Grover’s Algorithm: Single Solution By Michael Kontz

Application Grover’s algorithm can identify an item from a list of N elements in Grover’s algorithm can identify an item from a list of N elements in What’s this good for? What’s this good for? Unstructured database search Unstructured database search (virtual database) (virtual database) breaking DES (Data Encryption Standard)breaking DES (Data Encryption Standard) SAT (Satisfyability of boolean formula)SAT (Satisfyability of boolean formula) map coloring with 4 colorsmap coloring with 4 colors

Application: DES clear text + key = ciphertext clear text + key = ciphertext “attackatdawn” = “ojbevjewbvv” “attackatdawn” = “ojbevjewbvv” 56-bit key 56-bit key Best classical algorithm Best classical algorithm 36 quadrillion36 quadrillion Grover’s algorithm Grover’s algorithm 118 million118 million

Amplitude Amplification Overview Overview Start in an initial state that is equally every stateStart in an initial state that is equally every state Over time (iterations) amplify amplitude of solutionOver time (iterations) amplify amplitude of solution Measure (collapse system) when amplitutde^2 is greater than 0.5 (50%)Measure (collapse system) when amplitutde^2 is greater than 0.5 (50%)

Initial State Hadamard Gate Hadamard Gate Steps: N = 2^n Steps: N = 2^n begin in statebegin in state transform into equal superposition of all states using Hadamardtransform into equal superposition of all states using Hadamard

Oracle Oracle picks out which state to amplify Oracle picks out which state to amplify Black box: Black box: Oracle is unitary operator U O : Oracle is unitary operator U O :

Oracle Conjugate oracle with Hadamard transforms so only changes phase (sign) Conjugate oracle with Hadamard transforms so only changes phase (sign)

Algorithm Setup initial state Setup initial state Repeat these 4 steps times Repeat these 4 steps times Measure answer Measure answer

Inversion about mean Unitary operator describing phase shift:

Inversion about mean Unitary operator describing 2-4:

Inversion about mean

Complexity O(sqrt(N)) How many calls to oracle does it take to achieve amplitude^2 > 0.5? How many calls to oracle does it take to achieve amplitude^2 > 0.5? assume all states but one haveassume all states but one have other state isother state is each iterationeach iteration as long asas long as for large N this is true long enough for amplitude^2 > 0.5for large N this is true long enough for amplitude^2 > 0.5

Limitations Black box limitations Black box limitations Physical implementation (classical memory) implementations Physical implementation (classical memory) implementations