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Algorithms for quantum computers
Andrew ChildsDepartment of Combinatorics & Optimization
and Institute for Quantum ComputingUniversity of Waterloo
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What is a computer?
A means for performing calculations by following a sequence of instructions.
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Turing machines
In 1936, Alan Turing formulated what has become the standard mathematical model of computation.
The Turing machine is a mathematical abstraction of a concrete physical process.
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The Church-Turing thesis
Any calculation that can be performed efficiently by mechanical means can be performed efficiently by a Turing machine.
Strong Church-Turing Thesis
Consistent with everything we know about physics.
Any calculation that can be performed by mechanical means can be performed by a Turing machine.
Church-Turing Thesis
What about efficiency?
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Challenging the strong Church-Turing thesisQuantum mechanics seems to be hard for computers to simulate.
Do quantum systems naturally perform exponentially hard calculations?
A system of n quantum particles is described by 2n complex numbers. We don’t know how to predict the outcomes of experiments using less than an exponential amount of computation.
Can we harness this power?
“As far as I can tell, you can simulate this with a quantum system, with quantum computer elements. It’s not a Turing machine, but a machine of a different kind.”
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A simple experiment
laser 50%
50%
detectors
50/50 beamsplitter (half-silvered mirror)
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Interferometer
laser
50%?
50%?
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Interferometer
100%
0%
laser
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Phase shifts
laser
φ0
φ1
sin2 ϕ1−ϕ0
2
cos2 ϕ1−ϕ0
2
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Deutsch’s problem
Four possible functions:
x f1(x)
0 0
1 0
x f2(x)
0 1
1 1
x f3(x)
0 0
1 1
x f4(x)
0 1
1 0constant not constant
Classically, two function calls are required to solve this problem.
Given: A function f: {0,1} → {0,1}
Task: Determine whether f is constant.
(As a black box: You can call the function f, but you can’t read its source code.)
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Deutsch’s algorithm
laser
π f(0)
π f(1)
f is constant
f is not constant
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Factoring integers
In a nutshell: Quantum computers can efficiently detect periodicity. The periodicity of the powers of a number modulo N is closely related to the prime factorization of N.
Factoring integers is believed to be computationally difficult.
The security of modern electronic commerce relies on this assumption!
In 1994, Peter Shor showed that quantum computers can efficiently factor integers.
310741824049004372135075003588856793003734602284272754572016194882320644051808150455634682967172328678243791627283803341547107310850191954852900733
7724822783525742386454014691736602477652346609
1634733645809253848443133883865090859841783670033092312181110852389333100104508151212118167511579
×1900871281664822113126851573935413975471896789968515493666638539088027103802104498957191261465571
=
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Building a quantum computer
optics
superconducting circuits
trapped ions
quantum dots
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Experimental progress
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What are quantum computers good for?
Factoring integers
Other hard number theory problems(Pell’s equation, counting points on curves, ...)
Simulating quantum mechanics
Approximating topological invariants
Search
⋮ →
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A two-player game
Consider a two-player game between Andrea and Orlando where• Andrea goes first• players alternate moves• each player has two possible moves during their turn
(“left” or “right”)• there are a total of k turns• the winner after any given sequence of moves (n = 2k possibilities)
is given by a function f: {L, R}k → {0, 1}
Problem: How many times must we call the function f to decide whether Andrea has a winning move?
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and
or or
and and and and
or or or or or or or or
Example (k = 4):
Game trees
0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0
1 0 1 1 1 1 0 0
0 1 1 0
1 1
1
Orlando wins if he can make any move that gives 1 (OR)
Andrea wins if she can make any move that gives 0i.e., she only loses if all of her moves give 1 (AND)
Evaluate a tree of AND and OR gates:
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Classical algorithmsDeterministic strategy: May have to query all n = 2k leaves
In fact, this is optimal!
Randomized strategy:
• Pick a random move (L or R)• Recursively evaluate the corresponding subtree• If the result determines the evaluation (AND(0, x) = 0, OR(1, y) = 1)
then return the appropriate value• Otherwise, evaluate the other subtree to determine the value
Exercise: The expected number of leaves queried is about�1 +
√33
4
�k
= n0.753...
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A quantum algorithmExplore the game tree in quantum superposition (“quantum walk”)
This algorithm evaluates the game with arbitrarily good success probability in only about n0.5 queries!
Phase depends on the value of the game
nand
nand nand
nand nand nand nand
nand nand nand nand nand nand nand nand
0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0
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The future of quantum computing
Experimental challenge: robust control of quantum systems
Theoretical challenge: programming quantum computers
How can we build a scalable quantum computer?
How can we discover new fast quantum algorithms?