The race to demonstrate quantum supremacy, and onward to quantum computers

Abdullah Khalid

Quantum Information Scientist


5th November 2019

Computational Problems




Decision Problems


Input: integers P and Q

Output: integer R = P x Q


Is left most digit 1 in binary?

Decision Problems


Input: integer R

Ouput: prime numbers P and Q

             such that R = P x Q







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"






Email provider


Sampling Problems

Binomial sampling

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



Computer science

Computing Machine = Physical System

Computing Machines = Physical Systems

=> Computational complexity is determined by physical laws

Computational Complexity




Problem size (n)

Decision Problems Complexity Classes



Circuit model of classical computation

Width/Space = n

Depth/Time = m

Input: Any of 2n n-bit strings

Output: 1-bit string

Quantum Computers

Quantum Circuits



n-qubits => 2n outcomes

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

Quantum Computers: The transport analogy

Human computer

Super computer



Quantum computer

Solve computational problem = Go from A to B

Quantum computers CANNOT


do industrial optimization
predict the stock market
optimize airline schedules

NP-Complete problems


exponentially faster than classical computers.

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



But they don't!




Quantum computers can


simulate atoms, molecules and materials,


exponentially faster than classical computers.

Quantum computers can


maybe do machine learning and optimization


exponentially faster than classical computers.

Quantum computers can


break all currently used asymmetric encryption protocols


exponentially faster than classical computers.

Quantum vs Classical Computers

Breaking encryption on Quantum Computer


NP-Complete Problems on Classical/Quantum Computer

Breaking encryption on Classical Computer


Problem size (n)

Decision Problems Complexity Classes

BQP (bounded-error quantum polynomial time )

= set of problems efficiently solvable by a quantum computer

Quantum computers difficult to build due to engineering challenges

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

Quantum Random Circuit

Outputs = 2n Output strings, each with different probability

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


Quantum Random Circuit

Computation problem

Input: Circuit C

Output: a sample from the output probability distribution of C

Easy for quantum computer

Difficult for classical computer

Google's superconducting chip

53+1 qubits, depth = 20

How to verify a device which you claim can't even be classically simulated?

Linear cross-entropy benchmarking fidelity

  1. Collect finite samples x1, x2, x3, ...
  2. Calculate probabilities P(xi) of measuring xi given circuit C
  3. Calculate FXEB = 2n Avg(P(xi)) - 1

FXEB = 0 => uniform random distribution

FXEB = 1 => error-free quantum random circuit

Verification that device works

Is quantum supreme?

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