* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download The Future of Computer Science
Wave–particle duality wikipedia , lookup
Basil Hiley wikipedia , lookup
Bell test experiments wikipedia , lookup
Renormalization group wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Renormalization wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Particle in a box wikipedia , lookup
Double-slit experiment wikipedia , lookup
Quantum decoherence wikipedia , lookup
Scalar field theory wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Quantum dot wikipedia , lookup
Density matrix wikipedia , lookup
Quantum field theory wikipedia , lookup
Coherent states wikipedia , lookup
Hydrogen atom wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Path integral formulation wikipedia , lookup
Bell's theorem wikipedia , lookup
Quantum entanglement wikipedia , lookup
Quantum fiction wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Probability amplitude wikipedia , lookup
Quantum group wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum teleportation wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
Quantum key distribution wikipedia , lookup
Quantum machine learning wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum computing wikipedia , lookup
Canonical quantization wikipedia , lookup
Quantum state wikipedia , lookup
Quantum Computers and Beyond Scott Aaronson Associate Professor, EECS 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 it’s conjectured that thousands of interesting problems are inherently intractable for Turing machines… Is there any feasible way to solve these problems, consistent with the laws of physics? Relativity Computer DONE Zeno’s Computer Time (seconds) STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 Time Travel Computer S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465:631-647, 2009. arXiv:0808.2669. Answer Polynomial Size Circuit C “Closed Timelike Curve Register” R CTC R CR 0 0 0 “CausalityRespecting Register” Quantum Mechanics in 1 Slide “Like probability theory, but over the complex numbers” Probability Theory: Quantum Mechanics: s11 s1n p1 q1 s s p q nn n n1 n u11 u1n 1 1 u u nn n n1 n pi 0, n p i 1 i 1 Linear transformations that conserve 1-norm of probability vectors: Stochastic matrices i C, n i 1 2 i 1 Linear transformations that conserve 2-norm of amplitude vectors: Unitary matrices Interference “The source of all quantum weirdness” 1 2 1 2 1 1 1 2 12 20 1 01 11 2 2 2 Possible states of a single quantum bit, or qubit: 0 1 2 1 0 1 2 0 Quantum Computing “Quantum Mechanics on Steroids” Where we are: A QC has now factored 21 into 37, with high probability (Martín-López etn al. 2012) A general entangled state of n qubits requires ~2 amplitudes Scaling up is hard, because xofxdecoherence! But unless QM is wrong, theren doesn’t seem to be any x0,1 fundamental obstacle to specify: Presents an obvious practical problem when using conventional computers to simulate quantum mechanics Interesting 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 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 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!) So, is there any quantum algorithm for NP-complete problems that would exploit their structure? Quantum Adiabatic Algorithm (Farhi et al. 2000) Hi Hamiltonian with easilyprepared ground state Hf Ground state encodes solution to NP-complete problem Problem: “Eigenvalue gap” can be exponentially small Some of My Recent Research BosonSampling (with Alex Arkhipov): A proposal for a rudimentary optical quantum computer, which doesn’t seem useful for anything (e.g. breaking codes), but does seem hard to simulate using classical computers Computational Complexity of Decoding Hawking Radiation: Building on a striking recent proposal by Harlow and Hayden—that part of the resolution of the black hole information problem might be that reconstructing the infalling information from the Hawking radiation would require an exponentially long computation