Former inmate reveals existence of 'black jail'

BEIJING - "Black jails" run by security companies to intercept or confine "deviant petitioners" have not yet been swept out of the capital, according to Beijing police who said they recently dismantled one such jail in Changping district.

A courtyard house in Lingshang village, Beiqijia township of Changping, which illegally kept dozens of deviant petitioners, has been closed down. [Photo / China Daily]

Wang Wenjie, a publicity official with the municipal public security bureau, told China Daily on Tuesday that police in Changping caught several suspects who were running the illegal detention house after a released petitioner gave them information.
Police also rescued several petitioners detained there, Wang said.
The police officer, however, did not reveal how many suspects were detained or if the black jail, which was located somewhere in Lingshang village, Beiqijia township of Changping, had been closed down.
"All I can say is that the black jail is one individual case," Wang said, indicating there is no other illegal detention house in the city.
The issue of black jails shocked the nation last year when media reported that a local government had paid security companies to intercept and lock up - normally by force - "deviant petitioners".
Under the law, people from all over the country with grievances against governments can come to the capital to present their case to the State Bureau for Letters and Calls.
In September, Beijing Municipal Public Security Bureau detained Zhang Jun, chairman of Anyuanding Security Service Company, and Zhang Jie, general manager of the company, for "illegally detaining people and illegal business operations".
The company started business in 2004. In 2008 it began to help liaison officers from local governments in Beijing who wanted to stop people from petitioning, according to the police investigation.
China's latest version of Regulations on Letters and Visits does not define "deviant petitioners", but states if the letter-writer or visitor violates the laws or administrative regulations on processions and demonstrations, the public security body shall take necessary measures and impose an administrative penalty.
The Beijing News reported on Tuesday that information from a petitioner surnamed Zhou, who was released from the "black jail" in early July, led to the police operation.
Zhou, from Yancheng city, in East China's Jiangsu province, said as soon as she walked out of a government office in Beijing on July 4, several men blocked her way and dragged her into a minivan.
The men confiscated Zhou's mobile phone and drove her for an hour to a courtyard of three houses measuring about 60 square meters, the report said.
"There was no bed in my room. Elderly people and babies were sitting or lying on the ground," Zhou was quoted as saying. "It was hot, humid and airless."
She added that about 50 people from different places were detained in the courtyard with her. The houses were heavily guarded and it was "difficult to run away".
Zhou was released four days later for good behavior, but she said some people had been held there for months.
In last year's Anyuanding case, police took most "deviant petitioners" in Beijing to Majialou reception and rescue center, from where the petitioners were taken away by hired security personnel to black jails or sent home.
An owner of a grocery store near the Majialou center, surnamed Li, told China Daily she saw many petitioners being treated roughly.
"I once saw a petitioner pushed down from a truck. He almost had his leg run over," Li said.
Yu Jianrong, a professor at the Chinese Academy of Social Sciences, said the black jail case in Changping will not be the last one in Beijing.
"There is a chain of interests behind the control of petitioners," Yu said.
"The security company wants to make money from it, and the local government needs help from outside to maintain stability and fulfill their obligations."
Yu said only when the "system of maintaining social stability changes can the issue of black jails be eliminated completely".

BEIJING â€” Chinese Premier Wen Jiabao has met citizens in Beijing petitioning for redress over unpaid wages, home demolitions and land grabs, state media said Wednesday â€” the first such meeting in six decades.

Wenâ€™s visit to the State Bureau for Letters and Calls â€” where petitioners go to file complaints with authorities â€” highlighted the mounting anger felt by many Chinese living in a one-party state that enforces its will by diktat.

It was the first such trip by a premier since communist China was founded in 1949, the reports said, and it received widespread coverage in print and broadcast media.

But amid rising levels of public protest across China, Wenâ€™s conversations with unhappy citizens were slammed by Human Rights Watch as a political show.
Under a system dating from imperial times, Chinese people can petition government authorities in Beijing or provincial capitals over injustices or unresolved disputes such as illegal land grabs or police misconduct.

However, many of the millions of petitioners each year complain of official indifference to their concerns and occasionally lash out in frustration, while others report being detained by authorities in so-called â€œblack jailsâ€.

The premier â€” sometimes called â€œGrandpa Wenâ€ for his down-to-earth style â€” said during Mondayâ€™s visit that as long as petitionersâ€™ complaints were â€œreasonableâ€, they would be resolved.

â€œWe should use the power in our hands to serve the interests of the people, helping them to tackle difficulties in a responsible way,â€ Wen said, shaking hands with those filling in forms and queuing to submit their papers.

â€œLand is the lifeline of farmers,â€ Wen said.

The government is gearing up for its annual session of parliament, which will begin in early March, and Wenâ€™s visit to the petition office signals a government conscious of public anger.

Land disputes have become Chinaâ€™s most volatile social problem as officials and developers seek to cash in on the nationâ€™s property boom, sometimes forcing people out of their homes without proper compensation.

One family are locked in a standoff with police at their Beijing clothing shop, threatening to set themselves on fire after they were told the shop would be demolished to make way for a new subway line.

Some have used the Internet to rally support for their causes as the web, used by more than 450 million people in China, offers an outlet to expose abuses in a country where traditional media are heavily censored.

President Hu Jintao came under pressure last week during a state visit to the United States to do better on human rights, with President Barack Obama saying the â€œuniversal rights of every human beingâ€ had to be upheld.

Hu admitted â€œa lotâ€ needed to be done in China in terms of human rights but emphasised that â€œdifferent national circumstancesâ€ had to be considered.

Human Rights Watch criticised Wenâ€™s visit as a carefully orchestrated move that â€œwill be widely propagated to Chinese citizens through state media with the explicit message that the Chinese Communist Party leadership caresâ€.

â€œThe fact is that the system is broken and that petitioners are far too often subject to abuses even greater than those which prompted them to petition in the first place,â€ the New York-based group said in a statement.

HRW said the fact that millions of people file petitions each year was â€œproof positive of the lack of true rule of law in Chinaâ€.

Since year 2003, thousands of Chinese petitioners have been held for days, and months in what is commonly known as black jails. (An Alleyway in Hell-Human Rights Watch) These can be found anywhere from a state owned hotel, to a nursing home, to any common building you may find. Their purpose: to detain petitioners who come to Beijing or other provincial capitals seeking rectification for their complaints that were not resolved at lower levels of government. These secret and unlawful detention camps were created and used by local and provincial officials to detain innocent people and to put them through abuse in all forms. In black jails, petitioners are physically and psychologically abused, deprived of food, water, sleep, and medical care, in the aim of teaching them a lesson. These people, ranging from minors petitioning in the place of their parents, to the elderly population, are enclosed in sometimes-small and unsanitary spaces for several months and subjected to theft, blackmail and no communication with family or courts. There is constant surveillance and arbitrary beatings occurring for no reason at all, and upon an injury are not even allowed to se a doctor. What have these people done to deserve such nauseating treatment? (Hartley)

The underground system of black jails are run by officials all over China, right under the nose of the Chinese government, or so it seems. In past UN human rights council of Chinaâ€™s human rights record in June 2009, they completely denied the existence of black jails. The principal of black jails contradict the Chinese governments previous statements. They continually announce the fact that China respects all the rights and freedoms of its people. In 2004, they said at a UN conference, â€œThe state respects and preserves human rights.â€ In 2008, they launched a â€œPeople Firstâ€ policy promoting human rights. In 2009, they announced the â€œNational Human Rights Action Planâ€ stating â€œThe Chinese government unswervingly pushes forward the cause of human rights in China.â€ (An Alleyway in Hell-Human Rights Watch) The situation is just a little bit fishy.

Chinese and foreign journalists who have visited black jails, as well as academic researchers try to prove the facilityâ€™s existence. In essence, the detainees are petitioners coming from rural areas to Beijing or other capitals to find help from the government. They come for compensation for abuses to human rights or other laws such as: illegal land being stolen by higher officials, police torture, and other forms of corruption. These citizens are exercising their legal right to complain about these wrongdoings, however, the plot thickens. The government officials employ plainclothes thugs, commonly known as â€œretrieversâ€ to intercept and abduct them. The black jails are the symbol of failure of Chinaâ€™s petitioning system. This system, known in the past as â€œletters-and-visits practiceâ€ began during the Qing dynasty (1644-1911). This system mean that citizens could bring appeals for justice to central governments in Beijing or other official cities and seek compensation. (An Alleyway in Hell-Human Rights Watch)

Chinese scholars noticed a similarity between black jails and the Shourong Qiansong system abolished in 2003, which is suspect, since black jails began operation in that same year. The Shourong Qiansong was a detention system that started around the 1980s that legally gave permission to police to detain â€œundesirablesâ€ lacking official ID papers. (An Alleyway in Hell-Human Rights Watch) In other words, government officials and police had the power they needed to arrest anyone from petitioners to beggars if it suited them. Extralegal detention camps also target violators of the one child policy and followers of the Falungong movement. The Falungong movement is one of many systems of belief and practices that was outlawed by the government in 1999. Falungong petitioners have been subjected to torture, beatings, forced labor, organ harvesting, psychiatric abuses, and illegal imprisonment, in places similar to, if not black jails. (Beaumont)

Provincial and municipal level officials developed the perfect system in order to catch the petitioners: the intercept, abduct and detain system. Officials hire thugs as guards, who call themselves â€œretrieversâ€. Retrievers often recognize their provinceâ€™s most frequent petitioners on sight. Even if these innocent people lie about where they are from, their accent gives them away. Retrievers often dress scrubby as if they too are petitioning in order to fool them. Former detainees say that black jails are commonly found behind or under government ministry buildings, hotels, nursing homes, mental hospitals, drug rehabilitation centers, and even residential buildings, among others. (Hartley) To passersby a black jail does not look suspicious. One woman said that his was a simple bungalow, which from the outside wouldnâ€™t raise any suspicions. (Screams for help at Chinaâ€™s secret â€œBlack Jailsâ€) The black jail managers, usually letters and visits officials working in Beijing, hire guards and pay the hotel or institution in question to use the establishment. Daily payments are generally withing the range of 150 yuan to 300 yuan ($22-44 US). The rooms are then converted into temporary cells, fully equipped with iron bars and doors. There are also fenced in outdoor areas. In other parts of the building, there will be temporary offices for the retrievers and guards, as well as another area where new detainees are stripped of their ID, money, cell phones, and any other valuables. These facilities are all organized specifically to restrict detaineeâ€™s freedom of movement by using locked doors, barred windows, no access to communication, and 24 hour surveillance by the guards who are often armed with guns and clubs. As far as researchers are concerned, the only difference between black jail and real jail is that the detainees in black jail are actually innocent. (An Alleyway in Hell-Human Rights Watch)

Seven to fifty black jails exist. Tracking them is very difficult because they are secretive and operate on a temporary â€œas neededâ€ basis. Former detainees have brought journalists to their black jails, only to find them empty and abandoned. Local governments running black jails have relationships with police to ensure no interference by anyone of high power. Police actually play an active role in abuses against petitioners as they â€œturn a blind eye to retaliatory violence.â€ Beijing police sometimes double as retrievers. Police frequently believe that â€œpetitioners are disruptive to the stability of the countryâ€, probably because they are all complaining about all the corruption from police and higher officials. One detainee from Jiangsu province told Human Rights Watch that the municipal police showed up at the black jail he was being held at, made a â€œreportâ€, but then abruptly departed and never freed him. Another said that when he contacted the police for assistance, they explained that they have no power to intervene in the black jail facilities â€œ[Your detention] is the desire of upper levels [of government], and if you have opinions about that when this is over, consult the city government authorities or the mayor to solve. Anything that happens inside [black jails] we canâ€™t go to investigate and [we] canâ€™t solve any of your problems.â€ (An Alleyway in Hell-Human Rights Watch) During important dates on the Chinese Communist Partyâ€™s calendar, police forces prevent petitioners from reaching major cities while the ministry of public security â€œurges police to mobilize all of their resourcesâ€ in order to prevent petitioners already in the city from reaching central government offices and â€œdisturbing the social harmony and stabilityâ€ of Chinaâ€™s celebrations.

There are many different variations on the concept of black jails. One was called â€œRe-education through laborâ€ where police impose custodial sentences up to three years while depriving detainees of any due process of law. Another was â€œhouse arrestâ€ arbitrary detention at home that restricted and monitored Internet and phone use, along with 24-hour surveillance by security guards. The most common was the â€œStudy classesâ€. After the Cultural Revolution (1966-1976), government officials and security forces routinely detained individuals suspected of anti-government sentiments and put them in facilities described as â€œstudy classesâ€ however, they are far from that. These people were taken there without formal arrest or trial, and they were generally people targeted by the Chinese government because they were exposing official corruption. Study classes were devoted to educating petitioners about their alleged wrongdoings: basically turning the law around and tell them that they were doing something against the law. Zhang Jianping, a human rights activist said of study classes â€œThese black jails are clearly against the law. But local officials call them legal study classes, and that shows how they treat the law as just a tool for abusing rights.â€ (An Alleyway in Hell-Human Rights Watch)

Sometimes, the detainees must sign contracts stating they will never petition again in order to be freed. This is an act of deliberately stripping an innocent human being of their legal right. Others said they were forced to sign papers upon release from black jail which stated that they had not been detained but instead participated in a â€œstudy classâ€ of their own free will!

Here is a list of all the human rights violations involved in the black jail process: illegal abductions, physical violence, rape, denial of access to medical care, food and sleep deprivation, threats and mental assaults, theft, and extortion. It seems that with the elimination of black jails, there would be the elimination of more than one human rights abuse. Within the first four months of 2009, 15 detainees died an unnatural death. (An Alleyway in Hell-Human Rights Watch) There were cases where people were having convulsion and spitting out blood, however the guards said that they did not care. One person said that they told him that he could die in the black jails for all that they care. There are also violations of childrenâ€™s rights. One fifteen-year-old girl went to petition in the place of her grandmother and instead ended up in a black jail. Several former prisoners said that the guards also threaten detainees saying that if they petition in the future they can break your legs, get you locked up for 3 years in real jail, or they will kill your family. These underground facilities are responsible for countless deaths, illnesses, and family separations and should be stopped as soon as possible. Little is known about black jails, so in consequence, nothing is done about them. Awareness must be raised about the subject in order to start coming down on the black jails system.

What should be done? Human rights groups and the United Nations should continue asking the Chinese government to stop denying the existence of black jails. This assures that those operating black jails will go unpunished. It would also be wise to create a separate police force form the general Beijing police and have them go investigate and report on black jails, the detainees, and their conditions. These police should be backed by the inner Chinese government and should have the power to close them down. The government should also establish an independent commission to examine and evaluate the adequacy of their â€œletters and visitsâ€ petitioning system. These individuals should find a way for the system to effectively and efficiently address the citizenâ€™s problems. By improving the system, black jails will become used less and less. (An Alleyway in Hell-Human Rights Watch) Finally, we should raise awareness about black jails since it is a topic known to few people. We can only hope that the Chinese government recognizes these human rights violations and assist the citizens of their country.

How much harder would it be for India to launch an Democratic Naxal type campaign against the Communist government? There is so much descent in China that if it was properly channeled it can be used to bring down the government.


Structural and functional characterization of the C-terminal catalytic domain of SSV1 integrase.
The spindle-shaped virus SSV1 of the hyperthermophilic archaeon Sulfolobus shibatae encodes an integrase (SSV1 Int). Here, the crystal structure of the C-terminal catalytic domain of SSV1 Int is reported. This is the first structural study of an archaeal tyrosine recombinase. Structural comparison shows that the C-terminal domain of SSV1 Int possesses a core fold similar to those of tyrosine recombinases of both bacterial and eukaryal origin, apart from the lack of a conserved helix corresponding to αI of Cre, indicating conservation of these enzymes among all three domains of life. Five of the six catalytic residues cluster around a basic cleft on the surface of the structure and the nucleophile Tyr314 is located on a flexible loop that stretches away from the central cleft, supporting the possibility that SSV1 Int cleaves the target DNA in a trans mode. Biochemical analysis suggests that the N-terminal domain is responsible for the dimerization of SSV1 Int. The C-terminal domain is capable of DNA cleavage and ligation, but at efficiencies significantly lower than those of the full-length protein. In addition, neither the N-terminal domain alone nor the C-terminal domain alone shows a strong sequence preference in DNA binding. Therefore, recognition of the core-type sequence and efficient catalysis by SSV1 Int presumably requires covalent linkage and interdomain communication between the two domains.


REVIEW: Amnesia: The Dark Descent

Friday, June 1st, 2012

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Oh, Amnesia: The Dark Descent. I love you for what you do and hate that you do it so well. Amnesia is a psychological horror game for PC and Mac, available on Steam and Onlive. If your computer can’t run it, than buy it through Onlive as it relies on your internet connection, rather than your computer specs. The game is created by Frictional Games, who also made the Penumbra series as-well. The game is the only horror game that I’ve played that has scared me to the point that I stopped playing to calm myself. Resident Evil and Dead Space never came close to that.

Frictional, as an independent developer doesn’t make millions of dollars. If Amnesia didn’t sell very well than they would be forced out of business. Luckily the game was met with such high critical success, and strong word of mouth, that it sold well beyond their expectations. The game has become somewhat of a test of courage among the video game community; many people even post their ‘freak outs’ on youtube. People unfamiliar with the game would probably think that a game couldn’t be that scary. In my opinion they are wrong .

You play as Daniel, a young man who finds himself in an old run down castle unsure of how he got there, or who he even is. Right at the start of the game you find a letter written by yourself, addressed to yourself. You have forcefully given yourself amnesia and you’re tasked to kill the lord of the manor: Alexander Brennenburg. You’re not told why you have to kill him, but decide to explore the mansion in hopes of finding out. The story itself is passable and certainly better than Resident Evil. It reminds me of H.P. Lovecraft in a good way. You’ll find various notes around the castle that unveils its dark secrets. You’ll also have flashbacks too; you can still move around in the flashbacks as they are done by voice only.

The game is entirely in first person, which for a horror game is preferred. Third person dampers the immersion in horror games because you know very well that you’re controlling another person. You control your body with the directional pad, press shift to run and space to jump. Unlike, say, older Zeldas where a press of the button will swing your sword; you have to use the mouse and move it fully in order to achieve the same result. Say you want to hit something, you would take the mouse and move it from left to right like your own arm. This heightens immersion because you’ll be manipulating things closer to how you would in real life, than on a 360 or ps3 controller.

Outside of “puzzle important” items, you’ll have three items that are necessary to your survival. The tinder box, the lamp/oil, and your health restorer. The health restorer isn’t that important because you’ll pretty much die from most monster attacks. The weakest monster can kill you in a few hits, so if you’re in such a situation you’re probably already dead. Tinder boxes light up torches and the lamp is….well an oil lamp. You carry it around and it illuminates your surroundings. Tinder boxes are plentiful if you ration them, but oil is harder to come by. Light is a resource in this game as it’s the only thing that keeps you from going insane. When you’re in the light, your sanity slowly starts coming back but monsters can see you easier. Low sanity makes monsters easier to see you and blurs your vision. There are some small sanity effects but I won’t spoil them. The puzzles mostly involve finding an item and using it on a machine to start it up again. Puzzle hints are told through notes you find near the puzzle, so don’t skip on reading the text or you’ll be lost. The puzzles aren’t too hard but if you’re stuck on them, your immersion will drop and so will your feelings of fear.

There is no way to fight monsters so you must run and hide from them. You hide in the darkness and avoid looking at them to escape from them. If you look at them they find you easier, so you have to look away and hope they don’t find you. This creates intense terror, in a way that mowing down zombies in Resident Evil or Dead Space does not. It’s hard to feel scared by anything knowing full well that you can not only kill it, but kill it and 20 others just like it in a few minutes with your rocket launcher. I Don’t feel scared I feel excited when playing Resident Evil and that just shouldn’t be the case at all. The monsters don’t appear very frequently, but because of their immortality and strength they are a huge threat. In other games, zombies and necromorphs appear far too frequently to feel like a threat. Most of the time you’ll only vaguely see the monsters lurking in the shadows or hear it howling in the distance. You’ll never know just exactly where they are or how close they are to finding you unless you get too close.

There is one area of the game that is most assuredly not scary; but after being told what its intention is you’ll not want to stay in for long. The area is brightly light, has a fountain, and has calm relaxing music playing. If you enable the developer commentary (which you shouldn’t on your first play-through), you’ll find out that it exists to keep you from getting used to the terror of the game. If you’re exposed to terror for long periods of time, then it will slowly lose its effect. However,he most relaxing area of the game is perhaps the most devious of all!

Will you venture into the mist, not knowing what may lurk inside?

Music is one of the most important parts of the game as it has a significant impact on the atmosphere. The relaxing area has soft music playing with sounds of water flowing, reducing the tension. The sounds of the monsters moaning in the distance raises your heart rate, and makes you fear turning around the next corner. The music mostly encompasses the sounds of wind blowing, or the sounds of your own(and others…) footsteps. When you’re chased by a monster, you’ll hear intense music playing along with the screams of the beast and your own heavily beating heart.

The only major criticisms of the game that I have, is that it does its job too well and it’s a bit too short. If you’re immersed in the game and let it sink its hooks in you, you may quit playing it due to fear. It’s also only about 8 hours long give or take, which is short but understandable for an indie game. If my strongest criticism is that that it does its job too well and I desperately want more of it; I believe that’s more of an issue with the games industry than the game.

VERDICT

The horror genre has in recent years become a sub-genre of the shooter; with resident evil and dead space being the most prolific. Silent hill seems to be the last well known horror franchise that tries to be scary and that saddens me. The AAA horror games that come out, should be so scary and come out frequently enough that Amnesia shouldn’t be seen as the light at the end of the laughing stock of a tunnel that horror is today. But, that is exactly what it is. The feeling of fear is unique to each person though, what scares me might not scare you. If you’re not scared by Amnesia and are scared by Resident Evil, I don’t think you’re wrong in feeling that way. I personally believe the game is the best at what it does and puts most AAA horror games to shame. It is well worth the price of admission. I’d skip the game if you have a heart condition though. The game recommends playing in the dark with headphones on…which you may soon regret.


Chris & Producer Dan are joined by Comedian & Celtic Fan Chris Jones for the triumphant return of the podcast for it's long awaited second series.

The guys discuss all things Bhoys, including domestic domination, European woes and of course, the Glasgow derby.

Chris tells us all about his awful introduction to football, the joys of the Scottish Football pyramid and how there's no such thing as too much Buckfast.

Some of the regular features are back, including Unbelieavable Jeff, and the most difficult football quiz on the internet, The Hotseat. Also, listen out for the moment when three men thought that opening packs of football stickers on the air could possibly be passed off as 'entertainment'.


---
abstract: 'For a simple model of mutually interacting qubits it is shown how the errors induced by mutual interactions can be eliminated using concatenated coding. The model is solved exactly for arbitrary interaction strength, for two well-known codes, one and two levels deep: this allows one to see under which circumstances error amplitudes add coherently or incoherently. For deeper concatenation, approximate results are derived which make it possible to calculate an approximate “threshold” value for the product of interaction strength and free evolution time, below which the failure probability for an encoded qubit decreases exponentially with the depth of the encoding. The results suggest that concatenated coding could fully handle the errors arising from mutual interactions, at no extra cost, in terms of resources needed, from what would be required to deal with random environmental errors.'
address: 'Department of Physics, University of Arkansas, Fayetteville, AR 72701'
author:
- 'Julio Gea-Banacloche'
title: Error correction for mutually interacting qubits
---

\#1[|\#1]{} \#1[\#1|]{} \#1[\#1]{}

Introduction and model
======================

In previous papers [@myself1; @myself2] I have discussed how unwanted mutual interactions between qubits can be a source of error in a quantum computer (or quantum memory register) and studied some of the characteristic features of this type of error. It was found in [@myself1] that, if the characteristic interaction energy is $\hbar\delta$, the fidelity ${\cal F}$ of the register decreases with time $t$ and with the total number of qubits $N$ as $${\cal F} \sim e^{-N(\delta t)^2}
\label{zeroa}$$ Equation (\[zeroa\]) suggests that $(\delta t)^2$ can be interpreted as a sort of failure probability per qubit. In [@myself2] it was shown that the general scaling (\[zeroa\]) with $t$ holds even when the register is being acted on by “gates” such as one might use in the course of a quantum computation: that is, the error accumulates quasi-coherently in time, and there is no “quantum Zeno effect” associated with normal gate operation. (This suggests, in particular, that it is not necessary to simulate a full computation to estimate the magnitude of this effect; instead, it should be enough to calculate the survival probability of an appropriate initial state of the register under the action of the appropriate Hamiltonian [@garg].)

Mutual interactions are Hamiltonian and hence, in principle, reversible, and there exists, in fact, a variety of special techniques for eliminating, or greatly reducing, these errors: in particular, techniques similar to, and inspired by, the use of refocusing pulses in magnetic resonance [@lloyd; @cory]. Nonetheless, it is worthwhile to look at the question of how well these errors could be eliminated by the use of ordinary error-correction methods based on quantum error-correcting codes [@preskill; @knill1], and in particular on concatenated coding, if only because one would like to be able to make an informed choice between various available methods. Moreover, although pulse methods could in principle be used to deal with certain kinds of environmental noise, it seems very likely that, in large-scale implementations of quantum computers, error-correcting codes will have to be used, in any event, to deal with at least some of these random environmental errors. Assuming, then, that these codes would already be in place, it is natural to ask how much one could get from them.

While there is no question that the present theory of quantum error-correction codes covers mutually interacting qubits as a special case, this is not a case that is often explicitly discussed, and, in particular, it is not immediately clear whether, in order to obtain a threshold estimate for these kinds of errors one should add them coherently (adding error amplitudes) or incoherently (adding error probabilities). This is one of the issues explored in this paper, for the two best-known quantum error-correcting codes: the 5-qubit and 7-qubit codes, and their concatenated forms. The conclusion, at least for the kind of interaction Hamiltonian assumed, is that these codes could fully handle the errors arising from mutual interactions, almost certainly at no extra cost in terms of resources needed (e.g., depth of concatenation). This is an encouraging result which may not have been immediately obvious before.

The interaction to be considered here is one that results in pure phase shifts, conditioned on the relative state of neighboring qubits, according to a Hamiltonian $$H = \sum_{n=1}^{N-1} \hbar\delta \sigma_{zn}\sigma_{z,n+1}
\label{one}$$ Examples of physical systems leading to interactions of this form have been discussed in previous publications [@myself1; @myself2]; in [@myself2] it has also been shown that, for instance, changing $\sigma_z$ to $\sigma_x$ (bit errors instead or phase errors) does not modify substantially the scaling (\[zeroa\]).

In order to simplify the calculation, the (unrealistic) assumption is made that the qubits are divided into physical blocks, each of length $N_c$, which do not interact with each other; that is, the interaction Hamiltonian is applied separately to each such block. The blocks are intended to represent the “logical” qubits in a computer encoded for error correction. When the code is concatenated, the decoupling assumption is applied at the lowest (deepest) level: For instance, for a code concatenated once, a logical qubit is made of $N_c\times N_c$ physical qubits, and it is assumed that the $N_c$ lowest-level blocks do not interact with each other. While unrealistic, it is expected that this approximation should not modify the results very much, based on the numerical calculations made using a similar “decoupling of registers” approximation in ref. [@myself2].

In the following Section, the action of this Hamiltonian on a register encoded for error correction is analyzed, and then in Section III the analysis is extended to a concatenated code. The goal is to derive an expression for the “failure probability” per qubit, that is to say, the probability that the evolution under the Hamiltonian (\[one\]) may lead to a state which cannot be recovered by the error-correction protocol. Section II shows a few general features of this failure probability, such as its dependence on the initial state considered, and the way some error amplitudes add coherently and some do not, depending on the code considered. These results pave the way for the more complicated situation discussed in Section III. It is to be noted that the failure probability is calculated here for arbitrarily large interaction strength, for up to two-level-deep concatenated codes.

Results for simple distance 3 codes
===================================

Distance 3 codes can correct any of the three basic one-qubit errors represented by the three Pauli matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$. They will typically fail whenever two errors happen simultaneously in the same logical block.

The Hamiltonian (\[one\]) causes two phase errors in each physical block every time it is applied, but the actual damage it does to an encoded state turns out to depend on the code used, and on the state it acts upon. Two examples are considered in this section.

The evolution operator resulting from (\[one\]) is, for a single logical block, $$U(t) = \prod_{n=1}^{N_c-1} \left(\cos\delta t + i \sigma_{zn}\sigma_{z,n+1}
\sin\delta t\right)
\label{two}$$ since the Pauli matrices for different qubits commute. Thus, a simple perturbative estimate of the failure probability would be as follows: if it is assumed that each term of the form $\sigma_{zn}\sigma_{z,n+1}$ leads to a distinct (orthogonal) uncorrectible error, there are $N_c-1$ such terms in (\[two\]), and the sum of the squares of their amplitudes is $$\begin{aligned}
P_{pert}(t) &&= (N_c-1)\left(\cos^2 \delta t\right)^{N_c-1} \sin^2 \delta t
\nonumber\\
&&\simeq (N_c-1) (\delta t)^2 \label{three}\end{aligned}$$ for sufficiently small interaction strength and time $\delta t$.

As it turns out, this expression is not, in general, correct, even for small $\delta t$, because of two assumptions: that all terms of the form $\sigma_{zn}\sigma_{z,n+1}$ lead invariably to fatal (uncorrectible) errors, and that the different errors are orthogonal. This is explored in detail in what follows.

The 7-qubit code
----------------

Consider the 7-qubit code introduced by Steane [@steane]:

\[four\] $$\begin{aligned}
\ket{0_L} = {1\over\sqrt 8} \bigl(&&\ket{0000000}+ \ket{0001111} +
\ket{0110011}  \nonumber \\ &&+ \ket{0111100}+ \ket{1010101} +
\ket{1011010} \nonumber \\ &&+\ket{1100110} + \ket{1101001} \bigr)
\label{foura} \end{aligned}$$ $$\begin{aligned}
 \ket{1_L} = {1\over\sqrt 8}
\bigl(&&\ket{1111111}+ \ket{1110000} + \ket{1001100}
\nonumber \\
&&+\ket{1000011} + \ket{0101010} + \ket{0100101} \nonumber \\ &&+
\ket{0011001} + \ket{0010110} \bigr)
\label{fourb} \end{aligned}$$

Because the code is non-degenerate, all 7 states of the form $\sigma_{zn}\ket{0_L}$ (with $n=1,\dots,7$) are orthogonal to each other and to $\ket{0_L}$. Altogether, then, these eight states form an orthogonal basis of the space spanned by the eight kets appearing in (\[foura\]). But then, since any combination of products of $\sigma_{zn}$ acting on $\ket{0_L}$ does nothing but to change the relative signs in the superposition of these eight kets, it follows that it can always be written as a superposition of “one-error” states: $$U(t)\ket{0_L} = C_0\ket{0_L} + \sum_{n=1}^7 \sigma_{zn} \ket{0_L}
\label{five}$$ and this is always correctible. A similar reasoning applies to the state $\ket{1_L}$, which means that, if the initial state of the system is either $\ket{0_L}$ or $\ket{1_L}$, the evolution operator (\[two\]) does not introduce any fatal errors at all.

On the other hand, the initial state of the system is much more likely to be a coherent superposition of $\ket{0_L}$ and $\ket{1_L}$, and for such states the situation is different, because, although error correction will restore $\ket{0_L}$ and $\ket{1_L}$ separately, it will also introduce a phase difference between them. The following useful relations can be easily established from the explicit forms (\[four\]):

\[six\] $$\begin{aligned}
\sigma_{z1}\sigma_{z2}\ket{0_L} = \sigma_{z5}\sigma_{z6}\ket{0_L} &&= -\sigma_{z3}\ket{0_L}
\nonumber \\
\sigma_{z2}\sigma_{z3}\ket{0_L} = \sigma_{z4}\sigma_{z5}\ket{0_L} =
\sigma_{z6}\sigma_{z7}\ket{0_L} &&= -\sigma_{z1}\ket{0_L} \nonumber \\
\sigma_{z3}\sigma_{z4}\ket{0_L} &&= -\sigma_{z7}\ket{0_L}
\label{sixa} \end{aligned}$$ $$\begin{aligned}
\sigma_{z1}\sigma_{z2}\ket{1_L} = \sigma_{z5}\sigma_{z6}\ket{1_L} &&= \sigma_{z3}\ket{1_L}
\nonumber \\
\sigma_{z2}\sigma_{z3}\ket{1_L} = \sigma_{z4}\sigma_{z5}\ket{1_L} =
\sigma_{z6}\sigma_{z7}\ket{1_L} &&= \sigma_{z1}\ket{1_L} \nonumber \\
\sigma_{z3}\sigma_{z4}\ket{1_L} &&= \sigma_{z7}\ket{1_L}
\label{sixb} \end{aligned}$$

and it is easily seen from this that when error correction is applied to the state $\sigma_{zn}\sigma_{z,n+1} (\alpha\ket{0_L}+\beta\ket{1_L})$, the result will be the state $(-\alpha\ket{0_L}+\beta\ket{1_L})$, that is, a state with the wrong relative phase. Equations (\[six\]) also show that the assumption that different $\sigma_{zn}\sigma_{z,n+1}$ lead to orthogonal error states is incorrect: for instance, $\sigma_{z1}\sigma_{z2}$ and $\sigma_{z5}\sigma_{z6}$ lead to the same error state.

Using Eqs. (\[six\]) it is straightforward to derive the correct form of the approximate failure probability for small $\delta t$. When only the terms of first order in $\delta t$ are kept in the expression (\[two\]), acting on a general state $\alpha\ket{0_L}+\beta\ket{1_L}$, one obtains $$\delta t\,(3\sigma_{z1}+2\sigma_{z3}+\sigma_{z7})(-\alpha\ket{0_L}+\beta\ket{1_L})
\label{seven}$$ which means that the probability that after error correction one might end up with the state $-\alpha\ket{0_L}+\beta\ket{1_L}$ is $(3^2+2^2+1^2)(\delta t)^2 = 14 (\delta t)^2$. One might directly call this the failure probability (since the state $-\alpha\ket{0_L}+\beta\ket{1_L}$ is generally not equal to the initial state $\alpha\ket{0_L}+\beta\ket{1_L}$) or define the failure probability as including explicitly the square of the overlap between the initial and the final state (see Eq. (\[deffailprob\]), below, for the precise definition), in which case one finds $$P \simeq 14 (\delta t)^2 \left(1-\left(|\alpha|^2-|\beta|^2\right)^2 \right)
\label{eight}$$ where the $\simeq$ sign is a reminder that this is only valid for small $\delta t$. The failure probability (\[eight\]) is largest for the symmetric linear superpositions $(\ket{0_L}\pm\ket{1_L})/\sqrt{2}$. Note how it is different from the result (\[three\]) (for $N_c=7$), obtained under the assumption that the errors generated by different $\sigma_{zn}\sigma_{z,n+1}$ were all orthogonal, and it is also not equal to the same expression with $N_c-1$ replaced by $(N_c-1)^2$, which would correspond to a “worst case scenario” in which all the different errors added coherently. The coefficient $14$ falls instead in between the two extremes of 6 $(N_c-1)$ and 36 ($(N_c-1)^2$).

The definition of the failure probability $P$ used above is $$P=1-\left|\bra{\Psi_0}U(t)\ket{\Psi_0}\right|^2 - \sum_{n=1}^{N_c}
\left|\bra{\Psi_0}\sigma_{z,n}U(t)\ket{\Psi_0}\right|^2
\label{deffailprob}$$ that is, one minus the total probability that, after a time $t$, the system may be found in either the original state $\ket{\Psi_0}$, or one of the $N_c$ correctible states $\sigma_{z,n}\ket{\Psi_0}$ (there are, of course, other correctible states, of the form $\sigma_{x,n}\ket{\Psi_0}$ and $\sigma_{y,n}\ket{\Psi_0}$, but they cannot be accessed by this interaction operator).

Using this definition it is relatively straightforward to calculate $P$ exactly for arbitrarily large $\delta t$. The following relation is useful:

\[ten\] $$\bra{0_L} \sigma_{z,n_1},\ldots, \sigma_{z,n_k} U \ket{0_L} =
(-1)^k\bra{1_L} \sigma_{z,n_1},\ldots, \sigma_{z,n_k} U \ket{1_L}
\label{tena}$$ $$\bra{0_L} \sigma_{z,n_1},\ldots, \sigma_{z,n_k} U \ket{1_L} = 0
\label{tenb}$$

where $k=0,\ldots, 7$. This follows directly from the fact that $U$ involves always the product of an even number of $\sigma_z$ operators, and that the code words used in $\ket{0_L}$ are the complementaries of the ones used in $\ket{1_L}$.

For future use it is convenient to define

\[ten1\] $$f_0  = \left| \bra{0_L}  U \ket{0_L} \right|^2 = \left| \bra{1_L}  U \ket{1_L} \right|^2
\label{ten1a}$$ $$f_1  = \sum_{n=1}^7\left| \bra{0_L} \sigma_{z,n} U \ket{0_L} \right|^2 = \sum_{n=1}^7
\left| \bra{1_L}  U \ket{1_L} \right|^2
\label{ten1b}$$

Direct calculation then yields $$f_0 = 1-f_1 = \cos^2(\delta t) \cos^2(2\delta t) \cos^2(3 \delta t)
\label{ten2}$$ and $$P(t) = f_1
\left(1-\left(|\alpha|^2-|\beta|^2\right)^2 \right)
\label{eleven}$$ and it is easy to verify that (\[eight\]) is the lowest-order term in the expansion of (\[eleven\]) in powers of $\delta t$.

The failure probability (\[eleven\]) for the case $|\alpha|=|\beta|$ is plotted in Figure 1 (solid line) as a function of $\delta t$, from $\delta t = 0$ to $\pi/2$. From $0$ to $\pi$, the curve is symmetric about $\pi/2$, as is to be expected from the form of the evolution operator (\[two\]).

The 5-qubit code
----------------

In 1996 Laflamme et al. [@laflamme] introduced a 5-qubit code which can correct general one-qubit errors. A possible encoding is

\[twelve\] $$\begin{aligned}
\ket{0_L} = {1\over\sqrt 8} \bigl(&&-\ket{00000}+ \ket{01111} -
\ket{10011} + \ket{11100}  \nonumber \\ &&+ \ket{10101} +
\ket{11010} +\ket{00110} + \ket{01001} \bigr)
\label{twelvea} \end{aligned}$$ $$\begin{aligned}
 \ket{1_L} = {1\over\sqrt 8}
\bigl(&&-\ket{11111}+ \ket{10000} + \ket{01100} -\ket{00011}
\nonumber \\
&&- \ket{01010} - \ket{00101} + \ket{11001} + \ket{10110} \bigr)
\label{twelveb} \end{aligned}$$

The codewords can be obtained from the 7-qubit code by deleting any two qubits (in this case, the first two), but the pattern of signs is nontrivial.

Note that the logical 0 and 1 states are still superpositions of eight basis vectors each (where the basis vectors are simultaneous eigenstates of all the $\sigma_{z,n}$); hence, the ten one-error states $\sigma_{z,n}\ket{0_L}$ and $\sigma_{z,n}\ket{1_L}$, plus the two no-error states $\ket{0_L}$ and $\ket{1_L}$ are not enough to span the space resulting from all the possible combinations of the 16 basis kets. As a result, there are now no states which exhibit fully correctible evolution (unlike the $\ket{0_L}$ and $\ket{1_L}$ did for the 7-qubit code). However, we still have the phenomenon of non-orthogonal errors. In particular, we find that

\[thirteen\] $$\sigma_{z2}\sigma_{z3}\ket{0_L} =
\sigma_{z4}\sigma_{z5}\ket{0_L}
\label{thirteena}$$ $$\sigma_{z2}\sigma_{z3}\ket{1_L} =
\sigma_{z4}\sigma_{z5}\ket{1_L}
\label{thirteenb}$$

and neither of these expressions is equivalent to a linear combination of single-qubit errors acting on $\ket{0_L}$ or $\ket{1_L}$; in fact, they are orthogonal to all such states, and to the no-error states $\ket{0_L}$ and $\ket{1_L}$ as well.

The calculation of $P(t)$, as given by the definition (\[deffailprob\]) is a little more involved, although the results (\[ten\]) still hold. Defining as before

\[thirteen1\] $$f_0  = \left| \bra{0_L}  U \ket{0_L} \right|^2 = \left| \bra{1_L}  U \ket{1_L} \right|^2
\label{thirteen1a}$$ $$f_1  = \sum_{n=1}^5\left| \bra{0_L} \sigma_{z,n} U \ket{0_L} \right|^2 = \sum_{n=1}^5
\left| \bra{1_L}  U \ket{1_L} \right|^2
\label{thirteen1b}$$

one finds, thanks to Eqs. (\[ten\]), that $P(t)$ can still be written as $$P(t) = 1-f_0 - f_1 \left(|\alpha|^2-|\beta|^2\right)^2
\label{thirteen2}$$ but this time, unlike for the 7-qubit code, one does not have $f_0=1-f_1$. Instead,

\[thirteen3\] $$f_0 = \cos^4(\delta t) \cos^2(2\delta t)
\label{thirteen3a}$$ $$f_1 = \sin^2(\delta t) \left(1+\cos^2(\delta t)
\cos^2(2 \delta t) \right)
\label{thirteen3b}$$

The failure probability (\[thirteen2\]) for the case $|\alpha|=|\beta|$ is plotted in Figure 2 (solid line) as a function of $\delta t$, from $\delta t = 0$ to $\pi/2$. For small values of $\delta t$ Eq. (\[thirteen2\]) reduces to $$P \simeq (\delta t)^2 \left[6-2 \left(|\alpha|^2-|\beta|^2\right)^2 \right]
\label{fifteen}$$

The reason $f_0\ne 1-f_1$ for the 5-qubit code is that, as mentioned above, the states $\ket{0_L}$ and $\sigma_{z,n}\ket{0_L}$ do not form a complete basis of the space spanned by the eight kets which make up $\ket{0_L}$. For the 7-qubit code, instead, they do, and, since the operator $U$ acts entirely within that space (when the initial state is $\ket{0_L}$), any part of $U\ket{0_L}$ which is not accounted for by the original state $\ket{0_L}$ has to show up in one of the $\sigma_{z,n}\ket{0_L}$ (and the same for the initial state $\ket{1_L}$).

Concatenated codes
==================

A relatively straightforward way to obtain a code that can correct more than one-qubit errors is to concatenate a one-error correcting code, in principle, as many times (or levels deep) as necessary [@preskill; @knill1]. In a code concatenated once, each logical qubit consits of $N_c$ blocks of $N_c$ (lower-level) qubits each. This code can protect against general errors in any three lower-level qubits, and also against many other combinations of multiple lower-level errors. The minimum number of lower-level errors required to cause the code to fail is four, two in one upper level block (causing that whole block to fail) and two in a different one.

In this section I look first at the 7-qubit and 5-qubit codes concatenated once. A physical (lowest-level) qubit will be identified by two indices, $n$ and $m$, running from $1$ to $N_c$: the first index denotes the upper-level block it belongs to, and the second index the qubit’s position within that block. With the assumption that different blocks do not interact, the time-evolution operator (\[two\]) becomes $$U(t) = \prod_{n=1}^{N_c}\prod_{m=1}^{N_c-1} \left(\cos\delta t + i \sigma_{nm}\sigma_{n,m+1}
\sin\delta t\right)
\label{seventeen}$$ From now on, for simplicity, the subscript $z$ on $\sigma_z$ will be dropped, as only $\sigma_z$ is used.

The 7-qubit code
----------------

If all the errors generated by the individual terms in (\[seventeen\]) were orthogonal, it would be a relatively simple matter to calculate the probability of an uncorrectible error to lowest order. It would be given by the terms of the form $\sigma_{n_1m_1}
\sigma_{n_1,m_1+1}\sigma_{n_2m_2}\sigma_{n_2,m_2+1}$, with $n_1\ne n_2$ and $m_1, m_2$ arbitrary. There are $C(N_c,2)$ ways to choose the indices $n_1$ and $n_2$, and for each choice there are $(N_c-1)^2$ different ways to choose $m_1, m_2$, so there is a total of $${N_c\choose2}
\left(N_c-1 \right)^2
\label{eighteen}$$ terms in (\[seventeen\]) which contribute to the failure probability in lowest order, and all these terms are proportional to $(\delta t)^4$ (after squaring the probability amplitudes).

However, as has been shown already in the previous section, not all the terms enumerated in (\[eighteen\]) are mutually orthogonal. For the 7-qubit code, according to (\[six\]), there are two sets of values for $m_1$ (one set of three values, and one set of two values) which result in identical states. When the equivalent choices for $m_2$ are factored in and added, and the probability amplitudes of the corresponding states are squared, one finds, instead of the factor $(N_c-1)^2 = 36$ in (\[eighteen\]), a factor $$(3^2+2^2+1^2)(3^2+2^2+1^2) = 14^2 = 196
\label{nineteen}$$ As a result, to lowest order, the failure probability for this two-level code is given by $$\begin{aligned}
P^{(2)} &&\simeq 196
{7\choose2}
(\delta t)^4 \left(1-\left(|\alpha|^2-|\beta|^2\right)^2 \right)
\nonumber \\
&&= 4116 \, (\delta t)^4 \left(1-\left(|\alpha|^2-|\beta|^2\right)^2 \right)
\label{twenty}\end{aligned}$$ for a state $\alpha\ket{0_L}+\beta\ket{1_L}$. The dependence on $\alpha$ and $\beta$ exhibited by (\[twenty\]) may not be not immediately obvious at this point (it will be derived later), but it shows that, for the concatenated code also, the individual states $\ket{0_L}$ and $\ket{1_L}$ are fully correctible.

Note that the maximum error probability $P_{max}^{(2)}$ for the once-concatenated code, according to (\[twenty\]), is related to the corresponding one for the plain code, (\[eight\]), by the formula $$P_{max}^{(2)} =  {7\choose2} \left(P_{max}^{(1)}\right)^2
\label{twenty1}$$ and the right-hand side of (\[twenty1\]) is simply the probability that two blocks might fail [*independently*]{}, times the number of ways to choose two blocks. This is because, to this order (recall that these are only approximate results for small $\delta t$), errors across different blocks are orthogonal, ie., they add incoherently. This can also be seen from Equation (\[nineteen\]): within each block there is some coherent addition of error amplitudes (which results in the terms $3^2$ and $2^2$), but then the total error probabilities for each block are simply multiplied.

This is an important result which generalizes to higher orders, as discussed later in this Section (subsection C). The key point is that all the errors which would cause a particular block to fail are orthogonal to all the errors that would cause a [*different*]{} block to fail, [*to lowest order*]{}. This is because to lowest order all it takes is two errors in one block to cause that block to fail, and because the original code is nondegenerate, which means that all expectation values of the form $\bra{\xi_L}\sigma_i\sigma_j\ket{\eta_L}$ (where $\eta, \xi = 0,1$) within the same block are zero.

Equation (\[twenty1\]) does not, however, hold for arbitrarily large $\delta t$, when multiple (more than two) errors within the same block need to be taken into account. For a code concatenated only once it is still possible, albeit very cumbersome, to derive this failure probability for arbitrary $\delta t$. The definition (\[deffailprob\]) has to be changed to $$P=1-\left|\bra{\Psi_0}U(t)\ket{\Psi_0}\right|^2 - \sum_n
\left|\bra{\Psi_0}A_nU(t)\ket{\Psi_0}\right|^2
\label{twentyone}$$ where the $\{A_n\}$ represent a maximal set of correctible error operators leading to orthogonal states. The difficulty is that now, unlike in the previous Section, not all possible correctible errors lead to orthogonal states. For instance, any number of errors within one single block are now correctible, but certainly not all of these lead to orthogonal states. Thus, even though we started out with a nondegenerate code, [*the concatenated code is degenerate*]{} [@gottesman].

As far as I can tell, the set $\{A_n\}$ in (\[twentyone\]) must be constructed by inspection. For the 7-qubit code, concatenated once, I find that the maximum number of correctible errors acting on a single block and leading to orthogonal states is 15, which can be chosen to be $$\begin{aligned}
\sigma_{nm}\qquad (m=1,\ldots,7) \nonumber \\
\sigma_{n1}\sigma_{nm} \qquad (m=2,\ldots,7) \nonumber \\
\sigma_{n2}\sigma_{n3} \nonumber \\
\sigma_{n1}\sigma_{n4}\sigma_{n5}
\label{twentytwo}\end{aligned}$$ Of these, the first 7 are, of course, fully correctible at the lower level, whereas the others, upon correction at the lower level, lead to what look like “block” phase errors at the higher level.

The single-block errors (\[twentytwo\]) can be combined in a large number of ways among the different blocks to yield other, orthogonal, correctible errors. Fortunately, however, the actual number of such combinations which make a nonvanishing contribution to (\[twentyone\]) is relatively small. To see this, divide the error operators acting on a single block into those involving an even number of $\sigma$’s (denoted as $E_n$ if they act on the $n$-th block) and those involving an odd number of $\sigma$’s (denoted as $O_n$). Then, for a general multiblock error of the form $A=E_{n_1}\ldots E_{n_k}O_{m_1}\ldots O_{m_k}$, Eqs. (\[ten\]) for a single block imply that the expectation value $$\bra{0_L}O_{n_1}\ldots O_{n_k}E_{m_1}\ldots E_{m_l}U\ket{0_L}
\label{twentythree}$$ will only be nonzero if the corresponding combination $\bra{0_L}\sigma_{n_1}\ldots\sigma_{n_k}\ket{0_L}$ is nonzero for the single, non-concatenated, code. The same is true for the state $\ket{1_L}$, and there are no cross-terms between $\ket{0_L}$ and $\ket{1_L}$ (also as a direct consequence of the single-block Eqs. (\[ten\])). Thus, to evaluate (\[twentyone\]), it is sufficient to identify the combinations of numbers $\{n_1,\ldots,n_k\}$ which lead to a nonzero $\bra{0_L}\sigma_{n_1}\ldots\sigma_{n_k}\ket{0_L}$ for the 7-qubit code, and put odd errors, chosen from the list (\[twentytwo\]), in the blocks $n_1,\ldots,n_k$, augmented perhaps with even errors, always chosen from the list (\[twentytwo\]), in any other blocks, provided, of course, that the total error be correctible. This quickly reduces the total of different possibilities to a manageable number: for instance, since an even error in one block would cause that block to fail, one cannot have even errors in more than one block at a time.

For reference, for the 7-qubit code, the sets of values of $n_1,\ldots,n_k$ to be used are: $(1,2,3)$, $(1,4,5)$, $(1,6,7)$, $(2,4,6)$, $(2,5,7)$, $(3,4,7)$, $(3,5,6)$, the complementary set ($(4,5,6,7)$, $(2,3,6,7)$, and so on), and the set $(1,2,3,4,5,6,7)$. Then, possible error operators $A_n$ to be used in (\[twentyone\]) might be products such as such as $\sigma_{1,m_1}\sigma_{2,m_2}\sigma_{3,m_3}$, or $\sigma_{1,m_1}\sigma_{2,m_2}\sigma_{3,m_3}\sigma_{4,1}\sigma_{4,m_4}$, or $\sigma_{n,1}\sigma_{n,m}$. In terms of the functions $f_0$, $f_1$ defined in (\[ten1\]), the term(s) $\sum_{m_1,m_2,m_3} |\bra{0_L}
\sigma_{1,m_1}\sigma_{2,m_2}\sigma_{3,m_3}\ket{0_L}|^2$ contribute a term $f_0^4 f_1^3$ to the expression (\[twentyone\]); the term $\sum_{m_1,m_2,m_3,m_4} |\bra{0_L}
\sigma_{1,m_1}\sigma_{2,m_2}\sigma_{3,m_3}\sigma_{4,1}\sigma_{4,m_4}\ket{0_L}|^2$ contributes a term $f_0^3 f_1^4$ (since the product of two $\sigma$’s in the same block is equivalent, except for a sign, to a single $\sigma$ acting on the appropriate state, cf. Eqs. (\[six\])), and so on.

Collecting all the terms, and keeping track carefully of which ones have opposite signs in the states $\ket{0_L}$ and the states $\ket{1_L}$, one finds $$P^{(2)}(t) = \left(1-f_0^7-7f_0^6 f_1 - 28 f_0^4 f_1^3 - 7 f_0^3 f_1^4 - 21 f_0^2 f_1^5
\right)
\left(1-\left(|\alpha|^2-|\beta|^2\right)^2 \right)
\label{twentyfour}$$ with $f_0$ and $f_1$ given by (\[ten2\]), which does reduce to (\[twenty\]) for small $\delta t$.

The exact result (\[twentyfour\]) is plotted as a function of $\delta t$ in Figure 1 (dashed line), for $|\alpha|=|\beta|$. It is, perhaps, remarkable that it does not look all that different from the single-encoding result (\[eleven\]), except in the very small $\delta t$ region. It is also clear that Eq. (\[twenty1\]) does not hold for large $\delta t$, which means that for large $\delta t$ we do see the effects of errors across different blocks adding coherently.

The 5-qubit code
----------------

For the 5-qubit code, one again has to identify the appropriate maximal set of error operators leading to orthogonal errors. For a single block, one finds the following set of 15: $$\begin{aligned}
\sigma_{nm}\qquad (m=1,\ldots,5) \nonumber \\
\sigma_{n1}\sigma_{nm} \qquad (m=2,\ldots,5) \nonumber \\
\sigma_{n2}\sigma_{nm} \qquad (m=3,\ldots,5) \nonumber \\
\sigma_{n1}\sigma_{n2}\sigma_{nm} \qquad (m=3,\ldots,5)
\label{twentyfive}\end{aligned}$$

As in the previous subsection, it is found that only certain combinations of odd errors across different blocks lead to nonzero contributions to (\[twentyone\]). These turn out to be much fewer than before, being limited to $n_1,\ldots,n_k = (1,2,5), (1,3,4)$, and $(2,3,4,5)$.

The result, valid for arbitrarily large $\delta t$, turns out to be now $$\begin{aligned}
P^{(2)}(t) = 1  &&-(f_0^5+5f_0^4(1-f_0)+f_1^4 + 4f_0f_1^3(1-f_1)) \nonumber \\
&&-(4f_0f_1^3+6f_0^2 f_1^2-8f_0^2 f_1^3)\left(|\alpha|^2-|\beta|^2\right)^2
\label{twentysix}\end{aligned}$$ with $f_0$ and $f_1$ given by Eqs. (\[thirteen3\]). The factors of $1-f_0$ appearing in (\[twentysix\]) come from the sum of the single-block expectation values squared of $\sigma_{1}\sigma_{m}$, $m=2,\ldots,5$ and $\sigma_{2}\sigma_{m}$, $m=3,\ldots,5$; these seven error operators, acting on the singly encoded $\ket{0_L}$ or $\ket{1_L}$, generate the seven orthogonal states necessary to complete a basis of the space spanned by the 8 kets appearing in (\[twelve\]). The factors of $1-f_1$ come from the sum of the single-block expectation values squared of $\sigma_{1}\sigma_{2}\sigma_{m}$, $m=3,\ldots,5$: these 3 error operators turn out to be the complement of the 5 ones appearing in the definition (\[thirteenb\]) of $f_1$.

The limit of (\[twentysix\]) for small $\delta t$ is $$P^{(2)}(t) \simeq (\delta t)^4 \left[360 - 24 \left(|\alpha|^2-|\beta|^2\right)^2\right]
\label{thirty}$$ As usual, the failure probability is largest for the symmetric superposition state $(\ket{0_L}\pm\ket{1_L})/\sqrt{2}$. The corresponding coefficient, $360$, can be obtained also from the simple arguments used to derive the result (\[twenty\]) in the previous subsection: it equals $C(5,2)(2^2+1^2+1^2)^2$, where the $2^2$ stands for the amplitude square of the sum of the two error terms which lead to identical errors, as shown in Eq. (\[thirteen\]) (and the two $1^2$ stand for the other two orthogonal errors).

The exact result (\[twentysix\]) is plotted in Figure 2 (dashed line) as a function of $\delta t$, for $|\alpha|=|\beta|$. In this case, unlike for the 7-qubit code, the difference with the single-encoding formula (solid line, equation (\[thirteen2\])) is substantial; the twice-encoded qubit has a substantially smaller failure probability for most values of $\delta t$. The failure probability actually goes to zero at $\delta = \pi/2$, where Eq. (\[seventeen\]) yields $U = i\prod_{n=1}^5 \sigma_{n1}
\sigma_{n5}$. Upon inspection, it is found to be a peculiarity of the concatenated (two-levels deep) 5-qubit code that $$\prod_{n=1}^5 \sigma_{n1} \sigma_{n5} \left(\alpha \ket{0_L} + \beta \ket{1_L}
\right) = \sigma_{11} \sigma_{15} \sigma_{22} \sigma_{32} \sigma_{42} \sigma_{52}
\left(\alpha \ket{0_L} + \beta \ket{1_L}
\right)
\label{thirty2}$$ and the state on the right-hand side of (\[thirty2\]) is fully correctible, since it amounts to only one block failing.

Results for arbitrarily deep concatenation
------------------------------------------

Clearly, the exact calculation of $P^{(n)}$ for a code concatenated $n$ levels deep, with $n > 2$, becomes much too cumbersome to be feasible. However, the results in the previous subsections show that the leading term in $\delta t$ can be calculated using relatively simple arguments.

Consider, for instance, a code concatenated three levels deep. It will fail if at least two of the high-level qubits fail, but these high-level qubits are themselves two-level deep codes. One can choose the two highest-level qubits that fail in any of $C(N_c,2)$ ways, and for each of them the number of terms in the expansion of the evolution operator which lead to a qubit failure are those computed for the previous level. So, for the 7-qubit code, we can immediately iterate (\[twenty\]) to get $$\begin{aligned}
P^{(3)}_{max} &&\simeq {7\choose2} \left(4116 (\delta t)^4 \right)^2 \nonumber \\
&&\simeq {7\choose2} \left(P^{(2)}_{max} \right)^2
\label{thirtyone}\end{aligned}$$ where the subscript “max” indicates that we are looking at the failure probability for the state that maximizes it (the linear combination $(\ket{0_L}\pm\ket{1_L})/\sqrt{2}$), and Eq. (\[eight\]) has been used for $P^{(1)}$.

Equation (\[thirtyone\]) contains the essence of a recursion relation which yields the failure probability at the $n$-th level of concatenation ($n\ge 2$) as $$P^{(n)}_{max} \simeq {7\choose2}^{-1} \left({7\choose2} P^{(1)}_{max}\right)^{2^{n-1}}
\label{thirtytwo}$$ Figure 3 shows the approximate $P^{(n)}$, calculated from (\[thirtytwo\]), for $n=1,\ldots,4$ (solid lines), and also the exact results (\[eleven\]) and (\[twentyfour\]), for comparison. Based on these examples, it would seem that Equation (\[thirtytwo\]) overestimates the failure probability, which would make it a somewhat conservative estimate

As encoding depth $n$ increases, the failure probability (\[thirtytwo\]) will, for a given $\delta t$ increase or decrease depending on whether $C(7,2) P^{(1)}_{max}$ is greater than or less than one. This yields the “threshold” value for $(\delta t)^2$, below which virtually error-free operation can be achieved for sufficiently deep encoding. Using (\[eight\]), one finds the condition: $$(\delta t)^2 < {1\over 21\cdot 14} = 3.4\times 10^{-3}
\label{thirtythree}$$ (Figure 3 shows this as the point where all the curves cross.) This threshold (\[thirtythree\]) is comparable to some estimates for the correction of independent, random errors by concatenate codes, if one thinks of $(\delta t)^2$ as a sort of failure probability for a single physical qubit. As pointed out in the Introduction, this is a natural interpretation (in spite of the fact that two physical qubits at a time are modified by this interaction), because of the result (\[zeroa\]).

A similar calculation yields the threshold result for the 5-qubit code (using (\[fifteen\]) and C(5,2)=10): $$(\delta t)^2 < {1\over 10\cdot 6} = 1.7\times 10^{-2}
\label{thirtyfour}$$

Discussion
==========

The results in the previous section suggest that, in principle, the correction of even rather large undesired interactions between the qubits does not require special methods or resources beyond those needed to control independent, random errors arising from the interaction of the qubits with the environment. This observation is based, first, on the fact that estimates of the threshold for fault-tolerant computation with independent environmental errors tend to be rather more restrictive than either (\[thirtythree\]) or (\[thirtyfour\]) [@preskill; @knill1]; and, second, on the fact that for most proposed quantum computing systems the unwanted interactions betwen qubits would easily satisfy the threshold conditions (\[thirtythree\]) or (\[thirtyfour\]), provided that the time in between error corrections is not excessive.

As an example, consider a system of nuclear spins, in a solid, for instance (such as in the proposal by Kane [@kane]), separated by $\sim 150$ Å, and interacting via the magnetic dipole-dipole interaction, which would naturally lead to a Hamiltonian like (\[one\]). For such a system $\delta$ may be estimated as $$\delta \sim {\mu_0 \mu_N^2 \over 4 \pi \hbar d^3} \sim 10^{-2} \hbox{s}^{-1}
\label{thirtyfive}$$ Here $\mu_N$ is the nuclear magneton.

Without error correction $\delta t$ becomes already of the order of 1 after 100 s, and long calculations are impossible. But assume that one uses error correction and that complete error correction of an $n$-order encoded qubit requires a time $t = c_n \tau_{gate} $, where the time needed to perform an elementary logic gate for this system is about $\tau_{gate} = 10^{-5}$ s. Then the threshold is easily reached unless $c_n$ is exceptionally large, which would make error correction generally impossible anyway (a large $c_n$ would mean essentially that the computer architecture does not allow for a large degree of parallelism when correcting errors across different qubits).

Of course, the results presented here have been obtained only for a specific kind of interaction Hamiltonian and a specific geometry, and would need to be modified for other cases (e.g., two-dimensional geometries, with more nearest-neighbors per qubit, or to include non-nearest neighbor interactions). There is also the simplifying assumption made from the outset, of neglecting interactions beween qubits in different blocks. These complications might require recalculations for specific physical systems and hardware configurations, but I do not expect them to change substantially the threshold estimates of the previous Section.

For instance, for the kind of “linear” geometry considered here, including interactions between the last qubit of one block and the first one of the next block would not modify the failure probability to lowest order, since a single two-qubit error which straddles two blocks does not cause either block to fail; hence, to make any one block fail, more of these errors are required than of the kind of “internal” errors considered in this paper.

The key result of Section III concerning the incoherent addition of failure probabilities for different blocks, to lowest order, should be quite general: it depends only on the base code being a nondegenerate distance 3 code and the interaction Hamiltonian being a sum of pairwise products of elementary error operators for individual qubits.

The most important “real-life” complication which has been ignored here is, rather, the fact that, in practice, error correction will not be a clean and purely mathematical operation, but a physical one, involving interacting systems and therefore itself subject to error. The theory of fault-tolerant error correction has shown that even when this is accounted for it should still be possible to achieve “almost” perfect error correction, provided that the error rate per gate is below a certain threshold, but it is here where one finds the (possibly) very small error thresholds which constitute the main challenge for large-scale quantum computing.

I am grateful to R. Laflamme and D. A. Meyer for comments. This research has been sponsored by the National Science Foundation and by the Army Research Office.

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