast. On 1 October 2010, one man was sentenced to 20 years' imprisonment for attempting to import weapons and explosives, while the other was sentenced to four years' imprisonment for making a Portuguese property available for the purpose of terrorism.


title: Shor's algorithm

text: Shor's algorithm is a quantum computer algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor.


title: Shor's algorithm

text: On a quantum computer, to factor an integer formula_1, Shor's algorithm runs in polynomial time, meaning the time taken is polynomial in formula_2, the size of the integer given as input. Specifically, it takes quantum gates of order formula_3 using fast multiplication, or even formula_4 utilizing the asymptotically fastest multiplication algorithm currently known due to Harvey and Van Der Hoven, thus demonstrating that the integer factorization problem can be efficiently solved on a quantum computer and is consequently in the complexity class BQP. This is almost exponentially faster than the most efficient known classical factoring algorithm, the general number field sieve, which works in sub-exponential time: formula_5. The efficiency of Shor's algorithm is due to the efficiency of the quantum Fourier transform, and modular exponentiation by repeated squarings.


title: Shor's algorithm

text: If a quantum computer with a sufficient number of qubits could operate without succumbing to quantum noise and other quantum-decoherence phenomena, then Shor's algorithm could be used to break public-key cryptography schemes, such as


title: Shor's algorithm

text: RSA is based on the assumption that factoring large integers is computationally intractable. As far as is known, this assumption is valid for classical (non-quantum) computers; no classical algorithm is known that can factor integers in polynomial time. However, Shor's algorithm shows that factoring integers is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer. It was also a powerful motivator for the design and construction of quantum computers, and for the study of new quantum-computer algorithms. It has also facilitated research on new cryptosystems that are secure from quantum computers, collectively called post-quantum cryptography.


title: Shor's algorithm

text: In 2001, Shor's algorithm was demonstrated by a group at IBM, who factored formula_6 into formula_7, using an NMR implementation of a quantum computer with formula_8 qubits. After IBM's implementation, two independent groups implemented Shor's algorithm using photonic qubits, emphasizing that multi-qubit entanglement was observed when running the Shor's algorithm circuits. In 2012, the factorization of formula_6 was performed with solid-state qubits. Later, in 2012, the factorization of formula_10 was achieved. In 2019 an attempt was made to factor the number formula_11 using Shor's algorithm on an IBM Q System One, but the algorithm failed because of accumulating errors. Though larger numbers have been factored by quantum computers using other algorithms, these algorithms are similar to classical brute-force checking of factors, so unlike Shor's algorithm, they are not expected to ever perform better than classical factoring algorithms.


title: Shor's algorithm

text: The problem that we are trying to solve is, given a composite number formula_1, to find a non-trivial divisor of formula_1 (a divisor strictly between formula_14 and formula_1). Before attempting to find such a divisor, if there's any doubt whether formula_1 is composite or prime, one can use relatively quick primality-testing algorithms to verify that formula_1 is indeed composite, although this is not a part of Shor's algorithm.


title: Shor's algorithm

text: The aim of the algorithm is to find a non-trivial square root formula_18 of formula_14 modulo formula_1 that is different from formula_14 and formula_22, because then


title: Shor's algorithm

text: for a non-zero integer formula_24 that gives us two distinct non-trivial divisors formula_25 and formula_26 of formula_1.


title: Shor's algorithm

text: This idea is similar to other factoring algorithms, such as the quadratic sieve, and a more detailed explanation can be found in the Explanation section below. Before starting the algorithm, it is imperative to check formula_1 to be odd (otherwise formula_29 is a divisor) and not to be any power of an integer (otherwise that integer is a divisor), so as to guarantee the existence of a non-trivial square root formula_18 of formula_14 modulo formula_1.


title: Shor's algorithm

text: In turn, finding such a formula_18 is reduced to finding an element formula_34 as a parameter in an integer function, such that the function has an even period with a certain additional property (as explained below, it is required that the condition of Step 6 of the classical part does not hold). The quantum algorithm is used for finding the period of randomly chosen elements formula_34, as this is a difficult problem on a classical computer.


title: Shor's algorithm

text: For example: Given formula_36, formula_37, and formula_38, i.e.,formula_39 we have formula_40, where formula_41 and formula_42. For formula_1 that is a product of two distinct primes, formula_44 and formula_45, formula_46, which for formula_36 is formula_48, and formula_49 divides formula_48.


title: Shor's algorithm

text: The quantum circuits used for this algorithm are custom designed for each choice of formula_1 and each choice of the random formula_34 which have the relationship formula_53. Given value formula_54, a value formula_55 is chosen such that formula_56. Such a value of formula_57 implies that formula_58. The input and output qubit registers will store superpositions of values from formula_59 to formula_60. Therefore, these registers have formula_45 qubits each. Using what might appear to be twice as many qubits as necessary guarantees that there are at least formula_1 different values of formula_63 that produce the same formula_64, even as the period formula_49 approaches formula_66.


title: Shor's algorithm

text: = \left(\frac{1}{\sqrt{2}} \sum_{x_{1} = 0}^{1} | x_{1} \rangle \right) \otimes \cdots \otimes \left(\frac{1}{\sqrt{2}} \sum_{x_{q} = 0}^{1} | x_{q} \rangle \right).


title: Shor's algorithm

text: This initial state is a superposition of formula_68 states, and is obtained by generating formula_45 independent qubits, each an equal superposition of formula_59 and formula_14 states. We can accomplish this by initializing the qubits to the zero-position, and then applying a Hadamard gate to each of the formula_45 qubits, where formula_73. This process is called the Hadamard transform.


title: Shor's algorithm

text: Note: formula_53. This is still a superposition of formula_68 states. However, the formula_79 qubits, i.e, the formula_45 input qubits and formula_81 (formula_82) output qubits, are now entangled or not separable, as the state cannot be written as a tensor product of states of individual qubits.


title: Shor's algorithm

text: Importantly, the value containing the formula_49 we are looking for is now stored in the phase of the input qubits formula_63 as a result of "phase kickback", where using qubits as control inputs to unitary gates alters the relative phase of the control qubits.


title: Shor's algorithm

text: This is a superposition of many more than formula_68 states, but many fewer than formula_96 states, as there are fewer than formula_68 distinct values of formula_98. Let


title: Shor's algorithm

text: Then formula_113 is a unit vector in the complex plane (formula_114 is a root of unity, and formula_49 and formula_116 are integers), and the coefficient of formula_117 in the final state is


title: Shor's algorithm

text: Each term in this sum represents a "different path to the same result", and quantum interference occurs — constructive when the unit vectors formula_119 point in nearly the same direction in the complex plane, which requires that formula_113 point along the positive real axis.


title: Shor's algorithm

text: Analysis now shows that this probability is higher the closer the unit vector formula_113 is to the positive real axis, or the closer formula_128 is to an integer. Unless formula_49 is a power of formula_29, it will not be a factor of formula_68.


title: Shor's algorithm

text: Given these multiple conditions (and assuming formula_137 is irreducible), formula_139 is very likely to be the appropriate period formula_49, or at least a factor of it.


title: Shor's algorithm

text: The algorithm is composed of two parts. The first part of the algorithm turns the factoring problem into the problem of finding the period of a function and may be implemented classically. The second part finds the period using the quantum Fourier transform and is responsible for the quantum speedup.


title: Shor's algorithm

text: Shor's algorithm hinges on finding a non-trivial square root of formula_14 modulo formula_1; That is, a solution to


title: Shor's algorithm

text: If such formula_18 exists, we claim that formula_152 is a proper factor of formula_1, i.e., formula_154. In fact, if


title: Shor's algorithm

text: formula_155, then formula_1 divides formula_157, so that formula_158, which goes against the construction of formula_18. If, on the other hand, formula_160, then by Bézout's identity, there are integers formula_161 such that


title: Shor's algorithm

text: As formula_1 divides formula_166, we find that formula_1 divides formula_163, so that formula_169, again contradicting the construction of formula_18.


title: Shor's algorithm

text: Therefore, formula_171 is the required proper factor of formula_1. Similarly, it can be proven that formula_173 is also a proper factor of formula_1.


title: Shor's algorithm

text: For such a non-trivial square root of formula_14 modulo formula_1 to exist, notice that formula_177, and for any power of an odd prime formula_178, there is no non-trivial square root of formula_14 modulo formula_1: For any formula_181 either formula_182 or formula_183 has to be a multiple of formula_178.


title: Shor's algorithm

text: Therefore, for Shor's algorithm to work, we need formula_1 to be odd (otherwise formula_29 is a divisor) and not to be any power of an odd prime (otherwise that prime is a divisor). We can check that there are no integer roots formula_187 for formula_188, and if formula_1 is not a power of any integer, it is not a power of any odd prime. Here, the upper bound for the integer formula_190 that we need to check is determined by formula_191, since for formula_1 to be odd, formula_187 cannot be formula_29. This check, however, cannot rule out that formula_1 may be an odd prime itself, which can only be ruled out by primality-testing algorithms.


title: Shor's algorithm

text: Given that formula_1 is odd and not any power of an odd prime, based on the fundamental theorem of arithmetic, we may assume that formula_1 is the product of two coprime integers greater than formula_29 (formula_199 and formula_200). From the four combinations of choosing plus sign and minus sign in the integer equations formula_201, based on the Chinese remainder theorem and formula_202, there are at least four distinct square roots of formula_14 modulo formula_1, and therefore at least two distinct non-trivial square roots exist. In fact, they are the solutions to formula_205 and formula_206.


title: Shor's algorithm

text: The integers less than formula_1 and coprime with formula_1 form the multiplicative group of integers modulo formula_1, which is a finite abelian group formula_210. The size of this group is given by formula_211. By the end of step 3, we have an integer formula_34 in this group. As the group is finite, formula_34 must have a finite order formula_49, which is the smallest positive integer such that


title: Shor's algorithm

text: This is the order formula_49 of the finite cyclic subgroup ⟨"a"⟩ of the group formula_217, which is the smallest positive integer formula_49 for which formula_219. Since formula_220 and formula_221 are coprime, by Euler's totient theorem, formula_49 must exist, and divides formula_211, where formula_224 denotes Euler's totient function.


title: Shor's algorithm

text: Therefore, formula_1 divides formula_226 (also written formula_227). Suppose that we are able to obtain formula_49 and that it is even. (If formula_49 is odd, then by step 5, we have to restart the algorithm with a different random number formula_220) Now formula_231 is a square root of formula_14 modulo formula_1 that is different from formula_14. This is because formula_49 is the order of formula_34 modulo formula_1, so formula_238, or else the order of formula_34 in this group would be formula_240. If formula_241, then by step 6, we have to restart the algorithm with a different random number formula_34.


title: Shor's algorithm

text: Eventually, we must hit an formula_34 of order formula_49 in formula_210 such that formula_246. This is because such a formula_18 is a square root of formula_14 modulo formula_1 other than formula_14 and formula_22, whose existence is guaranteed by the Chinese remainder theorem, as the odd number formula_1 is not a prime power.


title: Shor's algorithm

text: Shor's period-finding algorithm relies heavily on the ability of a quantum computer to be in many states simultaneously.


title: Shor's algorithm

text: Physicists call this behavior a "superposition" of states. To compute the period of a function formula_102, we evaluate the function at all points simultaneously.


title: Shor's algorithm

text: Quantum physics does not allow us to access all this information directly, however. A measurement will yield only one of all possible values, destroying all others. If not for the no-cloning theorem, we could first measure formula_64 without measuring formula_63, and then make a few copies of the resulting state (which is a superposition of states all having the same formula_64). Measuring formula_63 on these states would provide different formula_63 values which give the same formula_64, leading to the period. Because we cannot make exact copies of a quantum state, this method does not work. Therefore, we have to carefully transform the superposition to another state that will return the correct answer with high probability. This is achieved by the quantum Fourier transform.


title: Shor's algorithm

text: Shor thus had to solve three "implementation" problems. All of them had to be implemented "fast", which means that they can be implemented with a number of quantum gates that is polynomial in formula_2.


title: Shor's algorithm

text: After all these transformations, a measurement will yield an approximation to the period formula_49. For simplicity assume that there is a formula_116 such that formula_128 is an integer. Then the probability to measure formula_116 is formula_14. To see this, we notice that then


title: Shor's algorithm

text: for all integers formula_18. Therefore, the sum whose square gives us the probability to measure formula_116 will be formula_272, as formula_18 takes roughly formula_272 values and thus the probability is formula_275. There are formula_49 possible values of formula_116 such that formula_128 is an integer, and also formula_49 possibilities for formula_280, so the probabilities sum to formula_14.


title: Shor's algorithm

text: The period-finding routine can be considered a variation of the more general quantum phase estimation algorithm to determine the eigenvalue of a unitary corresponding to an eigenvector. In the case of the period-finding routine used in Shor's Algorithm, the unitary in question is modular multiplication by the chosen base mod formula_221. While the computational basis formula_283 is not an eigenvector of this unitary, it is a uniform superposition of its formula_284 eigenvectors and thus the measurement will give the eigenvalue's phase for one of the eigenvectors. Since not all such phases can be used to extract the period, the retries of the subroutine may be necessary.


title: Shor's algorithm

text: The runtime bottleneck of Shor's algorithm is quantum modular exponentiation, which is by far slower than the quantum Fourier transform and classical pre-/post-processing. There are several approaches to constructing and optimizing circuits for modular exponentiation. The simplest and (currently) most practical approach is to mimic conventional arithmetic circuits with reversible gates, starting with ripple-carry adders. Knowing the base and the modulus of exponentiation facilitates further optimizations. Reversible circuits typically use on the order of formula_285 gates for formula_286 qubits. Alternative techniques asymptotically improve gate counts by using quantum Fourier transforms, but are not competitive with fewer than 600 qubits owing to high constants.


title: Shor's algorithm

text: Given a group formula_287 with order formula_44 and generator formula_289, suppose we know that formula_290, for some formula_291, and we wish to compute formula_49, which is the discrete logarithm: formula_293. Consider the abelian group formula_294, where each factor corresponds to modular addition of values. Now, consider the function


title: Shor's algorithm

text: This gives us an abelian hidden subgroup problem, as formula_102 corresponds to a group homomorphism. The kernel corresponds to the multiples of formula_297. So, if we can find the kernel, we can find formula_49. A quantum algorithm for solving this problem exists. This algorithm is, like the factor-finding algorithm, due to Peter Shor and both are implemented by creating a superposition through using Hadamard gates, followed by implementing formula_102 as a quantum transform, followed finally by a quantum Fourier transform. Due to this, the quantum algorithm for computing the discrete logarithm is also occasionally referred to as "Shor's Algorithm."


title: Shor's algorithm

text: The order-finding problem can also be viewed as a hidden subgroup problem. To see this, consider the group of integers under addition, and for a given formula_300 such that: formula_301, the function


title: Shor's algorithm

text: For any finite abelian group G, a quantum algorithm exists for solving the hidden subgroup for G in polynomial time.


title: Kayla Mueller

text: Kayla Jean Mueller (August 14, 1988 – February 6, 2015) was an American human rights activist and humanitarian aid worker from Prescott, Arizona. She was taken captive in August 2013 in Aleppo, Syria, after leaving a Doctors Without Borders hospital. Media had long reported that a 26-year-old American aid worker was being held by ISIS without naming her, at her family's request. In 2015, she was killed in uncertain circumstances. The operation that killed ISIS leader Abu Bakr al-Baghdadi was named Operation Kayla Mueller, in her honour.


title: Kayla Mueller

text: Mueller was a native of Prescott, Arizona. After graduating from Tri-City College Prep High School in 2007, she attended Northern Arizona University in Flagstaff to study political science.


title: Kayla Mueller

text: Mueller was a practicing Christian. As a college student, she was active in an ecumenical Christian campus ministry, United Christian Ministries. She also practiced Bhakti yoga.


title: Kayla Mueller

text: Mueller supported a variety of humanitarian aid and human rights initiatives. Her involvement in human rights activism and humanitarian aid included working in India with Tibetan refugees. Her work in the Middle East included volunteering for the pro-Palestinian activist group the International Solidarity Movement and helping African refugees in Israel with the African Refugees Development Center. Other humanitarian and activist causes Mueller was involved in at home and abroad were Vrindavan Food For Life, which provides free food, education, and medical care for those in need; and during college, Food Not Bombs.


title: Kayla Mueller

text: Mueller started working in southern Turkey in December 2012, where she was assisting Syrian refugees. On August 3, 2013, she traveled to the northern Syrian city of Aleppo with her boyfriend, a Syrian contractor hired to install communications equipment at the Doctors Without Borders hospital in Aleppo.


title: Kayla Mueller

text: Although Mueller had been working with international aid agency Support to Life in Turkey, this was not a work-related trip. Doctors Without Borders staff were "flabbergasted" at Mueller's arrival, fearing for her safety, as Syria was dangerous for international aid workers and amid a civil war. The following day, Doctors Without Borders staff tried to drive Mueller to a bus station so she could travel back to Turkey. The car was ambushed, and both Mueller and her boyfriend were abducted by ISIS. Her boyfriend was freed 20 days later.


title: Kayla Mueller

text: The US military and Mueller's family attempted to rescue Mueller several times, devoting considerable resources to the search. In July 2014, US special operations forces (from Delta Force and 75th Ranger Regiment) raided an abandoned oil refinery near Raqqa in an unsuccessful attempt to find reporter James Foley (whom ISIS later murdered) and other hostages. The commandos found evidence that the hostages had recently been held there, finding writing on the cell walls and hair believed to be Mueller's, but the refinery was empty.


title: Kayla Mueller

text: A Mueller family spokesman said that Mueller's parents "often communicated with the White House in trying to free their daughter." In the summer of 2014, as other options were exhausted, Mueller's parents asked President Barack Obama in a letter to consider trading her for Aafia Siddiqui, a convicted terrorist serving 86 years in federal prison; Siddiqui's release has been a long-sought demand of ISIS and al-Qaeda. The proposed exchange did not take place; the Obama administration had also rejected demands from other militant groups to exchange Siddiqui.


title: Kayla Mueller

text: Catherine Herridge of Fox News reported via anonymous sources that the location of Mueller and other American hostages was known by the White House in May 2014. However, a decision regarding a rescue mission was not made for seven weeks, because the White House had asked for further intelligence to be obtained. By that time, the hostages had been dispersed.


title: Kayla Mueller

text: In August 2015, the "New York Times" reported that Mueller had been forced into marriage to Abu Bakr al-Baghdadi, the so-called leader of ISIS, who raped her repeatedly. She had also been tortured.


title: Kayla Mueller

text: It had also been reported in May 2015 that Mueller was a "personal captive" of Abu Sayyaf. In August 2015, ABC News reported that Abu Sayyaf's widow, Umm Sayyaf, had confirmed that al-Baghdadi had been Mueller's primary abuser.


title: Kayla Mueller

text: Fellow captives who were released or escaped reported Mueller's strength and compassion. She was concerned for the safety of others, even intentionally passing up an escape attempt to give other women a better chance at freedom. After months of prolonged torture and abuse, ISIS members, who are known for their persecution of Christians, attempted to use her as propaganda by claiming she had abandoned the Christian faith in front of other prisoners who were men, but she denied it. Daniel Rye Ottosen, a Danish freelance photographer and fellow captive, said that the men in the room "...were impressed by the strength that she showed in front of us. That was very clear."


title: Kayla Mueller

text: On February 6, 2015, a media account affiliated with ISIS released a statement claiming that a female American hostage held by the group was killed by one of around a dozen Jordanian airstrikes in Raqqa. The statement came just days after the release of a video showing the burning alive of a Jordanian fighter pilot, Muath al-Kasasbeh, by ISIS and the subsequent execution of Sajida Mubarak Atrous al-Rishawi and other prisoners of Jordan. The statement was later translated by the SITE Intelligence Group, identifying the hostage as Mueller. Another statement by ISIS claimed Mueller was killed in an American airstrike.


title: Kayla Mueller

text: As part of the statement, ISIS published a photo of a damaged building, alleging that Mueller had been killed in a Jordanian airstrike of the building, where she had been left alone with no guards, but no proof of death was provided. The Pentagon agreed the building was one hit in the bombings, but disputed that Mueller or any other civilian had been inside at the time. The site had been bombed by the coalition twice before, and was targeted again because ISIS soldiers sometimes return to bombed sites, thinking the coalition would not bomb those sites again, according to Pentagon spokesman John Kirby. After this, Mueller's name was released by American and other media with the family's consent.


title: Kayla Mueller

text: On February 10, 2015, Mueller's family announced ISIS had confirmed her death to them in an e-mail, with three photographs of her dead body, bruised on the face and wearing a black hijab. National Security Council spokeswoman Bernadette Meehan said this message was authenticated by the intelligence community.


title: Kayla Mueller

text: On August 24, 2016, ABC News reported that Doctors Without Borders had declined to help negotiate her release. Doctors Without Borders then issued their version of the story in a statement (since deleted but archived) to the effect that they felt constrained in their available actions by the complexities of the larger situation and their general lack of expertise in hostage negotiation.


title: Kayla Mueller

text: In November 2019, the "New York Times" reported that Mueller may have been executed on the orders of Abu Bakr al-Baghdadi, according to reports from the wife of a close associate of his.


title: Kayla Mueller

text: On February 8, 2016, Umm Sayyaf was charged by American prosecutors in Virginia with providing material support to a foreign terrorist organization that resulted in a person's death. The federal charge carries a maximum sentence of life imprisonment. She remains in Iraqi custody on terrorism-related charges.


title: Kayla Mueller

text: In October 2020, Alexanda Kotey and El Shafee Elsheikh were extradited to the US and charged in Mueller's murder. Kotey pleaded guilty to four counts of hostage-taking resulting in death and was sentenced to life in prison without parole in 2022. Elsheikh was convicted at trial and also faces a mandatory life sentence.


title: Kayla Mueller

text: Mueller's parents reportedly implored ISIS to contact them, as they hoped their daughter might still be alive. “We have sent you a private message and ask that you respond to us privately,” Carl and Marsha Mueller said in a statement. They said they had not talked to the media, as ISIS warned them not to. Later, in an interview on "The Today Show", Carl Mueller expressed his frustration with the Obama administration over the way it conducted negotiations with their daughter's captors, and their policy of not paying ransom money for hostages—even though US citizens living abroad are required to pay taxes to the US. "We understand the policy about not paying ransom, but on the other hand, any parents out there would understand that you would want anything and everything done to bring your child home," Mueller said. "And we tried, and we asked. But they put policy in front of American citizens' lives. And it didn't get it changed." On February 6, 2020, Kayla Mueller's parents were publicly recognized by President Donald Trump during the 2020 State of the Union Address.


title: Kayla Mueller

text: Kayla Mueller was a member of an ecumenical Christian campus ministry, United Christian Ministries, which held ties to the Presbyterian Church. The denomination stated: "Each of us is called in our own way to make the world a better place and manifest the love of Christ to those around us. Kayla did that in a very profound and tangible way, and my hope is that she inspires others to alleviate suffering in the world and in their own communities."


title: Kayla Mueller

text: Hare Krishna Food For Life, for which Mueller was an active volunteer, stated: "Our prayers go out to her family and friends, a wonderful soul who was making a real difference in many people’s lives. Sure she has gone on to a better life in the spiritual realm."


title: Kayla Mueller

text: An American official cautioned that without proof of Mueller's death, the statement by ISIS could be a ploy to cause the Jordanians and the rest of the American-led coalition to refrain from any heavier airstrikes.


title: Kayla Mueller

text: Jordan's Foreign Minister Nasser Judeh called ISIS's claim "an old and sick trick" on Twitter. "So they behead innocent #US #UK & Japan hostages & BURN a brave #Jordan pilot ALIVE & now a hostage is killed by an airstrike? Sure! Sick!", he said. He further tweeted: "An old and sick trick used by terrorists and despots for decades: claiming that hostages human shields held captive are killed by air raids." Later upon confirmation of Mueller's death, he tweeted: "Saddened & angered to hear news confirming killing of #US hostage #kaylaMueller. Yet another ugly example of these terrorists' brutality."


title: Kayla Mueller

text: After many Western news outlets cast doubt on the claim of the hostage death and the extremists' ability to identify Jordanian and U.S.-made F-16s flying at high altitudes, Jordan dismissed the claim of a killed hostage as an ISIS publicity stunt and a lie, as the group is known for its propaganda techniques.


title: Kayla Mueller

text: After Mueller's family confirmed her death, President Barack Obama said "[Mueller] represents what is best about America, and expressed her deep pride in the freedoms that we Americans enjoy, and that so many others strive for around the world." United States Secretary of State John Kerry issued a statement saying "ISIS, and ISIS alone, is the reason Kayla is gone."


title: Kayla Mueller

text: The Pentagon declined to investigate whether Mueller was killed by the coalition airstrike. Policy dictates that the US only investigates reports of civilian casualties when they come from a "credible source", which ISIS is not.


title: Kayla Mueller

text: In October 2015, Mueller became the first person to be posthumously inducted into Northern Arizona University's college of Social and Behavioral Sciences Hall of Fame. She was recognized as an outstanding alumna for her humanitarian aid work.


title: Kayla Mueller

text: The military operation that killed ISIS leader Abu Bakr al-Baghdadi on October 27, 2019 was code-named after Kayla Mueller, according to several sources.


title: Kayla Mueller

text: On February 4, 2020, during the 2020 State of the Union Address, President Donald Trump personally recognized Mueller as part of his discussion of tackling ISIS insurgent forces and the military operations directed at taking down prominent members under his administration, directly referencing the Barisha raid in which ISIS leader Abu Bakr al-Baghdadi was killed. During this recognition, he also directly referenced Mueller's parents, Carl and Marsha Mueller, who were in attendance at the address. Several aspects of Mueller's life were mentioned, including her college career, her devotion to humanitarian aid and personal faith, as well as her capture, torture and eventual death at the hands of ISIS terrorists.


title: Kayla Mueller

text: Carl and Marsha Mueller spoke at the 2020 Republican National Convention, where they described their daughter's ordeal and praised the Trump administration's coordination of the raid that led to the killing of al-Baghdadi.


title: Tony La Russa

text: Anthony La Russa Jr. (; born October 4, 1944) is an American former professional baseball player, coach, and manager. His MLB career has spanned from 1963 to 2022, in several roles. He is the former manager of the St. Louis Cardinals, Oakland Athletics, and Chicago White Sox. In 33 years as a manager, La Russa guided his teams to three World Series titles, six league championships, and 13 division titles. His managerial total of 2,902 MLB wins is second only to Connie Mack's.


title: Tony La Russa

text: As a player, La Russa made his major league debut in 1963 and spent parts of five major league seasons with the Kansas City / Oakland Athletics, Atlanta Braves, and Chicago Cubs. After a shoulder injury during the 1964–65 off-season, he returned to college and received a degree from the University of South Florida before playing much of the remainder of his career in the minor leagues until retiring in 1977. He then earned a Juris Doctor degree from Florida State University.


title: Tony La Russa

text: Named manager of the White Sox in the middle of the 1979 season, La Russa guided the team to an American League West division title four seasons later. In the middle of the 1986 season, he was fired by the White Sox and hired less than three weeks later by the Athletics. La Russa led the A's to three consecutive American League championships from 1988 to 1990 and the 1989 World Series title. He left Oakland after the 1995 season to manage the Cardinals, whom he led the team to three National League championships and the and 2011 World Series titles. La Russa retired after winning the 2011 title and 34 seasons as a major league manager. Three months later, he accepted a position helping fellow former manager Joe Torre, the executive vice president for MLB operations. In 2014, he became the chief baseball officer for the Arizona Diamondbacks. In November 2019, he joined the Los Angeles Angels as a senior advisor of baseball operations. In the 2021 offseason, he was named the manager of the White Sox; he retired in 2022 due to health concerns.


title: Tony La Russa

text: In 2013, La Russa was unanimously elected to the Hall of Fame by the 16-member Veterans Committee. The induction ceremony was held at Cooperstown, New York, on July 27, 2014. On August 16, 2014, he was inducted into the St. Louis Cardinals Hall of Fame Museum.


title: Tony La Russa

text: Born in Tampa, Florida, on October 4, 1944, to Anthony and Olivia (Cuervo) La Russa, Anthony Jr.'s paternal grandparents had emigrated from Italy (Sicily) and his mother's family from Spain. He was raised in nearby Ybor City, Florida, where his parents had met while they were working in the local cigar factory.


title: Tony La Russa

text: The La Russa family moved to West Tampa, Florida, where Tony played American Legion baseball and PONY League baseball alongside teammate Lou Piniella.


title: Tony La Russa

text: After graduating from Jefferson High School in Tampa, La Russa was signed by the Kansas City Athletics in June 1962 as a middle infielder, with a clause that the Athletics pay for his college education at the University of South Florida.


title: Tony La Russa

text: La Russa made his major league debut with the Kansas City A's on May 10, 1963, after having played 76 games with A's affiliates Binghamton Triplets and Daytona Beach Islanders in 1962. He spent the entire 1963 season in the majors, as was required by his signing as a "bonus baby". He had suffered an off-season shoulder injury while playing softball with friends, and this limited him to only 34 games in 1963, in which he hit .250. The injured shoulder bothered him through the remainder of his playing career.


title: Tony La Russa

text: Over the next six seasons, La Russa spent most of his time in the minor leagues. He made it back up to the A's, which had since moved to Oakland, in 1968 and 1969. He spent the entire 1970 season with the A's, and then late in 1971 the A's traded him to the Atlanta Braves. His final big league playing stop was with the Chicago Cubs, where he appeared as a pinch runner in one game, on April 6, 1973, scoring the walk-off winning run. He also spent time in the organizations of the Pittsburgh Pirates, Chicago White Sox, and St. Louis Cardinals.


title: Tony La Russa

text: In total, he played 132 major-league games, 40 in the starting lineup. He went 35-for-176, for a batting average of .199. His 23 walks pushed his on-base percentage to .292. He had 7 RBI and scored 15 runs. He made 63 appearances at second base, 18 at shortstop, and two at third base, fielding .960 in 249 total chances and participating in 34 double plays.


title: Tony La Russa

text: Having started coursework following his A's signing in 1962, La Russa graduated from the University of South Florida in 1969 with a degree in Industrial Management. He earned a Juris Doctor (J.D.) degree from Florida State University College of Law in 1978. and was admitted to the Florida Bar on July 30, 1980. He is associated with a Sarasota law firm although he is not eligible to practice at this time. La Russa has been quoted as saying, "I decided I'd rather ride the buses in the minor leagues than practice law for a living." Shortly before graduating from FSU College of Law, La Russa spoke with one of his professors about his post-graduation plans, indicating to his professor that he had an opportunity to coach in the minor leagues and asking his professor what he should do. La Russa's professor responded, "Grow up, you're an adult now, you're going to be a lawyer."


title: Tony La Russa

text: He is one of a select number of major league managers in baseball history who have graduated from law school or passed a state bar exam; others include James Henry O'Rourke (Buffalo Bisons, 1881–84, Washington Senators, 1893), John Montgomery Ward (New York Giants, Brooklyn and Providence, late 1800s), Hughie Jennings (Detroit, 1907–20, New York Giants, 1924), Miller Huggins (St. Louis Cardinals and New York Yankees, 1913–29), Muddy Ruel (St. Louis Browns, 1947), Jack Hendricks (St. Louis Cardinals, 1918, Cincinnati, 1924–29), and Branch Rickey (St. Louis Browns, 1913–15, St. Louis Cardinals, 1919–25).


title: Tony La Russa

text: La Russa credits Loren Babe and Paul Richards of the White Sox organization for helping him to become a manager. The White Sox gave La Russa his first managerial opportunity in 1978 by naming him skipper of their Double-A affiliate, the Knoxville Sox of the Southern League. La Russa spent a half-season at Knoxville before being promoted to the White Sox coaching staff when owner Bill Veeck changed managers from Bob Lemon to Larry Doby. Doby was fired at the end of the season; Don Kessinger, former star shortstop of the crosstown Cubs, was named the White Sox' player-manager for 1979, and La Russa was named manager of the Triple-A Iowa Oaks of the American Association, choosing to manage in the minors after the White Sox had offered him his same major league coaching role.


title: Tony La Russa

text: The White Sox fired Kessinger with a 46–60 record two-thirds of the way through the 1979 season and replaced him with La Russa. The White Sox played .500 baseball (27–27) for the rest of the 1979 campaign. LaRussa, at 34, was the youngest manager in the major leagues. He credits Paul Richards with first inspiring him to believe he could succeed as a major league manager. La Russa was named American League Manager of the Year in 1983, when his club won the AL West but fell to the Baltimore Orioles in the American League Championship Series.


title: Tony La Russa

text: The White Sox fired La Russa after the club got off to a 26–38 start in 1986. In 1986, General Manager Roland Hemond, who had hired La Russa as White Sox Manager, was replaced by the White Sox' broadcaster Ken Harrelson as general manager. Harrelson then fired both La Russa and coach Dave Duncan during the season. Ironically, Harrelson and La Russa were teammates at Binghamton in 1962. In later years, White Sox Owner Jerry Reinsdorf expressed regret for allowing La Russa to be fired. He finished his White Sox career with a 522–510 regular season record and a 1–3 postseason record. Decades later, Hemond would say of La Russa: “Tony La Russa is one of the most brilliant managers that I ever encountered in my baseball career. He saw things other people didn’t see. There were some managers who thought he was out of line with what he was trying to do, but later on they had to respect him because it was working. There’s no question he changed the way the game is played.” La Russa is also commonly credited for the advent of the modern specialized bullpen.


title: Tony La Russa

text: La Russa had a break of less than three weeks before his old club, the Athletics, called him to take over as manager. LaRussa and Duncan both joined the A's, inheriting a team that was 31–52 and in 7th place. They went 45–34 the rest of the season to finish in 3rd place in 1986.


title: Tony La Russa

text: La Russa managed the Oakland A's to three consecutive American League Pennants and World Series appearances from 1988 to 1990. The A's swept the earthquake-delayed Bay Area 1989 World Series from the rival San Francisco Giants. The previous season, the A's had lost the 1988 World Series to manager Tommy Lasorda and the Los Angeles Dodgers. In their third consecutive appearance, the favored A's faced manager Lou Piniella, LaRussa's childhood teammate and friend, and his Cincinnati Reds in the 1990 World Series. La Russa's Athletics were swept by the Cincinnati Reds. In both 1988 and 1990 the A's had swept the Boston Red Sox 4–0 in the ALCS, after winning 100 and 99 games respectively. The 1988 World Series was made famous by the Kirk Gibson game-winning home run off of A's Hall of Fame closer Dennis Eckersley.


title: Tony La Russa

text: La Russa earned two American League Manager of the Year awards with the A's, in 1988 and 1992, giving him three AL awards, the latter after again winning the Western Division.


title: Tony La Russa

text: After the 1995 season, in which the A's finished 67–77, the Haas family, with whom La Russa had a close personal relationship, sold the team after the death of patriarch Walter A. Haas, Jr. In the off-season, La Russa left Oakland to take over for the fired Joe Torre as manager of the St. Louis Cardinals. La Russa had a 798–673 regular season record and a 19–13 postseason record with Oakland.


title: Tony La Russa

text: In his first campaign with the Cardinals in 1996, La Russa clinched the National League Central Division title (and also finished National League runner-up), a feat his clubs repeated in 2000, 2001, 2002 (his fourth Manager of the Year award), 2004, 2005, 2006, and 2009. The Cardinals also tied for the National League Central crown with the Houston Astros in 2001. He became the first manager to win the award four times. La Russa's fourth Manager of the Year award was arguably the most emotional; La Russa led the Cardinals to the National League Championship Series (where they would ultimately lose in five games to the San Francisco Giants) in a year in which the Cardinals were traumatized by the deaths of beloved Hall of Fame broadcaster Jack Buck and 33-year-old pitcher Darryl Kile just four days later. On September 10, 2003, he won his 2,000th career game as a manager against the Colorado Rockies, becoming the seventh to reach the mark in Major League Baseball.


title: Tony La Russa

text: The 2004 edition of the club played one of the finest seasons in Cardinals' history, as they won 105 games. After a regular season in which the Cardinals led the NL in runs scored (855) while allowing the fewest (659), La Russa's Cardinals defeated the Los Angeles Dodgers in the National League Division Series, 3 games to 1. St. Louis then took on the Houston Astros in the National League Championship Series. In a tense series with opposing pitcher Roger Clemens at the top of his game, Cardinals third baseman Scott Rolen hit a game-winning two-run home run off Clemens in Game 7 following Jim Edmonds' rally saving catch. This home run sent the Cardinals to the World Series for the first time since 1987. However, they were swept in four games by an historic Boston Red Sox team that had just surmounted a 3–0 deficit against the New York Yankees and captured their first championship since 1918.


title: Tony La Russa

text: When he came to St. Louis, La Russa wore number 10 to symbolize the team's drive to their 10th championship and pay tribute to Anderson, who wore number 10 while manager of the Cincinnati Reds. After winning the championship, he chose to continue wearing number 10 to pay tribute to Anderson.


title: Tony La Russa

text: It was as a player with the A's that La Russa first met catcher Dave Duncan, who would join his coaching staff in Chicago in 1983. The two worked together on every La Russa-managed team thereafter, and he often credits Duncan as playing a key role in his success.


title: Tony La Russa

text: La Russa led the Cardinals to the 2011 World Series, after defeating the Philadelphia Phillies in the NLDS (3–2), and then the Milwaukee Brewers in the NLCS (4–2). The Cardinals defeated the Texas Rangers in Game 7 of the World Series to win the franchise's 11th World Championship, and the third of La Russa's managerial career. He passed Bobby Cox for 2nd place on the all-time postseason wins list with his 68th win in Game 3. Three days following the World Series win, La Russa announced his retirement, ranking second all-time in postseason wins with 70, third all-time with 2,728 regular season wins, second with 5,097 games managed, and second with 33 years (tied) managing with John McGraw. He finished his Cardinals career with a 1408–1182 regular season record and 50–42 postseason record.


title: Tony La Russa

text: La Russa also became the first manager in Major League Baseball history to retire in the same season after winning a World Series title. Even though he had retired, La Russa managed the National League All Stars in the 2012 MLB All-Star Game for the final time as a member of the Cardinals. The National League won 8–0.


title: Tony La Russa

text: After serving in various executive roles for MLB and for several teams, La Russa was announced as the manager of the White Sox on October 29, 2020, replacing Rick Renteria. At 76, La Russa became the oldest manager in MLB. He also became the first manager in baseball history to return to managing after being elected to the National Baseball Hall of Fame as a manager. A. J. Hinch was known to be another finalist, but was hired by the Detroit Tigers instead when the White Sox decided on La Russa.


title: Tony La Russa

text: On June 6, 2021, La Russa passed John McGraw for second place in all-time managerial wins. La Russa led the White Sox to their first AL Central division title since 2008. They lost to the Houston Astros in the 2021 ALDS three games to one.


title: Tony La Russa

text: In 2022, at 77 years of age, La Russa was the oldest manager in major league baseball, four years older than Dusty Baker. On August 31, 2022, La Russa was ruled out indefinitely after undergoing tests on his heart. On October 3, La Russa announced that he was retiring for good, effective immediately.


title: Tony La Russa

text: Shortly after his retirement from the playing field, La Russa took a position with MLB assisting former managerial rival Joe Torre in matters of on-field discipline. He held this position for more than two seasons.


title: Tony La Russa

text: On May 17, 2014, La Russa accepted a position as Chief Baseball Officer for the Arizona Diamondbacks to oversee the entire baseball operations department.


title: Tony La Russa

text: After joining the Diamondbacks in 2014, La Russa was reunited with former assistants Duncan and McKay, and the general manager who hired him to manage the White Sox in 1979, Roland Hemond. On December 4, 2015, the Diamondbacks agreed to a six-year contract with free agent pitcher Zack Greinke worth a total of $206.5 million. At that time, it held the highest annual average value in MLB, exceeding $34.4 million per year, and was also the largest contract by total value in team history. La Russa was demoted to Chief Baseball Analyst/Advisor with the Diamondbacks following a disappointing 93-loss season in 2016, which also resulted in the firing of General Manager Dave Stewart and manager Chip Hale. Following the 2017 season, La Russa resigned.


title: Tony La Russa

text: In November 2017, the Boston Red Sox announced that La Russa had joined the team as vice president and special assistant to Dave Dombrowski, the president of baseball operations. In making the announcement, the Red Sox indicated that La Russa would assist with player development, serve as an advisor to the team's coaches at the major and minor league levels, and serve as a consultant for Alex Cora, the team's major league manager. La Russa worked with the Red Sox for two seasons, and after Dombrowski was released by the Red Sox during the 2019 season, the Los Angeles Angels hired La Russa as senior adviser for baseball operations in November 2019.


title: Tony La Russa

text: La Russa is second in major league history in victories as a manager (2,901), trailing only Connie Mack (3,731) and surpassing John McGraw (2,763) on June 6, 2021. He managed 5,097 games, joining Mack as the second manager or coach in American sports history to reach 5,000 games. In 2004, he became the sixth manager in history to win pennants with both American and National League teams; in 2006 he became the first manager ever to win multiple pennants in both leagues and the second manager to win the World Series in both leagues. La Russa has also joined Mack as the second manager to win World Series titles in three decades and to win pennants in four. He is one of only four managers to be named Manager of the Year in both of baseball's major leagues.


title: Tony La Russa

text: La Russa is the winningest manager in St. Louis Cardinals history, with 1,408 wins and 1,182 losses (.544) and one tie, managing the Cardinals from 1996 to 2011. He was 522–510–3 (.506) with the Chicago White Sox 1979–1986, leading the club to its first postseason appearance in 24 years in 1983, and 798–673 (.542) with the Oakland Athletics 1986–1995, winning three consecutive AL pennants from 1988 to 1990; he also holds the record for victories by an Athletics manager since the franchise relocated to Oakland in 1968.


title: Tony La Russa

text: La Russa became the leader in wins by Cardinals' managers on August 31, 2007, when the Cardinals defeated the Cincinnati Reds 8–5, passing Red Schoendienst (1,041–955) to take the title. He managed his 2,500th win against the Royals at Kansas City on June 21, 2009, becoming only the third manager to attain that win level after Mack and McGraw.


title: Tony La Russa

text: After the retirement of Atlanta Braves manager Bobby Cox in 2010, La Russa was the longest tenured manager in Major League Baseball, and with the resignation of longtime NBA head coach Jerry Sloan from the Utah Jazz on February 10, 2011, La Russa had been the longest tenured bench boss among all the Big Four sports leagues, until his retirement following his 2011 World Series victory with the Cardinals.


title: Tony La Russa

text: La Russa became eligible as a candidate for the Baseball Hall of Fame in 2013 through voting by the Expansion Era Committee for induction in 2014. On November 4, 2013, La Russa's inclusion on the Expansion Era ballot was announced with fellow former Cardinals Ted Simmons, Joe Torre and Dan Quisenberry. The Baseball Writers' Association of America's Historical Overview Committee listed the candidates on the ballot and voted the following December 8. La Russa was inducted into the 2014 Hall of Fame class as a manager.


title: Tony La Russa

text: In January 2014, the Cardinals announced La Russa among 22 former players and personnel to be inducted into the St. Louis Cardinals Hall of Fame Museum for the inaugural class of 2014.


title: Tony La Russa

text: In 2012, he became the second manager to manage the All-Star Game (2012) after retiring since John McGraw in 1933. With his All-Star Game 8–0 win in Kansas City, he became the first manager to win an All-Star Game in both leagues.


title: Tony La Russa

text: La Russa and his second wife, Elaine, are the founders of Tony La Russa's Animal Rescue Foundation, headquartered in Walnut Creek, California, which saves abandoned and injured animals as well as running programs to bring dogs and cats to abused children, hospital patients, seniors and shut-ins. La Russa is a vegetarian. The La Russas have two daughters, Bianca and Devon, and reside in Alamo, California.


title: Tony La Russa

text: La Russa has two older daughters, Andrea and Averie, from his first marriage to Luzette Sarcone. La Russa and Sarcone divorced in 1973 and Sarcone received full custody of their daughters. La Russa married Elaine Coker shortly after his divorce to Sarcone became official.


title: Tony La Russa

text: La Russa is also personal friends with celebrities outside the sports world, such as pianist and songwriter Bruce Hornsby, Bruce Springsteen, jazz bassist Christian McBride, and heavy metal musician Robb Flynn from the band Machine Head. In 2007, at a concert in San Francisco on La Russa's birthday, Hornsby played a comedic song he named "Hooray For Tony". The original song, titled "Hooray For Tom", is La Russa's favorite Hornsby song. In the "Hooray For Tony" version, Hornsby mentions the "Bash Brothers" Mark McGwire and José Canseco (from La Russa's days as the manager of the Oakland A's), Albert Pujols, Jim Edmonds, ARF, and La Russa's World Series Championships.


title: Tony La Russa

text: La Russa is a graduate of the University of South Florida (Bachelor of Arts in industrial management, 1969) and the Florida State University College of Law (Juris Doctor, 1978).


title: Tony La Russa

text: La Russa has Italian and Spanish ancestry, and speaks fluent Spanish. His father's parents were migrants from the Italian island of Sicily and his mother's family originated from Spain. Having a father who could speak Spanish and Italian and a mother who could speak Spanish, La Russa claimed in a 2008 interview that "Spanish was my first language". He was inducted into the National Italian American Sports Hall of Fame in 1998. La Russa was also inducted into the Hispanic Heritage Baseball Museum Hall of Fame on April 11, 2008, in a pregame ceremony at AT&T Park.


title: Tony La Russa

text: La Russa and his family had an uncredited extra role in the film "Angels in the Outfield". The La Russas also made an appearance in an episode of "Housecat Housecall", a reality show on Animal Planet presented by Purina Cat Chow, during the show's third season, which began on June 5, 2010. In 1980, La Russa appeared as a contestant on the game show "To Tell The Truth", and helped fool the celebrity panel.


title: Tony La Russa

text: In June 2010, La Russa was asked about a tea party protest taking place during his game against the Arizona Diamondbacks that criticized the Diamondbacks' position against the controversial new Arizona immigration statute. La Russa expressed support for the Tea Partiers' right to free speech to protest at the ballpark. He also stated, "I'm actually a supporter of what Arizona is doing... you know if people don't fix their problems they have to take care of it themselves".


title: Tony La Russa

text: La Russa and Pujols attended Glenn Beck's "Restoring Honor" rally on August 28, 2010, in Washington, D.C., while the Cardinals were in town for a series against the Washington Nationals. La Russa decided to attend after being told by Beck that the rally was not political in nature.


title: Tony La Russa

text: Throughout the 2011 season, La Russa struggled with shingles. He originally disclosed that it was conjunctivitis, however, on May 10, after a visit to the Mayo Clinic (Scottsdale, Arizona), he disclosed that he was dealing with a case of shingles, and had to take off a few days for treatment and rest. Because bench coach Joe Pettini was named 'acting manager' instead of 'interim manager', La Russa was credited for all wins and losses during his absence.


title: Tony La Russa

text: On June 4, 2009, the "St. Louis Post-Dispatch" reported that La Russa had sued the online social network platform Twitter the previous month for a fake page established under his name on the site. La Russa claimed that he had "suffered significant emotional distress (and) damage to reputation" because of the profile. The fake profile made several "distasteful references" to La Russa and his team, according to the suit. Twitter's terms of service forbids impersonation directly, stating that users "may not impersonate others through the Twitter service in a manner that does or is intended to mislead, confuse, or deceive others." Reports that Twitter had settled ("the first celebrity lawsuit against the 32-million-user site", according to the "Wall Street Journal") were rebuffed on the official Twitter blog. La Russa eventually filed a request to dismiss the suit, while Twitter, whether in regard to the suit or not, continued improving its privacy protections.


title: Tony La Russa

text: On March 22, 2007, La Russa was arrested in Jupiter, Florida, for suspicion of driving under the influence of alcohol. He was found asleep at the wheel of his running SUV with the car in park at an intersection. He was booked at the Palm Beach County Jail and blew a .093 blood alcohol content, above the legal limit of .08. Calling his arrest on the DUI charge an "embarrassment", La Russa apologized to "anyone who is close to me, members of the Cardinals organization, our fans." He was defended by the organization and some players, such as Albert Pujols. On November 28, 2007, La Russa pleaded guilty to DUI, saying that it was in the best interest of all concerned. In a statement released by his attorney, La Russa said, "I accept full responsibility for my conduct, and assure everyone that I have learned a very valuable lesson and that this will never occur again".


title: Tony La Russa

text: In February 2020, La Russa was charged with DUI again, this time in Phoenix; the case was filed on October 28, 2020, a day before he was rehired by the White Sox.


title: Tony La Russa

text: In 1980, La Russa appeared as the subject in a round of the Robin Ward-hosted version of "To Tell The Truth". Panelist Nipsey Russell seemed perplexed that there was another Chicago baseball team besides the Cubs. One of the imposters before the panel, when queried as to who was his best player at the moment, swiftly responded Todd Cruz. This resulted in him receiving votes. But he did not completely stump the panel.


title: Tony La Russa

text: In 2012, La Russa released his "New York Times" bestselling memoir, "One Last Strike", which recounts his legendary last season as manager of the St. Louis Cardinals and their remarkable journey to becoming the 2011 World Series Champions from ten and a half games back.


title: Tony La Russa

text: In 2005, La Russa was the focus of a book by sportswriter Buzz Bissinger. Bissinger's "Three Nights in August" delves into La Russa's role as manager during a 3-game series in 2003 between his Cardinals and manager Dusty Baker's Chicago Cubs, their longtime rivals. The book received much praise from both fans and critics, though some complained that Bissinger sets out to glorify La Russa's "old school" managerial style as a direct challenge to the statistical analysis theses of Michael Lewis's 2004 book "Moneyball".


title: Tony La Russa

text: As David Leonhardt of "The New York Times" wrote of the "stats vs. hunches" debate in an August 29, 2005 piece, ""what makes this fight truly comparable to those that periodically roil the world of art history or foreign policy is that the differences between the sides are not as great as the sniping between them suggests. La Russa spends much of his time jotting down information on index cards and studying statistics in his office.""


title: Tony La Russa

text: George Will's book "" likewise depicted La Russa and his long-time pitching coach Dave Duncan as making more use of statistical analysis than any other team in the major leagues (this book was published in 1990, more than a decade before the "Moneyball" revolution).


title: Tony La Russa

text: La Russa also provided the AI for a series of successful video games, "Tony La Russa Baseball" (1991–1997). The games won numerous awards and featured "new" statistics selected with La Russa (and provided by prominent sabermetrics authors John Thorn and Pete Palmer) as tools for players as they managed their teams.


title: Benign paroxysmal positional vertigo

text: Benign paroxysmal positional vertigo (BPPV) is a disorder arising from a problem in the inner ear. Symptoms are repeated, brief periods of vertigo with movement, characterized by a spinning sensation upon changes in the position of the head. This can occur with turning in bed or changing position. Each episode of vertigo typically lasts less than one minute. Nausea is commonly associated. BPPV is one of the most common causes of vertigo.


title: Benign paroxysmal positional vertigo

text: BPPV is a type of balance disorder along with labyrinthitis and Ménière's disease. It can result from a head injury or simply occur among those who are older. Often, a specific cause is not identified. When found, the underlying mechanism typically involves a small calcified otolith moving around loose in the inner ear. Diagnosis is typically made when the Dix–Hallpike test results in nystagmus (a specific movement pattern of the eyes) and other possible causes have been ruled out. In typical cases, medical imaging is not needed.


title: Benign paroxysmal positional vertigo

text: BPPV is easily treated with a number of simple movements such as the Epley maneuver ( in case of diagonal/rotational nystagmus ), the Lempert maneuver ( in case of horizontal nystagmus ), the deep head hanging maneuver ( in case of vertical nystagmus ) or sometimes the less effective Brandt–Daroff exercises. Medications, including antihistamines such as meclizine, may be used to help with nausea. There is tentative evidence that betahistine may help with vertigo, but its use is not generally needed. BPPV is not a serious medical condition, but may present serious risks of injury through falling or other spatial disorientation-induced accidents.


title: Benign paroxysmal positional vertigo

text: When untreated, it might resolve in days to months; however, it may recur in some people. One can needlessly suffer from BPPV for years despite there being a simple an very effective cure. Short-term self-resolution of BPPV is unlikely because the effective cure maneuvers induce strong vertigo which the patient will naturally resist and not accidentally perform.


title: Benign paroxysmal positional vertigo

text: The first medical description of the condition occurred in 1921 by Róbert Bárány. Approximately 2.4% of people are affected at some point in time. Among those who live until their 80s, 10% have been affected. BPPV affects females twice as often as males. Onset is typically in people between the ages of 50 and 70.


title: Benign paroxysmal positional vertigo

text: Many people will report a history of vertigo as a result of fast head movements. Many are also capable of describing the exact head movements that provoke their vertigo. Purely horizontal nystagmus and symptoms of vertigo lasting more than one minute can also indicate BPPV occurring in the horizontal semicircular canal.


title: Benign paroxysmal positional vertigo

text: The spinning sensation experienced from BPPV is usually triggered by movement of the head, will have a sudden onset, and can last anywhere from a few seconds to several minutes. The most common movements people report triggering a spinning sensation are tilting their heads upward in order to look at something and when rolling over in bed.


title: Benign paroxysmal positional vertigo

text: People with BPPV do not experience other neurological deficits such as numbness or weakness. If those symptoms are present, a more serious etiology, such as posterior circulation stroke or ischemia, must be considered.


title: Benign paroxysmal positional vertigo

text: The most significant symptom is nystagmus as it is essential to determine the kind of nystagmus ( horizontal, vertical, or diagonal ) to select the correct cure maneuver.


title: Benign paroxysmal positional vertigo

text: Within the labyrinth of the inner ear lie collections of calcium crystals known as otoconia or otoliths. In people with BPPV, the otoconia are dislodged from their usual position within the utricle, and over time, migrate into one of the three semicircular canals (the posterior canal is most commonly affected due to its anatomical position). When the head is reoriented relative to gravity, the gravity-dependent movement of the heavier otoconial debris (colloquially "ear rocks") within the affected semicircular canal causes abnormal (pathological) endolymph fluid displacement and a resultant sensation of vertigo. This more common condition is known as "canalithiasis". There is a direct link between the kind of nystagmus and which of the three semicircular canals is affected. With horizontal nystagmus ( left-right eye movement ) the horizontal ( also called lateral ) canal is affected, with vertical nystagmus ( up-down eye movement ) the superior ( also called anterior ) canal is affected, and with diagonal nystagmus ( diagonal or rotational eye movement ) the posterior canal is affected. Diagonal eye movement is easily confused with horizontal movement. This is important since it might result in selecting a wrong and thus ineffective cure maneuver.


title: Benign paroxysmal positional vertigo

text: In rare cases, the crystals themselves can adhere to a semicircular canal cupula, rendering it heavier than the surrounding endolymph. Upon reorientation of the head relative to gravity, the cupula is weighted down by the dense particles, thereby inducing an immediate and sustained excitation of semicircular canal afferent nerves. This condition is termed "cupulolithiasis".


title: Benign paroxysmal positional vertigo

text: There is evidence in the dental literature that malleting of an osteotome during closed sinus floor elevation, otherwise known as "osteotome sinus elevation" or "lift", transmits percussive and vibratory forces capable of detaching otoliths from their normal location and thereby leading to the symptoms of BPPV.


title: Benign paroxysmal positional vertigo

text: An episode of BPPV may be triggered by dehydration, such as that caused by diarrhea. For this reason, it commonly occurs in people with post-operative diarrhea induced by post-operative antibiotics.


title: Benign paroxysmal positional vertigo

text: BPPV is one of the most common vestibular disorders in people presenting with dizziness; a migraine is implicated in idiopathic cases. Proposed mechanisms linking the two are genetic factors and vascular damage to the labyrinth.


title: Benign paroxysmal positional vertigo

text: Although BPPV can occur at any age, it is most often seen in people older than the age of 60. Besides aging, there are no major risk factors known for BPPV, although previous episodes of head trauma, or the inner ear infection labyrinthitis, may predispose to the future development of BPPV.


title: Benign paroxysmal positional vertigo

text: The inside of the ear is composed of an organ called the vestibular labyrinth. The vestibular labyrinth includes three semicircular canals, which contain fluids and fine hairlike sensors that act as a monitor to the rotations of the head. Other important structures in the inner ear includes the otolith organs, the utricle and saccule, that contain calcium carbonate crystals(otoconia) that are sensitive to gravity.


title: Benign paroxysmal positional vertigo

text: The crystals may dislodge from the utricle (an otolith organ) and settle within the semicircular canals. When there is motion, the displaced otoconia shift within the endolymph of semicircular canals, causing an unbalanced (with respect to the opposite ear) stimulus, causing symptoms of BPPV.


title: Benign paroxysmal positional vertigo

text: The condition is diagnosed by the person's history, and by performing the Dix–Hallpike test or the roll test, or both. The patient can also be asked to induce vertigo by performing a movement that the patient knows to induce vertigo. The eyes of the patient can then easily be observed for which kind ( horizontal, vertical, or diagonal ) of nystagmus is present, to determine which semicircular canal ( horizontal, superior, or posterior ) is affected.


title: Benign paroxysmal positional vertigo

text: The Dix–Hallpike test is a common test performed by examiners to determine whether the posterior semicircular canal is involved. It involves a reorientation of the head to align the posterior semicircular canal (at its entrance to the ampulla) with the direction of gravity. This test will reproduce vertigo and nystagmus characteristic of posterior canal BPPV.


title: Benign paroxysmal positional vertigo

text: When performing the Dix–Hallpike test, people are lowered quickly to a supine position, with the neck extended by the person performing the maneuver. For some people, this maneuver may not be indicated, and a modification may be needed that also targets the posterior semicircular canal. Such people include those who are too anxious about eliciting the uncomfortable symptoms of vertigo, and those who may not have the range of motion necessary to comfortably be in a supine position. The modification involves the person moving from a seated position to side-lying "without" their head extending off the examination table, such as with Dix–Hallpike. The head is rotated 45 degrees away from the side being tested, and the eyes are examined for nystagmus. A positive test is indicated by the patient report of a reproduction of vertigo and clinician observation of nystagmus. Both the Dix–Hallpike and the side-lying testing position have yielded similar results, and as such the side-lying position can be used if the Dix–Hallpike cannot be performed easily.


title: Benign paroxysmal positional vertigo

text: The roll test can determine whether the horizontal semicircular canal is involved. The roll test requires the person to be in a supine position with their head in 30° of cervical flexion. Then the examiner quickly rotates the head 90° to the left side, and checks for vertigo and nystagmus. This is followed by gently bringing the head back to the starting position. The examiner then quickly rotates the head 90° to the right side and checks again for vertigo and nystagmus. In this roll test, the person may experience vertigo and nystagmus on both sides, but rotating toward the affected side will trigger a more intense vertigo. Similarly, when the head is rotated toward the affected side, the nystagmus will beat toward the ground and be more intense.


title: Benign paroxysmal positional vertigo

text: As mentioned above, both the Dix–Hallpike and roll test provoke the signs and symptoms in subjects with archetypal BPPV. The signs and symptoms people with BPPV experience are typically a short-lived vertigo and observed nystagmus. In some people, although rarely, vertigo can persist for years. Assessment of BPPV is best done by a medical health professional skilled in the management of dizziness disorders, commonly a physiotherapist, audiologist, or other physician.


title: Benign paroxysmal positional vertigo

text: The nystagmus associated with BPPV has several important characteristics that differentiate it from other types of nystagmus.


title: Benign paroxysmal positional vertigo

text: Although rare, CNS disorders can sometimes present as BPPV. A practitioner should be aware that if a person whose symptoms are consistent with BPPV, but does not show improvement or resolution after undergoing different particle repositioning maneuvers — detailed in the Treatment section below — need to have a detailed neurological assessment and imaging performed to help identify the pathological condition.


title: Benign paroxysmal positional vertigo

text: Vertigo, a distinct process sometimes confused with the broader term, dizziness, accounts for about six million clinic visits in the United States every year; between 17 and 42% of these people are eventually diagnosed with BPPV.


title: Benign paroxysmal positional vertigo

text: A number of maneuvers have been found to be effective including the Epley maneuver, the Semont maneuver, and to a lesser degree Brandt–Daroff exercises. Both the Epley and the Semont maneuvers are equally effective. None of these maneuvers addresses the presence of the particles (otoconia); rather it changes their location. The maneuvers aim to move these particles from some locations in the inner ear that cause symptoms such as vertigo and reposition them to where they do not cause these problems. These maneuvers are easily performed at home and online resources are available to patients. The Epley maneuver is the most popular of the maneuvers because it is designed to address particles in the posterior semicircular canal, the most common cause of BPPV. This might give the wrong impression that the Epley maneuver is the cure for all BPPV. Misdiagnosing which semicircular canal is affected, typically by confusing horizontal and diagonal nystagmus, or simply ignoring the identification of the affected canal, results regularly in no cure. Using the appropriate maneuver for the affected canal is essential. The maneuvers are uncomfortable for the patient as they induce strong vertigo and the patient might resist performing them. If the maneuver is not uncomfortable then most likely the wrong maneuver has been selected by a misdiagnosis of the affected semicircular canal. All the maneuvers consist of a series of steps in which the head is held in a specific position, typically for 30 to 60 seconds until nystagmus stops. Movement from one position to the position of the next step has to be done fluently to give the particles enough momentum to move. A position has to be held until nystagmus has completely resided, which indicates that the particles have stopped moving, before one proceeds to the next step.


title: Benign paroxysmal positional vertigo

text: The Epley maneuver employs gravity to move the calcium crystal build-up from the posterior semicercular canal ( resulting in diagonal nystagmus ) that causes the condition. This maneuver can be performed during a clinic visit by health professionals, or taught to people to perform at home, or both. Postural restriction after the Epley maneuver increases its effect somewhat.


title: Benign paroxysmal positional vertigo

text: When practiced at home, the Epley maneuver is more effective than the Semont maneuver. The most effective repositioning treatment for posterior canal BPPV is the therapist-performed Epley combined with home-practiced Epley maneuvers. Devices such as the DizzyFIX can help users conduct the Epley maneuver at home, and are available for the treatment of BPPV.


title: Benign paroxysmal positional vertigo

text: For the lateral (horizontal) canal, resulting in horizontal nystagmus, the Lempert maneuver has been used for productive results. It is unusual for the lateral canal to respond to the canalith repositioning procedure used for the posterior canal BPPV. Treatment is therefore geared toward moving the canalith from the lateral canal into the vestibule.


title: Benign paroxysmal positional vertigo

text: The roll maneuver or its variations are used, and involve rolling the person 360 degrees in a series of steps to reposition the particles. This maneuver is generally performed by a trained clinician who begins seated at the head of the examination table with the person supine There are four stages, each a minute apart, and at the third position the horizontal canal is oriented in a vertical position with the person's neck flexed and on forearm and elbows. When all four stages are completed, the head roll test is repeated, and if negative, treatment ceases.


title: Benign paroxysmal positional vertigo

text: For the superior ( also called anterior ) semicircular canal, resulting in vertical nystagmus, the Deep head hanging maneuver is used. The patient lays down on their back on a bed with their head overhanging the bed.


title: Benign paroxysmal positional vertigo

text: In the first step the head is turned as backward (hanging) as possible. In the following step, the patient remains lying but lifts their head with the chin close to the chest. In the last step, the patient sits upright with the head in the normal position. Before going to the next step, one has to wait until the nystagmus fully resides ( typical 30 to 60 seconds ) and progression from one step to the next has to happen in a fluid movement.


title: Benign paroxysmal positional vertigo

text: Some people will only need one treatment, but others may need multiple treatments, depending on the severity of their BPPV. In the Semont maneuver, as with the Epley maneuver, people are able to achieve canalith repositioning by themselves.


title: Benign paroxysmal positional vertigo

text: The Brandt–Daroff exercises may be prescribed by the clinician as a home treatment method, usually in conjunction with particle-repositioning maneuvers or in lieu of the particle-repositioning maneuver. The exercise is a form of habituation exercise, designed to allow the person to become accustomed to the position that causes the vertigo symptoms. The Brandt–Daroff exercises are performed in a similar fashion to the Semont maneuver; however, as the person rolls onto the unaffected side, the head is rotated toward the affected side. The exercise is typically performed 3 times a day with 5–10 repetitions each time, until symptoms of vertigo have resolved for at least 2 days.