Variable time amplitude amplification and quantum algorithms for linear algebra. Andris Ambainis University of Latvia
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1 Vrble tme mpltude mplfcton nd quntum lgorthms for lner lgebr Andrs Ambns Unversty of Ltv
2 Tlk outlne. ew verson of mpltude mplfcton;. Quntum lgorthm for testng f A s sngulr; 3. Quntum lgorthm for solvng A=b (s n HHL08).
3 Vrble tme mpltude mplfcton
4 Ampltude mplfcton [Brssrd, Hoyer, Mosc, Tpp, 00] Algorthm A tht succeeds wth probblty >0. Success s verfble. Incresng success probblty to 3/4: Clssclly: O(/). Quntumly: O(/ ).
5 Serch [Grover, 96]????? Fnd n object wth certn property. Check rndom object: success probblty /. Success probblty 3/4: O(/ )=O( ) repettons.
6 Vrble tme quntum lgorthms Algorthm tht stops t one of severl tmes T,, T k, wth probbltes p,, p k. A A A 3 A k T T T k
7 Ampltude mplfcton A U A- V A- V A Runnng tme:
8 Our result Let T v k p T Quntum lgorthm wth success probblty nd verge runnng tme T v quntum lgorthm wth success probblty /3 nd runnng tme
9 Bsc de A 3 outcomes: success, flure, dd not stop Amplfy success nd dd not stop. Amplfed verson A.
10 Bsc de () A A 3 outcomes: success, flure, dd not stop Amplfy success nd dd not stop. Amplfed verson A.
11 Dffcultes Ampltude mplfcton repeted k tmes; If one mplfcton loses fctor of c, then k mplfctons lose fctor of c k. We need very precse nlyss of mpltude mplfcton.
12 Testng f mtr s sngulr
13 Sngulrty testng Mtr A; Promse A s sngulr or ll sngulr vlues of A re t lest mn. Tsk: dstngush between the two cses.
14 The nturl lgorthm Replce A by A 0 If A hs sngulr vlue, B hs egenvlues. Implement B s Hmltonn. B Use egenvlue estmton for B. 0 A
15 Egenvlue estmton Input: Hmltonn B, stte : B =. Output: estmte for. Assume tht. To obtn estmte wth -, t suffces to use B for tme O(/).
16 Egenvlue estmton Input: Hmltonn B, rndom stte. Output: estmte for rndom, wth -.
17 The nturl lgorthm () B s ether sngulr or hs mn for ll egenvlues. To test B for sngulrty: Choose = mn /3; Egenvlue estmton on rndom. If B sngulr, mn /3, wth probblty /. If B nonsngulr, mn /3. Amplfy mn /3.
18 Runnng tme O(/ mn ) for egenvlue estmton; Amplfcton: repettons; Totl O(/ mn ). Cn we do better f most egenvlues re substntlly more thn mn?
19 Our mprovement Let,,, be the egenvlues of B. Theorem There s quntum lgorthm for sngulrty wth runnng tme where ~ O v v m, mn
20 The lgorthm Generte rndom stte. Run egenvlue estmton severl tmes, wth precson = /3, /6,, mn /3. If estmte stsfes >, stop, output 0. If ll estmtes stsfy, output =0.
21 Runnng tme If >0, the lgorthm stops fter frst egenvlue estmton wth < /. O(/) steps. If =0, the egenvlue estmton s run untl = mn /3. O(/ mn ) steps. Averge runnng tme: m mn,
22 Solvng systems of lner equtons
23 The problem Gven j nd b, fnd. Best clsscl lgorthm: O(.37 ). b b b
24 Obstcles to quntum lgorthm b b b Obstcle : tkes tme O( ) to red ll j. Soluton: query ccess to j. Grover: serch tems wth O() quntum queres. Obstcle : tkes tme O() to output ll.
25 Hrrow, Hssdm, Lloyd, 008 Output = b b b Mesurement wth probblty. Estmtng c +c ++c. Seems to be dffcult clssclly.
26 Hrrow, Hssdm, Lloyd, 008 Runnng tme for producng : O(log c ), but wth dependence on two other prmeters. b b b
27 Runnng tme. Sze of system O(log c ).. Tme to mplement A O() for sprse mtrces A, O() generlly. 3. Condton number of A. k m mn m nd mn bggest nd smllest sngulr vlues of A Tme O c log
28 Emple A = b, A rndom mtr; Lrgest sngulr vlue: O( ); Smllest sngulr vlue: Ω(/ ). Condton number: O(). Runnng tme of HHL: O( log c ) = O( log c ).
29 Emple A = b, A Lplcn of d-dmensonl grd; d, j j jd djcent A j j j d d j j j d, d d 0 otherwse Condton number: O( /d ). HHL runnng tme: O( log c ) = O( 4/d log c ).
30 Our result Theorem There s quntum lgorthm for genertng n tme O(k +o() log c ). [HHL, 008]: (k -o() ) tme requred, unless BQP=PSPACE.
31 The mn des Esy-to-prepre Soluton b b b
32 The mn des A b b b b A b b A b
33 The mn des A b b We desgn physcl system wth Hmltonn A. Untry e A. e A A - v egenvlue estmton.
34 The mn des Let v nd be the egenvectors nd egenvlues of A. c v b c v Implement quntum trnsformton v v b
35 Egenvlue estmton v v ' v ' v ' ' EE EE - ot untry. v ' fl C succ C v ' ' ' Soluton: perform Amplfy.
36 Our lgorthm Observton: suffces to hve such tht -. Run egenvlue estmton severl tmes, wth precson = /3, /6,, mn /3. Stop when <. Generte C C. Amplfy. succ ' ' fl
37 Open problem Wht problems cn we reduce to systems of lner equtons (wth s the nswer)? Emples: Serch; Perfect mtchngs n grph; Grph bprtteness. Bggest ssue: condton number.
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