Search via quantum walk J er emie Roland UC Berkeley Joint work - PowerPoint PPT Presentation

Search via quantum walk J´ er´ emie Roland UC Berkeley Joint work with eric Magniez 1 Ashwin Nayak 2 Miklos Santha 1 Fr´ ed´ 1 LRI-CNRS, France 2 Univ. Waterloo/Perimeter Institute, Canada STOC 2007 J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 1 / 14

Abstract search problem Available procedures The problem Setup (cost S): Input: pick a random x ∈ X a set of elements X Check (cost C): a set of marked elements check whether x ∈ M “ ” M ⊆ X ε = | M | Update (cost U): | X | Output: make a random walk P a marked element x ∈ M b b b t m b b b b y z n b here: assume P ergodic, symmetric x δ = e-v gap of P J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 2 / 14

Abstract search problem Available procedures The problem Setup (cost S): Input: pick a random x ∈ X a set of elements X Check (cost C): a set of marked elements check whether x ∈ M “ ” M ⊆ X ε = | M | Update (cost U): | X | Output: make a random walk P a marked element x ∈ M y z p xy p xz b b b x t m P = b b b b p xy p xz y z n b here: assume P ergodic, symmetric x δ = e-v gap of P J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 2 / 14

Three (classical) search algorithms b b b Random walk 1 t m Pick random x ∈ X (S) b b b b y z n Repeat T 1 × b x Check whether x ∈ M (C) Naive algorithm Repeat T 2 × Random walk (U) Repeat T 1 × “ ” T 1 = O ( 1 ε ) Cost: S + 1 ε ( 1 Pick random x ∈ X (S) δ U + C ) Check whether x ∈ M (C) Random walk 2 Cost: 1 ε ( S + C ) Pick random x ∈ X (S) Repeat T 1 T 2 × Idea: Use random walk! Check whether x ∈ M (C) “ ” T 2 × random walk T 2 = O ( 1 δ ) Random walk (U) ≈ pick random x Cost: S + 1 εδ ( U + C ) J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 3 / 14

Three (classical) search algorithms b b b Random walk 1 t m Pick random x ∈ X (S) b b b b y z n Repeat T 1 × b x Check whether x ∈ M (C) Naive algorithm Repeat T 2 × Random walk (U) Repeat T 1 × “ ” T 1 = O ( 1 ε ) Cost: S + 1 ε ( 1 Pick random x ∈ X (S) δ U + C ) Check whether x ∈ M (C) Random walk 2 Cost: 1 ε ( S + C ) Pick random x ∈ X (S) Repeat T 1 T 2 × Idea: Use random walk! Check whether x ∈ M (C) “ ” T 2 × random walk T 2 = O ( 1 δ ) Random walk (U) ≈ pick random x Cost: S + 1 εδ ( U + C ) J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 3 / 14

Three (classical) search algorithms b b b Random walk 1 t m Pick random x ∈ X (S) b b b b y z n Repeat T 1 × b x Check whether x ∈ M (C) Naive algorithm Repeat T 2 × Random walk (U) Repeat T 1 × “ ” T 1 = O ( 1 ε ) Cost: S + 1 ε ( 1 Pick random x ∈ X (S) δ U + C ) Check whether x ∈ M (C) Random walk 2 Cost: 1 ε ( S + C ) Pick random x ∈ X (S) Repeat T 1 T 2 × Idea: Use random walk! Check whether x ∈ M (C) “ ” T 2 × random walk T 2 = O ( 1 δ ) Random walk (U) ≈ pick random x Cost: S + 1 εδ ( U + C ) J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 3 / 14

Three (classical) search algorithms b b b Random walk 1 t m Pick random x ∈ X (S) b b b b y z n Repeat T 1 × b x Check whether x ∈ M (C) Naive algorithm Repeat T 2 × Random walk (U) Repeat T 1 × “ ” T 1 = O ( 1 ε ) Cost: S + 1 ε ( 1 Pick random x ∈ X (S) δ U + C ) Check whether x ∈ M (C) Random walk 2 Cost: 1 ε ( S + C ) Pick random x ∈ X (S) Repeat T 1 T 2 × Idea: Use random walk! Check whether x ∈ M (C) “ ” T 2 × random walk T 2 = O ( 1 δ ) Random walk (U) ≈ pick random x Cost: S + 1 εδ ( U + C ) J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 3 / 14

Quantum search problem Two related problems Available procedures Input: Setup (cost S): √ 1 prepare | π � = � x | x � a set of elements X | X | a set of marked elements Check (cost C): M ⊆ X reflection / marked elements if x ∈ M Output: � | x � ref M : | x � �→ otherwise −| x � Find a marked element 1 Update (cost U): x ∈ M apply quantum walk W Detect whether there is a 2 marked element ( M = ∅ ?) b b b t m b b b b y z n b x J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 4 / 14

Quantum search problem Two related problems Available procedures Input: Setup (cost S): √ 1 prepare | π � = � x | x � a set of elements X | X | a set of marked elements Check (cost C): M ⊆ X reflection / marked elements if x ∈ M Output: � | x � ref M : | x � �→ otherwise −| x � Find a marked element 1 Update (cost U): x ∈ M apply quantum walk W Detect whether there is a 2 marked element ( M = ∅ ?) b b b t m b b b b y z n b x J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 4 / 14

Grover’s algorithm √ 1 We start with | π � = � x ∈ X | x � | X | | M � √ 1 � Goal: prepare | M � = x ∈ M | x � | M | We use 2 reflections: | π � through | M ⊥ � : ref M ⊥ = − ref M (C) ϕ through | π � : ref π (S) | M ⊥ � Grover’s algorithm sin ϕ = � M | π � Prepare | π � (S) s | M | Repeat T 1 × = | X | apply ref M ⊥ √ ε (C) = apply ref π (S) Cost: T 1 ( S + C ) J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 5 / 14

Grover’s algorithm √ 1 We start with | π � = � x ∈ X | x � | X | | M � √ 1 � Goal: prepare | M � = x ∈ M | x � | M | We use 2 reflections: | π � through | M ⊥ � : ref M ⊥ = − ref M (C) ϕ through | π � : ref π (S) ϕ | M ⊥ � Grover’s algorithm sin ϕ = � M | π � Prepare | π � (S) s | M | Repeat T 1 × = | X | apply ref M ⊥ √ ε (C) = apply ref π (S) Cost: T 1 ( S + C ) J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 5 / 14

Grover’s algorithm √ 1 We start with | π � = � x ∈ X | x � | X | | M � √ 1 � Goal: prepare | M � = x ∈ M | x � | M | We use 2 reflections: 2 ϕ | π � through | M ⊥ � : ref M ⊥ = − ref M (C) ϕ through | π � : ref π (S) ϕ | M ⊥ � Grover’s algorithm sin ϕ = � M | π � Prepare | π � (S) s | M | Repeat T 1 × = | X | apply ref M ⊥ √ ε (C) = apply ref π (S) 1 Cost: √ ε ( S + C ) J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 5 / 14

Grover’s algorithm: Comments Quantum analogue of the naive algorithm “pick and check”. 1 1 √ ε ( S + C ) vs ε ( S + C ) = ⇒ Grover’s quadratic speed-up What if S is high? ⇒ Replace ref π by some quantum walk W ! = J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 6 / 14

From random to quantum walks [Szegedy’04] Quantum walk W ( P ) : State space: Pairs of neighbours | x �| y � = ⇒ Walk on edges ( x , y ) Two steps b b b t m Diffusion of y over the neighbours of x b b b b Diffusion of x over the neighbours of y | x �| y � y z n b x We use diffusions ` a la Grover , i.e., reflections √ p yx | y � Superposition over neighbours of x : | p x � = � y ref X : reflection through subspace X = {| x �| p x � : x ∈ X } Similarly for Y We define the quantum walk W as W = ref Y · ref X J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 7 / 14

From random to quantum walks [Szegedy’04] Quantum walk W ( P ) : State space: Pairs of neighbours | x �| y � = ⇒ Walk on edges ( x , y ) Two steps b b b t m Diffusion of y over the neighbours of x b b b b Diffusion of x over the neighbours of y | x �| y � y z n b x We use diffusions ` a la Grover , i.e., reflections √ p yx | y � Superposition over neighbours of x : | p x � = � y ref X : reflection through subspace X = {| x �| p x � : x ∈ X } Similarly for Y We define the quantum walk W as W = ref Y · ref X J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 7 / 14

From random to quantum walks [Szegedy’04] Quantum walk Random walk W = ref Y · ref X P = ( p xy ) E-v (on X ⊕ Y ) : e ± 2 i θ k E-v: λ k = cos θ k Stationary state ( θ 0 = 0) : Stationary dist. (cos θ 0 = 1) : √ π x | x �| p x � | π � = P π = ( π x ) x E-v gap: δ = 1 − | cos θ 1 | phase gap: ∆ = | 2 θ 1 | θ 3 θ 2 θ 1 δ b b b b b − 1 λ 3 λ 2 λ 1 1 J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 8 / 14

From random to quantum walks [Szegedy’04] Quantum walk Random walk W = ref Y · ref X P = ( p xy ) E-v (on X ⊕ Y ) : e ± 2 i θ k E-v: λ k = cos θ k Stationary state ( θ 0 = 0) : Stationary dist. (cos θ 0 = 1) : √ π x | x �| p x � | π � = P π = ( π x ) x E-v gap: δ = 1 − | cos θ 1 | phase gap: ∆ = | 2 θ 1 | 2 θ 3 b 2 θ 2 √ b ∆ = O ( δ ) 2 θ 1 b ⇓ ∆ quantum phase gap b � √ classical e-v gap π 0 � = O b b b J´ er´ emie Roland (UC Berkeley) Search via quantum walk STOC 2007 8 / 14