The Bethe Ansatz as a Quantum Circuit (2309.14430v2)
Abstract: The Bethe ansatz represents an analytical method enabling the exact solution of numerous models in condensed matter physics and statistical mechanics. When a global symmetry is present, the trial wavefunctions of the Bethe ansatz consist of plane wave superpositions. Previously, it has been shown that the Bethe ansatz can be recast as a deterministic quantum circuit. An analytical derivation of the quantum gates that form the circuit was lacking however. Here we present a comprehensive study of the transformation that brings the Bethe ansatz into a quantum circuit, which leads us to determine the analytical expression of the circuit gates. As a crucial step of the derivation, we present a simple set of diagrammatic rules that define a novel Matrix Product State network building Bethe wavefunctions. Remarkably, this provides a new perspective on the equivalence between the coordinate and algebraic versions of the Bethe ansatz.
- H. Bethe, Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71, 205 (1931).
- R. J. Baxter, Exactly solved models in statistical mechanics, in Integrable Systems in Statistical Mechanics, Series on Advances in Statistical Mechanics (World Scientific, 1982) pp. 5–63.
- G. Mussardo, Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics (Oxford University Press, 2020).
- C. N. Yang and C. P. Yang, One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System, Phys. Rev. 150, 327 (1966).
- E. H. Lieb and W. Liniger, Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, Phys. Rev. 130, 1605 (1963).
- L. D. Faddeev, How algebraic bethe ansatz works for integrable model, Les Houches School of Physics: Astrophysical Sources of Gravitational Radiation , 149 (1996), arXiv:hep-th/9605187 .
- J. I. Cirac and F. Verstraete, Renormalization and tensor product states in spin chains and lattices, J. Phys. A 42, 504004 (2009), arXiv:0910.1130 [cond-mat.str-el] .
- R. Orus, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals Phys. 349, 117 (2014), arXiv:1306.2164 [cond-mat.str-el] .
- F. C. Alcaraz and M. J. Lazo, The Bethe ansatz as a matrix product ansatz, J. Phys. A 37, L1 (2004), arXiv:cond-mat/0304170 .
- H. Katsura and I. Maruyama, Derivation of the matrix product ansatz for the Heisenberg chain from the algebraic Bethe ansatz, J. Phys. A 43, 175003 (2010), arXiv:0911.4215 [cond-mat.stat-mech] .
- A. Cervera-Lierta, Exact Ising model simulation on a quantum computer, Quantum 2, 114 (2018), arXiv:1807.07112 [quant-ph] .
- R. I. Nepomechie, Bethe ansatz on a quantum computer? (2020), arXiv:2010.01609 [quant-ph].
- B. Pozsgay, Excited state correlations of the finite heisenberg chain, Journal of Physics A: Mathematical and Theoretical 50, 074006 (2017), arXiv:1605.09347 [cond-mat.stat-mec] .
- E. K. Sklyanin, Quantum version of the method of inverse scattering problem, J. Soviet Math. 19, 1546 (1982).
- Although the dimensions of ΛΛ\Lambdaroman_Λ depend on k𝑘kitalic_k, we keep this dependence implicit and do not add the corresponding subscript. In this way we stress that the same principle defines ΛΛ\Lambdaroman_Λ for long and short gates.
- This equation corresponds to (17) of Sopena et al. (2022), with Gk−1subscript𝐺𝑘1G_{k-1}italic_G start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT identified with 𝒜ksubscript𝒜𝑘{\mathscr{A}}_{k}script_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and Gk−1superscriptsubscript𝐺𝑘1G_{k}^{-1}italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT with ℬk+1subscriptℬ𝑘1{\mathscr{B}}_{k+1}script_B start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT. The matrices Gksubscript𝐺𝑘G_{k}italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT were determined there by the recursion relation (16).
- V. E. Korepin, Calculation of norms of Bethe wave functions, Communications in Mathematical Physics 86, 391 (1982).
- R. Hernandez and J. M. Nieto, Correlation functions and the algebraic Bethe ansatz in the AdS/CFT correspondence, (2014), arXiv:hep-th/1403.6651.
- B. Kraus, Compressed Quantum Simulation of the Ising Model, Physical Review Letters 107, 250503 (2011), arXiv:1109.2455 [quant-ph] .
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