Capabilities and limitations of quantum computers

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Capabilities and limitations of quantum computers
1 November 1999 ECC 99

• Michele Mosca

mmosca_at_cacr.math.uwaterloo.ca
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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)

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Overview

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

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Computing Model
Acyclic circuits of reversible gates
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Information and Physics
Realisations are getting smaller and faster
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A small computer
NOT
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A small computer
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A small computer
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A closer look
?NOT
?NOT
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A closer look
?NOT
?NOT
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In general
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In general
F(x)

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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?
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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
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Efficient algorithms
Shor
Find .
,
Find .
Generalisations
Find .
,
Find .
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Further generalisation
Hidden Subgroup Problem
Find
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Another algorithm
Hidden Affine Functions
Find using only m evaluations of
(instead of n1) (D,BV,CEMM,H,M)
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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)
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Counting
Estimate with accuracy
Use only applications of
. (BBHT,BHT,M,BHMT, amplitude estimation)
(vs. applications classically)
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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, )

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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?

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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

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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