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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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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

- 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

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

© 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.

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

Quantum black box

Oracle Model- Boolean function f:{0,1}n → {0,1}
- Count only the number of queries, not computational steps

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

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

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))

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

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]

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++

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

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)

I guess we are out of time…

Questions?

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