Introduction
Cross-plane Thermal Conductivity of 1-D Superlattices
Previous experimental work has shown that thermal conductivity can be significantly reduced using nanostructures and nanocompositions due to increased boundary scattering. Thus, studying a simple system to understand its physics and developing a predictive model to calculate its thermal conductivity can help in the search for materials with even lower thermal conductivity.
Low-rank Representation of the Electronic Structure Hamiltonian
The utility of this low-level decomposition can be realized in a variety of quantum algorithms, ranging from near-term target algorithms to fault-tolerant schemes (qubitization [6, 7]) to state measurements [8, 9]. ] and even classical algorithms such as quantum Monte Carlo [10].
Long-time Dynamics Using Tensor Networks
For example, recent works include the construction of IFs (or IF-like objects such as the auxiliary density operator as defined in QUAPI, the process tensor or the time-line collector network) for translational invariant systems [63], harmonic baths with linear coupling using analytical solutions [57, 64] and other systems with star geometries [65, 66]. Combining the two results suggests (but does not prove) that 𝑆 ∼ O (𝜉), which is consistent with the intuition that entanglement across some boundary must depend only on O (𝜉) sites near the boundary.
Thermal Transport in 1-D Superlattices via the Boltzmann Trans-
Introduction
However, it has never been verified that the Boltzmann transport equation (BTE), which by construction describes phonons as particles, cannot realize a minimum of thermal conductivity in 1-D superlattices with respect to the superlattice period. We then introduce our algorithm for solving the thermal conductivity for 1-D superlattices, incorporating spectral phonon properties and considering more general interfaces.
Review of Phonon Band Structure in Crystals
In Einstein's phonon model, all phonons are assumed to have the same energy, and the density of states is where 𝑁 is the number of atoms and thus modes in the 1-D system. 𝑅®ℓ is the lattice vector pointing to ℓth unit cell, 𝑚𝑖 is the mass of the 𝑖th atom in the unit cell and 𝜓.
Introduction to the Boltzmann Transport Equation (BTE)
2.6) where the last term is the collision term obtained by considering the interactions of the single particle captured by the first hierarchical equation with the other remaining particles. Phonons are determined by atomistic material properties and are described by a dispersion relation𝜔(®𝑘), where𝜔 is the frequency of the phonons.
Prior Work on Superlattice Thermal Conductivity
The interface's TBR therefore also varies with the thicknesses of the materials on either side of it. Alternatively, one can calculate the thermal conductivity of the superlattice using ab initio harmonic and anharmonic force constants calculated using density functional perturbation theory (DFPT.
Solving the Spectral BTE for 1-D Superlattices
Δ𝑇(𝑥ˆ) is the deviation of the local temperature with respect to the equilibrium temperature, and Δ𝑇𝐿 and Δ𝑇𝑅 are the temperatures specified at the left and right boundaries, respectively. We define the partial specularity for the frequency bin 𝜔 as 𝑝𝜔 with respect to the diffusive phonon mode distribution as .
Calculations for Toy Models
After solving for the coefficients 𝐵®(1), 𝑃®(1), 𝐵®(2) and 𝑃®(2), the thermal conductivity can be calculated by inserting them into Eq. In the case of frequency filters, as the thermal conductivity decreases, it first decreases and then reaches a plateau. As 𝐿 decreases, the thermal conductivity will initially decrease as phonons bounce off the interface more often.
We find that the thermal conductivity trend with respect to temperature varies significantly qualitatively depending on the properties of the interface.
Discussion
These rotations are determined by the angles of the Givens rotations 𝑟𝑝 𝑞(𝜃𝑝 𝑞) which are used to perform the QR decomposition 𝑈(ℓ). The influence functionals [43] (IF) represent the dynamics of an arbitrary bath and its interaction with the subsystem [43] and can be considered as a reweighting of the path integral of the subsystem. The accuracy of the transverse contraction scheme depends on the entanglement in the time-like direction.
In diagram (c), we combine the three arms of the tensor into a single arm.
Low-Rank Approximation of Electronic Structure Hamiltonian . 45
Background of Related Technical Concepts
For electronic structure calculations we are interested in solutions of the electronic part of the. One can represent the quantum state using Slater determinants containing the single-particle orbitals along column, for each of the 𝑚 electrons. Here𝑝, 𝑞index one of the𝑁 orbitals,𝑎†, 𝑎 are fermionic creation and annihilation operators, and 𝑛 are number operators.
However, the number of groups alone does not accurately reflect the number of circuit iterations required to converge to the expected values.
Low-Rank Decomposition
Due to the commutation relations of the creation and annihilation operators, 𝑉0 obeys an eightfold symmetry: ℎ 𝑟 𝑠= ℎ𝑞 𝑝 𝑠𝑟 = ℎ𝑟 𝑠𝑞 𝑝 = ℎ ℎ𝑠𝑟 𝑝𝑞. To perform the decomposition, we use a modified Cholesky decomposition scheme that recursively increases 𝐿, the number of Cholesky vectors L(ℓ) involved in the sum, until the maximum decomposition error is within an error threshold 𝜀𝐶 𝐷 [33 -37]. The computational cost of the modified Cholesky decomposition scheme is known to scale asymptotically as O (𝑁3) within the AO basis for a fixed error threshold [36–38].
Again, due to the eightfold symmetry of the Hamiltonian, this is also a real symmetry and can be diagonalized.
Accuracy of Low-Rank Approximation
The doubly nested decomposition of the unitarily coupled cluster (uCC) operator is slightly more complicated due to its antisymmetry (𝑡𝑎 𝑏𝑖 𝑗 =𝑡𝑗 𝑖 𝑏 𝑎 =−𝑡𝑏 𝑎𝑖 𝑗 =−𝑡𝑎 𝑏 𝑗 𝑖), and is elaborated in the Additional information of ref. The structure of the iron-sulfur nucleus and the number of active orbitals and electrons for each complex are summarized in the diagram below. Left) linear scaling of the number of 𝐿 vectors with base size 𝑁, in the low approximation of 𝐻. Middle) sublinear scaling of the mean eigenvalue number h𝜌ℓi.
One can obtain a better estimate of the energy of the target eigenstate by using a first order correction.
Quantum Resource Estimates
Finally, using a fermionic exchange network, the component corresponding to the evolution of the pairwise operator𝑉(ℓ) can be applied to 𝜌2ℓ. From our analysis of the decay steps, we expect the Trotterized exponential Hamiltonian to have a gate number of 𝑁𝑔𝑎𝑡 𝑒 ∼ O (𝑁3) for fixed molecular size and increasing basis size. We expect to see O(𝑁2) scaling for asymptotically large molecular sizes, but none of the systems studied here reach that limit.) Moreover, the divergence from the linear scaling we saw earlier with increasing alkane chain length in hℓi is no longer as noticeable due to tails in the distribution at 𝜌ℓ).
This means that the sizes of the rotation matrices are reduced and the eigenvalue equation becomes
Discussion
We investigate its approximation with respect to the bond dimensions and time-like entanglement in the tensor network description. However, IF is simply a definite integral of the space-time dynamics that can be obtained numerically. The thick dashed line corresponds to reference dynamics from direct time evolution of the density matrix.
Assuming normalization of the state, the maximum entanglement entropy (Eq. A.2) that the MPS can have is 𝑆𝑚 𝑎𝑥 =log(𝐷).
Time Evolution Using Low-Rank Influence Functional for Gen-
Introduction
However, the cost of computing the memory core without approximation is comparable to determining the dynamics of the full system. This analytical expression takes the form of the Boltzmann weight of a complex valued Hamiltonian defined in the time direction with pairwise interactions between time points [ 44 , 45 ]. In summary, there exists a variety of time evolution methods for open quantum systems that rely on simplifying the calculation of the bath dynamics (using classical, semi-classical or perturbative methods) to reduce computational cost.
In the context of the NZM equation, more spatio-temporal dynamics would correspond to memory cores with a shorter lifetime.
Theory
The overall evolution of the system with 𝐾 bathing modes and 𝑁 time steps thus corresponds to the two-dimensional tensor lattice diagram shown in the figure. The cost of shrinking the columns together in the process of constructing the final influence functional is O (𝑁 𝐷2 . . 𝜌), where 𝐷𝐼 is the bond dimension of the MPS column bath (defined along the time direction). If the bath extends to the left and right of the place of interest, we calculate the IF.
In our tensor network diagrams, we denote tensors in canonical form using triangles, pointing in the direction of the uncontracted leg.
Results
The plots show that for small bath sizes the error in the iterative compression scheme is dominated by the lack of compressibility of the final IF. First, compared to using the continuous bath density, the error of the analytical IF dynamics is increased, although it is still somewhat compressible. 4.8(b), we show the time-averaged error of the IF dynamics as a function of the number of bath sites.
As the bath size increases, the EE IF decreases and approaches some finite value.
Conclusions
Millis, “Alleviation of the dynamical sign problem in the real-time evolution of quantum many-body problems,” Phys. Royer, “Combination of projection superoperators and cumulant expansions in open quantum dynamics with initial correlations and fluctuation of Hamiltonians and environments,” Phys. Pollock, “Exploiting the causal tensor network structure of quantum processes to efficiently simulate non-Markovian path integrals,” Phys.
White, “Spectral functions in one-dimensional quantum systems at finite temperature using the density matrix renormalization group,” Phys.
High-dimensional Tensors
This represents vectorizing the tensors into a vector whose magnitude is the product of all the dimensions of the original legs. Transforming a vector into any number of legs follows the same idea, but requires that the original dimension of the leg be equal to the product of the dimensions of the new legs. For higher dimensional tensors, depending on the desired connection of the decomposed tensors, there is no generally optimal decomposition algorithm.
Other decomposition methods include CP and Tucker decomposition, which rely on alternating least squares (ALS) to minimize the error of the decomposition relative to the original tensor.
Many-body Quantum States
If you measure qubit 𝐵, depending on the outcome (0 or 1), the state of 𝐴 is also known: the states of the qubits are perfectly correlated with each other. The matrix has only rank one, which means that |Ψ𝑥 is actually just a product of one-qubit states of 𝐴and𝐵,. The correlations between parts 𝐴 and 𝐵 of the system have nothing to do with that, and that is what entanglement refers to.
From a linear algebra point of view, we can understand the difference between the two as performing decompositions along different tensor axes.
Tensor Networks
It is clear that the memory associated with the system state representation scales exponentially with the system size 𝑛. After applying a block, the updated part of the MPS is compressed in the link dimension and placed in the appropriate left/right canonical form while right/left is erased. Sierra, “Equivalence of the variational matrix product state method and density matrix renormalization set applied to spin chains,” Europhys.
Head-Gordon, “Highly correlated computations with a polynomial cost algorithm: a study on the density matrix renormalization group,” J.
