We present our response to the commentary piece by Makri {\it et al.} [arXiv:2410.08239], which raises critiques of our work [Nat. Commun. 15, 8087 (2024)]. In our paper, we considered various settings of open-quantum system dynamics, including non-commuting, non-diagonalizable system-bath coupling, and bosonic/spin/fermionic baths. For these, we showed a direct and explicit relationship between the discrete-time memory kernel ($\mathcal K$) of the generalized quantum master equation (GQME) and the discrete-time influence functions ($I$) of the path integrals. As an application of this, we showed one can construct $\mathcal K$ without projection-free dynamics inputs that conventional methods require, and we also presented a quantum sensing protocol that characterizes the bath spectral density from reduced system dynamics. As the Comment focused on the relationship between ($\mathcal K$) and $I$ in one specific setup (i.e., commuting, diagonalizable system-bath coupling with a bosonic bath), we focus on that aspect in this response. In summary, we could not find a set of equations that explicitly connect $I$ and $\mathcal K$ from Makri's 2020 paper [J. Chem. Theory Comput. 16, 4038
A recent article by Ivander, Lindoy and Lee [Nature Communications 15, 8087 (2024)] claims to discover the relationship between the generalized quantum master equation (GQME) and the path integral for a system coupled to a harmonic bath. However, this relationship was already established in 2020 by Makri in the context of the small matrix decomposition of the path integral (SMatPI) [J. Chem. Theory and Comput. 16, 4038 (2020)]. The procedure that this article uses in its Supplementary Information (SI) to obtain the various matrices follows the SMatPI decomposition steps for the alternative Trotter ordering. The absence of endpoint effects in the kernel matrices of the discretized GQME expression for the reduced density matrix (RDM) is the consequence of a crude GQME discretization and is not consistent with the SMatPI decomposition of an auxiliary matrix presented in the SI. This form is identical to the transfer tensor method (TTM) of Cerrillo and Cao [Phys. Rev. Lett. 112, 110401 (2014)]. Further, the Dyck path section of this article follows precisely the diagrammatic analysis developed by Wang and Cai in a recent paper [Communications in Computational Physics 36, 389 (2024)]. W
The appearance and disappearance of coprocessors by integration into the CPU, the success or failure of coprocessors are examined by summarizing their characteristics from the mainframes of the 1960s. The coprocessors most particularly reviewed are the IBM 360 and CDC-6600 I/O processors, the Intel 8087 math coprocessor, the Cell processor, the Intel Xeon Phi coprocessors, the GPUs, the FPGAs, and the coprocessors of manycores SW26010 and Pezy SC-2 used in high-ranked supercomputers in the TOP500 or Green500. The conditions for a coprocessor to be viable in the medium or long-term are defined.
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