Spectral Gap Amplification
Authors: 
R. D. Somma and S. BoixoAuthors Info & Claims
R. D. Somma and S. BoixoAuthors Info & ClaimsAbstract
Many problems can be solved by preparing a specific eigenstate of some Hamiltonian $H$. The generic cost of quantum algorithms for these problems is determined by the inverse spectral gap of $H$ for that eigenstate and the cost of evolving with $H$ for some fixed time. The goal of spectral gap amplification is to construct a Hamiltonian $H'$ with the same eigenstate as $H$ but a bigger spectral gap, requiring that constant-time evolutions with $H'$ and $H$ are implemented with nearly the same cost. We show that a quadratic spectral gap amplification is possible when $H$ satisfies a frustration-free property and give $H'$ for these cases. This results in quantum speedups for optimization problems. It also yields improved constructions for adiabatic simulations of quantum circuits and for the preparation of projected entangled pair states, which play an important role in quantum many-body physics. Defining a suitable black-box model, we establish that the quadratic amplification is optimal for frustration-free Hamiltonians and that no spectral gap amplification is possible, in general, if the frustration-free property is removed. A corollary is that finding a similarity transformation between a stoquastic Hamiltonian and the corresponding stochastic matrix is hard in the black-box model, setting limits to the power of some classical methods that simulate quantum adiabatic evolutions.
References
[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.
Index Terms
- Spectral Gap Amplification
Index terms have been assigned to the content through auto-classification.
Recommendations
Non-commuting two-local Hamiltonians for quantum error suppression
Physical constraints make it challenging to implement and control many-body interactions. For this reason, designing quantum information processes with Hamiltonians consisting of only one- and two-local terms is a worthwhile challenge. Enabling error ...
Complexity Classification of Local Hamiltonian Problems
The calculation of ground-state energies of physical systems can be formalized as the $k$-local Hamiltonian problem, which is a natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful ...
Anchored Parallel Repetition for Nonlocal Games
We introduce a simple transformation on two-player nonlocal games, called “anchoring,” and prove an exponential-decay parallel repetition theorem for all anchored games in the setting of quantum entangled players. This transformation is inspired in part ...
Comments
Information & Contributors
Information
Published In
© 2013, Society for Industrial and Applied Mathematics.
Publisher
Society for Industrial and Applied Mathematics
United States
Publication History
Published: 01 January 2013
Author Tags
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Other Metrics
Citations
View Options
View options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in