Quantum Factoring Michele Mosca The Fifth Canadian Summer
- Slides: 76
Quantum Factoring Michele Mosca The Fifth Canadian Summer School on Quantum Information August 3, 2005
Quantum Algorithms should exploit quantum parallelism and quantum interference. l We have already seen some elementary algorithms. l
Quantum Algorithms 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 l
Extracting phase information with the Hadamard operation
Overview Quantum Phase Estimation l Eigenvalue Kick-back l Eigenvalue estimation and orderfinding/factoring l Shor’s approach l Discrete Logarithm and Hidden Subgroup Problem (if there’s time) l
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 following then we can do the
Quantum Phase Estimation l So if following then we can do the
Quantum Phase Estimation l Generalizing this network (and reversing the order of the qubits at the end) gives us 2 a network with O(n ) gates that implements
Discrete Fourier Transform l l 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 l Thus we have illustrated how to implement (the inverse of) the quantum Fourier n transform in a Hilbert space of dimension 2
Estimating arbitrary l l What if form is not necessarily of the for some integer x? The QFT will map superposition where to a
Quantum Phase Estimation l For any real l With high probability
Eigenvalue kick-back l Recall the “trick”:
Eigenvalue kick-back l Consider a unitary operation U with eigenvalue and eigenvector
Eigenvalue kick-back
Eigenvalue kick-back l As a relative phase, measurable becomes
Eigenvalue kick-back l If we exponentiate U, we get multiples of
Eigenvalue kick-back
Eigenvalue kick-back
Phase estimation
Eigenvalue estimation
Eigenvalue estimation
Eigenvalue estimation l Given with eigenvector and eigenvalue we thus have an algorithm that maps
Eigenvalue kick-back l Given with eigenvectors respective eigenvalues an algorithm that maps and therefore and we thus have
Eigenvalue kick-back l Measuring the first register of is equivalent to measuring probability i. e. with
Example 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 l 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 l Then l We can easily implement, for example, l l 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. l 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 l
Example Let l We can easily implement l l The eigenvectors of include
Example
Example
Eigenvalue kick-back l Given with eigenvectors respective eigenvalues an algorithm that maps and therefore and we thus have
Eigenvalue Estimation
Eigenvalue kick-back l Measuring the first register of is equivalent to measuring probability with
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 If r is even, then let so that l is non-trivial
Shor’s approach 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 l
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 analysis CEMM analysis
Equivalence of Shor&CEMM Shor analysis CEMM analysis
Discrete Logarithm Problem Consider two elements group G satisfying Find s. from a
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 (Simon’s Problem) is an eigenvector of
Other applications of Abelian HSP 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. l
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 Consider the symmetric group l Sn is the set of permutations of n elements l Let G be an n-vertex graph l Let l Define l Then where l
Graph automorphism problem 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 NPcomplete. l
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/0102014 • 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
- Quantum physics vs quantum mechanics
- Quantum physics vs mechanics
- Gcf of 36 and 90
- Protozorios
- Mosca theorem
- Olimpiadi mosca 1980
- Emilia mosca
- Mosca npm
- Mosca florida
- Vamos arando dijo la mosca
- Chupador esponjoso
- Gấu đi như thế nào
- Bảng số nguyên tố
- Thiếu nhi thế giới liên hoan
- Tỉ lệ cơ thể trẻ em
- Fecboak
- Các châu lục và đại dương trên thế giới
- Thế nào là hệ số cao nhất
- Hệ hô hấp
- Tư thế ngồi viết
- Hình ảnh bộ gõ cơ thể búng tay
- đặc điểm cơ thể của người tối cổ
- Cách giải mật thư tọa độ
- Tư thế worm breton là gì
- Tư thế ngồi viết
- ưu thế lai là gì
- Thẻ vin
- Cái miệng xinh xinh thế chỉ nói điều hay thôi
- Thể thơ truyền thống
- Các châu lục và đại dương trên thế giới
- Từ ngữ thể hiện lòng nhân hậu
- Diễn thế sinh thái là
- Frameset trong html5
- Thế nào là giọng cùng tên
- Làm thế nào để 102-1=99
- Chúa sống lại
- Sự nuôi và dạy con của hổ
- đại từ thay thế
- Quá trình desamine hóa có thể tạo ra
- Vẽ hình chiếu vuông góc của vật thể sau
- Công thức tiính động năng
- Thế nào là mạng điện lắp đặt kiểu nổi
- Dạng đột biến một nhiễm là
- Lời thề hippocrates
- Vẽ hình chiếu đứng bằng cạnh của vật thể
- Bổ thể
- độ dài liên kết
- Môn thể thao bắt đầu bằng chữ f
- Sự nuôi và dạy con của hổ
- điện thế nghỉ
- Một số thể thơ truyền thống
- Nguyên nhân của sự mỏi cơ sinh 8
- Trời xanh đây là của chúng ta thể thơ
- Michele colucci
- Michele weiss
- Michele cirelli
- Michele thomas
- Michele liuzzi psicologo
- Gian michele innocenti
- Michele rubinelli monaco
- Michele chaban
- Michele de pasquale
- Michele cirelli
- Cest un enfant pas bien dans sa vie
- Michele floris
- Michele blago
- Ambienti folino
- Michele pavanello
- Michele travi arezzo
- Michele liguori
- Michele chaban
- Michele pestalozzi
- Michele battisti
- Michele hemberg
- Michele graziadei
- Gennaro autuori e michele del giudice
- Michele amorena