Quantum Factoring Michele Mosca The Fifth Canadian Summer School on Quantum Information August 3, 2005

Quantum Algorithms l Quantum Algorithms should exploit quantum parallelism and quantum interference. l We have already seen some elementary algorithms.

Quantum Algorithms l These algorithms have been computing essentially classical functions on quantum superpositions l This encoded information in the phases of the basis states: measuring basis states would provide little useful information l 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 l Quantum Phase Estimation l Eigenvalue Kick-back l Eigenvalue estimation and order- finding/factoring l Shor’s approach l Discrete Logarithm and Hidden Subgroup Problem (if there’s time)

Quantum Phase Estimation l Suppose we wish to estimate a number given the quantum state l Note that in binary we can express

Quantum Phase Estimation l Since for any integer k, we have

Quantum Phase Estimation l If then we can do the following

Useful identity l We can show that

Quantum Phase Estimation l So if then we can do the following

Quantum Phase Estimation l So if then we can do the following

Quantum Phase Estimation l Generalizing this network (and reversing the order of the qubits at the end) gives us a network with O(n 2 ) gates that implements

Discrete Fourier Transform l The discrete Fourier transform maps vectors of dimension N by transforming the elementary vector according to l The quantum Fourier transform maps vectors in a Hilbert space of dimension N according to

Discrete Fourier Transform l Thus we have illustrated how to implement (the inverse of) the quantum Fourier transform in a Hilbert space of dimension 2 n

Estimating arbitrary l What if is not necessarily of the formfor some integer x? l The QFT will mapto a superposition where

l For any real Quantum Phase Estimation l With high probability

l Recall the “trick”: Eigenvalue kick-back

l Consider a unitary operation U with eigenvalue and eigenvector Eigenvalue kick-back

l As a relative phase, becomes measurable

l If we exponentiate U, we get multiples of Eigenvalue kick-back

Phase estimation

Eigenvalue estimation

l Given with eigenvectorand eigenvalue we thus have an algorithm that maps

Eigenvalue kick-back l Given with eigenvectorsand respective eigenvalues we thus have an algorithm that maps and therefore

Eigenvalue kick-back l Measuring the first register of is equivalent to measuring with probability i.e.

Example l Suppose we have a group and we wish to find the order of (I.e. the smallest positive such that ) l If we can efficiently do arithmetic in the group, then we can realize a unitary operator that maps l Notice that l This means that the eigenvalues of are of the formwhere 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 l Let l Then l We can easily implement, for example, l The eigenvectors of include

Example

Eigenvalue Kickback

Quantum Factoring l 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 l Let l We can easily implement l The eigenvectors of include

Example

Eigenvalue kick-back l Given with eigenvectorsand respective eigenvalues we thus have an algorithm that maps and therefore

Eigenvalue Estimation

Eigenvalue kick-back l 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 don’t know k). (using continued fractions)

(aside: how does factoring reduce to order-finding??) l 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 l If r is even, then let so that

Shor’s approach l This eigenvalue estimation approach is not the original approach discovered by Shor l Kitaev developed an eigenvalue estimation approach (to the more general “Hidden Stabilizer Problem”) l We’ve presented the CEMM version here

Discrete Fourier Transform l 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 l The quantum Fourier transform maps vectors in a Hilbert space of dimension N according to

Shor’s Factoring Algorithm

Network for Shor’s Factoring Algorithm

Eigenvalue Estimation Factoring Algorithm

Network for Eigenvalue Estimation Factoring Algorithm

Equivalence of Shor&CEMM Shor analysisCEMM analysis

Equivalence of Shor&CEMM Shor analysisCEMM analysis

Consider two elementsfrom 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

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 is an eigenvector of (Simon’s Problem)

Other applications of Abelian HSP l Any finite Abelian group G is the direct sum of finite cyclic groups l But finding generators satisfying is not always easy, e.g. for it’s as hard as factoring N l 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: Deutsch’s Problem: or Order finding: any group

Example: Discrete Log of to base : any group

Examples: Self-shift equivalences:

What about non-Abelian HSP l Consider the symmetric group l S n is the set of permutations of n elements l Let G be an n-vertex graph l Let l Define l Then where

Graph automorphism problem l So the hidden subgroup of is the automorphism group of G l 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/ Roetteler,Beth quant-ph/ Ivanyos,Magniez,Santha arxiv.org/abs/quant-ph/ Friedl,Ivanyos,Magniez,Santha,Sen quant-ph/ (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/