Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson (MIT  UT Austin) NYSC, West Virginia, June 24, 2016

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… = And Turing machines have limitations—on what they can compute at all, and certainly on what they can compute efficiently

P Efficiently solvable NP Efficiently verifiable NP- complete NP-hard All NP problems are efficiently reducible to these Graph connectivity Primality testing Matrix determinant Linear programming … Matrix permanent Halting problem … Steiner tree Coin balancing Maximum cut Satisfiability Maximum clique … Factoring …

As Dana discussed, 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”

Separable state: Two qubits: |a| 2 +|b| 2 +|c| 2 +|d| 2 =1 Example: What happens when you measure?|a| 2, |b| 2, |c| 2, |d| 2 What if you measure (say) the first qubit only?

No—we call it “entangled” Is this state separable? “Spooky Action at a Distance”? No-Communication Theorem vs. The Bell Inequality

The No-Cloning Theorem: No physical procedure can copy an unknown quantum state

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 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

NP NP-complete P Factoring BQP Bounded-Error Quantum Polynomial-Time

Factoring is in BQP, but not believed to be NP-complete! 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… (We showed that a fast, exact classical simulation would “collapse the polynomial hierarchy to the third level”) Experimentally demonstrated with 6 photons by a group in Bristol, UK

Quantum Computing and Black Holes Hawking 1970s: Black holes radiate The radiation seems thermal (uncorrelated with whatever fell in). But if quantum mechanics is true, then it can’t be! Susskind, ‘t Hooft 1990s: “Black-hole complementarity.” Idea that quantum states emerging from black hole are somehow “the same states” as the ones trapped inside, just measured in a different way

The Firewall Paradox [Almheiri et al. 2012] If the black hole interior is “built” out of the same qubits coming out as Hawking radiation, then why can’t we do something to those Hawking qubits (after waiting ~10 67 years for enough to come out), then dive into the black hole, and see that we’ve completely destroyed the spacetime geometry in the interior? Entanglement among Hawking photons detected!

Harlow-Hayden 2013: Argued that, to do the experiment on the Hawking radiation that would produce a “firewall” in the interior, would require an amount of processing time exponential in the number of qubits—meaning for a black hole the mass of our sun! In which case, long before one had made a dent in the problem, the black hole would’ve already evaporated… Their evidence used a theorem I proved as a grad student in 2002: given a “black box” function with N outputs and >>N inputs, any quantum algorithm needs at least ~N 1/5 steps to find two inputs that both map to the same output (improved to ~N 1/3 by Yaoyun Shi, which is optimal)

Quantum computers are the most powerful kind of computer allowed by the currently-known laws of physics There’s a realistic prospect of building them Contrary to what you read, even quantum computers would have limits But those limits might help protect the geometry of spacetime! Summary