# The Power of Quantum Systems on a Line

- 365 Downloads
- 52 Citations

## Abstract

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 one-dimensional 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 QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one-dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness 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 Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses.

## Keywords

Ground State Energy Turing Machine Transition Rule Quantum Circuit Qubit State## Preview

Unable to display preview. Download preview PDF.

## References

- ACdf88.Apolloni B., Carvalho C., de Falco D.: Quantum stochastic optimization. Stochastic Processes and their Applications
**33**(5), 233–244 (1988)MathSciNetGoogle Scholar - ACdf90.Apolloni, B., Cesa-Bianchi, N., de Falco, D.: A numerical implementation of “quantum annealing”. In:
*Stochastic Processes, Physics and Geometry: Proceedings of the Ascona-Locarno Conference*. River Edge, NJ: World Scientific. 1990, pp. 97–111Google Scholar - AGIK07.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. 373–383Google Scholar - AGK07.Aharonov, D., Gottesman, D., Kempe, J.:
*The power of quantum systems on a line*. http://arXiv.org/abs/0705.4077v2 [quant-ph], 2007 - AvDK+04.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. 42–51Google Scholar - Bar82.Barahona F.: On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen.
**15**, 3241–3253 (1982)CrossRefADSMathSciNetGoogle Scholar - BY86.Binder K., Young A.P.: Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys.
**58**, 801–976 (1986)CrossRefADSGoogle Scholar - CFP02.Childs A., Farhi E., Preskill J.: Robustness of adiabatic quantum computation. Phys. Rev. A
**65**, 012322 (2002)CrossRefADSGoogle Scholar - DRS07.Deift P., Ruskai M.B., Spitzer W.: Improved gap estimates for simulating quantum circuits by adiabatic evolution. Quant Infor. Proc.
**6**(2), 121–125 (2007)MATHCrossRefMathSciNetGoogle Scholar - Fey85.Feynman R.: Quantum mechanical computers. Optics News
**11**, 11–21 (1985)CrossRefGoogle Scholar - FGG+01.Farhi E., Goldstone J., Gutmann S., Lapan J., Lundgren A., Preda D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science
**292**(5516), 472–476 (2001)CrossRefADSMathSciNetGoogle Scholar - FGGS00.Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.:
*Quantum computation by adiabatic evolution*. http://arXiv.org/list/quant-ph/0001106, 2000 - Fis95.Fisher D.S.: Critical behavior of random transverse-field Ising spin chains. Phys. Rev. B
**51**(10), 6411–6461 (1995)CrossRefADSGoogle Scholar - Has.Hastings, M.: Personal communicationGoogle Scholar
- Has07.Hastings, M.: An area law for one dimensional quantum systems. JSTAT. P08024 (2007)Google Scholar
- HT.Hastings, M., Terhal, B.: Personal communicationGoogle Scholar
- Ira07.Irani, S.:
*The complexity of quantum systems on a one-dimensional chain*. http://arXiv.org/abs/0705.4067v2[quant-ph], 2007 - JFS06.Jordan S.P., Farhi E., Shor P.W.: Error-correcting codes for adiabatic quantum computation. Phys. Rev. A
**74**, 052322 (2006)CrossRefADSMathSciNetGoogle Scholar - JWZ07.Janzing, D., Wocjan, P., Zhang, S.:
*A single-shot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation*. http://arXiv.org/abs/0710.1615v2[quant-ph], 2007 - Kay08.Kay A.: The computational power of symmetric hamiltonians. Phys. Rev. A.
**78**, 012346 (2008)CrossRefADSGoogle Scholar - KKR06.Kempe J., Kitaev A., Regev O.: The complexity of the Local Hamiltonian problem. SIAM J. Comp.
**35**(5), 1070–1097 (2006)MATHCrossRefMathSciNetGoogle Scholar - KSV02.Kitaev, A.Y., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. Providence, RI: Amer. Math. Soc., 2002Google Scholar
- Nag08.Nagaj, D.:
*Local Hamiltonians in Quantum Computation*. PhD thesis, MIT. http://arXiv.org/abs/0808.2117v1[quant-ph], 2008 - NW08.Nagaj D., Wocjan P.: Hamiltonian quantum cellular automata in 1d. Phys. Rev. A
**78**, 032311 (2008)CrossRefADSGoogle Scholar - Osb06.Osborne T.: Efficient approximation of the dynamics of one-dimensional quantum spin systems. Phys. Rev. Lett.
**97**, 157202 (2006)CrossRefADSMathSciNetGoogle Scholar - Osb07.Osborne T.: Ground state of a class of noncritical one-dimensional quantum spin systems can be approximated efficiently. Phys. Rev. A
**75**, 042306 (2007)CrossRefADSMathSciNetGoogle Scholar - OT05.Oliveira R., Terhal B.: The complexity of quantum spin systems on a two-dimensional square lattice. Quant. Inf. Comp.
**8**(10), 0900–0924 (2008)MathSciNetGoogle Scholar - Sch05.Schollwöck U.: The density-matrix renormalization group. Rev. Mod. Phys.
**77**, 259–316 (2005)CrossRefADSGoogle Scholar - SFW06.Shepherd D.J., Franz T., Werner R.F.: Universally programmable quantum cellular automaton. Phys. Rev. Lett.
**97**, 020502 (2006)CrossRefADSGoogle Scholar - Suz76.Suzuki M.: Relationship between d-dimensional quantal spin systems and (d+1)-dimensional ising systems. Prog. Theor. Phys.
**56**(5), 1454–1469 (1976)MATHCrossRefADSGoogle Scholar - vEB90.van Emde Boas, P.:
*Handbook of Theoretical Computer Science*. volume A, Chapter 1. Cambridge, MA: MIT Press, 1990, pp. 1–66Google Scholar - VV86.Valiant L.G., Vazirani V.V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci.
**47**(3), 85–93 (1986)MATHCrossRefMathSciNetGoogle Scholar - Wat95.Watrous, J.: On one-dimensional quantum cellular automata. In:
*Proc. 36th Annual IEEE Symp. on Foundations of Computer Science (FOCS)*, Los Alamitos, CA: IEEE Comp. Sci, 1995, pp. 528–537Google Scholar - Whi92.White S.R.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett.
**69**, 2863 (1992)CrossRefADSGoogle Scholar - Whi93.White S.R.: Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B
**48**, 10345 (1993)CrossRefADSGoogle Scholar