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Quantum Factoring

Michele Mosca

The Fifth Canadian Summer Schoolon Quantum

Information August 3, 2005

Quantum Algorithms

- Quantum Algorithms should exploit quantum

parallelism and quantum interference. - We have already seen some elementary algorithms.

Quantum Algorithms

- These algorithms have been computing essentially

classical functions on quantum superpositions - This encoded information in the phases of the

basis states measuring basis states would

provide little useful information - But a simple quantum transformation translated

the phase information into information that was

measurable in the computational basis

Extracting phase information with the Hadamard

operation

Overview

- Quantum Phase Estimation
- Eigenvalue Kick-back
- Eigenvalue estimation and order-finding/factoring
- Shors approach
- Discrete Logarithm and Hidden Subgroup Problem

(if theres time)

Quantum Phase Estimation

- Suppose we wish to estimate a number given

the quantum state

- Note that in binary we can express

Quantum Phase Estimation

- Since for any integer k, we have

Quantum Phase Estimation

- If then we can do the following

Useful identity

- We can show that

Quantum Phase Estimation

- So if then we can do the following

Quantum Phase Estimation

- So if then we can do the following

Quantum Phase Estimation

- Generalizing this network (and reversing the

order of the qubits at the end) gives us a

network with O(n2) gates that implements

Discrete Fourier Transform

- The discrete Fourier transform maps vectors of

dimension N by transforming the elementary

vector according to

- The quantum Fourier transform maps vectors in a

Hilbert space of dimension N according to

Discrete Fourier Transform

- Thus we have illustrated how to implement (the

inverse of) the quantum Fourier transform in a

Hilbert space of dimension 2n

Estimating arbitrary

- What if is not necessarily of the form for

some integer x?

- The QFT will map to a superposition

where

Quantum Phase Estimation

- For any real

- With high probability

Eigenvalue kick-back

- Recall the trick

Eigenvalue kick-back

- Consider a unitary operation U with eigenvalue

and eigenvector

Eigenvalue kick-back

Eigenvalue kick-back

- As a relative phase, becomes measurable

Eigenvalue kick-back

- If we exponentiate U, we get multiples of

Eigenvalue kick-back

Eigenvalue kick-back

Phase estimation

Eigenvalue estimation

Eigenvalue estimation

Eigenvalue estimation

- Given with eigenvector and eigenvalue we

thus have an algorithm that maps

Eigenvalue kick-back

- Given with eigenvectors and respective

eigenvalues we thus have an algorithm that

maps

and therefore

Eigenvalue kick-back

- Measuring the first register of

is equivalent to measuring with probability

Example

- Suppose we have a group and we wish to find

the order of (I.e. the smallest

positive such that ) - If we can efficiently do arithmetic in the group,

then we can realize a unitary operator

that maps - Notice that

- This means that the eigenvalues of are of

the form where k is an integer

(Aside more on reversible computing)

If we know how to efficiently compute and

then we can efficiently and reversibly map

(Aside more on reversible computing)

And therefore we can efficiently map

Example

- Let
- Then
- We can easily implement, for example,

- The eigenvectors of include

Example

Example

Example

Example

Example

Eigenvalue Kickback

Eigenvalue Kickback

Eigenvalue Kickback

Eigenvalue Kickback

Quantum Factoring

- The security of many public key cryptosystems

used in industry today relies on the difficulty

of factoring large numbers into smaller factors. - Factoring the integer N into smaller factors can

be reduced to the following task

Given integer a, find the smallest positive

integer r so that

Example

- Let
- We can easily implement

- The eigenvectors of include

Example

Example

Eigenvalue kick-back

- Given with eigenvectors and respective

eigenvalues we thus have an algorithm that

maps

and therefore

Eigenvalue Estimation

Eigenvalue kick-back

- Measuring the first register of

is equivalent to measuring with probability

Finding r

For most integers k, a good estimate of (with

error at most ) allows us to determine r

(even if we dont know k). (using continued

fractions)

(aside how does factoring reduce to

order-finding??)

- The most common approach for factoring integers

is the difference of squares technique - Randomly find two integers x and y satisfying
- So N divides
- Hope that is non-trivial
- If r is even, then let
- so that

Shors approach

- This eigenvalue estimation approach is not the

original approach discovered by Shor - Kitaev developed an eigenvalue estimation

approach (to the more general Hidden Stabilizer

Problem) - Weve presented the CEMM version here

Discrete Fourier Transform

- The discrete Fourier transform maps uniform

periodic states, say with period r dividing N,

and offset w, to a periodic state with period N/r.

Discrete Fourier Transform

- The quantum Fourier transform maps vectors in a

Hilbert space of dimension N according to

Shors Factoring Algorithm

Network for Shors Factoring Algorithm

Eigenvalue Estimation Factoring Algorithm

Network for Eigenvalue Estimation Factoring

Algorithm

Equivalence of ShorCEMM

- Shor analysis CEMM analysis

Equivalence of ShorCEMM

- Shor analysis CEMM analysis

Discrete Logarithm Problem

Consider two elements from a group G

satisfying Find s.

Discrete Logarithm Problem

We know has eigenvectors

Discrete Logarithm Problem

Thus has the same eigenvectors but

with eigenvalues exponentiated to the power of s

Discrete Logarithm Problem

Discrete Logarithm Problem

Given k and ks, we can compute s mod r (provided

k and r are coprime)

Abelian Hidden Subgroup Problem

Find generators for

Network for AHS

AHS Algorithm in standard basis

AHS for in eigenbasis

(Simons Problem)

is an eigenvector of

Other applications of Abelian HSP

- Any finite Abelian group G is the direct sum of

finite cyclic groups - But finding generators

satisfying is not always easy, e.g. for

its as hard as factoring N - Given any polynomial sized set of generators, we

can use the Abelian HSP algorithm to find new

generators that decompose G into a direct sum of

finite cyclic groups.

Examples

Deutschs Problem

or

Order finding

any group

Example

Discrete Log of to base

any group

Examples

Self-shift equivalences

What about non-Abelian HSP

- Consider the symmetric group
- Sn is the set of permutations of n elements
- Let G be an n-vertex graph
- Let
- Define
- Then
- where

Graph automorphism problem

- So the hidden subgroup of is the

automorphism group of G - This is a difficult problem in NP that is

believed not to be in BPP and yet not

NP-complete.

Other

- Progress on the Hidden Subgroup Problem in

non-Abelian groups (not an exhaustive list) - Ettinger, Hoyer arxiv.gov/abs/quant-ph/9807029
- Roetteler,Beth quant-ph/9812070
- Ivanyos,Magniez,Santha arxiv.org/abs/quant-ph/0102

014 - Friedl,Ivanyos,Magniez,Santha,Sen

quant-ph/0211091 (Hidden Translation and Orbit

Coset in Quantum Computing) they show e.g. that

the HSP can be solved for solvable groups with

bounded exponent and of bounded derived series - Moore,Rockmore,Russell,Schulman, quant-ph/0211124

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