Quantum algorithms based on quantum walks . J er emie Roland - PowerPoint PPT Presentation

. Quantum algorithms based on quantum walks . J´ er´ emie Roland Universit´ e Libre de Bruxelles Quantum Information & Communication J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 1 / 39

Outline Preliminaries ▶ Classical random walks ▶ Three search algorithms based on random walks Search algorithms based on quantum walks ▶ From random to quantum walks [Szegedy’04] ▶ Grover’s algorithm: Complete graph [Grover’95] ▶ Element Distinctness: Johnson graph [Ambainis’04] ▶ Generalized search algorithm via quantum walk [Magniez,Nayak,Roland,Santha’07] ▶ Quantum hitting time: Detecting vs finding [Szegedy’04],[Krovi,Magniez,Ozols,Roland’10] Other algorithms based on quantum walks ▶ Exponential speed-up: ”Glued trees” [Childs,Cleve,Deotto,Farhi,Gutmann,Spielman’03] ▶ Scattering-based algorithms: Formula evaluation [Farhi,Goldstone,Gutmann’07] ▶ Universal quantum computation by quantum walks [Childs’09,Childs,Gossett,Webb’12] Conclusion J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 2 / 39

Random walk on a graph . Stochastic matrix P = ( p xy ) . p xy ̸ = 0 only if ( x , y ) is an edge Eigenvalues 1 = λ 0 > λ 1 ≥ . . . ≥ λ n − 1 > − 1 (Assume P ergodic, symmetric) Stationary distribution: π P = π ( π uniform if P symmetric) Eigenvalue gap δ = 1 − λ 1 . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 5 / 39

Mixing time . Definition: Mixing time . Mixing time MT ( P ) : Number of steps necessary to approach π MT ( P ) ≤ 1 δ , where δ = 1 − λ 1 is the eigenvalue gap . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 6 / 39

Hitting time . Definition: Hitting time . Let M be a set of marked vertices Assume we start from a random vertex x ∼ π Hitting time HT ( P , M ) : Expected number of steps to reach m ∈ M εδ , where ε = | M | 1 HT ( P , M ) ≤ | X | . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 7 / 39

Abstract search problem (classical) . The problem . Available procedures . Input: . Setup (cost S): a set of elements X pick a random x ∈ X with unknown subset of Check (cost C): marked elements M ⊆ X check whether x ∈ M ( ) ε = | M | | X | Update (cost U): Output: make a random walk P . a marked element x ∈ M . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 9 / 39

Three (classical) search algorithms . Naive algorithm . . Random walk I Repeat ( 1 ε ) × . Pick random x ∈ X (S) Pick random x ∈ X (S) Repeat ( 1 ε ) × Check whether x ∈ M (C) ▶ Check whether x ∈ M (C) Cost: 1 ε ( S + C ) ▶ Repeat ( 1 . δ ) × ⋆ Random walk (U) . Idea: Use random walk! Cost: S + 1 ε ( 1 δ U + C ) . . MT × random walk ≈ pick random x . (mixing time MT ≤ 1 . δ ) ) Random walk II . Pick random x ∈ X (S) Repeat HT × (hitting time HT ≤ 1 εδ ) ) ▶ Check whether x ∈ M (C) ▶ Random walk (U) Cost: S + HT ( U + C ) . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 10 / 39

Observations Naive algorithm ▶ Samples a new independent vertex at each step ⇒ Equivalent to walk on complete graph with U = S ▶ p xy = 1 n for all x , y ▶ Eigenvalues: λ 0 = 1 and λ i = 0 for all i ̸ = 0 Random walk I ▶ Repetitions of an ergodic walk approximates the walk on the complete graph ▶ Mathematically: For large T , λ T i → 0 whenever | λ i | < 1 J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 11 / 39

Abstract search problem (quantum) . . Two related problems Available procedures . . Input: Setup (cost S): 1 √ ∑ prepare | π ⟩ = x | x ⟩ a set of elements X | X | with unknown subset of Check (cost C): marked elements M ⊆ X reflection / marked elements Output: { | x ⟩ if x ∈ M ref M : | x ⟩ �→ . . −| x ⟩ otherwise Find a marked element x ∈ M 1 Update (cost U): . . Detect whether there is a 2 apply quantum walk W marked element ( M = ∅ ?) . . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 13 / 39

Grover’s algorithm [Grover’95] 1 √ We start with | π ⟩ = ∑ x ∈ X | x ⟩ | X | | M ⟩ √ 1 Goal: prepare | M ⟩ = ∑ x ∈ M | x ⟩ | M | We use 2 reflections: 2 ϕ ▶ through | M ⊥ ⟩ : ref M ⊥ = − ref M (C) | π ⟩ ▶ through | π ⟩ : ref π ϕ (S) ϕ | M ⊥ ⟩ . Grover’s algorithm . sin ϕ = ⟨ M | π ⟩ Prepare | π ⟩ (S) √ | M | Repeat T × = | X | √ ε ▶ apply ref M ⊥ (C) = ▶ apply ref π (S) 1 Cost: T √ ε ( S + C ) . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 15 / 39

Grover’s algorithm: Observations 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 (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 16 / 39

From random to quantum walks [Szegedy’04] . Random walk P on edges ( x , y ) . Acts on two registers: position x and coin y ▶ Flip the coin y over the neighbours of x Walk in two steps: ▶ Swap x and y . . Quantum analogue W ( P ) . Acts on two registers | x ⟩| y ⟩ y ′ √ p y ′ x | y ′ ⟩ ▶ reflection of | y ⟩ through | p x ⟩ = ∑ Walk in two steps: ▶ Swap the | x ⟩ and | y ⟩ registers . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 18 / 39

Spectral correspondance [Szegedy’04] . . Random walk Quantum walk . . P = ( p xy ) W ( P ) = S WAP · ref X E-v: e ± i θ k E-v: λ k = cos θ k Stationary dist. (cos θ 0 = 1) : Stationary state ( θ 0 = 0) : √ π x | x ⟩| p x ⟩ π = ( π x ) | π ⟩ = ∑ x √ E-v gap: δ = 1 − | cos θ 1 | phase gap: ∆ = | θ 1 | = Θ( δ ) . . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 19 / 39

Back to Grover’s algorithm . . Grover’s algorithm Use quantum walk? . . Prepare | π ⟩ (S) Replace ref π by Repeat O ( 1 √ ε ) × − → T × quantum walk??? Reflection / marked ref M (C) ( ) T = O ( 1 1 ∆ ) = O ( δ ) √ . Reflection / uniform ref π (S) . 1 Cost: √ ε ( S + C ) Tentative quantum walk . . Prepare | π ⟩ (S) . Repeat O ( 1 √ ε ) × What if S is high??? . Reflection / marked ref M (C) Repeat O ( 1 δ ) × √ ▶ Quantum walk (U) √ ε ( 1 1 Cost: S + δ U + C ) √ . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 21 / 39

b b b b b b b b b b b b b Use repetitions of quantum walk instead of reflection? . . Random walk P : eigenvalues . Quantum walk W : eigenvalues . µ k = e 2 i θ k λ k = cos θ k | λ k | ≤ 1 − δ | 1 − µ k | ≤ ∆ θ 3 µ 3 θ 2 µ 2 P W θ 1 µ 1 ∆ − 1 λ 3 λ 2 λ 1 1 − 1 1 δ × ( λ k ) T ≈ 0 T = O ( 1 δ ) Problem: ( µ k ) T − T →∞ − 1 − − − → P T ref π − 1 0 1 − 1 1 P T ≈ walk on complete graph W T does not simulate ref π ! J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 22 / 39

b b b b b b b b b b b b b b Ambainis’ quantum walk [Ambainis’04] C π ∆ W T W C C ∆ × C ∆ ∆ − → π 0 π 0 If W has eigenvalues e i θ k , W T has eigenvalues e iT θ k Suppose that ∃ C ≤ π such that ∀ k ̸ = 0: ∆ ≤ θ k ≤ π ∆ − π ∆ C ≤ θ k ≤ − ∆ or C ∆ , then W T has eigenvalue gap ∆ ′ = C If T = C J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 23 / 39

Quantum walk for Element Distinctness [Ambainis’04] . Idea . Replace ref π by W T in Grover’s algorithm, with T = O ( 1 δ ) √ . Works under some assumptions: W T must have constant gap Ω( C ) ⇒ OK for Johnson graphs (Element Distinctness) Unique solution Properties: Finds a marked element Cost S + 1 √ ε ( 1 √ U + C ) δ . What about other graphs? . J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 24 / 39

b b b b b b b b b Indirect simulation of the reflection . Idea . Using W , simulate ref π to use Grover’s algorithm . W ref π ∆ − 1 1 − 1 1 W | π ⟩ = | π ⟩ ref π | π ⟩ = | π ⟩ W | ψ k ⟩ = e i θ k | ψ k ⟩ ref π | ψ k ⟩ = −| ψ k ⟩ We need a procedure to discriminate between eigenstates | ψ k ⟩ with | θ k | ≥ ∆ | π ⟩ with θ 0 = 0. We use quantum phase estimation! [Kitaev’95, Cleve et al ’98] J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 25 / 39

Using phase estimation [Magniez,Santha,Roland,Nayak’07] √ Discriminating between phases 0 and ≥ ∆ has a cost O ( 1 / ∆) = O ( 1 / δ ) ⇒ We obtain = Search algorithm via quantum walk from any ergodic Markov chain Total cost: S + 1 √ ε ( 1 √ U + C ) δ finds marked elements No assumption on the number of marked elements J´ er´ emie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 26 / 39