Proof emerges that a quantum computer can outperform a classical one
economist.com
economist.com
—————————————————————————————————————————————————————————
The classes of complexity of interest here are P, NP, BPP, and BQP.
— P corresponds to the problems that can be solved by a classical computer (deterministic Turing machine) in polynomial time. For example finding a shortest path between nodes in a graph is in this complexity class.
— NP corresponds to the problems where “verifying a solution” is in P. In other words, it's “easy” to check if a solution is actually valid. We know that P ⊆ NP which means that when it's easy to find a solution, it's also easy to check that a solution is valid (but we don't know if the opposite is true). This class contains interesting problems such as prime factorization on which all the current Web's security depends. It's easy to check if a factorization is valid, but it's hypothesized that it's hard to actually find it. Now the interesting part is that we're not sure it's actually true, so the security house of cards might actually tumble at some point (if it hasn't already secretly).
— BPP contains all the problems that you can solve in polynomial time (so like P) but with a chance of less than 1/3 of being incorrect! In practice, we run the algorithm several times to reduce the probability of being wrong to less than 2^(−ck) (c is a positive constant, k is the number of times it's run).
— BQP is probably the most interesting class here. It's just like BPP but on a quantum computer. Now the interesting part is that we don't know (yet) an efficient way of finding the prime factorization on a classical computer. But on a quantum computer, we have the infamous Shor's algorithm which can solve this problem in polynomial time with a bounded error!
—————————————————————————————————————————————————————————
The current knowledge about complexity classes is that P ⊆ BPP ⊆ BQP. We don't know how NP and BQP relate to each other.
What does this mean? We don't know if quantum computers can solve harder problems theoretically. But they seem to do sometimes in practice to the best of our knowledge. Coming back to the Shor's algorithm example, we don't know any classical algorithm that can solve the prime factorization problem as efficiently. But this doesn't mean that such an algorithm doesn't exist!
Now you might think that all of this is only theoretical considerations that only mathematicians care about. After all, we don't care if theoretically there might exist efficient classical algorithms for supposedly hard problems if we don't find them anyway right? This is a valid observation. If big enough quantum computers become a reality one day, we'll be able to solve some hard problems right away. However in the meantime, we might totally find efficient classical algorithms making quantum computers less useful. Actually, it has already happened last year[1]! A recommendation problem that we thought was intractable by classical means (but easy to solve with quantum computers) turned out to be solvable more easily than expected by a classical computer. This classical algorithm draws inspiration from quantum algorithms however. So if anything, quantum computing will force us to look at problems from another perspective and solve them in novel ways.
For a more detailed and better explanation than mine, you can read post of Scott Aaronson https://www.scottaaronson.com/blog/?p=4317 (HN discussion: https://news.ycombinator.com/item?id=21053405 (595 points, 5 days ago, 207 comments))
What limitations does it have, other than not having enough qubits to run Shor's algorithm (Scott Aaronson mentions it would require thousands of logical qubits, translating to possibly millions of physical qubits due to error-correction requirements)?
Scott's FAQ says: "The key difference with Google’s effort is that they have a fully programmable device—one that you can program with an arbitrary sequence of nearest-neighbor 2-qubit gates, just by sending the appropriate signals from your classical computer."
Unfortunately, I don't know what "arbitrary sequence of nearest-neighbor 2-qubit gates" means and what kinds of algorithms are out of reach for that method.
The classical equivalent is (very handwavily) a processor that can perform all the ops from a reasonable small ISA, but each op has 1% chance of failing silently.
Hence, any quantum algorithm can be put on this device in principle, but you are limited to stuff that uses only 50 qubits and does not mind the 1% error. There are not really any useful things you can do in the presence of (accumulating) 1% errors, beside proof of concept contrived problems.
Would appreciate informed commentary on the scaling characteristic of the current approaches.
It seems reasonable to question whether that the scaling challenge itself will prove to be as formidable as solving certain problems. Is it?
Given a sufficient time horizon, large swathes of crypto will be broken by such a computer and has many people worried. NIST is working on algorithms resistant to quantum attacks, but we're still far off.
These have existed for years already
If hashes are safe, then why not simply base our signatures and other stuff on them? The only thing that may be missing is actual encryption / decryption, but that can be done by XORing with “one-time” pads built from peeling onion layers from hashes.
Quantum computing affects the distribution of symmetric keys using public key encryption (and so affects a large amount of encryption as used in people's daily lives) which is vulnerable to quantum computers with sufficient bits.
The only problem is that it seems to also have forward secrecy built in because once you’ve revealed the next onion layer, anyone could have constructed the transcript up to that point. So you can only be sure that the author signed something if you are sure they didn’t reveal the next private key to anyone else (to verify it).
https://crypto.stackexchange.com/questions/35809/whats-wrong...
Here is a basic explanation:
https://cryptography.fandom.com/wiki/Snuffle
Note however that you would need a nonce / salt transmitted with each message because otherwise decrypting messages encrypted with the same key would just be as easy as XORing them!
Still that’s plenty useful! It can be used for blockchains and crypto currency.
With the 1TB keysizes, what you are thinking of is Berstein et al.'s tongue in cheek paper "Post-quantum RSA": https://cr.yp.to/papers/pqrsa-20170419.pdf
If you mean brute forcing an encrypted message, then yeah: that could take until the heat death of the universe for complex passphrases. Infact you might even need several universes of time to crack some passphrases.
Yes, it's an adiabatic quantum computer, not different from D-Wave's device.
Calling it a quantum computer is like bunching an electromechanical differential analysers with modern CPUs because both use electricity.
Your initial comment also had an imprecision that is pretty common: 1. Adiabatic quantum computers and circuit model quantum computers are equivalent models. 2. Dwave never constructed an adiabatic quantum computer, rather something they called a quantum annealer. These are interesting devices inspired by the model of adiabatic computers, but we do not have a reason to think they are better than classical computers.
do you mean literally exponentially? or do you mean "a lot" faster?
But keep in mind that QC does not mean you can solve _any_ problem exponentially faster than a binary computer. If something is NP-Complete (or any of the less familiar “hard problem” classes) then it’s still gonna be hard for the quantum computer.
So you’ll get an exponential speed up for some specific problems like quantum circuit sampling, but it can’t help you solve the Traveling Salesman Problem any faster unless we find a new algorithm—and even then that would imply P=NP.
There’s still lots that QC could bring to the table even if it only works for some problems, but it’s not expected to completely replace classical computers—more like augment them with some new super powers.
You can also read the actual paper: https://drive.google.com/file/u/1/d/19lv8p1fB47z1pEZVlfDXhop...
But did anyone think this wasn't going to be the case? Quantum circuit solving is just one example of this phenomenon; physically-based rendering is another: it's much faster to stick a camera in a room and measure the resulting image than it is to simulate light transport of the identical scene in the computer using an unbiased rendering algorithm.
I'm sure there are other examples: any physical phenomenon that is "NP-like" will have this property where setting up the (physical) system and measuring the end result after it has "settled" is exponentially faster than simulating the entire system in a classical computer and then measuring the result.
If you built a programmable room for creating a large variety of material and lighting setups under computer control, that might be comparable?
However, here are the breakthroughs necessary for scalable QC: make the quantum operations about 10 times more precise and the qubits about 10 times more long lived (currently the information stored in the qubits decays a bit too fast). Then make it possible to manufacture not just 50, but a couple of thousand of the qubits. At that point you will be able to simulate important chemistry (for drug discovery and material design) and important field theories (for high energy physics and cosmology).
Breaking RSA will be possible later on when we get to millions of qubits, but it is not that big of a deal as we can switch to other crypto schemes (but recordings of important messages will be decrypted).
To your other question about signal processing: quantum computing is providing technologies that would be very useful in metrology and the creation of crazy sensitive sensors (sensing various fields or displacements or taking pictures at crazy resolutions and signal to noise ratios).
One little aside, John Ambrose Fleming[1] is credited for inventing the first electronic tube (a vacuum tube based diode) around 1904 that initiated the development of modern (tube based) electronics.
However, the work at Thomas Edison’s laboratories in previous decades on the electric light bulb were almost certainly important prerequisites to Fleming’s work. He even worked for Edison for a period of time.
You get an extremely fast Fourier Transform, but for signal processing the problem is how to enter the data and how to extract the data.
In the Shor's algorithm the initial vale is easy to construct so you don't need O(N) time just to enter the data, and the output that you need to read is only a value, not all the values for every frequency, so you don't need O(N) time just to extract the data. https://en.wikipedia.org/wiki/Shor%27s_algorithm
[1] I prefer to call it QFFT because the method is more similar to the FFT than to the FT, but most of the people disagree with me.
How many can fit in a server rack, now and in the near future?