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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

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The Quantum 7 Dwarves

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The quantum 7 dwarves l.jpg

The Quantum 7 Dwarves

Alexandra Kolla

Gatis Midrijanis

UCB CS252

2006


Seven dwarves l.jpg

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


7 dwarves l.jpg

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


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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


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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.


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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


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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


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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


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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


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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))


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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


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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]


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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++


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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


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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)


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I guess we are out of time…

Questions?


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