The Quantum 7 Dwarves

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?