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Capabilities and limitations of quantum computers

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Title: Quantum Algorithms Author: Artur Ekert Last modified by: Mike Mosca Created Date: 6/3/1997 12:46:02 AM Document presentation format: Letter Paper (8.5x11 in) – PowerPoint PPT presentation

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Title: Capabilities and limitations of quantum computers


1
Capabilities and limitations of quantum computers
1 November 1999 ECC 99
  • Michele Mosca

mmosca_at_cacr.math.uwaterloo.ca
2
What Im not talking about
  • Quantum Communication Theory (reduce the
    complexity of distributed computation tasks ask
    Alain Tapp)
  • Quantum Information Security (quantum key
    exchange security based on uncertainty principle
    and not computational assumptions)

3
Overview
  • A small computer
  • A quantum computer
  • Fast quantum algorithms
  • Limitations
  • Are they realistic?

4
Computing Model
Acyclic circuits of reversible gates
5
Information and Physics
Realisations are getting smaller and faster
6
A small computer
NOT
7
A small computer
8
A small computer
9
A closer look
?NOT
?NOT
10
A closer look
?NOT
?NOT
11
In general
12
In general
F(x)

13
Quantum computers
Note that it becomes exponentially difficult
(classically) to keep track of an n-qubit system
after t operations, but to implement quantumly
only requires n qubits and t steps! (Feynman
82, Deutsch 85)
Can we exploit this apparent computational
advantage?
14
Efficient algorithms
(Deutsch 85)
Find
using only 1 evaluation of
(Deutsch, CEMM, Tapp implemented in NMR by
JonesM, Chuang et al.)
BernsteinVazirani, Simon came up with
relativized separations between P and QP
15
Efficient algorithms
Shor
Find .
,
Find .
Generalisations
Find .
,
Find .
16
Further generalisation
Hidden Subgroup Problem
Find
17
Another algorithm
Hidden Affine Functions
Find using only m evaluations of
(instead of n1) (D,BV,CEMM,H,M)
18
Searching and Counting
Find
Suppose algorithm succeeds with probability
(e.g. ). We can iterate and
times to find such an . i.e.
SQUARE ROOT speed-p (Grover, BBHT,BH, amplitude
amplification)
19
Counting
Estimate with accuracy
Use only applications of
. (BBHT,BHT,M,BHMT, amplitude estimation)
(vs. applications classically)
20
Limitations
No luck with
  • Square root speed up for serial algorithms
  • Graph automorphism/isomorphism
  • Short vectors in a lattice
  • NP-complete problems (e.g. minimum codeword,
    graph colouring, subset sum, )

21
What about implementations?
  • 1-7 qubits using NMR technology
  • 1-2 qubits using ion traps
  • 1-2 qubits using various other quantum
    technologies
  • Scaling is very hard!
  • Is the problem technical or fundamental?

22
Technical or Fundamental?
  • Noise, decoherence, imprecision are detrimental
  • Similar problems exist in classical systems
  • Theory of linear error correction and fault
    tolerant computing can be generalised to the
    quantum setting (Shor, Steane, etc.)
  • Using reasonable physical models, there exist
    fault-tolerant schemes for scalable quantum
    computing

23
Summary
  • Quantum Computers are a natural generalisation of
    classical computers
  • Quantum algorithms Factoring, Discrete log,
    Hidden Subgroup, Hidden Affine Functions,
    Searching, Counting
  • Small implementations exist
  • Scaling is difficult, but seems to be a
    technological (not fundamental) problem
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