Upper bounds for query complexity inspired by the Elitzur-Vaidman - PowerPoint PPT Presentation

Upper bounds for query complexity inspired by the Elitzur-Vaidman bomb tester Cedric Yen-Yu Lin, Han-Hsuan Lin Center for Theoretical Physics MIT QIP 2015 January 12, 2015 arXiv:1410.0932 Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 1 / 39

Overview Bomb Query Complexity 1 Elitzur-Vaidman bomb tester Bomb query complexity B ( f ) Main result: B ( f ) = Θ( Q ( f ) 2 ) Algorithms 2 Introduction: O ( N ) bomb query algorithm for OR Main theorem 2: constructing q. algorithms from c. ones Applications: graph problems Summary and open problems 3 Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 2 / 39

Section 1 Bomb Query Complexity Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 3 / 39

Elitzur-Vaidman Bomb Tester [EV93] A collection of bombs, some of which are duds Live: Explodes on contact with photon Dud: No interaction with photon Can we tell them apart without blowing ourselves up? Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 4 / 39

Elitzur-Vaidman Bomb Tester [EV93] We can put a bomb in an Mach-Zehnder interferometer: If D2 detects a photon, then we know the bomb is live, even though it has not exploded. Image source: A. G. White et al., PRA 58, 605 (1998). Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 5 / 39

EV bomb in circuit model We can rewrite the Elitzur-Vaidman bomb in the circuit model: • | 0 � explode if 1 I or X Live bomb: X in the above diagram Dud: I in the above diagram Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 6 / 39

Quantum Zeno Effect [KWH+95] � cos θ � − sin θ Let R ( θ ) = exp ( i θ X ) = . sin θ cos θ | 0 � R ( θ ) • R ( θ ) • . . . | 0 � | 0 � I or X I or X π/ ( 2 θ ) times in total Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 7 / 39

Quantum Zeno Effect [KWH+95] � cos θ � − sin θ Let R ( θ ) = exp ( i θ X ) = . sin θ cos θ | 0 � R ( θ ) • R ( θ ) • | 1 � . . . | 0 � | 0 � I I π/ ( 2 θ ) times in total If dud: Ctrl- I does nothing, so | 0 � gets rotated to | 1 � . Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 8 / 39

Quantum Zeno Effect [KWH+95] � cos θ � − sin θ Let R ( θ ) = exp ( i θ X ) = . sin θ cos θ | 0 � R ( θ ) • R ( θ ) • | 0 � . . . | 0 � | 0 � X X π/ ( 2 θ ) times in total If live: First register is projected back to | 0 � on each measurement. Probability of explosion: Θ( θ 2 ) × Θ( 1 /θ ) = Θ( θ ) . Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 9 / 39

Quantum Zeno Effect [KWH+95] � cos θ � − sin θ Let R ( θ ) = exp ( i θ X ) = . sin θ cos θ | 0 � R ( θ ) • R ( θ ) • . . . | 0 � | 0 � I or X I or X π/ ( 2 θ ) times in total Probability of explosion: Θ( θ ) Number of queries: Θ( 1 /θ ) Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 10 / 39

Quantum Query Quantum query | r � | r ⊕ x i � O x | i � | i � Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 11 / 39

Quantum Query vs Bomb Query Quantum query | r � | r ⊕ x i � O x | i � | i � Bomb query | c � • | c � | 0 � explodes if c · x i = 1 bomb O x | i � | i � Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 11 / 39

Bomb Query | c � • | c � | 0 � explodes if c · x i = 1 bomb O x | i � | i � Differences from quantum query: Extra control register c . The record register, where we store the query result, must contain 0 as input. We must measure the query result after each query; if the result is 1, the bomb explodes and the algorithm fails. Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 12 / 39

Bomb Query | c � • | c � | 0 � explodes if c · x i = 1 bomb O x | i � | i � equivalent to | c � • | c � | i � P x , 0 ( 1 − c · x i ) | i � where � � P x , 0 = | i �� i | , Ctrl − P x , 0 = I − | 1 , i �� 1 , i | x i = 0 x i = 1 Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 13 / 39

Bomb Query Complexity • • • P x , 0 P x , 0 P x , 0 U 0 U 1 U 2 U 3 . . . Call the minimum number of bomb queries needed to determine f with bounded error, with probability of explosion ≤ ǫ , the bomb query complexity B ǫ ( f ) . Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 14 / 39

Main Theorem Theorem B ǫ ( f ) = Θ( Q ( f ) 2 /ǫ ) . Upper bound: Quantum Zeno effect. Lower bound: Adversary method. Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 15 / 39

B ǫ ( f ) = O ( Q ( f ) 2 /ǫ ) : Proof We can simulate each quantum query using Θ( 1 /θ ) bomb queries: | r � | r ⊕ x i � X | 0 � R ( θ ) • • R ( − θ ) • | 0 � (discard) P x , 0 P x , 0 | i � | i � repeat π/ 2 θ times repeat π/ 2 θ times Total probability of explosion: Θ( θ ) · Q ( f ) = Θ( ǫ ) , if θ = Θ( ǫ/ Q ( f )) . Total number of bomb queries: Θ( 1 /θ ) · Q ( f ) = O ( Q ( f ) 2 /ǫ ) . Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 16 / 39

B ǫ ( f ) = Ω( Q ( f ) 2 /ǫ ) : Proof The proof uses the general-weight adversary method [HLS07]. We know [Rei09,Rei11,LMR+11] that the general-weight adversary bound tightly characterizes quantum query complexity: Adv ± ( f ) = Θ( Q ( f )) . By modifying the proof of the general-weight adversary bound, we can show that B ǫ ( f ) = Ω( Adv ± ( f ) 2 /ǫ ) . This implies that B ǫ ( f ) = Ω( Q ( f ) 2 /ǫ ) . Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 17 / 39

Section 2 Algorithms Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 18 / 39

O ( N ) Bomb Query Algorithm for OR There are N bombs, want to check if any are live. Check each bomb using Θ( ǫ − 1 ) queries, or O ( N /ǫ ) queries in total. Each live bomb has Θ( ǫ ) chance of exploding. Each dud has no chance of exploding. Since we can stop at the first live bomb, the total chance of failure is only Θ( ǫ ) . Therefore B ǫ ( OR ) = O ( N /ǫ ) . Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 19 / 39

√ Since B ( OR ) = O ( N ) , Q ( OR ) = O ( N ) . This is a nonconstructive proof of the existence of Grover’s algorithm! Can we generalize this further? Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 20 / 39

Main Theorem 2 Theorem Suppose there is a classical randomized algorithm A that computes f ( x ) using at most T queries. Moreover, suppose there is an algorithm G that predicts the results of each query A makes (0 or 1), making at most an expected G mistakes. √ Then B ( f ) = O ( TG ) , and Q ( f ) = O ( TG ) . Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 21 / 39

Main Theorem 2 Theorem Suppose there is a classical randomized algorithm A that computes f ( x ) using at most T queries. Moreover, suppose there is an algorithm G that predicts the results of each query A makes (0 or 1), making at most an expected G mistakes. √ Then B ( f ) = O ( TG ) , and Q ( f ) = O ( TG ) . √ For example, for OR we have T = N and G = 1, so Q ( f ) = O ( N ) . Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 21 / 39

Bomb algorithm with B ( f ) = O ( TG ) For each classical query, check whether G correctly predicts the query result of A using Θ( G /ǫ ) bomb queries. If G guesses incorrectly then the probability of explosion is O ( ǫ/ G ) ; otherwise it is zero. (This actually requires defining an equivalent symmetric variant of the bomb query complexity.) The total probability of explosion is O ( ǫ/ G ) · G = O ( ǫ ) , and the number of bomb queries used is O ( G /ǫ ) · T = O ( TG /ǫ ) . Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 22 / 39

√ Explicit q. algorithm with Q ( f ) = O ( TG ) Repeat until all queries of A are determined: Use G to predict all remaining queries of A , under assumption it 1 makes no mistakes. � Search for the location d j of first mistake, using O ( d j − d j − 1 ) 2 quantum queries. This determines the actual query results up to the d j -th query that 3 A would have made. Kothari’s algorithm for oracle identification [Kot14] actually already uses these steps above. Cedric Lin, Han-Hsuan Lin (MIT) Upper bounds inspired by EV bomb tester January 12, 2015 23 / 39