Papers
Topics
Authors
Recent
2000 character limit reached

Exact Spin Correlators of Integrable Quantum Circuits from Algebraic Geometry (2405.16070v2)

Published 25 May 2024 in quant-ph, cond-mat.stat-mech, hep-th, and nlin.SI

Abstract: We calculate the correlation functions of strings of spin operators for integrable quantum circuits exactly. These observables can be used for calibration of quantum simulation platforms. We use algebraic Bethe Ansatz, in combination with computational algebraic geometry to obtain analytic results for medium-size (around 10-20 qubits) quantum circuits. The results are rational functions of the quantum circuit parameters. We obtain analytic results for such correlation functions both in the real space and Fourier space. In the real space, we analyze the short time and long time limit of the correlation functions. In Fourier space, we obtain analytic results in different parameter regimes, which exhibit qualitatively different behaviors. Using these analytic results, one can easily generate numerical data to arbitrary precision.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (52)
  1. B. Sutherland, Beautiful Models, World Scientific Publishing Company (2004).
  2. V. Korepin, N. Bogoliubov and A. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press (1993).
  3. Yang-Yang Thermodynamics on an Atom Chip, Phys. Rev. Lett. 100(9), 090402 (2008), 10.1103/PhysRevLett.100.090402, 0709.1899.
  4. Multispinon Continua at Zero and Finite Temperature in a Near-Ideal Heisenberg Chain, Phys. Rev. Lett. 111(13), 137205 (2013), 10.1103/PhysRevLett.111.137205, 1307.4071.
  5. Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry, Science 327, 177 (2010), 10.1126/science.1180085, 1103.3694.
  6. J. Wu, M. Kormos and Q. Si, Finite-temperature spin dynamics in a perturbed quantum critical ising chain with an E8subscript𝐸8{E}_{8}italic_E start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT symmetry, Phys. Rev. Lett. 113, 247201 (2014), 10.1103/PhysRevLett.113.247201.
  7. Z. Zhang et al., Observation of E8 Particles in an Ising Chain Antiferromagnet, Phys. Rev. B 101(22), 220411 (2020), 10.1103/PhysRevB.101.220411, 2005.13772.
  8. E8subscript𝐸8{E}_{8}italic_E start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT Spectra of Quasi-One-Dimensional Antiferromagnet BaCo2⁢V2⁢O8subscriptBaCo2subscriptV2subscriptO8{\mathrm{BaCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}roman_BaCo start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_O start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT under Transverse Field, Phys. Rev. Lett. 127, 077201 (2021), 10.1103/PhysRevLett.127.077201.
  9. J. Yang, X. Wang and J. Wu, Magnetic excitations in the one-dimensional Heisenberg-Ising model with external fields and their experimental realizations, Journal of Physics A Mathematical General 56(1), 013001 (2023), 10.1088/1751-8121/acad48, 2301.05823.
  10. A quantum Newton’s cradle, Nature 440, 900 (2006), 10.1038/nature04693.
  11. Experimental observation of a generalized Gibbs ensemble, Science 348, 207 (2015), 10.1126/science.1257026, 1411.7185.
  12. Generalized HydroDynamics on an Atom Chip, Phys. Rev. Lett. 122(9), 090601 (2019), 10.1103/PhysRevLett.122.090601, 1810.07170.
  13. Formation of robust bound states of interacting microwave photons, Nature 612, 240 (2022), 10.1038/s41586-022-05348-y, 2206.05254.
  14. M. Vanicat, L. Zadnik and T. Prosen, Integrable Trotterization: Local Conservation Laws and Boundary Driving, Phys. Rev. Lett. 121(3), 030606 (2018), 10.1103/PhysRevLett.121.030606, 1712.00431.
  15. Conserved charges in the quantum simulation of integrable spin chains, J. Phys. A 56(16), 165301 (2023), 10.1088/1751-8121/acc369, 2208.00576.
  16. R. J. Baxter, Exactly solved models in statistical mechanics, London: Academic Press Inc (1982).
  17. Integrable Digital Quantum Simulation: Generalized Gibbs Ensembles and Trotter Transitions, Phys. Rev. Lett. 130(26), 260401 (2023), 10.1103/PhysRevLett.130.260401, 2212.06455.
  18. I. L. Aleiner, Bethe ansatz solutions for certain periodic quantum circuits, Ann. Phys. 433, 168593 (2021), 10.1016/j.aop.2021.168593, 2107.05715.
  19. Quantum error mitigation, Rev. Mod. Phys. 95(4), 045005 (2023), 10.1103/RevModPhys.95.045005, 2210.00921.
  20. W. Hao, R. I. Nepomechie and A. J. Sommese, Completeness of solutions of Bethe’s equations, Phys. Rev E. 88(5), 052113 (2013), 10.1103/PhysRevE.88.052113, 1308.4645.
  21. C. Marboe and D. Volin, Fast analytic solver of rational Bethe equations, J. Phys. A 50(20), 204002 (2017), 10.1088/1751-8121/aa6b88, 1608.06504.
  22. E. Granet and J. L. Jacobsen, On zero-remainder conditions in the Bethe ansatz, JHEP 03, 178 (2020), 10.1007/JHEP03(2020)178, 1910.07797.
  23. On Generalized Q𝑄Qitalic_Q-systems, JHEP 03, 177 (2020), 10.1007/JHEP03(2020)177, 1910.07805.
  24. R. I. Nepomechie, Q-systems with boundary parameters (2019), 10.1088/1751-8121/ab9386, 1912.12702.
  25. R. I. Nepomechie, The Am(1)superscriptsubscript𝐴𝑚1A_{m}^{(1)}italic_A start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT Q-system, Mod. Phys. Lett. A 35(31), 2050260 (2020), 10.1142/S0217732320502600, 2003.06823.
  26. R. I. Nepomechie, Wronskian-type formula for inhomogeneous TQ-equations, In International Workshop on Classical and Quantum Integrable Systems, 10.1134/S004057792009007X (2020), 2005.08326.
  27. J. Gu, Y. Jiang and M. Sperling, Rational Q𝑄Qitalic_Q-systems, Higgsing and mirror symmetry, SciPost Phys. 14(3), 034 (2023), 10.21468/SciPostPhys.14.3.034, 2208.10047.
  28. Spin-s𝑠sitalic_s Rational Q𝑄Qitalic_Q-system, SciPost Phys. 16, 113 (2023), https://scipost.org/10.21468/SciPostPhys.16.4.113, 2303.07640.
  29. J. Hou, Y. Jiang and Y. Miao, Rational Q-systems at Root of Unity I. Closed Chains (2023), 2310.14966.
  30. Y. Jiang and Y. Zhang, Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE, JHEP 03, 087 (2018), 10.1007/JHEP03(2018)087, 1710.04693.
  31. J. Lykke Jacobsen, Y. Jiang and Y. Zhang, Torus partition function of the six-vertex model from algebraic geometry, JHEP 03, 152 (2019), 10.1007/JHEP03(2019)152, 1812.00447.
  32. Cylinder partition function of the 6-vertex model from algebraic geometry, JHEP 06, 169 (2020), 10.1007/JHEP06(2020)169, 2002.09019.
  33. Y. Jiang, R. Wen and Y. Zhang, Exact quench dynamics from algebraic geometry, Phys. Rev. E 108(2), 024128 (2023), 10.1103/PhysRevE.108.024128, 2109.10568.
  34. Geometric algebra and algebraic geometry of loop and Potts models, JHEP 05, 068 (2022), 10.1007/JHEP05(2022)068, 2202.02986.
  35. C. Destri and H. De Vega, Light-cone lattice approach to fermionic theories in 2D: The massive Thirring model, Nucl. Phys. B 290, 363 (1987), 10.1016/0550-3213(87)90193-3.
  36. C. Destri and H. J. de Vega, New thermodynamic bethe ansatz equations without strings, Phys. Rev. Lett. 69(16), 2313 (1992), 10.1103/PhysRevLett.69.2313.
  37. C. Destri and H. J. de Vega, Unified approach to Thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories, Nucl. Phys. B 438(3), 413 (1995), DOI: 10.1016/0550-3213(94)00547-R.
  38. N. A. Slavnov, Algebraic Bethe ansatz 10.48550/arXiv.1804.07350, 1804.07350.
  39. Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices, Nucl. Phys. B 435, 430 (1995), 10.1016/0550-3213(94)00448-N, hep-th/9410042.
  40. V. E. Korepin, Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86, 391 (1982), 10.1007/BF01212176.
  41. A. G. Izergin, Partition function of the six-vertex model in a finite volume, Soviet Physics Doklady 32, 878 (1987).
  42. A. G. Izergin, D. A. Coker and V. E. Korepin, Determinant formula for the six-vertex model, J. Phys. A 25(16), 4315 (1992), 10.1088/0305-4470/25/16/010.
  43. J. Mossel and J.-S. Caux, Relaxation dynamics in the gapped XXZ spin-1/2 chain, New Journal of Physics 12(5), 055028 (2010), 10.1088/1367-2630/12/5/055028, 1002.3988.
  44. J. Salas and A. D. Sokal, Transfer matrices and partition function zeros for antiferromagnetic Potts models. 1. General theory and square lattice chromatic polynomial, J. Statist. Phys. 104, 609 (2001), 10.1023/A:1010376605067, cond-mat/0004330.
  45. K. Fabricius and B. M. McCoy, Bethe’s equation is incomplete for the XXZ model at roots of unity, J. Statist. Phys. 103, 647 (2001), 10.1023/A:1010380116927, cond-mat/0009279.
  46. K. Fabricius and B. M. McCoy, Completing Bethe’s equations at roots of unity, J. Statist. Phys. 104, 573 (2001), 10.1023/A:1010372504158, cond-mat/0012501.
  47. E. Ilievski, Popcorn Drude weights from quantum symmetry, J. Phys. A 55(50), 504005 (2022), 10.1088/1751-8121/acaa77, 2208.01446.
  48. D. A. Cox, J. B. Little and D. O’Shea, Using Algebraic Geometry, vol. 185 of Graduate Texts in Mathematics, Springer, first edn., 10.1007/978-1-4757-6911-1 (1998).
  49. Singular 4-3-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de (2022).
  50. Towards massively parallel computations in algebraic geometry (2018), 1808.09727.
  51. J. Böhm, H. Xu and Y. Zhang, Massively Parallel Methods for Computing Gröbner bases, to appear on arXiv (2024).
  52. Handbook of mathematical functions with formulas, graphs and mathematical tables (national bureau of standards applied mathematics series no. 55), Journal of Applied Mechanics 32, 239 (1965).
Citations (4)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now

Whiteboard

Paper to Video (Beta)

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Tweets

Sign up for free to view the 2 tweets with 13 likes about this paper.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up for Free