Dynamical implementation of canonical second quantization on a quantum computer

We develop theoretical methods for the implementation of creation and destruction operators in separate registers of a quantum computer, allowing for a transparent and dynamical creation and destruction of particle modes in second quantization in problems with variable particle number. We establish theorems for the commutation (anticommutation) relations on a finite memory bank and provide the needed symmetrizing and antisymmetrizing operators. Finally, we provide formulas in terms of these operators for unitary evolution under conventional two- and four-body Hamiltonian terms, as well as terms varying the particle number. In this formalism, the number of qubits needed to codify <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>n</a:mi></a:math> particles with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>N</c:mi><c:mi>p</c:mi></c:msub></c:math> modes each is of order <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi mathvariant="script">O</e:mi><e:mrow><e:mo stretchy="false">(</e:mo><e:mi>n</e:mi><e:msub><e:mi>log</e:mi><e:mn>2</e:mn></e:msub><e:msub><e:mi>N</e:mi><e:mi>p</e:mi></e:msub><e:mo stretchy="false">)</e:mo></e:mrow></e:math>. Such scaling is more efficient than the Jordan-Wigner transformation which requires <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>O</j:mi><j:mo stretchy="false">(</j:mo><j:msub><j:mi>N</j:mi><j:mi>p</j:mi></j:msub><j:mo stretchy="false">)</j:mo></j:math> qubits, whenever there are a modest number of particles with a large number of states available to each (and less advantageous for a large number of particles with few states available to each). Published by the American Physical Society 2024