The Power of Quantum Systems on a Line
 Dorit Aharonov,
 Daniel Gottesman,
 Sandy Irani,
 Julia Kempe
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We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a onedimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMAcomplete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, onedimensional MAX2SAT with nearest neighbor constraints, is in P. The proof of the QMAcompleteness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum ChurchTuring thesis and that quantum computers cannot efficiently solve QMAcomplete problems) that there are onedimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being onedimensional spin glasses.
 Apolloni, B., Carvalho, C., de Falco, D. (1988) Quantum stochastic optimization. Stochastic Processes and their Applications 33: pp. 233244
 Apolloni, B., CesaBianchi, N., de Falco, D.: A numerical implementation of “quantum annealing”. In: Stochastic Processes, Physics and Geometry: Proceedings of the AsconaLocarno Conference. River Edge, NJ: World Scientific. 1990, pp. 97–111
 Aharonov, D., Gottesman, D., Irani, S., Kempe, J.: The power of quantum systems on a line. In: FOCS. Proc. 48 ^{th} Ann. IEEE, Symp on Foundations of Computer Science, Los Alamitos, CA: IEEE Comp. Soc., 2007, pp. 373–383
 Aharonov, D., Gottesman, D., Kempe, J.: The power of quantum systems on a line. http://arXiv.org/abs/0705.4077v2 [quantph], 2007
 Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. In: Proc. 45th FOCS, Los Alamitos, CA: IEEE Comp. Soc., 2004, pp. 42–51
 Barahona, F. (1982) On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15: pp. 32413253 CrossRef
 Binder, K., Young, A.P. (1986) Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58: pp. 801976 CrossRef
 Childs, A., Farhi, E., Preskill, J. (2002) Robustness of adiabatic quantum computation. Phys. Rev. A 65: pp. 012322 CrossRef
 Deift, P., Ruskai, M.B., Spitzer, W. (2007) Improved gap estimates for simulating quantum circuits by adiabatic evolution. Quant Infor. Proc. 6: pp. 121125 CrossRef
 Feynman, R. (1985) Quantum mechanical computers. Optics News 11: pp. 1121 CrossRef
 Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D. (2001) A quantum adiabatic evolution algorithm applied to random instances of an NPcomplete problem. Science 292: pp. 472476 CrossRef
 Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. http://arXiv.org/list/quantph/0001106, 2000
 Fisher, D.S. (1995) Critical behavior of random transversefield Ising spin chains. Phys. Rev. B 51: pp. 64116461 CrossRef
 Hastings, M.: Personal communication
 Hastings, M.: An area law for one dimensional quantum systems. JSTAT. P08024 (2007)
 Hastings, M., Terhal, B.: Personal communication
 Irani, S.: The complexity of quantum systems on a onedimensional chain. http://arXiv.org/abs/0705.4067v2[quantph], 2007
 Jordan, S.P., Farhi, E., Shor, P.W. (2006) Errorcorrecting codes for adiabatic quantum computation. Phys. Rev. A 74: pp. 052322 CrossRef
 Janzing, D., Wocjan, P., Zhang, S.: A singleshot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation. http://arXiv.org/abs/0710.1615v2[quantph], 2007
 Kay, A. (2008) The computational power of symmetric hamiltonians. Phys. Rev. A. 78: pp. 012346 CrossRef
 Kempe, J., Kitaev, A., Regev, O. (2006) The complexity of the Local Hamiltonian problem. SIAM J. Comp. 35: pp. 10701097 CrossRef
 Kitaev, A.Y., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. Providence, RI: Amer. Math. Soc., 2002
 Nagaj, D.: Local Hamiltonians in Quantum Computation. PhD thesis, MIT. http://arXiv.org/abs/0808.2117v1[quantph], 2008
 Nagaj, D., Wocjan, P. (2008) Hamiltonian quantum cellular automata in 1d. Phys. Rev. A 78: pp. 032311 CrossRef
 Osborne, T. (2006) Efficient approximation of the dynamics of onedimensional quantum spin systems. Phys. Rev. Lett. 97: pp. 157202 CrossRef
 Osborne, T. (2007) Ground state of a class of noncritical onedimensional quantum spin systems can be approximated efficiently. Phys. Rev. A 75: pp. 042306 CrossRef
 Oliveira, R., Terhal, B. (2008) The complexity of quantum spin systems on a twodimensional square lattice. Quant. Inf. Comp. 8: pp. 09000924
 Schollwöck, U. (2005) The densitymatrix renormalization group. Rev. Mod. Phys. 77: pp. 259316 CrossRef
 Shepherd, D.J., Franz, T., Werner, R.F. (2006) Universally programmable quantum cellular automaton. Phys. Rev. Lett. 97: pp. 020502 CrossRef
 Suzuki, M. (1976) Relationship between ddimensional quantal spin systems and (d+1)dimensional ising systems. Prog. Theor. Phys. 56: pp. 14541469 CrossRef
 van Emde Boas, P.: Handbook of Theoretical Computer Science. volume A, Chapter 1. Cambridge, MA: MIT Press, 1990, pp. 1–66
 Valiant, L.G., Vazirani, V.V. (1986) NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47: pp. 8593 CrossRef
 Watrous, J.: On onedimensional quantum cellular automata. In: Proc. 36th Annual IEEE Symp. on Foundations of Computer Science (FOCS), Los Alamitos, CA: IEEE Comp. Sci, 1995, pp. 528–537
 White, S.R. (1992) Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69: pp. 2863 CrossRef
 White, S.R. (1993) Densitymatrix algorithms for quantum renormalization groups. Phys. Rev. B 48: pp. 10345 CrossRef
 Title
 The Power of Quantum Systems on a Line
 Journal

Communications in Mathematical Physics
Volume 287, Issue 1 , pp 4165
 Cover Date
 20090401
 DOI
 10.1007/s0022000807103
 Print ISSN
 00103616
 Online ISSN
 14320916
 Publisher
 SpringerVerlag
 Additional Links
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 Authors

 Dorit Aharonov ^{(1)}
 Daniel Gottesman ^{(2)}
 Sandy Irani ^{(3)}
 Julia Kempe ^{(4)}
 Author Affiliations

 1. School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel
 2. Perimeter Institute, Waterloo, Canada
 3. Computer Science Department, University of California, Irvine, CA, 92697, USA
 4. School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel