Finding is as easy as detecting for quantum walks J´ er´ emie Roland Hari Krovi Fr´ ed´ eric Magniez Maris Ozols LIAFA [ICALP’2010, arxiv:1002.2419] J´ er´ emie Roland (NEC Labs) QIP 2011 1 / 14
Spatial search on a graph Setup The problem Move the robot to a Graph G on n vertices X marked vertex x ∈ M Marked vertices: unknown M ⊆ X Complexity: # moves Vertex register: “robot” position Edges: legal moves J´ er´ emie Roland (NEC Labs) QIP 2011 2 / 14
Search via random walk Markov chain on the graph Algorithm Stochastic matrix P = ( p xy ) Start from random x ∼ π p xy � = 0 only if ( x , y ) is an edge Apply P until x is marked Stationary distribution π ( π P = π ) Definition: Hitting time HT ( P , M ) Expected # steps of P until x ∈ M J´ er´ emie Roland (NEC Labs) QIP 2011 3 / 14
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 ) J´ er´ emie Roland (NEC Labs) QIP 2011 4 / 14
Algorithmic applications Grover Search [Grover’95] ◮ Search for a 1 in an n -bit string ◮ G : complete graph Quantum: √ n ◮ Classical: 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 Quantum: n 1 . 3 ◮ Classical: n 2 Others [Buhrman, ˇ ◮ Matrix Multiplication Testing Spalek’06] ◮ Commutativity testing [Magniez,Nayak’05] J´ er´ emie Roland (NEC Labs) QIP 2011 5 / 14
Our main result Theorem Let P be a reversible, ergodic Markov chain π be the (unique) stationary distribution of P ǫ = 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 ǫ is known � HT ( P , M ) × log n steps otherwise Quadratic speed-up for any reversible P ! J´ er´ emie Roland (NEC Labs) QIP 2011 6 / 14
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 (NEC Labs) QIP 2011 7 / 14
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 (NEC Labs) QIP 2011 8 / 14
Absorbing walk Recall: � P UU � P UM Reversible, ergodic Markov chain P P = P MU P MM (unique) stationary distribution π Set of marked elements M : Absorbing walk P ′ � P UU � P UM P ′ = Same as P but self-loops for marked vertices 0 I Stationary distribution π M : π restricted to marked vertices k | π �| 2 |� v ′ = “# steps of P ′ to map π �→ π M ” Hitting time HT ( P , M ) = � λ ′ k � = 1 1 − λ ′ k J´ er´ emie Roland (NEC Labs) QIP 2011 9 / 14
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 J´ er´ emie Roland (NEC Labs) QIP 2011 10 / 14
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 ǫ 1 − s ( 1 − ǫ ) ◮ Similarly, | π ( s ) � = cos φ ( s ) | π U � + sin φ ( s ) | π M � ◮ Rotates from | π � = √ 1 − ǫ | π U � + √ ǫ | 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 J´ er´ emie Roland (NEC Labs) QIP 2011 11 / 14
The algorithm General idea Using quantum phase estimation [Kitaev’95][Cleve,Ekert,Macchiavello,Mosca’98] ◮ 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 ) � Algorithm (known ǫ ) 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 J´ er´ emie Roland (NEC Labs) QIP 2011 12 / 14
Interpolated hitting time “Interpolated hitting time” |� v k ( s ) | π �| 2 HT ( s ) = � = “# steps of P ( s ) to map π �→ π ( s ) ” λ k ( s ) � = 1 1 − λ k ( s ) HT ( s ) = sin 4 φ ( s ) · HT ( P , M ) We show: Proof: By computing the derivatives of P ( s ) and HT ( s ) � � Therefore: Algorithm has cost HT ( s ∗ ) ≤ HT ( P , M ) Case of unknown ǫ : Dichotomic search for s ∗ J´ er´ emie Roland (NEC Labs) QIP 2011 13 / 14
Conclusion Our contribution There exists a quantum algorithm that finds an element in M within ◮ � HT ( P , M ) steps, if ǫ is known ◮ � HT ( P , M ) × log n steps, otherwise Application: 2D-grid, finding an element within ◮ √ n log n steps, if ǫ 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: J´ er´ emie Roland (NEC Labs) QIP 2011 14 / 14
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