What is "Qi power"? Has it been scientifically investigated? How can we achieve this?
It is the idea that you can physically use spirit energy. If you've ever watched Bleach (you should if you haven't. It's awesome), you'll hear them referring to it as reiatsu.
It's totally a myth of course. What old school practitioners of martial arts understood as spirit energy is actually just good old physics and/or body mechanics. When "building up qi", all you're really doing is relaxing your muscles which means there is more "snap" in your technique. Or when you "use qi to root yourself into the ground", all you're really doing is acting like dead weight. If you've ever tried to pick up a friend when his muscles are tense and again when his muscles are limp (dead weight), you'll understand the difference.
You can improve your "qi" by increasing your fast-twitch muscles.
Study on acupuncture: http://consensus.nih.gov/1997/1997Acupuncture107html.htm There's also this hilarious fight between what seems to be an MMA fighter and a chi master: http://www.youtube.com/watch?v=djUKqxGWj_Y
What do we mean when we say Qi?
It's not possible to scientifically study Qi until people agree on what it is. If it's good body mechanics, then that's perfectly reasonable. Nobody would argue that some people use their body well and can punch hard because they know how to relax and efficiently use their muscles.
But if Qi is supposed to be something supernatural, then it very quickly becomes absurd. People claim they can perform no-touch knockouts with Qi, but they always fail when asked to do it on someone who is not their student. That tells us that their students are playing along--usually due to unconscious social pressure. Note that in that video, Dillman says that Qi is a radio wave. If it's a radio wave, then we can test that, study it, see if it's present, and examine where it comes from. In fact Qi is not a radio wave--Dillman is just making up BS.
Qi debunked--over and over and over
Proponents of Qi will always find a way to keep believing, despite every instance of Qi being shown to be bunk. Skeptical people keep testing claims of Qi and keep finding them to be plainly false. Over and over, whenever we scientifically examine Qi, it turns out to be a parlor trick. Scientists aren't interested in claims so vague that we don't even know what Qi is or what it does. To scientifically study it, first decide what you're trying to prove:
Qi is similar to the medieval idea of humours, or astrology. They are both clumsy attempts to describe very real phenomena (people getting sick, people getting knocked out with a punch to the neck) with naive and ultimately incorrect frameworks. It turns out that the planets don't even move the way that astrology describes, and that phlegmatic humours were just a shot in the dark that turned out to be wrong, and that Qi was just a way to vaguely describe medical and martial phenomena that people didn't understand.
The word "Energy"
It's perfectly reasonable to talk about energy in the context of martial arts. Wrestlers and mixed martial artists use it to simply mean "resistance" or "oomph". It's silly to think that the word is anything more than a metaphor or out-of-context hand-waving. It doesn't mean anything like what scientists mean when they talk about electrical energy. It's silly to claim that the word relates to any specific, tangible, yet mystical force.
I think the main thing Qi does is subconsciously boost your own willpower. This may be simplified, but it works like a placebo. If someone believes they have an increased energy about them that will allow them to defeat their opponent, their thought goes less into fear and doubt, more into trusting their body and their knowledge/technique.
When training at home, i find that my spiritual excersises from tai chi have absolutely no effect on my performance. At first i relied on these techniques before training, but now i know the precise way to use each muscle and joint, i do not waste my time.
Being that most people seem to be incredibly biased on the subject, i doubt there is any credible scientific work into the subject. As for 'How can we achive this?', have no doubt in your technique and train for a lifetime. Keep in mind the martial arts grandmasters who claim to use Qi have been training their entire lives and have the skill and willpower to outmatch someone twice their physical strength.
Your Qi or Ki or Chi or Energy or Willpower is really how much energy [as in actual electrical charge] you are willing to put into your body's muscles before deciding that you have lost and will not win the arm wrestle, or will not be able to hold your ground when being pushed, i could go on.
One more thing i have to mention, on my first kung fu lesson, i watched a video of myself, my brother and two friends [with our eyes closed] being pushed forward or backwards by the sifu without him touching us, we were told this was Qi. His hand was very close but not close enough to make contact with the skin or hairs so i think this is a combination or two things, - the fact you are expecting to be pushed in one direction -your mind picking up signals such as heat, sounds at a negligible amount you would not normally pay them any attention and/or [i would like to believe, because it's cool] minute interference in your electric circuits due to the radiation given off by another persons electric circuit in close proximity. Either way it is nothing supernatural.
I used to believe in Qi when i started martial arts until i realized it was, as i say, a placebo.
An interesting article on how Qi came west
A few chinese (born and trained) tai chi guys I've worked with have said "When the old people talked about Qi they meant leg strength".
Qi is your life force energy. When you die, your life force leaves the body and the skin and flesh become whitish in hue.
You Qi is strengthened in a number of ways (I'm only a piker here.). The breathing methods in traditional karate (as well as other martial arts, yoga, etc.) are a primary way of building up your Qi. Diet, rest, and exercise are also elements of building up strong Qi. Again, I have only a layman's understanding.
The martial arts applications of Qi progress from increased health, to increased physical strength, to levels where the practitioner can mentally direct Qi to areas of the body for protection or to generate force by itself (advanced).
Martial arts exercises, starting with basic techniques, builds Qi through the proper breathing and alignment of the body, which then facilitates the circulation of Qi throughout the body.... leading to the enhancements described above....
Qi, as pure energy, is problematic to sense or measure. As one poster comments, empirical evidence of Qi can be seen in the pink color in one's palm's, the breaking of bricks and boards (especially good vid of the chinese doing this.), and the resistance of the body to what would be certain injury (chinese vids again).
Let us try to combine the descriptions an experience of a 3000 year old culture with the scientific accievments of our time:
To my understanding Qi means the energy or a fighter has available. That is to say, if I am tired or exhausted my Qi is low, but when I am well rested I have a lot of Qi.
That would fit the answers others have provided like "Qi is the power of your kick".
It also would also provide a scientific prove that greter Qi can be attained trough a healthy life style and MA practice, the latter allowing you to create more powerful techniques through more precise execution.
Also certain breathing techniques are known to calm your body, thereby removing stress, thereby allowing a regeneration and increased energy.
Finally, from an ethymological viewpoint, I the last syllable of "energy" sounds a lot like "Qi" (Chi).
Put your left hand out facing up. With your right pointer and middle finger move swiftly up and down 2 inches away from left palm. You can feel the qi , well at least something you didn't know you could.