Exact Ising model simulation on a quantum computer

Alba Cervera-Lierta

Barcelona Supercomputing Center (BSC), Barcelona, Spain
Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain

We present an exact simulation of a one-dimensional transverse Ising spin chain with a quantum computer. We construct an efficient quantum circuit that diagonalizes the Ising Hamiltonian and allows to obtain all eigenstates of the model by just preparing the computational basis states. With an explicit example of that circuit for $n=4$ spins, we compute the expected value of the ground state transverse magnetization, the time evolution simulation and provide a method to also simulate thermal evolution. All circuits are run in IBM and Rigetti quantum devices to test and compare them qualitatively.

In this work, it is presented a quantum circuit that diagonalizes exactly the 1D antiferromagnetic Ising Hamiltonian. With this circuit, it is possible to simulate time and temperature evolution since we have access to the whole model spectrum by just preparing a product state. As an example, it is provided an explicit circuit for four spins which is run in IBM's and Rigetti's quantum devices. As the Ising model can be solved analitically and this circuit can be extenend to higher number of qubits, it can also be used to benchmark quantum computers.

► BibTeX data

► References

[1] D. P. DiVincenzo, Fortschritte der Physik 48, 771 (2000).
arXiv:quant-ph/0002077

[2] IBM Quantum Experience, https:/​/​www.research.ibm.com/​ibm-q/​.
https:/​/​www.research.ibm.com/​ibm-q/​

[3] R. Smith, M. J. Curtis and W. J. Zeng, arXiv:1608.03355 [quant-ph] (2016).
arXiv:1608.03355

[4] D. Alsina and J. I. Latorre, Phys. Rev. A 94, 012314 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.012314

[5] Y. Wang, Y. Li, Z. Yin and B. Zeng, npj Quantum Information 4, 46 (2018).
https:/​/​doi.org/​10.1038/​s41534-018-0095-x

[6] J. S. Devitt, Phys. Rev. A 94, 032329 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.032329

[7] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
https:/​/​doi.org/​10.1007/​BF02650179

[8] M. H. Kalos, Phys. Rev. 128, 1791 (1962).
https:/​/​doi.org/​10.1103/​PhysRev.128.1791

[9] B.L. Hammond, W. A. Lester Jr. and P.J. Reynolds, MonteCarlo Methods in Ab Initio Quantum Chemistry, World Scientific, Singapore (1994).
https:/​/​doi.org/​10.1142/​1170

[10] N. S. Blunt, T. W. Rogers, J. S. Spencer and W. M. C. Foulkes, Phys. Rev. B 89, 245124 (2014).
https:/​/​doi.org/​10.1103/​PhysRevB.89.245124

[11] R. Orús, Ann. Phys. 349, 117 (2014).
https:/​/​doi.org/​10.1016/​j.aop.2014.06.013

[12] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).
https:/​/​doi.org/​10.1103/​PhysRevLett.91.147902

[13] G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A 64, 022319 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.022319

[14] D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak and M. Troyer, Phys. Rev. A 92, 062318 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.062318

[15] Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy and S. Boixo, Phys. Rev. Appl. 9, 044036 (2018).
https:/​/​doi.org/​10.1103/​PhysRevApplied.9.044036

[16] B. Kraus, Phys. Rev. Lett. 107, 250503 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.250503

[17] M. Hebenstreit, D. Alsina, J. I. Latorre and B. Kraus, Phys. Rev. A 95, 052339 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.052339

[18] F. Verstraete, J. I. Cirac and J. I. Latorre, Phys. Rev. A 79, 032316 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.79.032316

[19] P. Schmoll and R. Orús, Phys. Rev. B 95, 045112 (2017).
https:/​/​doi.org/​10.1103/​PhysRevB.95.045112

[20] H. Bethe, Z. Phys. 71, 205 (1931).
https:/​/​doi.org/​10.1007/​BF01341708

[21] V. Murg, V. E. Korepin and F. Verstraete, Phys. Rev. B 86, 045125 (2012).
https:/​/​doi.org/​10.1103/​PhysRevB.86.045125

[22] E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16, 407 (1961).
https:/​/​doi.org/​10.1016/​0003-4916(61)90115-4

[23] S. Katsura, Phys. Rev. 127, 1508 (1962).
https:/​/​doi.org/​10.1103/​PhysRev.127.1508

[24] P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928).
https:/​/​doi.org/​10.1007/​BF01331938

[25] A. J. Ferris, Phys. Rev. Lett. 113, 010401 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.010401

[26] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge (1999).
https:/​/​doi.org/​10.1017/​CBO9780511973765

[27] Device specifications: https:/​/​github.com/​Qiskit/​qiskit-backend-information/​tree/​master/​backend.
https:/​/​github.com/​Qiskit/​qiskit-backend-information/​tree/​master/​backends

[28] Official announce of IBM ``Teach Me QISKit" award winnerhttps:/​/​www.ibm.com/​blogs/​research/​2018/​06/​teach-qiskit-winner/​.
https:/​/​www.ibm.com/​blogs/​research/​2018/​06/​teach-qiskit-winner/​

[29] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin and H. Weinfurter, Phys. Rev. A 52 3457 (1995).
https:/​/​doi.org/​10.1103/​PhysRevA.52.3457

Cited by

[1] Manoranjan Swain, Amit Rai, Bikash K. Behera, and Prasanta K. Panigrahi, "Experimental demonstration of the violations of Mermin’s and Svetlichny’s inequalities for W and GHZ states", Quantum Information Processing 18 7, 218 (2019).

[2] Bartłomiej Gardas, Marek M. Rams, and Jacek Dziarmaga, "Quantum neural networks to simulate many-body quantum systems", Physical Review B 98 18, 184304 (2018).

[3] Harshavardhan Reddy Nareddula, Bikash K. Behera, and Prasanta K. Panigrahi, "Quantum Cost Efficient Scheme for Violating the Holevo Bound and Cloning in the Presence of Deutschian Closed Timelike Curves", arXiv:1901.00379.

[4] K. M. Anandu, Muhammad Shaharukh, Bikash K. Behera, and Prasanta K. Panigrahi, "Demonstration of teleportation-based error correction in the IBM quantum computer", arXiv:1902.01692.

[5] Amandeep Singh Bhatia and Mandeep Kaur Saggi, "Implementing Entangled States on a Quantum Computer", arXiv:1811.09833.

[6] Alakesh Baishya, Lingraj Kumar, Bikash K. Behera, and Prasanta K. Panigrahi, "Experimental Demonstration of Force Driven Quantum Harmonic Oscillator in IBM Quantum Computer", arXiv:1906.01436.

[7] Adam Smith, M. S. Kim, Frank Pollmann, and Johannes Knolle, "Simulating quantum many-body dynamics on a current digital quantum computer", arXiv:1906.06343.

The above citations are from Crossref's cited-by service (last updated 2019-07-15 10:44:36) and SAO/NASA ADS (last updated 2019-07-15 10:44:37). The list may be incomplete as not all publishers provide suitable and complete citation data.