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

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

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

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(n 2 ) for computing det(M) over finite fields or reals Ω(n 2 ) 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(n 2 ) [Freivalds] Quantumly, O(n 7/4 ), Ω(n 3/2 ) [BS05] Triangle finding in a graph (by adjacency matrix) O(n 13/10 ) [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(nm 2 ) steps Quantum Viterbi – O(nm 3/2 ) steps Optimal in each parameter Ω(n) (for DNA) Ω(m 3/2 ) (for speech recognition)

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