Quantum Computing. Continuous Quantum Hidden Subgroup Algorithms. This work is in collaboration with. Louis H. Kauffman. Samuel J. Lomonaco, Jr.

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1 Quantum Computing amuel J. Lomonaco, Jr. Dept. of Comp. ci.. & Electrical Engineering University of Maryland Baltimore County Baltimore, MD WebPage: Overview Four Talks A Rosetta tone for Quantum Computation Three Quantum s Elementary Quantum Hidden ubgroup s An Entangled Tale of Quantum Entanglement Advanced Lecture 3 Continuous Quantum Hidden ubgroup s amuel J. Lomonaco, Jr. Dept. of Comp. ci. & Electrical Engineering University of Maryland Baltimore County Baltimore, MD Lomonaco@UMBC.EDU WebPage: Defense Advanced Research Projects Agency (DARPA) & Air Force Research Laboratory, Air Force Materiel Command, UAF Agreement Number F This work is supported by: This work is in collaboration with Louis H. auffman The Defense Advance Research Projects Agency (DARPA) & Air Force Research Laboratory (AFRL), Air Force Materiel Command, UAF Agreement Number F The National Institute for tandards and Technology (NIT) The Mathematical ciences Research Institute (MRI). L-O-O-P The L-O-O-P Fund.

2 Existing Quantum s ome Existing HA s Hidden ubgroup s hor-like s Amplitude amplification Grover-Like s Quantum s imulating Quantum ystems ipser s Adiabatic s??? Hidden subgroup algorithms Deutsch-Jozsa imon hor Legendre symbol Hallgen arious Non arious Non-abel.. s Others We will now discuss the following ix HA s Continuous hor on Wandering hor Lift of hor to HA on Circle Dual hor HA HA for Functional Integrals These ix s Can Be Found in the Following Three Papers Lomonaco & auffman, Quantum Hidden ubgroup s: A Mathematical Perspective, AM, CONM/35, (22), ph/295 Lomonaco & auffman, A Continuous ariable hor, ph/24 Lomonaco & auffman, Continuous Quantum Hidden ubgroup s, ph/3484 Ambient Target Group et Def. A Map : A Hidden ubgroup subgroup structure if there exist s A subgroup of A, and An injection ι : A/ s. t. the diagram Hidden Natural A urjection ν is commutative. A/ Hidden ubgroup tructure is said to have hidden Hidden ubgroup et of Right Cosets ι 2

3 Hidden Epimorphism Hidden ubgroup tructure (Cont.) Hidden Quotient Group A ν If is an invariant subgroup of H is a group, and epimorphism A/ ν : A A/ ι A/ A, then is an Origin of QH s hor s Quantum factoring algorithm reduces the task of factoring an integer N to the task of finding the period P of a function Z Zmod N n a n mod N itaev observed that finding the period P is equivalent to finding the subgroup PZ Z, i.e., the kernel of. Quantum Hidden ubgroup s The Hidden ubgroup Problem (HP) Quantum Deutsch-Jozsa imon hor Factoring Ambient Gp A Hidden ubgp { } 2 2 P Given a map : A with hidden subgroup structure, determine the hidden subgroup of the ambient group A. An algorithm solving this problem is called a hidden subgroup algorithm (HA) The First of the Three Papers Lomonaco & auffman, Quantum Hidden ubgroup s: A Mathematical Perspective, AM, CONM/35, (22). ph/295 The Quantum Hidden ubgroup Paper hows how to create a Meta 3

4 An Analogy Autos before Henry Ford Quantum ersion of Henry Ford s Assembly Line Autos after Henry Ford Three Methods for Creating New Quantum s Two Ways to Create New Quantum s Lifting and Pushing Given : A Lifted Gp Amb. Gp Approx Gp L η A ι ν H ι Target et Lift Push A 3rd Way to Create New Quantum s Duality ummary 3 Ways to create New Quantum s Amb. Gp A QH Alg Lifting Dual Gp A Φ Dual Dual QH Alg Pushing Duality 4

5 Hidden ubgroup s ome Past s Wandering hor Lomonaco & auffman, Quantum Hidden ubgroup s: A Mathematical Perspective, AM, CONM/35, (22). ph/295 Continuous hor Lomonaco & auffman, A Continuous ariable hor, ph/24 Free Abel Finite Rk Amb GP hor Transv Approx Gp Wandering hor A ι ν Q ι Approx Map Target et Push Continuous hor Three Recent QH s Ambient Group Add. Gp of Reals A A quantum algorithm on the Circle ey Idea: Lifting of discrete algorithms to a continuous groups A quantum algorithm dual to hor s algorithm A highly speculative quantum algorithm for functional integrals? Quantum algorithm for the Jones polynomial hor s Alg Lifting QH Alg on Duality QH Alg on / Pushing Dual of hor s Alg Road Map Lifting QH Algs for Functional Integrals Q / Q ~ ~ Lifting & Duality hor Lift of hor Dual Lifted Dual hor Dual 5

6 A Momentary Digression A Lifting of hor s Quantum Factoring to Integers Fourier Analysis on the Circle The Circle as a Group The Circle as a Group The circle group can be viewed as A multiplicative group,, i.e., as the unit circle in the complex plane where reals. 2π ix { e : x } 2 π ix 2 π iy 2 i( x + y) e ie e π denotes the additive group of The circle group can also be viewed as An additive group,, i.e., as / reals mod x+ ymod where denotes the additive group of integers. The Character Group The Character Groups of A A The character group of The character group is defined as of an abelian group A Hom ( A, Circle ) { χ : A Circle: χ a morphism} with group operation (in multiplicative notation), ( χ iχ )( a) χ ( a) iχ ( a) 2 2 or (in additive notation) as ( χ + χ )( a) χ ( a) + χ ( a) 2 2 The Character Groups of The character group of is and 2π { χ x n e x } is / : : / / 2π / { χ n : xe : n } Discrete { χ : x nxmod: n } n / Continuous 6

7 Fourier Analysis on the Circle / The Fourier transform of is defined as the map given by : f f : / f ( n) dxe f ( x) The inverse Fourier transform is defined as 2π f ( x) e f ( n) n Needed Mathematical Machinery Dirac Delta function δ ( x) P P n δ x P P n ( x) δ on / For P a non-zero integer, we will also need on / the generalized function H / denotes the rigged Hilbert space on with orthonormal basis / { x : x / }, i.e., xy δ ( x y) The elements of of the form Rigged Hilbert pace H / dx f ( x ) x are formal integrals Finally, let sums H with orthonormal basis denote the space of formal an n : an n n { n : n } A Lifting of hor s Quantum Factoring to Integers ~ Q Lifting & Duality hor Lift of hor 7

8 Periodic Functions on tep. Initialize : Let be periodic function with hidden minimum period. Objective: Find P P tep. Apply tep 2. Apply H H / F - H H 2 e πin i n n n n U : n u n u+ ( n) 2 n ( n) n tep 3. Apply F dx x e ( n) H H 3 / n P 2π + dx x e np + n n n i( np n) x ( ) P in Px in x dx x e e n n n P in x δp ( ) ( ) n P P in x e ( n ) n P P n P n dx x x e n n n n Ω P P ( ) tep 4. Measure 3 P n n Ω P P n with respect to the observable O Qy dy y y Q m / Q to produce a random eigenvalue and then proceed to find the corresponding n/ P using the continued fraction recursion. (We assume 2 ) Q 2P The Actual (Un-Lifted) hor The Actual Un-Lifted hor Make the following approximations by selecting a sufficiently large integer Q : { : } Q k k < Q r / Q mod : r,,, Q Q : : Q is only approximately periodic! 8

9 Run the algorithm in H Q H and measure the observable Q r r O QQ r r Q A Quantum Hidden ubgroup on the Circle The Dual on the Circle Q ~ / Lifting & Duality hor Lift of hor Dual Lifted Dual Rigged Hilbert pace H / denotes the rigged Hilbert space on with orthonormal basis / { : / } x x, i.e., xy δ ( x y) Finally, let H denote the space of formal sums an n : an n n with orthonormal basis The elements of H / integrals of the form dx f ( x ) x are formal { n : n } 9

10 Periodic Admissible Functions on / f : / Let be an admissible periodic function of minimum rational period α / Proposition: Let α a (with ) / a gcd ( a 2, a 2) be a period of f. Then /a is also 2 a period of. f Remark: Hence, the minimum rational period is always the reciprocal of an integer modulo. tep. Initialize tep. Apply tep 2. Apply H H F - H H dxe x dx x 2πixi / U : x u x u+ ( x) 2 dx x ( x) tep 3. Apply F 3 n n ( ) dx e n x ( ) H H n dxe x Letting x m m x, we have a m+ a a dx e ( x) dx e ( x) m m a a a m 2π in xm + a m m m dx e xm + a a 2π inm a a e dxe ( x) m But Thus, a m 3 e 2π inm a a aδ n moda ( a) if n moda otherwise ( ) n dxe x n / a δn moda n / a iax ( ) n dxe x a dxe ( x) a Ω tep 4. Measure ( ) 3 a Ω a with respect to the observable O n nn n to produce a random eigenvalue a

11 The corresponding discrete algorithm The ic Dual of hor s Quantum Factoring Q / Q ~ ~ Lifting & Duality hor Lift of hor Dual Lifted Dual hor Dual We now create a corresponding discrete algorithm The approximations are: { : } Q k k < P r / Q mod : r,,, Q Q : : Q is only approximately periodic! Run the algorithm in H Q H and measure the observable Q O kk k k Quantum s based on Feynman Functional integrals Caveat Emptor The following algorithm is highly speculative. In the spirit of Feynman, the following quantum algorithm is based on functional integrals whose existence is difficult to determine,, let alone approximate.

12 The pace all continuous paths 2 which are L with respect to the inner product is a vector space over respect to xiy dsx( s) iy( s) ( λ ) ( ) [ ] x :, with x ( s) λx( s) x + y ( s) x( s) + y( s) n The Problem to be olved : Let be a functional with a hidden subspace of such that ( ) ( ) x + v x v Objective. Create a quantum algorithm that finds the hidden subspace. The Ambient Rigged Hilbert pace Let H be the rigged Hilbert space with orthonormal basis, { x : x } Parenthetical Remark Please note that can be written as the following disjoint union: ( ) v + v and with bracket product δ ( ) x y x y tep. Initialize tep. Apply D H H F - D 2π ix i x e x x x tep 3. Apply F 3 D ( ) ixi y y Dxe y x ix i y ( ) D yy Dxe x tep 2. Apply U : x u x u+ ( x) 2 D x x ( x) 2

13 But ixiy ixiy Dxe ( x) Dv Dxe ( x) v + 2πiv ( + x) iy ( ) Dv Dxe v+ x D ( ) iviy ix iy ve Dxe x However, o, 3 D D δ( ) ivi y ve u y u ( ) iviy ix iy Dyy Dve Dxe x n ixi y Dyy Duδ( y u) Dxe ( x) n D ( u) ( ) ixiu uu Dxe x D uu Ω tep 4. Measure 3 D uu Ω ( u) with respect to the observable A D w w w w to produce a random element of Question Can the above path integral quantum algorithm be modified in such a way as to create a quantum algorithm for the Jones polynomial? I.e., can it be modified by replacing by the space of gauge connections,, and by making suitable modifications? where ( ) D A ( A) W ( A) W ( A) W is the Wilson loop ( A) tr( Pexp ( A) ) Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium, amuel J. Lomonaco, Jr. (editor), AM PAPM/58, (22). The End 3

14 Quantum Computation and Information, amuel J. Lomonaco, Jr. and Howard E. Brandt (editors), AM CONM/35, (22). 4