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Quantum computing random walks and adiabatic computation 5346768

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Outline Quantum random walks: discrete model – motivation, overview, results hitting time on the hypercube routing a quantum random walk search algorithm implementations decoherence Adiabatic computation: model and equivalence to the quantum circuit model universality on a 2D lattice, relation to Kitaev’s QMA Hamiltonian J. Kempe: "Quantum Random Walks Hit Exponentially Faster"

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Quantum computing random walks and adiabatic computation 5346768

  1. 1. New tools for Quantum Computing – Random walks and Adiabatic computation Julia Kempe CNRS & Dept. of Computer Science Univ. de Paris-Sud, Orsay, France 2 1 + 2 1 Speed -up Hi+1j Hij Hij+1 H’ij H H H H H H H H 000 010 100 101 111011 RESQ-Meeting, MPI, Munich, May 12, 2003
  2. 2. Outline Quantum random walks: • discrete model – motivation, overview, results • hitting time on the hypercube • routing • a quantum random walk search algorithm • implementations • decoherence Adiabatic computation: • model and equivalence to the quantum circuit model • universality on a 2D lattice, relation to Kitaev’s QMA Hamiltonian J. Kempe: "Quantum Random Walks Hit Exponentially Faster", quant-ph/0205083 N. Shenvi, J. Kempe, and K.B. Whaley: "A Quantum Random Walk Search Algorithm", Phys. Rev. A, to appear, quant-ph/0210064 D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, O. Regev: "Universality of Adiabatic Quantum Computation with Two-Body Interactions", in preparation 000 010 100 101 111011 Hi+1j Hij Hij+1 H’ij H H H H H H H H
  3. 3. Motivation - Quantum algorithms • Deutsch-Jozsa algorithm (’92): determines if a function constant or 2-1 with only one query • Simon ’s algorithm (’94): period finding • Shor (’95): efficient factoring • general problem (factoring, discrete log) = hidden subgroup: – Input: function f: G ® G s.t. f(x)=f(x+H) where H< G – Output: H (generators) – efficient quantum algorithm if G - Abelian or « special » • Grover (’96): Search of one entry in a database of size N with queries (Classical lower bound W(N)) • Pell’s equation (’02): (Hallgren): stronger than )( NO
  4. 4. Motivation - Quantum algorithms • Deutsch-Jozsa algorithm (’92): determines if a function constant or 2-1 with only one query • Simon ’s algorithm (’94): period finding • Shor (’95): efficient factoring • general problem (factoring, discrete log) = hidden subgroup: – Input: function f: G ® G s.t. f(x)=f(x+H) where H< G – Output: H (generators) – efficient quantum algorithm if G - Abelian or « special » • Grover (’96): Search of one entry in a database of size N with queries (Classical lower bound W(N)) • Pell’s equation (’02): (Hallgren): stronger than )( NO QFT! QFT! QFT! QFT! QFT!
  5. 5. QFT omnipresent in quantum algorithms Search for new quantum algorithms: • (Nearly) all efficient quantum algrithms are based on (2) applications of the QFT • New algorithmic tools???
  6. 6. QFT omnipresent in quantum algorithms Search for new quantum algorithms: • (Nearly) all efficient quantum algrithms are based on (2) applications of the QFT • New algorithmic tools??? Better amenable implementation physical system
  7. 7. New tools Quantum random walks • Continuous: Farhi, Gutmann – PRA’98, – Exponential oracle speed-up: Childs, Cleve, Deotto, Farhi, Gutmann, Spielman - STOC’03, based on Childs et al. ’01 – Simulable by quantum circuit: Aharonov and Ta-Shma – STOC’03 • Discrete: Watrous CCC’99, Aharonov, Ambainis, Kempe, Vazirani -STOC’01, Ambainis, Bach, Nayak, Vishwanath, Watrous - STOC’01, … … !!! Adiabatic computation • Model: Farhi, Goldstone, Gutmann, Sipser – quant-ph‘00 • More examples, different paths, … Farhi et al.’00-02, Childs et al. ‘00,… • Failure in certain scenarios, simulable by quantum circuit van Dam, Mosca, Vazirani – FOCS’01
  8. 8. Quantum Random Walks
  9. 9. Markov chains as an algorithmic tool Markov chains for algorithms: – rapidly mixing Markov chains - standard technique to solve combinatorial optimisation and counting problems – at the base of numerous efficient approximation algorithms for optimisation and #P-complete counting problems (ex. approximating the permanent, M.Jerrum & A.Sinclair) Idea: construct a Markov chain (simple, local transitions only) – (1) whose stationary distribution gives the solution to the problem Þ Mixing time – or (2) which hits the desired solution Þ Hitting time « Quantum » Markov chains ?
  10. 10. Example: Random walk for 2SAT Input: Boolean formula F (conjunction of clauses of 2 variables) in X1, … , Xn (ex. ) Question: Is F satisfaisable? (ex. YES, FFT is satisfying assignment) Algorithm: 1) initialise the variables u.a. random (T- ¨true¨, F-¨false¨) 2) if all clauses satisfied – STOP, otherwise: 3) chose a non-satisfied clause, chose one of its two variables and flip its value; return to 2) )()()()( 31323121 XXXXXXXX ¬Ú¬ÙÚÙÚ¬Ù¬Ú=F
  11. 11. Example: Random walk for 2SAT Algorithm: 1) initialise the variables u.a. random (T- ¨true¨, F-¨false¨) 2) if all clauses satisfied – STOP, otherwise: 3) chose a non-satisfied clause, chose one of its two variables and flip its value; return to 2) FFT TFT FFFFTT FTFTFFTTT TTF 0 1 2 3 STOP>1/2 >1/2 >1/2 <1/2 <1/2 Hamming distance Random walk on a line with n+1 vertices ! After t=2n2 repetitions (« Hitting time ») the succes probability is >1/2 (if F satisfiable).
  12. 12. Random Walks... • Random walk on the line: Mixing time=Hitting time =O(n2) • 3SAT - “biased” random walk with exponential hitting time • in general : local, simple Markov chain on exponential domain 0 1 >1/3 <2/3 2 >1/3 <2/3<2/3 >1/3 3 4 >1/3 <2/3 5 <2/3 >1/3 STOP 1/21/2 O(n2)
  13. 13. 1/21/2 2 1 + 2 1 ? The discrete time quantum walk Classical Quantum A. Ambainis, D. Aharonov, J. K. , U. Vazirani, “Quantum walks on graphs”, STOC’01 A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, J. Watrous, “One-dim. quantum walks”, STOC’01
  14. 14. Quantum random walk • Meyer [‘97]: All local, translationary invariant unitary matrices are simple translations. Incorporate coin-flip into walk! “R” “L”
  15. 15. The discrete time quantum walk • Classical random walk • Flip coin for direction • Walk conditioned on outcome • Quantum “coined” walk: {|®ñ,|¬ñ} Ä – ”flip” direction coin C (e.g. H) – perform controlled shift S : |®ñ Þ “R” |¬ñ Þ “L” “R” “L” U collapses to the classical random walk if we measure directions or positions at every step! ( )U C I S= Ä !
  16. 16. General Graphs {|®ñ, |®ñ ,|®ñ} ÄHV ( )U C I S= Ä ! ( , ) blue ( , ) red ( , ) black u u u u v u v u v S v v v= Ä + Ä + Äå å å Can be implemented efficiently if the corresponding classical walk can be implemented efficiently. |®ñá®| |®ñá®| |®ñá®| Ex:
  17. 17. Quantum random walks induces probability-dist. Pt(i) on sites (after measuring) Example: |0ñ |1ñ |2ñ |-1ñ U 0t ® Ä 0® Ä ( ) ( )1 1 2 0 0 2H shift ¾¾® ® + ¬ Ä + ® - ¬ Ä - ¾¾® ® Ä + ¬ Ä + ® Ä - ¬ Ä - ( ) 0 1 1H shift ¾¾® ® + ¬ Ä ¾¾® ® Ä + ¬ Ä - ( ) ( )2 0 22H shift ¾¾® ® + ¬ Ä + ® Ä - ® - ¬ Ä - ¾¾® Interference, cancellation 3 1 1 3 ...3 H ¬ Ä + ® Ä - ® Ä - - ¬ Ä - ¾¾® Probability distribution after 3 steps: Classical: Quantum: -3 31-1 1/8 3/8 3/8 1/8 -3 31-1 1/12 3/4 1/12 1/12
  18. 18. Quantum random walks induces probability-dist. Pt(i) on sites (after measuring) Convergence? NO! U is unitary Þ reversible! (no stationary distrib.) Def. “averaged distribution” Qt (Cesaro limit): Theorem: Qt converges to a stationary distribution, . Total variation distance: Mixing time: te - time t until Dt<e. |0ñ |1ñ |2ñ |-1ñ U 0t ® Ä t t s s 0 1 Q (v) P v t ( ) = = å t t lim Q p ®¥ ® tπ (i) π(i)t 1 2 t i Q Q ÎW D = - = -å
  19. 19. Results on mixing time* Cycle: – quantum walk converges towards uniform distribution – Mixing time: • classical: e = q(N2 log(1/e)) • quantum: e =O(N log N / e3) τ τ *D. Aharonov,A. Ambainis, J.K., U.Vazirani, “Quantum walks on graphs”, STOC’01 with «Warmstart» 1 ( log log( ))O N Net e =
  20. 20. Results on mixing time* Cycle: – quantum walk converges towards uniform distribution – Mixing time: • classical: e = q(N2 log(1/e)) • quantum: e =O(N log N / e3) Similar results in higher dimensions, for Cayley graphs, graphs on abelian groups, walks with different coins,… τ τ *D. Aharonov, A. Ambainis, J.K., U.Vazirani, “Quantum walks on graphs”, STOC’01 with «Warmstart» 1 ( log log( ))O N Net e =
  21. 21. Results on mixing time* Cycle: – quantum walk converges towards uniform distribution – Mixing time: • classical: e = q(N2 log(1/e)) • quantum: e =O(N log N / e3) Similar results in higher dimensions, for Cayley graphs, graphs on abelian groups, walks with different coins,… Observation: classically: mixing time depends on second largest eigenvalue of M quantum: mixing time depends on all eigenvalues of U τ τ *D. Aharonov, A. Ambainis, J.K., U.Vazirani, “Quantum walks on graphs”, STOC’01 with «Warmstart» 1 ( log log( ))O N Net e =
  22. 22. Results on mixing time* Conductance-type lower bound for mixing time of any quantum walk on bounded degree graph: capacitance flow conductance: Theorem (Jerrum,Sinclair’89): 1 ( ) d t = W F( ) ( )2 1 O 1tW F £ £ F Classical: Quantum: d-max.degree *D. Aharonov, A. Ambainis, J.K., U.Vazirani, “Quantum walks on graphs”, STOC’01 X u u X C p Î = å , , X u v u u X v X F p p Î Ï = å X GÌ 0 1 2 min X X X G XC F C< < £ F = 2 2(1 ) 2 2 l F £ - £ F At most quadratic speed-up!
  23. 23. Quantum Hitting Time on Hypercube • Space: 000 010 100 101 111011 ({ 1 ,..., } { : {0,1} })n n z zÄ Î C. Moore and A.Russel, “Quantum Walks on the Hypercube”, RANDOM’02 { , , }« Ä!
  24. 24. Quantum Hitting Time on Hypercube • Space: • Walk: – Conditional Shift – Coin C (respects permutational symmetry of hypercube) 000 010 100 101 111011 ({ 1 ,..., } { : {0,1} })n n z zÄ Î : iS i z i z eÄ ® Ä Å !00..01 00..0i i e = ... a b b b a b b C b b a æ ö ç ÷ ç ÷= ç ÷ ç ÷ç ÷ è ø ! " # " $ 2 21a bn n= - = C. Moore and A.Russel, “Quantum Walks on the Hypercube”, RANDOM’02 { , , }« Ä!
  25. 25. Quantum Hitting Time on Hypercube • Space: • Walk: – Conditional Shift – Coin C (respects permutational symmetry of hypercube) • Initial state: 000 010 100 101 111011 { , , }« Ä! ({ 1 ,..., } { : {0,1} })n n z zÄ Î : iS i z i z eÄ ® Ä Å !00..01 00..0i i e = ... a b b b a b b C b b a æ ö ç ÷ ç ÷= ç ÷ ç ÷ç ÷ è ø ! " # " $ 2 21a bn n= - = 1 1 (0) 00...0 n i i n y = = Äå Symmetric superposition over all directions “Grover coin” C. Moore and A.Russel, “Quantum Walks on the Hypercube”, RANDOM’02
  26. 26. Hitting time? • Dilemma: constant measurement of position will collapse U to the classical walk… Two options: One-shot q-hitting-time (T,p): – Measure only at time T – “Hits” desired target-state x with probability >p Concurrent q-hitting-time (T,p): – Partial measurement (“Am I at x/Am I not at x?”) at all times
  27. 27. Results on hitting time* • Classical: from v to opposite v’ hitting-time exponential • Quantum: – One-shot hitting-time from v to v’ (T,p) *JK. quant-ph/0205083 n 1T=2 (1 ( ))O n+ 000 010 100 101 111011 3 log T= n p=1- O 2 n n p æ ö Þ ç ÷ è ø
  28. 28. Results on hitting time* • Classical: from v to opposite v’ hitting-time exponential • Quantum: – One-shot hitting-time from v to v’ (T,p) Need to know with accuracy when to measure, success »1 in linear time! – Concurrent hitting-time from v to v’ (T,p) No information on when to measure needed, with amplification success »1 in T=O(n2)! *J.K.’02 T ( ), ( ) 2 2 n O n n O nb bp pé ù Î - +ê úë û 1-2 log n p=1-O n b æ ö Þ ç ÷ è ø 1 2b <(T-n) even, 000 010 100 101 111011 ( )O n! and 3 log T= n p=1- O 2 n n p æ ö Þ ç ÷ è ø ( )1T= n p= 2 n p Þ W n 1T=2 (1 ( ))O n+
  29. 29. “Details” • Use symmetry to calculate eigenvalues/eigenvectors of unmeasured walk U • “Assymptotics” to calculate hitting probability at T Þ one-shot hitting time (T,p) • For concurrent hitting time give a lower bound on hitting probability in terms of unmeasured walk U: Lemma: 2 t(hit at t|not stopped before t)=p b 2 t(hit v' if measured at t)=p a 2 3 n 2 1 (log )O n npa = - 1t t tb a a -³ - ( ) 22 2T T T 2 1 t=0 t=0 t=0 1 1 (hit before T)= 1T t t t tp O n T T T a b b a a - æ ö æ ö ³ ³ - = =ç ÷ ç ÷ è ø è ø å å å *J.K.’02
  30. 30. Robustness of starting state • Polynomial hitting time to opposite corner, how long from other sites (or to sites close to corner)? • “close” initial states give similar polynomial behavior • Upper bound: Region around v of polynomial hitting time to v’ at most (otherwise we could find search algorithm that beats the lower bound for quantum searching) 000 010 100 101 111011 ( )2n O ( )2n W *J.K.’02
  31. 31. Routing From to in hypercube network w.h.p. h(x,y)=|xÅy|=d=O(n) route on subcube supported on xÅy (d-dimensional) 000 010 100 101 110 011 111 1 2 nx = x x ...x 1 2 ny = y y ...y 000 010 100 110 *J.K.’02
  32. 32. Routing From to in hypercube network w.h.p. h(x,y)=|xÅy|=d=O(n) route on subcube supported on xÅy (d-dimensional) 000 010 100 101 110 011 111 1 2 nx = x x ...x 1 2 ny = y y ...y Classical deterministic strategy: flip d bits one by one in f • fast (d-steps), requires only knowledge of xÅy • vulnerable (one faulty edge or node on path -> failure) • adversary can launch efficient attack 000 010 100 110 *J.K.’02
  33. 33. Routing 000 010 100 101 110 011 111 Classical deterministic strategy: flip d bits one by one in fi • fast (d-steps), requires only knowledge of xÅy • vulnerable (one faulty edge or node on path -> failure) • adversary can launch efficient attack Classical randomized strategy: flip d bits one by one in random • fast (d-steps), requires knowledge of x and y • robust against random noise (failure of subexponential # nodes/edges) • adversary: efficient attack knowing x or y 000 010 100 110 1/2 1/2 *J.K.’02
  34. 34. Routing 000 010 100 101 110 011 111 Classical deterministic strategy: flip d bits one by one in fi • fast (d-steps), requires only knowledge of xÅy • vulnerable (one faulty edge or node on path -> failure) • adversary can launch efficient attack Classical randomized strategy: flip d bits one by one in random • fast (d-steps), requires knowledge of x and y • robust against random noise (failure of subexponential # nodes/edges) • adversary: efficient attack knowing x or y Classical fully randomized strategy: random walk on subc • knowledge of only xÅy required • robust against adversary (=random noise) • slow!!! (2d -steps) 000 010 100 110 1/2 1/2 *J.K.’02
  35. 35. Routing 000 010 100 101 110 011 000 010 100 110 111 2 1 + 2 1 Classical deterministic strategy: flip d bits one by one in fi • fast (d-steps), requires only knowledge of xÅy • vulnerable (one faulty edge or node on path -> failure) • adversary can launch efficient attack Classical randomized strategy: flip d bits one by one in random • fast (d-steps), requires knowledge of x and y • robust against random noise (failure of subexponential # nodes/edges) • adversary: efficient attack knowing x or y Classical fully randomized strategy: random walk on subc • knowledge of only xÅy required • robust against adversary (=random noise) • slow!!! (2d -steps) Quantum random walk: knowledge of only xÅy needed • robust against adversary (=random noise) • fast!!! (O(d)-steps) Routing strategy for future quantum netsJK. ‘02
  36. 36. Quantum Random walk search algorithm* Idea: one “marked” state |wñ with different coin C’ (*) can be obtained from standard search Oracle *N. Shenvi, J. K., K.B. Whaley: “A quantum random walk search algorithm”, Phys. Rev. A, 2003 000 010 100 101 111011 ( ( ) ' )U C I w w C w w S= Ä - + Ä ! C fw C’ x 0 … 0 ( )wf xÅ Coin- space x ,( )w x wf x d= (*) C=Grover coin, C’= -Id, but other “marked” coins are possible
  37. 37. Quantum Random walk search algorithm* Idea: one “marked” state |wñ with different coin C’ (*) can be obtained from standard search Oracle Algorithm: • Start in |sñ (uniform superposition, eigenstate of the unperturbed walk ) • Perform random walk with “marked” coin T times Claim: After steps the probability to measure |wñ is p=1/2-O(1/n). *N. Shenvi, J. K., K.B. Whaley: “A quantum random walk search algorithm”, Phys. Rev. A, 2003 000 010 100 101 111011 ( ( ) ' )U C I w w C w w S= Ä - + Ä ! C fw C’ x 0 … 0 ( )wf xÅ Coin- space x ,( )w x wf x d= 1 2 2 2 2 n T N p p- = = ( )U C I S= Ä ! (*) C=Grover coin, C’= -Id, but other “marked” coins are possible
  38. 38. Quantum Random walk search algorithm* *N. Shenvi, J. K., K.B. Whaley: “A quantum random walk search algorithm”, Phys. Rev. A, 2003 Proof: • U has only two “relevant” eigenstates with eigenvalues (closest to 1) • up to exponentially small correction: where (nearly) two dimensional dynamics • angle • starting in after Remarks: • topology of the hypercube, only RW-operations – easier to implement? • only approximately 2-dimensional, contains traces of underlying topology • similar algorithms work on d-dim grids (numerical simulations) w± i e w± 1 2 2 i s sw± » ± ! n-1 1 - 2 w » 1 2w s »! 1 ( ) 2 s w w» + - 2 T p w = 1 ( ) 2 T U s i i sw w» - + - » ! h(w,x) p 1/2 0 1 2 3 s! s!
  39. 39. Implementations Easier than full fledged quantum computer… • Ion trap positions = motional states of ion (Travaglione&Milburn’02):coin = internal electronic states of ion Raman beam = flip, displacement beam = shift • Optical cavity positions = “truncated” mom. eigenst. of mode (Sanders et al.’03): coin = atomic states • Optical lattices neutral atom in periodic potential coupled to (Dur et al.’02): internal states |®ñ and |¬ñ walk in position space, coin-flip = (uniform) laser pulses J.K: “Quantum random walks – an introductory overview”, Cont. Phys. ‘03
  40. 40. Implementations Easier than full fledged quantum computer… • Optical lattices neutral atom in periodic potential coupled to (Dur et al.’02): internal states |®ñ and |¬ñ walk in position space, coin-flip = (uniform) laser pulses J.K: “Quantum random walks – an introductory overview”, Cont. Phys. ‘03 |®ñ |®ñ|®ñ|¬ñ |¬ñ|¬ñ hundreds of steps realisable with current Uniform laser pulse for H
  41. 41. Applications: “decoherence object” No decoherence in coin-space -> polynomial hitting time Complete decoherence on coin -> exponential hitting time “…in between…”? Partial decoherence n T n T decoherence T poly exp decoherence T poly exp or
  42. 42. Other graphs nG … … n-level binary tree Example*: *A.Childs, E.Farhi, S. Gutman, QIC’02, quant-ph/0103020 Childs, Cleve, Deotto, Farhi, Gutman, Spielman, STOC’03,quant-ph/0209131 start hit 1 1/3 2/3 2 2/3 1/3 n 1/3 2/3 n+1 2/3 2/3 … 1/3 2/3 1/3 2/3 Reduces to (assymetric) walk on the line (classic Classical: O(exp(n)) hitting time
  43. 43. Other graphs nG … … n-level binary tree Example*: *A.Childs, E.Farhi, S. Gutman, QIC’02, quant-ph/0103020 **Childs, Cleve, Deotto, Farhi, Gutman, Spielman, STOC’03,quant-ph/0209131 start hit Classical: O(exp(n)) hitting time Based on continuous time walk – oracle query speed-up by exponential amount**
  44. 44. Other graphs nG … … Example*: start hit 1 2 n n+1… Reduces to assymetric walk on the line (quantum) Classical: O(exp(n)) hitting time 0 1 1 0 æ ö ç ÷ è ø 1 2 2 3 3 2 2 1 3 3 æ ö ç ÷ ç ÷ ç ÷ -ç ÷ è ø 1 2 2 3 3 2 2 1 3 3 æ ö ç ÷ ç ÷ ç ÷ -ç ÷ è ø 1 2 2 3 3 2 2 1 3 3 æ ö -ç ÷ ç ÷ ç ÷ ç ÷ è ø 0 1 1 0 æ ö ç ÷ è ø {|®ñ,|¬ñ} Ä Quantum: (numeric) poly(n) hitting time (N.Shenvi & JK.) 1 2 2 1 2 1 2 3 2 2 1 - -æ ö ç ÷ - -ç ÷ ç ÷- -è ø
  45. 45. Continuous-time random walk Classical: Transition matrix H-generator (rows sum to zero, “local”) Continuous-time process: Example (4-cycle): exp( )M H= ( ) exp( )M t tH= 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 æ ö ç ÷ ç ÷= ç ÷ ç ÷ç ÷ è ø M 2 0 2 0 0 2 0 2 g g g g g g g g g g g g -æ ö ç ÷ -ç ÷= ç ÷- ç ÷ç ÷-è ø H 0 1 2 3
  46. 46. Continuous-time random walk Classical: Transition matrix H-generator (rows sum to zero, “local”) Continuous-time process: Quantum (Farhi&Gutmann): “quantization” •no coin-space •non-zero transition probability to non-neighboring sites: Results: On Gn the continuous-time quantum random walk starting in one corner has an amplitude of 1/poly(n) on the opposite corner after a time poly(n). exp( )M H= ( ) exp( )M t tH= ( ) exp( )U t itH= 2 2 ( ) 1 ... 2 U i H i H e e e= + + +
  47. 47. Open questions • More applications!!!!! • Other graphs, when rapid hitting? • Connection discrete – continuous walk
  48. 48. Adiabatic Quantum Comput
  49. 49. Idea* • Start in the groundstate of a Hamiltonian H0 (easy to prepare) • Encode problem as a Hamiltonian HP (groundstate gives soln.) • Adiabatically evolve from H0 to HP *E. Farhi, J. Goldstone, S. Gutmann, M. Sipser: “Quantum Computation by Adiabatic Evolution”, q-p/’00 0 P t tH(t)=(1- )H + HT T Adiabatic theorem: H(s) parametrized Hamiltonia g(s) gap between ground- and firs If then final state arbitrarily clo 2 2 ( ) (s) ( ) d H s ds g s t >>
  50. 50. Results* • Start in the groundstate of a Hamiltonian H0 (easy to prepare) • Encode problem as a Hamiltonian HP (groundstate gives soln.) • Adiabatically evolve from H0 to HP Encoding of optimization problems: minimize some fn. f of z=z1z2… zn e.g. Grover fw(z)=1- dz,w • Gives Grover speed-up*E. Farhi, J. Goldstone, S. Gutmann, M. Sipser: “Quantum Computation by Adiabatic Evolution”, q-p/’00 0 P t tH(t)=(1- )H + HT T PH ( ) z f z z z= å ( )0 1 1 H 2 n i x i I s = = -å
  51. 51. Universality* Quantum circuit model = adiabatic computation with local interactions on 2D lattice Idea: encode “quantum computation” into ground- state of local Hamiltonian HC (Kitaev-QMA) *D. Aharonov, W. van Dam, J. K., Z. Landau, S. Lloyd, O. Regev: "Universality of Adiabatic Quantum Computation with Two-Body Interactions", in preparation Hi+1j Hij Hij+1 H’ij H H H H H H H H @ 0 0 0 0 outy 0 CH( ) (1 ) H Ht tt T T= - + L gates time T=poly(L)
  52. 52. Kitaev’s local QMA-Hamiltonian* U1 0 0 0 0 t 0 1 2 3 L U2 U3 UL L t 1 1 0 1 ... 00...0 L+1 t t U U U th - = = Äå Quantum circuit: Ground state of Hc: n qubits logT qubits *A.Y. Kitaev, A.H. Shen, M.N. Vialyi: “Classical and Quantum Computation” (book), AMS’02 D. Aharonov, T. Naveh: “QNP-a survey”, quant-ph/0210077 J.K., O. Regev: “3-local Hamiltonian is QMA-complete”, QIC’03, quant-ph/0302079
  53. 53. Kitaev’s local QMA-Hamiltonian* U1 0 0 0 0 t 0 1 2 3 L U2 U3 UT L t 1 1 0 1 ... 00...0 L+1 t t U U U th - = = Äå Quantum circuit: Ground state of Hc: ( ) L Ü C t=1 1 H 1 1 1 1 2 t tt t t t U t t U t t= I Ä + I Ä - - - Ä - - Ä -å n qubits logL qubits Circuit Hamiltonian HC (logL-local): groundstate CH 0h = CH 0³ Initial Hamiltonian H0: (unique) groundstate – in n L 0 i i=1 t=1 H ( 1 1) 0 0t t t t t= Ä = = + I Äå å 0H 00...0 0 0tÄ = = *A.Y. Kitaev, A.H. Shen, M.N. Vialyi: “Classical and Quantum Computation” (book), AMS’02 D. Aharonov, T. Naveh: “QNP-a survey”, quant-ph/0210077 J.K., O. Regev: “3-local Hamiltonian is QMA-complete”, QIC’03, quant-ph/0302079
  54. 54. Construction* L t 1 1 0 1 ... 00...0 L 1 t t U U U th - = = Ä + å *D. Aharonov, W. van Dam, J. K., Z. Landau, S. Lloyd, O. Regev: "Universality of Adiabatic Quantum Computation with Two-Body Interactions", in preparation 0 C t tH(t)=(1- )H + HT T Claim 1: The minimal gap of H(t) is >1/18L T=poly(L) Claim 2: H0, HC can be made 2-local on a l 8-level systems at polynomial cost 0 00...0 0ty = Ä = adiabat time T=poly(L) • • measure t-qubits – if “L” – o/w repeat (O(L) times) L 1 1... 00...0f LU U U Ly -= Ä Adiabatic algorithm:
  55. 55. The gap n L 0 i i=1 t=1 H ( 1 1) 0 0t t t t t= Ä = = + I Äå å L C t=1 Ü 1 H ( 1 1 2 1 1 )t t t t t t U t t U t t = I Ä + I Ä - - - Ä - - Ä - å 0 C t tH(t)=(1- )H + HT T 0H 00...0 0 0tÄ = = CH 0h = L 1 1 t=1 W= ...t tU U U t t- Äå ' Ü 0 0 0H =W H W=H Ü W 00...0 0 00...0 0t tÄ = = Ä = ' Ü C C 1 1 H =W H W= ( 1 1 2 1 1 ) L t t t t t t t t t = I Ä + - - - - - - å Ü 1 W 00...0 L t th = = Äå Hamiltonians: Ground states: Change of Basis: unitary
  56. 56. The gap 0 C t tH(t)=(1- )H + HT T 00...0 0in ty = Ä = ' C 1 1 H = ( 1 1 1 1 ) 2 L t t t t t t t t t = I Ä + - - - - - -å ' C 1 2 1 2 0 ... 0 1 2 1 1 2 ... 0 0 ... ... ... ...H ... ... 1 2 1 1 2 0 0 1 2 1 2 -æ ö ç ÷- -ç ÷ ç ÷= ç ÷ - -ç ÷ ç ÷-è ø Hamiltonians: n L ' 0 i i=1 t=1 H ( 1 1) 0 0t t t t= Ä = = + I Äå å State of the system stays in the space spanned b00...0 tÄ ' 0 0 0 0 ... 0 0 1 0 ... 0 0 0 2 ... 0H ... ... ... ... ... 0 0 0 ... L æ ö ç ÷ ç ÷ ç ÷= ç ÷ ç ÷ ç ÷ è ø Gersgorin-Theorem, Perron-Frobenius ->random wal line! Conductance bounds,… Gap > 1/18L2
  57. 57. From log-local to 2D-local From log-local to 5-local: 1..10...0t Þ L C t=1 Ü 1 H ( 1 1 2 1 1 )t t t t t t U t t U t t = I Ä + I Ä - - - Ä - - Ä - å , 1 10 10 t t t t + Þ 1, , 1 1 110 100 t t t t t - + - Þ 1, 1 1 10 10 t t t t - - - Þ 1, , 1 1 100 110 t t t t t - + - Þ unary representation on L qubits “penalize” invalid representations of t: 1 P , 1 1 H 01 01 L j j j - + = = å t L-t
  58. 58. From log-local to 2D-local From log-local to 5-local: 1..10...0t Þ L C t=1 Ü 1 H ( 1 1 2 1 1 )t t t t t t U t t U t t = I Ä + I Ä - - - Ä - - Ä - å , 1 10 10 t t t t + Þ 1, , 1 1 110 100 t t t t t - + - Þ 1, 1 1 10 10 t t t t - - - Þ 1, , 1 1 100 110 t t t t t - + - Þ unary representation on L qubits “penalize” invalid representations of t: 1 P , 1 1 H 01 01 L j j j - + = = å t L-t U1 0 0 0 0 t 0 1 2 3 6 U2 U3 U6 4 5 U4 U5 1 2 3 64 5
  59. 59. From log-local to 2D-local From log-local to 5-local: 1..10...0t Þ L C t=1 Ü 1 H ( 1 1 2 1 1 )t t t t t t U t t U t t = I Ä + I Ä - - - Ä - - Ä - å , 1 10 10 t t t t + Þ 1, , 1 1 110 100 t t t t t - + - Þ 1, 1 1 10 10 t t t t - - - Þ 1, , 1 1 100 110 t t t t t - + - Þ unary representation on L qubits “penalize” invalid representations of t: 1 P , 1 1 H 01 01 L j j j - + = = å t L-t U1 0 0 0 0 t 0 1 2 3 6 U2 U3 U6 4 5 U4 U5 1 2 3 64 5
  60. 60. From log-local to 2D-local From log-local to 5-local: 1..10...0t Þ L C t=1 Ü 1 H ( 1 1 2 1 1 )t t t t t t U t t U t t = I Ä + I Ä - - - Ä - - Ä - å , 1 10 10 t t t t + Þ 1, , 1 1 110 100 t t t t t - + - Þ 1, 1 1 10 10 t t t t - - - Þ 1, , 1 1 100 110 t t t t t - + - Þ unary representation on L qubits “penalize” invalid representations of t: 1 P , 1 1 H 01 01 L j j j - + = = å t L-t U1 0 0 0 0 t 0 1 2 3 6 U2 U3 U6 4 5 U4 U5 1 2 3 64 5
  61. 61. From log-local to 2D-local From 5-local to 2D-local: put circuit into U1 0 0 0 0 0 1 2 3 6 U2 U3 U6 4 5 U4 U5 U1 0 0 0 0 0 1 2 3 6 U2 U3 U6 4 5 U4 U5 Ui either UÄI or IÄU or C[Z] (controlled Z) Enlarge and “interleave” time qubits: One qubit gate: “trigger” “copy into time qubit and perform gate” “copy” - 6-level system
  62. 62. From log-local to 2D-local From 5-local to 2D-local: put circuit into U1 0 0 0 0 0 1 2 3 6 U2 U3 U6 4 5 U4 U5 U1 0 0 0 0 0 1 2 3 6 U2 U3 U6 4 5 U4 U5 Ui either UÄI or IÄU or C[Z] (controlled Z) Enlarge and “interleave” time qubits: Two qubit gate: - 8-level system
  63. 63. STOP!!!!!
  64. 64. Conductance-bound Quantum: d-max.degree Cut (X,X) of G, boundary Idea: start with state concentrated in X and show that at each time step “leakage” into X is bounded by . Then after steps And hence 1 ( ) d t = W F { : v X}XB v X edge= Î $ ® 1 2 ' min ' dX X G B X£ F = F £ F XB X (1 )XB X VX et e³ -et 1 ' et æ ö = Wç ÷ Fè ø

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