Adiabatic quantum state generation

SIAM Journal on Computing

Volumn 37, Issue 1, 2007, Pages 47-82

  Aharonov, Dorit ^a   Ta Shma, Amnon ^b  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

Adiabatic theorem; Hamiltonians; Markov chains; Quantum algorithm; Quantum computation; Quantum sampling; Spectral gap; State generation; Statistical zero knowledge; Zeno effect

Indexed keywords

ALGORITHMS; GRAPH THEORY; HAMILTONIANS; MARKOV PROCESSES; MATHEMATICAL MODELS; PROBLEM SOLVING;

ADIABATIC THEOREM; QUANTUM ALGORITHMS; QUANTUM SAMPLING; SPECTRAL GAP; STATE GENERATION; STATISTICAL ZERO KNOWLEDGE; ZENO EFFECT;

QUANTUM COMPUTERS;

EID: 40049091589     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/060648829     Document Type: Conference Paper

References (48)

1. S. AARONSON, Quantum lower bound for the collision problem, in Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, Montreal, Canada, 2002, pp. 635-642.
2. D. AHARONOV, Adiabatic quantum computation: Universality and tools, talk at Mathematical Sciences Research Institute, 2002.
3. D. AHARONOV AND A. TA-SHMA, Adiabatic Quantum state generation and statistical zero knowledge, in Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, 2003, pp. 20-29. A longer version is available online from http://arxiv.org/abs/quant- ph/0210077.
4. D. AHARONOV, W. VAN DAM, J. KEMPE, Z. LANDAU, S. LLOYD, AND O. REGEV, Adiabatic quantum computation is equivalent to standard quantum computation, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy, 2004, pp. 42-51.
5. D. AHARONOV AND O. REGEV, A lattice problem in quantum NP, in Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, Cambridge, MA, 2003, pp. 210-219.
6. D. AHARONOV AND O. REGEV, Lattice problems in NP ∩ coNP, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy, 2004, pp. 362-371.
7. N. ALON, Eigenvalues and expanders, Combinatorica, 6 (1986), pp. 83-96.
8. A. AMBAINIS AND O. REGEV, An elementary proof of the quantum adiabatic theorem, http://arxiv.org/abs/quant-ph/0411152 (2006).
9. D. APPLEGATE AND R. KANNAN, Sampling and integration of near log-concave functions, in Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, New Orleans, LA, 1991, pp. 156-163.
10. Y. AVRON AND A. ELGART, Adiabatic theorem without a gap condition, Comm. Math. Phys., 203 (1999), pp. 445-463.
11. R. B. BOPPANA, J. HASTAD, AND S. ZACHOS, Does coNP have short interactive proofs?, Inform. Process. Lett., 25 (1987), pp. 127-132.
12. R. BUBLEY AND M. DYER, Faster random generation of linear extensions, in Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, 1998, pp. 350-354.
13. A. M. CHILDS, R. CLEVE, E. DEOTTO, E. FARHI, S. GUTMANN, AND D. A. SPIELMAN, Exponential algorithmic speedup by quantum walk, in Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, 2003, pp. 59-68. Also available online from http://arxiv.org/abs/quant-ph/0209131.
14. A. M. CHILDS, E. DEOTTO, E. FARHI, J. GOLDSTONE, S. GUTMANN, AND A. J. LANDAHL, Quantum search by measurement, Phys. Rev. A, 66 (2002), 032314.
15. A. M. CHILDS, E. FARHI, J. GOLDSTONE, AND S. GUTMANN, Finding cliques by quantum adiabatic evolution, http://arxiv.org/abs/quant-ph/0012104 (2000).
16. A. M. CHILDS, E. FARHI, AND S. GUTMANN, An example of the difference between quantum and classical random walks, http://arxiv.org/abs/quant-ph/0103020 (2001).
17. W. VAN DAM AND S. HALLGREN, Efficient quantum algorithms for shifted quadratic character problems, http://arxiv.org/abs/quant-ph/0011067 (2001).
18. W. VAN DAM, M. MOSCA, AND U. V. VAZIRANI, HOW powerful is adiabatic quantum computation?, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, Las Vegas, NV, 2001, pp. 279-287.
19. W. VAN DAM AND U. VAZIRANI, More on the Power of Adiabatic Computation, manuscript, 2001.
20. E. FARHI, J. GOLDSTONE, AND S. GUTMANN, A numerical study of the performance of a quantum adiabatic evolution algorithm for satisfiability, http://arxiv.org/abs/quantph/0007071 (2000).
21. E. FARHI, J. GOLDSTONE, S. GUTMANN, J. LAPAN, A. LUNDGREN, AND D. PREDA, A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem, Science, 292 (2001), pp. 472-476.
22. E. FARHI, J. GOLDSTONE, S. GUTMANN, AND M. SIPSER, Quantum computation by adiabatic evolution, http://arxiv.org/abs/quant-ph/0001106 (2000).
23. E. FARHI AND S. GUTMANN, Quantum computation and decision trees, http://arxiv.org/abs/quant-ph/9706062 (1998).
24. L. FORTNOW, The complexity of perfect zero knowledge, in Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, New York, NY, 1987, pp. 204-209.
25. O. GOLDREICH AND E. KUSHILEVITZ, A perfect zero-knowledge proof system for a problem equivalent to the discrete logarithm, J. Cryptology, 6 (1993), pp. 97-116.
26. O. GOLDREICH AND S. GOLDWASSER, On the limits of nonapproximability of lattice problems, in Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, Dallas, TX, 1998, pp. 1-9.
27. O. GOLDREICH, A. SAHAI, AND S. VADHAN, Honest-verifier statistical zero-knowledge equals general statistical zero-knowledge, in Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, Dallas, TX, 1998, pp. 399-408.
28. S. GOLDWASSER, S. MICALI, AND C. RACKOFF, The knowledge complexity of interactive proof systems, SIAM J. Comput., 18 (1989), pp. 186-208.
29. M. GRÖTSCHEL AND L. LOVÁSZ, Combinatorial optimization, in Handbook of Combinatorics, Vol. 1, 2, Elsevier, Amsterdam, 1995, pp. 1541-1597.
30. L. GROVER AND T. RUDOLPH, Creating superpositions that correspond to efficiently integrable probability distributions, http://arxiv.org/abs/quant-ph/0208112 (2002).
31. S. HALLGREN, Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem, in Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, Montreal, Canada, 2002, pp. 653-658.
32. M. JERRUM, A. SINCLAIR, AND E. VIGODA, A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries, in Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, Heraklion, Greece, 2001, pp. 712-721.
33. M. JERRUM AND A. SINCLAIR, The Markov chain Monte Carlo method: An approach to approximate counting and integration, in Approximation Algorithms for NP-Hard Problems, D. S. Hochbaum, ed., PWS, Boston, MA, 1996.
34. T. KATO, On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Japan, 5 (1951), pp. 435-439.
35. A. YU. KITAEV, Quantum measurements and the Abelian stabilizer problem, http://arxiv.org/abs/quant-ph/9511026 (1995).
36. E. KNILL, private communication.
37. J. KOBLER, U. SCHONING, AND J. TURAN, The Graph Isomorphism Problem, Birkhäuser Boston, Boston, MA, 1993.
38. L. LOVÁSZ, Random walks on graphs: A survey, in Combinatorics, Paul Erdös is Eighty, Vol. 2, D. Miklos, V. T. Sos, and T. Szonyi, eds., Janos Bolyai Mathematical Society, Budapest, Hungary, 1996, pp. 353-398.
39. A. MESSIAH, Quantum Mechanics, John Wiley and Sons, New York, 1958.
40. M. A. NIELSEN AND I. CHUANG, Quantum Computation and Information, Cambridge University Press, Cambridge, UK, 2000.
41. T. OKAMOTO, On relationships between statistical zero knowledge proofs, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, PA, 1996, pp. 649-658.
42. A. PERES, Quantum Theory: Concepts and Methods, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995.
43. B. REICHARDT, The quantum adiabatic optimization algorithm and local minima, in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, Chicago, IL, 2004, pp. 502-510.
44. A. SAHAI AND S. P. VADHAN, A complete promise problem for statistical zero-knowledge, in Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, FL, 1997, pp. 448-457.
45. P. W. SHOR, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput., 26 (1997), pp. 1484-1509.
46. W. L. SPITZER AND S. STARR, Improved bounds on the spectral gap above frustration free ground states of quantum spin chains, http.//arxiv.org/abs/math-ph/0212029 (2002).
47. S. VADHAN, A Study of Statistical Zero Knowledge Proofs, Ph.D. thesis, MIT, Cambridge, MA, 1999.
48. J. WATROUS, Quantum algorithms for solvable groups, in Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, Heraklion, Greece, 2001, pp. 60-67.