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Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians

Alejandro Kunold

Abstract

Based oh the properties of Lie algebras, in this work we develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix and the Hamiltonian superoperator takes the form $I\otimes H-H^\top \otimes I$ where $I$ is the identity matrix and $H$ is the standard Hamiltonian. It is proven that this particular form belongs to a wider class of ways of linearizing the von Neumann equation that can be categorized by the algebra from which they originated. Particular attention is payed to Hermitian algebras that yield real density matrix coefficients substantially simplifying the quantum tomography of the state vector. Based on this ideas, a quantum algorithm to simulate the dynamics of the density matrix is proposed. It is shown that this method, along with the unique properties of the algebra formed by Pauli strings allows to avoid the use of Trotterization hence considerably reducing the circuit depth. Even though we have used the special case of the algebra formed by the Pauli strings, the algorithm can be readily adapted to other algebras. The algorithm is demonstrated for two toy Hamiltonians using the IBM noisy quantum circuit simulator.

Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians

Abstract

Based oh the properties of Lie algebras, in this work we develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix and the Hamiltonian superoperator takes the form where is the identity matrix and is the standard Hamiltonian. It is proven that this particular form belongs to a wider class of ways of linearizing the von Neumann equation that can be categorized by the algebra from which they originated. Particular attention is payed to Hermitian algebras that yield real density matrix coefficients substantially simplifying the quantum tomography of the state vector. Based on this ideas, a quantum algorithm to simulate the dynamics of the density matrix is proposed. It is shown that this method, along with the unique properties of the algebra formed by Pauli strings allows to avoid the use of Trotterization hence considerably reducing the circuit depth. Even though we have used the special case of the algebra formed by the Pauli strings, the algorithm can be readily adapted to other algebras. The algorithm is demonstrated for two toy Hamiltonians using the IBM noisy quantum circuit simulator.
Paper Structure (10 sections, 113 equations, 7 figures)

This paper contains 10 sections, 113 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Vectorization of the density matrix.
  3. Time evolution
  4. Algorithm
  5. Quantum circuit
  6. Example 1: one spin 1/2 particle subject to a time varying magnetic field
  7. Example 2: two spin 1/2 particles coupled through exchange interaction subject to a oscillating magnetic field.
  8. Conclusions
  9. Useful identities for the $\mathfrak{q}_n\otimes \mathfrak{q}_n$
  10. Useful properties of the Pauli strings algebra

Figures (7)

  • Figure 1: Main quantum algorithm to determine the dynamics of the density matrix coefficients $\rho_1(t)$,$\dots$, $\rho_n(t)$. (b) Controlled $M\left(\boldsymbol{\alpha}\right)$ gate expressed as a sequence of differential time steps. (c) One time step controlled differential $M\left(d\boldsymbol{\alpha}\right)$ gate expressed as the series of Hamiltonian gates generated by the structure constants of the Pauli strings.
  • Figure 2: Controlled $M\left(\boldsymbol{\alpha}\right)$ gate expressed as a sequence of differential time steps.
  • Figure 3: One time step controlled differential $M\left(d\boldsymbol{\alpha}\right)$ gate expressed as the series of Hamiltonian gates generated by the structure constants of the Pauli strings.
  • Figure 4: Density matrix coefficients as functions of time for the magnetic resonance Hamiltonian. The plots from the classical (continuous lines) and quantum (circles) computations are shown.
  • Figure 5: Expected values of the lowest energy level population, highest energy level population and spin projection along the $z$ axis. The classical and quantum computations are shown as solid lines and circles, respectively.
  • ...and 2 more figures