Duality quantum computer and the efficient quantum simulations

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• 作者：Shi-Jie Wei ; Gui-Lu Long
• 关键词：Duality computer ; Duality quantum computer ; Duality computing mode ; Quantum divider ; Quantum combiner ; Duality parallelism ; Quantum simulation ; Linear combination of unitary operators
• 刊名：Quantum Information Processing
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：15
• 期：3
• 页码：1189-1212
• 全文大小：652 KB
• 参考文献：1.Brandt, H.E., Myers, J.M., Lomonaco Jr., S.J.: Aspects of entangled translucent eavesdropping in quantum cryptography. Phys. Rev. A. 56, 4456 (1997)ADS CrossRef
  2.Myers, J.M., Brandt, H.E.: Converting a positive operator-valued measure to a design for a measuring instrument on the laboratory bench. Meas. Sci. Technol. 8, 1222 (1997)ADS CrossRef
  3.Brandt, H.E.: Qubit devices and the issue of quantum decoherence. Prog. Quant. Eletron. 22, 257–370 (1999)ADS CrossRef
  4.Brandt, H.E.: Positive operator valued measure in quantum information processing. Am. J. Phys. 67, 434–439 (1999)ADS MathSciNet CrossRef MATH
  5.Brandt, H.E.: Secrecy capacity in the four-state protocol of quantum key distribution. J. Math. Phys. 43, 4526–4530 (2002)ADS MathSciNet CrossRef MATH
  6.Brandt, H.E.: Quantum-cryptographic entangling probe. Phys. Rev. A. 71, 042312 (2005)ADS MathSciNet CrossRef MATH
  7.Brandt, H.E.: Quantum computational geodesics. J. Mod. Opt. 56, 2112–2117 (2009)ADS CrossRef MATH
  8.Brandt, H.E.: Geodesic derivative in quantum circuit complexity analysis. J. Mod. Opt. 57, 1972–1978 (2010)ADS CrossRef MATH
  9.Brandt, H.E.: Aspects of the Riemannian geometry of quantum computation. Int. J. Mod. Phys. B. 26, 1243004 (2012)ADS MathSciNet CrossRef MATH
  10.Long, G.L.: General quantum interference principle and duality computer. Commun. Theor. Phys. 45, 825–844 (2006); Also see arXiv:​quant-ph/​0512120 . It was briefly mentioned in an abstract (5111–53) (Tracking No. FN03-FN02-32) submitted to SPIE conference Fluctuations and Noise in Photonics and Quantum Optics in 18 Oct 2002
  11.Gudder, S.: Mathematical theory of duality quantum computers. Quantum Inf. Process. 6, 37–48 (2007)MathSciNet CrossRef MATH
  12.Long, G.L.: Mathematical theory of the duality computer in the density matrix formalism. Quantum Inf. Process. 6(1), 49–54 (2007)MathSciNet CrossRef MATH
  13.Zou, X.F., Qiu, D.W., Wu, L.H., Li, L.J., Li, L.Z.: On mathematical theory of the duality computers. Quantum Inf. Process. 8, 37–50 (2009)MathSciNet CrossRef MATH
  14.Cui, J.X., Zhou, T., Long, G.L.: Density matrix formalism of duality quantum computer and the solution of zero-wave-function paradox. Quantum Inf. Process. 11, 317–323 (2012)MathSciNet CrossRef MATH
  15.Long, G.L.: Duality quantum computing and duality quantum information processing. Int. J. Theor. Phys. 50, 1305–1318 (2011)MathSciNet CrossRef MATH
  16.Long, G.L., Liu, Y.: Duality computing in quantum computers. Commun. Theor. Phys. 50, 1303–1306 (2008)ADS CrossRef
  17.Long, G.L., Liu, Y., Wang, C.: Allowable generalized quantum gates. Commun. Theor. Phys. 51, 65–67 (2009)ADS CrossRef MATH
  18.Cao, H.X., Li, L., Chen, Z.L., Zhang, Y., Guo, Z.H.: Restricted allowable generalized quantum gates. Chin. Sci. Bull. 55, 2122–2125 (2010)CrossRef
  19.Wang, Y.Q., Du, H.K., Dou, Y.N.: Note on generalized quantum gates and quantum operations. Int. J. Theor. Phys. 47, 2268–2278 (2008)MathSciNet CrossRef MATH
  20.Gudder, S.: Duality quantum computers and quantum operations. Int. J. Theor. Phys. 47, 268–279 (2008). http://​www.​math.​du.​edu/​data/​preprints/​m0611.​pdf
  21.Du, H.K., Wang, Y.Q., Xu, J.L.: Applications of the generalized Lders theorem. J. Math. Phys. 49, 013507 (2008)ADS MathSciNet CrossRef MATH
  22.Zhang, Y., Cao, H.X., Li, L.: Realization of allowable qeneralized quantum gates. Sci. China Phys. Mech. Astron. 53, 1878–1883 (2010)ADS CrossRef
  23.Long, G.L., Liu, Y.: Duality quantum computing. Front. Comput. Sci. 2, 167–178 (2008)MathSciNet CrossRef
  24.Long, G.L., Liu, Y.: General principle of quantum interference and the duality quantum computer. Rep. Prog. Phys. 28, 410–431 (2008). (in Chinese)
  25.Li, C.Y., Li, J.L.: Allowable generalized quantum gates using nonlinear quantum optics. Commun. Theor. Phys. 53, 75–77 (2010)ADS CrossRef MATH
  26.Liu, Y., Zhang, W.H., Zhang, C.L., Long, G.L.: Quantum computation with nonlinear optics. Commun. Theor. Phys. 49, 107–110 (2008)ADS CrossRef
  27.Wang, W.Y., Shang, B., Wang, C., Long, G.L.: Prime factorization in the duality computer. Commun. Theor. Phys. 47, 471–473 (2007)ADS CrossRef
  28.Chen, Z.L., Cao, H.X.: A note on the extreme points of positive quantum operations. Int. J. Theor. Phys. 48, 1669–1671 (2010)MathSciNet CrossRef MATH
  29.Hao, L., Liu, D., Long, G.L.: An N/4 fixed-point duality quantum search algorithm. Sci. China Phys. Mech. Astron. 53, 1765–1768 (2010)ADS CrossRef
  30.Liu, Y.: Deleting a marked state in quantum database in a duality computing mode. Chin. Sci. Bull. 58, 2927–2931 (2013)CrossRef
  31.Hao, L., Liu, D., Long, G.L.: An N4 fixed-point duality quantum search algorithm. Sci. China Phys. Mech. Astron. 53, 1765–1768 (2010)ADS CrossRef
  32.Cui, J.X., Zhou, T., Long, G.L.: An optimal expression of a Kraus operator as a linear combination of unitary matrices. J. Phys. A Math. Theor. 45, 444011 (2012)ADS MathSciNet CrossRef MATH
  33.Liu, Y., Cui, J.X.: Realization of Kraus operators and POVM measurements using a duality quantum computer. Chin. Sci. Bull. 59, 2298–2301 (2014)CrossRef
  34.Cao, H.X., Chen, Z.L., Guo, Z.H., et al.: Complex duality quantum computers acting on pure and mixed states. Sci. China Phys. Mech. Astron. 55, 2452–2462 (2012)ADS CrossRef
  35.Cao, H.X., Long, G.L., Guo, Z.H., et al.: Mathematical theory of generalized duality quantum computers acting on vector-states. Int. J. Theor. Phys. 52, 1751–1767 (2013)MathSciNet CrossRef MATH
  36.Li, C.Y., Li, J.L.: Allowable generalized quantum gates using nonlinear quantum optics. Commun. Theor. Phys. 53, 75–77 (2010)ADS CrossRef MATH
  37.Wu, Z.Q., Zhang, S.F., Zhu, C.X.: Remarks on generalized quantum gates. Hacettepe J. Math. Stat. 43, 451–460 (2014)MathSciNet CrossRef MATH
  38.Chen, L., Cao, H.X., Meng, H.X.: Generalized duality quantum computers acting on mixed states. Quantum Inf. Process. (2015). doi:10.​1007/​s11128-015-1112-z MathSciNet MATH
  39.Hao, L., Long, G.L.: Experimental implementation of a fixed-point duality quantum search algorithm in the nuclear magnetic resonance quantum system. Sci. China Phys. Mech. Astron. 54, 936–941 (2011)ADS CrossRef
  40.Zheng, C., Hao, L., Long, G.L.: Observation of a fast evolution in a parity-time–symmetric system. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 371, 20120053 (2013)ADS MathSciNet CrossRef
  41.Aaronson, S.: Quantum computing, postselection, and probabilistic polynomial-time. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 461, 3473–3482 (2005)ADS MathSciNet CrossRef MATH
  42.Childs, A.M., Wiebe, N.: Hamiltonian simulation using linear combinations of unitary operations. Quantum Inf. Comput. 12(11–12), 901–924 (2012)MathSciNet MATH
  43.Berry, D.W., Childs, A.M., Cleve, R., Kothari, R., Somma, R.D.: Simulating Hamiltonian dynamics with a truncated Taylor series. Phys. Rev. Lett. 114, 090502 (2015)ADS CrossRef
  44.Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)ADS CrossRef
  45.Dieks, D.: Communication by EPR devices. Phys. Lett. A 92, 271–272 (1982)ADS CrossRef
  46.Yao, S., Liang, H., Gui-Lu, L.: Why can we copy classical information? Chin. Phys. Lett. 28, 010306 (2011)CrossRef
  47.Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467 (1982)MathSciNet CrossRef
  48.Benioff, P.: The computer as a physical system: a microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. J. Stat. Phys. 22, 563–591 (1980)ADS MathSciNet CrossRef
  49.Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997)MathSciNet CrossRef MATH
  50.Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325–328 (1997)ADS CrossRef
  51.Long, G.L.: Grover algorithm with zero theoretical failure rate. Phys. Rev. A 64, 022307 (2001)ADS CrossRef
  52.Toyama, F.M., van Dijk, W., Nogami, Y.: Quantum search with certainty based on modified Grover algorithms: optimum choice of parameters. Quantum Inf. Proc. 12, 1897–1914 (2013)ADS MathSciNet CrossRef MATH
  53.Lloyd, S.: Universal quantum simulators. Science 273, 1073 (1996)ADS MathSciNet CrossRef MATH
  54.Lu, Y., Feng, G.R., Li, Y.S., Long, G.L.: Experimental digital quantum simulation of temporal-spatial dynamics of interacting fermion system. Sci. Bull. 60, 241–248 (2015)CrossRef
  55.Sornborger, A.T.: Quantum simulation of tunneling in small systems. Sci. Rep. 2, 597 (2012)ADS CrossRef
  56.Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by quantum walk. In: Proceedings of the 35th ACM Symposium on Theory of Computing, pp. 59–68 (2003)
  57.Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proceedings of the 35th ACM Symposium on Theory of Computing, pp. 20–29 (2003)
  58.Feng, G.R., Xu, G.F., Long, G.L.: Experimental realization of nonadiabatic holonomic quantum computation. Phys. Rev. Lett. 110, 190501 (2013)ADS CrossRef
  59.Feng, G.R., Lu, Y., Hao, L., Zhang, F.H., Long, G.L.: Experimental simulation of quantum tunneling in small systems. Sci. Rep. 3, 2232 (2013)ADS
  60.Suzuki, M.: General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys. 32, 400 (1991)ADS MathSciNet CrossRef MATH
  61.Blanes, S., Casas, F., Ros, J.: Extrapolation of symplectic integrators. Celest. Mech. Dyn. Astr. 75, 149 (1999)ADS MathSciNet CrossRef MATH
  62.Berry, D.W., Childs, A.M., Cleve, R., Kothari, R., Somma, R.D.: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, New York, pp. 283–292. ACM Press, New York (2014)
  63.Shor, P.W.: Why haven’t more quantum algorithms been found? J. ACM (JACM) 50, 87–90 (2003)MathSciNet CrossRef MATH
  64.Ray, P., Chakrabarti, B.K., Chakrabarti, A.: Sherrington–Kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations. Phys. Rev. B. 39, 11828 (1989)ADS CrossRef
  65.Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E. 58, 53555363 (1998)CrossRef
  66.Das, A., Chakrabarti, B.K.: Colloquium: quantum annealing and analog quantum computation. Rev. Mod. Phys. 80, 1061 (2008)ADS MathSciNet CrossRef MATH
  67.Denchev, V. S., Boixo, S., Isakov, S. V., Ding, N., Babbush, R., Smelyanskiy, V., Neven, H.: What is the computational value of finite range tunneling? arXiv preprint arXiv:​1512.​02206 (2015)
  68.Johnson, M.W., Amin, M.H.S., Gildert, S., Lanting, T., Hamze, F., Dickson, N., Chapple, E.M.: Quantum annealing with manufactured spins. Nature 473, 194–198 (2011)ADS CrossRef
  
• 作者单位：Shi-Jie Wei (1)
  Gui-Lu Long (1) (2) (3)
  
  1. State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
  2. Tsinghua National Laboratory for Information Science and Technology, Beijing, 100084, China
  3. Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Physics
  Mathematics
  Engineering, general
  Computer Science, general
  Characterization and Evaluation Materials
  
• 出版者：Springer Netherlands
• ISSN：1573-1332

文摘

Duality quantum computing is a new mode of a quantum computer to simulate a moving quantum computer passing through a multi-slit. It exploits the particle wave duality property for computing. A quantum computer with n qubits and a qudit simulates a moving quantum computer with n qubits passing through a d-slit. Duality quantum computing can realize an arbitrary sum of unitaries and therefore a general quantum operator, which is called a generalized quantum gate. All linear bounded operators can be realized by the generalized quantum gates, and unitary operators are just the extreme points of the set of generalized quantum gates. Duality quantum computing provides flexibility and a clear physical picture in designing quantum algorithms, and serves as a powerful bridge between quantum and classical algorithms. In this paper, after a brief review of the theory of duality quantum computing, we will concentrate on the applications of duality quantum computing in simulations of Hamiltonian systems. We will show that duality quantum computing can efficiently simulate quantum systems by providing descriptions of the recent efficient quantum simulation algorithm of Childs and Wiebe (Quantum Inf Comput 12(11–12):901–924, 2012) for the fast simulation of quantum systems with a sparse Hamiltonian, and the quantum simulation algorithm by Berry et al. (Phys Rev Lett 114:090502, 2015), which provides exponential improvement in precision for simulating systems with a sparse Hamiltonian.