Quantum Computing and the Limits of the Efficiently Computable

1. Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson (UT Austin) Information Theory and Applications, San Diego February 13, 2017

2. GOLDBACH CONJECTURE: TRUE NEXT QUESTION Things we never see… Warp drive Übercomputer Perpetuum mobile 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? What are the ultimate physical limits on what can be feasibly computed? And do those limits have any implications for physics?

3. Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique… Matrix permanentHalting problem… FactoringGraph isomorphism… Graph connectivityPrimality testingMatrix determinantLinear programming… NP-hardAll NP problems are efficiently reducible to these NP-complete NPEfficiently verifiable PEfficiently solvable

4. Does P=NP? The (literally) $1,000,000 question

5. 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

8. Zeno’s Computer STEP 1 STEP 2 Time (seconds) 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) 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 Quantum computing: “The power of 2n complex numbers working for YOU”

10. Quantum Mechanics in One Slide Probability Theory: Quantum Mechanics: Linear transformations that conserve 1-norm of probability vectors:Stochastic matrices Linear transformations that conserve 2-norm of amplitude vectors:Unitary matrices

11. Journalists Beware:A quantum computer is NOT like 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

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

13. Can QCs Actually Be Built? Where we are now: A quantum computer can factor 21 into 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 building a QC will be fundamentally impossible 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

14. 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

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

16. Some of My Own Research…

17. “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 (with Lijie Chen): Group at Google is planning a system with 40-50 good superconducting qubits in the near future; we’re thinking about what to do with it that’s classically hard

18. Can n qubits really contain ~2n classical bits? Machine-learning responses… Theorem (A. 2004): Given an n-qubit state |, suppose we only care about approximating Pr[Mi accepts |] to , for measurements M1,…,MK. We can summarize this information using a classical string of length Proof uses boosting-like iterative improvement procedure Theorem (A. 2006): Suppose we only care about approximating Pr[Mi accepts |] to  for most Mi’s (a 1- fraction). We can learn this information w.h.p. by observing randomly-chosen sample measurements Proof uses fat-shattering dimension

19. | Theorem (A. 2017): We can learn Pr[Mi accepts |] to within , for every two-outcome measurement M1,…,MK, w.h.p., if given this many copies of the n-qubit state | : The upshot: We can summarize an n-qubit state’s behavior on exp(n) binary measurements in very succinct ways—because of the limitations of quantum measurement, there’s “less to a quantum state than meets the eye”

20. 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 Even quantum computers would have nontrivial limits—which might be the limits of what’s efficiently computable in reality Summary