Part 1. Grover s s Algorithm. Quantum Computing. The Grover, Shor,, and Deutsch-Jozsa. Algorithms. This work is supported by: Samuel J. Lomonaco, Jr.

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1 uantum Computing Samuel J. Lomonaco, Jr. Dept. o Comp. Sci.. & Electrical Engineering University o Maryland Baltimore County Baltimore, MD 5 Lomonaco@UMBC.EDU Webage: Overview Four Talks Elementary A Rosetta Stone or uantum Computation Three uantum Algorithms uantum idden Subgroup Algorithms An Entangled Tale o uantum Entanglement Lecture Three uantum Algorithms The Grover, Shor,, and Deutsch-Jozsa Algorithms Samuel J. Lomonaco, Jr. Dept. o Comper Science & Electrical Engineering University o Maryland Baltimore County Baltimore, MD 5 Lomonaco@UMBC.EDU Webage: Advanced This work is supported by: The Deense Advance Research rojects Agency (DARA & Air Force Research Laboratory (AFRL, Air Force Materiel Command, USAF Agreement umber F The ational Institute or Standards and Technology (IST. The Mathematical Sciences Research Institute (MSRI. art Grover s s Algorithm L-O-O- The L-O-O- L Fund.

2 Grover s s Algorithm Searching or a eedle in a aystack Samuel J. Lomonaco, Jr. Dept. o Comper Science & Electrical Engineering University o Maryland Baltimore County Baltimore, MD 5 Lomonaco@UMBC.EDU Webage: This work is supported by: The Deense Advance Research rojects Agency (DARA & Air Force Research Laboratory (AFRL, Air Force Materiel Command, USAF Agreement umber F The ational Institute or Standards and Technology (IST. The Mathematical Sciences Research Institute (MSRI. L-O-O- The L-O-O- L Fund. Lomonaco, Samuel J., Jr., Grover s s uantum Search Algorithm,, in AMS SAM/58, (, pages 8 9. Grover s s Algorithm Finding a eedle in a aystack Consider a large unstructured database consisting o = n records labeled in random order by the integers integers,,,, E.g., the database could be stored as a linked list ead Record Record #%*!*# %#*!#* On average, we have to retrieve inding the label x. / ence, the average computational work is computational steps Record *!#%*# labels beore O( This is a practical Searching problem that appears a hone in many Book guises For example, consider a city phone book containing phone numbers. Find the name associated with the phone number x = ( The best classical algorithm or inding the associated name, say Jane Doe,, would search through / phone numbers on average beore inding the name Jane Doe. In other words, it would take on average computational steps. O More ormally, we are given a unction { } { } :,,,,, called an Oracle,, such that ( x i x = x ( YES = Otherwise YES ( O By calling it an Oracle,, we mean that we do not have immediate access to all argument- unction pairs (,. The Oracle is simply a blackbox,, which we can query as many times as we like by inputting a number x, and then observing the resulting output ( x. But each such query comes with an associated computational $$cost$$. x x ( x

3 The Search roblem or an Unstructured Database is: To ind the record labeled as x with the minimum amount o computational $$work$$, i.e., with the minimum number o queries o the oracle. Another Example Consider a plaintext/ciphertext attack by brute orce key search on a message encrypted with the Data Encryption Standard (DES, where the key K is a 56 bit number Given the plaintext/ciphertext pair laintext Ciphertext TheStolenGoldIsiddenAt xjepwvziderkqldievmsfk crack the entire cipher by encrypting the plaintext TheStolenGoldIsiddenAt with each o the keys,,,. 56 -, in turn, until the key K is ound that produces the ciphertext xjepwvziderkqldievmsfk In other words, i (, C denotes the available plaintext/ciphertext pair, and i then the Oracle is ( K K denotes the key such that (, DES K = = C i K = K Otherwise Mission Impossible Assignment Your Mission Impossible assignment, should you choose to accept it, is to devise an algorithm which inds the label in steps. O( As always, should you ail, your lecturer will disavow any association with your activities. x Mission Impossible Assignment Lov Grover has accepted this Mission Impossible challenge, and has successully created an algorithm which inds the label x in O( steps, with a total computational work o O( log The uantum Mechanical erspective -D D ilbert space with orthonormal basis {,} n -D D ilbert space with orthonormal basis {,,,, } n = Oracle is given as a blackbox unitary transormation U U x y x ( x y 3

4 The uantum Mechanical erspective From the Oracle U x y x ( x y we construct the unitary transormation x x i x = x I ( x = ( x x = x x otherwise as ollows x ( x x U I I ote that x is an Inversion I = I x x, since x I x x x= x x x x i x= x = x otherwise Also please note that or any unit length ket I = I is an inversion about the hyperplane to, i.e., A A Mirror Image Transormation ilbert Space Let A A Mirror Relection Right Right yperplane Mirror Let roperties o I χ χ Deinition. Let, be unit length kets in s.t... Then (, χ { χ, } = Span = a + b a b is a vector space over lying in with a real inner product induced by the bracket in. ence, i, χ are linearly independent, then is a -D Euclidean plane lying in. Inversion by I roperties o I (, χ { χ, } = Span = a + b a b I & I χ leave the plane invariant roposition. The plane is invariant under the transormations I, I χ, i.e., I I χ S = S S = S I χ χ I 4

5 Let be a ket in perpendicular to, and let L denote the line in passing through the origin and to. Then I : S S Line L And moreover, i is a unit length vector in which is to, then I = I I And inally, i is a unit length ket in, and i U : is a unitary transormation, then Relection in line L UI U = I U Summary Given two unit kets and χ living in with χ real, there is a -D D plane (i.e., -D inner product space over living in spanned by and χ s.t. I ( S = S and I χ S = S S I a relection in is or the line L an inversion about I = I or all unitary transs UI U = I U U Overview o Grover s s Algorithm Step. (Initialization Step. = j k j= Loop until ( π k= π /4sin / 4 = I I x k k+ Step. Measure with respect to the standard basis,,, to obtain the marked unknown state x with probability / What s s Going On? It s s All About this icture Let be the adamard transorm given by where The Method in Lov s Madness : n = ( ( = L x L x x 5

6 Thus, we now have living in the -D D plane Step. x (, S = Span x (Initialization This step creates a superposition o all states, i.e., = j = j= Step. The Iteration Loop Each iteration rotates (in closer to x. Let x and be unit length kets in which are to x and, respectively. Let = Angle( x, L x L x x L x L Step. With each iteration (Cont. x k+ = x where = = x ( I I I I = I I = I I = I I is the product o two inversions. k x x x x Theorem But, So, L x L x x L& L Linesin L L = point O Re Re = Rot = L L Angle ( L, L = cos x + sin x k So, = cos[ ( + ] + sin [( + ] k x k x = cos [( + ] + sin [( + ] k k x k x We seek to iterate until sin k + possible. [ ] But what is? is as large as In other words, we seek the smallest positive integer k = K such that ( k + is as close as possible to π /. This turns out to be k K π / ( = = L x L But what is? Recall that = Angle( x, We ind is complementary to x x by noting that the angle = Angle x, i.e., + = π / (, 6

7 Since = π( x + + Since ence, the number o iterations in O( Step. is π we have = x = cos = cos = sin ence, ( = sin / / π k = K = 4 and π /4sin ( / But each iteration uses the adamard transorm n = ( at the computational cost o O ( lg Since Step. is the computationaly dominant part o Grover s s algorithm, it ollows that the computational time complexity o this algorithm is O( lg The probability that the measurement perormed in Step. o Grover s s algorithm will successully retrieve the unknown label is given by x The robability o Success [( K ] robsuccess = sin + cos = Grover s s Algorithm Step. (Initialization Step. = j k j= Loop until ( π k= π /4sin / 4 = I I x k k+ Step. Measure with respect to the standard basis,,, to obtain the marked unknown state x with probability / The End o Grover s s Algorithm art Shor s Algorithm 7

8 Shor s Factoring Algorithm Samuel J. Lomonaco, Jr. Dept. o Comper Science & Electrical Engineering University o Maryland Baltimore County Baltimore, MD 5 Lomonaco@UMBC.EDU Webage: This work is supported by: The Deense Advance Research rojects Agency (DARA & Air Force Research Laboratory (AFRL, Air Force Materiel Command, USAF Agreement umber F The ational Institute or Standards and Technology (IST. The Mathematical Sciences Research Institute (MSRI. L-O-O- The L-O-O- L Fund. This talk is loosely based on the paper Lomonaco, Samuel J., Jr., Shor s uantum Factoring Algorithm,, in AMS SAM/58, (, pages A Crytanalyst s Dream I m m going to crack the RSA Cryto System by inding a Superast Factoring Algorithm!! I ll be rich & amous!!! Code Breaker roblem. Given an integer which is the product o two unknown primes p & q,, i.e., =pq, ind p and q, i.e., actor. Step. Step. Simpliied Shor s Algorithm Choose an integer a s.t. o the unction: x x = a mod gcd a, = Step 3. 3 I is not even, then goto Step. Step 4. 4 Use a uantum Computer to determine the period I is even, then / / a = a a + mod So use the Euclidean algorithm to compute / / ( a and cd( a +, gcd, g Step 5. 5 Simpliied I the above gcd s are & or q & p, then we have actored. I not, goto Step. p q 8

9 Step. Step. Simpliied Shor s Algorithm Choose an integer a s.t. gcd a, = Step 3. 3 I is not even, then goto Step. Step 4. 4 Use a uantum Computer to determine the period o the unction: I ( x = a x mod is even, then / / a = a a + mod So use the Euclidean algorithm to compute / / ( a and cd( a +, gcd, g uantum art o Algorithm {,,, } = ± ± The Integers roblem. Given a periodic unction : Find the period o. Choose a suiciently large positive integer, and restrict to the set S = {,,,, } and ocus on the restricted unction : S Simpliication. To avoid minor technicalities, we assume that is a multiple o, i.e.,, Choose an integer n s.t. < n & Max(< n Construct two n-qubit registers, i.e., Reg and Reg. Reg Reg = an an a bn bn b Arguments o Arguments o n a a a = a Convention. n n j= j For example, = 3 j The -oint Fourier Transorm ω = rimitive -th root o unity, e.g., e πi/ S {,,,, } = The Fourier Transorm is: where F : S : S y = x ω x Remark. We will implement transormation. F xy as a unitary Step.. Initialize = F Step.. Apply to Reg F I i j ω j j= j= = = j Step.. Let U be a unitary transormation ence, that Reg takes now j holds to all the j integers ( j. Apply U,,,., - in superposition U j x ( x j= j= Reg Reg = j= j ( j = j + j ( j + j j= j= j= j= = j + j ( j = j j + ( j j= j= 9

10 Step.3. Reg Reg = j j + ( j j= j= r {,,,, } Measure Reg rob(j rob(j =r =r =/ =/ Reg Reg = j+ r ( r j = Whoosh Whoosh! F Step.4. Apply to Reg F I ( j+ r k j+ r r ω k r j= j= k= ( j+ r k ω k But =! k= j = = ω rk k= j = k ( ω j ( r k ( r But! k j ( ω ence, j = / i y= mod / = otherwise Reg Re = rk ω k ( r k λ : λ=,,, = rλ ω λ ( r λ = g Step.4. r λ Reg Reg = ω λ ( r λ = Measure Reg rob( rob(λ=λ Reg Reg = λ ( r ence, we have obtained λ or some =λ =/ =/ Whoosh Whoosh! λ { },,, Repeat steps through 5 until there are enough multiples o / to recover /, and hence. Remark. We should remind the reader that the above description is a simpliication o Shor s actoring algorithm which contains most o the key eatures o the actual algorithm. A more detailed and thorough can be ound in the ollowing two papers: Shor s quantum actoring algorithm,, in uantum Computation, AMS SAM/58, (, pages 6-79 uantum idden Subgroup Algorithms: A Mathematical espective,, in uantum Computation and Inormation, AMS COM/35, (, pages

11 The End o Shor s Algorithm art 3 The Deutsch-Jozsa Algorithm The Deutsch-Jozsa Algorithm Samuel J. Lomonaco, Jr. Dept. o Comp. Sci. & Electrical Engineering University o Maryland Baltimore County Baltimore, MD 5 Lomonaco@UMBC.EDU Webage: Deense Advanced Research rojects Agency (DARA & Air Force Research Laboratory, Air Force Materiel Command, USAF Agreement umber F This work is supported by: The Deense Advance Research rojects Agency (DARA & Air Force Research Laboratory (AFRL, Air Force Materiel Command, USAF Agreement umber F The ational Institute or Standards and Technology (IST The Mathematical Sciences Research Institute (MSRI. L-O-O- The L-O-O- Fund. Deutsch s s Algorithm Deinition.. A coin is air (or balanced i it has heads on one side and tails on the other side. It is unair (or constant i either it has tails on both sides, or heads on both sides. Side T T Side Fair (Balanced Side Side T Side T Side Unair (Constant Side Side Observation Observation. In the classical world, we need to observe both sides o the coin to determine whether or not it is air? But what about in the quantum world? Fair (Balanced Unair (Constant

12 We represent a coin mathematically as a Boolean unction: Side { } { } :,, Side T Let then The Unitary Implementation o U be the unitary transormation x U x y x ( x y U ( ( x x Moreover, U ( + ( ( ( + ( (. is air,, i.e., balanced ( ( ( ( Outpu t = i + =±. is unair,, i.e., constant ( ( ( + ( Output = + i =± Case. Case. ( ( So I we only make one observation, i.e., i we observe the let register, then we can determine whether or not is air or unair. uantum Computation: A Grand Mathematical Mathematical Challenge or the Twenty-First Century Century and the Millennium, Samuel J. Lomonaco, Jr. (editor, AMS SAM/58, (. The End

13 uantum Computation and Inormation, Samuel J. Lomonaco, Jr. and oward E. Brandt (editors, AMS COM/35, (. 3