Geek out on mathematics of quantum technologies – I’m a Professor of Mathematics – AMA!

Hello Professor Singh, thanks for stopping by for an AMA.

1) What's one question you'd like to be asked about quantum tech that you usually aren't, and how would you answer it?

2) Considering that (as I understand it), quantum computers will be able to bypass a lot of current computer security measures, should ordinary people be doing anything today to protect their most valuable data (in preparation for others' future ability to decode it)?

Thanks for the interesting questions.

1. Is quantum tech overhyped and likely to be a "vapor-ware"?
   A: Quantum technology is much broader than quantum computer. In fact, we already use quantum technology in medical scans (MRI), lasers, semiconductors etc. These are called quantum tech 1.0. The current wave we are focusing on is quantum tech 2.0. Even this is broader than quantum computers, and I expect that many technologies such as quantum sensors, imaging, and communication have a much higher chance of an earlier success, even if quantum computers stay in the more distant horizon.

2. I should qualify this by saying this is not my domain of expertise. While RSA and elliptic curve encryption can be broken by more easily by quantum computers, post quantum or quantum-resistant cryptography already exists, and is likely to become more widely available. I would keep an eye out for these. Again, not an expert so I cannot recommend a specific one.

hey prof, if i potentially wanted to re-learn maths from scratch, since they squashed every bit of curiosity out of me in school - i would say my knowledge ends somewhere near fractions - what literature or channels would you recommend?

3blue1brown (youtube)! Curiosity is the fuel of mathematics, unfortunately too often squeezed out in a procedural and uninspiring education system. Best of luck for your journey!

Hi Prof P Singh!

I've heard that noise is an issue for quantum systems. While there are efforts to reduce noise or understand it, I've also heard of efforts to using the noise inherent in these systems (last I heard was by some Harvard researchers). In the end, which effort would be more plausible ?

Very interesting question.

In the long run, if it is not possible to systematically overcome the noise (by physical approaches or algorithmic, e.g. error correction), I think a lot of the currently touted advantages of quantum computing may disappear since many of the current algorithms (Shor's factorization for decrypting cryptography for e.g.) are contingent on large scale fault-tolerant quantum computers being available.

The algorithms that utilize noise inherent in the system are designed (at least for now) keeping NISQ devices, i.e. Noisy Intermediate-Scale Quantum computers, in mind. I think these are great. The initial motivation, of course, is that using these we might get quantum advantage much sooner on quantum computers currently available. But it is very much possible that fault-tolerance proves a pipe-dream, or is too far in the future, and these algorithms end up playing a much larger role that initially envisaged.

I may have missed the boat, perhaps you come back to this later in the week:

1. How have you / the team been using entanglement in your algorithms - the way I understand it, an entangled bit will immediate reflect changes made to it's "partner". This seems to me to have huge implications for compute speeds.

2. Follow-up on that... would computation with an entangled pair be considered "FTL" communication?

I'm naive to the space but it is wayyy too cool.

Entanglement is a crucial aspect of computational speedup in most quantum algorithms, but not due to the reason that you mentioned.

A quantum gate is the basic building block of a quantum circuit/algorithm just like logical gates are for classical computing. No gates are instantaneous, whether entanglement is involved or not and in general your measurement of a particle will also take a finite time. Certainly faster physics = fast gates, which makes computation faster. However, this is a linear improvement: i.e. if all gates at 10x faster, your computation will finish 10x faster.

What makes quantum computers really impressive is that N entangled qubits have 2^N degrees of freedom - this means you need 2^N numbers (on a classical computer) to describe a state of N qubits OR you can prepare a state of N entangled qubits to describe 2^N numbers. When this can be exploited properly (it is not straightforward, nor always possible), a problem that requires 2^100 > 10^30 bits = 10^20 GBs to store and work with on a classical computer can be dealt with using only 100 qubits. This is exponential speedup!

Answer below addresses point 2 very well.

Any good papers, videos, or books you recommend for someone who knows basic quantum mechanics, "regular" programming, and a lot of math (analysis) to start learning quantum computing (not necessarily seriously, just for fun)?

Quantum Computation and Quantum Information by Nielsen and Chuang is an excellent book, which should also be reasonably accessible for someone with your background. For fun, you could jump straight into Qiskit tutorials/introductions and refer to Nielsen and Chuang, when things become too "weird" to understand.

Hello Professor Singh.

Can you give examples of some mathematical algorithms for solving real-life problems that would benefit from quantum computing?

Some of the algorithms are (1) Grover's search algorithm (although there is some contention around it providing the claimed speedup) which could make database searches substantially faster, (2) Hamiltonian simulation algorithms that allow you to simulate properties of atoms, molecules, and materials, and might help design of new drugs, new meta-materials, understanding of biological processes (e.g. how birds navigate using Earth's magnetic field), etc. (3) Quantum Linear Algebra - e.g. Quantum eigendecomposition, SVD, and linear equation solvers - these are expected to have very wide applications, including to quantum machine learning.

Eeek! I love this!

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Two questions!

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As someone who loves math but isn't trained beyond intro Calculus courses from high school, where should I start in rekindling my love of math?

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Second, for someone in the above situation (read, me), what's something going on in recent research that we should be aware of, but probably aren't due to lack of coverage in the media?

It depends on how much time you have and what area of mathematics interests you. I suggest going through youtube channels like 3blue1brown to build intuition, explore interests and magazines such as Quanta to read about research level ideas explained in accessible terms.

Linear Algebra is probably the most versatile tool, so I would suggest starting there. MIT OpenCourseWare videos by Gilbert Strang on this topic are good, as are his lectures on differential equations.

For your second question, I would point you to Quanta magazine again. There is a lot going on, but they do a much better job than I can do here.

You are a mathematician, but your work seems to be more in the field of theoretical physics. Do you collaborate more with other mathematicians or with theoretical physicists? Do you find that you approach problems, emphasize different aspects of your work, or write papers in a significantly different manner compared to theoretical physicists?

I collaborate with mathematicians, physicists, chemists and quantum computing researchers. Generally speaking I find that mathematicians have a higher emphasis on making equations precise (sometimes at expense of readability), claims more rigorous, and the applications more universal but accompanied with artificial/toy examples. With chemists and physicists, often the emphasis is on getting the examples realistic (often at the expense of the broader relevance of their ideas to other systems/equations), the techniques themselves can be so specialised as to be hard to generalise, and a lot of ad-hoc "approximations" are involved that make mathematicians very uncomfortable.

Is it possible to prove quantum concepts e.g. string theory?

How did you become interested in physics?

Thank you.

Edit: added you

Quantum mechanics by itself is among the most highly tested theories and it has essentially always come out correct wherever measured. However, there are other theories built on top of this which are quantum mechanical in nature, e.g. approaches for quantum gravity, string theory, etc. that are not verified. I am not an expert on these, but from what I understand string theory currently has an issue with verifiability.

Through Feynman's lectures in physics. Very inspiring.

Hello, Professor Singh! Thank you for taking the time to do this!

Now, I'm sorry to disappoint, but I'm far from familiar with your field and expertise. To that end, I was wondering how you would explain the appeal to someone who knows nothing about it? If so, I'd love to hear what you have to say.

In the last century, transistors have revolutionized everything around us, allowing the computing revolution. That's what allows us to search Google, type here on Reddit, listen to Spotify etc on our laptops, phones and smart watches, take and view photos and videos, and share them on WhatsApp and Instagram. It has also revolutionized design of buildings and bridges - we can assess the effectiveness of a design before constructing the building - as well as pretty much everything else including computer processors, digital cameras, bluetooth earphones, cars and airplanes: this is the "digital lab". Transistors are a quantum 1.0 technology. Other quantum 1.0 technology include MRI scans and digital cameras.

Quantum computers will revolutionize everything all over again: our ability to compute will be exponentially larger. Suddenly we will have a much more substantial ability to understand chemistry of large biomolecules - which means revolutionary drugs and treatments (much as MRI was revolutionary) - we might be able to create meta-materials with amazing properties, catalysts, e.g. for Hydrogen capture for the next generation of fuel cell based transport, etc. etc. Essentially, our "digital lab", which has served us extremely well will be turbocharged. User faced applications are also likely to benefit hugely from the computational advantage - e.g. through quantum machine learning.

But quantum computers are only one part of quantum 2.0. Others include single pixel cameras, super sensitive and miniaturised quantum sensors, including wearable MRI etc.

Hello Mr Singh, as I understand, the Quantum behaviours do not change unless they are observed, however on observation there is a chance if contamination and randomness...

How can this paradoxical situation be solved?

This is at the heart of the measurement problem in quantum mechanics, with multiple different interpretations ranging from the Copenhagen to the Many World (Everettian) interpretations. It is an unsolved problem in the sense that there isn't any consensus on one of these interpretations, and (currently) these are untestable, so they lie in the realm of metaphysics. I'm more comfortable with the notion of decoherence in the many world interpretation - this seems mathematically more palatable, even if physically uncomfortable. Sean Caroll claims that this is the only honest quantum explanation.

Why is making quantum computing hardware so .. err .. hard and why do pictures of quantum computers always look like they are cryogenic chambers ? I suppose it's because they need to operate at low temperatures but my question is why ?

Quantum computing works when we can use the "quantum nature" of the qubits. It turns out that this quantum nature is destroyed whenever qubits interact with the environment around - this is called decoherence and it is the main limiting factor for quantum computers. One way to reduce this is to reduce the number of particles, their kinetic energy (i.e. temperature) and stray electromagnetic waves in the environment by using vacuum and by using ultra low temperatures. That's why cryogenic chambers are required.

Other approaches such as topological quantum computers are being investigated that might prove more stable to such environmental effects.

Hello Professor.

If Quantum Computers had a breakthrough tomorrow and become reliable, efficient and readily available, how do you think ordinary people outside of the industry would feel this progress the most?

Possibly the biggest immediate advantage to ordinary people could be quantum machine learning.

Other advantages such as drug design, material design etc will still take years if not decades to flow to ordinary folk.

Negatives: if it becomes available to rogue entities only? pandemonium! All secured web-based communication (including banking, payments) will become insecure overnight and much of it would have to be shut down for safety.

What do you think about current tooling around writing and reasoning about programs for quantum computing currently, and what do you think could be better or would be needed as it becomes more viable?

Currently most quantum algorithms are still being described at circuits level, which is quite frustrating - for classical computers we would typically describe algorithms with pseudocode and implement in high level programming languages (e.g. Python, C/C++). While high level quantum programming languages are being developed, they don't seem to be widely adopted. I think this makes it difficult to reason about quantum algorithms at a higher, abstract level. To some extent this is changing in recent works on quantum numerical linear algebra.

Is there a chance quantum computation will "complete" mathematics and physics by a mistake?

Not sure what you mean by complete. It will certainly help us explore aspects of physics well beyond our current abilities, and might help practically in computational aspects of mathematics. But apart from that, I don't think it has a fundamental implication for (pure) mathematics.

Another one for you. We all know that there are some implications for cryptography from breakthroughs in quantum computing. However, have there been any real instances of folks using quantum computing to decipher any encrypted information (encrypted using a sensible algorithm) ? If yes, while having an implication for privacy, would such capabilities have real world implications for cryptocurrency, if any ? Thanks for your time!

None that I am aware of. Of course, even once it is available, those capable of breaking encryption might not want to advertise it. But given our current understanding of the algorithms required for this, the state of fault-tolerance available, and the number of qubits available, it seems extremely unlikely that anyone anywhere currently has this capability.

Once the capability is available, there is a real threat to cryptocurrency based on RSA or elliptic curve cryptography. To be safe, these should move to post-quantum or quantum-resistant cryptography.

Are you saying that there is an opportunity to launch QCoin based on post quantum/quantum-resitant cryptography ? :)

Yes. There are companies like Arqit already doing this for communication. I am not sure if there are Qcoins out there, but there's a good case for their existence (*if* there is any case for any cryptocurrency).

I know this is a bit controversial and honestly this maths stuff makes me believe people like this are superhuman. But I really wish we hadn’t adopted this pointless ‘assistant’ professor title in the U.K. What does that even mean? You either have a chair, or you don’t. There is no little chair. I just don’t get it. Is the title that important to some people? Is reader not enough if you can’t live with the indignity of being a lecturer? Personally when I see that title used it makes me think of think of people that exited their PhD with an MPhil and still put it on their email signature.

From what I understand, the main reason for the adoption of the "assistant" and "associate" professor titles in the UK has been mutual intelligibility with the US and increasingly with other parts of the world fashioned after the US academic setup. Internally within UK this does not seem to be a problem, at least I have not come across anyone too bothered with it, but when you go on international conferences, people from other countries have no clue what a reader is, and what they understand of "lecturer" is also very different.

Good day professor.

Do you have any advice for other people that are teaching math that you'd wish you'd known a out sooner? I mostly wonder about things like dealing with administration, handling students, funding, or anything one might not anticipate.

You need to guard your research time *very* carefully. Up till postdoctoral level this doesn't seem like a problem. Once you get a permanent position, admin and a hundred micro commitments start eating up all your time, and you will have no time and energy left if you are not careful.

Hi! Are quantum computers going to replace traditional computers in our homes anytime soon? And what would this mean if everyone had access to a quantum computer?

No. Firstly, it will take a long time for quantum computers to surpass traditional computers at *any* task (anywhere from 3-5 years if you are extremely optimistic, to 5-20 years for more realistic estimates). Secondly, even when they have exceeded traditional computers at many tasks, they are expected to be used as QPUs (quantum processing units), which similar to GPUs don't replace CPU based computing but enhance it. Lastly, it is likely that for a very long time these quantum computers are only accessed remotely through the cloud - so that long after quantum computers are available and widely used, your home computer may still be a traditional computer.

I should add that everyone already has access to quantum computers via the cloud, e.g. IBM-Q. They are just not very accurate and large scale quantum computers yet. So I expect that, just like Google collab, some access to quantum computers will be available to everyone, even if not to the most cutting edge ones.

What is the probability that the Riemann hypothesis will be proved in the near future?

No clue! Sorry.

I should add that it is much harder to say in pure mathematics whether one is making progress, often till one has arrived. In that sense, it is quite different from sciences and engineering, where there can be a more tangible sense of progress.

Hey thanks!

Why was it so important for someone to produce that lengthy proof that 1 + 1 = 2?

We would have been alright without that?

thanks!

While I am not Prof. Singh, I'll take a stab at answering this.
Assuming you're talking about Principia Mathematica, the goal of the text was not to prove that 1 + 1 = 2. It was actually an attempt to describe the set of rules from which all of mathematics could be derived from, and how to do so. This was proven impossible in later years, but the work was still very influential in philosophy, logic, and mathematics. 1 + 1 = 2 IS proven in the text, but only as a milestone towards the authors' larger goals.

This is a good summary! To add to this, the development of new mathematical concepts and techniques is itself a worthwhile cause, even if the original objective is not met. Often these find uses in future works, as motivation for other techniques, or even pitfalls and paradoxes.