1 / 16

The Quantum 7 Dwarves

The Quantum 7 Dwarves. Alexandra Kolla Gatis Midrijanis UCB CS252 2006. Seven Dwarves. Key time consuming problems for next decade by Phillip Colella High-end simulation in physical sciences

gad
Download Presentation

The Quantum 7 Dwarves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. The Quantum 7 Dwarves Alexandra Kolla Gatis Midrijanis UCB CS252 2006

  2. Seven Dwarves • Key time consuming problems for next decade by Phillip Colella • High-end simulation in physical sciences • Representive codes may vary over time, but these numberical methods will be important for a long time • For us: help to understand what styles of architectures we need

  3. 7 Dwarves • 1,2,6 - structured and unstructured grid problems, particle methods • 3 - Fast Fourier Transform • 4, 5 – Linear Algebra (dense and sparse) • 7 Monte Carlo • + search + sorting + HMM+ others

  4. Why Quantum Computing? • Provides a method by bypassing the end of Moore’s Law • Provides a way of utilizing the inevitable appearance of quantum phenomena. • Factoring (break RSA), simulations of quantum mechanical systems... More efficient on quantum computer than on any classical • Cryptography: doesn’t require assumptions about factoring

  5. Quantum circuit model © C. Nielsen Quantum Classical Unit: bit Unit: qubit • Prepare n-qubit input in the computational basis. 1. Prepare n-bit input 2. Unitary 1- and 2-qubit quantum logic gates 2. 1- and 2-bit logic gates 3. Readout partial information about qubits 3. Readout value of bits External control by a classical computer.

  6. How to compute classical functions on quantum computers © C. Nielsen Use the quantum analogue of classical reversible computation. The quantum NAND The quantum fanout Classical circuit Quantum circuit

  7. Classical black box Quantum black box Oracle Model • Boolean function f:{0,1}n → {0,1} • Count only the number of queries, not computational steps

  8. Quantum Lower Bounds • Very hard to show circuit lower bounds (even classically) • Show oracle lower bounds • There is no quantum speed-up for reading and outputing n bit string • There is no is exponential speed-up for unstructured search

  9. Dwarves 1,2,6 • Not a good match for quantum computing • Even reading N*N grid needs Ω(N*N) operations • But: there is known quantum speed-up for simulating differential operators

  10. 3rd Dwarf – Spectral Methods • Data is represented in the frequency, essential for digital signal processing • Classicaly, Θ(N*log N) operations • Quantumly, O((log N)2) gates for simple QFT circuit • Using parallel Fourier state computation and estimation [CW00], O(log N*(log log N)2 *log(log log N))

  11. 4th,5th Dwarf – Linear Algebra 1/2 • Determinant of n*n matrix M [A+05] • We don’t know quantum speed-up • Ω(n2) for computing det(M) over finite fields or reals • Ω(n2) for checking if det(M)=0 over finite field • Ω(n) for checking if det(M)=0 over reals

  12. Linear Algebra 2/2 • Verification of matrix product • Classically, Θ(n2) [Freivalds] • Quantumly, O(n7/4), Ω(n3/2) [BS05] • Triangle finding in a graph (by adjacency matrix) • O(n13/10) [MSS05, BDH+05]

  13. 7th Dwarf - Monte Carlo • Throw darts u.a.r. • Know the area of map • Estimate size of country • Want a = area of the red country • Clasiscally, Θ(1)/a throws • Quantumly, Θ(1)/√a ! [BHT’98] • Grover’s search++

  14. Hidden Markov Models 1/2 • Markov process with unknown paramaters • Used to solve many problems like speech recognition or bioinformatics • One canonical type of problems solved by HMM: • we know Markov process • we know ouput sequence • we want to know most likely path, ie. Viterbi path • Classically – Viterbi algorithm

  15. Hidden Markov Models 2/2 • Output string of length n • m-states Markov process • Viterbi algorithm has O(nm2) steps • Quantum Viterbi – O(nm3/2) steps • Optimal in each parameter • Ω(n) (for DNA) • Ω(m3/2) (for speech recognition)

  16. I guess we are out of time… Questions?

More Related