Grover Search and its Applications Bung-Soo Choi bschoi3@gmail.com
Contents Basics of Quantum Computation General Properties of Grover Search Idea Analsis Weight Decision Smmetric To Weights Asmmetric To Weights Multiple Weights Conclusion
Unit of Information 0 Basis, logical value ZERO Basis, logical value ONE = a 0 + b Unit of Information b to basis Bit(Classical Unit of Information) hen a =, b = 0 Þ = 0 or a = 0, b = Þ = Qubit(Quantum Unit of Information) = a 0 + b, a, b Î Complex Number, a + b = ( ) Example) = 0 + i
Basics of Quantum Computation Superposition A qubit can represent to basis states simultaneousl Example) To Qubits for Four Basis Ä = ( 0 + ) Ä ( 0 + ) n qubits can represent Än = å N - k n k = 0 Exponential Reduction of Resource for State Space!!!! = 4 0 Ä 0 + 0 Ä + Ä 0 + Ä = ( 0 0 + 0 + 0 + 4 ) = ( 00 + 0 + 0 + ) 4 ( )
Basics of Quantum Computation Entanglement Space-like long distance correlation ith nosignaling condition Example) An Entangled State for to qubits Ä = Ä + Ä ( 0 0 ) If the state of first qubit is 0, then the state of second qubit MUST be 0 No signaling condition: No a to communicate faster than light!!!
Basics of Quantum Computation Interference To phases of a same basis can be interfered a 0 + b 0 = ( a + b ) 0
Basics of Quantum Computation Operation on quantum state Unitar Operator for Evolving Quantum State + U Þ, hereuu = I init next Measurement Operator(Projection Operator) a 0 b 0 0 = + M = ( + ) M ì ï 0, P = a = í ïî, P = b k m ì, if k = m ü = í ý î0, otheriseþ If measure b, e can get the folloing state ith corresponding probabilit
Basics of Quantum Computation Quantum Parallelism
Basics of Quantum Computation General Vie of Quantum Computation Initialize All Input Qubits Make a Superposed State for All Search Space Appl Necessar Unitar Operators ith Oracle Calles Measure Final State If the final measurement output corresponds to the target solution, it stops. Otherise, it redo all steps from beginning
Basics of Quantum Computation Bounded Quantum Probabilistic (BQP) Classes Unless the quantum computation is sure success, the success probabilit is not unit Hence, e have to redo the quantum computation several times
Contents Basics of Quantum Computation General Properties of Grover Search Idea Analsis Weight Decision Smmetric To Weights Asmmetric To Weights Multiple Weights Conclusion
Search Problem Finds a certain x t here F(x t )= and x t is n- bit number x i = n = N Classical Oracle Quer Complexit is Quantum Oracle Quer Complexit is Quadratic Speedup!!! O N O ( ) ( N )
Basic Idea of Grover Search Lov Grover found a quantum search (996) Basic Idea Prepare and Initialize all input space Appl Grover operator until onl target input survive Each Grover operator checks all function values for all input values simultaneousl Single Time Step for All Evaluations Each Grover operator increases the phase amplitude of target input, and decreases the phase amplitudes of all other nontarget inputs Phase Amplitude increases linearl Measure the final state, and hence onl the target can be measured The measurement probabilit is the square of the phase amplitude
Schematic circuit for the quantum search algorithm O( N ) For Search space 0 Än n H Ä G G G Measure For Oracle Workspace 0 Oracle Phase x (- ) f x ( ) x n H Ä 0 0 x - x fo r x > 0 H Än
Some Insights Phase x oracle (- ) f x ( ) x n H Ä 0 0 x - x fo r x > 0 H Än 0 0 üï ý Þ - x - x fo r x > 0 ïþ ( 0 0 I ) x, x ¹ x t üï ý Þ - - x, x = xt ïþ ( I x t x t ) Ä n ( 0 0 - ) Þ ( - ) Ä n H I H I \ G = ( -I)( I - x x ) t t
Some Insights ( I - x x ) t t ( - I ) : Inversion about the target : Inversion about the average \ G= Inversion about the average * Inversion about the target
Example of Grover Search Original Amplitudes Negate Amplitude Average of all Amplitudes Flip all Amplitudes around Avg
Analsis Initial State: N - = å N x= 0 x Let s = x t ns = - å x xt N ¹ x q q N - sin = cos = N N q q = sin s + cos ns
Analsis Basis Vector: æ ns ö è s ø æ q ö ç cos = q sin è ø æ q q ö æ 0 ö ç cosg sin g I - xt xt = ( ) - I = è0 -ø q q sin cos ç g - g è ø æ q q ö ç cos g -sin g æcos( q ) -sin ( q ) ö G = = R ( q ) q q sin ( q ) cos( q ) sin cos ç g g è ø è ø ( )
G Analsis æ q q öæ q ö æ æ q öö cos sin cos cos q + ç g - g ç ç è ø = = q q q q sin cos ç sin ç æ ö ç g g ç sin q + è øè ø è è ø ø \ G k æ æ q öö ç cosç kq + è ø = ç æ q ö ç sinç kq + è è ø ø
k Psucess s G sin k Analsis æ q ö = = ç q + è ø æ q ö sin ç kq + =>, P sucess => è ø q p p kq + = k = - q q sin =. N q q q q If N is large, 0, sin 0, sin, \» N N k p N 4 = - \O( N )
Contents General Properties of Grover Search Idea Analsis Weight Decision Smmetric To Weights Asmmetric To Weights Multiple Weights Conclusion
Smmetric To Weight Decision Exact quantum algorithm to distinguish Boolean functions of different eights, Samuel L Braunstein, Bung-Soo Choi, Subhamo Maitra, and Subhroshekhar Ghosh, Journal of Phsics A: Mathematical and Theoretical, 40(9) 844-845 4, http://dx.doi.org/0.088/75-83/40/9/07
Motivation We can exploit the quantum computation for crptanalsis Crptanalsis: Analze the securit of secure functions Usuall, it takes large volume of computation, and hence computationall hard to crack the secure functions In this ork, as the basic level, e can consider to check the eight of Boolean functions
Definitions Weight of a Boolean function f = # of solutions / # of all inputs Smmetric Weight Condition {, + =, 0< < <} Smmetric Weight Decision Problem Decide the exact eight of f ith the smmetric eight condition Limited Weight Decision Problem æ kp ö = sin = è k + ø cos kp ö è k + ø æ
Grover Search in Hilbert Space b b,0 = sin s + cos ns G = -I ( q ) I ( f) s,0 s,4 ( p, p ),3 ( p, p ), i I ( q ) º I - ( - e q ) b ( p, p ) q = f = p b b, ( p, p ) G = -I ( p ) I ( p ) s,0 = ( - I)( I - s s ),0,0 b b /,0 ns b b, k = sin(k + ) s + cos(k + ) ns
Grover Search in Bloch Sphere,0 æ sin b ö = 0 ç cos b è - ø q f i( + ) G = -e R (-q ) R s (-f ),0 X,,3 ( p, p ) ( p, p ) ( p, p ) b b,4 b b Z s b æ 0 ö = 0 ç + è ø, k æ sin((k + ) b) ö = 0 ç - cos((k + ) b) è ø, ( p, p ) æ sin b ö 0 = ç cos b è - ø,0 ns æ 0 ö = 0 ç - è ø
+ = s æ 0 ö = 0 ç + è ø ns æ 0 ö = 0 ç - è ø
= /4 s æ 0 ö = 0 ç + è ø Z X æ p ö ç sin 3 ç 0 = ç p ç - cos è 3 ø 0 ns æ 0 ö = 0 ç - è ø
= 3/4 æ p ö ç sin 3 ç 0 = ç p ç - cos è 3 ø 0 X s æ 0 ö = 0 ç + è ø Z ns æ 0 ö = 0 ç - è ø
One Quer is Sufficient When = ¼ and = ¾, One Quer is Sufficient If the final measurement output is one of solutions, = If the final measurement output is one of non-solutions, =
Sure Success ith One Quer If ¼ <= <= and is knon before, One Quer is Sufficient to find an One of Solutions b Finding To phases Z X D.P. Chi and J. Kim, Quantum database search b a single quer in First NASA International Conference on Quantum Computing and Quantum Communications, Palm Springs, 998, edited b C.P. Williams (Springer, Berlin, 998)
Limited Weight-Decision Algorithm = = = kp k + sin ( ) kp k + ( k + ) p k + cos ( ) sin ( ) æ sin( kp ) ö æ 0 ö æ 0 ö = 0 = 0 = 0 ç - cos( kp ) ç - cos( kp ) ç (-) è ø è ø è ø, k k + æ sin(( k + ) p ) ö æ 0 ö æ 0 ö 0 0 0, k = = = ç k - cos(( k + ) p ) ç - cos(( k + ) p ) ç (-) è ø è ø è ø Limited Weight-Decision Algorithm Appl k Grover operators to Measure, k in the computation basis Let the measured result be,0 If k is even and f( )=, = else = ˆx ˆx If k is odd and f( )=, = else = ˆx
When To Steps are Sufficient + Z = S,0 +X -X b,0 - Z = NS
When To Steps are Sufficient R S ( p ),0 + Z = S,0 B = R ( p) S,0 +X -X,0 b A = R ( p) S,0 - Z = NS
When To Steps are Sufficient R ( q ) R S ( p ),0,0 + Z = S Line f Line A A = R ( q ) R ( p ) S,0,0,0 B = R ( p) S,0 +X -X,0 b A = R ( p) S,0 B = R ( q ) R S ( p),0,0 - Z = NS Line B Line -f
When To Steps are Sufficient R ( p ) R ( q ) R ( p ) S,0 S,0 Line B + Z = S Line f Line A A = R ( p) R ( q ) R ( p) 3 S S,0,0 A = R ( q ) R ( p ) S,0,0,0 B = R ( p) S,0 +X -X,0 b A = R ( p) S,0 B = R ( p) R ( q ) R ( p) 3 S S,0,0 B = R ( q ) R S ( p),0,0 Line A - Z = NS Line B Line -f
When To Steps are Sufficient B = R ( q ) R ( p ) R ( q ) R ( p ) 4 S S,0,0,0 + Z = S Line f Line B Line A A = R ( p) R ( q ) R ( p) 3 S S,0,0 A = R ( q ) R ( p ) S,0,0,0 B = R ( p) S,0 +X -X,0 b A = R ( p) S,0 B = R ( p) R ( q ) R ( p) 3 S S,0,0 B = R ( q ) R S ( p),0,0 Line A - Z = NS Line B Line -f A = R ( q ) R ( p ) R ( q ) R ( p ) 4 S S,0,0,0
æ æ,0,0 X = sin b sin(k - 3) b 0 = ç cos(k 3) b è - - ø ö, k - Z Z = X tan b - ö ç 0 ç cos b è - ø B b Z Even K cos(k - ) b = X tan b - cos b non - solution A Z = - X tan b - R ( p ) Z, k - æ - sin(k - 3) b = ç 0 ç cos(k 3) b è - - ø ö = A R æ =,0, k - ( q ) R ( p ) Z, k - cos(k - ) b - cos b ç sin b = ç ç - cos(k - ) b - cos b ç cos b è ø B = R ( p ) A Z æ - cos(k - ) b + cos b ç sin b = ç - ç - cos(k - ) b - cos b ç cos b è ø ö ö
Odd K R ( p ) Z, k- æ -sin(k - 3) b æ,0 sin b X ö = ç 0 ç cos b è - ø = ç 0 ç cos(k 3) b è - - ø,0 Z = - X tan b + ö Z A b Z solution = X tan b -, k- cos(k - ) b cos b æ sin(k - 3) b = ç 0 ç cos(k 3) b è - - ø Z = X tan b + ö A = = R,0, k- ( q ) R ( p ) Z, k- æ cos b + cos(k - ) b ç sin b = ç ç cos b - cos(k - ) b ç cos b è ø B = R ( p ) A B æ - cos b - cos(k - ) b Z ö ç sin b = ç - ç cos b - cos(k - ) b ç cos b è ø ö
-, k cosq = Phase Conditions for A k (-) cos b - cos b cos(k - ) b sin b sin(k - ) b q k cos(k - ) b - (-) cos b æ sin(k - 3) b ö ç sin b R ( q,0 ) R z ( p ) 0 ç = ç k cos(k 3) b è - - cos(k ) b ( ) cos ø ç - - - - b ç cos b è ø = sinq sin(k - ) b æ Based on Even K ö
Phase Conditions for q R,0 k æ -cos(k - ) b + (-) cos b ç sin b æ 0 ö ( q) ç - = 0 k k ç -cos(k - ) b ( ) cos ( ) - - b è - - ø ç cos b è ø cosq = k k (-) sin b ( sin q - (-) sin b ) k cos b cos b - (-) (k - ) b ö
Smmetric Weight-Decision Algorithm Smmetric Weight Decision Algorithm,0 = 0, i = 0 If Otherise k satisfies While Än p 5 sin, k =, k - p k p sin ( ) < sin ( ) k - k + k - k p < b p b - + + b = p k k i < ( k - ) do { + = - I ( p ) I ( p ), i = i + }, i s, i,0 = -I (-q ) I ( p ),, k - s, k -,0 = -I (-q ) I ( p ) -, k s, k,0 Measure, k Let the result be in the computational basis ˆx If k is odd and f ( x ˆ) = then = else = If k is even and ( ˆ) then else = f x = =
Performance When k is given, Classical Loer Bound At least O(k ) Quantum Upper Bound At most O(k)
Conclusions and Open Problems For General Weight Condition, an Exact Quantum Algorithm is possible When General Weights? Onl the condition: 0< < < When Multiple Weights?,, 3, What is the Loer Bound for Weight Decision Problems?
Contents General Properties of Grover Search Idea Analsis Weight Decision Smmetric To Weights Asmmetric To Weights Multiple Weights Conclusion
Asmmetric To Weight Decision Quantum Algorithm for the Asmmetric Weight Decision Problem and its Generalization to Multiple Weights, Bung-Soo Choi and Samuel L. Braunstein To Appear on Quantum Information Processing, http://dx.doi.org/0.007/s8-00-087-9
Motivation Extend the Smmetric Case to Asmmetric Case Problem 0< + <
Modif the Oracle Function b Adding to qubits more to reduce the problem into the smmetric case =n /N, =n /N Basic Idea To satisf + =, e modif the Oracle as f '( x) ì f ( x), 0 x < N, ï f ( x), N x < N, = í ï, N x < N + l, ï î0, N + l x < 4N
Basic Idea n + ' = 4N l n + ' = 4N l ' + ' = Þ = - + l N ( n n )
Conclusion B adding more input space, e can reduce the asmmetric eight decision problem into the smmetric eight decision problem For adding more inputs, e just need onl to qubits
Contents General Properties of Grover Search Idea Analsis Weight Decision Smmetric To Weights Asmmetric To Weights Multiple Weights Conclusion
Motivation Given a Boolean function f, decide exactl the eight of f here {0 < < < m < }.
Basic Idea Multiple Weight Decision Algorithm Let S = {,,..., m}. WHILE ( S = ) { min = smallest eight from S. max = largest eight from S. Asmmetric Weight Decision Algorithm( min, max ). S = S non_selected eight. } Return the exact eight as S.
Analsis Classical Quer Complexit is O(N) Overall Complexit is O( m N ) When m N, the proposed algorithm orks faster than classical one
Contents General Properties of Grover Search Idea Analsis Weight Decision Smmetric To Weights Asmmetric To Weights Multiple Weights Conclusion
Conclusion The Speedup of Grover search is quadratic, not exponential. Hoever, the applications based on Grover search is ver ide In this talk, e just discuss applications of Grover search for Weight Decision Problem of Boolean function Smmetric Weight Asmmetric Weight Multiple Weights Since lots of classical algorithms are based on Search algorithm, Grover search can be utilized for them ith shoing quadratic speedup