Languages: QCL http://www.itp.tuwien.ac.at/~oemer/qcl.html Quipper http://www.mathstat.dal.ca/~selinger/quipper/ Algos: http://math.nist.gov/quantum/zoo/
For your information, most often the solution to a problem that needs to be solved is not limited by computing resources but the right approach and thought process behind. Throwing more computational power after a problem almost always does nothing beneficial. I am saying this strictly in the context of being in 2015. We need to catch up with innovative ways to solve many problems...raw computational computing power is way fast enough for 99% of problems.
Quantum computers can't exist Quantum computers are an illusion similar to middle ages chemical processes that were claimed to transmute pigs ears into gold. They are something that people wish existed because if such a a physical device with such "perfect" properties could exist it would give people wished for unlimited knowledge. The experiments and reports of progress are similar to experiments proving the existence of the ether in the late 19th century, There were anomalies and alternative explanations of experimental results were ignored, Eventually Einstein's special relativity showed that the ether did not exist - was an illusion. The experiments and theories that builders of quantum believe is called Von Neumann MacLane quantum logic. Its main problem is that it replaces quantum behavior that is a strange interaction of waves and particles like nothing else with formal mathematics. The quantum computer languages are just virtual machines with instructions that some people find attractive similar to the old APL programming language that had matrix operations as primitives or Matlab that has primitives good for doing algebra. The primitive operations are simulated by some computer language close to the underlying Von Neumann architecture. People now mostly use matrix operation libraries instead of APL because it is more efficient. There is a long history of attempts to disprove quantum logic that have not been published because of an almost religious like believe in formal logic. Even famous philosopher of science and British Knight Karl Popper could not get his disproof published (try searching for "Popper and quantum logic"). Einstein student David Bohm sympathized with Popper by writing "My advice to you is Never entangle yourself with buzz saws, cobras, and Von Neumann's articles on physics." The history is interesting because in the 1940s Einstein and Niels Bohr conversations with Von Neumann probably led to Von Neumann's computer architecture. My reply is long because I am working on my quantum logic talk for some summer conferences in Europe. ~
"In a feat of technical mastery, condensed-matter physicists have managed to detect the elusive third constituent of an electron — its 'orbiton'. The achievement could help to resolve a long-standing mystery about the origin of high-temperature superconductivity, and aid in the construction of quantum computers. ...Orbitons could also aid the quest to build a quantum computer — which would use the quantum properties of particles to perform calculations more quickly than its classical counterpart. “That seems be the direction this will go in the future — encoding and manipulating information in both spinons and orbitons,” says Boothroyd. A major stumbling block for quantum computing has been that quantum effects are typically destroyed before calculations can be performed. “The advantage here is that orbital transitions are extremely fast, taking just femtoseconds,” he says. “That’s so fast that it may create a better chance for making a realistic quantum computer..." http://www.nature.com/news/not-quite-so-elementary-my-dear-electron-1.10471
We are well aware of the paradigm shift from Hilbert Space to a categorial approach to QFT, for example, n-category, probably in the form of higher topos (or perhaps even infinity-categories) theory is almost certainly needed. Some peope are also trying monoidal category theory. There are also attempts at using homotopy type theory to formalizing mathematics and foundational problems in general, e.g., http://link.springer.com/article/10.1007/s11229-015-0781-6. People can read about category theory here: https://golem.ph.utexas.edu/category/
"...Scientists have already built qubits, but if Willett’s topological version — which would store information in the braided paths of particles — is realized, it has the potential to be much more stable than existing prototypes. Experts say it could become the most promising foundation on which to build a full-scale quantum computer. The key to building a quantum computer is increasing the number of qubits that can be linked together. Despite the investment of vast resources over the past 20 years, the extreme fragility of existing qubits has so far restricted efforts to network them and has even fueled uncertainty about whether the technology will ever materialize. Topological qubits, however, would offer a fundamental advantage: Although they would rely on a rare and extraordinarily finicky quantum state (one so difficult to conjure that at present, only Willett can consistently do it), once formed, they theoretically would behave like sturdy knots — resistant to the disturbances that wreck the delicate properties of every other kind of qubit...." http://www.wired.com/2014/05/quantum-computing-topological-qubit/
The potential in Quantum Computing is beyond stunning though... Our desktop computers promised a whole lot more than they delivered for a long time but they are coming through nicely nowadays. Maybe Quantum will work out similarly?