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

Things we never see… Warp drive Perpetuum mobile GOLDBACH CONJECTURE: TRUE NEXT QUESTION 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?

Moore’s Law

Extrapolating: Robot uprising?

But even a killer robot would still be “merely” a Turing machine, operating on principles laid down in the 1930s… =

P: Polynomial Time Class of all “decision problems” (infinite sets of yes-or-no questions) solvable by a Turing machine, using a number of steps that scales at most like the size of the question raised to some fixed power Example: Given this map, is there a route from Charlottesville to Bartow? But Turing machines have fundamental limits—even more so, if you need the answer in a reasonable amount of time!

NP: Nondeterministic Polynomial Time Class of all decision problems for which a “yes” answer can be verified in polynomial time, if you’re given a witness or proof for it Example: Does have a divisor ending in 7?

NP-hard: If you can solve it, then you can solve every NP problem NP-complete: NP-hard and in NP Example: Is there a tour that visits each city once?

Does P=NP? The (literally) $1,000,000 question If there actually were a machine with [running time] ~Kn (or even only with ~Kn 2 ), this would have consequences of the greatest magnitude. —Gödel to von Neumann, 1956

Most computer scientists believe that P  NP But if so, there’s a further question: is there any way to solve NP-complete problems in polynomial time, consistent with the laws of physics?

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”

Relativity Computer DONE

Zeno’s Computer STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 Time (seconds)

Time Travel Computer S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465: , arXiv:

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)

The Famous Double-Slit Experiment Probability of landing in “dark patch” = |amplitude| 2 = |amplitude Slit1 + amplitude Slit2 | 2 = 0 Yet if you close one of the slits, the photon can appear in that previously dark patch!!

If we observe, we see |0  with probability |a| 2 |1  with probability |b| 2 Also, the object collapses to whichever outcome we see 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

To modify a state we can multiply the vector of amplitudes by a unitary matrix—one that preserves

We’re seeing interference of amplitudes—the source of “quantum weirdness”

A general entangled state of n qubits requires ~2 n amplitudes to specify: Quantum Computing Presents an obvious practical problem when using conventional computers to simulate quantum mechanics Feynman 1981: So then why not turn things around, and build computers that themselves exploit superposition? Shor 1994: Such a computer could do more than simulate QM—e.g., it could factor integers in polynomial time Interesting Where we are: A QC has now factored 21 into 3  7, with high probability (Martín-López et al. 2012) Scaling up is hard, because of decoherence! But unless QM is wrong, there doesn’t seem to be any fundamental obstacle

But factoring is not believed to be NP-complete! And today, we don’t believe quantum computers can solve NP-complete problems in polynomial time in general (though not surprisingly, we can’t prove it) Bennett et al. 1997: “Quantum magic” won’t be enough If you throw away the problem structure, and just consider an abstract “landscape” of 2 n possible solutions, then even a quantum computer needs ~2 n/2 steps to find the correct one (That bound is actually achievable, using Grover’s algorithm!) If there’s a fast quantum algorithm for NP-complete problems, it will have to exploit their structure somehow

The “Adiabatic Optimization” Approach to Solving NP-Hard Problems with a Quantum Computer HiHi Operation with easily- prepared lowest energy state HfHf Operation whose lowest-energy state encodes solution to NP-hard problem

Problem: “Eigenvalue gap” can be exponentially small Hope: “Quantum tunneling” could give speedups over classical optimization methods for finding local optima Remains unclear whether you can get a practical speedup this way over the best classical algorithms. We might just have to build QCs and test it!

BosonSampling (with Alex Arkhipov): A proposal for a rudimentary photonic quantum computer, which doesn’t seem useful for anything (e.g. breaking codes), but does seem hard to simulate using classical computers Some Examples of My Research on Computational Complexity and Physics (We showed that a fast, exact classical simulation would “collapse the polynomial hierarchy to the third level”) Experimentally demonstrated (with 3-4 photons…) by groups in Brisbane, Oxford, Vienna, and Rome!

Computational Complexity of Decoding Hawking Radiation Firewall Paradox (2012): Hypothetical experiment that involves waiting outside a black hole for ~10 70 years, collecting all the Hawking photons it emits, doing a quantum computation on them, then jumping into the black hole to observe that your computation “nonlocally destroyed” the structure of spacetime inside the black hole Harlow-Hayden (2013): Argument that the requisite computation would take exponential time (~2 10^70 years) even for a QC—by which time the black hole has already fully evaporated! Recently, I strengthened Harlow and Hayden’s argument, to show that performing the computation is generically at least as hard as inverting cryptographic “one-way functions”

Quantum computing is one of the most exciting things in science—but the reasons are a little different from what the press says Even a quantum computers couldn’t solve all problems in an instant (though they’d provide amazing speedups for a few problems, like factoring and quantum simulation, and maybe broader speedups) And building them is hard (though the real shock for physics would be if they weren’t someday possible) On the other hand, one thing quantum computing has already done, is create a bridge between computer science and physics, carrying amazing insights in both directions Summary