Capturing long-range memory structures with tree-geometry process tensors (2312.04624v2)
Abstract: We introduce a class of quantum non-Markovian processes -- dubbed process trees -- that exhibit polynomially decaying temporal correlations and memory distributed across time scales. This class of processes is described by a tensor network with tree-like geometry whose component tensors are (1) causality-preserving maps (superprocesses) and (2) locality-preserving temporal change of scale transformations. We show that the long-range correlations in this class of processes tends to originate almost entirely from memory effects, and can accommodate genuinely quantum power-law correlations in time. Importantly, this class allows efficient computation of multi-time correlation functions. To showcase the potential utility of this model-agnostic class for numerical simulation of physical models, we show how it can efficiently approximate the strong memory dynamics of the paradigmatic spin-boson model, in terms of arbitrary multitime features. In contrast to an equivalently costly matrix product operator (MPO) representation, the ansatz produces a fiducial characterization of the relevant physics. Finally, leveraging 2D tensor network renormalization group methods, we detail an algorithm for deriving a process tree from an underlying Hamiltonian, via the Feynmann-Vernon influence functional. Our work lays the foundation for the development of more efficient numerical techniques in the field of strongly interacting open quantum systems, as well as the theoretical development of a temporal renormalization group scheme.
- J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
- M. Srednicki, Phys. Rev. E 50, 888 (1994).
- L. P. Kadanoff, Statistical physics: statics, dynamics and renormalization, repr ed. (World Scientific, Singapore, 2007).
- J. P. Crutchfield, Nature Physics 8, 17 (2012).
- S. Sachdev, Physics World 12, 33 (1999).
- G. Vidal, Phys. Rev. Lett. 101, 110501 (2008).
- G. Evenbly and G. Vidal, Phys. Rev. B 79, 144108 (2009).
- S. Milz and K. Modi, PRX Quantum 2, 030201 (2021).
- N. Dowling and K. Modi, “An Operational Metric for Quantum Chaos and the Corresponding Spatiotemporal Entanglement Structure,” (2022), arXiv:2210.14926 [quant-ph] .
- K. Le Hur, Annals of Physics 323, 2208 (2008).
- K. Le Hur, “Quantum phase transitions in spin-boson systems: Dissipation and light phenomena,” in Understanding Quantum Phase Transitions, edited by L. D. Carr (CRC Press, New York, 2010) pp. 217–240.
- G. Evenbly and G. Vidal, Journal of Statistical Physics 145, 891 (2011).
- F. G. Brandao and M. Horodecki, Communications in mathematical physics 333, 761 (2015).
- I. Vilkoviskiy and D. A. Abanin, “A bound on approximating non-markovian dynamics by tensor networks in the time domain,” (2023), arXiv:2307.15592 [quant-ph] .
- A. Rivas and S. F. van Huelga, Open Quantum Systems (Springer-Verlag, 2012).
- G. Chiribella and D. Ebler, Nat. Commun. 10, 1472 (2019).
- G. A. L. White, F. A. Pollock, L. C. L. Hollenberg, C. D. Hill, and K. Modi, “From many-body to many-time physics,” (2021), arXiv:2107.13934 [quant-ph] .
- M. B. Hastings and R. Mahajan, Phys. Rev. A 91, 032306 (2015).
- M. R. Jørgensen and F. A. Pollock, Phys. Rev. Lett. 123, 240602 (2019).
- V. Link, H.-H. Tu, and W. T. Strunz, “Open quantum system dynamics from infinite tensor network contraction,” (2023), arXiv:2307.01802 [quant-ph] .
- J. C. Bridgeman and C. T. Chubb, J. Phys. A: Math. Theor. 50, 223001 (2017).
- R. Orús, Nature Reviews Physics 1, 538 (2019).
- One can always add a finite-dimensional ancilla to the SE𝑆𝐸SEitalic_S italic_E system so that the total isolated system evolves unitarily (Stinespring dilation).
- P. Taranto, M. T. Quintino, M. Murao, and S. Milz, “Characterising the hierarchy of multi-time quantum processes with classical memory,” (2023), arXiv:2307.11905 [quant-ph] .
- E. P. Butler, G. E. Fux, C. Ortega-Taberner, B. W. Lovett, J. Keeling, and P. R. Eastham, “Optimizing performance of quantum operations with non-markovian decoherence: the tortoise or the hare?” (2023), arXiv:2303.16002 [quant-ph] .
- P. Strasberg and M. G. Díaz, Phys. Rev. A 100, 022120 (2019).
- One could also consider a fine-graining superprocess that maps an intervention slot to more than two slots. The process ansatz obtained from such higher-order fine-graining maps has all the same structural properties as the one presented in this paper. We chose one-to-two slot intervention maps in order to generate a simple binary tree tensor network.).
- We do not claim that this choice will capture all physics. There may be superprocesses that do not preserve the identity operations, that capture interesting physics.
- B. Swingle, Phys. Rev. D 86, 065007 (2012).
- R. Bhatia, Positive Definite Matrices (Princeton University Press, Princeton, 2007).
- M. M. Wolf, “Quantum channels & operations guided tour,” https://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf, accessed: 2022-6-21.
- M. M. Wilde, Quantum Information Theory, 2nd ed. (Cambridge University Press, 2017).
- M. R. Jørgensen and F. A. Pollock, Phys. Rev. A 102, 052206 (2020).
- The TEMPO collaboration, “OQuPy: A Python 3 package to efficiently compute non-Markovian open quantum systems.” (2020).
- Dong C. Liu and Jorge Nocedal, Mathematical Programming 45, 503 (1989).
- J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, “JAX: composable transformations of Python+NumPy programs,” (2018).
- J. Gray, Journal of Open Source Software 3, 819 (2018).
- D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” (2017), arXiv:1412.6980 [cs.LG] .
- D. Burgarth, P. Facchi, R. Hillier, and M. Ligabò, “Taming the rotating wave approximation,” (2023), arXiv:2301.02269 [quant-ph] .
- F. A. Pollock and K. Modi, Quantum 2, 76 (2018).
- C. Guo, “Memory complexity of quantum processes,” (2022), arXiv:2203.01492 [quant-ph] .
- M. Cygorek, J. Keeling, B. W. Lovett, and E. M. Gauger, “Sublinear scaling in non-markovian open quantum systems simulations,” (2023), arXiv:2304.05291 [quant-ph] .
- R. Movassagh and J. Schenker, Phys. Rev. X 11, 041001 (2021).
- This group is realized by a canonical injective map (i,j)→a→𝑖𝑗𝑎(i,j)\rightarrow a( italic_i , italic_j ) → italic_a such as the ones underlying the reshape() functions of Python and MATLAB.
- R. N. C. Pfeifer, G. Evenbly, S. Singh, and G. Vidal, “Ncon: A tensor network contractor for matlab,” (2015), arXiv:1402.0939 [physics.comp-ph] .
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.