Quantum simulation of chiral phase transitions. (English) Zbl 1522.81008
Summary: The Nambu-Jona-Lasinio (NJL) model has been widely studied for investigating the chiral phase structure of strongly interacting matter. The study of the thermodynamics of field theories within the framework of Lattice Field Theory is limited by the sign problem, which prevents Monte Carlo evaluation of the functional integral at a finite chemical potential. Using the quantum imaginary time evolution (QITE) algorithm, we construct a quantum simulation for the \((1 + 1)\) dimensional NJL model at finite temperature and finite chemical potential. We observe consistency among digital quantum simulation, exact diagonalization and analytical solution, indicating further applications of quantum computing in simulating QCD thermodynamics.
MSC:
| 81-08 | Computational methods for problems pertaining to quantum theory |
| 81V05 | Strong interaction, including quantum chromodynamics |
| 81T25 | Quantum field theory on lattices |
| 65C05 | Monte Carlo methods |
Keywords:
finite temperature or finite density; phase diagram or equation of state; phase transitionsReferences:
| [1] | Chiesa, A., Quantum hardware simulating four-dimensional inelastic neutron scattering, Nature Phys., 15, 455 (2019) · doi:10.1038/s41567-019-0437-4 |
| [2] | Smith, A.; Kim, MS; Pollmann, F.; Knolle, J., Simulating quantum many-body dynamics on a current digital quantum computer, npj Quantum Inf., 5, 106 (2019) · doi:10.1038/s41534-019-0217-0 |
| [3] | J. Zhang at al., Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature551 (2017) 601. |
| [4] | Islam, R., Emergence and frustration of magnetism with variable-range interactions in a quantum simulator, Science, 340, 583 (2013) · doi:10.1126/science.1232296 |
| [5] | A. Francis, J.K. Freericks and A.F. Kemper, Quantum computation of magnon spectra, Phys. Rev. B101 (2020) 014411. |
| [6] | Feynman, RP, Simulating physics with computers, Int. J. Theor. Phys., 21, 467 (1982) · doi:10.1007/BF02650179 |
| [7] | Lloyd, S., Universal quantum simulators, Science, 273, 1073 (1996) · Zbl 1226.81059 · doi:10.1126/science.273.5278.1073 |
| [8] | W.A. De Jong, M. Metcalf, J. Mulligan, M. Płoskoń, F. Ringer and X. Yao, Quantum simulation of open quantum systems in heavy-ion collisions, Phys. Rev. D104 (2021) 051501 [arXiv:2010.03571] [INSPIRE]. |
| [9] | W.A. de Jong, K. Lee, J. Mulligan, M. Płoskoń, F. Ringer and X. Yao, Quantum simulation of non-equilibrium dynamics and thermalization in the Schwinger model, arXiv:2106.08394 [INSPIRE]. |
| [10] | A. Wallraff at al., Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature431 (2004) 162. |
| [11] | Majer, J., Coupling superconducting qubits via a cavity bus, Nature, 449, 443 (2007) · doi:10.1038/nature06184 |
| [12] | Jordan, SP; Lee, KSM; Preskill, J., Quantum Algorithms for Quantum Field Theories, Science, 336, 1130 (2012) · doi:10.1126/science.1217069 |
| [13] | E. Zohar, J.I. Cirac and B. Reznik, Simulating Compact Quantum Electrodynamics with ultracold atoms: Probing confinement and nonperturbative effects, Phys. Rev. Lett.109 (2012) 125302 [arXiv:1204.6574] [INSPIRE]. |
| [14] | E. Zohar, J.I. Cirac and B. Reznik, Cold-Atom Quantum Simulator for SU(2) Yang-Mills Lattice Gauge Theory, Phys. Rev. Lett.110 (2013) 125304 [arXiv:1211.2241] [INSPIRE]. |
| [15] | D. Banerjee et al., Atomic Quantum Simulation of U(N) and SU(N) Non-Abelian Lattice Gauge Theories, Phys. Rev. Lett.110 (2013) 125303 [arXiv:1211.2242] [INSPIRE]. |
| [16] | D. Banerjee et al., Atomic Quantum Simulation of Dynamical Gauge Fields coupled to Fermionic Matter: From String Breaking to Evolution after a Quench, Phys. Rev. Lett.109 (2012) 175302 [arXiv:1205.6366] [INSPIRE]. |
| [17] | Wiese, U-J, Ultracold Quantum Gases and Lattice Systems: Quantum Simulation of Lattice Gauge Theories, Annalen Phys., 525, 777 (2013) · Zbl 1279.81069 · doi:10.1002/andp.201300104 |
| [18] | Wiese, U-J, Towards Quantum Simulating QCD, Nucl. Phys. A, 931, 246 (2014) · doi:10.1016/j.nuclphysa.2014.09.102 |
| [19] | S.P. Jordan, K.S.M. Lee and J. Preskill, Quantum Algorithms for Fermionic Quantum Field Theories, arXiv:1404.7115 [INSPIRE]. |
| [20] | L. García-Álvarez et al., Fermion-Fermion Scattering in Quantum Field Theory with Superconducting Circuits, Phys. Rev. Lett.114 (2015) 070502 [arXiv:1404.2868] [INSPIRE]. |
| [21] | Marcos, D., Two-dimensional Lattice Gauge Theories with Superconducting Quantum Circuits, Annals Phys., 351, 634 (2014) · Zbl 1360.81282 · doi:10.1016/j.aop.2014.09.011 |
| [22] | A. Bazavov, Y. Meurice, S.-W. Tsai, J. Unmuth-Yockey and J. Zhang, Gauge-invariant implementation of the Abelian Higgs model on optical lattices, Phys. Rev. D92 (2015) 076003 [arXiv:1503.08354] [INSPIRE]. |
| [23] | E. Zohar, J.I. Cirac and B. Reznik, Quantum Simulations of Lattice Gauge Theories using Ultracold Atoms in Optical Lattices, Rept. Prog. Phys.79 (2016) 014401 [arXiv:1503.02312] [INSPIRE]. |
| [24] | A. Mezzacapo, E. Rico, C. Sabín, I.L. Egusquiza, L. Lamata and E. Solano, Non-Abelian SU(2) Lattice Gauge Theories in Superconducting Circuits, Phys. Rev. Lett.115 (2015) 240502 [arXiv:1505.04720] [INSPIRE]. |
| [25] | Dalmonte, M.; Montangero, S., Lattice gauge theory simulations in the quantum information era, Contemp. Phys., 57, 388 (2016) · doi:10.1080/00107514.2016.1151199 |
| [26] | E. Zohar, A. Farace, B. Reznik and J.I. Cirac, Digital lattice gauge theories, Phys. Rev. A95 (2017) 023604 [arXiv:1607.08121] [INSPIRE]. |
| [27] | Martinez, EA, Real-time dynamics of lattice gauge theories with a few-qubit quantum computer, Nature, 534, 516 (2016) · doi:10.1038/nature18318 |
| [28] | A. Bermudez, G. Aarts and M. Müller, Quantum sensors for the generating functional of interacting quantum field theories, Phys. Rev. X7 (2017) 041012 [arXiv:1704.02877] [INSPIRE]. |
| [29] | Gambetta, JM; Chow, JM; Steffen, M., Building logical qubits in a superconducting quantum computing system, npj Quantum Inf., 3, 1 (2017) · doi:10.1038/s41534-016-0004-0 |
| [30] | Krinner, L.; Stewart, M.; Pazmiño, A.; Kwon, J.; Schneble, D., Spontaneous emission of matter waves from a tunable open quantum system, Nature, 559, 589 (2018) · doi:10.1038/s41586-018-0348-z |
| [31] | A. Macridin, P. Spentzouris, J. Amundson and R. Harnik, Electron-Phonon Systems on a Universal Quantum Computer, Phys. Rev. Lett.121 (2018) 110504 [arXiv:1802.07347] [INSPIRE]. |
| [32] | T.V. Zache, F. Hebenstreit, F. Jendrzejewski, M.K. Oberthaler, J. Berges and P. Hauke, Quantum simulation of lattice gauge theories using Wilson fermions, Quantum Science and Technology3 (2018) 034010. |
| [33] | J. Zhang, J. Unmuth-Yockey, J. Zeiher, A. Bazavov, S.W. Tsai and Y. Meurice, Quantum simulation of the universal features of the Polyakov loop, Phys. Rev. Lett.121 (2018) 223201 [arXiv:1803.11166] [INSPIRE]. |
| [34] | N. Klco et al., Quantum-classical computation of Schwinger model dynamics using quantum computers, Phys. Rev. A98 (2018) 032331 [arXiv:1803.03326] [INSPIRE]. |
| [35] | N. Klco and M.J. Savage, Digitization of scalar fields for quantum computing, Phys. Rev. A99 (2019) 052335 [arXiv:1808.10378] [INSPIRE]. |
| [36] | E. Gustafson, Y. Meurice and J. Unmuth-Yockey, Quantum simulation of scattering in the quantum Ising model, Phys. Rev. D99 (2019) 094503 [arXiv:1901.05944] [INSPIRE]. |
| [37] | NuQS collaboration, σ Models on Quantum Computers, Phys. Rev. Lett.123 (2019) 090501 [arXiv:1903.06577] [INSPIRE]. |
| [38] | G. Magnifico, M. Dalmonte, P. Facchi, S. Pascazio, F.V. Pepe and E. Ercolessi, Real Time Dynamics and Confinement in the ℤ_nSchwinger-Weyl lattice model for 1 + 1 QED, Quantum4 (2020) 281 [arXiv:1909.04821] [INSPIRE]. |
| [39] | Jordan, SP; Lee, KSM; Preskill, J., Quantum Computation of Scattering in Scalar Quantum Field Theories, Quant. Inf. Comput., 14, 1014 (2014) |
| [40] | H.-H. Lü et al., Simulations of Subatomic Many-Body Physics on a Quantum Frequency ProceSSOR, Phys. Rev. A100 (2019) 012320 [arXiv:1810.03959] [INSPIRE]. |
| [41] | N. Klco and M.J. Savage, Minimally entangled state preparation of localized wave functions on quantum computers, Phys. Rev. A102 (2020) 012612 [arXiv:1904.10440] [INSPIRE]. |
| [42] | H. Lamm and S. Lawrence, Simulation of Nonequilibrium Dynamics on a Quantum Computer, Phys. Rev. Lett.121 (2018) 170501 [arXiv:1806.06649] [INSPIRE]. |
| [43] | N. Klco, J.R. Stryker and M.J. Savage, SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers, Phys. Rev. D101 (2020) 074512 [arXiv:1908.06935] [INSPIRE]. |
| [44] | NuQS collaboration, Gluon Field Digitization for Quantum Computers, Phys. Rev. D100 (2019) 114501 [arXiv:1906.11213] [INSPIRE]. |
| [45] | N. Mueller, A. Tarasov and R. Venugopalan, Deeply inelastic scattering structure functions on a hybrid quantum computer, Phys. Rev. D102 (2020) 016007 [arXiv:1908.07051] [INSPIRE]. |
| [46] | NuQS collaboration, Parton physics on a quantum computer, Phys. Rev. Res.2 (2020) 013272 [arXiv:1908.10439] [INSPIRE]. |
| [47] | B. Chakraborty, M. Honda, T. Izubuchi, Y. Kikuchi and A. Tomiya, Classically emulated digital quantum simulation of the Schwinger model with a topological term via adiabatic state preparation, Phys. Rev. D105 (2022) 094503 [arXiv:2001.00485] [INSPIRE]. |
| [48] | Bermudez, A.; Tirrito, E.; Rizzi, M.; Lewenstein, M.; Hands, S., Gross-Neveu-Wilson model and correlated symmetry-protected topological phases, Annals Phys., 399, 149 (2018) · Zbl 1404.81195 · doi:10.1016/j.aop.2018.10.007 |
| [49] | L. Ziegler, E. Tirrito, M. Lewenstein, S. Hands and A. Bermudez, Correlated Chern insulators in two-dimensional Raman lattices: a cold-atom regularization of strongly-coupled four-Fermi field theories, arXiv:2011.08744 [INSPIRE]. · Zbl 1514.81212 |
| [50] | L. Ziegler, E. Tirrito, M. Lewenstein, S. Hands and A. Bermudez, Large-N Chern insulators: Lattice field theory and quantum simulation approaches to correlation effects in the quantum anomalous Hall effect, Annals Phys.439 (2022) 168763 [arXiv:2111.04485] [INSPIRE]. · Zbl 1514.81212 |
| [51] | Arute, F., Hartree-Fock on a superconducting qubit quantum computer, Science, 369, 1084 (2020) · Zbl 1478.81010 · doi:10.1126/science.abb9811 |
| [52] | Ma, H.; Govoni, M.; Galli, G., Quantum simulations of materials on near-term quantum computers, npj Computational Materials, 6, 85 (2020) · doi:10.1038/s41524-020-00353-z |
| [53] | Kandala, A.; Temme, K.; Córcoles, AD; Mezzacapo, A.; Chow, JM; Gambetta, JM, Error mitigation extends the computational reach of a noisy quantum processor, Nature, 567, 491 (2019) · doi:10.1038/s41586-019-1040-7 |
| [54] | P.J.J. O’Malley et al., Scalable quantum simulation of molecular energies, Phys. Rev. X6 (2016) 031007. |
| [55] | Kandala, A.; Mezzacapo, A.; Temme, K.; Takita, M.; Brink, M.; Chow, JM; Gambetta, JM, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature, 549, 242 (2017) · doi:10.1038/nature23879 |
| [56] | Peruzzo, A.; McClean, J.; Shadbolt, P.; Yung, M-H; Zhou, X-Q; Love, PJ; Aspuru-Guzik, A.; O’Brien, JL, A variational eigenvalue solver on a photonic quantum processor, Nature Commun., 5, 4213 (2014) · doi:10.1038/ncomms5213 |
| [57] | J.I. Colless et al., Computation of molecular spectra on a quantum processor with an error-resilient algorithm, Phys. Rev. X8 (2018) 011021. |
| [58] | Bauer, B.; Bravyi, S.; Motta, M.; Chan, GK-L, Quantum algorithms for quantum chemistry and quantum materials science, Chem. Rev., 120, 12685 (2020) · doi:10.1021/acs.chemrev.9b00829 |
| [59] | B.M. Terhal and D.P. DiVincenzo, On the problem of equilibration and the computation of correlation functions on a quantum computer, Phys. Rev. A61 (2000) 22301 [quant-ph/9810063] [INSPIRE]. |
| [60] | D. Poulin and P. Wocjan, Sampling from the thermal quantum gibbs state and evaluating partition functions with a quantum computer, Phys. Rev. Lett.103 (2009) 220502. |
| [61] | A. Riera, C. Gogolin and J. Eisert, Thermalization in nature and on a quantum computer, Phys. Rev. Lett.108 (2012) 080402. |
| [62] | Temme, K.; Osborne, TJ; Vollbrecht, KG; Poulin, D.; Verstraete, F., Quantum metropolis sampling, Nature, 471, 87 (2011) · doi:10.1038/nature09770 |
| [63] | Yung, M-H; Aspuru-Guzik, A., A quantum-quantum metropolis algorithm, Proc. Nat. Acad. Sci., 109, 754 (2012) · doi:10.1073/pnas.1111758109 |
| [64] | QuNu collaboration, Partonic collinear structure by quantum computing, Phys. Rev. D105 (2022) L111502 [arXiv:2106.03865] [INSPIRE]. |
| [65] | D.-B. Zhang, H. Xing, H. Yan, E. Wang and S.-L. Zhu, Selected topics of quantum computing for Nucl. Phys., Chin. Phys. B30 (2021) 020306 [arXiv:2011.01431] [INSPIRE]. |
| [66] | S.-N. Sun, M. Motta, R.N. Tazhigulov, A.T. Tan, G.K.-L. Chan and A.J. Minnich, Quantum computation of finite-temperature static and dynamical properties of spin systems using quantum imaginary time evolution, PRX Quantum2 (2021) 010317. |
| [67] | Motta, M., Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Nature Phys., 16, 205 (2020) · doi:10.1038/s41567-019-0704-4 |
| [68] | McArdle, S.; Jones, T.; Endo, S.; Li, Y.; Benjamin, SC; Yuan, X., Variational ansatz-based quantum simulation of imaginary time evolution, npj Quantum Inf., 5, 75 (2019) · doi:10.1038/s41534-019-0187-2 |
| [69] | Yuan, X.; Endo, S.; Zhao, Q.; Li, Y.; Benjamin, SC, Theory of variational quantum simulation, Quantum, 3, 191 (2019) · doi:10.22331/q-2019-10-07-191 |
| [70] | M.J.S. Beach, R.G. Melko, T. Grover and T.H. Hsieh, Making trotters sprint: A variational imaginary time ansatz for quantum many-body systems, Phys. Rev. B100 (2019) 094434. |
| [71] | H. Nishi, T. Kosugi and Y.-i. Matsushita, Implementation of quantum imaginary-time evolution method on nisq devices by introducing nonlocal approximation, npj Quantum Inf.7 (2021). |
| [72] | Gomes, N.; Zhang, F.; Berthusen, NF; Wang, C-Z; Ho, K-M; Orth, PP; Yao, Y., Efficient step-merged quantum imaginary time evolution algorithm for quantum chemistry, J. Chem. Theor. Comput., 16, 6256 (2020) · doi:10.1021/acs.jctc.0c00666 |
| [73] | K. Yeter-Aydeniz, G. Siopsis and R.C. Pooser, Scattering in the Ising model with the quantum Lanczos algorithm, New J. Phys.23 (2021) 043033 [arXiv:2008.08763] [INSPIRE]. |
| [74] | Yeter-Aydeniz, K.; Pooser, RC; Siopsis, G., Practical quantum computation of chemical and nuclear energy levels using quantum imaginary time evolution and lanczos algorithms, npj Quantum Inf., 6, 63 (2020) · doi:10.1038/s41534-020-00290-1 |
| [75] | J.-L. Ville et al., Leveraging Randomized Compiling for the QITE Algorithm, arXiv:2104.08785 [INSPIRE]. |
| [76] | K. Fukushima, Phase diagrams in the three-flavor Nambu-Jona-Lasinio model with the Polyakov loop, Phys. Rev. D77 (2008) 114028 [Erratum ibid.78 (2008) 039902] [arXiv:0803.3318] [INSPIRE]. |
| [77] | Costa, P.; de Sousa, C.; Ruivo, M.; Kalinovsky, Y., The qcd critical end point in the SU(3) Nambu-Jona-Lasinio model, Phys. Lett. B, 647, 431 (2007) · doi:10.1016/j.physletb.2007.02.045 |
| [78] | W.-j. Fu, Z. Zhang and Y.-x. Liu, 2 + 1 flavor Polyakov-Nambu-Jona-Lasinio model at finite temperature and nonzero chemical potential, Phys. Rev. D77 (2008) 014006 [arXiv:0711.0154] [INSPIRE]. |
| [79] | S.-S. Xu, Z.-F. Cui, B. Wang, Y.-M. Shi, Y.-C. Yang and H.-S. Zong, Chiral phase transition with a chiral chemical potential in the framework of Dyson-Schwinger equations, Phys. Rev. D91 (2015) 056003 [arXiv:1505.00316] [INSPIRE]. |
| [80] | Jiang, Y.; Luo, L-J; Zong, H-S, A model study of quark number susceptibility at finite temperature beyond rainbow-ladder approximation, JHEP, 02, 066 (2011) · Zbl 1294.81352 · doi:10.1007/JHEP02(2011)066 |
| [81] | A.-M. Zhao, Z.-F. Cui, Y. Jiang and H.-S. Zong, Nonlinear susceptibilities under the framework of Dyson-Schwinger equations, Phys. Rev. D90 (2014) 114031 [arXiv:1412.6884] [INSPIRE]. |
| [82] | C. Shi, Y.-l. Wang, Y. Jiang, Z.-f. Cui and H.-S. Zong, Locate qcd critical end point in a continuum model study, JHEP07 (2014) 014. |
| [83] | C. Ratti and R. Bellwied, The Deconfinement Transition of QCD: Theory Meets Experiment, Lecture Notes in Physics, Springer Cham, Switzerland (2021) [DOI]. |
| [84] | K. Holland and U.-J. Wiese, The Center symmetry and its spontaneous breakdown at high temperatures, hep-ph/0011193 [INSPIRE]. · Zbl 0987.81607 |
| [85] | K. Rajagopal, The Chiral phase transition in QCD: Critical phenomena and long wavelength pion oscillations, hep-ph/9504310 [INSPIRE]. |
| [86] | Borsányi, S., The QCD equation of state with dynamical quarks, JHEP, 11, 077 (2010) · Zbl 1294.81269 · doi:10.1007/JHEP11(2010)077 |
| [87] | HotQCD collaboration, Chiral crossover in QCD at zero and non-zero chemical potentials, Phys. Lett. B795 (2019) 15 [arXiv:1812.08235] [INSPIRE]. |
| [88] | S. Borsányi et al., QCD Crossover at Finite Chemical Potential from Lattice Simulations, Phys. Rev. Lett.125 (2020) 052001 [arXiv:2002.02821] [INSPIRE]. |
| [89] | HotQCD collaboration, Chiral Phase Transition Temperature in (2 + 1)-Flavor QCD, Phys. Rev. Lett.123 (2019) 062002 [arXiv:1903.04801] [INSPIRE]. |
| [90] | H.T. Ding, S.T. Li, S. Mukherjee, A. Tomiya, X.D. Wang and Y. Zhang, Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD, Phys. Rev. Lett.126 (2021) 082001 [arXiv:2010.14836] [INSPIRE]. |
| [91] | Philipsen, O., Lattice QCD at finite temperature and density, Eur. Phys. J. ST, 152, 29 (2007) · doi:10.1140/epjst/e2007-00376-3 |
| [92] | P. de Forcrand and O. Philipsen, The QCD phase diagram for small densities from imaginary chemical potential, Nucl. Phys. B642 (2002) 290 [hep-lat/0205016] [INSPIRE]. · Zbl 0998.81535 |
| [93] | M. D’Elia and M.-P. Lombardo, Finite density QCD via imaginary chemical potential, Phys. Rev. D67 (2003) 014505 [hep-lat/0209146] [INSPIRE]. |
| [94] | L.-K. Wu, X.-Q. Luo and H.-S. Chen, Phase structure of lattice QCD with two flavors of Wilson quarks at finite temperature and chemical potential, Phys. Rev. D76 (2007) 034505 [hep-lat/0611035] [INSPIRE]. |
| [95] | M. D’Elia, F. Di Renzo and M.P. Lombardo, The Strongly interacting quark gluon plasma, and the critical behaviour of QCD at imaginary mu, Phys. Rev. D76 (2007) 114509 [arXiv:0705.3814] [INSPIRE]. |
| [96] | S. Conradi and M. D’Elia, Imaginary chemical potentials and the phase of the fermionic determinant, Phys. Rev. D76 (2007) 074501 [arXiv:0707.1987] [INSPIRE]. |
| [97] | de Forcrand, P.; Philipsen, O., The Chiral critical point of N_f = 3 QCD at finite density to the order (μ/T)^4, JHEP, 11, 012 (2008) · doi:10.1088/1126-6708/2008/11/012 |
| [98] | M. D’Elia and F. Sanfilippo, Thermodynamics of two flavor QCD from imaginary chemical potentials, Phys. Rev. D80 (2009) 014502 [arXiv:0904.1400] [INSPIRE]. |
| [99] | Moscicki, JT; Wos, M.; Lamanna, M.; de Forcrand, P.; Philipsen, O., Lattice QCD Thermodynamics on the Grid, Comput. Phys. Commun., 181, 1715 (2010) · Zbl 1219.81245 · doi:10.1016/j.cpc.2010.06.027 |
| [100] | C.R. Allton et al., The QCD thermal phase transition in the presence of a small chemical potential, Phys. Rev. D66 (2002) 074507 [hep-lat/0204010] [INSPIRE]. · Zbl 1080.81622 |
| [101] | C.R. Allton et al., Thermodynamics of two flavor QCD to sixth order in quark chemical potential, Phys. Rev. D71 (2005) 054508 [hep-lat/0501030] [INSPIRE]. |
| [102] | R.V. Gavai and S. Gupta, QCD at finite chemical potential with six time slices, Phys. Rev. D78 (2008) 114503 [arXiv:0806.2233] [INSPIRE]. |
| [103] | MILC collaboration, QCD equation of state at non-zero chemical potential, PoSLATTICE2008 (2008) 171 [arXiv:0910.0276] [INSPIRE]. |
| [104] | O. Kaczmarek et al., Phase boundary for the chiral transition in (2 + 1)-flavor QCD at small values of the chemical potential, Phys. Rev. D83 (2011) 014504 [arXiv:1011.3130] [INSPIRE]. |
| [105] | D.E. Kharzeev and Y. Kikuchi, Real-time chiral dynamics from a digital quantum simulation, Phys. Rev. Res.2 (2020) 023342 [arXiv:2001.00698] [INSPIRE]. |
| [106] | Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I, Phys. Rev.122 (1961) 345 [INSPIRE]. |
| [107] | Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II, Phys. Rev.124 (1961) 246 [INSPIRE]. |
| [108] | Lu, Y.; Du, Y-L; Cui, Z-F; Zong, H-S, Critical behaviors near the (tri-)critical end point of qcd within the njl model, Eur. Phys. J. C, 75, 495 (2015) · doi:10.1140/epjc/s10052-015-3720-2 |
| [109] | Du, Y-L; Lu, Y.; Xu, S-S; Cui, Z-F; Shi, C.; Zong, H-S, Susceptibilities and critical exponents within the Nambu-Jona-Lasinio model, Int. J. Mod. Phys. A, 30, 1550199 (2015) · Zbl 1337.81124 · doi:10.1142/S0217751X15501997 |
| [110] | Cui, Z-F; Hou, F-Y; Shi, Y-M; Wang, Y-L; Zong, H-S, Progress in vacuum susceptibilities and their applications to the chiral phase transition of QCD, Annals Phys., 358, 172 (2015) · Zbl 1343.81212 · doi:10.1016/j.aop.2015.03.025 |
| [111] | Y.-l. Du, Z.-f. Cui, Y.-h. Xia and H.-s. Zong, Discussions on the crossover property within the Nambu-Jona-Lasinio model, Phys. Rev. D88 (2013) 114019 [arXiv:1312.1796] [INSPIRE]. |
| [112] | S. Shi, Y.-C. Yang, Y.-H. Xia, Z.-F. Cui, X.-J. Liu and H.-S. Zong, Dynamical chiral symmetry breaking in the NJL model with a constant external magnetic field, Phys. Rev. D91 (2015) 036006 [arXiv:1503.00452] [INSPIRE]. |
| [113] | D.P. Menezes, M. Benghi Pinto, S.S. Avancini and C. Providencia, Quark matter under strong magnetic fields in the SU(3) Nambu-Jona-Lasinio Model, Phys. Rev. C80 (2009) 065805 [arXiv:0907.2607] [INSPIRE]. |
| [114] | D.P. Menezes, M. Benghi Pinto, S.S. Avancini, A. Perez Martinez and C. Providencia, Quark matter under strong magnetic fields in the Nambu-Jona-Lasinio Model, Phys. Rev. C79 (2009) 035807 [arXiv:0811.3361] [INSPIRE]. |
| [115] | S. Ghosh, S. Mandal and S. Chakrabarty, Chiral properties of QCD vacuum in magnetars- A Nambu-Jona-Lasinio model with semi-classical approximation, Phys. Rev. C75 (2007) 015805 [astro-ph/0507127] [INSPIRE]. |
| [116] | G.S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S.D. Katz and A. Schafer, QCD quark condensate in external magnetic fields, Phys. Rev. D86 (2012) 071502 [arXiv:1206.4205] [INSPIRE]. · Zbl 1309.81258 |
| [117] | D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D10 (1974) 3235 [INSPIRE]. |
| [118] | M. Thies, Phase structure of the (1 + 1)-dimensional Nambu-Jona-Lasinio model with isospin, Phys. Rev. D101 (2020) 014010 [arXiv:1911.11439] [INSPIRE]. |
| [119] | Jordan, P.; Wigner, E., Über das paulische äquivalenzverbot, Z. Phys., 47, 631 (1928) · JFM 54.0983.03 · doi:10.1007/BF01331938 |
| [120] | Gross, DJ; Pisarski, RD; Yaffe, LG, QCD and Instantons at Finite Temperature, Rev. Mod. Phys., 53, 43 (1981) · doi:10.1103/RevModPhys.53.43 |
| [121] | L.D. McLerran and B. Svetitsky, Quark Liberation at High Temperature: A Monte Carlo Study of SU(2) Gauge Theory, Phys. Rev. D24 (1981) 450 [INSPIRE]. |
| [122] | Polyakov, A., Thermal properties of gauge fields and quark liberation, Phys. Lett. B, 72, 477 (1978) · doi:10.1016/0370-2693(78)90737-2 |
| [123] | Z. Fang, Y.-L. Wu and L. Zhang, Chiral Phase Transition with 2 + 1 quark flavors in an improved soft-wall AdS/QCD Model, Phys. Rev. D98 (2018) 114003 [arXiv:1805.05019] [INSPIRE]. |
| [124] | F. Burger, E.-M. Ilgenfritz, M.P. Lombardo and A. Trunin, Chiral observables and topology in hot QCD with two families of quarks, Phys. Rev. D98 (2018) 094501 [arXiv:1805.06001] [INSPIRE]. |
| [125] | J. Walecka, A theory of highly Condens. Mat., Annals Phys.83 (1974) 491. |
| [126] | H. Ohata and H. Suganuma, Clear correlation between monopoles and the chiral condensate in SU(3) QCD, Phys. Rev. D103 (2021) 054505 [arXiv:2012.03537] [INSPIRE]. |
| [127] | N. Carabba and E. Meggiolaro, Study of some local and global U(1) axial condensates in QCD at finite temperature, Phys. Rev. D105 (2022) 054034 [arXiv:2106.10074] [INSPIRE]. |
| [128] | J.I. Kapusta and C. Gale, Finite-Temperature Field Theory: Principles and Applications, 2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K. (2006) [DOI]. · Zbl 1121.70002 |
| [129] | M. Buballa, NJL model analysis of quark matter at large density, Phys. Rept.407 (2005) 205 [hep-ph/0402234] [INSPIRE]. |
| [130] | L. Susskind, Lattice Fermions, Phys. Rev. D16 (1977) 3031 [INSPIRE]. |
| [131] | J.B. Kogut and L. Susskind, Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys. Rev. D11 (1975) 395 [INSPIRE]. |
| [132] | Borsányi, S.; Fodor, Z.; Hölbling, C.; Katz, SD; Krieg, S.; Szabo, KK, Full result for the QCD equation of state with 2 + 1 flavors, Phys. Lett. B, 730, 99 (2014) · doi:10.1016/j.physletb.2014.01.007 |
| [133] | Y. Aoki, Z. Fodor, S.D. Katz and K.K. Szabo, The Equation of state in lattice QCD: With physical quark masses towards the continuum limit, JHEP01 (2006) 089 [hep-lat/0510084] [INSPIRE]. |
| [134] | HotQCD collaboration, Equation of state in (2 + 1)-flavor QCD, Phys. Rev. D90 (2014) 094503 [arXiv:1407.6387] [INSPIRE]. |
| [135] | C. Aubin, T. Blum, C. Tu, M. Golterman, C. Jung and S. Peris, Light quark vacuum polarization at the physical point and contribution to the muon g − 2, Phys. Rev. D101 (2020) 014503 [arXiv:1905.09307] [INSPIRE]. |
| [136] | D. Chakrabarti, A.K. De and A. Harindranath, Fermions on the light front transverse lattice, Phys. Rev. D67 (2003) 076004 [hep-th/0211145] [INSPIRE]. |
| [137] | Trotter, HF, On the product of semi-groups of operators, Proc. Am. Math. Soc., 10, 545 (1959) · Zbl 0099.10401 · doi:10.1090/S0002-9939-1959-0108732-6 |
| [138] | Suzuki, M., Generalized trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Commun. Math. Phys., 51, 183 (1976) · Zbl 0341.47028 · doi:10.1007/BF01609348 |
| [139] | Chowdhury, AN; Somma, RD, Quantum algorithms for gibbs sampling and hitting-time estimation, Quant. Inf. Comput., 17, 41 (2017) |
| [140] | Brandão, FGSL; Kastoryano, MJ, Finite Correlation Length Implies Efficient Preparation of Quantum Thermal States, Commun. Math. Phys., 365, 1 (2019) · Zbl 1406.81017 · doi:10.1007/s00220-018-3150-8 |
| [141] | R.S. Smith, M.J. Curtis and W.J. Zeng, A practical quantum instruction set architecture, arXiv:1608.03355. |
| [142] | J.S. Kottmann et al., TEQUILA: a platform for rapid development of quantum algorithms, Quantum Sci. Technol.6 (2021) 024009. |
| [143] | N.H. Stair and F.A. Evangelista, Qforte: an efficient state simulator and quantum algorithms library for molecular electronic structure, arXiv:2108.04413. |
| [144] | G. Aleksandrowicz et al., Qiskit: An Open-source Framework for Quantum Computing. |
| [145] | A.J. McCaskey, D.I. Lyakh, E.F. Dumitrescu, S.S. Powers and T.S. Humble, XACC: a system-level software infrastructure for heterogeneous quantum-classical computing, Quantum Sci. Technol.5 (2020) 024002. |
| [146] | C. Developers, Cirq, (2021) [Full list of authors on Github: https://github.com/quantumlib/Cirq/graphs/contributors]. |
| [147] | A. Anand et al., A Quantum Computing View on Unitary Coupled Cluster Theory, arXiv:2109.15176 [INSPIRE]. |
| [148] | K. Bharti et al., Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys.94 (2022) 015004 [arXiv:2101.08448] [INSPIRE]. |
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