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Dynamical Simulations of Schrödinger's Equation via Rank-Adaptive Tensor Decompositions

N. Anders Petersson, Chase Hodges-Heilmann, Stefanie Günther

Abstract

Classical simulations of quantum computing devices generally become intractable as the number of qubits increases. This is due to the exponential growth of the quantum state vector and the associated increase in computational effort. However, when entanglement within the system is limited, rank-adaptive tensor decomposition techniques can be employed to mitigate the exponential scaling. This paper broadens the application of tensor decomposition methods to dynamical simulations of Schrödinger's equation where the Hamiltonian is time-dependent, e.g., to study quantum computing devices subject to time-dependent control pulses. We focus on the tensor-train and Tucker-tensor decompositions that both support low-rank representations, and present an overview of the TDVP, TDVP-2, and BUG, time-integration algorithms for capturing quantum dynamics. The effectiveness of the tensor decomposition approaches is evaluated on representative time-independent and time-dependent Hamiltonian models, with emphasis on how the computational effort scales with the required accuracy and the number of sub-systems in the composite system.

Dynamical Simulations of Schrödinger's Equation via Rank-Adaptive Tensor Decompositions

Abstract

Classical simulations of quantum computing devices generally become intractable as the number of qubits increases. This is due to the exponential growth of the quantum state vector and the associated increase in computational effort. However, when entanglement within the system is limited, rank-adaptive tensor decomposition techniques can be employed to mitigate the exponential scaling. This paper broadens the application of tensor decomposition methods to dynamical simulations of Schrödinger's equation where the Hamiltonian is time-dependent, e.g., to study quantum computing devices subject to time-dependent control pulses. We focus on the tensor-train and Tucker-tensor decompositions that both support low-rank representations, and present an overview of the TDVP, TDVP-2, and BUG, time-integration algorithms for capturing quantum dynamics. The effectiveness of the tensor decomposition approaches is evaluated on representative time-independent and time-dependent Hamiltonian models, with emphasis on how the computational effort scales with the required accuracy and the number of sub-systems in the composite system.
Paper Structure (19 sections, 1 theorem, 78 equations, 10 figures, 1 algorithm)

This paper contains 19 sections, 1 theorem, 78 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Quantum simulations
  3. Matrix-vector formulation
  4. Tensor trains
  5. Matrix product operators
  6. The time-dependent variational principle
  7. The TDVP algorithm
  8. The TDVP-2 algorithm
  9. The basis update and Galerkin (BUG) algorithm
  10. BUG for matrices and Tucker tensors
  11. BUG for tensor trains
  12. Numerical Examples
  13. Time-independent Hamiltonians
  14. Time-Dependent Hamiltonians
  15. Coupled systems with optimized control pulses
  16. ...and 4 more sections

Key Result

Lemma 1

Assume that the basis vectors $\{|\sigma_k\rangle\}_{k=1}^N$ are mutually orthogonal and normalized. Let the set of left state vectors $\{|\psi^L_j(m_j)\rangle\}_{m_j=1}^{b_j}$ be defined by eq:psi-left, where the order-3 tensors $A_{[k]}$ satisfy the left-normalization condition in eq:left-right-no Similarly, let the set of right state vectors $\{|\psi^R_j(m'_j)\rangle\}_{m'_j=1}^{b_j}$ be define

Figures (10)

  • Figure 1: Left: A tensor diagram of a tensor train (MPS) of an order-4 tensor of size $d_1\times d_2\times d_3 \times d_4$, is represented by four cores (circles), each of order three. The vertical legs represent the physical indices and the horizontal legs connecting the cores correspond to contractions over virtual indices. The sizes of the first and last cores are $1\times d_1 \times b_1$ and $b_{N-1}\times d_N\times 1$, respectively; horizontal legs are suppressed on all singleton dimensions. Right: the contraction of an MPS by an order $4\times 4$ matrix product operator (MPO), represented by four (square) cores of order four.
  • Figure 2: TDVP: evaluating the action of the effective Hamiltonian on the state in a 4-site system. The MPS of the state is in the top row, the MPO of the Hamiltonian in the middle row, and the resulting action is represented by the open legs in the bottom row. Left: the single-site Hamiltonian acting on the (green) core $M_{[2]}$, connected with dashed legs. Right: the bond-centered Hamiltonian acting on the (light blue) bond-matrix $C_{[1]}$, connected with dashed legs between the first and second core.
  • Figure 3: TDVP-2: evaluating the action of the effective two-site Hamiltonian on the merged (green rectangular) core. The MPS of the state is in the top row, the MPO of the Hamiltonian in the middle row, and the resulting action is represented by the four open legs in the bottom row.
  • Figure 4: MPS-BUG: application of $\hat{H}^{eff}_2$ to $\widetilde{A}_{[2]}$ (green, top row), giving the slope $\hat{H}^{eff}_2{\widetilde{A}}_{[2]}$ (3 solid legs into empty space, bottom row).
  • Figure 5: Transverse Ising model with $J=1$ and $g=\{0, 0.5\}$, starting from the ground state, integrated with TDVP-2 and MPS-BUG to time $T=5$. Left: run times as functions of the number of qubits. Right: Maximum bond dimensions.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Lemma 1
  • proof
  • Remark 1