Global variational quantum circuits for arbitrary symmetric state preparation

Quantum states that are symmetric under particle exchange play a crucial role in fields such as quantum metrology and quantum error correction. We use a variational circuit composed of global one-axis twisting and global rotations to efficiently prepare arbitrary symmetric states, i.e., any superposition of Dicke states. The circuit does not require local addressability or ancilla qubits and thus can be readily implemented in a variety of experimental platforms including trapped-ion quantum simulators and cavity QED systems. We provide analytic and numerical evidence that any <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math>-qubit symmetric state can be prepared in <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mn>2</b:mn><b:mi>N</b:mi><b:mo>/</b:mo><b:mn>3</b:mn></b:mrow></b:math> steps. We demonstrate the utility of our protocol by preparing (i) metrologically useful <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>N</c:mi></c:math>-qubit Dicke states of up to <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>N</d:mi><d:mo>=</d:mo><d:mn>300</d:mn></d:mrow></d:math> qubits in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi mathvariant="script">O</e:mi><e:mo>(</e:mo><e:mn>1</e:mn><e:mo>)</e:mo></e:mrow></e:math> gate steps with theoretical infidelities <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mn>1</g:mn><g:mo>−</g:mo><g:mi mathvariant="script">F</g:mi><g:mo><</g:mo><g:msup><g:mn>10</g:mn><g:mrow><g:mo>−</g:mo><g:mn>3</g:mn></g:mrow></g:msup></g:mrow></g:math>, (ii) the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>N</i:mi><i:mo>=</i:mo><i:mn>9</i:mn></i:mrow></i:math> Ruskai codewords in <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>P</j:mi><j:mo>=</j:mo><j:mn>4</j:mn></j:mrow></j:math> gate steps with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mn>1</k:mn><k:mo>−</k:mo><k:mi mathvariant="script">F</k:mi><k:mo><</k:mo><k:msup><k:mn>10</k:mn><k:mrow><k:mo>−</k:mo><k:mn>4</k:mn></k:mrow></k:msup></k:mrow></k:math>, and (iii) the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>N</m:mi><m:mo>=</m:mo><m:mn>13</m:mn></m:mrow></m:math> Gross codewords in <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mi>P</n:mi><n:mo>=</n:mo><n:mn>7</n:mn></n:mrow></n:math> gate steps with <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mrow><o:mn>1</o:mn><o:mo>−</o:mo><o:mi mathvariant="script">F</o:mi><o:mo><</o:mo><o:msup><o:mn>10</o:mn><o:mrow><o:mo>−</o:mo><o:mn>4</o:mn></o:mrow></o:msup></o:mrow></o:math>. Focusing on trapped-ion platforms, for the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mrow><q:mi>N</q:mi><q:mo>=</q:mo><q:mn>9</q:mn></q:mrow></q:math> Ruskai and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:mrow><r:mi>N</r:mi><r:mo>=</r:mo><r:mn>13</r:mn></r:mrow></r:math> Gross codewords we estimate that the protocol achieves fidelities <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:mrow><s:mo>≳</s:mo><s:mn>95</s:mn><s:mo>%</s:mo></s:mrow></s:math> in the presence of typical experimental noise levels, thus providing a pathway to the preparation of a wide range of useful highly entangled quantum states.