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## Quantum Computing

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**Quantum Computing**Scott Aaronson (UT Austin) www.scottaaronson.com October 28, 2016**The field of quantum computing and information arguably**started here at UT in the early 1980s—with David Deutsch, William Wootters, Wojciech Zurek, Ben Schumacher (who coined “qubit”), and John A. Wheeler Together with colleagues in physics, ECE, and math, as well as ARL and TACC, we’re now seeking to build up a new quantum information presence at UT, with grad students, postdocs, and additional faculty hires**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?**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**Does P=NP?**The (literally) $1,000,000 question**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?**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 Time (seconds) 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) 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”**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**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**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**Can QCs Actually Be Built?**Where we are now: A quantum computer can factor 21 into 37, 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**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)**Hi Hf Hamiltonian with easily-prepared ground state Ground state encodes solution to NP-complete problem Problem: “Eigenvalue gap” can be exponentially small**“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) Some of My Recent Research Experimentally demonstrated with 6 photons by O’Brien 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”)**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