[Paper Review] Time-Optimal Generation of Cluster States

Robert A. Fisher, Haidong Yuan+2 more|DSpace@MIT (Massachusetts Institute of Technology)|Mar 24, 2009

Stochastic processes and statistical mechanics3 citations

TL;DR

This paper demonstrates that time-optimal generation of cluster states can be achieved using optimal control theory, revealing that standard Ising evolution is not always fastest. By analyzing geodesics on the Bloch sphere and leveraging non-orthogonal drift and control Hamiltonians, the authors show speedups—e.g., 23% faster for $K_3$—by designing tailored pulse sequences that outperform naive evolution.

ABSTRACT

The definition of a cluster state naturally suggests an implementation scheme: find a physical system with an Ising coupling topology identical to that of the target state, and evolve freely for a time of 1/2J. Using the tools of optimal control theory, we address the question of whether or not this implementation is time-optimal. We present some examples where it is not and provide an explanation in terms of geodesics on the Bloch-sphere.

Motivation & Objective

• To determine whether the standard Ising evolution method for generating cluster states is time-optimal.
• To investigate if control fields can accelerate cluster state preparation beyond the naive evolution time of $1/(2J)$.
• To explore the role of non-orthogonality between drift and control Hamiltonians in enabling time savings.
• To extend analytical insights from the $K_3$ case to larger graphs using numerical optimization.
• To establish minimal preparation times for various cluster state topologies under realistic control settings.

Proposed method

• Applied optimal control theory to minimize time for cluster state preparation under local and global control settings.
• Used the GRAPE algorithm for numerical optimization of control pulses, achieving high-fidelity state transfer.
• Reduced the problem to a $d imes d$ effective Hamiltonian using symmetry-adapted bases for larger graphs.
• Analyzed geodesic paths on the Bloch sphere to explain time savings in the $K_3$ case via non-orthogonal drift and control terms.
• Employed the Trotter decomposition to decompose time evolution into alternating rotations about orthogonal axes, identifying optimal trajectories.
• Validated results across multiple graph topologies, including $K_3$, $K_4$, $C_4$, $K_5$, $K_6$, $K_7$, and $G_{2,3}$, using numerical simulations.

Experimental results

Research questions

• RQ1 Is the standard Ising evolution for cluster state preparation time-optimal?
• RQ2 Can non-orthogonal drift and control Hamiltonians enable faster cluster state generation than the $1/(2J)$ evolution time?
• RQ3 What is the minimal time required to prepare cluster states on various graph topologies, such as $K_3$, $K_4$, and $K_7$?
• RQ4 How do symmetry-adapted bases and reduced-dimensional Hamiltonians facilitate numerical optimization of time-optimal controls?
• RQ5 Does the speedup observed in the $K_3$ case generalize to larger, more complex cluster state graphs?

Key findings

• For the $K_3$ graph, the minimal preparation time is approximately $0.77 imes rac{1}{2J}$, representing a 23% speedup over the standard $1/(2J)$ evolution.
• The time-optimal solution for $K_3$ arises from non-orthogonal drift and control Hamiltonians, enabling a geodesic path on the Bloch sphere that minimizes rotation angle.
• Numerical optimization using GRAPE confirms that both local and global control settings yield identical minimal times for 4-qubit graphs like $K_4$, $C_4$, and $L_3$.
• For $K_4$, the minimal time is $0.91 imes rac{1}{2J}$, indicating a modest but measurable improvement over the standard method.
• Larger graphs such as $K_5$, $K_7$, and $G_{2,3}$ show significant speedups—$0.70 imes rac{1}{2J}$ and $0.60 imes rac{1}{2J}$ respectively—when drift and control Hamiltonians are non-orthogonal.
• The minimal time for $K_7$ is $0.60 imes rac{1}{2J}$, demonstrating that speedups scale with the complexity and non-orthogonality of the system's Hamiltonian structure.