I really think that some caches, in particular caches that are up trees, should have their own classifications and not be called "Traditional". If a cache requires specialist knowledge and/or equipment, then it should be classified so. Caches that are up trees and require canoes are the first that leap to mind for this.
Also, there seems to be a massive variety in puzzle caches too and some multicaches are more like puzzles/vice versa. There are some multicaches that claim to be multi-virtuals too and get very complicated. There are some multis that require up to 19 places to be visited to get to one cache, instead of being a series or a Wherigo. A cache that's a route/exploration would be good to know too.
Plus, challenge caches should have their own symbol? Like a trophy.
Am I alone in thinking that reclassification/additional cache types could be really helpful?
Plus, in terms of having solved the co-ords for a puzzle/multi and added the waypoint, it would be cool if the website could change the white symbol ring on the map to another colour, so you know it's pending you going to get it.
Tree climbs, canoes, etc. are covered by attributes.
It would be nice to have challenge caches with their own symbol/cache type. That's been discussed before (and pretty much agreed upon).
These factors are also reflected in the terrain rating. If you aren't ready to take on such terrain, then stick to caches with lower terrain ratings.
I agree; they should be reflected in the terrain rating.
But not all COs rate them correctly. Not when the know-it-all beginner rates their cache up a tree at 1.5, because "the walk to the tree is flat".
If the finders could rate terrain (especially) and difficulty and this be averaged out and shown to future finders, it would overcome these COs who interpret things their way. It would also make caching safer; indicating the true terrain; not what the CO rates it as. For safety's sake alone; this feature should be added, at least for terrain. Some COs low ratings for terrain are dangerous.
If you correct the coords (on the app or the website), the maps show a puzzle-piece icon at those coordinates.
I think Attributes and D/T ratings cover these scenarios adequately - assuming they are applied correctly!
Agreed, there are puzzles that could/should be classified as multis, and vice versa; and there do seem to be regional differences; but I don't see how additional cache types will help here.
It would help if geocachers know the difference better. Basically, a multi-cache is a puzzle which you start searching from the posted coordinates. A mystery cache is a puzzle with bogus-coordinates you are not supposed to visit. This is the basic principle with some exceptions but did you know this?
Yes, having challenges as separate cache types is something I've been hoping for for a long time.
As for the rest issues covered in this topic, you can't fix every wrong doing with a system patch. If a CO sets incorrect attributes, it's his fault, not the site administrators. What should be done in such situation is mentioning the wrong assignment in the logs.
Basically, a multi-cache is a puzzle which you start searching from the posted coordinates. A mystery cache is a puzzle with bogus-coordinates you are not supposed to visit. This is the basic principle with some exceptions but did you know this?
I have to disagree with this one. A multi is a "look and calculate" type of cache. You should be able to easily calculate final or next stage once you find necessary data (finding the data might be hard though - e.g. hidden in a well camouflaged container).
A puzzle is a cache where you (usually) have to solve problems or bring additional knowledge.
"Go to N00 00.000 E00 00.000, read the description on the tablet. Who is it about? Us the birth date of that person in the following formula (the birth date isn't on the tablet)" - A puzzle.
"Go to N00 00.000 E00 00.000, there is a mystery box which you have to open to sign the logbook" - A puzzle.
Of course there are many grey areas but for me the above is the main diffrentiation.
This is not something you may agree or disagree. You have to know the difference to understand how to play and make caches. There is no need to add more categories when there is already difficulties to understand what the current types are. You have seen some caches and made your own interpretation based on what you have seen but have you checked what the guidelines are telling about them?
Or quite probably a Traditional/Multi/Letterbox with a "Field Puzzle" attribute.
The 'Field Puzzle' is the most 'mysterious' attribute in this case, in my opinion.
Suppose I make a cache where there is a container at starting point with a puzzle inside required to solve to find the final container. The puzzle is not an easy one, you do not need to use external knowledge resources (like internet) to solve it but it might help a lot. If I add a 'Field Puzzle' attribute, can it still be a multi-cache or should it be a mystery cache regardless of an attribute added?
Multi: The cache can be found by reading the cache page and following the instructions in the field.
Mystery: The cache cannot be found without calculation, or research that goes beyond reading the cache page.
However, I think we could all find plenty of examples of caches that have been classified incorrectly.
This is one exception not always remembered even by revievers. It the cache requires research that goes beyond reading the cache page, like using internet, it is a mystery cache. For example you can not make Intercache style multi-cache because it requires constant on-line connection. QR-codes have been causing problems. Some reviewers allows them as a multi-cache and some doesn't.
My thoughts, if I elaborate on my initial point in regards to tree climbing etc. is that surely, there is a way, that Traditional Caches could have a subsection for specialist equipment/knowledge.
If this isn't possible, then is there a search function that can exclude tree climbing attributes for example and you can view a map that's actually achievable? Currently, the main map lets you vanish traditional caches etc but only as a whole.
Sort by Treeain rating. The tree climbing caches should be quite high (like 4+) on that rating.
But, this isn't doable once you're looking at the map and you can't edit that. So, say you're looking at a certain area, you can't now say, actually, don't show me these high terrains, you have to create a new search.
Why is it designed this way? Surely, by looking at the map, you should be able to filter everything.
This is all by and by, because you can't filter on the app, which is where a lot of cache finding and choosing happens. A subcategory would allow the app and the website, to clearly show a tree climb/canoe etc.
Currently, you'd need to use pocket queries to filter out caches with specific attributes set.
You can do it by using the Search page and just input your city. Then additional filter options are available and you can show the results on the map.
If this isn't possible, then is there a search function that can exclude tree climbing attributes for example and you can view a map that's actually achievable?
As IceColdUK has posted, what you want is a pocket query. I use them almost exclusively myself for caching. You can fiddle with attributes and ratings, and map the results.
So what you really want, rather than new cache categories, is the ability to filter/search by attributes - I think that will get much more support and is more likely to be implemented.
I would like that 100%, including on the app.
But I stick by my original point too, being able to see on the standard map that a cache involves climbing, without having to go into it, would be really helpful. I know for many people, they probably don't have many trees for climbing around, but I'm on the border of town/city and fields/woods, there's a surprising number around. Some people love climbing ones, I've seen posts on Fb about it and that would be handy for them too.
I don't feel we really need to dumb-down the hobby any more than it is. All caches are rated by their difficulty and terrain.
If you're simply looking to stay clear of those caches that are difficult for you, D/T usually works.
For example, what kind of climbing ?
"Climb a tree" could be anything from a 4 , simply hand-climbing a tree, and 5, technical rope use.
How would this "category" differentiate the two? Next month would someone propose sub-categories since the categories now need some work ?
See how things start getting silly ?
You could already get close (if COs would place attributes on their caches), by using PQs.
There's also the matter of the style of cache. For example, if a tree-climbing cache today was listed as a Multi and was later listed as a "Tree-cache", you lose that multi information. Likewise with a puzzle cache up a tree, etc. That's where the "Tree climbing" attribute comes in, because it can be added to a cache and we can have both the cache type and tree climbing information alongside each other.
It sounds like what's really needed is an easier way to search based on attributes.
Attributes can solve issues if they are easily searchable and added correctly. From the main map, this isn't doable, so unless you're a seasoned hand and you know that search actually brings up something different, then you might not stumble across it. You also might not see the filters option. I've brought a few people to caching and they really struggle to find the website/app/whole deal accessible. PQ's are a big complicated mess that no-one wants to touch, if they've actually heard of them.
It looks like you're about to get your wish on point 2. From the latest Release Notes thread, from the 25th of April the Browse map (what you call the main map) will be replaced by the Search map when you click on View Map from the Play menu at the top of the page or click on View Larger Map from a cache page, with the Browse map only accessible as a toggle from the Search map.
All caches are rated by their difficulty and terrain.
How would this "category" differentiate the two?
Difficulty/Terrain too much depends on a cache owner estimation, it is too ambiguous in my opinion. But there are more attributes to choose from, although the icons do not suggest so. There is a 'Difficult climbing' attribute, it can be used with 'Tree climbing required' together to flag up the climbing requiring higher skills. Also, there is a 'Climbing gear required' attribute, with 'Tree climbing required' it can be used for really challenging trees.
But I miss a kind of 'Dangerous cache' attribute. There is a 'Dangerous area' attribute but it is not the same, I suppose. I mean an attribute which says the cache is placed in such manner that getting to it without sufficient skills and equipment may lead to serious injuries or even death.
You can use the dangerous area attribute always for this purpose unless the cache itself, not the surrounding area, exposes a danger like electric schock or explosion, if you open or access the container wrong way. There is no dangerous cache attribute, but in this case it would be kind to explain the situation in the description and still use the dangerous area attibute.
I think "dangerous" would be too broad a brush. There are many reasons why a cache could be dangerous for someone ill-prepared and the best way to deal with that is to know what the specific danger is. A P&G could be dangerous if it's close to a busy road, for example. Some of the more common dangers are covered by specific attributes, such as cliffs/falling rocks, poisonous plants, dangerous animals, thorns, ticks, hunting and abandoned mines, while significant hike, difficult climbing, tree climbing, may require swimming, may require wading, watch for livestock and abandoned structure could present hazards to some who aren't prepared. Some of the negative attributes can indicate hazards too, like not recommended at night and no kids. Most of the special equipment required (boat, climbing gear, scuba gear, flashlight, snow shoes, cross country skis) can infer hazards. If there was an attribute I'd most like to have on some of my hides which isn't currently available, it'd be slippery surface.
It is possible that I understand dangerous area differently. For example: there is a forest with roads quite often traveled by walkers, runners, bikers etc. I can hardly name it a dangerous area, in daylight especially. And there is a cache near the road, placed on a tree at about 5 meters. It does not need special equipment because thick branches are available from the cache to ground level almost. But it is a dangerous cache for me because the risk of falling from the cache height is not marginal and it may end badly.
Yes, there are many attributes covering dangers but there is no common attribute which will warn on the first sight: this cache hunting can be harmful if you are not well prepared, look closely at another attributes and cache description to know more. I someone tries to avoid any kind of potentially harmful actions, one look will be enough to ignore the cache.
Do you fall from the tree because the cache container gives some nasty surprise or is it just the area where you are seeking the cache which makes it dangerous? Normally the danger comes from the area, not the cache itself.
That is why I wrote my understanding of area is different. A single standing tree is not area for me, it may stand within area. I know it may only be a semantic difference but if I ask some people not involved in geocaching the answer would be the same. A 'dangerous location' or even 'dangerous place' would be more meaningful in this case.
All of my caches have some element of danger. They're all in bushland so at the very least is the possibility of encountering snakes and other wildlife like leeches, ticks, spiders and biting insects. Many are off-track and are close to two or three metre dropoffs where you could easily break an arm or a leg if you're not careful, likewise there are loose rocks and slippery leaf litter that can lead to a fall. Some have cliffs nearby and on those I have both the cliffs/falling rocks attribute and the no kids attribute. Some involve long and/or steep hikes which could be dangerous in hot weather (dehydration and heat stroke can be deadly) or for someone with a poor level of fitness. Even my EarthCache is dangerous if attempted at high tide or when the seas are high, and the rocks around the headland can develop a very slipperly algal growth after prolonged rain.
Whether a cache is dangerous or not can also depend on the individual. For a tall skinny twenty-year-old, climbing a tree or scrambling over boulders would pose little risk, but for someone my age (64) with stubby little arms and legs and a bit of excess girth it's a different story. If I see a cache has a tree climb attribute I'll take a ladder.
The other problem I see with a blanket "dangerous" attribute is that its absence could be construed by some to mean there are no dangers at all, which in itself is a very dangerous assumption to make. A cache in a park hidden at the base of a tree might seem innocuous enough, until a branch falls and hits the cacher on the head. All trees can be dangerous even if you don't try to climb them. Rain can make a paved area slippery. Wind can topple things over. Rivers, creeks and stormwater drains can suddenly flood. Lightning can strike. Motor vehicles are an ever-present danger on urban hides. The whole world is a dangerous place.
A "dangerous" attribute by itself really doesn't convey much information, as for anyone concerned about safety, the very next question has to be "What are the dangers and how can they be managed?" And that question should be asked of any cache whether it has a dangerous attribute or not.
I should be more specific here. If simple entrance into area can be dangerous whatever purpose you are there - looking for a particular cache or only passing without any geocaching on mind - it is a dangerous area, a deep swamp for example. If you can relatively safely pass area, even close to the cache, but without looking for it - it is not a dangerous area (and as I wrote before, area is wider than single tree for me). I agree, of course, one can see something dangerous where someone else does not, it is subjective view and always be. But going back to the tree example I put before - if I made a cache like this and add 'dangerous area' attribute, because I consider it dangerous, I am almost sure there would be questions 'what is so dangerous here, it is such a quiet neighborhood?' or something like that.
I'm not a fan of multi caches, so before heading out I skim read those pages and exclude the ones with multiple waypoints and complex calculations which I know I'll not enjoy in the field. Officially categorising puzzle types would be impossible: , many of the best I've managed to solve use multiple tricks or entirely novel ones, working out those tricks is actually the attraction of the cache type. By the way, the type is 'unknown', not actually 'puzzle' , I believe I read somewhere the name is a catch-all for those listings which do not neatly fit the other categories, so the definition of 'unknown type' might be 'not a trad., multi, Wherigo or virtual'.
The cache listing is how the cache owner decides to present their cache on Groundspeak, if they don't want to announce it is a tree climb in the title or wording, and a cacher fails to read the attributas or past logs or gallery photos or terrain rating where the tree climb is shown and mentioned, that's a failure to prepare ( which is , as we know, a preparation to fail !) rather than something which needs to be addressed by a complex set of cache types which would inevitably cause more disputes over what exactly is a puzzle or a multi or a field puzzle or a dangerous place .... Those awkward shades of meaning are perfectly well served by a couple of paragraphs of words describing the cache.
As adults we take responsibility for our actions, part of that is assessing risk, and what we personally find an acceptable level of risk. I cannot tell you what is safe for you: I've chickened out of tree climbs other folk have had their 10 year old children do ( I claim my problem is having have a highly developed imagination, which when I think of the climb vividly shows me broken and bloody at the bottom of the tree ...) but then I have scrambled up rock or into confined spaces plenty of others have bottled out of. Part of the joy of caching for me is that my limitations have been challenged and sometimes overcome, but if I get to a GZ and look at the terrain and think, actually, no, that's beyond what I can safely get to, so what ? I did the sensible thing and learned something about myself.
Personally, I'd have a 'dangerous area' attribute on every cache in every city : obviously ridiculous, but muggers, traffic, pollution, , late night drunks, carjackers , gang warfare, knife crime , urban rioting etc etc, at different seasons, times and moods the city can be a dangerous place. You are far, far more likely to die in a car crash en route to a cache than you are to die attempting to find that cache. The only log I have on any of my caches where a cacher was in danger was when someone tried to threaten a cacher into handing over his 'phone while he was doing a stage of my urban earthcache, the only cache I have in the city.
GSAK (as usual) has this covered, with two different symbols on its cache page map display, one for the original co-ords (sometimes needed for parking, trailhead etc) and a more prominent one for the final calculated location. It also shows a yellow triangle on wider maps and the cache list ,and is sortable by that corrected co-ordinate flag. I have over 350 solved but not yoet found caches in my GSAK database, easily searched for and downloaded to my device.
More and more, I'm thinking there are two entirely different factions developing within groundspeaks version of geocaching: there's the rural, adventurous type, largely (but not solely) involving GPS, website, outdoorsy , walking/hiking/boating , where numbers are not the main aim, decent size caches are preferred, and folk are reliant on their own skills and resources. On the other hand is the urban app type of caching, which is drive by, smartphone, predominantly micro (that's all that fits behind a road sign) quick easy find, statistic driven, and altogether less involving sort of caching, where participants may become conditioned to expect a quick find at every attempt with no planning or forethought required.
I suggested that a few years ago when the "new search" came out.
With the exception of Challenge Caches, I don't believe we need any new cache types. In fact, with the appropriate attribute (something like 'has multiple waypoints'), Multi-Cache could be eliminated.
However, IMO there are several problems with attributes.
Attributes used for searching should be relevant for how a large number of people want to search. For example, the often discussed 'Part of a Power Trail' attribute. Lots of people want that, but we don't have it. So, attributes are not as relevant as they could be for searching.
Attributes are also used for information. 'Watch for Livestock' may be a good example. It informs people that there may be livestock. I have a hard time envisioning why someone would want to search for a cache with this attribute (outside of a challenge cache). Are there other informational attributes that people want?
Some attributes are overlapping with others, and the definitions are ambiguous. There needs to be well defined definitions. The discussion about dangerous area shows that. And what the heck is 'Truck Driver/RV'?
New caches are being published without any attributes.
Old caches don't have attributes.
There is no way to crowd source attributes on caches.
Yep. Challenge caches seem distinct from the other cache types, and have outgrown the "catch-all" mystery/puzzle type.
In fact, with the appropriate attribute (something like 'has multiple waypoints'), Multi-Cache could be eliminated.
What type do you envision multi-caches moving too? Do you see them being merged with mystery/puzzle caches?
And what the heck is 'Truck Driver/RV'?
I think it refers to locations that are accessible from the freeway by vehicles with long wheelbases (i.e., trucks and RVs).
I think that's incredibly dismissive to sideline people as "statistic driven" just because they're urban/smartphone cachers.
What you're saying that now Geocaching is becoming more accessible to everyone because it's accessible via an app, is that everyone wants an easy find. It's not true, but in a city, it is a lot more limited towards the smaller caches. There is still variety. London has a huge variety. You can still go for a 6-hour walk that includes hills, trees and a beach. Plus, there's a ridiculous amount of history in a city and not in a forest or up a hill, so, there's a lot more limit to what can be written in a description, whereas in a city, every building and street has a story to tell. I love that as I go out geocaching I discover parts of the city I've never found before or connect different parts together because normally I don't walk/bus between the sections, I go underground. It is an adventure, I'm just not looking at grass or leaves.
I cannot climb trees or mountains etc, I have chronic health issues that stop this and I like to head out not having read every minute detail of a cache. I like to head out and see what I find. I am suggesting that when I'm looking at a map, I'd like to not see Tree climbing caches, or be able to tell from afar/by filtering that it's not possible.
Also, I just want to say that behind a road sign is anywhere from micro magnetic - 35mm film pot - small tupperware, plus caches on road signs are not always "behind".
I'm not serious about removing multi-caches. I just wanted to point out that in addition to adding new cache types, we could also eliminate some, by adding enough attributes. The majority of multi-cache's I have done are what I would call traditionals with multiple waypoints. Go to these coordinates, to get more coordinates to follow. If there is a puzzle, or math, or character substitution involved, they could be mystery/puzzle caches, with a multiple waypoints attribute.
You're not alone, but I don't agree with you. Yes, those things you're talking about can cause confusion, but I think that's because different things are different. I don't think any kind of formal classification for the cases you're talking about will make these complications any easier to navigate. Putting a geocache in a new category doesn't really help when the example is sufficiently rare that no one recognizes the category when it's used. If the CO doesn't tell you what you need to know in the description, there's no reason to think he'd categorize it correctly if there were appropriate categories.
It's been a long-held tradition that with a traditional, you should be able to just navigate to the published coordinates and find a cache. The app even formalises this by saying to only look at the description if you get stuck. I think trying to implement a multi as a traditional with multiple waypoints would cause problems like people stuffing a signed scrap of paper into the first waypoint object and claiming the find (and perhaps adding an NM for a "missing log" as well).
There's no need to read "minute details" of a cache.
Without PQs though, you would need to look at D/T ... but that's just the pop up box when you click on any cache on the map.
Most your finds are 2T and below (but you did do a 3 today ...). Curious, are you saying you're finding tree-climbing caches rated 2T ?
By the help center, a 2T cache's physical effort needed is "The hike is less than 2 miles (3 km) along well-defined paths with no significant elevation change or overgrowth".
You have chosen to ignore six paragraphs of my post which explain how to avoid your problem, and why it should not be a problem in the first place, in order to tell me what you think I'm saying in the final paragraph, which you seem to interpret as a personal affront.
I've seen too many nasty disputes here to think it is worthwhile adding any more to this thread.
Just because I didn't quote them, doesn't mean I ignored them. You quoted the original post and we've had a lot of discussion since then. Plus you referenced an additional site assuming people have knowledge of them during a discussion of this website. I enjoyed your comment about imagination, I have rather too much too.
Of course not, I'm saying that I set out along a route and suddenly up pops a tree when everything else has been casual little bisons on fences or small tupperware behind a tree etc. Or we look to see which way has a good little run of caches and then discover that most of them are up trees. They're going to sit on the map providing false optimism. I'd like to be able to look at a map and not see these, or see them differently. As we've all settled on, an attributes search/hide option would be the best way to do this. Some people like climbing trees and actively going looking, others keep feet firmly on the ground. Currently, it put everyone as equally able to climb. I'll only ever have tree climbs logged if I go with someone who can climb or we take a cherry picker.
The three terrain was because you had to walk 13m along a country lane to a bus stop or so from the nearest place to you could pause with the car. Would've been a lot longer of a walk in the road without a car.
I don't agree, I would put it as "Currently, it puts the onus on you to check the cache page before you go hunting for it", which might be a bit of a hassle but it's perfectly doable.
Having said that I do agree (as do many others) that there should be an ability to search by attributes, and this has been requested many times in the past but never implemented. I did hold out some slight hope that with the advent of the new search map the powers that be might see their way to implementing it, but it doesn't seem so.
For the time being your best bet is to check the cache details for anything with a T rating of 4 or 4.5 or above, that ought to get the vast majority of climbs (anything requiring equipment will be a T5 if it's been correctly classified), and is unlikely to throw up many caches with that rating due to long walks or steep hills in the part of the country where you cache.
Alternatively you could hook up with a climber and give it a go (there are quite a few in London/SE England).
The basic idea behind this thinking is there's some hideous problem with walking up to a cache -- or even a string of caches -- and realizing it's up in a tree, so you can't get it. I think that's wrong headed. If you're just casually geocaching, it is so simple to just shrug your shoulders and move on to the next cache. If you can't do that, what are you going to do when you can't find a cache that's on the ground? If, on the other hand, you're carefully planning your route and it will be spoiled because this cache is up a tree, then it's equally simple to read the description while you're planning so you can adjust the route before you're disappointed.
I'm a little different than you: some trees I can climb, some I can't. So unless they invent a "climbable by dprovan" attribute, I always have to go to GZ before I know whether I'll be able to get the cache. Instead of being disappointed when I can't, I just accept it as one more of many reasons I can't find a cache.


This is a shortcut to describe a swelling.
- PULSATION: Is the swelling pulsatile?
- TENDERNESS: Is the swelling tender?
- TEMPERATURE: Is there any local rise in temperature?
- Plane/Mobility: Where is the swelling lie on in relation of underlying structures such as bone, muscle, skin?
- SWALLOWING: Does the swelling moves with swallowing?
- PROTRUDE TONGUE: Does the swelling move with protruding the tongue?
- PEMBERTEN SIGN: Ask the patient to put both his/her hands on his/her head, wait for 10 seconds and look for any pressure signs/cyanosis on his/her face. Its significant is that, if it is positive, it suggest that there is something (such as retrosternal extension of goitre) compressing the iminent vein.
- TRACHEA POSITION: Is the trachea centrally placed?
- COUGH IMPULSE: Does the swelling become more prominent on coughing?
- REDUCIBILITY: Is the swelling reducible (on its own or on manual reduction)?
You may add other special tests on your examination if it is relevant. But the above are the important points that you must not miss.


HomeCongressional Champions Call for Bold Leadership to Eliminate Deadly Hepatitis B and C Epidemics in U.S.
Congressional Champions Call for Bold Leadership to Eliminate Deadly Hepatitis B and C Epidemics in U.S.
CONGRESSIONAL CHAMPIONS CALL FOR BOLD LEADERSHIP TO ELIMINATE DEADLY HEPATITIS B AND HEPATITIS C EPIDEMICS IN U.S.
WHAT: This World Hepatitis Day, Congressman Mike Honda (D-Silicon Valley) and Congressman Hank Johnson (D-GA) will join with community leaders, in solidarity with those impacted by viral hepatitis around the world, to host a press conference calling on Congressional colleagues to urgently commit bold leadership and resources to eliminate the hepatitis B and C epidemics in the United States. This event is one of many worldwide and across the nation calling attention to the largest communicable – yet preventable – epidemics in the United States.
WHY: World Hepatitis Day is commemorated annually on July 28th to raise awareness of the estimated 400 million people worldwide affected by hepatitis B and/or C, and the 4000 lives lost each day to these epidemics. In the United States an estimated 6 million people are affected by one of these chronic, communicable, potentially life-threatening conditions, which are the largest drivers of liver cancer domestically. A shocking 65-75% of these individuals do not know they are infected, given the largely asymptomatic nature of hepatitis B and C. Despite very limited surveillance capacity, over 21,200 deaths were attributed to hepatitis B and hepatitis C in 2013 in the U.S. Tragically, exemplifying our failure to adequately address hepatitis B – an entirely preventable condition thanks to a safe, effective vaccine – infants continue to be born hepatitis B-positive. In these cases, 90% will develop chronic viral infection dramatically increasing the odds of developing cirrhosis, end-stage liver disease, or liver cancer. As the opioid/heroin and overdose crises grip the nation, an emerging wave of the hepatitis C epidemic has followed in their wake, particularly among young people under 30 years old in Appalachia and neighboring regions. In just the past few years, novel curative treatments were released, and for the first time, there is truly an opportunity to eliminate hepatitis C in the U.S. given a genuine, urgent commitment of leadership and resources proportional to the immense need in order to drive effective intervention.


text:
\section{Introduction}
Recent years have seen enormous progress toward large-scale quantum computation.
A central goal of this effort is the implementation of a type of quantum computation that solves computational problems of practical relevance faster than any classical computer. However, the noisy nature of quantum gates and the high overhead cost of noise reduction and error correction
limit near term devices to shallow circuits~\cite{Pre18}.

\Acp{VQA} have been proposed to bring us a step closer to this goal.
Here, an optimization problem is captured by a loss function given by expectation values of observables w.r.t.\ states generated from a parametrized quantum circuit.
Then a classical computer trains the quantum circuit by optimizing the expectation value over the circuit's parameters. Figure~\ref{fig:sketch} illustrates a possible \ac{VQA} routine.
Popular candidates to be used on near term devices are \acp{QAOA} \cite{FarGolGut14}
and \acp{VQE} \cite{PerMcCSha14}; see Ref.~\cite{Cerezo20VariationalQuantumAlgorithms} for a review.

\acp{VQE} are proposed, for instance, to solve electronic structure problems, which are central to quantum chemistry and material science.
Proposals of \acp{QAOA} include improved algorithms for quadratic optimization problems over binary variables such as the problem of finding the maximum cut of a graph (\class{MaxCut}).
For hybrid classical-quantum computation to be successful, two challenges need to be overcome.
First, one needs to find parameterized quantum circuits that have the expressive power to yield a sufficiently good approximation to the optimal solution of relevant optimization problems (i.e., the model mismatch is small).
Second, the classical optimization over the parameters of the quantum circuit needs to be solved quickly enough and with sufficient accuracy.
We will focus on this second challenge.

\begin{figure}

	\includegraphics[width=1\linewidth]{sketch}
	\caption{Sketch of a \ac{VQA} optimization routine.
		This work addresses the complexity of the classical optimization part (red).
	}
	\label{fig:sketch}
\end{figure}

For the classical optimization several heuristic approaches are known, most of which are based on gradient descent ideas and higher order methods.
This is convenient, as with the parameter shift rule \cite{Schuld19EvaluatingAnalyticGradients}- the gradient can be calculated efficiently.
Methods include standard BFGS optimization and extensions \cite{Byrd1995LimitedMemoryAlgorithm} and
natural gradient descent \cite{Stokes2020QuantumNaturalGradient}, which has a favorable performance for at least certain easy instances \cite{Wierichs2020AvoidingLocalMinima}.
Second order methods require significant overhead in the number of measurements but can yield better accuracy \cite{Mari2020EstimatingTheGradient}.
Quantum analytic descent \cite{Koczor2020QuantumAnalyticDescent} uses certain classical approximations of the objective function in order to reduce the number of quantum circuit evaluations at the cost of a higher classical computation effort.

However, it has also been shown recently that there are certain obstacles that need to be overcome to render the classical optimization successful.
The training landscape can have so-called \emph{barren plateaus} where the loss function is effectively constant and hence yields a vanishing gradient, which prevents efficient training.
This phenomenon can be caused, for example by random initializations \cite{McClean2018BarrenPlateausIn} and nonlocality of the observable defining the loss function \cite{Cerezo20CostFunctionDependent}.
Also, sources of randomness given by noise in the gate implementations can cause similar effects \cite{Wang2020Noise-InducedBarren}.
Moreover, the problem of barren plateaus cannot be fully resolved by higher order methods \cite{Cerezo20ImpactOfBarren}.

In this work, we show that the existence of persistent local minima can also render the training of \acp{VQA} infeasible.
For this purpose, we encode the \class{NP}-hard \class{MaxCut}\ problem into the corresponding classical optimization task for several versions of \acp{VQA},
which have many far from optimal local minima.

Specifically, we obtain hardness results concerning the optimization in four different settings:
(i) We use an oracle description of a quantum computer and show that the classical optimization of \acs{VQA} is an \class{NP}-hard problem, even if it needs to be solved only within constant relative precision.
Next, we remove the oracle from the problem formulation by focusing on classically tractable systems where the underlying ground state problem is efficiently solvable.
Here, we consider quantum systems where (ii) the Hilbert space dimension scales polynomially in the number of parameters (i.e., logarithmically many qubits) or (iii) is composed of free fermions.
(iv) If the setup is restricted to the \ac{QAOA} type, we show that our hardness results also hold.

\subsection{Connection to complexity theory}
The decision version of \ac{VQA} optimization is in the complexity class \class{QCMA}, problems that can be verified with a classical proof on a quantum computer.
The class \class{QMA}, which allows for the proof to be a quantum state, contains \class{QCMA}.
Much about the relationship between classical \class{MA}, \class{QCMA}, and \class{QMA}\ is still unknown.
Notably, finding the ground state energy of a local Hamiltonian is \class{QMA}-hard \cite{KitSheVya02,Kempe04TheComplexityOf}.
This means that if $\class{QCMA}\neq \class{QMA}$, then \ac{VQA} algorithms will not be able to solve the local Hamiltonian problem, but only problems contained in \class{QCMA}.
Our results imply that even if the relevant energy eigenstates are contained in the \ac{VQA} ansatz, the classical optimization may still be at least as difficult as solving \class{NP}\ problems (Section~\ref{sec:VQAopt_q}).

\subsection{Notation}
We use the notation $[n]\coloneqq \{1,\dots,n\}$.
The Pauli matrices are denoted by $\sigma_x$, $\sigma_y$, and $\sigma_z$.
An operator $X$ acting on subsystem $j$ of a larger quantum system is denoted by $X^{(j)}$ -
e.g., $\sigma_x^{(1)}$ is the Pauli-$x$ matrix acting on subsystem $1$.
By $\|X\|$ we refer to the operator norm of operator $X$.

The number of edges of the graph with the adjacency matrix $A$ is denoted by $\left|E(A)\right|$.
By $\MaxCut(A)$ we denote the solution of \class{MaxCut}\ for an adjacency matrix $A$; see Problem~\ref{p:maxcut}.

Throughout, we consider only adjacency matrices $A$ of undirected, unweighted graphs with at least one edge; i.e., $A\in \{0,1\}^{d\times d}$\ is a nonzero symmetric binary matrix with a vanishing diagonal.

\section{A continuous \texorpdfstring{\NoCaseChange{\class{MaxCut}}}{MaxCut} optimization}
We introduce a continuous, trigonometric problem which we show to be \class{NP}-hard to optimize and approximate.
This is related to earlier work on the optimization of trigonometric functions~\cite{Pfister2018BoundingMultivariateTrigonometric} for which \class{NP}-hardness is known. For the specific class of functions, we also show the existence of an approximation ratio explicitly.
Below, we use this problem to obtain hardness results for various \ac{VQA} versions.

	\begin{problem}[\class{MaxCut}]\label{p:maxcut}\hfill\vspace{-.8\baselineskip} \\
		\begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont]
			\item [Instance]
			The adjacency matrix $A\in \{0,1\}^{d\times d}$\ of an unweighted undirected graph.

			\item [Task]
			Find $S\subset [n]$ that maximizes $\sum_{i\in S,j\in [n]\setminus S} A_{i,j}$.
		\end{description}
	\end{problem}
	{\class{MaxCut}} is famously known to be \class{NP}-hard.
	Additionally \class{MaxCut}\ is \class{APX} -hard, meaning that every polynomial time approximation algorithm
	there exist some instances, where the approximation ratio $\alpha$, the ratio between the algorithmic solution and the optimal solution, is bounded by $\alpha\leq \alpha_{\max}<1$, assuming that $\P\neq \class{NP}$.
	It was shown that if the unique games conjecture is true, then the best approximation ratio of a polynomial algorithm is $\alpha_{\max}=\min_{0<\theta<\pi}\frac{\theta/\pi}{(1-\cos(\theta)/2)}\approx 0.8786$~\cite{khot_optimal_2007}, which is also what the best known algorithms can guarantee~\cite{goemans_improved_1995}. Without the use this conjecture, it has been proven that $\alpha_{\max}\leq\frac{16}{17}\approx0.941$~\cite{hastad_optimal_2001}.
For our purposes we define a continuous, trigonometric version of \class{MaxCut}.
Minima of real valued functions are given by real numbers that may not have an efficient numerical representation.
However, it is commonly said that a minimization problem is solved if it is solved to exponential precision, which is the convention we will also be using throughout this paper. The intuitive notion is that the hardness does not come from the difficulty of representing the minimum.
	\begin{problem}[Continuous \class{MaxCut}]\label{p:shift_maxcut}\hfill\vspace{-.8\baselineskip} \\
		\begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont]
			\item [Instance]
			The adjacency matrix $A\in \{0,1\}^{d\times d}$\ of an unweighted graph.

			\item [Task]
			Find $\vec\phi\in[0,2\pi)^d$ that minimizes
			\begin{equation}
				\mu (\vec \phi)\coloneqq \frac{1}{4}\sum_{i,j=1}^dA_{i,j}[\cos(\phi_i)\cos(\phi_j)-1].
			\end{equation}
		\end{description}
	\end{problem}
\begin{lemma}\label{lem:continuous_MaxCut}
	Problem~\ref{p:shift_maxcut} is \class{NP}-hard.
	Moreover, if $\P\neq\class{NP}$,
	for every polynomial time algorithm there exists an approximation ratio, which is at most

	that
	of \class{MaxCut}.
\end{lemma}
\begin{proof}
 We will show that it suffices to look at $\vec\phi$ from the discrete subset $\{0,\pi\}^d$.
 For this purpose, we analyze the dependence of $\mu$ on one coordinate $\phi_i$ of $\vec \phi$.
 Denoting the vector obtained from $\vec\phi$ by replacing $\phi_i$ with $x$

 by $\left.\vec\phi\right|_{\phi_i=x}$,
 we write $\mu$ as
 \begin{align}
 \mu(\left. \vec \phi\right|_{\phi_i=x})
 =
 \cos(x) \left[\frac{1}{2}\sum_{j:\; j\neq i} A_{i,j}\cos(\phi_j)\right]+C\,,
 \end{align}

 where we have used that $A_{i,i} = 0$ for all $i$, $A=A^T$ and $C$ is independent of $x$.

 Since the only dependence on $x$ is given by the cosine,
 it follows that for any $\vec \phi\in [0,2\pi)^d$
 \begin{equation}
 \mu(\vec\phi)\geq\min\{\mu(\left.\vec\phi\right|_{\phi_i=0}),\mu(\left.\vec\phi\right|_{\phi_i=\pi})\}\,.
 \end{equation}
 This observation implies that we can iteratively choose each $\phi_i$ to be in $\{0,\pi\}$ for $i\in[d]$, without increasing $\mu$. Therefore, an algorithm returning continuous values of $\phi_i$ can be turned discrete in polynomial time without reducing the approximating power.
 The discrete problem can be written as a quadratic unconstrained binary optimization
 \begin{equation}
 \begin{aligned}
 \min_{\vec\phi\in \{0,\pi\}^d}\mu(\vec \phi)
 &=
 \frac{1}{4}\min_{\vec v \in\{-1,1\}^d} \sum_{i, j=1}^dA_{i,j}(v_iv_j-1)\\

 &=\frac{1}{4}
 \min_{\vec v \in\{-1,1\}^d} \sum_{i,j=1}^d A_{i,j}\left(-2\delta_{v_i\neq v_j}\right)\\
 &=-\max_{S \subset [d]} \sum_{i\in S,j\in [d]\setminus S} A_{i,j} \, ,
 \end{aligned}
 \end{equation}
 where $\delta_{v_i\neq v_j}=1-\delta_{v_i,v_j}$ and we used again that $A$ is symmetric.
 Therefore, we obtain a  (many-one) reduction of {\class{MaxCut}} to Problem~\ref{p:shift_maxcut}, implying Problem~\ref{p:shift_maxcut} is \class{NP}-hard, which finishes the proof.
\end{proof}

We note that the derivative and the Hessian of $\mu$ is given by
\begin{equation}
\begin{aligned}
\frac{\partial\mu(\vec\phi)}{\partial \phi_i}
&=-\frac 12 \sin(\phi_i)\sum_{j:\; j\neq i} A_{i,j}\cos(\phi_j)\\
\frac{\partial^2\mu(\vec\phi)}{\partial \phi_i\partial \phi_k}
&=-\frac 12\delta_{i,k} \cos(\phi_i)\sum_{j:\; j\neq i} A_{i,j}\cos(\phi_j)\\
&+\frac{A_{i,j}}{2} \sin(\phi_i)\sin(\phi_k) \, .
\end{aligned}
\end{equation}
At the relevant discrete set $\vec\phi\in\{0,\pi\}^d$, the derivative vanishes and the Hessian is diagonal,
meaning a point in the set describes a local minimum whenever changing any single $\phi_i$ increases the objective function.
The same minima (in the discrete \class{MaxCut}\ formulation) are also achieved by a greedy algorithm:
start with a random bipartition of the vertex set, then repeatably change a single vertex assignment if it increases the cut until the cut cannot be increased any further by this update rule.
This algorithm has an approximation ratio of $\alpha=\frac{1}{2}$, meaning it only guarantees to approximate \class{MaxCut}\ to half its optimal solution.
If Problem~\ref{p:shift_maxcut} is solved with gradient based methods, any local minimum can be the final result, therefore gradient based algorithms also only have an approximation ratio of $\alpha=\frac 1 2$, which is significantly worse than what modern \class{MaxCut}\ solvers can achieve~\cite{goemans_improved_1995}.

\section{\texorpdfstring{\Aclp{VQA}}{VQAs}}
\Ac{VQA} is a general framework of hybrid quantum computers, where classically tunable parameters $\vec \phi$ of a unitary circuit are used to minimize the expectation value of an observable.
First, in Section~\ref{sec:VQAopt_q}, we consider such computing schemes, where the quantum part is composed of qubits.
In order to show that the classical simulation is hard, we assume oracle access to an idealized quantum device.
Next, in Section~\ref{sec:VQEopt_poly}, we show that the problem is also hard for \ac{VQA} settings with small Hilbert space dimensions (or logarithmically many qubits) so that the oracle can be replaced by efficient classical simulation.
In Section~\ref{sec:VQAopt_QAOA} we use the same setting, but consider \acp{QAOA} instances instead.
Last, in Section~\ref{sec:VQAopt_ff}, we analyze \acp{VQA} in free Fermionic systems, where the oracle can be replaced by efficient free fermionic calculations.

\subsection{\texorpdfstring{\Ac{VQA}}{VQA} optimization with quantum computer access}
\label{sec:VQAopt_q}
The common application of \acp{VQA} is within quantum computing, where a quantum computer is used to estimate the expectation value and a classical algorithm chooses the circuit parameters of the quantum computer.
For the classical optimization, we describe the information obtained from the quantum computer with oracle calls made by the classical algorithm.

\begin{problem}[\Ac{VQA} minimization, oracular formulation]\label{p:VQA_O}\hfill
	\vspace{-.8\baselineskip}
	\begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont]

		\item[Instance]
		A set of generators $\{H_i \}_{i \in \{1,\dots, L\}}$
		and an observable $O$ acting on $\H=(\mathbb{C}^2)^{\otimes N}$,
		given in terms of their Pauli basis representation.

		\item[Oracle access]
		We set $\ket{\Psi(\vec \phi)}\coloneqq U_L(\phi_L)\cdots U_1(\phi_1)\ket{\vec 0}$ with $U_i(\phi)=\ensuremath\mathrm{e}^{-\i H_i \phi}$.
		 The oracle $\mc O$ returns
		$\OEx{\vec \phi} \coloneqq \sandwich{\Psi(\vec \phi)}{O}{\Psi(\vec \phi)}$, for a given $\vec \phi$, up to any desired polynomial additive error.

		\item[Task]
		Find $\vec \phi \in \mathbb{R}^L$ that minimizes $\OEx{\vec \phi}$ provided access to $\mc O$.
	\end{description}
\end{problem}
We use the oracle to outsource difficult computations, which is similar to how a quantum computer would in a physical implementation. The motivation of our oracle is that Problem~\ref{p:VQA_O} captures the complexity of only the classical optimization effort in hybrid quantum computations.
The oracle can be seen as postselecting on the successful runs only, therefore making the return deterministic.

\begin{proposition}[Hardness of \ac{VQA} optimization, oracular formulation]
	\label{thm:VQA1}
	Assuming $\P\neq \class{NP}$ there is no deterministic classical algorithm that solves Problem~\ref{p:VQA_O} in polynomial time.
\end{proposition}

It is straightforward to show that Problem~\ref{p:VQA_O} is \class{NP}-hard to solve.
Essentially,
we use a diagonal observable for which the ground state problem is \class{NP}-hard and use unitaries to reach every computational basis state.
\begin{proof}
	We prove the proposition via a reduction of Problem~\ref{p:shift_maxcut} to Problem~\ref{p:VQA_O}. For this, let $N=d$ and let $O$ the usual Ising Hamiltonian encoding of \class{MaxCut}
	\begin{align}
	O&\coloneqq\frac{1}{4}\sum_{i,j=1}^dA_{i,j}(\sigma_z^{(i)}\sigma^{(j)}_z-1)\, .
	\end{align}
	We use $L=d$ layers with
	\begin{align}
		H_i&\coloneqq \frac{\sigma_y^{(i)}}{2} \quad,\quad i\in [d] \,,
	\end{align}
	as generators.
	By direct calculation we find that
	\begin{equation}
	\begin{aligned}
	\OEx{\vec \phi}&=\bra{\Psi(\vec\phi)}O\ket{\Psi(\vec \phi)}\\
	\label{eq:oracle_replacement}
	&=\frac{1}{4}\sum_{i,j=1}^dA_{i,j}\left[\cos(\phi_i)\cos(\phi_j)-1\right]\\
	&=\mu(\vec\phi)
	\, ,
	\end{aligned}
	\end{equation}
	which is the objective function of Problem~\ref{p:shift_maxcut}.
\end{proof}

To analyze the overall approximation power of an algorithm we define the \emph{approximation error} for an instance as
\begin{align}
\delta\coloneqq \frac{\braket{O}_a-\lambda_{\min}(O)}{\lambda_{\max}(O)-\lambda_{\min}(O)}\, ,
\end{align}
where $\lambda_{\min}(O)$ is the smallest eigenvalue of the observable $O$ and $\lambda_{\max}(O)$ is the largest; the expectation value of the final output of the algorithm is $\braket{O}_a\geq \lambda_{\min}(O)$.
We normalize by the \emph{spectral width}
\begin{align}
\sw(O)\coloneqq\lambda_{\max}(O)-\lambda_{\min}(O)\,,
\end{align} as this ensures that $\delta\in [0,1]$.
There are two error contributions:
(i) the \emph{model mismatch} $\delta_m$ is the approximation error resulting from the ansatz class being unable to represent the ground state and
(ii) the \emph{optimization error} $\delta_o$ is the error due to the classical algorithm not converging to the optimal solution within the class.
That is,
\begin{align}
\delta&=
\frac{\braket{O}_{\min}-\lambda_{\min}(O)}{\sw(O)}+\frac{\braket{O}_a-\braket{O}_{\min}}{\sw(O)}\\
&=\qquad\delta_{\mathrm{m}}\qquad+\qquad\delta_o
\, ,
\end{align}
where $\braket{O}_{\min}$ refers to the smallest expectation value over the ansatz class, i.e., the global minimum over the circuit parameters.
Since we are interested in classical algorithms, we define an optimization error, in a similar manner to how approximation ratios are defined for \class{NP}\ optimization problems (the complexity class \class{APX}), over all considered instances.
\begin{definition}[Optimization error]\label{def:approx_val}
	The \emph{optimization error} of an optimization algorithm $\Delta\in [0,1]$ is the smallest number such that
	\begin{align}
	\Delta \geq \frac{\braket{O}_a-\braket{O}_{\min}}{\sw(O)}
	\end{align}
	for all considered \ac{VQA} instances.
\end{definition}

\begin{corollary}\label{cor:apvalexp}
	If $\P\neq\class{NP}$,
	then there exists no polynomial time algorithm which can guarantee any optimization error $\Delta<1$ for all \acp{VQA} defined by Problem~\ref{p:VQA_O}.
\end{corollary}

\begin{proof}
	We prove this statement by relating the optimization error of a \ac{VQA} to the approximation ratio of \class{MaxCut}\ and by introducing a boosting technique to amplify errors in the setting of Problem~\ref{p:VQA_O}.

	From the proof of Proposition~\ref{thm:VQA1} we obtain $\sw(O)=\MaxCut(A)$ and the optimal solution is also $\left|\braket{O}_{\min} \right|=\MaxCut(A)$, as there is no model mismatch.
	From the algorithm we get $\braket{O}_{a}=\mu_a(A)}%{\MaxCut_a(A)$, where $\mu_a(A)}%{\MaxCut_a(A)$ is the approximation of the continuous \class{MaxCut}\ problem (Problem~\ref{p:shift_maxcut}) and, therefore, an approximation to \class{MaxCut}\ itself.
	With this argument it follows that
	\begin{equation}
		\Delta\geq1-\alpha\, ,
	\end{equation}
	where $\alpha$ is the approximation ratio related to Problem~\ref{p:shift_maxcut} of the algorithm.
	To boost this result we introduce a variable $k$ and choose operators for $k\times d$ qubits
	\begin{align}
		\tilde O&=(-1)^{k-1}O^{\otimes k} \, ,\\
		\tilde U(\vec \phi)&=U(\vec \phi)^{\otimes k} \, .
	\end{align}
	We can verify that the generators and $\tilde O$ only have $\func{poly}(d)$ many terms for constant $k$. For the expectation value this gives
\begin{equation}
	\begin{aligned}
		\braket{\tilde O(\vec \phi)}&=(-1)^{k-1}\bra{\Psi(\vec \phi)}^{\otimes k}O^{\otimes k}\ket{\Psi(\vec \phi)}^{\otimes k}\\
		&=-\left|\OEx{\vec\phi}\right|^k=-|\mu(\vec\phi)|^k \, ,
	\end{aligned}
\end{equation}
	where the introduced sign ensures that the problem remains a minimization for all $k$.
	We obtain $\sw(\tilde O)=\bigl|\braket{\tilde O}_{\min}\bigr|=\MaxCut(A)^k$.
	This yields
\begin{equation}
	\begin{aligned}
	\Delta&\geq \sup_A \frac{\left|\MaxCut(A)^k-\left|\braket{\tilde O}_a\right|\right|}{\MaxCut(A)^k}\\
	&=\sup_A \left(1-\frac{|\mu_a(A)}%{\MaxCut_a(A)|^k}{\MaxCut(A)^k}\right)\\
	&= (1-\alpha^k)\,,
	\end{aligned}
\end{equation}
	Therefore, no optimization error strictly smaller than $\Delta<1$ can exist for all instances (if $\P\neq \class{NP}$), as this would mean in return, the algorithm could solve Problem~\ref{p:shift_maxcut} to arbitrary precision.
\end{proof}

\subsection{Logarithmic number of qubits --- polynomial Hilbert space dimension}
\label{sec:VQEopt_poly}
We can improve on the previous result by allowing only $N\in O(\log(d))$ many qubits, where $d$ is the input length of the \class{MaxCut}\ instance.
This drastically reduces the system's size and complexity. Notably, since the Hilbert space is now only of polynomial dimension, both the calculation of expectation values and the ground state problem can be computed efficiently.
Yet we show that \ac{VQA} optimization is still \class{NP}-hard.
This means that the classical optimization does not merely inherit the hardness of the ground state problem but rather is intrinsically difficult.
Since the expectation value is efficiently numerically simulatable, we do not require oracle access to a quantum computer to analyze the problem.
Also, for convenience, instead of the Pauli-basis we use the computational basis of the Hilbert space $\H$ of dimension $\dim(\H)=2^N\eqqcolon n$.
This gives the following problem description.

\begin{problem}[\Ac{VQA} minimization problem]\label{p:VQA}\hfil
\begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont]
\item [Instance]
An initial state $\ket{\Psi_0}\in \mathbb{C}^n$, a set of generators $\{H_i \}_{i \in \{1,\dots, L\}}\subset \Herm(\mathbb{C}^n)$, where $L$ is the number of layers and an observable $O\subset \Herm(\mathbb{C}^n)$.

\item[Task]
For $\ket{\Psi(\vec \phi)}\coloneqq U_L(\phi_L)\cdots U_1(\phi_1)\ket{\Psi_0}$ with $U_i(\phi)=\ensuremath\mathrm{e}^{-\i H_i \phi}$,
find a $\vec \phi \in \mathbb{R}^L$ that minimizes $\OEx{\vec\phi}\coloneqq~\sandwich{\Psi(\vec \phi)}{O}{\Psi(\vec \phi)}$.
		\end{description}
\end{problem}

\begin{theorem}[Hardness of \ac{VQA} optimization]\label{thm:VQA2}
	\Ac{VQA} optimization (Problem~\ref{p:VQA}) is \class{NP}-hard.
\end{theorem}

\begin{proof}
We prove the theorem via a many-one reduction from Problem~\ref{p:shift_maxcut}
Let $A\in \{0,1\}^{d\times d}$\ be the adjacency matrix of an unweighted graph.
On the Hilbert space $\H=\mathbb{C}^{2d}$
we first define an observable in the standard basis as
\begin{align}\label{eq:defOprime}
	O'&\coloneqq \frac{d}{8}\cdot A\otimes \begin{pmatrix} 1&1\\mathds{1}&1	\end{pmatrix}\,,
\end{align}
where $\otimes$ denotes the Kronecker product. For the actual observable we modify the diagonal as
\begin{align}\label{eq:Observable_loc_VQA}
	O_{i,j}=\left\{\begin{matrix}
	O'_{i,j}& i\neq j\\
	-\sum_{\alpha=1}^{2d} O'_{\alpha,j} & i=j\\
	\end{matrix}\right.\,.
\end{align}
The initial state and generators are chosen as
	\begin{align}
	\ket{\Psi_0}&\coloneqq \frac{1}{\sqrt{2d}} \sum_{j=1}^{2d} \ket{j} \, ,\\\label{eq:VQA_U_small}
	H_i&\coloneqq \ketbra{2i-1}{2i-1}- \ketbra{2i}{2i}\,,
	\end{align}
	where we take $L=d$ layers.
	As the parametrized state we obtain
\begin{equation}
	\begin{aligned}
	\ket{\Psi(\vec\phi)}
	&\coloneqq
	U_{d}(\phi_d)\dots U_1(\phi_1) \ket{\Psi_0}
	\\
	&=
	\frac{1}{\sqrt{2d}}\sum_{j=1}^{2d}\left( \ensuremath\mathrm{e}^{-\i \phi_j}\ket{2j-1}+\ensuremath\mathrm{e}^{\i \phi_j}\ket{2j} \right)
\end{aligned}
\end{equation}
and
\begin{equation}\label{eq:VQA}
	\begin{aligned}
	\OEx{\vec \phi}&=\bra{\Psi(\vec \phi)}O\ket{\Psi(\vec \phi)}
	\\
	&=\frac{1}{16} \sum_{s,p\in\{+,-\}}\sum_{i,j=1}^d\ensuremath\mathrm{e}^{\i s \phi_i }A_{i,j} \ensuremath\mathrm{e}^{-\i p \phi_j }-\frac{1}{4}\sum_{i,j=1}^d A_{i,j} \\
	&=\frac{1}{8}\sum_{i,j=1}^d A_{i,j} \left(\cos(\phi_i-\phi_j)+\cos(\phi_i+\phi_j)-2\right)\\
	&=\frac{1}{4}\sum_{i,j=1}^d A_{i,j}\left[\cos(\phi_i)\cos(\phi_j)-1\right]\\
	&=\mu(\vec\phi)
	\end{aligned}
\end{equation}
as corresponding expectation value.
	This completes the reduction of Problem~\ref{p:shift_maxcut} to Problem~\ref{p:VQA}.
\end{proof}
From this result, \class{NP}-completeness follows for the decision version.
\begin{problem}[\Ac{VQA} minimization, decision version]\label{p:VQA_d}\hfill
	\begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont]
	\item[Instance] An initial state $\ket{\Psi_0}\in \mathbb{C}^n$, a set of generators $\{H_i \}_{i \in \{1,\dots, L\}}\subset \Herm(\mathbb{C}^n)$, where $L$ are the number of layers, an observable $O\in\Herm(\mathbb{C}^n)$ and a threshold $a\in \mathbb{R}$.

	\item[Task]
	For $\ket{\Psi_{\vec \phi}}\coloneqq U_L(\phi_L)\cdots U_1(\phi_1)\ket{\Psi_0}$ with $U_i(\phi)~=~\ensuremath\mathrm{e}^{-\i H_i \phi}$,
	determine wheter there exists
	$\vec \phi \in \mathbb{R}^d$ for which $\sandwich{\Psi(\vec \phi)}{O}{\Psi(\vec \phi)}\leq a$.
\end{description}
\end{problem}

\begin{corollary}
	Problem~\ref{p:VQA_d} is \class{NP}-complete.
\end{corollary}
\begin{proof}
	As calculating the expectation value of observable on polynomial dimensional Hilbert spaces is in \P, $\vec \phi$ is a valid proof for the \emph{yes} instances, which can be verified in polynomial time and is therefore in \class{NP}. Together with hardness of problem~\ref{p:VQA}, this means Problem~\ref{p:VQA_d} is \class{NP}-complete.
\end{proof}
We now show that $L=1$ layer is sufficient to show hardness.
For this purpose we will use certain properties of Hamiltonian spectra.

\begin{definition}[Approximate ergodic energy spectrum]
Let $\epsilon>0$.
We call a set $\{E_i\}_{i\in n}\subset \mathbb{R}$ an \emph{$\epsilon$-approximate ergodic energy spectrum} if for all
$\vec \phi \in [0,2\pi)^n$ there exists $t\in \mathbb{R}_0^+$
such that
\begin{equation}
	\modnorm{\phi_i - E_i t}
	\leq
	\epsilon
\end{equation}
for all $i\in [n]$,
where
$\modnorm{x} \coloneqq
\inf_{k\in \mathbb{Z}} |x-2\pi k| \in [0,\pi]
$.
\end{definition}

Generic energy spectra are exactly ($\epsilon=0$) ergodic.
For our purpose we want to show that there are also efficiently expressible approximate ergodic energy spectra.

\begin{lemma}[Approximate ergodic energy spectra]\label{l:ph_to_en}\hfill
\\
Let $m\in \mathbb{N}$.
Then
\begin{equation}
E_i\coloneqq \frac{2\pi}{m^i}
\end{equation}
with $i \in [n]$ defines an $\epsilon$-approximate ergodic energy spectrum with
\begin{equation}
	\epsilon = \frac {4\pi} m\, .
\end{equation}

\end{lemma}

We provide a proof in Appendix~\ref{ap:erd_en}. The chosen energies can be expressed with $n\times \lceil\log_2(m)\rceil$ bits of precision. With this property we can show the following theorem.

\begin{theorem}\label{t:unitary}
	\Ac{VQA} optimization (Problem~\ref{p:VQA}) is \class{NP}-hard for $L=1$ layer.
\end{theorem}

	\begin{proof}

	For the single layer, we choose the generators as a linear combination  of terms from Eq.~\eqref{eq:VQA_U_small}
	\begin{align}\label{eq:Ham_ev}
	H=\sum_{j=1}^d E_j \left( \ket{2j-1}\bra{2j-1}- \ket{2j}\bra{2j}\right)\,
	\end{align}
	meaning $U(\phi)=\exp(-\i \phi H)=U(\vec E \phi)$. The initial state and $O$ remain identical. This leads to the expectation value
	\begin{align}
		\OEx{\phi}=\sum_{i,j=1}^d A_{i,j}\left[\cos(E_i \phi)\cos(E_j \phi)-1\right]=\mu(\vec E\phi)\,.
	\end{align}
If $\{E_i\}_{i}$ are chosen as in Lemma~\ref{l:ph_to_en} then $\OEx{\phi}$ approximates $\mu(\vec\phi)$ with $\vec \phi = \vec E \phi$ to arbitrary precision,
which we have shown to be \class{NP}-hard to optimize in Lemma~\ref{lem:continuous_MaxCut}.
\end{proof}

By viewing the \ac{VQA} in Theorem~\ref{t:unitary} as a continuous time evolution for logarithmically many qubits, we obtain the following result (we are unaware of this statement being explicitly proven before).

\begin{corollary}
For a system with logarithmically many qubits, we consider the expectation value of a (unitarily) time evolved observable $\langle O(t)\rangle$, starting from some initial state.
Minimizing the expectation value over $t\in \mathbb{R}_0^+$ is then \class{NP}-hard.
\end{corollary}

\subsection{\texorpdfstring{\Aclp{QAOA}}{QAOA} for a logarithmic number of qubits}
\label{sec:VQAopt_QAOA}
\Acp{QAOA} can be seen as certain types of \acp{VQA}, which are inspired by adiabatic computation, where a slow enough transition between two Hamiltonians $H_b$ and $H_c$ guarantees remaining in the ground state as long as the Hamiltonians are gapped and level crossings are avoided~\cite{Albash2018AdiabaticQuantumComputation}.
\Acp{QAOA} capture a time-discrete version of this approach by alternatingly applying the time evolutions of the Hamiltonians.
Accordingly, parameter vectors $\vec \beta,\vec \gamma \in \mathbb{R}^{L}$ need to be chosen, which define how long each Hamiltonian is applied.

We demonstrate that the hardness of \ac{VQA} optimization for logarithmically many qubits also translates to \ac{QAOA} problems.

Formally, the problem is as follows.
\begin{problem}[\Ac{QAOA} minimization problem]\label{p:QAOA}\hfil
	\begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont]
		\item[Instance] Two Hamiltonians $H_b,H_c\in \Herm(\mathbb{C}^n)$ and the number of layers $L$ in unary notation\footnote{This means that the length of the input scales linearly with $L$.}.

		\item[Task] For a tunable state $\ket{\Psi(\vec \beta, \vec \gamma)}\coloneqq U_b(\beta_L)U_c(\gamma_L)\cdots U_b(\beta_1)U_c(\gamma_1) \ket{\Psi_0}$, where $\ket{\Psi_0}$ is the ground state of $H_b$, $U_b(\beta)=\ensuremath\mathrm{e}^{-\i H_b \beta}$ and $U_c(\gamma)=\ensuremath\mathrm{e}^{-\i H_c \gamma}$,
		find $\vec \beta,\vec \gamma \in \mathbb{R}^d$ which minimize $\OEx{\vec \beta, \vec \gamma}\coloneqq\sandwich{\Psi(\vec \beta, \vec \gamma)}{H_c}{\Psi(\vec \beta, \vec \gamma)}$.
	\end{description}
\end{problem}

\begin{theorem}[Hardness of optimization in \acp{QAOA}]\label{thm:QAOA}
	Problem~\ref{p:QAOA} is \class{NP}-hard for $L=1$ layer.
\end{theorem}
\begin{proof}
	We will perform a reduction from single layer \ac{VQA} to \ac{QAOA}, which implies that Problem~\ref{p:QAOA} is \class{NP}-hard.

	We consider the Hilbert space $\H = \mathbb{C}^{2d+1}$.
	For $H_b$ we take
	\begin{align}
	H_b&=\mathrm{diag}(E_1,-E_1,E_2,-E_2,\dots ,E_d,-E_d,-1)\, ,
	\end{align}
	where $|E_i|<1$ for all $i\in[d]$.  For $H_c$
	\begin{align}
	H_c&=O\oplus 0 +\tau\left(\ket{+_{2d}}\bra{2d+1}+\ket{2d+1}\bra{+_{2d}}\right)\, ,
	\end{align}
	where $\ket{+_{2d}}=\sum_{j=1}^{2d}\ket{j}/\sqrt{2d}$, $\tau$ is some real constant that we adjust later and the observable $O$ is as defined in Eq.~\eqref{eq:Observable_loc_VQA}. $O\oplus 0$ refers to $O$ being embedded in the first $2d$ computational states in the Hilbert space.
	By design, $\lambda_{\min}(H_b)=-1$ is the ground state energy with ground state $\ket{2d+1}$.
	\\
	For the state we obtain
\begin{equation}
	\begin{aligned}
	\ket{\Psi(\gamma)}&\coloneqq U_c(\gamma)\ket{2d+1}\\
	&=\cos(\tau\gamma)\ket{2d+1}+\i\sin(\tau\gamma)\ket{+_{2d}}\\
		\end{aligned}
	\end{equation}
	after applying the first Hamiltonian, where we used that $\ket{+_{2d}}$ is an eigenstate of $O\oplus 0$. This gives the final state
	\begin{equation*}
	\begin{aligned}
	\ket{\Psi(\beta,\gamma)}&=U_b(\beta)U_c(\gamma)\ket{2d+1}
	\\
	&=
	\cos(\tau\gamma)\ensuremath\mathrm{e}^{\i\beta}\ket{2d+1}\\&+\i\sin(\tau\gamma)\frac{1}{\sqrt{2d}}\sum_{j=1}^d\ensuremath\mathrm{e}^{-\i E_j\beta}\ket{2j-1}+\ensuremath\mathrm{e}^{\i E_j\beta}\ket{2j}\, .
	\end{aligned}
\end{equation*}
	From this variational state we derive the expectation value
	\begin{equation}\nonumber
		\begin{aligned}
	\OEx{\beta,\gamma}&=\bra{\Psi(\beta,\gamma)} H_c\ket{\Psi(\beta,\gamma)}\\
	&=\sin^2(\tau\gamma)f(\beta)+2\tau\cos(\tau\gamma)\sin(\tau\gamma)g(\beta)
	\end{aligned}
	\end{equation}
	with
	\begin{align}\label{eq:O_QAOA1}
	f(\beta)&=\frac{1}{4}\sum_{i,j}(A_{i,j}\left[\cos(E_i\beta)\cos(E_j\beta)-1\right]\, ,\\
	g(\beta)&=-\frac{\sin(\beta)}{d}\sum_{i=1}^d\cos(E_j\beta)\,.
	\end{align}
	For $\tau\ll1$ the contribution of $g$ becomes insignificant and $\gamma=\frac{\pi}{2\tau}$ minimizes the objective function as $f(\beta)\leq0$, meaning the problem is equivalent to minimizing $f(\beta)=\mu(\vec E \beta )$ which approximates $\mu(\vec\phi)$ to arbitrary precision  if $\vec E$ is chosen as in Lemma~\ref{l:ph_to_en} and therefore gives a reduction from Problem~\ref{p:shift_maxcut}.
\end{proof}

In the proof of Theorem~\ref{thm:QAOA}, the energies of $H_b$ span many orders of magnitude. This means a potential quantum computer needs to be incredibly precise to implement such a \ac{QAOA}.
We will show, that this is not required, but that also for very simple spectra, hardness results can be obtained.
\begin{theorem}\label{thm:QAOA2}
	QAOA optimization (Problem~\ref{p:QAOA}) is \class{NP}-hard for periodic optimization $\vec \beta,\vec\gamma\in [0,2\pi)^L$, even if we restrict $\|H_b\|\leq3$ and $\|H_c\|\leq3$ (i.e $E_i\in \{-3,-2,\dots,3\}$).
\end{theorem}

\begin{proof}[Proof outline]
	In Appendix~\ref{ap:multiLQAOA}, we construct explicit Hamiltonians $H_b$ and $H_c$ from an adjacency matrix $A$, where the solution is $\left<H_c\right>_{\min}=1-\frac{2\MaxCut(A)}{\left|E(A)\right|}$ and $\|H_c\|=1$. We do this by embedding a modified version of Problem~\ref{p:shift_maxcut} into the \ac{QAOA} circuit in such a way, that deviations from the intended structure are penalized by increasing the expectation value.
\end{proof}
From this we can derive bounds on the optimization errors for \ac{VQA} in this restricted setting. Here, we are unable to use the same boosting technique to increase the hardness result further.
\begin{corollary}
	All polynomial time algorithms for \Ac{QAOA} and therefore \ac{VQA} optimization (Problems~\ref{p:QAOA} and \ref{p:VQA}) have an optimization error $\Delta\geq \frac{1-\alpha_{\max}}{2}$, where $\alpha_{\max}$ is the approximation ratio of \class{MaxCut}.
\end{corollary}
\begin{proof}
	For the Hamiltonians in proof of Theorem~\ref{thm:QAOA2}, we have $\sw(H_c)=2$ and the lowest achievable expectation value is $\left<H_c\right>_{\min}=1-\frac{2\MaxCut(A)}{\left|E(A)\right|}$, where $\left|E(A)\right|$ is the number of edges of the the graph. From this we can calculate an upper limit on the possible guaranteed precision of an optimization algorithm for all instances
	\begin{equation}
	\begin{aligned}
	\Delta\geq&\sup_A\left(\frac{\left|\left<H_c\right>_{\min}-\left<H_c\right>_{\mathrm{a}}\right|}{\sw(H_c)}\right)\\
	&\geq\frac{1}{2}\sup_A\left|1-\frac{2\MaxCut(A)}{\left|E(A)\right|}-\left(1-\frac{2|\mu_a(A)}%{\MaxCut_a(A)|}{\left|E(A)\right|}\right)\right|\\
	&=\frac{1}{2}\sup_A\left(\frac{2\MaxCut(A)}{\left|E(A)\right|}-\frac{2|\mu_a(A)}%{\MaxCut_a(A)|}{\left|E(A)\right|}\right)\\
	&=\frac{1}{2}(1-\alpha)\frac{2\MaxCut(A)}{\left|E(A)\right|}\\
	&\geq \frac{1-\alpha}{2}\, ,
	\end{aligned}
	\end{equation}
	where  the supremum goes over all adjacency matrices and $|\mu_a(A)}%{\MaxCut_a(A)|$ is the approximation of \class{MaxCut}\ from the algorithm and $\alpha$ is the approximation ratio of the algorithm;
	in the last step we used that $\MaxCut(A)\geq \frac{\left|E(A)\right|}{2}$.
	This means that if $\P\neq\class{NP}$, any polynomial time algorithm is only able to guarantee \ac{QAOA} and therefore general \ac{VQA} minimization to an optimization error $\Delta\geq\frac{1-\alpha_{\max}}{2}$.
\end{proof}
For gradient based methods this means $\Delta\geq 1/4$ for logarithmically many qubits, as $\alpha = 1/2$ was shown.

\subsection{Free fermionic models}
\label{sec:VQAopt_ff}
Free fermionic models are a certain class of fermionic many-body systems that are without actual particle-particle interactions.
They are especially interesting for us, as they can be simulated efficiently for so-called Gaussian input states and observables.

Fermionic creation and annihilation operators are denoted by $c_j^\dagger$ and $c_j$.
They satisfy the anticommutation relations $\{c_i^\dagger,c_j\}=\delta_{i,j}$ and $\{c_i,c_j\}=0$ for all $i,j$.
We call an operator \emph{quadratic} or \emph{Gaussian} if it is a quadratic polynomial in the creation and annihilation operators.
We will consider \emph{(balanced) quadratic observables} of the form
\begin{equation}\label{eq:quadraticO}
	H = \sum_{i,j} h_{i,j}\, c^{\dagger}_{i}c_{j}
\end{equation}
and will call $h$ the \emph{coefficient matrix} of $H$, which is Hermitian.
Also, in the following, we denote operators by capital and their respective coefficient matrices by lowercase letters.

A quantum state is \emph{Gaussian} if it can be arbitrarily well approximated by a thermal state of a quadratic Hamiltonian.
For a Hamiltonian $H$ we denote its ground state by
\begin{equation}\label{eq:gs}
	\rho[H]
=
\lim_{\beta\rightarrow \infty }\frac{\ensuremath\mathrm{e}^{-\beta H}}{\Tr[\ensuremath\mathrm{e}^{-\beta H}]}\,.
\end{equation}
From this we can define the \ac{VQA} problem in the free fermionic setting.
\begin{problem}[\Ac{VQA} minimization problem, free fermions]\label{p:VQA_fermionic}\hfill\vspace{-.8\baselineskip}
\begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont]
\item [Instance]
Coefficient matrices $h_{0}, h_1, \dots, h_L, o \in \Herm(\mathbb{C}^n)$.
\item[Task]
The coefficient matrices define quadratic observables $H_0, H_1, \dots, H_L$ and $O$ via \eqref{eq:quadraticO} and
$\rho_0= \rho[H_0]$.
For the evolved state
\[\rho(\vec \phi)\coloneqq U_L(\phi_L)\cdots U_1(\phi_1)\rho_{0}U^\dagger_1(\phi_1)\cdots U^\dagger_L(\phi_L)\,,\]
with $U_i(\phi)=\ensuremath\mathrm{e}^{-\i H_i \phi}$,
find a $\vec \phi \in \mathbb{R}^L$ that minimizes $\OEx{\vec\phi}\coloneqq \Tr[O\rho(\vec\phi)]$.

		\end{description}
\end{problem}

\begin{theorem}
	Problem~\ref{p:VQA_fermionic} is \class{NP}-hard, even if the initial state $\rho_{0}$ is pure.
\end{theorem}

\begin{proof}
	We prove the theorem via a reduction of Problem~\ref{p:shift_maxcut} to Problem~\ref{p:VQA_fermionic}.
	Therefore, we consider a Hermitian adjacency matrix $A \in \{0,1\}^{d\times d}$.

	For the \ac{VQA} setup, we use  $n=d\times2 $ fermionic modes $c_{i}$ with $i \in [2d]$ and $L=d$ layers.
	To encode Problem~\ref{p:shift_maxcut} we define $h_{0},\{h_i\}_{i\in [L]},o\in \mathrm{Herm}(\mathbb{C}^{2d\times 2d})$ as follows:
	\begin{align}
	h_0&=\left(\mathds{1}-\frac{\vec 1}{n}\right),
	\\
	h_i&=\vec E_i\otimes \left(\begin{matrix} 1&0\\0&-1\end{matrix}\right)\ ,
	\quad i\in[d]\, ,
	\end{align}
	where $\vec 1_{a,b}=1$ and $E_{i;a,b}=\delta_{i,a,b}$ (Kronecker delta) for all $i,a,b$.
	The coefficient matrix $o$ is given by the matrix $O$ defined in Eqs.~\eqref{eq:defOprime} and \eqref{eq:Observable_loc_VQA}, which is used for the encoding of the adjacency matrix $A$.

	We define $\Gamma_{i,j}\coloneqq\Tr(c_j^\dagger c_i\rho_0)$ to be the correlation matrix of $\rho_0$, which can be evaluated to  $\Gamma=\vec 1/(2d)$ using the identity \eqref{eq:ap_cov}.
	As the eigenvalues of $h_0$ are $\lambda=(-1,1,\cdots,1)$, $\rho_{0}$ describes a pure state, cp.\ Appendix~\ref{ap:therm_st}.
	From Eq.~\eqref{eq:ap_heisev} we obtain the coefficient matrix of $O(\vec\phi)$ in the Heisenberg picture as
	\begin{align}
		o(\vec \phi)=\ensuremath\mathrm{e}^{\i h_d\phi_d}\cdots\ensuremath\mathrm{e}^{\i h_1\phi_1}o\,\ensuremath\mathrm{e}^{-\i h_1\phi_1}\cdots\ensuremath\mathrm{e}^{-\i h_d\phi_d}\,.
	\end{align}
	With these prerequisites we can derive the following expectation value:
	\begin{equation}
	\begin{aligned}
		\OEx{\vec\phi}&=\Tr(O(\vec\phi)\rho_{0})\\
		&=\Tr\left(\sum_{i,j=1}^{2d}o(\vec\phi)_{i,j} c_i^\dagger c_j \rho_{0}\right)\\
		&=\sum_{i,j}o(\vec\phi)_{i,j} \Gamma_{j,i}\\

		&=\frac{1}{2d}\sum_{i,j}o(\vec\phi)_{i,j}\\
		&=\frac{1}{4}A_{i,j}\left(\cos(\phi_i)\cos(\phi_j)-1\right)=\mu(\vec\phi)\, ,
	\end{aligned}
	\end{equation}
	where the last step analogously follows Eq.~\eqref{eq:VQA}.
	As this gives the objective function from Problem~\ref{p:shift_maxcut}, this completes the desired reduction.
\end{proof}

\section{Conclusion and outlook}
Our results show that classical training poses challenge in \ac{VQA} based hybrid quantum computations.
Not only is optimizing \ac{VQA} algorithms \class{NP}-hard, but also that no polynomial time algorithm can have an optimization error $\Delta<1$ in all instances (assuming $\P\neq \class{NP}$).
Additionally, for significantly simpler systems, such as those composed of logarithmically many qubits or free fermions, the hardness results already hold.
This also shows that hardness does not merely derive from the ground state problem. We have extended these results further to optimization on a single layer of gates, to continuous unitary time evolution and to \ac{QAOA} problems.

We encoded \class{NP}-hard problems into local extrema of the optimization landscape of \ac{VQA} problems.
Gradient descent type optimization and also higher order methods can converge to any local minimum, determined mostly by the initialization. From this we could explicitly show, that even for logarithmically many qubits, these methods have an approximation error of $\Delta\geq \frac{1}{4}$. For our particular \ac{VQA}, this is significantly worse than what modern efficient \class{MaxCut}\ solvers can guarantee.
This emphasizes the need for effective initialization procedures for \ac{VQA} algorithms and poses the challenge of finding non-local heuristics for \ac{VQA} optimization to overcome the problem of these persistent local minima to reach smaller optimization errors.

In order to put our results into perspective, we briefly compare them to other hardness results for relevant optimization problems.
For instance, optimization within the \ac{DMRG} method is \class{NP}-hard \cite{Eisert06ComputationalDifficultyOf}.
However, hardness holds only for errors scaling as the inverse of the bond dimension and there are variants where convergence can be rigorously guaranteed \cite{LanVazVid15}.
\ac{VQA} optimization is arguably more similar to the optimization in the Hartree-Fock method.
Despite being \class{NP}-hard \cite{Schuch09ComputationalComplexityOf} it is widely used in many practical calculations.
It is our hope that this work helps of identify and overcome optimization challenges also for practically relevant \ac{VQA} problems.

\section*{Acknowledgments}

We thank David Wierichs, Sevag Gharibian, Raphael Brieger and Thomas Wagner for helpful comments on our manuscript and Jens Watty,  Christian Gogolin and David Gross for fruitful discussions on the nature of \acp{VQE} and \acp{QAOA}. We also thank the anonymous Referee B for valuable comments, which have helped us to improve this paper.  This  work  was  supported  by  the  Deutsche Forschungsgemeinschaft   (DFG,   German   Research Foundation) via the Emmy Noether program (Grant No. 441423094) and by the German Federal Ministry of Education and Research (BMBF) within the funding program ``Quantum technologies—From basic research to  market'' in  the  joint  project  MANIQU  (Grant No. 13N15578).

\section*{Appendices}

redpajama_set_name: RedPajamaArXiv


The Commission for Teaching of Computer Science, KfL for short, is an institution of the Department of Computer Science. It supports the Department and the Faculty in organizing and coordinating the courses of study in computer science and thus has an important advisory function.
The Commission for Teaching develops recommendations for the coordination and further development of courses of study (BA/MA Computer Science, Teaching Degree Computer Science, SSE, MIT). It advsises on examination regulations and the development of curricula and course contents. Moreover, it is concerned with strategic development of teaching and learning, including new teaching formats.
The commission offers recommendations on the above and other topics which serve as a basis for departmental decisions.
Furthermore, the KfL ensures the quality of the courses of study at the strategic level, while the academic advisors, examination boards, and the Central Examination Office, provide support and make decisions in individual cases.


text:
Australia is Getting Over-the-Counter CBD in 2021…But Not Much
Starting next year, Australians will be able to access CBD
Australia is now officially allowing for medical cannabis to be purchased from a pharmacy without even getting a prescription. However, it won't happen right away, as there are currently no approved products available, and some would have liked to see more from this law.
According to the Therapeutic Goods Administration, who pushed this through, not only will there be a wait before cannabis is available, the only product that will legally be sold is CBD oil with trace amounts—specifically 1 percent—of CBD.
As of February 1 of 2021, CBD oil will be available for sale—that is if there is any product for sale by that time. Up to 150 mg a day could be offered over the counter to help with things like sleeping, pain, and anxiety.
Shifting The Views On Cannabis
This change was made when cannabis was changed from a schedule four drug, which needs a prescription, to a schedule three, which is available over the counter.
"It's the biggest milestone since legalisation in 2016 and it's going to dramatically change access for patients in Australia," said Cassandra Hunt, managing director of Fresh Leaf Analytics. "Previously, they've had to go to a GP and they've had to talk to the GP about what they're trying to treat, and then the GP has had to get special permission in order for them to be able to access a prescription."
However, she added that because of all the registration and red tape involved, it could be a while before over-the-counter cannabis is readily available.
"In order for companies to get these products registered or available over-the-counter in pharmacies, they have to be registered as an S3 medicine," she said. "What that requires is quite significant data about the safety and efficacy and quality of the products."
This is in line with Australia's propensity for heavy regulation, and also because the industry is so new. Regulators want time to learn more about CBD and ensure the quality of their products.
"That data, given the industry is so young, isn't easy to come by," Hunt added.
She also mentioned that part of the idea behind this move could be to push cannabis further away from the illicit market.
"If you couple that with making it more easily accessible through pharmacies, that should hopefully make a big difference for people who've been getting relief through those channels, but maybe can't rely on those products being good quality."
"Sometimes those products say they have something in them and that's not in there, so you can't really rely on the fact that you're getting a good product.If you couple that with making it more easily accessible through pharmacies, that should hopefully make a big difference for people who've been getting relief through those channels, but maybe can't rely on those products being good quality."
The country is now waiting to see what comes of this new ruling, and how long it takes providers in the legal industry to get up and running and able to enter the market.
Addison Herron-Wheeler
Addison Herron-Wheeler is co-publisher and owner of OUT FRONT Magazine, and web editor of New Noise Magazine. She covers cannabis and heavy metal, and is author of Wicked Woman: Women in Metal from the 1960s to Now and Respirator, a collection of short stories.
The Foggy Forest says:
Very informative article 🙂 "Australians will be able to access CBD next year", does the law of age restriction implemented there?
High Times Greats: Frank Zappa
Study Finds One-Third Of Canadian Cancer Patients Use Cannabis
The next step is distributing those funds as planned.
Another study indicates that cannabis may not be harmful to patients undergoing organ transplants.
byThomas Edward
UFC Decides Marijuana No Longer Violates Anti-Doping Rules
Under the new policy, MMA fighters will no longer be penalized for off-duty cannabis use.
Indiana State Senator Files Two Cannabis Reform Bills
An Indiana lawmaker is making another push for cannabis law reform.

redpajama_set_name: RedPajamaCommonCrawl


text:
Busta may refer to:
 Busta Rhymes (born 1972), American hip hop recording artist, actor, record producer and record executive
 Busta (surname)
 Busta Voe, a sea inlet on Mainland, Shetland Islands
 Mouth Moods an album with the track named Busta

redpajama_set_name: RedPajamaWikipedia


text:
This Season's Most Notable NBA Coaches
Sports NBA Basketball
We'd hate to see how the league would be without them
Without a leader, a team would get nowhere, no matter how talented its individual players are. There always needs to be someone to call the shots and to make one set of skills complement the other. This is the very reason why NBA coaches are so admired and applauded.
This NBA season was no exception. From every side of the United States, teams played to the best of their abilities with the guidance of their coaches.
Rick Carlisle, Dallas Mavericks
If there was anyone that knows how to stretch what he has been given, it's the Dallas Maverick's Rick Carlisle. He has turned what is arguably a rather standard set of players into a team that can contend with some of the best in the NBA. He's a system coach that knows what set of plays works best for his players.
Erik Spoelstra, Miami Heat
Arguably, no other coach in the NBA can handle changes as well as Erik Spoelstra. Where others might have faltered and fallen short, Spoelstra has only shined. He almost took an injury-stricken team to the Playoffs. Despite the expectations that Miami would not do well this year, Spoelstra took them far. He also has a knack for choosing players in the D League that contribute significantly to the team's overall statistics.
Brad Stevens, Boston Celtics
Brad Stevens is one of the youngest coaches in the NBA, but he is also one of the most successful. He carried the Boston Celtics to the number one seed in the Eastern Conference entering the Playoffs this year. He knows how to hone the talents of standard players to turn them into real and notable contributors.
Steve Kerr, Golden State Warriors
In order to know the game, you need to see it from all sides possible and no know understands this as much as Steve Kerr. He played alongside Michael Jordan, took the mic as a commentator and arguably has in-game knowledge that no one else can match. He has a critical eye to his plays, has a talent for figuring out what's wrong and adding what's missing.
Gregg Popovich, San Antonio Spurs
No one in the league has numbers like Gregg Popovich. That's five championships, three Coach of the Year awards and almost 1,100 overall wins. If you want the poster guy of success, Popovich is it. What makes him so great, however, is his consistency. Over and over and over again, he has proven that he can drive results from players and a team.
Catch the NBA playoffs on the NBA LEAGUE PASS. Globe subscribers are eligible for exclusive NBA packages available in three service offerings: NBA499 for a 30-day premium access, NBA299 for 7 days and NBA50 for a 24-hour access. Globe users must text the preferred NBA promo keyword (NBA499, NBA299, NBA50) to 8080 to begin the registration process.


text:
Stațiunea McMurdo este o stațiune americană antarctică situată în sudul insulei Ross. Stațiunea este situată pe teritoriul Zonei Antarctice Neozeelandeză, teritoriu revendicat de Noua Zeelandă, care posedă stațiunea Scott la 5 kilometri distanță de McMurdo.

Istoria
Baza a fost construită în 1956, atunci numindu-se Naval Air Facility McMurdo, loc descoperit de exploratorul englez Robert Falcon Scott.

Descriere
Ea constă dintr-un port, trei aeroporturi, un helioport, mai mult de o sută de clădiri, avînd singurul loc de culte din Antarctica, Capela Zăpezilor. Stațiunea numără în jur de 1000 de persoane vara, dar mai puțin de 300 în timpul iernii, fiind cea mai mare comunitate din Antarctida (stațiunea poate găzdui până la 1.258 de persoane). Marea parte a populației activează în domeniul științelor.

Acolo se află Centrul Albert P. Crary (CSEC), Observation Hill și o dependență a National Science Foundation.

Stațiunea McMurdo este punctul de lansare a numeroase baloane stratosferice, care beneficiază de circulația foarte regulată a vânturilor în Antarctida pentru a putea efectua turul acestui continent inainte de a fi recuperate aproape de punctul lor de lansare.

Referințe

Legături externe

  McMurdo Station on Google Maps
  National Science Foundation page about McMurdo Station
  Virtual Tour - McMurdo Station Antarctica
  Life and work at McMurdo Station – from USA Today
  Information (including flight records) about NASA's balloon launches at Williams Field

Avanposturi în Antarctida


text:
Q: Should there be a direct resource route for any nested route? I have a class User and a class Task. My resource route for tasks are nested inside the User resource and tasks belong_to users and users have_many tasks.
resources :users, except: [:new, :edit] do
  resources :tasks, except: [:new, :edit]
end

In my rspec "requests"  test I am trying to POST a new task with an associated user but it looks like it's trying to hit a direct /tasks route eventually which it does not find. The test passes if my routes file looks like this for example:
  resources :users, except: [:new, :edit] do
    resources :tasks, except: [:new, :edit]
  end

  resources :tasks, except: [:new, :edit]

The test looks like this (there is a before(:each) that creates a @user1):
  describe "POST /users/1/tasks" do
    it "creates a new task for user1" do
        post user_tasks_path(@user1, task: {description: "post new task", due_date: "2/1/15", user_id: @user1.id})
        expect(response).to have_http_status(201)
        puts response.body
    end
  end

A: No, it's not compulsory to have direct resource for nested ones.
Now, regarding your spec. I'm not sure it's trying to reach /tasks, but please mind the syntax for the post method:
post(action, *args)

That said, you should be doing:
post(user_tasks_path(@user1), task: {})

instead of:
post(user_tasks_path(@user1, task: {}))

HTH.

redpajama_set_name: RedPajamaStackExchange


With the support of people like you, OpenMedia was founded just four years ago to engage citizens in key decisions that affect the Internet's future. In the face of telecom giants and governments that want to control the Internet, we've watched as the pro-Internet community has come together and grown into an empowered international movement.
You've taken a stand against the Trans-Pacific Partnership's Internet trap, spoken out to Protect Internet Freedom, and shared your stories and messages to bureaucrats behind the invasive TPP negotiations. Now, we want to hear your stories about how you became engaged with OpenMedia.
Why did you become involved with OpenMedia?
What motivates you to stay vigilant?
As we ramp up to prepare for our December Allies Drive, we'll be taking your stories to heart and sharing them with the pro-Internet community. OpenMedia's story is incomplete without hearing from you, so please take a moment to tell us your story now.
We'll hope you'll share your story with us, so that we can show our supporters that the pro-Internet community is worth fighting for.


text:
Tag Archives: policy
The new Bartleby – "It's a policy".
Since you are all erudite and well-read, Gentle Readers, I know you are familiar with Melville's story "Bartleby the Scrivener." Shall I risk insulting the sensibities of the intelligensia amongst you with a recap and a quote? I shall, of course, as I am unable to restrain my English Professor self.
So, in "Bartleby," Melville has this narrator who claims to be a very busy and important lawyer, but his employees are all hilariously bad – there's Turkey, who only works until lunch, when he gets drunk, and his counterpart, Nippers, who is a martyr to indigestion, and thus never does anything useful in the morning. These guys are dreadful enough, but then the lawyer hires Bartleby, who makes them look like paragons.
At first, Bartleby is an apparently productive employee, but then his boss makes the fatal mistake of asking him to produce some work quickly – this is a lawyer's office, in the days before photocopiers, after all.
In this very attitude did I sit when I called to him, rapidly stating
what it was I wanted him to do–namely, to examine a small paper with
me. Imagine my surprise, nay, my consternation, when without moving
from his privacy, Bartleby in a singularly mild, firm voice, replied, "I
would prefer not to."
Bartleby's preferring not to escalates, as he passively refuses more and more things; he prefers not to work, but when he is fired he prefers not to leave the premises. He never says no, just states his preference, in a monument to what I suppose post-modern sensibility would call passive-aggression. His refusals are all the more powerful for being unstated.
The story is full of subtle irony and humour, which I found pleasantly surprising when I read it, my experience wading through Moby Dick as an undergraduate not leaving me with high expectations for Melville's readability. The bit where the lawyer is confronted by the new tenant after he moves office to get away from Bartleby is laugh out loud funny, in a restrained 19th century way. If you need to refresh your memory of it (hur), those angels of e-text at Project Gutenberg have it available.
Today's tale arises from the Bartelby-esque (or perhaps Bartlebian) behaviour of that monolith of higher education, the unnamed beaurocrats who invent stupid-ass rules which are placed under the umbrella term "policy". It also leads me to the conclusion that if I had been in Melville's story, I would have punched the original Bartleby in the face.
Here, then, is the scenario. Our Department, in its wisdom, has generated, at the expense of hours of faulty committee time arguing over comma placement, and the nature of the examples, and Roberts Rules of Order, a number of documents, which we might call academic policies, procedures and standards. Two of these documents explain in detail what our expectations are with regard to written expression and citation, others are generic descriptions that apply to specific courses. All well and good. I admire these documents, especially since I know how much effort it took to produce them.
Up until now, these documents were distributed to students in what we might, I suppose, describe as the "old fashioned" medium of print on paper, although it wasn't handwritten by recalicitrant clerks, but rather mass-produced by that technological wonder, the photocopier. Ah, but then someone noticed that photocopying and distributing umpteen thousand copies of a handout every semester was costing quite a lot of money. Not to mention the trees. So, it was decided that we would distribute these items to students "electronically," which sounds not only economic, but technology-forward. Like we were hip, cutting edge and internets-savvy.
This would be all very well, except 90% of my colleagues are about as internets-savvy as a stale donut. And then there's the issue of what, exactly, electronic distribution means. Pointing students to a web site that has these documents on it? Well, that might work. How about the library? Well, that's a possibility, except that a) the library website is a dog to navigate, and it is programmed by illiterate monkeys who will, naturally, when you give them 2 documents to link, link Document A with text entitled "Document B" and vice versa.
What about Blackboard, then? Don't get me started. Or the Departmental website? I mean, we have this web space on the institution's site, and it looks all official, and even says "English Department." What better place to put documents outlining rules and expectations for students in order to make them look official and enforceable?
Gentle Reader, your naive optimism makes me laugh. For as long as I can remember, which okay, is pretty much 2 years in the case of Faculty Meetings, on account of the mind-numbingness of them, this action has been suggested. No one disagrees with the idea in principle. Yes, we have a site. Yes, official departmental documents would seem to be the kind of thing that ought to go on the site. Can we do that then? Eventually.
Yesterday, at a meeting in which all of this was rehashed yet again, it fell to me to be the person to make the request to Bartleby, who in this case is represented by our Associate Head of Department. The Committee, said I, has asked me, in light of the fact that we are not printing these handouts anymore, and in light of the fact that the ones on the library website are FUBAR, to ask if we can have them on our website. It is a job of as much as 10 minutes, and in view of that, I am even volunteering my own precious time to accomplish this Herculean task. Point me, I said, in the direction of a password, and all shall be accomplished.
You know what happened. I encountered Bartleby, who now hides behind "policy" to express his preference, but the passive resistance is exactly the same.
There are plans, to put this information on the site, "eventually," but there are problems, at a policy level, with the "content management system". Moreover, while Faculty may determine what content is put on the site, it is against policy to allow faculty access to the site. Implementation is tightly controlled, although, obviously, the implementers don't have any control over content, because that has to be determined by faculty, through a process which is shrouded in mystery.
It's astounding, and frustrating, and naturally one's first reaction is to fly into the kind of rage that is expressed by profanity and foaming at the mouth, but the genius of Bartleby, as Melville points out, is that none of this has any effect.
"I prefer not to," he replied in a flute-like tone. It seemed to me that while I had been addressing him, he carefully revolved every statement that I made; fully comprehended the meaning; could not gainsay the irresistible conclusion; but, at the same time, some paramount consideration prevailed with him to reply as he did.
"You are decided, then, not to comply with my request—a request made according to common usage and common sense?"
He briefly gave me to understand that on that point my judgment was sound. Yes: his decision was irreversible.
Like the lawyer in the story, then, I am stymied. No effort on my part will make Bartleby so much as dream of admitting that 2 years is an unconscionable amount of time to wait for a task that can be completed in 10 minutes. I will resort to working around the problem because in the face of his preferences, common sense and I are powerless.
This entry was posted in stupidity and tagged Bartleby, frustration, policy on January 15, 2009 by whatladder.


honestly, really the only single thing that bothers me is the sloweness of edge.. i don't think theres anything else you can really complain about. sure tiny things here and there can use improvement (ringtones, landscape keyboard during txts etc), but really i think the only thing thats really going to take away from the enjoyment of the phone is that I wont be able to surf the internet during a carride as fast as I would on my verizon phone.
I think people put down folks who make "baseless claims". Being a completely independent community, this forum will always be open to constructive criticism. This is not me speaking, but an example of constructive criticism would be, "I really like to use AOL IM on my SideKick and I don't understand why Apple did not include iChat".
I think the data speed is the only thing i'm going to miss as well. I have an 8525 running WM6 and i'm getting 894 kbit/sec in the cleveland area.
I'm not a heavy data user.. it's nice though when i want to kill some time and watch Cingular Video or You Tube.
as a stockholder, i'd rather not see bad reviews.
I posted about it in another thread, but I just read an article in my local paper that made me happy and sad at the same time.
On one hand it bashed the phone thus decreasing the competition for it. On the other hand they listed every single location to buy one.
I am finding a lot of the criticism comes from lazy reporting. These guys are simply not working hard.


I can count reliably with numbers from 1 to 20, placing them in order.
I can add or subtract two single digit numbers counting on or back to find the answer.
I can record using marks that I can interpret and explain.