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Entanglement Trajectory and its Boundary

Ruge Lin

TL;DR

A novel approach to investigating entanglement in the context of quantum computing by analyzing reduced density matrices at different stages of a quantum algorithm's execution and representing the dominant eigenvalue and von Neumann entropy on a graph, creating an "entanglement trajectory".

Abstract

In this article, we present a novel approach to investigating entanglement in the context of quantum computing. Our methodology involves analyzing reduced density matrices at different stages of a quantum algorithm's execution and representing the dominant eigenvalue and von Neumann entropy on a graph, creating an "entanglement trajectory." To establish the trajectory's boundaries, we employ random matrix theory. Through the examination of examples such as quantum adiabatic computation, the Grover algorithm, and the Shor algorithm, we demonstrate that the entanglement trajectory remains within the established boundaries, exhibiting unique characteristics for each example. Moreover, we show that these boundaries and features can be extended to trajectories defined by alternative entropy measures. The entanglement trajectory serves as an invariant property of a quantum system, maintaining consistency across varying situations and definitions of entanglement. Numerical simulations accompanying this research are available via open access.

Entanglement Trajectory and its Boundary

TL;DR

A novel approach to investigating entanglement in the context of quantum computing by analyzing reduced density matrices at different stages of a quantum algorithm's execution and representing the dominant eigenvalue and von Neumann entropy on a graph, creating an "entanglement trajectory".

Abstract

In this article, we present a novel approach to investigating entanglement in the context of quantum computing. Our methodology involves analyzing reduced density matrices at different stages of a quantum algorithm's execution and representing the dominant eigenvalue and von Neumann entropy on a graph, creating an "entanglement trajectory." To establish the trajectory's boundaries, we employ random matrix theory. Through the examination of examples such as quantum adiabatic computation, the Grover algorithm, and the Shor algorithm, we demonstrate that the entanglement trajectory remains within the established boundaries, exhibiting unique characteristics for each example. Moreover, we show that these boundaries and features can be extended to trajectories defined by alternative entropy measures. The entanglement trajectory serves as an invariant property of a quantum system, maintaining consistency across varying situations and definitions of entanglement. Numerical simulations accompanying this research are available via open access.
Paper Structure (18 sections, 2 theorems, 58 equations, 22 figures, 1 table)

This paper contains 18 sections, 2 theorems, 58 equations, 22 figures, 1 table.

Table of Contents

  1. Introduction
  2. Examples from quantum computation
  3. Adiabatic quantum computation
  4. Grover algorithm
  5. Semi-classical Shor algorithm
  6. Mathematics Background
  7. Marchenko Pastur distribution
  8. Dominant eigenvalue of decentralized Wishart matrices
  9. Asymptotic Analysis
  10. Average entropy of a subsystem
  11. Average entropy of a subsystem (general case)
  12. Relation with previous examples
  13. $k$-almost Prime states and their unions
  14. General Entanglement Trajectory
  15. Entanglement trajectory in the momentum space
  16. ...and 3 more sections

Key Result

Theorem 1

Marchenko Pastur distribution, (MPD) If $X$ denotes an $\alpha \times \beta$ random matrix whose entries are i.i.d $\mathcal{N}_{\mathbb{C}}\left(0,\sigma\right)$ random variables. Let $Y_{\beta}=\frac{1}{\beta}XX^{\dagger}$. It is a Wishart matrix. And let $\lambda_0, ..., \lambda_{\alpha-1}$ be ei and with

Figures (22)

  • Figure 1: The numerical boundary versus the analytical boundary constrained by $f_1$, $f_2$ and $f_3$. The numerical boundary is calculated as follow: take $\lambda_0=0.4$, its lower bound is given by $-0.4\ln\left(0.4\right)-0.4\ln\left(0.4\right)-0.4\ln\left(0.2\right)$ and its upper bound is given by $-0.4\ln\left(0.4\right)-31\times\frac{1-0.4}{31}\ln\left(\frac{1-0.4}{31}\right)$.
  • Figure 2: The entanglement trajectories for solving three instances of EC problem with $n=12$, we took three random bi-partition (where the subsystem $A$ is a random half of qubits) of each instance. Different instances are plotted with markers $+$, $\times$ and $\star$.
  • Figure 3: The quantum circuit for Grover algorithm to solve one instance of EC problem with $n=10$ and $7$ clauses. The number of bits and clauses are chosen such that two Grover examples share the same number of qubits and can be traced over subsystems of the same size. We sample $\rho_A$ in each iteration before and after the central multi-Toffoli gate and the diffusion operator as indicated with dashed lines. In this circuit, we have $t=25$.
  • Figure 4: The quantum circuit of Grover algorithm to find the pre-image of a Hash function with a given cipher-text. The oracle consists of a unitary that encodes the Hash function, the inverse of this unitary, with a multi-Toffoli gate in the middle that encodes the cipher-text, which is $10100011$ in this figure. We sample $\rho_A$ in each iteration before and after the multi-Toffoli gate and the diffusion operator as indicated with dashed lines. This cipher-text has $2$ pre-images, and the number of Grover iterations needed is $t=8$.
  • Figure 5: The entanglement trajectory of Grover circuits to solve three instances (with $n=10$ and $7$ clauses) of EC problem (plotted in green markers $+$, $\times$ and $\star$) and three cipher-texts (with $1$, $2$ and $3$ pre-images) of the same Hash function (plotted in violet markers $+$, $\times$ and $\star$).
  • ...and 17 more figures

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • proof