1. Quantum Computing and the Limits of the Efficiently Computable
Scott Aaronson (UT Austin)
Cornell Messenger Lecture, November 28, 2017
Papers and slides at

2. Things we never see… Warp drive Perpetuum mobile Übercomputer
GOLDBACH CONJECTURE: TRUE
NEXT QUESTION
Warp drive
Perpetuum mobile
Übercomputer
The (seeming) impossibility of the first two machines reflects fundamental principles of physics—Special Relativity and the Second Law respectively
The starting point for this talk is, there are certain technologies we never see that would be REALLY cool if we had them. The first is warp drive. Where is it? The second is perpetual-motion machines – the ultimate solution to the world’s energy problems. The third is what I like to call the Ubercomputer. This is a machine where you feed it any well-posed mathematical question and it instantly tells you the answer. Currently, even with the fastest computers today, if you ask them to prove a hard theorem, they could do it eventually, but it might take longer than the age of the universe. That’s why there are still human mathematicians.
In this talk, I want to convince you that the impossibility of ubercomputers is also something physicists should think about, and also something that may have implications for physics.
So what about the third one? What are the ultimate physical limits on what can be feasibly computed? And do those limits have any implications for physics?

3. NP Efficiently verifiable
NP-hard All NP problems are efficiently reducible to these
Matrix permanent Halting problem …
Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique …
NP-complete
NP Efficiently verifiable
Factoring Graph isomorphism …
Here’s a rough map of the world. At the bottom is P, which includes everything we know how to solve quickly with today’s computers. Containing it is NP, the class of problems where we could recognize an answer if we saw it, and at the top of NP is this huge family of NP-complete problems. There are plenty of problems that are even harder than NP-complete – one famous example is the halting problem, to determine whether a given computer program will ever stop running. Very interestingly, there are also problems believed to be intermediate between P and NP-complete. One example is factoring. These intermediate problems are extremely important for quantum computing, as we’ll see later, and they’re also important for cryptography.
Graph connectivity Primality testing Matrix determinant Linear programming …
P Efficiently solvable

4. The (literally) $1,000,000 question
Does P=NP?
The (literally) $1,000,000 question
The big question is whether P=NP. Literally a million dollar question – if you solve it, you get a million dollars from the Clay Math Institute.
In my opinion, it’s the most important of all 7 Clay problems – since if P=NP, then probably you could not only solve that one problem, but also the other six. For you would simply program your computer to find the proofs for you.
I should mention, because of this blog I write, I get claims to solve the P vs. NP problem in my inbox every other week or so. The most recent *relatively-serious* claim was this summer, when this guy Vinay Deolalikar got all over the news claiming to have proved P!=NP. I was on vacation, but eventually it got to the point where I said, listen, if he’s right, I’ll supplement his million-dollar prize by $200,000. I took a lot of flak for that, but in case you’re wondering, the end result was I didn’t have to pay. This is still an open problem, one of the hardest and most profound open problems in mathematics.

5. The Extended Church-Turing Thesis (ECT)
An important presupposition underlying P vs. NP is the
The Extended Church-Turing Thesis (ECT)
“Any physically-realistic computing device can be simulated by a deterministic or probabilistic Turing machine, with at most polynomial overhead in time and memory”
But how sure are we of this thesis? What would a challenge to it look like?

6. Old proposal: Dip two glass plates with pegs between them into soapy water.
Let the soap bubbles form a minimum Steiner tree connecting the pegs—thereby solving a known NP-hard problem “instantaneously”

7. Relativity Computer DONE
But while we’re waiting for scalable quantum computers, we can also base computers on that other great theory of the 20th century, relativity! The idea here is simple: you start your computer working on some really hard problem, and leave it on earth. Then you get on a spaceship and accelerate to close to the speed of light. When you get back to earth, billions of years have passed on Earth and all your friends are long dead, but at least you’ve got the answer to your computational problem. I don’t know why more people don’t try it!

8. STEP 1 Zeno’s Computer STEP 2 Time (seconds) STEP 3 STEP 4
Another of my favorites is Zeno’s computer. The idea here is also simple: this is a computer that would execute the first step in one second, the next step in half a second, the next in a quarter second, and so on, so that after two seconds it’s done an infinite amount of computation. Incidentally, do any of you know why that WOULDN’T work?
The problem is that, once you get down to the Planck time of 10^{-43} seconds, you’d need so much energy to run your computer that fast that, according to our best current theories, you’d exceed what’s called the Schwarzschild radius, and your computer would collapse to a black hole. You don’t want that to happen.
STEP 3
STEP 4
STEP 5

9. Ah, but what about quantum computing? (you knew it was coming)
Quantum mechanics: “Probability theory with minus signs”
(Nature seems to prefer it that way)

10. The Famous Double-Slit Experiment
Another of my favorites is Zeno’s computer. The idea here is also simple: this is a computer that would execute the first step in one second, the next step in half a second, the next in a quarter second, and so on, so that after two seconds it’s done an infinite amount of computation. Incidentally, do any of you know why that WOULDN’T work?
The problem is that, once you get down to the Planck time of 10^{-43} seconds, you’d need so much energy to run your computer that fast that, according to our best current theories, you’d exceed what’s called the Schwarzschild radius, and your computer would collapse to a black hole. You don’t want that to happen.
Probability of landing in “dark patch” = |amplitude|2 = |amplitudeSlit1 + amplitudeSlit2|2 = 0
Yet if you close one of the slits, the photon can appear in that previously dark patch!

11. A bit more precisely: the key claim of quantum mechanics is that, if an object can be in two distinguishable states, call them |0 or |1, then it can also be in a superposition a|0 + b|1
Here a and b are complex numbers called amplitudes satisfying |a|2+|b|2=1
If we observe, we see |0 with probability |a|2 |1 with probability |b|2
Also, the object collapses to whichever outcome we see

12. Quantum Computing
A general entangled state of n qubits requires ~2n amplitudes to specify:
Presents an obvious practical problem when using conventional computers to simulate quantum mechanics
As zany as this sounds, Deutsch’s speculations are part of what gave rise to the modern field of quantum computing. So, what’s the idea of quantum computing?
Well, a general entangled state of n qubits requires 2^n amplitudes to specify, since you need to give an amplitude for every configuration of all n of the bits. That’s a staggering amount of information! It suggests that Nature, off to the side somewhere, needs to write down 2^1000 numbers just to keep track of 1000 particles. And that presents an obvious practical problem when people try to use conventional computers to SIMULATE quantum mechanics – they have all sorts of approximate techniques, but even then, something like 10% of supercomputer cycles today are used, basically, for simulating quantum mechanics.
In 1981, Richard Feynman said, if Nature is going to all this work, then why not turn it around, and build computers that THEMSELVES exploit superposition? What would such computers be useful for? Well, at least one thing: simulating quantum physics! As tautological as that sounds, I predict that if QCs ever become practical, simulating quantum physics will actually be the main thing that they’re used for. That actually has *tremendous* applications to materials science, drug design, understanding high-temperature superconductivity, etc.
But of course, what got everyone excited about this field was Peter Shor’s discovery, in 1994, that a quantum computer would be good for MORE than just simulating quantum physics. It could also be used to factor integers in polynomial time, and thereby break almost all of the public-key cryptography currently used on the Internet. (Interesting!)
Where we are: After 18 years and more than a billion dollars, I’m proud to say that a quantum computer recently factored 21 into 3*7, with high probability. (For a long time, it was only 15.) Scaling up is incredibly hard because of decoherence – the external environment, as it were, constantly trying to measure the QC’s state and collapse it down to classical. With classical computers, it took more than 100 years from Charles Babbage until the invention of the transistor. Who knows how long it will take in this case? But unless quantum mechanics itself is wrong, there doesn’t seem to be any fundamental obstacle to scaling this up. On the contrary, we now know that, IF the decoherence can be kept below some finite but nonzero level, then there are very clever error-correcting codes that can render its remaining effects insignificant. So, I’m optimistic that if civilization lasts long enough, we’ll eventually have practical quantum computers.
Feynman 1981: So why not turn things around, and build computers that themselves exploit QM?
What would such a computer be good for? For one thing, simulating quantum mechanics itself!

13. Any hope for a speedup rides on the magic of quantum interference
Journalists Beware: A quantum computer is NOT like a massively-parallel classical computer!
Exponentially many possible outcomes, but you only get to observe one of them
Any hope for a speedup rides on the magic of quantum interference

14. BQP (Bounded-Error Quantum Polynomial-Time): The class of problems solvable efficiently by a quantum computer, defined by Bernstein and Vazirani in 1993
Interesting
Shor 1994: Factoring integers is in BQP NP
NP-complete P
Factoring
BQP

15. Can QCs Actually Be Built?
Where we are now with implementing Shor’s factoring algorithm: 21 = 37 with high probability
Why is scaling up so hard? Because of decoherence: unwanted interaction between a QC and its external environment, “prematurely measuring” the quantum state
A few skeptics, in CS and physics, even argue that QC beyond some size is fundamentally impossible
As zany as this sounds, Deutsch’s speculations are part of what gave rise to the modern field of quantum computing. So, what’s the idea of quantum computing?
Well, a general entangled state of n qubits requires 2^n amplitudes to specify, since you need to give an amplitude for every configuration of all n of the bits. That’s a staggering amount of information! It suggests that Nature, off to the side somewhere, needs to write down 2^1000 numbers just to keep track of 1000 particles. And that presents an obvious practical problem when people try to use conventional computers to SIMULATE quantum mechanics – they have all sorts of approximate techniques, but even then, something like 10% of supercomputer cycles today are used, basically, for simulating quantum mechanics.
In 1981, Richard Feynman said, if Nature is going to all this work, then why not turn it around, and build computers that THEMSELVES exploit superposition? What would such computers be useful for? Well, at least one thing: simulating quantum physics! As tautological as that sounds, I predict that if QCs ever become practical, simulating quantum physics will actually be the main thing that they’re used for. That actually has *tremendous* applications to materials science, drug design, understanding high-temperature superconductivity, etc.
But of course, what got everyone excited about this field was Peter Shor’s discovery, in 1994, that a quantum computer would be good for MORE than just simulating quantum physics. It could also be used to factor integers in polynomial time, and thereby break almost all of the public-key cryptography currently used on the Internet. (Interesting!)
Where we are: After 18 years and more than a billion dollars, I’m proud to say that a quantum computer recently factored 21 into 3*7, with high probability. (For a long time, it was only 15.) Scaling up is incredibly hard because of decoherence – the external environment, as it were, constantly trying to measure the QC’s state and collapse it down to classical. With classical computers, it took more than 100 years from Charles Babbage until the invention of the transistor. Who knows how long it will take in this case? But unless quantum mechanics itself is wrong, there doesn’t seem to be any fundamental obstacle to scaling this up. On the contrary, we now know that, IF the decoherence can be kept below some finite but nonzero level, then there are very clever error-correcting codes that can render its remaining effects insignificant. So, I’m optimistic that if civilization lasts long enough, we’ll eventually have practical quantum computers.
The #1 application of QC, in my mind: disproving those people!
What makes many of us optimistic of eventual success: the Quantum Fault-Tolerance Theorem

16. Can QCs Solve NP-complete Problems in Polynomial Time?
We don’t know—but most of us conjecture not
Grover’s Algorithm (1996): Searches any space of N possible solutions in only about N steps, assuming you can check the solutions in superposition
Bennett, Bernstein, Brassard, Vazirani: For “black-box” searching, Grover’s algorithm is the best you can do, even with a quantum computer
So if there were a fast quantum algorithm for NP-complete problems, it would have to exploit their structure somehow

17. Operation with easily-prepared lowest energy state
The “Adiabatic” Approach to Solving NP-Complete Problems with a Quantum Computer Hi Hf
Operation with easily-prepared lowest energy state
Operation whose lowest-energy state encodes solution to NP search problem

18. Problem: “Eigenvalue gap” can be exponentially small
Hope: “Quantum tunneling” could give speedups over classical optimization methods for finding global optima
Remains unclear how much useful speedup adiabatic optimization can give over the best classical heuristics, even assuming perfect hardware (unlike D-Wave)
Problem: “Eigenvalue gap” can be exponentially small

19. “Quantum Supremacy”
Getting a clear quantum speedup for some task—not necessarily a useful one
BosonSampling (with Alex Arkhipov): A proposal for a simple optical quantum computer to sample a distribution that (we think) can’t be sampled efficiently classically
Experimentally demonstrated with 6 photons by group at Bristol
Random Quantum Circuit Sampling: Groups at Google and IBM are building systems with ~50 high-quality superconducting qubits this year; we’ve been studying what to do with them that’s classically hard

20. Complexity of Decoding Hawking Radiation
Hawking famously asked in the 1970s how information can escape from a black hole, as it must if QM is universally valid
His question led to the proposal of black hole complementarity (Susskind, ‘t Hooft 1990s)
More broadly: We’ve been able to use ideas from quantum computing theory to get new insights into condensed-matter physics, quantum gravity, and even classical computer science (e.g. “quantum proofs for classical theorems”)
But then the “firewall paradox” (AMPS 2012) said that, by doing a suitable measurement on the Hawking radiation, you could destroy the spacetime geometry inside the black hole!
Harlow and Hayden 2013: Yes, but that measurement would probably require performing an exponentially long quantum computation! (For a solar-mass black hole: ~210^67 years)
I’ve improved Harlow and Hayden’s argument to base it on “standard” hardness assumptions in cryptography

21. Summary
Quantum computers are the most powerful kind of computer allowed by currently known laws of physics
The first clear quantum speedups may be achieved in a year
Useful speedups will take longer, but are also a serious prospect
Contrary to what you read, we expect exponential speedups only for certain special problems! (And polynomial speedups more broadly)
The limits of QCs seem subtler than any sci-fi writer would’ve had the imagination to invent
But those limits could help protect our cryptography in a world with QCs—not to mention the geometry of spacetime!