Spectral gap amplification

SIAM Journal on Computing

Volumn 42, Issue 2, 2013, Pages 593-610

  Somma, R D ^a   Boixo, S ^b  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

^b UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

Adiabatic quantum computing; Quantum algorithms; Quantum Monte Carlo

Indexed keywords

ADIABATIC SIMULATION; OPTIMIZATION PROBLEMS; PROJECTED ENTANGLED PAIR STATE; QUANTUM ALGORITHMS; QUANTUM COMPUTING; QUANTUM MONTE CARLO; SIMILARITY TRANSFORMATION; STOCHASTIC MATRICES;

AMPLIFICATION; COSTS; INVERSE PROBLEMS; OPTIMIZATION; QUANTUM COMPUTERS; QUANTUM ENTANGLEMENT; STOCHASTIC MODELS;

HAMILTONIANS;

EID: 84880100362     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/120871997     Document Type: Article

References (44)

1. D. AHARONOV, D. GOTTESMAN, S. IRANI, AND J. KEMPE, The power of quantum systems on a line, Comm. Math. Phys., 287 (2009), pp. 41-65.
2. D. AHARONOV AND A. TA-SHMA, Adiabatic quantum state generation and statistical zero knowledge, in Proceedings of the 35th ACM Symposium on Theory of Computing, 2003, pp. 20-29.
3. D. AHARONOV, W. VAN DAM, J. KEMPE, Z. LANDAU, S. LLOYD, AND O. REGEV, Adiabatic quantum computation is equivalent to standard quantum computation, SIAM J. Comput., 37 (2007), pp. 166-194.
4. P. AMARA, D. HSU, AND J. E. STRAUB, Global energy minimum searches using an approximate solution of the imaginary time Schroedinger equation, J. Phys. Chem., 97 (1993), pp. 6715-6721.
5. A. AMBAINIS, Quantum walk algorithm for element distinctness, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004, pp. 22-31.
6. J. B. ANDERSON, Quantum Monte Carlo, Oxford University Press, Oxford, UK, 2007.
7. C. H. BENNETT, E. BERNSTEIN, G. BRASSARD, AND U. VAZIRANI, Strengths and weaknesses of quantum computing, SIAM J. Comput., 26 (1997), pp. 1510-1523.
8. D. BERRY, G. AHOKAS, R. CLEVE, AND B. SANDERS, Efficient quantum algorithms for simulating sparse Hamiltonians, Comm. Math. Phys., 270 (2007), pp. 359-371.
9. S. BOIXO, E. KNILL, AND R. D. SOMMA, Eigenpath traversal by phase randomization, Quantum Inf. Comput., 9 (2009), pp. 833-855.
10. S. BOIXO, E. KNILL, AND R. D. SOMMA, Fast Quantum Algorithms for Traversing Paths of Eigenstates, preprint, arXiv:1005.3034v1 [quant-ph], 2010.
11. S. BOIXO AND R. D. SOMMA, Necessary condition for the quantum adiabatic approximation, Phys. Rev. A, 81 (2010), 032308.
12. S. BRAVYI, D. P. DIVICENZO, R. OLIVEIRA, AND B. M. TERHAL, The complexity of stoquastic local Hamiltonian problems, Quantum Inf. Comput., 8 (2008), pp. 361-385.
13. S. BRAVYI AND B. TERHAL, Complexity of stoquastic frustration-free Hamiltonians, SIAM J. Comput., 39 (2009), pp. 1462-1485.
14. A. CHILDS AND J. GOLDSTONE, Spatial search by quantum walk, Phys. Rev. A, 70 (2004), 022314.
15. A. CHILDS AND R. KOTHARI, Simulating sparse Hamiltonians with star decompositions, in Theory of Quantum Computation, Communication, and Cryptography, Springer, Berlin, 2011, pp. 94-103.
16. 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.
17. N. DE BEAUDRAP, M. OHLIGER, T. OSBORNE, AND J. EISERT, Solving frustration-free spin systems, Phys. Rev. Lett., 105 (2010), 060504.
18. E. FARHI, J. GOLDSTONE, AND S. GUTMANN, Quantum Adiabatic Evolution Algorithms versus Simulated Annealing, preprint, arXiv:quant-ph/0201031v1, 2002.
19. 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.
20. E. FARHI, J. GOLDSTONE, S. GUTMANN, AND M. SIPSER, Quantum Computation by Adiabatic Evolution, preprint, arXiv:quant-ph/0001106v1, 2000.
21. E. FARHI AND S. GUTMANN, Analog analogue of a digital quantum computation, Phys. Rev. A, 57 (1998), pp. 2403-2406.
22. A. B. FINNILA, M. A. GOMEZ, C. SEBENIK, C. STENSON, AND J. D. DOLL, Quantum annealing: A new method for minimizing multidimensional functions, Chem. Phys. Lett., 219 (1994), pp. 343-348.
23. S. JANSEN, M. RUSKAI, AND R. SEILER, Bounds for the adiabatic approximation with applications to quantum computation, J. Math. Phys., 48 (2007), pp. 102111-102115.
24. T. KADOWAKI AND H. NISHIMORI, Quantum annealing in the transverse Ising model, Phys. Rev. E, 58 (1998), pp. 5355-5363.
25. A. YU. KITAEV, Quantum Measurements and the Abelian Stabilizer Problem, preprint, arXiv:quant-ph/9511026v1, 1995.
26. A. YU. KITAEV, A. SHEN, AND M. VYALYI, Classical and Quantum Computation, American Mathematical Society, Providence, RI, 2002.
27. A. MESSIAH, Quantum Mechanics, Vol. II, North-Holland, Amsterdam, 1962.
28. A. MIZEL, D. A. LIDAR, AND M. MITCHELL, Simple proof of the equivalence between adiabatic quantum computation and the circuit model, Phys. Rev. Lett., 99 (2007), 070502.
29. A. PAPAGEORGIOU AND C. ZHANG, On the efficiency of quantum algorithms for Hamiltonian simulation, Quantum Inf. Process., 11 (2012), pp. 541-561.
30. D. PEREZ-GARCIA, F. VERSTRAETE, M. M. WOLF, AND J. I. CIRAC, PEPS as unique ground states of local Hamiltonians, Quantum Inf. Comput., 8 (2008), pp. 650-663.
31. S. SACHDEV, Quantum Phase Transitions, Cambridge University Press, Cambridge, UK, 2001.
32. M. SCHWARZ, K. TEMME, AND F. VERSTRAETE, Preparing projected entangled pair states on a quantum computer, Phys. Rev. Lett., 108 (2012), 110502.
33. R. SOMMA, C. BATISTA, AND G. ORTIZ, Quantum approach to classical statistical mechanics, Phys. Rev. Lett., 99 (2007), 030603.
34. R. SOMMA AND G. ORTIZ, Quantum approach to classical thermodynamics and optimization, in Quantum Quenching, Annealing, and Computation, Lecture Notes in Phys. 802, Springer, Berlin, Heidelberg, 2010, pp. 1-20.
35. R. SOMMA, G. ORTIZ, J. GUBERNATIS, E. KNILL, AND R. LAFLAMME, Simulating physical phenomena by quantum networks, Phys. Rev. A, 65 (2002), 042323.
36. R. D. SOMMA, S. BOIXO, H. BARNUM, AND E. KNILL, Quantum simulations of classical annealing processes, Phys. Rev. Lett., 101 (2008), pp. 130504.
37. M. SUZUKI, Quantum Monte Carlo Methods in Condensed Matter Physics, World Scientific, Singapore, 1998.
38. M. SZEGEDY, Quantum speed-up of Markov chain based algorithms, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004, pp. 32-41.
39. F. VERSTRAETE AND J. CIRAC, Renormalization Algorithms for Quantum-Many Body Systems in Two and Higher Dimensions, preprint, arXiv:cond-mat/0407066v1 [cond-mat.str-el].
40. F. VERSTRAETE, M. WOLF, D. PEREZ-GARCIA, AND J. CIRAC, Criticality, the area law, and the computational power of projected entangled pair states, Phys. Rev. Lett., 96 (2006), 220601.
41. V. VIZING, On an estimate of the chromatic class of a p-graph, Diskret. Analiz, 3 (1964), pp. 25-30.
42. N. WIEBE, D. BERRY, P. HØYER, AND B. C. SANDERS, Higher order decompositions of ordered operator exponentials, J. Phys. A, 43 (2010), 065203.
43. P. WOCJAN AND A. ABEYESINGHE, Speedup via quantum sampling, Phys. Rev. A, 78 (2008), 042336.
44. M. M. WOLF, G. ORTIZ, F. VERSTRAETE, AND J. I. CIRAC, Quantum phase transitions in matrix product systems, Phys. Rev. Lett., 97 (2006), 110403.