Scott Aaronson (UT Austin) Bazaarvoice May 24, 2017 Quantum Computing ZOOM THRU ZOOM THRU Thanks so much for inviting me! When I typed “quantum computer” into Google Image Search, that’s the first picture that came up. That’s apparently what they look like. (I should warn you that I’m a theorist rather than an engineer.) Scott Aaronson (UT Austin) Bazaarvoice May 24, 2017

The field of quantum computing and information arguably started here in Austin in the early 1980s—with David Deutsch and other students, faculty, and postdocs in physics 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. Together with colleagues, we’re now seeking to build up a new quantum computing and information presence at UT Austin

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?

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

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 RULES: If a system can be in two distinguishable states, labeled |0 and |1, it can also be in a superposition, written |0 + |1 Here  and  are complex numbers called amplitudes, which satisfy ||2+||2=1. A 2-state superposition is called a qubit. If we observe, we see |0 with probability ||2 and |1 with probability ||2. But if the qubit is isolated, it evolves by rules different from those of classical probability. In the 1980s, Feynman, Deutsch, and others noticed that a system of n qubits seems to take ~2n steps to simulate on a classical computer, because of the phenomenon of entanglement between the qubits. They had the amazing idea of building a quantum computer to overcome that problem

Popularizers Beware: A quantum computer is NOT like a massively-parallel classical computer! Exponentially many possible answers, but you only get to observe one of them Any hope for a speedup rides on choreographing an interference pattern that boosts the amplitude of the right answer

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

Key point: 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 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!) If there’s a fast quantum algorithm for NP-complete problems, it will have to exploit their structure somehow

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

Landscapeology Adiabatic algorithm can find global minimum exponentially faster than simulated annealing (though maybe other classical algorithms do better) Simulated annealing can find global minimum exponentially faster than adiabatic algorithm (!) Simulated annealing and adiabatic algorithm both need exponential time to find global minimum

Some of My Recent Research “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 can’t be sampled efficiently classically (unless P#P=BPPNP) Experimentally demonstrated with 6 photons by group at Bristol Random Quantum Circuit Sampling: Martinis group at Google is planning a system with 40-50 high-quality superconducting qubits in the near future; we’re thinking about what to do with it that’s classically hard

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) 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” crypto assumptions (injective OWFs) 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”)