2025/12/30 by Anonymous, Vladimir M. Stojanovic, Tommaso Calarco +1
Computer Science · Physics and Astronomy · #Quantum Information and Cryptography #Nonlinear Photonic Systems #Spectroscopy and Quantum Chemical Studies
paper · pdf · doi:10.1103/t2zk-4nxh
We explore the feasibility of realizing Dicke states in qubit arrays with always-on isotropic Heisenberg coupling between adjacent qubits, assuming a single Zeeman-type control acting in the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>z</a:mi> </a:math> direction on an actuator qubit. The Lie-algebraic criteria of controllability imply that such an array is not completely controllable, but satisfies the conditions for subspace controllability on any subspace with a fixed number of excitations. Therefore, a qubit array described by the model under consideration is state-to-state controllable for an arbitrary choice of initial and final states that have the same Hamming weight. This limited controllability is exploited here for the time-efficient dynamical generation of an <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>a</b:mi> </b:math> -excitation Dicke state <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mrow> <c:mo>|</c:mo> </c:mrow> <c:msubsup> <c:mi>D</c:mi> <c:mi>a</c:mi> <c:mi>N</c:mi> </c:msubsup> <c:mrow> <c:mo>〉</c:mo> </c:mrow> </c:mrow> </c:math> ( <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>a</d:mi> <d:mo>=</d:mo> <d:mn>1</d:mn> <d:mo>,</d:mo> <d:mn>2</d:mn> <d:mo>,</d:mo> <d:mspace width="0.16em"/> <d:mo>...</d:mo> <d:mo>,</d:mo> <d:mspace width="0.16em"/> <d:mi>N</d:mi> <d:mo>−</d:mo> <d:mn>1</d:mn> </d:mrow> </d:math> ) in a linear array with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>N</g:mi> </g:math> qubits starting from a generic Hamming-weight- <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>a</h:mi> </h:math> product state. To dynamically generate the desired Dicke states—including <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>W</i:mi> </i:math> states <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mrow> <j:mo>|</j:mo> </j:mrow> <j:msub> <j:mi>W</j:mi> <j:mi>N</j:mi> </j:msub> <j:mrow> <j:mo>〉</j:mo> </j:mrow> </j:mrow> </j:math> as their special ( <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mi>a</k:mi> <k:mo>=</k:mo> <k:mn>1</k:mn> </k:mrow> </k:math> ) case—in the shortest possible time with a single local <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mi>Z</l:mi> </l:math> control, we employ an optimal-control scheme based on the (dCRAB) algorithm. We optimize the target-state fidelity over the expansion coefficients of smoothly varying control fields in a truncated random Fourier basis; this is done by combining Nelder-Mead-type local optimizations with the multistart-based clustering algorithm that facilitates searches for global extrema. In this manner, we obtain the optimal-control fields for Dicke-state preparation in arrays with up to nine qubits. Based on our numerical results, we find that the shortest possible state-preparation times scale as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">O</m:mi> <m:mo>(</m:mo> <m:msup> <m:mi>N</m:mi> <m:mrow> <m:mn>2.08</m:mn> </m:mrow> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:math> for <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mi>W</o:mi> </o:math> states and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mi mathvariant="script">O</p:mi> <p:mo>(</p:mo> <p:msup> <p:mi>N</p:mi> <p:mrow> <p:mn>1.78</p:mn> </p:mrow> </p:msup> <p:mo>)</p:mo> </p:mrow> </p:math> for <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:mrow> <r:mi>a</r:mi> <r:mo>=</r:mo> <r:mn>2</r:mn> </r:mrow> </r:math> Dicke states. Finally, we demonstrate the robustness of our dCRAB-based state-engineering scheme against various types of imperfections of relevance for its anticipated experimental implementation.