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Quantum Random Walks

Combinatorial and Computational Aspects of Statistical Physics/ Random Graphs and Structures Cambridge, September 5, 2002. Quantum Random Walks. Julia Kempe Computer Science Division and Department of Chemistry, University of California, Berkeley &

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Quantum Random Walks

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  1. Combinatorial and Computational Aspects of Statistical Physics/ Random Graphs and Structures Cambridge, September 5, 2002 Quantum Random Walks Julia Kempe Computer Science Division and Department of Chemistry, University of California, Berkeley & CNRS & LRI, Université de Paris-Sud, France

  2. Towards nanotechnology Gordon Moore 1965 Size of the components Number of components Speed Theoretical limitations reached in 2020 !!! Apparition of quantum phenomena prevent or use quantum effects ?

  3. Information is physical! Use the laws of quantum mechanics for the basic components of an information processing machine! • Quantum computing • Quantum cryptography • Quantum information • …

  4. Main applications • Cryptography • Protocol of unconditionally secure secret key distribution [Bennett, Brassard 84] Implementation : ~ 100 km • Quantum information • Teleportation [B, B, Crépeau, Jozsa, Peres, Wooters 93] Implementation[Bouwmeester, Pan, Mattle, Eibl, Weinfurter, Zeilinger 97] • Algorithms • Factoring, discrete logarithm, ... [Shor 94] • Database search [Grover 96] Num. of qubits ?1995 : 2, 1998 : 3, 2002 : 8 [Chuang (IBM)] - 10 [Los Alamos]

  5. The qubit Classical bit:b{0,1} Probabilistic bit:probability distribution dR+{0,1}such that||d||1 =1. d=(p,1-p)with p [0,1] Quantum bit:|C{0,1}such that|| |||2=1. |=  |0 +  |1 with | |2+ | |2=1 (Dirac notation)

  6. Qubit evolution • Measure:reads and modifies |0 | |2  |0 +  |1 Measure |1 | |2  Superposition  Probability distribution • Unitary transformation: U C22such thatUU†=Id U |’ = U | | unitary  reversible: U† | U|

  7. Example Superposition: Measure: |0 1/3 Measure | |1 2/3

  8. Example Superposition: Measure: Unitary transformations: • NOT:|0 |1 • Hadamard: |0 1/3 Measure | |1 2/3 U |’ = U | | H

  9. Quantum computer: n qubits • nqubits  tensor product| C{0,1}nsuch that|| |||2=1.  |=x{0,1}n x|xwith x|x|2 =1 • Measure • Partial Measure |x|2 x{0,1}n x|x |x Measure Second bit = 0 (||2 + | |2 )  |00+  |01+ |10+ |11 Measure

  10. Quantum computer: n qubits • nqubits  tensor product|C{0,1}nsuch that|| |||2=1.  |=x{0,1}nx|xwith x|x|2 =1 • Measure • Partial Measure • Unitary transformation |U|withU U(2n) ex: XOR= |x|2 x{0,1}n x|x |x Measure Second bit = 0 (||2 + | |2 )  |00+  |01+ |10+ |11 Measure |00 |01 |10 |11 |i |i |XOR(i,j) |j +

  11. Quantum computing a function Let f:{0,1}n {0,1}m x  f(x) Reversible: Rf :{0,1}n+m {0,1}n+m (x,y)  (x,yf(x)) Quantum: UfU(2n+m): Cn+m Cn+m |x|y |x|yf(x) 

  12. Simplest Quantum Algorithm:Deutsch’s Problem Input: function f:{0,1}{0,1} (in black box) Question:fconstant (f(0)=f(1)) or balanced (f(0)f(1)) ? Quantum black box (reversible): Algorithm: one query only!!! |x |x f |y |yf(x) |0 H H Measure f |1 H |0 -constant |1 -balanced

  13. Simplest Quantum Algorithm:Deutsch’s Problem Input: function f:{0,1}{0,1} (in black box) Question:fconstant (f(0)=f(1)) or balanced (f(0)f(1)) ? Quantum black box (reversible): Algorithm: one query only!!! |x |x f |y |yf(x) |0 H H Measure f |1 H |0 -constant |1 -balanced =0 if f constant =0 if f balanced

  14. Universal computation Classical circuit model: Quantum circuit model: • evaluates booleanfunctions • can be constructed fromuniversal local gates(ex.: NAND, COPY)  0 0 0 1 0 … 1  bits  • unitary transformations U |0 |0 |1 |1 |0 |0 U qubits Measure

  15. Quantum circuits

  16. Quantum Circuits Quantum circuits can simulate classical circuits efficiently (with polynomial overhead) • Classical circuits can be efficiently simulated by classical reversible circuits; universal reversible gate – e.g. Toffoli-gate • Toffoli-gate can be generated with local unitary gates on a quantum computer -> Classical circuits  Quantum circuits

  17. Quantum algorithms • Deutsch-Jozsa algorithm (’92): determines if a function (black box) is constant or 2-1 with only one query • Simon ’s algorithm (’94): period finding

  18. Quantum algorithms • Deutsch-Jozsa algorithm (’92): determines if a function (black box) is 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 »

  19. Quantum algorithms • Deutsch-Jozsa algorithm (’92): determines if a function (black box) is 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 is (N)) (quantum lower bound)

  20. Discrete Quantum Walks • Discrete-time walks on finite graphs • (Mixing Time) *: • Dorit Aharonov (Hebrew University) • Andris Ambainis(IAS, Princeton) • J. K.(LRI, Orsay&UC Berkeley) • Umesh Vazirani (UC Berkeley) • (*STOC’01) • Polynomial hitting time on the Hypercube: • J. K. ( ’02+) • hitting time on other graphs (numerical & Analytical studies): • Neil Shenvi and J. K. (in preparation ‘02) • Mixing on the Hypercube: • C. Moore and A. Russel (quant-ph’01)

  21. Markov chains Markov chains for algorithms: Idea: construct a Markov chain (simple, local transitions only, efficiently implementable) • (1) whose stationary distribution gives the solution to the problem  Mixing time • or (2) which hits the desired solution  Hitting time « Quantum » Markov chains ?

  22. Example: Random walk for 2SAT Input: Boolean formula  (conjunction of clauses of 2 variables) in X1, … , Xn (ex. ) Question: Is  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)

  23. 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 0 STOP >1/2 TFT FTT FFF Hamming distance 1 <1/2 >1/2 TTT TFF FTF 2 <1/2 >1/2 TTF 3 Random walk on a line with n+1 vertices ! After t=2n2 repetitions (« Hitting time ») the succes probability is >1/2 (if  satisfiable).

  24. Random Walks... • 3SAT - “biased” random walk with exponential hitting time • in general : local, simple Markov chain on exponential domain <2/3 <2/3 <2/3 <2/3 <2/3 (fastest known 3-SAT algorithm based on random walk [Schöning’99, Hofmeister, Schöning & Watanabe’02]) STOP 0 1 2 3 4 5 >1/3 >1/3 >1/3 >1/3 >1/3

  25. Random Walks... • Random walk on the line:Mixing time=Hitting time =O(n2) stationary dist.=uniform • Questions: • Stationary distribution? (ergodic –> independent of initial state?) • Mixing time? • Hitting time? • Methods:spectral gap, conductance, Log Sobolev, coupling, ……… 1/2 1/2 O(n2)

  26. Classical/quantum random walks Classical Transition matrix: translationally invariant Dt(i)-distribution after time t stationary distribution measure of “closeness”: total variation distance mixing time  - time until <const. 1/2 1/2 O(n2)

  27. Classical/quantum random walks Classical Quantum Transition matrix: translationally invariant Dt(i)-distribution after time t stationary distribution measure of “closeness”: total variation distance mixing time  - time until <const. + 1/2 1/2 O(n2) ? unitary? reversible? local translationally invariant

  28. Quantum random walk • “Classical” Markov process: Quantum??? Unitary??? • Meyer [‘97]: All local, translationary invariant unitary matrices are simple translations. “R” “L”

  29. Classical random walk “R” “L” • Incorporate “coin-flip” into walk! • Classical walk in two steps: {,}  ={(,0),(,0),(,1),…,(,n-1),(,n-1)} • flip direction coin C= • perform controlled shift S:   “R”   “L” • M=S•C • Trace out (ignore, average over) the direction-space

  30.  Classical random walk   …   {,}  ={(,0),(,0),(,1),…,(,n-1),(,n-1)} • M=S•C • Trace out (ignore, average over) the direction-space • perform controlled shift :   “R” •   “L” • S = • flip direction coin … …

  31. Quantum random walk • Meyer [‘97]: All local, translationary invariant unitary matrices are simple translations. • “coined” walk in two steps: {|,|}  • ”flip” direction coin ( ) • perform controlled shift : |  “R” |  “L” “R” “L” H H unitary “walk” U U “collapses” to the classical random walk if we measure directions or positions at every step!

  32. Quantum random walk “R” “L” {,}  ={(,0),(,0),(,1),…,(,n-1),(,n-1)} • M=S•C • After t steps measure • Trace out (ignore, average over) the direction-space • perform controlled shift :   “R” •   “L” • S = • flip direction coin … …

  33. |0 |n-1 Quantum random walks |1 |2 Example: start in induces probability-dist. Pt(i) on the sites (after measurement) Convergence? NO! U is unitary  reversible! (no stationary distrib.) Def. “averaged distribution” Qt (Cesaro limit): Theorem: Qt converges to a stationary distribution.

  34. |0 |n-1 Stationary distribution |1 |2 Theorem: Qt converges to a stationary distribution. • Calculate eigenvectors/eigenvalues of U • Expand initial state: • State at time t: • Stationary distribution: if

  35. |0 |n-1 Stationary distribution |1 |2 Theorem: Qt converges to a stationary distribution. • Stationary distribution: • uniform if G non-degenerate ( ): • If G also abelian -> stationary distribution uniform: characters of the abelian group (unit norm)

  36. Observations • Classically: real eigenvalues • Quantum: complex eigenvalues • Classically: “behavior” depends ( ) on second largest eigenvalue • Quantum: all eigenvalues equally important • Ex: mixing time determined by convergence of i.e. by (minimum gap)

  37. |0 |n-1 Results on mixing time* |1 |2 Cycle: • quantum walk converges towards uniform distribution • Mixing time: • classical: = (N2 log(1/)) • quantum:=O(N log N / 3) • Total variation distance: Similar results in higher dimensions, for Cayley graphs, graphs on abelian groups, walks with different coins,… *D. Aharonov,A. Ambainis, J.K., U.Vazirani-STOC’01

  38. |0 |n-1 Results on mixing time* |1 |2 Cycle: • quantum:=O(N log N / 3)  « Warmstart » to get logarithmic -dependence: • Initialize in • Run quantum walk for steps -> measure (node v) • Restart new walk in (d-random) • Repeat k-times Resulting distribution is -close to the stationary distribution (works if stationary distribution is independent of initial state) *D. Aharonov,A. Ambainis, J.K., U.Vazirani-STOC’01

  39. 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): Classical: Quantum: d-max.degree *D. Aharonov,A. Ambainis, J.K., U.Vazirani-STOC’01

  40. Conductance 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

  41. Quantum Hitting Time on Hypercube • Space:

  42. Quantum Hitting Time on Hypercube • Space: • Walk: • Conditional Shift • Coin C (respects permutational symmetry of hypercube)

  43. Quantum Hitting Time on Hypercube • Space: • Walk: • Conditional Shift • Coin C (respects permutational symmetry of hypercube) • Initial state: Symmetric superposition over all directions Mixing time:classical: quantum: (coupon collector) (Moore&Russel’01)

  44. 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 • Stop walk if x is hit. Probability >p to hit x before time T

  45. Results on hitting time* • Classical:from v to opposite v’ hitting-time • Quantum: • One-shot hitting-time from v to v’ (T,p) and (T-n) even, *J.K.’02

  46. Results on hitting time* • Classical:from v to opposite v’ hitting-time • 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)! and (T-n) even, *J.K.’02

  47. “Details” • Use symmetry to calculate eigenvalues/eigenvectors of unmeasured walk U • “Assymptotics” to calculate hitting probability at T  one-shot hitting time (T,p) • For concurrent hitting time give a lower bound on hitting probability in terms of unmeasured walk U: Lemma:

  48. Robustness of initial condition • 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 (Grover))

  49. Open graphs Example*: … … start hit n-level binary tree Reduces to assymetric walk on the line (classically and quantum). 2/3 2/3 2/3 2/3 1/3 1/3 … 1 2 n n+1 1/3 1/3 1/3 2/3 2/3 2/3 *A.Childs, E.Farhi, S. Gutman, quant-ph/01…

  50. Open graphs Example*: Classical: O(exp(n)) hitting time … … start hit Quantum: (numeric) poly(n) hitting time (N.Shenvi & J.K.’02) n-level binary tree Reduces to assymetric walk on the line (classically and quantum). 2/3 2/3 2/3 2/3 1/3 1/3 … 1 2 n n+1 1/3 1/3 1/3 2/3 2/3 2/3 *A.Childs, E.Farhi, S. Gutman, quant-ph/01…

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