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by Levina
Save Constitution Campaign – Remember Emergency and Shah Bano case?
An year long Save Constitution campaign is to begin today, lead by the scion of Congress party, Mr.Rahul Gandhi. It is really a capper that this comes from Congress party which created a Kakistocracy rather than a democracy in India. Kakistocracy, the word has been derived from a Greek word which means government by the worst persons; a form of government in which the worst persons are in power. Rings a bell?
Over seven decades, Indian democracy was in shambles and its economic growth dragged at a pace of tortoise, yet the party responsible for this plight has the audacity to launch a save constitution drive. Ostensibly the march which Rahul Gandhi would be leading today is to voice its protest against the ill treatment of Dalits in India by BJP. The real reason many say is Congress's inability to break the ice with the dalit community; In 2014 out of the 84 seats congress could manage just 3-4 seats, while majority of the seats were won by BJP. The success of this protest can only be determined by the number of seats congress will win in 2019. Or will it?
The long list of amendments made by Congress
Congress has so far amended the constitution more than a hundred times, under the garb of "taking a fresh look at it".
Home Minister GB Pant,under Nehru Government observed: We are rehabilitating the Constitution, not tampering with it.
Out of the many amendments made under the Indira Gandhi government in 1976, one of them was to reduce the President's powers, turning him into a mere figurehead. This would have happened earlier had the first president of India, Dr. Rajendra Prasad, not fought with Nehru for preserving the powers of the chair. Rajendra Prasad bravely put it in public that the Constitution gave the President the power to overrule the PM's advice. Unfortunately nothing has been done so far to restore this amendment to what it was in the original constitution.
The media houses which jump the wagon to support congress and its endeavors, have conveniently forgotten that it was under Jawahar Lal Nehru that an amendment was brought about to curb the press's freedom. It was done under the pretense of keeping the press "clean", which was clearly a sign of times to come; it didn't take long for the next congress leader Indira Gandhi to impose censorship on media during the emergency.
In the words of Time magazine "Part of the Indian press, said he, is dirty, indulges in "vulgarity, indecency and falsehood." To teach it manners, Nehru proposed an amendment to India's constitution that would impose severe restrictions".
It did not end with Nehru and Indira Gandhi, Rajeev Gandhi had proposed a Postal bill, which was an attempt at censoring and intercepting all mails. Had President Zail Singh not refused to sign the bill, this draconian law would have come into effect.
In 2012, it was under Congress led government that sec 66(A) IT act was proposed, this was a blatant attempt to curb the rise of dissent on social media. Experts observed that under the proposed IT Act the penalties for the same offences were higher as compared to those in the IPC. For example threatening to defame through email would have attracted a penalty of 3 years imprisonment under the IT Act while if the same offence if committed verbally would have attracted just 2 years. But the sinister reason behind suggesting such a flummoxing punishment was– the offences punishable with a jail term of 3 years or more are 'cognisable' under law, and ergo a police officer can make the arrest without a warrant. An epiphany moment anyone? The supreme court of India decided to strike it down in 2017 and let the netizens continue to have their freedom of speech intact.
Shah Bano Case:
Beyond indirectly curbing the right to speech, there have been many more controversial amendments under the rule of Congress party which the populace of India should never forget. One of the most notable among them is the controversial Shah Bano act, 1986, when a law was enacted by the Rajiv Gandhi government under pressure from the Muslim clergy to overturn the Supreme Court ruling.
The voice of a 62 year old woman, who was supported by the Supreme Court, to have the right to alimony was stifled. The law brought about by the government was in line with Muslim personal law, under which maintenance to a divorced woman is limited only to the period of iddat, or three menstrual cycles/three lunar months after divorce. Could there have been a more regressive approach towards a woman's rights? The capper is, same party today proclaims itself to be the beacon holder of women empowerment in the country. With this law the chances of having Uniform civil code too was trashed.
It is no surprise to the populace of India that parties in the opposition are picking up on non-existent issues in an attempt to improve their chances of a better performance in the elections of 2019. But with every attempt to malign Modi and his government falling flat on its face and biting the dust, opposition must now associate itself with something more credible.
There's a very famous saying "when you point a finger at someone, three fingers point back at you". Introspection time for congress party and cohort.
Posted in General, Levina, Politics
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Discussion in 'Staff Member' started by Cyrup, Nov 8, 2015.
Here you can apply for the position of Helper (staff member) on the Factory server!
All positions are voluntary. You will not be 'paid' (either in ranks, or currency or rewards of any form).
You cannot 'buy' or bribe your way into a staff member rank. That's not cool bro!
Do not post a thread with your application. Use the form.
you will only receive a response to your application if you are accepted to interview stages.
You must be at least 13 due to COPPA.
It is recommended you are available frequently through the week.
other languages aren't required, but they may be helpful.
What is the Helper rank?
The primary role of a Helper is to moderate chat, gain experience to become a Moderator (if you wish to do so), provide information, friendship and guidance to new and existing players, and generally keep the peace in a less serious way than a Moderator would do.
The Helper rank is in no way less 'equal' to that of a Moderator or Admin - we don't consider them to be inferior in any way - but it's a rank with fewer permissions and powers than a Moderator. The Helper rank is right for you if you wish to support and 'help' the server in keeping the peace, but without greater responsibilities and the specialised position of a Mod (i.e., you don't have to decide who gets to play on the server and who doesn't due to their bad actions). Remember, you won't just be a Helper for one part of the server (e.g. Strike or Gameshow), but the entire server.
To be a Helper, you need to have a good understanding of the server (so you can help players), an open learning attitude, a will to help whenever possible, a level of maturity and friendliness to all people - regardless of what situation you are caught in.
Consider the Helper as a 'normal' player but with the ability to keep things under control.
What is the Moderator rank?
The primary function of the Moderator is to enforce the rules and guidelines of the server, decide a player's ability to play or not play on the server (banning, muting, and for however long), and act as a role model for behaviour, support, and guidance. They communicate with Admins on major issues - e.g. hoards of spamming players, players that are out of their control, and so forth.Typically our Moderators should have experience in moderating - whether it be on Factory as a Helper or otherwise. You can become a Moderator only after you have been a Helper for at least 5 weeks.
It's a position with responsibility, and may not be for everyone, but it is very rewarding (in the psychological, feeling-good kinda way). Therefore you will find there are less Moderators than Helpers, as it's a very respected role which people who are excellent at how they do it can only retain.
If you are denied: You simply won't receive a reply. You may reapply again in the next intake period.
an interview will be conducted over Discord. We will not add you as a contact on Skype for privacy reasons. You will need to use Discord as a Mod anyway.
after being a Helper for at least 5 weeks you may apply for the position of Moderator - if you wish.
we want you to be detailed, but mention only what is necessary. we don't really need to know about why you started using minecraft, but knowing why you want to be a Helper is good!
Be honest. If you're not, it might lose you your rank when we find out something you lied about. It can affect your performance as a staff member.
Helper applications are open again! Click the link in the information above once you've read the application guidelines.


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Social media has become an integral piece of the internet marketing and optimization puzzle in recent years.
Social media goes hand-in-hand with content marketing and search engine optimization. With 56% of the population admitting to having at least one social media profile, it's time for website owners to take it seriously.
The Hermit offers social marketing strategies for all of the top social media outlets today. If they use it, we know how to market it.
The mention of social media marketing usually brings Facebook to mind first. The platform used to be for students with college email addresses only, but that changed quickly. Now it seems like everyone has "friended" at least one family member.
But that's a great thing for businesses on Facebook. Over half the US population used Facebook and 68% of those said the network positively influenced their purchasing decisions (2011 statistics). Similar to Google, Facebook has a proprietary algorithm, dubbed EdgeRank, and evolves, just like Google's own algorithm. This affects the visibility of your brand to your fans. SEO Hermit has a dedicated team posting to your company's Facebook page, keeping you relevant and visible.
Twitter delivers users' thoughts 140 characters at a time. The overload of material on Twitter (170 billion tweets & counting!) can make it difficult for small brands to be heard. Twitter is a global conversation, though, and fans do talk back.
Through the use of hashtags, companies can participate in relevant topics to educate and gain the trust of their potential clients. Twitter has 200 million active users each month, so it's worth gaining some of that market share.
Google Plus Marketing
If you're thinking about search engine optimization, Google search rankings are probably your #1 target for domination. Google Plus is Google's answer to the major social platforms like Facebook and Twitter. But they have an edge on those guys because you can consolidate virtually your entire life into Google products: email, social connections, entertainment, documents, and phone, just to name a few.
Google has really started to put emphasis on their new social platform in recent months. Maintaining an active status with your Google Plus company page is now a requirement for Google to view your online reputation favorably. A robust Google Plus presence is imperative.
With 1 billion active users each month, YouTube is a surprisingly effective tool for communicating with your audience. YouTube rounds out Google's social platform and is a must for each and every one of our SEO clients.
One of the golden rules of marketing is "Show, don't tell" and that applies to the internet, too. We have helped clients offering both product & services find a way to show their customers why they're the best in their field.
Ready to Take Your Business Social?
Let the SEO Hermit handle it for you!
LinkedIn is the place where professionals gather. Businesses have a shot at introducing themselves, hiring terrific employees, and participating in relevant conversations within LinkedIn Groups, all in one place. LinkedIn Company Pages are a must to complete your company's web branding profile, and bonus points for maintaining an active page. Over 7 million people are hitting this site every day (compared with just over 4 million unique daily visitors just 2 years ago), and they may be looking for what you have to offer!
Consider that the average Pinner spends between $140 and $180 when purchasing an item they saw on their feed, compared to the average Facebooker spending a maximum of only $80 (ReadWrite.com). Pinterest admittedly targets a niche — the audience is almost 80% adult women and products are best showcased here, typically. But some service companies we work with have done well on the platform. One thing's for sure: Pinterest shows hockey stick-like growth, which has outperformed even Facebook. We'd say that's not worth ignoring.
Snapchat Marketing
Snapchat is one of the newest social media platforms on the block. Snapchat is a picture-based messaging platform where users can send and receive photos to other users on the app. The catch? The photos expire within 1-10 seconds, depending on the sender's settings. Users can also post photos on their "Story" which remain visible for 24 hours to all of their Snapchat friends.
Very few brands are utilizing Snapchat, but don't let that discourage you from marketing your business on it. Some of the biggest brands are on Snapchat, including CNN, CoverGirl, Amazon, the San Francisco Giants and many more. No matter what industry you're in, Snapchat is an untapped goldmine for brand recognition and building a loyal following.
Tumblr serves as both a blogging platform and a social networking website, essentially having a built-in audience for your brand! It allows for users to upload multimedia and other content to short-form blogs (otherwise known as microblogs). Tumblr has been around almost as long as Twitter, but it does not have the same character and layout constraints that you see with Twitter, and many other leading social platforms, which makes it very easy to customize and brand your posts.
Currently, 53% of its users are between the ages of 18-34, making it a great tool to reach millennials. Major fashion and publishing brands such as Calvin Klein, GQ, Disney, and Coca-Cola are already taking advantage of this fact!
Get Started With Our Social Media Marketing
The above is just a shortlist of the most popular social media platforms existing today. Not all of them will work for every one of our clients, and there may be others not mentioned that are better suited for your niche. It's virtually impossible to keep up with your social media fans while you're trying to run your business. Contact The Hermit to discuss a tailored social media plan that's right for you.
What are you waiting for?Contact us today!

redpajama_set_name: RedPajamaCommonCrawl


First, we make sure that we have everything we need to begin the project. This includes your domain and hosting details, if necessary, as well as your creative brief and all the website business requirements.
Next, we organize all the information for the website into a hierarchical and meaningful structure so that it is easy for users to find what they need, and it is easy for web owners to publish their story.
The next phase is where we perform our magic and turn your beautiful prototype and interface into a fully working website. It's all very technical here, so we won't bore you with all the details. Needless to say we spend a lot of time indoors during this process.
Lastly, we put your website through our 42-point pre-flight check and then launch it into cyberspace so that the rest of the world can visit it on computers, tablets, and smartphones.


The Tom Green County Detention Facility is a new 215,647sf direct-supervision adult detention facility located in San Angelo, Texas. The base building contains 552 beds in various housing modalities including separation cells, single cells, multiple occupancy cells, and dormitories. The facility includes a variety of support spaces including intake, release, visitation, public lobby, jail administration offices, food service, laundry, truck dock, warehouse, medical, control rooms, exercise areas, multi-purpose rooms and maintenance areas. Of the 13 identified building areas, two areas are two-story and the balance are one-story.


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Ryan Reynolds Explains How He Got Brad Pitt To Cameo In Deadpool 2
David Pountain Jul 23, 2018 10:11 pm 2018-11-05T13:48:45-05:00
It's strange to think that Brad Pitt's blink-and-you'll-miss-it cameo in Deadpool 2 could end up being the only moment of screen time that the movie megastar will get in a cinematic release this year, with the actor seemingly saving himself for 2019's Ad Astra and Once Upon a Time in Hollywood.
You'd think there'd be plenty of producers ready to throw checks in the star's direction, and yet all that Pitt has brought to theaters in these past seven months has been this split-second appearance in the latest Wade Wilson actioner – a role which the A-lister apparently agreed to do for a very low salary and a cup of coffee.
It's a story that Deadpool 2 co-writer Rhett Reese brought up to Ryan Reynolds when the cast and crew were questioned by Entertainment Tonight, prompting the actor to reflect on Pitt's cushy deal.
"I was told all he wants is a cup of coffee and I said, 'Like a franchise or just one individual cup of coffee?' And I was told one individual cup of coffee, which was really his way of saying, 'I'm doing it for nothing.' And it was a total solid and the nicest thing anyone could do."
Pitt was cast as the appropriately named Vanisher, a member of Wade's ill-fated X-Force team who's neither seen nor heard until moments before his untimely death by electrocution. And as Reynolds explains, that's what he loved about the cameo.
"I just loved it, because what's more irresponsible than taking one of the biggest movie stars in the world and giving him a role that is utterly wordless and invisible with an exception of three frames of film? Yeah, that's kind of amazing."
New Deadpool 2 BTS Photos Show Us How Cable Was Brought To Life
While Reynolds undoubtedly has a lot of famous friends, director David Leitch argued that it was the cheeky spirit of the Merc with a Mouth's movies that really helped to reel in Pitt.
"I knew both of those guys and I worked with them before, but they would have never done a cameo in a movie I directed if it wasn't Deadpool. There's such a… beloved feeling for the things you guys created in the first one that it was an easy ask."
If you're keen to get a closer look at Brad Pitt's electrifying cameo – or want to check for yourself if that really was Matt Damon in an earlier scene – it's just a short wait until Deadpool 2: The Super Duper Cut hits Digital on August 7th and Blu-ray and DVD on August 21st. Also, if you haven't done so already, be sure to enter our contest for a chance to win a copy.


Orientations vary from Saturdays, Sundays, and very occasional weekday evenings, but we hold them every 4-6 weeks.
Come when you are ready!
Cohousing, housing cooperatives, cohouseholding, communes and ecovillages - Intentional communities, transition towns, nonprofit affordable housing, senior "villages" -- What's the difference and why does it matter?
What's living one of these communities really like? Should I join an existing community or start my own? Where do I find openings? How do cooperative communities really work? Who's in charge? You'll get answers and links to a world of resources at our EBCOHO orientation.
Very informative and well-presented. Nice group of people, too (of course!).
Doors open at 9:45 for a little social time and set up; Program runs from 10:00 to 11:30, including a little more time for networking, wrapping up by 12. Light refreshments available, but lunch is not included.
Member introductions, goals and connections.
How to use EBCOHO and other online resources.
Community-building game or exercise, and time for open sharing on your experience and interests.
Raines Cohen, Cohousing Coach and community organizer, will present and facilitate.
The orientation fee of $20 per household helps cover ongoing EBCOHO expenses. Bring a friend and save up to $5 each; supporting members (and members of supporting groups) pay just $10.
Your contribution is refunded if you become a supporting member on the spot. (Supporting members get our thanks, a free Cohousing book or Communities Magazine subscription, plus discounts on many of our events).
If you RSVP for multiple people, MeetUp will tell you the price is per person, but relax... it's really just per household. Bring your visiting relatives or housemates. You can pay at the door by cash, check, or credit card. Get your better half or thirds(!) to join you.
If you can't come out with us on a Sunday morning -- fear not! Keep eye on the EBCOHO calendar for what's coming up, weekend and weeknight orientations, (un)common meals, socials, movies, and other ways to connect. We hope to see you soon.
Get not just this meeting but a featured profile, discounts on future events for the next year, early notice of openings, a cohousing book and more.


Adding a new tier to the i heart Wines family, i heart Regional Wines are carefully crafted from some of the most prestigious wine regions around the world- in this case Marlborough, New Zealand.
Bright flavours of citrus and gooseberry are complimented by hints of grapefruit and passionfruit on the nose. The vibrant acidity is balanced by luscious tropical fruit giving it a refreshing finish.
Having Pasta? Try with a creamy carbonara or an herby pesto, the crisp acidity and zesty lemon flavours of this wine cut through the creaminess of these dishes perfectly.
Or seafood? Pair with a seafood risotto for a match made in heaven.
How about Chicken? Goes beautifully with chicken pie.
To be enjoyed on its own too!
Marlborough enjoys some of the longest daylight hours in New Zealand.


We have big news | RevoLusion Consultants, Inc.
News From RevoLusion Consultants, Inc.
Even greater success is on the horizon for RevoLusion Consultants, Inc.! Our people are reaching new targets every day. Visit here to stay informed.


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The provincial government is facing cost pressures in designing a planned overpass of CN rail tracks at a level crossing on Hwy16 west of Terrace. (File photo)
CN level crossing overpass project likely over budget
Province cites complexity of construction plan for rail crosing between Prince Rupert and Terrace
Rod Link
A project to build an overpass over the CN level crossing on Highway 16, 50km west of Terrace, first announced in 2015, is likely to cost more than first anticipated with a construction start not scheduled until next year.
It had a $37 million price tag — $17.5 million from the federal government with the remaining $19.5 million coming from the province — when announced in July 2015 during the run up to the federal election of that year but has missed several earlier planned construction start dates.
"This has been a complex project from its inception, with design work, environmental reviews, and stakeholder consultations requiring significant time and effort from the project team," a statement from the provincial infrastructure and highways ministry now indicates.
"At this time, the budget is still in review and the project team is actively working through the assessment."
While the federal money remains committed, this is a provincial government project and it is now assessing cost pressures.
"Federal funding of $17.5 million has been committed to this project and the project team is currently reviewing the budget. The province considers all options, including changes in project scope, when reviewing budget pressures," the ministry added.
READ MORE: Mile 28 level crossing and Highway 16 improvements
The ministry cited grading and rock blasting needed for new approaches to the overpass and the structural design of the overpass itself as factors making the project a complex one.
"Stakeholder consultations have been ongoing with CN Rail, regional districts, First Nations, and the Ministry of Environment," the ministry continued.
"Discussions have focused on ensuring the project does not have a negative impact on the fish habitat in the Skeena River and surrounding riverside vegetation, or the mountain goat habitat and other wildlife in the area. Since work will be taking place above an active railway, careful coordination with CN Rail is required to determine work windows and blasting operations."
The level crossing, which features a twisty 's' curve for traffic crossing over CN's tracks, is the last of its kind along Hwy16 within B.C. and replacing it with an overpass was touted as a safety measure and in recognition of the growing volume of rail traffic to and from Prince Rupert's port operations.
The initial design of the project calls for two new kilometres of Hwy16 in addition to the overpass itself.
CN is making no financial contribution to the project.
RELATED: Highway 16's Car Wash Rock addressed by NCRD
newsroom@thenorthernview.com
CN RailHighway 16
Martin Mars unlikely to see firefighting action in Port Alberni
Children born to spies in Canada should not be handed citizenship: Ottawa


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Q: Dynamic create environment and run Cypress parallel We are using Cypress to run our end-2-end-tests in GitLab. Before we run the test we create a dynamic environment. A dynamic environment is an environment which is created with docker-compoe inside the gitlab runner which executes the cypress tests. After the dynamic environment is up fire the tests against this dynamic environment. Everything happens in one gitlab-runner so no external deployment to a test environment takes place.
Now we want to move forward and parallelize the cypress run. Its documented here https://docs.cypress.io/guides/guides/parallelization and it is working under the assumption that the environment is already there. It creates several GitLab runners and cypress takes care for distribution the scenarios between the runners.
The question is, how to set up a dynamic environment with GitLab which can be shared between GitLab runners? Is it only possible with a dummy deployment to a Kubernetes environment which is prepared for this user case? Do I need create a dynamic environment in each runner? Or any other hints?

A: Our solution now looks like this.
We start the environment from docker-compose in each running gitlab-job. This of course reduces the time-saving of parallel execution of tests since gitlab runners are blocked longer. But since it happens in parallel and we are not short on runners its OK for us. An other advantage was that we could keep our current setup with only minimal modifications. In the end we reduced our "run all end 2 end tests" - execution time by about 30%

redpajama_set_name: RedPajamaStackExchange


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What is McDonald's net worth?
By this GOBankingRates metric, McDonald's has a net worth of $36.53 billion.
McDonald's Net Worth.
What Is McDonald's Worth?
Fiscal Year 2021 Profit $7.55 billion
GOBankingRates' Evaluation of McDonald's Net Worth $36.53 billion
What is the profit on a Big Mac?
It stated that the cost of raw materials making a Bic Mac was accumulated to be $0.80. And as far as I know, a Big Mac is priced around $4. Therefore, McDonalds make around $3.20 per Big Mac, if you only take raw materials into account.
How much does 1 McDonalds store make a year?
McDonald's has close to 14,000 stores nationwide. If you've already been doing the math in your head, you'll realize each location makes an average of $2.7 million annually, according to QSR data available from 2018, the most recent we can find individualized U.S. data for.
How much do McDonalds owners make a year?
Average McDonald's Owner yearly pay in the United States is approximately $300,000, which is 376% above the national average.
What is McDonald's net worth? – Related Questions
How much of the cost of a Big Mac is labor?
In general, McDonald's franchisees pay about 20 percent in labor costs, according to Richard Adams, a consultant out of San Diego who works with McDonald's operators. Thus, doubling those salaries would push that Big Mac cost up 80 cents.
How much does average McDonald's make a day in profit?
On average, McDonalds makes $75 million revenue in a day! With a turn over of $27 billion annually, it is the 90th largest economy in the world!
SEE ALSO How can I make 500000 fast in GTA?
Are McDonalds profits up 59%?
Conversation. McDonald's, now with higher prices, topped $23 billion in revenue in 2021. Profit soared 59 percent from a year earlier, to $7.5 billion.
How much does a burger cost McDonald's to make?
While food and packaging costs just 45 cents for a double cheese burger, franchisees also have to pay for rent, labor and utilities. In total, a promotional price of just $1 leaves store operators with a measly 6 cents of profit, according to a franchisee in Florida.
How many burgers does one Mcdonalds sell in an hour?
McDonald's sells more than 75 hamburgers every second
This is roughly 4,500 burgers every minute, 270,000 every hour, 6.48 million every day, and 2.36 billion burgers every year.
Who sets prices at mcdonalds?
One reason they're just fine with it has been those prices increases and consumers' apparent willingness to pay them. McDonald's franchisees have full control of their prices and use Deloitte to recommend pricing on a local level, Lewis said, based on a restaurant's costs.
Where does McDonald's make the most money?
While the company is a huge player in the fast-food market, its biggest revenue source is renting its locations to franchisees. As a result, we can think of McDonald's as a real estate empire, more than a fast-food chain. This model provides McDonald's with a steady revenue stream from rent and royalties on food sales.
What is sold the most at mcdonalds?
The Most Popular Thing at McDonald's Is…
Ever since the first McDonald's menu from when the chained opened in the 1950s, french fries have been a star. The famous Golden Arches sell upward of 9 million pounds of fries per day!
SEE ALSO How much can you make on Pococha?
What is the busiest McDonald's?
It's actually in Russia. There's more than 200 McDonald's outlets in Russia, but the one in Pushkin Square in Moscow holds the distinction as being the busiest in the world.
Does McDonald's make money off their food?
The company makes money by leveraging its product, fast food, to franchisees who have to lease properties, often at large markups, that are owned by McDonald's. Franchisees are lured by the impressive margins that make McDonald's franchises an almost guaranteed moneymaker.
Who owns the most McDonald's franchises?
(English: "Golden Arches Holdings, Inc") is a company that owns the master franchise of the fast food restaurant chain McDonald's in 20 countries within Latin America and the Caribbean. It is the largest McDonald's franchisee in the world in terms of system-wide sales and number of restaurants.
Who does mcdonalds get their meat from?
Lopez Foods provides McDonald's U.S. Supply Chain with USDA-inspected beef used to make the patties served in restaurants.
Which fast food restaurant makes the most money?
What is the #1 fast food?
What's the #1 fast food chain in the world?
McDonald's is the world's largest fast-food restaurant chain and one of the best-known brand names. The company has more than 39,000 locations in about 100 countries.
What jobs pay the most per hour?
Highest-Paying Hourly Jobs
U.S. Marine Corps officer.
Tattoo artist.
Political speechwriter.
Health coach.
Wedding planner.
Interior designer.
Police officer.
Commercial pilot.
What profession makes $100 an hour?
The top three jobs that pay $100 an hour are anesthesiologist, life coach, and simultaneous interpreter. The median hourly wage in the United States is $16 an hour. Most of these jobs require a college degree, certifications, and training.
SEE ALSO What are the best stocks to make quick money?
How much does Max Verstappen earn from Red Bull?
How much does Microsoft make in profit?


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Q: Help with proof : $Cl_F$ with only one equivalance class implies $D_F$ is a PID I am self-studying a set of legitimately downloaded notes on algebraic number theory. They are somewhat akin to "Ireland and Rosen," Ch. 12.
I would appreciate help in understanding a proof (in the direction) that if $Cl_F$ consists of one equivalence class, then $D_F$, the ring of integers of $F$, is a PID. $F$ is a number field.
For convenience, I've numbered the four questions I would like help with.
Using the definition of equivalence in this context, if there is only one equivalence class, any two ideals of $D_F$ are equivalent and there exist $\alpha, \beta \in D_F$ such that
$\alpha I = \beta D_F$,
where $I$ is any ideal in $D_F$. We want to show $I$ is a principal ideal. So far so good. (1) Although the notes has $\alpha, \beta \in I$, I think they ought to be in $D_F$. Is this correct?
Thus $\alpha I = (\beta)$.
The proof goes on: Let $\omega = \beta \alpha^{- 1} \in F$.
Now I get stuck:
Then (2) $\omega I = \beta I \subseteq I$. I don't see how to get the equality.
Based on this, we know $\omega \in D_F$. This is clear since this type of assertion, based on the inclusion, was previously proved.
Then I also can't get the final assertion, (3) that this implies $(\omega) = I$.
Lastly, I would appreciate any guidance or references as to (4) in general what the product, e.g., $\gamma J$ means, where $J$ is an ideal.
As always, thanks for your patience and help.

A: Hint $\rm\ \ a I = (b)\:\Rightarrow\: b\in aI\:\Rightarrow\: b = ai\:$ thus $\rm\:aI = a(i)\:\Rightarrow\: I = (i)\:$ by cancelling $\rm\:a\ne 0$.
Principal ideals are invertible, so cancellable, thus we can cancel $\rm\,(a)\ne 0\:$ from $\rm\,(a)I = (a)(i)\,$ above.  If you're not familiar with invertible or fractional ideals then you can easily prove this directly: suppose $\rm\:aI = aJ\ne 0.\:$ To show $\rm\:I\subseteq J\:$ note $\rm\:i\in I\:\Rightarrow\:ai\in aI\subseteq aJ,\:$ so $\rm\:ai = aj\:$ for some $\rm\:j\in J.\:$ Cancelling $\rm\:a\ne 0\:$ yields $\rm\:i = j\in J,\:$ so $\rm\:I\subseteq J.\:$ By symmetry, $\rm\:J\subseteq I,\:$ therefore $\rm\:I = J.$


Holiday.gr - Lampi, a trip to a myth!
Very close to the town of Kos is Lampi beach. It has been awarded with the Blue Flag from the European Union. It has shallow and crystal clear waters that make it ideal for those with no swimming skills. It is an organized beach with all the facilities for water sports.
There are hotels of all categories, restaurants and bars. It is a developing area with all the qualifications to become a cosmopolitan summer resort.
Been to Lampi ? Share your experiences!


Katte Masoor Dal Recipe/ Sour Lentil Curry Recipe is a delicacy from Chhattisgarh, simple to make and yet tasty as well as healthy.
Serve Khatte Masoor Dal with rice and you will be eating just what Folks from the "Rice Bowl of India" eat.
Like the rest of the country, Chhattisgarh cuisine is influenced by the cuisine of the neighbouring states. Though the staple food is wheat, jowar and maize, rice is also widely consumed. Rice is used to make food as well as savoury food and hence the name as the "Rice Bowl of India". In Chhattisgarh cuisine, the usage of spices and oil is sparing. However protein-rich lentils especially Toor dal also called Arhar Dal are enjoyed.
For January 2018, we are making recipes from Chhattisgarh cuisine for "Ssshhhh Secretly Cooking." Amrita Iyer is my partner, who choose masoor and rice as my secret ingredients.
I choose to make Khatte Massor Dal with rice crepes but finally had to settle for steamed rice. This recipe I copied from here.
Curds used to make the masoor to make it more flavourful, tasty, and tangy. The masoor dal is so delicious, flavourful and filling and the dal does not need any side dish to complete it.
In the past, I made Fara (Phara) for Blogging Marathon and I had a list of the dishes that I wanted to make but unfortunately, I cannot find it now. Anyway will get down to what I want to try later.
So the main ingredients are whole masoor, curds. The rest you can add or reduce as you wish.
You can use tamarind also to make the dal but then it tastes different from this one. Trust me I make the tamarind one on a regular basis.
Wash and rinse the masoor pick stones out if any, and then pressure cook for 2 whistles on high flame and keep for about 10 minutes on low flame. Let the pressure come down naturally.
Meanwhile, slice the onions and chop tomato fine.
In a pan/kadhai/wok heat, the oil add the jeera and once they splutter, add the hing.
Add onion and sauté for a minute.
Add ginger-garlic paste and cook until onion becomes light golden brown.
Next add the add chilli powder, coriander powder and turmeric powder.
Mix well and sauté for a minute and add tomato and cook until are mashed well.
Cook until the oil comes out.
Add cooked dal and 1-2 cups water and bring it to boil.
Add salt and beaten curd mix well and bring it to a boil.
Garam masala and simmer for 5 minutes on low flame.
Switch off the flame and cover the dal.
Let it sit covered for 5-10 minutes then serve.
Serve hot with steamed rice or roti and sides of pickle, papad and a simple stir-fry.
A delicacy from Chhattisgarh, simple to make and yet tasty as well as healthy.
Comfort meal.. looks so tempting.
That looks delicious. I love masoor dal and gravies that are curd based so this dish is just right for me. Mildly spiced and full of flavours as I can see from the recipe. A Great share.
Dal looks lovely.. I love any kind of dal preparation.
I make masoor dal often but definitely will have to try the khatta version. Must be so tasty… but then I love yogurt in anything.
This is an interesting recipe and the dal looks scrumptious! Ive made number of dals but hadn't thought of using curd in any of them! Must try this soon!
I Love gravies and this looks so tempting. No doubt about the taste of it.
Would love to enjoy this katte masoor dal with some rice and papads, delicious dish.
Different method of dal preparation , never tasted masoor dhal…….
Heatlhy version of dhal with curd and I love the curry colour.. Can dig bowl of hot steamed rice with this curry..
You hit it on the nail. Tastes great with rice.
Such a comforting and delicious Khatte Masoor, protein-packed dishes are welcome in my kitchen.


August Hill Winery is owned by Mark & Teri Wenzel. Mark watched his grandfather, August Engelhaupt, carefully tend the scenic piece of land that is now the location of the AHW vineyard and winery. Mark shares the passion that Grandpa Augie had for growing premium crops. When he discovered that French-hybrid grapes grow well in our climate, the concept for August Hill Winery was born.
The AHW motto is "From the heart to the hand, a passion for creation." We take this seriously at August Hill. We are passionate about creating wines with the upmost care and attention to detail. From harvesting and crushing, to fermentation and barrel-aging, all phases of grape production and the winemaking process are carefully monitored by our winemaker, Mark Wenzel.
Caramel Apple – Enjoy refreshing flavors and aromas of crisp, juicy apples and delicious, creamy caramel in this light, sweet wine.
Niagara – A sweet, fruity, and very aromatic white wine with a strong American grape flavor.
Ravat 51 – A dry, medium-bodied wine filled with notes of rose, pineapple, lemon, orange, and moonflower. Similar to a dry German Riesling, this wine is from Vignoles grapes.
Vignoles – semi-dry, light-bodied wine with tropical fruit aromas and flavors. Vignoles is a French-American hybrid grape. This wine is similar in style to a Riesling.
Sweet Catawba – A light, sweet, and fruity blush wine. This wine is made from American Grapes and has great fruit flavor. Catawba pairs well with light fish dishes, cheese and crackers, or friends who just want to sip and relax.
Augie's Blackberry – Fruit-forward port style wine made from fresh blackberries. Named after the grandfather of August Hill winemaker, because Augie had a passion for farming and eating fruit such as blackberries.
Moscato – This slightly sparkled wine is elegant with a fruity bouquet of peaches and honey and has an excellent balance between its level of sweetness and acidity. It is a wine intended for pure and simple pleasure.
Berlyn – A light, semi-dry red wine made from Frontenac and Chambourcin grapes grown in Illinois.
Ginocchio XI – This wine is made from a premium blend of Zinfandel, Norton, and Chancellor grapes. Aged in oak barrels, this medium-bodied dry wine has notes of raspberry, black pepper, cherry, vanilla, and cedar.
Trapolino – A medium-bodied semi-sweet red table wine that is very approachable and fruity. Excellent selection for first-time red wine drinkers.
Chambourcin – A smooth and timeless dry, red table wine made from Illinois-grown Chambourcin grapes. This medium-bodied wine is similar in style to a light pinot noir with its upfront fruit character and aromas and flavors of blueberry and dark chocolate.
Almond Infusion – This light sweet bubbling wine is infused with lively almond aromas and natural flavors. This is a great wine to introduce you to bubbles because the sweetness and flavors make it very easy to drink.
Mardi Gras – Mardi Gras! Celebration! We have taken one of our most popular wines, Sweet Catawba, and added bubbles. This sweet, bubbly wine is sure to bring out the fun all year long.
Cherry Lime Infusion – This light, sweet bubbling wine is an infusion of lively cherry and lime aromas and natural flavors. Lime flavor has been added to the base cherry wine to balance the sweetness and acidity.
Illinois Sparkling Co.'s traditional method sparkling wines are also served at this booth. Click here for the complete listing.
The Vintage Illinois designated driver program offers complimentary non-alcoholic beverages to all designated drivers. Upon entering, the DD receives a special color wrist band and two tickets for an alternative beverage to be enjoyed during the festival. VI thanks all our DD's!


text:
Universal quantum Hamiltonians
Toby Cubitt1 Ashley Montanaro2 and Stephen Piddock2
1 Department of Computer Science, University College London, UK
2 School of Mathematics, University of Bristol, UK.
Quantum many-body systems exhibit an extremely diverse range of phases and physical phenomena. Here, we prove that the entire physics of any other quantum many-body system is replicated in certain simple, "universal" spin-lattice models. We first characterise precisely what it means for one quantum many-body system to replicate the entire physics of another. We then show that certain very simple spin-lattice models are universal in this very strong sense. Examples include the Heisenberg and XY models on a 2D square lattice (with non-uniform coupling strengths). We go on to fully classify all two-qubit interactions, determining which are universal and which can only simulate more restricted classes of models. Our results put the practical field of analogue Hamiltonian simulation on a rigorous footing and take a significant step towards justifying why error correction may not be required for this application of quantum information technology.
Contents\@mkbothCONTENTSCONTENTS
\@afterheading\@starttoc
List of Theorems
Definition (Analogue Hamiltonian simulation)
Theorem (follows from [JR52] , Theorem 4 and [Mar67] , Theorem 2)
Theorem (Encodings)
Definition (Standard encoding)
Definition (Local encoding)
Lemma (Lemma 3.3 of [AE11] )
Corollary (Product-preserving encodings)
Lemma (First-order simulation [BH17] )
Lemma (Second-order simulation [BH17] )
Lemma (Third-order simulation [BH17] )
Theorem (essentially [OT08] )
Definition (Spatial sparsity [OT08] )
List of Figures \@mkbothLIST OF FIGURESLIST OF FIGURES
Part I Extended overview
The properties of any physical system are captured in its Hamiltonian, which describes all the possible energy configurations of the system. Amongst the workhorses of theoretical many-body and condensed matter physics are spin-lattice Hamiltonians, in which the degrees of freedom are quantum spins arranged on a lattice, and the overall Hamiltonian is built up from few-body interactions between these spins. Although these are idealised, toy models of real materials, different spin-lattice Hamiltonians are able to model a wide variety of different quantum phases and many-body phenomena: phase transitions [Sac07] , frustration [Die13] , spontaneous symmetry-breaking [ADZ12] , gauge symmetries [Kog79] , quantum magnetism [SRFB08] , spin liquids [ZKN16] , topological order [Kit03] , and more. In this work, we prove that there exist particular, simple spin models that are universal: they can replicate to any desired accuracy the entire physics of any other quantum many-body system (including systems composed not only of spins, but also bosons and fermions). This implies, in particular, that the ground state, full energy spectrum and associated excited states, all observables, correlation functions, thermal properties, time-evolution, and also any local noise processes are reproduced by the universal model.
Note that this is a very different notion of "universality" from that of universality classes in condensed matter and statistical physics [Car96] . Universality classes capture the fact that, if we repeatedly "zoom out" or course-grain the microscopic degrees of freedom of a many-body system, models that are microscopically different become increasingly similar (converge to the same limit under this "renormalisation group flow"), and their macroscopic properties turn out to fall into one of a small handful of possible classes. The "universality" we are concerned with here [lCC16] has a completely different and unrelated meaning. It is closer to the notion of universality familiar from computing. A universal computer can carry out any possible computation, including simulating completely different types of computer. Universal models are able to produce any many-body physics phenomena, including reproducing the physics of completely different many-body models.
One might expect that universal models must be very complicated for their phase diagram to encompass all possible many-body physics. In fact, some of the models we show to be universal are amongst the simplest possible. In particular, we prove that allowing only the strengths of the local interactions to vary, the Heisenberg model on a 2D square lattice of spin-1/2 particles (qubits) with nearest-neighbour interactions and non-uniform coupling strengths is universal. This is an important and somewhat surprising example, as it is a 2D model with the simplest possible local degrees of freedom (qubits), short-range, two-body interactions, and the largest possible local symmetry (full SU(2) invariance). Yet our results prove that by varying only the coupling strengths, the 2D Heisenberg model can replicate in a complete and rigorous sense the entire physics of any other many-body model, including models with higher spatial dimensions, long-range interactions, other symmetries, higher-dimensional spins, and even bosons and fermions.
In addition to the new relationships this establishes between apparently very different quantum many-body models, with implications for our fundamental understanding of quantum many-body physics, there are also potential practical applications of our results in the field of analogue quantum simulation. There is substantial interest nowadays in using one quantum many-body system to simulate the physics of another, and one of the most important applications of quantum computers is anticipated to be the simulation of quantum systems [BMK10, GAN14, CZ12] .
Two quite different notions of Hamiltonian simulation are studied in the literature. The first concerns simulating the time-dynamics of a Hamiltonian on a quantum computer using an algorithm originally proposed by Lloyd [Llo96] , and refined and improved in the decades since [BACS07, BCC+14, BCK15, LC16] . This is the quantum computing equivalent of running a numerical simulation on a classical computer. However, it requires a scalable, fault-tolerant, digital quantum computer. Except for small-scale proof-of-principle demonstrations, this is beyond the reach of current technology.
The second notion, called "physical" or "analogue" – in the sense of "analogous" – Hamiltonian simulation, involves directly engineering the Hamiltonian of interest and studying its properties experimentally. (Akin to building a model of an aerofoil and studying it in a wind tunnel.) This form of Hamiltonian simulation is already being performed in the laboratory using a variety of technologies, including optical lattices, ion traps, superconducting circuits and others [STH+99, Nat12, GAN14] . Just as it is easier to study a scale model of an aerofoil in a wind tunnel than an entire aeroplane, the advantage of artificially engineering a Hamiltonian that models a material of interest, rather than studying that material directly, is that it is typically easier to measure and manipulate the artificially-engineered system. It is possible to measure the state of a single atom in an optical lattice [SWE+10, BGP+09, GZHC09] ; it is substantially harder to measure e.g. the state of a single electron spin in a particular 2D layer within a cuprate superconductor.
Many important theoretical questions regarding analogue quantum simulation remain open, despite its practical significance and experimental success [STH+99, Nat12, GAN14] . Which systems can simulate which others? How can we characterise the effect of errors on an analogue quantum simulator? (Highlighted in the 2012 review article [CZ12] as one of the key questions in this field.) On a basic level, what should the general definition of analogue quantum simulation itself be? The notion of universality we develop here enables us to answer all these questions.
1 Background and previous work
This computationally-inspired notion of physical universality has its origins in earlier work on "completeness" of the partition function of certain classical statistical mechanics models [VdNDB08, KZ12a, DlCDVdNB09, DlCDBMD09, KZ12b, XDlCD+11] . Recent results by one of us and De las Cuevas built on those ideas to establish the more stringent notion of universality for classical spin systems [lCC16] . Related, more practically-focused notions have also been explored in recent work motivated by classical Hamiltonian engineering experiments [LHZ15] . Here we consider the richer and more complex setting of quantum Hamiltonians, which requires completely different techniques.
Pinning down precisely what it means for one many-body model to simulate the complete physics of another requires applying mathematical theory of spectrum-preserving maps and Jordan algebras developed between the 1950s and 80s, which has to our knowledge not arisen before in quantum information theory. Using this, we are able to derive from basic operational considerations the precise conditions under which one quantum many-body Hamiltonian exhibits exactly the same physics as another. This is a necessary precursor to establishing our main result – showing for the first time that there exist universal quantum Hamiltonians – and allows us to prove these new physics results with full mathematical rigour (see part II for full technical details). However, to our knowledge no general understanding – or even definition – of when one Hamiltonian simulates the complete physics of another existed in the literature. Thus this precursor to our main result may be of significance in itself.
For our explicit constructions that establish the existence of universal Hamiltonians, we are able to draw on a long literature in the field of Hamiltonian complexity [KKR06, OT08, AGIK09, BL08, CM16, BH17] , studying the computational complexity of estimating ground state energies by mapping the problem to ever-simpler quantum many-body systems. The computational complexity of the ground state energy problem for the Heisenberg model with arbitrarily varying local fields was shown to be the maximum possible (QMA-complete [KSV02] ) by Schuch and Verstraete [SV09] . The availability of local fields breaks the symmetry of the model, simplifying the analysis. The complexity of the pure Heisenberg model without local fields was not known until very recently, when it was shown by two of us [CM16] to also be QMA-complete.
These results per se only concern the ground state energy, and moreover only the computational complexity aspects of this single quantity; they do not need to address any of the physics of the resulting Hamiltonians beyond the ground state energy. Nonetheless, the "perturbative gadget" techniques developed to prove Hamiltonian complexity results [KKR06, OT08] turn out to be highly useful in constructing the full physical simulations required for our results. By combining our new and precise understanding of physical Hamiltonian simulation with these perturbative gadget techniques, we are able to design new "gadgets" that transform one many-body Hamiltonian into another whilst preserving its entire physics and local structure, as required to construct universal models.
In this way, we are able to show how certain quantum many-body models can be transformed step-by-step into any other many-body Hamiltonian, thereby establishing that these models are universal. On a high level, as discussed in [lCC16] for the classical case, this process can in some sense be viewed as the "opposite" of a renormalisation group flow: depending on the initial microscopic parameter settings, universal Hamiltonians can "flow" under this sequence of transformations to any other many-body Hamiltonian, which can have very different physical characteristics. Note that our result is constructive: it provides an efficiently computable algorithm that, given a description of any quantum many-body Hamiltonian, produces the parameter settings of the universal model that simulate this Hamiltonian.
In fact, we go beyond exhibiting individual examples, and completely classify the simulation power of all two-qubit interactions. From this, we see that essentially all many-body models in 2D (or higher) with two-qubit interactions and individually tunable coupling strengths are universal (see part II). The 2D Heisenberg model, and also the 2D XY model (which has local U(1) invariance), arise as important specific cases of this.
2 Hamiltonian simulation
We start by establishing precisely what it means for one quantum many-body system to simulate another. Any non-trivial simulation of one Hamiltonian H with another H′ will involve encoding the first within the second in some way. We want this encoding H′=E(H) to "replicate all the physics" of the original H system. To reproduce all static, dynamic and thermodynamic properties of H, the encoding E needs to fulfil a long list of operational requirements:
Clearly E(H) should be a valid Hamiltonian: E(H)=E(H)†.
E should reproduce the complete energy spectrum of H: spec(E(H))=spec(H). More generally, E(M) should preserve the outcomes (eigenvalues) of any measurement M: spec(E(M))=spec(M).
Individual interactions in the Hamiltonian should be encoded separately: E(∑iαihi)=∑iαiE(hi). Otherwise, one would have to solve the full many-body Hamiltonian in order to encode it, in which case there is little point simulating it in the first place.
There should exist a corresponding encoding of states, Estate, such that measurements on states are simulated correctly: for any observable A, Tr(E(A)Estate(ρ))=Tr(Aρ).
E should preserve the partition function (potentially up to a physically unimportant constant rescaling): ZH′(β)=Tr(e−βE(H))=cTr(e−βH)=cZH(β).
Time-evolution according to E(H) should simulate time-evolution according to H.
Errors or noise on the E(H) system should correspond to errors or noise on the H system.
We prove (see part II) that, remarkably, the very basic requirements (iii), (ii) and (i) already imply that all other operational requirements are satisfied too. Furthermore, any encoding map E that satisfies them must have a particularly simple mathematical form:
E(H)=U(H⊕p⊕¯H⊕q)U† (1)
for some unitary U and non-negative integers p, q such that p+q≥1. (¯H denotes complex conjugation of H.)
This characterisation of Hamiltonian encodings holds if the simulation is to exactly replicate all the physics of the original. But in practice no simulation will ever be exact. What if the simulator Hamiltonian H′ only replicates the physics of the original Hamiltonian H up to some approximation? As long as this approximation can be made arbitrarily accurate, H′ will be able to replicate the entire physics of H to any desired precision.
Furthermore, it is clearly sufficient if H′ replicates the physics of H for energies below some energy cut-off Δ if that cut-off can be made arbitrarily large (see Figure 1). Due to energy conservation, any initial state with energy less than the energy cut-off will not be affected by the high-energy sector. Indeed, as long as the cut-off is larger than the maximum energy eigenvalue of H, this means that H′ can simulate all possible states of H. This also holds for all thermodynamic properties; any error in the partition function due to the high-energy sector is exponentially suppressed as a function of the cut-off. In practice, one is often only interested in low-temperature properties of a quantum many-body Hamiltonian, as these are the properties relevant to quantum phases and quantum phase transitions. In that case, the energy cut-off does not even need to be large, merely sufficiently above the lowest excitation energy. We therefore want to simulate H in the low-energy subspace of H′.
λ1
λN
ϵ
λ′1
λ′N
λ′N+1
Figure 1: Simulating one Hamiltonian within the low-energy space of another. H′ (on right) simulates H (on left) to precision (η,ϵ) below energy cut-off Δ.
Finally, for a good simulation we would also like the encoding to be local, in the sense that each subsystem of the original Hamiltonian corresponds to a distinct subset of particles in the simulator Hamiltonian. This will enable us to map local observables on the original system to local observables on the simulator system, as well as to efficiently prepare states of the simulator system.
By making all the above mathematically precise, we show that this necessarily leads to the following rigorous notion of Hamiltonian simulation (also see Figure 1):
A many-body Hamiltonian H′ simulates a Hamiltonian H to precision (η,ϵ) below an energy cut-off Δ if there exists a local encoding E(H)=V(H⊗P+¯H⊗Q)V†, where V=⨂iVi for some isometries Vi acting on 0 or 1 qudits of the original system each, and P and Q are locally orthogonal projectors, such that:
There exists an encoding ˜E(H)=˜V(H⊗P+¯H⊗Q)˜V† such that ˜E(\mathbbm1)=P≤Δ(H′) and ∥˜V−V∥≤η;
∥H′≤Δ−˜E(H)∥≤ϵ.
Here, P≤Δ(H′) denotes the projector onto the subspace spanned by eigenvectors of H′ with eigenvalues below Δ, and we write H′≤Δ=P≤Δ(H′)H′.
The first requirement in Definition (i) states that, to good approximation (within error η), the local encoding E approximates an encoding ˜E onto low-energy states of H′. The second requirement, 2(ii), says that the map ˜E gives a good simulation of H to within error ϵ.
This definition, which we show follows from physical requirements, turns out to be a refinement of a definition of simulation introduced in prior work [BH17] in the context of Hamiltonian complexity theory. There are two important differences. We allow the encoding map E to be anything that satisfies the physical requirements (iii), (ii) and (i) from the previous section, which can be more complicated than a single isometry. On the other hand, we restrict E to be local, since we require simulations to preserve locality. Note that if η=ϵ=0 and Δ→∞, the simulation is exact. Increasing the accuracy of the simulation will typically require expending more "effort", e.g. by increasing the energy of the interactions.
We are usually interested in simulating entire quantum many-body models, rather than individual Hamiltonians. By "model", we mean very generally here any family of Hamiltonians. In the many-body models usually encountered in physics, these Hamiltonians are typically related to one another in some way. For example, the 2D Heisenberg model consists of all Hamiltonians with nearest-neighbour Heisenberg interactions on a 2D square lattice of some given size.
When we say that a model A can simulate another model B, we mean it in the following strong sense: any Hamiltonian H on n qudits (d-dimensional spins) from model B can be simulated by some Hamiltonian H′ on m qudits from model A, and this simulation can be done to any precision η,ϵ with as large an energy cut-off Δ as desired. The simulation is efficient if each qudit of the original system is encoded into a constant number of qudits in the simulator (each Vi in Definition maps to O(1) qudits), H′ is efficiently computable from H, and the energy overhead and number of qubits of the simulation scales at most polynomially (∥H′∥=poly(n,1/η,1/ϵ,Δ) and m=poly(n,1/η,1/ϵ,Δ)).
Figure 2: Schematic illustrating simulation of one Hamiltonian with another. Each logical (red) qubit is encoded within 4 physical (blue) qubits, forced into their ground space by strong pairwise interactions. Interactions between the physical qubits implement effective interactions between the logical qubits. An error on a physical qubit only affects one logical qubit.
3 Implications of simulation
We arrived at a rigorous notion of Hamiltonian simulation by requiring the simulation to approximate the entire physics to arbitrary accuracy. This is clearly very strong. Just as exact simulation preserves all physical properties perfectly, approximate simulation preserves all physical properties approximately. First, all energy levels are preserved up to any desired precision ϵ. Second, by locality of E, for any local observable A on the original system there is a local observable A′ on the simulator and a local map Estate(ρ) such that applying A′ to Estate(ρ) perfectly reproduces the effect of A applied to ρ. This applies to all local observables, all order parameters (including topological order), and all correlation functions. Thus all these static properties of the original Hamiltonian are reproduced by the simulation.
Third, Gibbs states of the original system correspond to Gibbs states of the simulator, and the partition function of H is reproduced by H′, up to a physically irrelevant constant rescaling and an error that can be exponentially suppressed by increasing the energy cut-off Δ and improving the precision ϵ. More precisely, if the original and simulator Hamiltonians have local dimension d, then
|ZH′(β)−(p+q)ZH(β)|(p+q)ZH(β)≤dm−ne−βΔ(p+q)e−β∥H∥+(eϵβ−1). (2)
Since it is able to reproduce the partition function to any desired precision, all thermodynamic properties of the original Hamiltonian are reproduced by the simulation.
Finally, all dynamical properties are also reproduced to any desired precision. More precisely, the error in the simulated time-evolution grows only linearly in time (which is optimal without active error correction), and can be suppressed to any desired level by improving the approximation accuracy ϵ and η:
∥e−iH′tEstate(ρ)eiH′t−Estate(e−iHtρeiHt)∥1=O(tϵ+η). (3)
We can also derive some important consequences for simulation errors and fault-tolerance. A recurring criticism of analogue Hamiltonian simulation is that, because it does not implement any error-correction, errors will accumulate over time and swamp the simulation. A common counter-argument is that any real physical system is itself always subject to noise and errors. If the properties of its Hamiltonian are sensitive to noise, the behaviour of the real physical system will also include the effects of the noise, and it is that which we wish to simulate. There is truth to both sides. In the absence of error-correction, errors will accumulate over time. It is also true that the same will happen in the original physical system, so this may not matter. But only if noise and errors in the simulation mimic the noise and errors experienced by the real physical system we are trying to simulate.
Fully justifying this would require modelling the noise and error processes in the physical system, and showing that the natural noise and error processes occurring in the particular simulator being used do indeed faithfully reproduce the same effects. Even then, the validity of this argument rests on the validity of the noise model. Ultimately, determining whether or not a simulation is accurate always comes down to testing its predictions in the laboratory. But with our precise definition of Hamiltonian simulation in hand, we can take a significant step towards justifying generally why lack of error correction may not be an issue. Most natural noise models are local: physical errors tend to act on neighbouring particles, not across the entire system. The definition of Hamiltonian simulation arrived at in the previous section immediately implies that local errors in the original system correspond to local errors in the simulator.
But we go further than this. We prove that, under a reasonable physical assumption, a local error affecting the simulator system approximates arbitrarily well the encoded version of some local error on the original system. More precisely, if we take the energy cut-off Δ to be large enough, errors on the simulator system are unlikely to take the simulated state out of the low-energy space of H′. Assume that this happens with probability at most δ, for some δ≤η. Then for any noise operation N′ acting on ℓ qudits of the simulator system, there is always some noise operation N on at most ℓ qudits of the original system (which we can easily write down) such that, for any state ρ, the effect of N′ on the simulator approximates (again, to any desired precision) the effect of N on the system being simulated. Or, to state this mathematically:
Estate(N(ρ))=N′(Estate(ρ))+O(√η) (4)
where N and N′ are superoperators. The fact we can prove the result this way around is crucial: it shows that any local noise and errors that might occur in our simulator simply reproduce the effects of local noise and errors in the original physical system. This is much stronger than merely showing that errors on the original system can be simulated.
4 Universal Hamiltonians
The notion of Hamiltonian simulation we have arrived at is extremely demanding. It is not a priori clear whether any interesting simulations exist at all. In fact, not only do such simulations exist, we prove that there are even universal quantum simulators. A model is "universal" if it can simulate any Hamiltonian whatsoever, in the strong sense of simulation discussed above. Depending on the target Hamiltonian, this simulation may or may not be efficient. In general, the simulation will be efficient for target Hamiltonians with local interactions in the same (or lower) spatial dimension. Whereas, whilst universal models can also simulate Hamiltonians in higher spatial dimensions with only modest (polynomial) system-size overhead, this comes at an exponential cost in energy.
Remarkably, even certain simple 2D quantum spin-lattice models are universal. To show this, we in fact prove a still stronger result. We completely classify all two-qubit interactions (i.e. nontrivial interactions between two spin-1/2 particles) according to their simulation ability. This classification tells us which two-qubit interactions are universal. The universal class turns out to be identical to the class of QMA-complete two-qubit interactions from quantum complexity theory [CM16] , where QMA is the quantum analogue of the complexity class NP [KSV02] .
The classification also shows that there are two other classes of two-qubit interaction, with successively weaker simulation ability. Combining our Hamiltonian simulation results with previous work [BH17] , we find that there is a class of two-qubit interactions that can simulate any stoquastic Hamiltonian, i.e. any Hamiltonian whose off-diagonal entries in the standard basis are non-positive. This is the class of Hamiltonians believed not to suffer from the sign-problem in numerical Monte-Carlo calculations. Another class is able, by previous work [lCC16] , to simulate any classical Hamiltonian, i.e. any Hamiltonian that is diagonal in the standard basis.
The 2D Heisenberg- and XY-models (with non-uniform coupling strengths) are important examples which we show fall into the first category, hence are universal simulators. The 2D (quantum) Ising model with transverse fields falls into the second category, so can simulate any other stoquastic Hamiltonian [BH17] . The 2D classical Ising model with fields falls into the third category, so is an example of a universal classical Hamiltonian simulator [lCC16] .
Heisenberg interactions
XY interactions
2-local Pauli interactions with no σy's
Arbitrary (2k+1)-local terms with no σy's
Arbitrary real 2k-local qubit Hamiltonian
Arbitrary k-local qubit Hamiltonian
Bosons
Fermions
Figure 3: Part of the sequence of simulations used in this work. An arrow from one box to another indicates that a Hamiltonian of the first type can simulate a Hamiltonian of the second type.
The universality proof involves chaining together a number of steps, some of which are shown in Figure 3. In fact, most of the technical difficulty lies in proving universality of the Heisenberg and XY interactions. Once these are shown to be universal, it is relatively straightforward to use recently developed techniques [CM16, PM17] to show that any other Hamiltonian from the universal category can simulate one of these two. Hence, by universality of the Heisenberg or XY interactions, such Hamiltonians can also simulate any other Hamiltonian. We now sketch the universality proof for these two interactions. (See part II for full technical details.)
The Heisenberg interaction hHeis=σx⊗σx+σy⊗σy+σz⊗σz (where σx,y,z are the Pauli matrices) has full local rotational symmetry. Mathematically, this is equivalent to invariance under arbitrary simultaneous local unitary rotations U⊗U. The XY interaction hXY=σx⊗σx+σy⊗σy is invariant under arbitrary rotations in the z-plane, i.e. U⊗U with U=eiθσz for any angle θ. Any Hamiltonian composed of just one of these types of interaction inherits the corresponding symmetry. Thus all its eigenspaces also necessarily have this symmetry. Yet if it is to be universal, it must simulate Hamiltonians that do not have this symmetry.
Before addressing symmetry, however, there is a more elementary obstacle to overcome. All matrix elements of hHeis or hXY are real numbers (in the standard basis). Thus any Hamiltonian built out of these interactions is also real. Yet if it is to be universal, it must simulate Hamiltonians with complex matrix elements.
A simple encoding overcomes this restriction, by adding an additional qubit and encoding the real and imaginary parts of H separately, controlled on the state of the ancilla qubit. The Hamiltonian H′=Re(H)⊕Im(H) is clearly real and is easily seen to be an encoding of H, since H′=H⊗|+y⟩⟨+y|+¯H⊗|−y⟩⟨−y|, where |±y⟩=(|0⟩±i|1⟩)/√2. To make this encoding local, it can be adjusted to a simulation where there is an ancilla qubit for each qubit of the system, but these ancillas are forced by additional strong local interactions to be in the two dimensional subspace spanned by |+y⟩⊗n,|−y⟩⊗n.
To overcome the symmetry restriction, we develop more complicated simulations based around the use of "perturbative gadgets" (a technique originally introduced to prove QMA-completeness results in Hamiltonian complexity theory [KKR06, OT08] ). In a perturbative gadget, a heavily weighted term CH0 (for some large constant C) dominates the overall Hamiltonian H′=CH0+H1 such that the low-energy part of H′ is approximately just the ground space of H0. Within this low-energy subspace, an effective Hamiltonian is generated by H1 and can be calculated using a rigorous version of perturbation theory [BH17] . The first-order term in the perturbative expansion is given by H1 projected into the ground space of H0, as one might expect. But if this term vanishes, then the more complicated form of higher order terms may be exploited to generate more interesting effective interactions.
For most of our simulations, H0 is used to project a system of ancilla qubits into a fixed state, such that the effective Hamiltonian that this generates couples the remaining qubits. This type of gadget is known in the Hamiltonian complexity literature as a mediator qubit gadget [OT08] , because the ancilla qubits are seen to "mediate" an effective interaction between the other qubits in the system.
But in order to break the symmetry of the Heisenberg and XY interactions, it is necessary for the encoded Hamiltonian to act not on the physical qubits of the system, but on qubits encoded into a subspace of multiple physical qubits. To achieve this, we design a four-qubit gadget where the strong H0 term, consisting of equally weighted interactions across all pairs of qubits, has a two-fold degenerate ground space. This two-dimensional space can be used to encode a qubit. This gadget is used repeatedly to encode all qubits of the systems separately, as illustrated in Figure 2. We then add less heavily weighted interactions acting between qubits in different gadgets, in order to generate effective interactions between the encoded qubits. These interactions are calculated using a precise version of second-order perturbation theory, which accounts rigorously for the approximation errors resulting from neglecting the higher-order terms [BH17] . Combined with a new mediator gadget, together with previously known gadgets [OT08] which allow many-body interactions to be simulated using two-body interactions, this suffices to show that the Heisenberg and XY interactions can simulate all real local Hamiltonians, and hence all local Hamiltonians using the complex-to-real encoding described above.
In order to show that Hamiltonians with arbitrary long-range interactions can be simulated with a 2D lattice model, there is a final step: embedding an arbitrary interaction pattern within a square lattice. This can be achieved by effectively drawing the long-range interactions as lines on the lattice, and using further perturbative gadgets to remove crossings between lines [OT08] . This step requires multiple rounds of perturbation theory, which can result in the final Hamiltonian containing local interaction strengths that scale exponentially in the number of particles. Thus the final simulation, whilst efficient in terms of the number of particles and interactions, is not necessarily efficient in terms of energy cost for arbitrary Hamiltonians. For example, we do not know how to construct an energy-efficient simulation of a 3D lattice Hamiltonian using a 2D lattice model, nor do we necessarily expect it to be possible. However, full efficiency is recovered when the original Hamiltonian is spatially sparse [OT08] (a class which encompasses all 2D lattice Hamiltonians). Finally, if we want to simulate indistinguishable particles, one can verify that standard techniques for mapping fermions or bosons to spin systems give the required simulations. (See part II for details.)
We close by highlighting some of the limitations of our results, and possible future directions. One should note that whilst our strong notion of simulation preserves locality in the sense that a few-particle observable in the original system will correspond to a few-particle observable in the simulator, simulating e.g. a 3D system in a 2D system necessarily means that the corresponding observables in the simulation will not always be on nearby particles. Also, to simulate higher-dimensional systems in 2D, our constructions require very large coupling strengths.
From the analogue Hamiltonian engineering perspective, our results show that surprisingly simple types of interactions suffice for building a universal Hamiltonian simulator. Together with the ability to prepare simple initial states, these would even suffice to construct a universal quantum computer, or to perform universal adiabatic quantum computation. (However, error correction and fault-tolerance, which are essential for scalable quantum computation, would require additional active control.) The converse point of view is that, as these apparently restrictive models turn out to be universal, simulating them on a quantum computer may be more difficult than previously thought. Furthermore, our mathematical constructions require extremely precise control over the strengths of individual local interactions across many orders of magnitude. Though some degree of control is possible in state-of-the-art experiments [Nat12, GAN14] , the requirements of our current universal models are beyond what is currently feasible. On the other hand, it is already possible to engineer more complex interactions than those we have shown to be universal. Now we have shown for that universal models exist, and need not be extremely complex, it may be possible to construct other universal models tailored to particular experimental setups.
From the fundamental physics perspective, an important limitation of our current results is that the models we show to be universal are not translationally invariant. (The same is also true of the earlier classical results [lCC16] .) Although we show there are universal models in which all interactions have an identical form, our proofs rely heavily on the fact that the strengths of these interactions can differ from site to site. Classic results showing that local symmetries together with translational-invariance can restrict the possible physics [MW66, Hoh67] suggest breaking translational-invariance may be crucial for universality. On the other hand, much of the intuition behind our proofs comes from Hamiltonian complexity, where recent results have shown that translational-invariance is no obstacle [GI09, BCO16] .
In light of our results, determining the precise boundary between simplicity and universality in quantum many-body physics is now an important open question for future research.
Part II Technical content
6 Notation and terminology
As usual, B(H) denotes the set of linear operators acting on a Hilbert space H. For conciseness, we sometimes also use the notation Mn for the set of all n×n matrices with complex entries. Hermn denotes the subset of all n×n Hermitian matrices. \mathbbm1 denotes the identity matrix. For integer n, [n] denotes the set {1,…,n}.
If R,R′ are rings, a ring homomorphism ϕ:R→R′ is a map that is both additive and multiplicative: ∀a,b∈R:ϕ(ab)=ϕ(a)ϕ(b) and ϕ(a+b)=ϕ(a)+ϕ(b). Similarly, a ring anti-homomorphism is an additive map that is anti-multiplicative: ϕ(ab)=ϕ(b)ϕ(a). If ϕ(\mathbbm1)=\mathbbm1, we say the map is unital.
For a ring R, the corresponding Jordan ring Rj is the ring obtained from R by replacing multiplication with Jordan multiplication {ab}:=ab+ba. A Jordan homomorphism ϕ on R is an additive map such that ∀a,b∈R:ϕ(ab+ba)=ϕ(a)ϕ(b)+ϕ(b)ϕ(a). If R is not of characteristic 2, this is equivalent to the constraint that ∀a∈R:ϕ(a2)=ϕ(a)2. Note that any ring homomorphism is a Jordan homomorphism, but the converse is not necessarily true.
spec(A) denotes the spectrum of A∈Mn, i.e. the set of values λ∈C such that A−λ\mathbbm1 is not invertible. (This of course coincides with the set of eigenvalues, ignoring multiplicities.) We say that ϕ:Mn→Mm is invertibility-preserving if ϕ(A) is invertible in Mm for all invertible A∈Mn. We say that ϕ is spectrum-preserving if spec(ϕ(A))=spec(A) for all A∈Mn.
For an arbitrary Hamiltonian H∈B(Cd), we let P≤Δ(H) denote the orthogonal projector onto the subspace S≤Δ(H):=span{|ψ⟩:H|ψ⟩=λ|ψ⟩,λ≤Δ}. We also let H′|≤Δ(H) denote the restriction of some other arbitrary Hamiltonian H′ to S≤Δ(H), and write H|≤Δ:=H|≤Δ(H) and H≤Δ:=HP≤Δ(H).
We say that a Hamiltonian H∈B((Cd)⊗n) is k-local if it can be written as a sum of terms such that each hi acts non-trivially on at most k subsystems of (Cd)⊗n. That is, hi∈B((Cd)⊗k) and H=∑ihi⊗\mathbbm1 where the identity in each term in the sum acts on the subsystems where that hi does not. An operator on a composite Hilbert space "acts trivially" on the subsystems where it acts as identity, and "acts non-trivially" on the remaining subsystems. We will often employ a standard abuse of notation, and implicitly exend operators on subsystems to the full Hilbert without explicitly writing the tensor product with identity, allowing us e.g. to write simply H=∑hi. We say that H is local if it is k-local for some k that does not depend on n111Technically, this makes sense only for families of Hamiltonians H, where we consider n to be growing..
We let X, Y, Z denote the Pauli matrices and often follow the condensed-matter convention of writing XX for X⊗X etc. For example, XX+YY+ZZ is short for X⊗X+Y⊗Y+Z⊗Z and is known as the Heisenberg (exchange) interaction. The XY interaction is XX+YY.
Let M be a k-qudit Hermitian matrix. We say that U∈SU(d) locally diagonalises M if U⊗kM(U†)⊗k is diagonal. We say that a set S of Hermitian matrices is simultaneously locally diagonalisable if there exists U∈SU(d) such that U locally diagonalises M for all M∈S. Note that matrices in S may act on different numbers of qudits, so can be of different sizes.
We will often be interested in families of Hamiltonians. For a subset S of interactions (Hermitian matrices on a fixed number of qudits), we define the family of S-Hamiltonians to be the set of Hamiltonians which can be written as a sum of interaction terms where each term is either picked from S, with an arbitrary positive or negative real weight, or is an arbitrarily weighted identity term. For example, H is a {ZZ}-Hamiltonian if it can be written in the form H=α\mathbbm1+∑i<jβijZiZj for some α,βij∈R. A model is a (possibly infinite) family of Hamiltonians. Typically the Hamiltonians in a model will be related in some way, e.g. all Hamiltonians with nearest-neighbour Heisenberg interactions on an arbitrarily large 2D lattice (the "2D Heisenberg model").
7 Hamiltonian encodings
Any non-trivial simulation of one Hamiltonian with another will involve encoding the first within the second in some way. Write H′=E(H) for some "encoding" map E that encodes a Hamiltonian H into some Hamiltonian H′. Any such encoding should fulfil at least the following basic requirements. First, any observable on the original system should correspond to an observable on the simulator system. Second, the set of possible values of any encoded observable should be the same as for the corresponding original observable. In particular, the energy spectrum of the Hamiltonian should be preserved. Third, the encoding of a probabilistic mixture of observables should be the same as a probabilistic mixture of the encodings of the observables.
To see why this last requirement holds, imagine that we are asked to encode observable A with probability p, and observable B with probability 1−p. Then, for any state ρ on the simulator system, the expected value of the encoded observable acting on ρ should be the same as the corresponding probabilistic mixture of the expected values of the encoded observables A and B acting on ρ. In order for this to hold for all states ρ, we need the mixture of observables pA+(1−p)B to be encoded as the corresponding probabilistic mixture of encodings of A and B.
These operational requirements correspond to the following mathematical requirements on the encoding map E:
E(A)=E(A)† for all A∈Hermn.
spec(E(A))=spec(A) for all A∈Hermn.
E(pA+(1−p)B)=pE(A)+(1−p)E(B) for all A,B∈Hermn and all p∈[0,1].
Of course, there are many other desiderata that we would like E to satisfy, such as preserving the partition function, measurement outcomes, time-evolution, local errors, and others. For the Hamiltonian itself, we almost certainly want E to not only be convex, but also real-linear: E(∑iαihi)=∑iαiE(hi), so that a Hamiltonian expressed as a sum of terms can be encoded by encoding the terms separately. However, we will see later that meeting just the above three basic requirements necessarily implies also meeting all these other operational requirements (which we will make precise).
It turns out there is a simple and elegant characterisation of what such encodings have to look like. To prove this, we will need the following theorem concerning Jordan ring homomorphisms.
For any n≥2, any Jordan homomorphism of the Jordan ring Hermn can be extended in one and only one way to a homomorphism of the matrix ring Mn. ■
Theorem was shown by Jacobson and Rickart for n≥3 [JR52] , and by Martindale for n=2 [Mar67] , in each case in a far more general setting than we need here.
Any unital, invertibility-preserving, real-linear map ϕ:Hermn→Hermm is a Jordan homomorphism. ■
The argument is standard (see e.g. [HŠ03] ).
ϕ(H−λ\mathbbm1)=ϕ(H)−λ\mathbbm1, thus spec(ϕ(H))⊆spec(H) since ϕ is invertibility-preserving. In particular, spec(ϕ(P))∈{0,1} for every projector P. Since ϕ(P) is also Hermitian, this implies ϕ(P) is a projector.
By the spectral decomposition, any H∈Hermn can be decomposed as H=∑iλiPi where Pi are mutually orthogonal projectors and λi∈R. For i≠j, Pi+Pj is a projector, thus ϕ(Pi+Pj) is a projector and (ϕ(Pi+Pj))2=ϕ(Pi)+ϕ(Pj), so that ϕ(Pi)ϕ(Pj)+ϕ(Pi)ϕ(Pj)=0. Therefore, ϕ(H)2=∑iλ2iϕ(Pi)2+∑i≠jλiλjϕ(Pi)ϕ(Pj)=∑iλ2iϕ(Pi)=ϕ(H2). □
For any map E:Hermn→Hermm, the following are equivalent:
For all A,B∈Hermn, and all p∈[0,1]:
E(A)=E(A)†
spec(E(A))=spec(A)
E(pA+(1−p)B)=pE(A)+(1−p)E(B).
There exists a unique extension E′:Mn→Mm such that E′(H)=E(H) for all H∈Hermn and, for all A,B∈Mn and x∈R:
E′(\mathbbm1)=\mathbbm1
E′(A†)=E′(A)†
E′(A+B)=E′(A)+E′(B)
E′(AB)=E′(A)E′(B)
E′(xA)=xE′(A).
There exists a unique extension E′:Mn→Mm such that E′(H)=E(H) for all H∈Hermn with E′ of the form
E′(M)=U(M⊕p⊕¯M⊕q)U† (5)
for some non-negative integers p, q and unitary U∈Mm, where M⊕p:=⨁pi=1M and ¯M denotes complex conjugation.
We call a map E satisfying (iii), (ii) and (i) an encoding. ■
Note that (iii) is basis-independent, despite the occurrence of complex conjugation; taking the complex conjugation with respect to a different basis is equivalent to modifying U, which just gives another encoding. Given that E′ is unique, for the remainder of the paper we simply identify E′ with E. In particular, this allows us to assume that E is of the form specified in Item (iii). The characterisation creftype 5 can equivalently be written as
E′(M)=U(M⊗P+¯M⊗Q)U† (6)
for some orthogonal projectors P and Q such that P+Q=\mathbbm1; this alternative form will sometimes be useful below. We think of the system on which P and Q act as an ancilla, and often label this "extra" subsystem by the letter E.
(i) ⇒ (ii):
We first show that E is a Jordan homomorphism. Item (i)a states that E preserves Hermn, and Item (i)b implies that E is unital and invertibility-preserving on Hermn, with E(0)=0. We next check that E(0)=0 together with Item (i)c are equivalent to real-linearity of E. For any λ<0, setting p=λ/(λ−1), B=pA/(p−1) and using Item (i)c gives
0=E(0)=pE(A)+(1−p)E(pA/(p−1))⇔λE(A)=E(λA). (7)
Apply creftype 7 to λA to get E(λ2A)=λ2E(A), showing that E is homogeneous for all real scalars. Additivity follows by combining Item (i)c and homogeneity: E(A+B)=E(2A)/2+E(2B)/2=E(A)+E(B). Therefore E is also real-linear so by Lemma E is a Jordan homomorphism.
By Theorem , there exists a unique homomorphism E′:Mn→Mm such that E′(H)=E(H) for all H∈Hermn. As E′ agrees with E on Hermn, it satisfies (ii)a. As E′ is a homomorphism, it satisfies (ii)d and (ii)c by definition; this also implies that E′(xA)=E′(x\mathbbm1)E′(A)=E(x\mathbbm1)E′(A)=xE′(A) for any x∈R, so (ii)e holds.
We finally prove (ii)b. It is sufficient to show that E′(i\mathbbm1)†=−E′(i\mathbbm1), because if this holds we can expand any matrix A∈Mn as A=B+iC for some Hermitian matrices B and C to obtain
E′(A†) =E′(B−iC)=E′(B)−E′(C)E′(i\mathbbm1)=E′(B)†+E′(C)†E′(i\mathbbm1)† (8)
=E′(B+iC)†=E′(A)†. (9)
To show E′(i\mathbbm1)†=−E′(i\mathbbm1), we first write i\mathbbm1 as a linear combination of products of Hermitian matrices. That this can be done is an immediate consequence of the fact that Mn is the enveloping associative ring of Hermn. However, it can also be seen explicitly by writing
i|j⟩⟨j|=|j⟩⟨j|(i|j⟩⟨k|−i|k⟩⟨j|)(|j⟩⟨k|+|k⟩⟨j|) (10)
for any j, and some k≠j; summing this product over j, we obtain i\mathbbm1. Thus we can write i\mathbbm1=∑jAjBjCj for Hermitian matrices Aj, Bj, Cj. By taking adjoints on both sides, it follows that −i\mathbbm1=∑jCjBjAj. So we have
E′(i\mathbbm1)† =E′(∑jAjBjCj)†=(∑jE(Aj)E(Bj)E(Cj))† (11)
=∑jE(Cj)E(Bj)E(Aj)=E′(∑jCjBjAj) (12)
=E′(−i\mathbbm1)=−E′(i\mathbbm1). (13)
(ii) ⇒ (iii):
Existence and uniqueness of E′ were already shown in the previous part. In the proof of the remaining claim, for readability we just use E to denote this unique extension. First define the complex structure J:=E(i\mathbbm1)≡E(i) (where the latter notation is a convenient shorthand). We have
J2=E(i)E(i)=E(i2)=E(−1)=−\mathbbm1, (14)
thus J has eigenvalues ±i. Furthermore,
J†=E(i)†=E(i†)=−E(i)=−J, (15)
so J is anti-Hermitian, hence diagonalisable by a unitary transformation.
For any A∈Hermn, we have
JE(A)=E(i)E(A)=E(iA)=E(Ai)=E(A)J, (16)
so that [E(A),J]=0. Thus E(A) and J are simultaneously diagonalisable for all A. H=H+⊕H− therefore decomposes into a direct sum of the ±i eigenspaces of J, on which E(A)=A+⊕A− acts invariantly.
Now, restricting to either of these invariant subspaces,
E(A)|±=A± (17)
E(iA)|±=JA±=±iA± (18)
E(AB)|±=E(A)E(B)|±=A±B± (19)
E(A†)|±=E(A)†|±=A†±. (20)
Thus E=E+⊕E− decomposes into a direct sum of a *-representation E+(A):=E(A)|+ and an anti-*-representation222By "anti-*-representation" we mean an anti-linear algebra homomorphism, not a *-antihomomorphism (which would be a linear map preserving adjoints that reverses the order of multiplication). E−(A):=E(A)|−. Since for any vector |ψ⟩∈Cm, E±(\mathbbm1)|ψ⟩=\mathbbm1|ψ⟩=|ψ⟩, these (anti-)*-representations are necessarily non-degenerate.
By a standard result on the representations of finite-dimensional C*-algebras [Dav91, Corollary III.1.2] , any non-degenerate *-representation of Mn is unitarily equivalent to a direct sum of identity representations. If ϕ is an anti-*-homomorphism, let φ(A):=¯¯¯¯¯¯¯¯¯¯¯¯ϕ(A). Then φ(iA)=¯¯¯¯¯¯¯¯¯¯¯¯¯ϕ(iA)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯−iϕ(A)=iφ(A), φ(A+B)=φ(A)+φ(B), φ(A†)=φ(A)†, and φ(AB)=φ(A)φ(B). Thus ϕ(A)=¯¯¯¯¯¯¯¯¯¯¯¯φ(A) where φ is a *-homomorphism. Therefore, any non-degenerate anti-*-representation is unitarily equivalent to a direct sum of complex conjugates of identity representations, which completes the argument.
(iii) ⇒ (i) can readily be verified directly. □
The above theorem characterises encodings of observables. This immediately tells us how to encode physical systems themselves, expressed as Hamiltonians: since the Hamiltonian itself is an observable, the encoding map must have the same characterisation.
It is easy to see from the characterisation in Item (iii) of the Theorem that any encoding preserves all interesting physical properties of the original Hamiltonian. For example, the set of eigenvalues is preserved, up to possibly duplicating each eigenvalue the same number of times, implying preservation of the partition function (up to an unimportant constant factor). It is also easy to see that any encoding E properly encodes arbitrary quantum channels: if {Ek:∑kE†kEk=\mathbbm1} are the Kraus operators of the channel, then
∑kE(Ek)†E(Ek)=\mathbbm1. (21)
7.1 A map on states, Estate
We now show that, for any encoding E, there exists a corresponding map Estate that encodes quantum states ρ such that encoded observables E(A) applied to encoded states Estate(ρ) have correct expectation values.
First, note that for any observable A and any state ρ′ on the simulator system, we have
Tr(E(A)ρ′) =Tr[U(A⊗P+¯A⊗Q)U†ρ′] (22)
=Tr[(A⊗\mathbbm1)(\mathbbm1⊗P)U†ρ′U)]+Tr[(¯A⊗\mathbbm1)(\mathbbm1⊗Q)U†ρ′U] (23)
=Tr[AF(ρ′)]+Tr[¯A¯¯¯¯¯¯¯¯¯¯¯¯B(ρ′))]=Tr(Aρ) (24)
F(ρ′)=TrE[(\mathbbm1⊗P)U†ρ′U],B(ρ′)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯TrE[(\mathbbm1⊗Q)U†ρ′U], (25)
ρ=F(ρ′)+B(ρ′) (26)
and we label the second subsystem E as discussed after creftype 6. Note that F(ρ′) and B(ρ′) are both positive but not necessarily normalised, but ρ is normalised.
Therefore any map Estate(ρ) on states ρ such that ρ=F(Estate(ρ))+B(Estate(ρ)) will preserve measurement outcomes appropriately. One natural choice is
Estate(ρ)={U(ρ⊗σ)U† for some σ such that Pσ=σif P≠0U(¯ρ⊗σ)U† for some σ such that Qσ=σotherwise. (27)
Then in the former case F(Estate(ρ))=ρ, B(Estate(ρ))=0; and in the latter case the roles of F and B are reversed.
We now show that Estate simulates time-evolution correctly too. We have
F(e−iE(H)tρ′eiE(H)t) =e−iHtF(ρ′)eiHt, (28)
B(e−iE(H)tρ′eiE(H)t) =eiHtB(ρ′)e−iHt. (29)
This is why they are labelled with the letters F and B: the F part evolves forwards in time while the B part evolves backwards in time. Taking ρ′=Estate(ρ), we have proven the following result.
For any encoding E, the corresponding map Estate satisfies the following:
Tr(E(A)Estate(ρ))=Tr(Aρ)
For any time t,
e−iE(H)tEstate(ρ)eiE(H)t={Estate(e−iHtρeiHt)if p≥1Estate(eiHtρe−iHt)if p=0. (30)
It is worth highlighting the last point. We see that if p≥1, evolving according to E(H) for time t simulates evolving according to H for time t, as we would expect; but that if p=0, we simulate evolution according to H for time −t. That is, if our encoding only includes copies of ¯H, we simulate evolution backwards in time. To avoid this issue, we define the concept of a standard encoding as one where p≥1, and hence which is able to simulate evolution forward in time.
An encoding E(M)=U(M⊕p⊕¯M⊕q)U† is a standard encoding if p≥1. ■
7.1.1 Gibbs-preserving state mappings
The choice of Estate in creftype 27 is convenient, as it allows us to use the same mapping E for both the Hamiltonian and for observables. However, it does not map Gibbs states e−βH/Tr(e−βH) of the original system to Gibbs states e−β′H′/Tr(e−β′H′) of the simulator. If we have limited ability to manipulate or prepare states of the simulator, it may be difficult to prepare a state of the form creftype 27. At equilibrium, the system will naturally be in a Gibbs state. From this perspective, it would be more natural if the state mapping identified Gibbs states of the original system with Gibbs states of the simulator.
An alternative choice of Estate does map Gibbs states to Gibbs states:
Estate(ρ)=E(ρ)Tr[E(ρ)]=1p+qU(ρ⊗P+¯ρ⊗Q)U† (31)
where p=Tr(P) and q=Tr(Q). However, to obtain the correct measurement outcome probabilities, we now need to choose a slightly different mapping for observables:333The Hamiltonian is of course also an observable. With this choice of state mapping, to construct the simulator Hamiltonian we must still use the mapping H′=E(H). But if we want to measure the Hamiltonian – i.e. carry out the measurement on the simulator that corresponds to measuring the energy of the original system – we must measure Emeas(H).
Emeas(A)={p+qpU(A⊗P)U† if P≠0p+qqU(¯A⊗Q)U† otherwise. (32)
For simplicity, in the remainder of the paper we will state and prove our results for the choice of state mapping Estate from creftype 27, so that both Hamiltonians and observables are encoded by E. However, our results also go through with the appropriate minor modifications for the choice of Gibbs-preserving E


text:
package lekt09_services;

import androidx.appcompat.app.AppCompatActivity;
import android.content.Intent;
import android.os.Bundle;
import android.view.View;
import android.view.View.OnClickListener;
import android.widget.Button;
import android.widget.ScrollView;
import android.widget.TableLayout;
import android.widget.TextView;

/**
 * @author Jacob Nordfalk
 */
public class BenytService extends AppCompatActivity implements OnClickListener {

  Button knap1, knap2, knap3, knap4, knap5;

  @Override
  protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);

    TableLayout tl = new TableLayout(this);
    TextView tv = new TextView(this);
    tv.setText("Når en service er startet vil systemet prøve at holde programmet "
            + "i hukommelsen.\n"
            + "Hvis det programmet alligevel ryger ud vil systemet genstarte processen "
            + "og reinstantiere servicen igen hurtigst muligt.\n"
            + "Hvis ens program SKAL blive i hukommelsen skal det starte en forgrundsservice, "
            + "der er knyttet til en notifikation som brugeren kan se.\n"
            + "Nederste knap kalder System.exit() for at simulere at processen afsluttes, "
            + "hvorefter systemet straks vil genstarte processen og reinstantiere servicen.");
    tl.addView(tv);

    knap1 = new Button(this);
    knap1.setText("Start service");
    tl.addView(knap1);

    knap2 = new Button(this);
    knap2.setText("Stop service");
    tl.addView(knap2);

    knap3 = new Button(this);
    knap3.setText("Start forgrundsservice");
    tl.addView(knap3);

    knap4 = new Button(this);
    knap4.setText("Stop forgrundsservice");
    tl.addView(knap4);

    knap5 = new Button(this);
    knap5.setText("Stop proces");
    tl.addView(knap5);

    ScrollView sv = new ScrollView(this);
    sv.addView(tl);
    setContentView(sv);

    knap1.setOnClickListener(this);
    knap2.setOnClickListener(this);
    knap3.setOnClickListener(this);
    knap4.setOnClickListener(this);
    knap5.setOnClickListener(this);
  }

  public void onClick(View klikketPå) {

    if (klikketPå == knap1) {
      Intent i = new Intent(this, BaggrundsService.class);
      i.putExtra("nøgle", "en værdi fra BenytService");
      startService(i);
    } else if (klikketPå == knap2) {
      stopService(new Intent(this, BaggrundsService.class));
    } else if (klikketPå == knap3) {
      startService(new Intent(this, ForgrundsService.class));
    } else if (klikketPå == knap4) {
      stopService(new Intent(this, ForgrundsService.class));
    } else if (klikketPå == knap5) {
      System.exit(0);
    }
  }
}

redpajama_set_name: RedPajamaGithub