Abdullah Khalid

Quantum Information Scientist

HUAIC

5th November 2019

Decision

Search

Optimization

Input: integers P and Q

Output: integer R = P x Q

Algorithms

Is left most digit 1 in binary?

Input: integer R

Ouput: prime numbers P and Q

such that R = P x Q

Algorithms

R = 21

digits = 2

R = 498556150811

digits = 12

- Dixon's algorithm
- Continued fraction factorization
- Quadratic sieve
- Rational sieve
- General number field sieve
- Shanks's square forms factorization

Is left most digit of bigger prime 1 in binary?

"Top secret info"

"Top secret info"

"hf72h18v82ja9"

You

You

Military

Bank

Email provider

Military

Input: n, p

Output: a sample from the binomial probability distribution

A random number generator

Extended Church-Turing Thesis [1]

Any algorithmic process can be simulated efficiently using a probabilistic Turing machine.

Church-Turing Thesis

Any algorithmic process can be computed using a Turing machine.

[1] They didn't actually say it.

Physicist's Extended Church-Turing Thesis

Every finitely realizable physical system can be perfectly simulated by a universal computing machine operating by finite means - Deutsch 1985

Chemistry

Biology

Computer science

Computing Machine = Physical System

Computing Machines = Physical Systems

=> Computational complexity is determined by physical laws

Multiplication

Factorization

Resources

Problem size (n)

Factoring

Multiplication

Width/Space = n

Depth/Time = m

Input: Any of 2^{n} n-bit strings

Output: 1-bit string

Text

Text

n-qubits => 2^{n} outcomes

- Solves same computational problems as classical computer
- Algorithm is different because of architecture difference

Human computer

Super computer

Quantitative

Qualitative

Quantum computer

Solve computational problem = Go from A to B

predict the stock market

optimize airline schedules

If this was true, quantum computers could solve NP-complete problems efficiently

But they don't!

Breaking encryption on Quantum Computer

Text

NP-Complete Problems on Classical/Quantum Computer

Breaking encryption on Classical Computer

Resources

Problem size (n)

BQP (bounded-error quantum polynomial time )

= set of problems efficiently solvable by a quantum computer

Nature | Vol 574 | 24 OCTOBER 2019

Quantum Supremacy | Quantum Advantage | |
---|---|---|

A devices that: | Solves any problem faster than classical | Solves useful problems faster than classical |

Requires | Non-universal quantum computational device | Universal Quantum Computer |

Quantum error correction | ||

Analogies | Fission experiments | Nuclear power stations |

Wright brothers flight | Commercial/military airplanes |

Depth/Time = m

Width/Space = n

Outputs = 2^{n} Output strings, each with different probability

Form a random circuit using √ X , √ Y , √ W , and a combination of iSWAP and C-Z.

,Computation problem

Input: Circuit C

Output: a sample from the output probability distribution of C

Easy for quantum computer

Difficult for classical computer

53+1 qubits, depth = 20

- Collect finite samples x
_{1}, x_{2}, x_{3}, ... - Calculate probabilities P(x
_{i}) of measuring x_{i}given circuit C - Calculate F
_{XEB}= 2^{n}Avg(P(x_{i})) - 1

F_{XEB} = 0 => uniform random distribution

F_{XEB} = 1 => error-free quantum random circuit

Google's claim: 10,000 years on a state of the art supercomputer, using the best classical algorithm they could think of.

IBM's claim: Sorry, 2.5 days only using our better classical algorithm!

Quantum device performance: 600 seconds to sample 3 million times