# Quantum Computing and the Limits of the Efficiently Computable

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Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson MIT.

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Quantum Computing and the Limits of the Efficiently Computable
Scott Aaronson MIT

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 So what about the third one? 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.

NP P NP-hard NP-complete
Matrix permanent Halting problem … Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique … NP-complete NP 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

The (literally) \$1,000,000 question
Does P=NP? The (literally) \$1,000,000 question If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude. —Gödel to von Neumann, 1956 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.

Extended Church-Turing Thesis
An important presupposition underlying P vs. NP is the... Extended Church-Turing Thesis “Any physically-realistic computing device can be simulated by a deterministic or probabilistic Turing machine, with at most polynomial overhead in time and memory” So how sure are we of this thesis? Have there been serious challenges to it?

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”

Other Approaches Protein folding: Can also get stuck at local optima (e.g., Mad Cow Disease) DNA computers: Just massively parallel classical computers

Ah, but what about quantum computing?
Quantum mechanics: “Probability theory with minus signs” (Nature seems to prefer it that way) In the 1980s, Feynman, Deutsch, and others noticed that quantum systems with n particles seemed to take ~2n time to simulate—and had the amazing idea of building a “quantum computer” to overcome that problem Actually building a QC: Damn hard, because of decoherence. (But seems possible in principle!)

Any hope for a speedup rides on the magic of quantum interference
Science Popularizers Beware: A quantum computer is NOT a massively-parallel classical computer! Exponentially-many basis states, but you only get to observe one of them Any hope for a speedup rides on the magic of quantum interference

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

Remember: factoring isn’t thought to be NP-complete!
Today, we don’t believe BQP contains all of NP (though not surprisingly, we can’t prove that it doesn’t) Bennett et al. 1997: “Quantum magic” won’t be enough If you throw away the problem structure, and just consider an abstract “landscape” of 2n possible solutions, then even a quantum computer needs ~2n/2 steps to find the correct one (That bound is actually achievable, using Grover’s algorithm!) So, is there any quantum algorithm for NP-complete problems that would exploit their structure?

Quantum Adiabatic Algorithm (Farhi et al. 2000)
Hf Hamiltonian with easily-prepared ground state Ground state encodes solution to NP-complete problem Problem: “Eigenvalue gap” can be exponentially small

Nonlinear variants of the Schrödinger Equation
Abrams & Lloyd 1998: If quantum mechanics were nonlinear, one could exploit that to solve NP-complete problems in polynomial time 1 solution to NP-complete problem No solutions

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!

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

Closed Timelike Curves (CTCs)
Here’s a polynomial-time algorithm to solve NP-complete problems (only drawback is that it requires time travel): Read an integer x{0,…,2n-1} from the future If x encodes a valid solution, then output x Otherwise, output (x+1) mod 2n If valid solutions exist, then the only fixed-points of the above program input and output them Building on work of Deutsch, [A.-Watrous 2008] defined a formal model of CTC computation, and showed that in both the classical and quantum cases, it has exactly the power of PSPACE (believed to be even larger than NP)

“The No-SuperSearch Postulate” There is no physical means to solve NP-complete problems in polynomial time. Includes PNP as a special case, but is stronger No longer a purely mathematical conjecture, but also a claim about the laws of physics If true, would “explain” why adiabatic systems have small spectral gaps, the Schrödinger equation is linear, CTCs don’t exist...

Lesson: If you can’t observe the answer, it doesn’t count!
Question: What exactly does it mean to “solve” an NP-complete problem? Example: It’s been known for decades that, if you send n identical photons through a network of beamsplitters, the amplitude for the photons to reach some final state is given by the permanent of an nn matrix of complex numbers Lesson: If you can’t observe the answer, it doesn’t count! Recently, Alex Arkhipov and I gave the first evidence that even the observed output distribution of such a linear-optical network would be hard to simulate on a classical computer—but the argument was necessarily more subtle But the permanent is #P-complete (believed even harder than NP-complete)! So how can Nature do such a thing? Resolution: The amplitudes aren’t directly observable, and require exponentially-many probabilistic trials to estimate

Conclusion One could imagine worse research agendas than the following: Prove P≠NP (better yet, prove factoring is classically hard, implying P≠BQP) Prove NPBQP—i.e., that not even quantum computers can solve NP-complete problems Build a scalable quantum computer (or even more interesting, show that it’s impossible) Determine whether all of physics can be simulated by a quantum computer “Derive” as much physics as one can from No-SuperSearch and other impossibility principles

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