Finding is as easy as detecting for quantum walks

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1 Finding is as easy as detecting for quantum walks Jérémie Roland Hari Krovi Frédéric Magniez Maris Ozols LIAFA [ICALP 2010, arxiv: ] Jérémie Roland (NEC Labs) QIP / 14

2 Spatial search on a graph Setup Graph G on n vertices X Marked vertices: unknown M X Vertex register: robot position Edges: legal moves The problem Move the robot to a marked vertex x M Complexity: # moves Jérémie Roland (NEC Labs) QIP / 14

3 Search via random walk Markov chain on the graph Stochastic matrix P = (p xy ) p xy 0 only if (x, y) is an edge Stationary distribution π (πp = π) Algorithm Start from random x π Apply P until x is marked Definition: Hitting time HT(P, M) Expected # steps of P until x M Jérémie Roland (NEC Labs) QIP / 14

4 Quantum case: Related work Quantum walks Complete graph [Grover 95] Hypercube [Shenvi,Kempe,Wayley 03] Johnson Graph [Ambainis 04] 2D-grid [Ambainis,Kempe,Rivosh 05] Quantum analogue W(P) of Markov chain P [Szegedy 04] Quantum hitting time Detecting marked elements: HT(P, M) [Szegedy 04] Finding marked elements for state-transitive P and M = 1: HT(P, M) [Tulsi 08][Magniez,Nayak,Richter,Santha 09] Question Is finding as easy as detecting for quantum walks? QHT(P, M)? = HT(P, M) Jérémie Roland (NEC Labs) QIP / 14

5 Algorithmic applications Grover Search [Grover 95] Search for a 1 in an n-bit string G: complete graph Classical: n Quantum: n Extends to G hypercube and unique marked element ( M = 1) Element Distinctness [Ambainis 04] Search for equal elements in a set of n elements G: Johnson graph Classical: n Quantum: n 2/3 Triangle Finding [Magniez,Santha,Szegedy 05] Search for a triangle in a graph with n vertices G: Johnson graph Classical: n 2 Quantum: n 1.3 Others Matrix Multiplication Testing [Buhrman,Špalek 06] Commutativity testing [Magniez,Nayak 05] Jérémie Roland (NEC Labs) QIP / 14

6 Our main result Theorem Let P be a reversible, ergodic Markov chain π be the (unique) stationary distribution of P ɛ = Pr π (M) be the probability of marked elements Then, there exists a quantum algorithm that finds an element in M within HT(P, M) steps if ɛ is known HT(P, M) log n steps otherwise Quadratic speed-up for any reversible P! Jérémie Roland (NEC Labs) QIP / 14

7 From random to quantum walks [Szegedy 04] Random walk P on edges (x, y) Acts on two registers: position x and coin y Walk in two steps: Quantum analogue W(P) Acts on two registers x y Flip the coin y over the neighbours of x Swap x and y reflection of y through px = Walk in two steps: y py x y Swap the x and y registers Jérémie Roland (NEC Labs) QIP / 14

8 Spectral correspondance Random walk P = (p xy ) E-v: λ k = cos θ k Stationary dist. (cos θ 0 = 1): π = (π x ) E-v gap: δ = 1 cos θ 1 [Szegedy 04] Quantum walk W(P) = SWAP ref X E-v: e ±iθ k Stationary state (θ 0 = 0): π = x πx x p x phase gap: = θ 1 = Θ( δ) Jérémie Roland (NEC Labs) QIP / 14

9 Absorbing walk Recall: ( ) PUU P Reversible, ergodic Markov chain P P = UM P MU P MM (unique) stationary distribution π Set of marked elements M: Absorbing walk P Same as P but self-loops for marked vertices P = Stationary distribution π M : π restricted to marked vertices Hitting time HT(P, M) = λ k 1 v k π 2 1 λ k ( ) PUU P UM 0 I = # steps of P to map π π M Jérémie Roland (NEC Labs) QIP / 14

10 Quantum analogues of P and P Absorbing walk P HT(P, M) iterations of W(P ) make π deviate by angle Ω(1) Good for detecting if M is non-empty [Szegedy 04] But: state may remain far from marked elements Can be fixed for state-transitive P, M = 1 Difficult analysis, less intuition [Tulsi 08][Magniez,Nayak,Richter,Santha 09] Original walk P Extends Grover s algorithm for any graph Good for finding [Ambainis 04][Magniez,Nayak,Roland,Santha 07] But: in general, # steps can be HT(P, M) New approach: mixture of P and P Finds marked elements for any reversible P, and any M Better intuition, simpler analysis Jérémie Roland (NEC Labs) QIP / 14

11 Interpolation between P and P P(s) = (1 s)p + sp Unmarked vertices: apply P Marked vertices: apply P with probability 1 s, otherwise self-loop Stationary distribution π(s) = (cos 2 φ(s))π U + (sin 2 φ(s))π M where φ(s) = arcsin ɛ 1 s(1 ɛ) Similarly, π(s) = cos φ(s) πu + sin φ(s) π M Rotates from π = 1 ɛ πu + ɛ π M to π M Reminiscent of adiabatic quantum computing Indeed, we can also design an adiabatic algorithm [Krovi,Ozols,R. 10, PRA] Note: Interpolation at the classical level Jérémie Roland (NEC Labs) QIP / 14

12 The algorithm General idea Using quantum phase estimation We can measure in the eigenbasis of W(P(s)) At a cost HT(s) (see later) W(P(s)) has unique 1-eigenvector π(s) Measuring phase 0 projects onto π(s) [Kitaev 95][Cleve,Ekert,Macchiavello,Mosca 98] Algorithm (known ɛ) Prepare π Project onto π(s ) = 1 2 ( π U + π M ) succeeds with prob. 1/2 Measure current vertex marked with prob. 1/2 Jérémie Roland (NEC Labs) QIP / 14

13 Interpolated hitting time Interpolated hitting time HT(s) = λ k (s) 1 v k (s) π 2 1 λ k (s) = # steps of P(s) to map π π(s) We show: HT(s) = sin 4 φ(s) HT(P, M) Proof: By computing the derivatives of P(s) and HT(s) Therefore: Algorithm has cost HT(s ) HT(P, M) Case of unknown ɛ: Dichotomic search for s Jérémie Roland (NEC Labs) QIP / 14

otherwise Open problems Hitting time Can we beat the quadratic improvement?

14 Conclusion Our contribution There exists a quantum algorithm that finds an element in M within HT(P, M) steps, if ɛ is known HT(P, M) log n steps, otherwise Application: 2D-grid, finding an element within n log n steps, if ɛ is known n log n steps, otherwise Open problems Hitting time Can we beat the quadratic improvement? Mixing time Can we also mix quadratically faster using quantum walks? Very few results for Cayley graphs [Aharonov,Ambainis,Kempe,Vazirani 01] Support: Jérémie Roland (NEC Labs) QIP / 14

Quantum algorithms based on quantum walks. Jérémie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre / 39

Quantum algorithms based on quantum walks Jérémie Roland Université Libre de Bruxelles Quantum Information & Communication Jérémie Roland (ULB - QuIC) Quantum walks Grenoble, Novembre 2012 1 / 39 Outline