Why haven't quantum computers factored 21 yet?
So how many gates are we talking to factor some "cryptographically useful" number?
That is a hard question to answer for two reasons. First, there is no bright line that delineates "cryptographically useful". And second, the exact design of a QC that could do such a calculation is not yet known. It's kind of like trying to estimate how many traditional gates would be needed to build a "semantically useful" neural network back in 1985.
But the answer is almost certainly in the millions.
[UPDATE] There is a third reason this is hard to predict: for quantum error correction, there is a tradeoff between the error rate in the raw qbit and the number of gates needed to build a reliable error-corrected virtual qbit. The lower the error rate in the raw qbit, the fewer gates are needed. And there is no way to know at this point what kind of raw error rates can be achieved.
Is there some pathway that makes quantum computers useful this century?
This century has 75 years left in it, and that is an eternity in tech-time. 75 years ago the state of the art in classical computers was (I'll be generous here) the Univac[1]. Figuring out how much less powerful it was than a modern computer makes an interesting exercise, especially if you do it in terms of ops/watt. I haven't done the math, but it's many, many, many orders of magnitude. If the same progress can be achieved in quantum computing, then pre-quantum encryption is definitely toast by 2100. And it pretty much took only one breakthrough, the transistor, to achieve the improvement in classical computing that we enjoy today. We still don't have the equivalent of that for QC, but who knows when or if it will happen. Everything seems impossible until someone figures it out for the first time.
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[1] https://en.wikipedia.org/wiki/UNIVAC_I#Technical_description
What made computing-at-scale possible wasn't the transistor, it was the precursor technologies that made transistor manufacturing possible - precise control of semiconductor doping, and precision optical lithography.
Without those the transistor would have remained a lab curiosity.
QC has no hint of any equivalent breakthrough tech waiting to kick start a revolution. There are plenty of maybe-perhaps technologies like Diamond Defects and Photonics, but packing density and connectivity are always going to be huge problems, in addition to noise and error rate issues.
Basically you need high densities to do anything truly useful, but error rates have to go down as packing densities go up - which is stretching optimism a little.
Silicon is a very forgiving technology in comparison. As long as your logic levels have a decent headroom over the noise floor, and you allow for switching transients (...the hard part) your circuit will be deterministic and you can keep packing more and more circuitry into smaller and smaller spaces. (Subject to lithography precision.)
Of course it's not that simple, but it is basically just extremely complex and sophisticated plumbing of electron flows.
Current takes on QC are the opposite. There's a lot more noise than signal, and adding more complexity makes the problem worse in non-linear ways.
What are the steps?
If I were doing this work, I'd look at a rich virtual environment like Minecraft or simcity or something like that. But it could also be coq or a code development environment.
Meanwhile, even after the infamous LK-99 fiasco (which gripped this forum almost more than anywhere else) was exposed as an overblown nothingburger, I still had seemingly-intelligent people telling me with all seriousness that the superconductor breakthrough had a 50% chance of happening within the next year. People are absolutely, terminally terrible at estimating the odds of future events that are surrounded by hype.
no one knew how to even convincingly fake a natural language interaction.
There was some decent attempts at the turing test given limited subject matter long before LLM’s. As in people looking at the conversation where unsure if one of the parties was a computer. It’s really interesting to read some of those transcripts.
LLM’s actually score worse one some of those tests. Of course they do a huge range of other things, but it’s worth understanding both their strengths and many weaknesses.
Indeed, and at the same the breakthroughs are vastly outnumbered by ideas which had plausible sounding counterarguments which turned out to be correct. Which is to say, the burden of proof is on the people making claims that something implausible-sounding is plausible.
Is there some pathway that makes quantum computers useful this century?This century has 75 years left in it, and that is an eternity in tech-time.
As a comparison, we went from first heavier than air flight to man walking on the moon in only 66 years.
to man walking on the moon in only 66 years
And that was before Epoch (1969, unix time started in 1970). We went from calculator to AI in 55 years, which is, actually, extremely long. It took exactly the time to miniaturize CPUs enough that you would hold as many gates in a GPU as neurones in a human’s brain. The moment we could give enough transistors to a single program, AI appeared. It’s like it’s just an emergent behavior.
We went from calculator to AI in 55 years, which is, actually, extremely long.
I think it is insanely fast.
Think about it: that planet has been here for billions of years. Modern humanity has been here for 200,000 years, give or take. It took 199700 years and change to get to a working steam engine. 266 years later men were walking on the moon and another 55 years and we had a good facsimile of what an AI looks like in practice. That's insane progress. The next 75 years are going to be very interesting, assuming we don't fuck it all up, the chances of which are right now probably 50/50 or so.
Science fiction has been predicting what an AI would be like for over a hundred years, there was even one in a movie in 1927. We're so far from what we dream that, to me, it feels like a mere leaf blowing in the wind compared to the Wright Flyer.
Sci-fi is fanciful and doesn’t take into account psychology. What we got is the local maxima of what entrepreneurs think they can build and what people are willing to pay for.
Sci-fi is not a prediction. It is a hypothetical vision for what humanity could be in a distant future. The writer doesn’t have to grapple with limitations of physics (note FTL travel is frequently a plot device, not a plausible technology) or limitations about what product-market-fit the market will adopt.
And, of course, sci-fi dates are rarely close or accurate. That’s probably by design (most Star Trek space technologies would be unbelievable if the timeline was 2030, but more easily believable if you add a few thousand years for innovation).
Star Trek communicator
As a trekkie this was a dream come true.
Unfortunately we still don't have a tricorder yet (despite Elisabeth Holmes' promise).
But we do have the apps and the games, they didn't have these in star trek. My phone is loaded with these (apps, not games)
The Wright Flyer was a complete aircraft but small, awkward and not very practical. But it had all of the parts and that was the bit that mattered.
LLMs are not a 'complete AI' at all, they are just a very slick imitation of one through a completely different pathway. Useful, but not AI (at least, not to me). Meanwhile, a very large fraction of the users of OpenAI, Claude etc all think that AI has arrived and from that perspective it is mostly the tech crowd that is disappointed. For the rest of the people the thing is nothing short of magic compared to what they were able to do with a computer not so long ago. And for people like translators it is a massive threat to their jobs, assuming they still have one.
It is both revolutionary and a letdown, depending on your viewpoint and expectations.
walking on the moon in only 66 years.
Yet it has been 53 years since we have been able to send a manned mission to the moon . No other program has or likely to come close in the next 13 years including the current US one. By 2038 the moon landings would be closer to Wright brothers than future us.
The curve of progress is only smooth and exponential when you squint hard .
It is a narrow few decades of exponential growth hardly can reasonably be expected to last for 100+ years .
It is for the same reason you cannot keep doubling grains on a chess board just because you did it 10-20 steps quickly.
Fusion power, quantum computing are all always two decades away for a reason despite the money being spent . AI has gone through 3-4 golden ages in living memory and yet too many keep believing this one would last.
Reality is when the conditions are right, I.e. all the ground work has been done for decades or centuries before there can be rapid innovation for a short(few decades at best) time
Yet it has been 53 years since we have been able to send a manned mission to the moon
A near total lack of demand explains that impressive stall.
Even if the shuttle had worked out as well as its designers hoped, was envisioned as a major retreat, while sucking all the dollars out of the room.
And today, the market for lunar landings is still very small.
I think what it shows is that many technologies might have come earlier from a research and development standpoint, if we had enough money to burn. But that was an unusual situation.
It is not like Fusion or Quantum Computing has lacked serious or continuous funding over the last 20-30 years.
Foundational model development is a classic current example. The returns are diminishing significantly, despite the tens of billions each quarter being thrown at the problem.
No other R&D effort in our history has this much resources being allocated to it, perhaps including even the Moon landings.
However the ability to allocate resources has limits. Big tech can spend few hundred billion a year a number that would have been unimaginable even a decade ago, but even they cannot spend few trillion dollars a year.
No other program has or likely to come close in the next 13 years including the current US one.
The Chinese are planning manned lunar landings in 2029-2030, and this is not a pipe dream, they've been systematically working at this for several decades now. They have already completed 6 out of 8 preparatory missions plus placed comms satellites in lunar orbit, and the final two are scheduled for 2026 and 2028.
https://en.wikipedia.org/wiki/Chinese_Lunar_Exploration_Prog...
Perhaps milestones are being set to be competing with Artemis. When NASA gets delayed or reduced in scope, CNSA might reset to more achievable date.
That is just engineering risk on dates, there are other class of risks in geopolitics or economics etc.
Bottom line I am skeptical that a successful landing and return can be attempted in 2030. 2035 is a more realistic target I think.
So how many gates are we talking to factor some "cryptographically useful" number?
Table 5 of[1] estimates 7 billion Toffoli gates to factor 2048 bit RSA integers.
Is there some pathway that makes quantum computers useful this century?
The pathway to doing billions of gates is quantum error correction.[1] estimates distance 25 surface codes would be sufficient for those 7 billion gates (given the physical assumptions it lists). This amplifies the qubit count from 1400 logical qubits to a million physical noisy qubits.
Samuel Jacques had a pretty good talk at PQCrypto this year, and he speculates about timelines in it[2].
(I'm the author of this blog post and of[1].)
The operations all consist of saying, connect these 3 bits and do a reversible operation on them all together. Same as assembly, "add these two registers and store the sum in the first one..." You didn't need to introduce any new bits.
You only need to introduce new bits for steps that cannot be reversibly done, in assembly you get around this by being able to overwrite a register: in quantum, that requires an explicit measurement in the computational basis to figure out how you want to do stuff to zero it; zeroing a bit is not a unitary operation. But if you can encode the circuit in Toffoli gates which are perfectly reversible, you don't have to delete any bits after that encoding (but you may have to introduce extra bits to get to that encoding, like using Toffoli to build “x AND y” requires an extra z bit that effectively gets discarded at the end of the computation when everything is done and nobody cares what that bit holds, but it holds the information you would need to reverse that logical AND).
But yeah it's just number of operations that you need to run the algorithm, versus the number of registers that you need to run the algorithm, they're just two different numbers.
The big thing that could change the numbers is more reliable qbits. Most of the calculations so far are done with qbits right at the edge of where error correction works (about 5x better than current qbits). if you get another 10x in qbit quality you probably drop the required qbits by ~100-1000x.
Note that the magic of quantum error correction (exponential improvement in the error rate goes both ways): if you could get another 9 in qubit fidelity, you get a much larger improvement in qubit numbers. On the other hand, if you need to split your computation over several systems, things get much worse.
As a layman the pathway seems to exist behind multiple massive materials science breakthroughs
You can do useful and valuable quantum chemistry calculations already with few 100s of qubits with that low error rates, while post-quantum algorithms are becoming more common everyday removing incentives to build crypto cracking quantum computers.
I think the quantum computing will advance fastest in directions that are not easy to use in cryptography.
In some special problems hybrid methods start giving gains in 100 qubits or below.
Gate count estimates for performing quantum chemistry on small quantum computers https://arxiv.org/pdf/1312.1695
A Perspective on Quantum Computing Applications in Quantum Chemistry using 25--100 Logical Qubits https://arxiv.org/pdf/2506.19337
Randomness is something which I feel is a pretty weird phenomenon. I am definitely one of those 'God doesn't play with dice' types.
Randomness is also something that we call things when actually it's random from a subjective perspective. If we knew more about a system the randomness just falls away. E.g. if we knew the exact physical properties of a dice roll we could probably predict it better than random.
What if it's the case that quantum mechanics is similar. I.e. that what we think of as randomness isn't really randomness but only appears that way to the best of what we can observe. If this is the case, and if our algorithms rely on some sort of genuine randomness inherent in the universe, then doesn't that suggest there's a problem? Perhaps part of the errors we see in quantum mechanics arise from just something fundamental to the universe being different to our model.
I don't think this is that far fetched given the large holes that our current understanding of physics have as to predicting the universe. It just seems that in the realm of quantum mechanics this isn't the case, apparently because experiments have verified things. However, I think there really is something in the proof being in the pudding (provide a practical use case).
You are probably talking about the Copenhagen interpretation, involving superposition.
Personally, I don't think this is the final theory.
Any theory using calculus, cannot be considered discrete, so is therefore not quantized, and not possibly "physical".
Gerard 't Hooft has more to say on this if you want to hear something from a nobel laureate on the subject.
I think what I've just said foots with your calculus comment, and also a Wolfram-like interpretation is closer to "truth" and your point on discretisation.
Why do you think discretisation/quantisation is necessary for the "physical"?
What can I search for to find his comments on this subject?
Why do you think discretisation/quantisation is necessary for the "physical"?
We are trying to explain, the physical reality we find outselves in, so, if the universe is fundamentally quantized, it must be discrete, as continuous math would reify infinities.
What can I search for to find his comments on this subject?
You could check Curt Jaimungal's youtube, Hooft was on it recently.
But if you are up for an existential crisis, just google “hidden variable theories”
I mean no disrespect, but I don’t think it’s a particularly useful activity to speculate on physics if you don’t know the basic equations.
In addition, selling information to a government on how to break either system would be more valuable than the amount of bitcoin you would able to sell before exchanges stop accepting deposits or the price crashes.
In addition, selling information to a government on how to break either system would be more valuable
Honest question because one can find such claims very often on forums like HN:
Does there really exist a "feasible" way how some "lone hacker" could sell such information to some government and become insanely rich?
I know that people who apparently have some deep knowledge about how exploit markets work claimed on HN that "if you have to ask how/where to solve your exploit (i.e. you have the respective contacts), you are very likely not able to".
This latter observation seems a lot more plausible to me than the claim often found on HN that some "lone individual" would be able to monetize on it if he found a way how to break ECDSA or RSA by selling it to some government.
So your best bet would probably be to try to sell as many BTC as possible then give away the solution for free to your/a government.
On a quantum computer, my understanding is that Shor's algorithm could potentially target both problems, though.
So a hypothetical classic algorithm that breaks the RSA is also highly likely to break the ECDSA.
That being said, NFS is almost thirty years old so maybe the NSA doesn't have anything better still.
(Quick aside: the amount of optimization that has gone into this factoring-21 circuit is probably unrepresentative of what would be possible when factoring big numbers. I think a more plausible amount of optimization would produce a circuit with 500x the cost of the factoring-15 circuit… but a 100x overhead is sufficient to make my point. Regardless, special thanks to Noah Shutty for running expensive computer searches to find the conditional-multiplication-by-4-mod-21 subroutine used by this circuit.)
Third, notice that the only remaining multiplication is a multiplication by 4. Because 15 is one less than a power of 2, multiplying by 2 modulo 15 can be implemented using a circular shift. A multiplication by 4 is just two multiplications by 2, so it can also be implemented by a circular shift. This is a very rare property for a modular multiplication to have, and here it reduces what should be an expensive operation into a pair of conditional swaps.Aside: multiplication by 16 mod 21 is the inverse of multiplying by 4 mod 21, and the circuits are reversible, so multiplying by 16 uses the same number of Toffolis as multiplying by 4.
I couldn't really find anything explaining the significance of this. The only info I found said that "4 mod 21 = 4" (but I don't know if it was AI slop or not).
Is "multiplying by 4 mod 21" something distinct to quantum computing?
For instance the following are equivalent:
2 = 6 mod 4
6 = 2 mod 4
This 'mod 4' can also appear in parentheses or in some other way, but it must appear at the end. Like I said it is not an operator rather it denotes that the entire preceding statement takes place in the appropriate quotient space.
So it is not (multiplying by (4 mod 21)) but ((multiplying by 4) mod 21)
For example under mod 21 a half can actually be represented by 11. Try it. Times any even number by 11 and you’ll see you halved it.
Take any number that’s a multiple of 4 and times it by 16 under mod 21. You now have that number divided by 4.
Etc.
Absolutely nothing to do with quantum computers.
From the article it sounds like we will still be safe for 20+ years. On the other hand 15 was just extraordinarily easy, progress after 21 will be much quicker. And we never know which breakthroughs might come in the next decades that speed up progress.
progress after 21 will be much quicker
Can you provide a quick verification for that?
But 22 and 24 are in the same boat as 21 here. All three of them require computing only factors that are not one, all three are not one less than a factor of 2. You need slightly more multiplications (and thus more gates) as the numbers get larger, but that only grows linearly. Maybe the conditional multiplications required get slightly more expensive to implement, but I wouldn't expect a 100x cost blowup from that. Error correction is still an issue, potentially making a linear complexity increase quadratic, but qubit counts in quantum computers also increase at an exponential rate
The practical problem is that ‘noise’ between gates seems to increase exponentially, so practically it may actually be impossible to construct anything with more than a handful of gates for the foreseeable (possibly indefinite?) future.
It’s essentially the crypto version of Fusion.
so we should totally be able to factor 21 (or larger numbers)…. When?
Just because we solve one problem doesn't imply all the problems in QC are also instantly solved. I guess it does if you assume noise is the only problem and once is it solved the engineering is trivial. That is not the case. Even assuming all foundational problems have been solved, figuring out how actually engineer and also mass produce large numbers of gates, will take a while.
As the article pointed out, going from 15 to 21 requires a 100x increase in gates.
As the article that you posted under says:
Because of the large cost of quantum factoring numbers (that aren’t 15), factoring isn’t yet a good benchmark for tracking the progress of quantum computers. If you want to stay abreast of progress in quantum computing, you should be paying attention to the arrival quantum error correction (such as surface codes getting more reliable as their size is increased) and to architectures solving core scaling challenges (such as lost neutral atoms being continuously replaced).
Do you have a citation?
As it turns out, that's a big if, but the bigness of the if is about hardware implementation. The theory behind it is just basic quantum mechanics
Is this what you can conjure Saruman?
I have a strong belief that new mathematical tools and methods can be developed that can make it "easy" to break a lot of modern cryptography primitives without ever using a quantum computer.
The goal of cryptography is to make something as close to theoretically unbreakable as possible. That means even theoretical vulnerabilities are taken seriously.
For ECC and RSA and related algorithms we have a theoretical and physically plausible pathway toward a practical machine that could break them. That means many cryptographers consider them theoretically broken even if such a machine does not exist and may not exist for a long time. The math works even if we can’t build it yet.
So it’s considered prudent to go ahead and upgrade now while no QC exists. That way if some major advance does arrive we are ready.
Nobody’s talking seriously about replacing SHA2, AES, ChaCha, etc because there is no physically plausible theoretically valid path to a machine that can break these in, say, less than many millions of years. AFAIK there is no proof that such a path does not exist but nobody has found one, hence they are considered unbroken.
Note that cryptography is not the only or even the most useful application of QC. Things like physical stimulation of quantum systems, protein folding, machine learning, etc. could be more useful. Like digital computers there’s probably a ton of uses we don’t know about because we need to tinker with the machine to figure them out.
Things like physical stimulation of quantum systems, protein folding, machine learning, etc. could be more useful
is there still more to do in protein folding after AlphaFold?
https://www.isomorphiclabs.com/articles/alphafold-3-predicts...
QC might be directly applicable to AI training too. It may be possible to compute the optimal model over a data set in linear time. It could allow training that is faster and consumes a tiny fraction of the energy current brute force methods need.
I would expect this to become relevant later than crypto, though, because you need larger data sizes for things to get interesting.
However, the main focus is on Key Exchange. Why? Well, Key Exchange is the clever bit where we don't say our secrets out loud. Using a KEX two parties Alice and Bob agree a secret but neither of them utters it. Break that and you can learn the secret, which was used to encrypt everything else - for any conversation, including conversations which you recorded any time in the past, such as today.
If future bad guys did have a Quantum Computer the Key Exchange lets them read existing conversations they've tapped but today can't read, whereas breaking say the signing algorithm wouldn't let them somehow go back in time and sign things now because that's not how time works. So that's why the focus on KEX. Once such a thing exists or clearly is soon to deliver it's important to solve a lot of other problems such as signing, but for KEX that's already too late.
Digital computers were much easier than that. Make it smaller, make a larger number of it, and you're set.
Quantum computers complexity goes up with ~ n^2 (or possibly ~ e^n) where n is the number of qbits
At the same time, things like d-wave might be the most 'quantum' we might get in the practical sense
There are no theoretical reasons QEC can't exist. In fact it already does. Is it already good enough for universal fault tolerance? No. But then again no one said it would. We are slowly getting closer every year.
In his book, Dyakonov offers zero solid reasons other than "it's hard" and thus likely not possible. That's just an opinion.
Most digital algorithms would explode in terms of hardware needed, for increasing N, if we didn't distribute that computation in time, as well as across elements.
This would still require more hardware than digital circuits (which can be made reversible for energy efficiency, but that is rarely done).
While still reducing the number of operation components, and reusing them.
If your program has a compilation process that requires you to already know the answer to the problem you're trying to solve, then what they did was not factorization, but "print 3" with extra steps.
That said, we are REALLY far off from having a useful quantum computer. Jensen was probably being conservative when he said 20-30 years away, hence the immediate pressure he received form the investor community to reverse his statement followed by the flood of ridiculous press releases from the usual companies claiming to be 2-3 years away.
There are papers that claim to have factored 21 with a quantum computer. For example, here’s one from 2021[1]. But, as far as I know, all such experiments are guilty of using optimizations that imply the code generating the circuit had access to information equivalent to knowing the factors.
A while ago I generated a gigabit RSA public key. It is available at[3]. From what I remember, the format is: 4-byte little-endian key size in bytes, then little-endian key, then little-endian inverse of key mod 256**bytes. The public exponent is 3.
[1] https://eprint.iacr.org/2017/351.pdf
I think a more plausible amount of optimization would produce a circuit with 500x the cost of the factoring-15 circuit
I don't get this part
If author already produced "115x", how can optimizations make it worse?
He said that factoring and cryptography applications are red herrings. It's not what most people in the field are working on.
The practical application lies in simulating processes where quantum effects are actually directly relevant, such as quantum chemistry: https://en.wikipedia.org/wiki/Quantum_chemistry