Bloch oscillation with a diatomic tight-binding model on quantum computers

We aim to explore a more efficient way to simulate few-body dynamics on quantum computers. Instead of mapping the second quantization of the system Hamiltonian to qubit Pauli gate representation via the Jordan-Wigner transform, we propose to use the few-body Hamiltonian matrix under the state-vector basis representation, which is more economical on the required number of quantum registers. For a single-particle excitation state on a one-dimensional chain, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="normal">Γ</a:mi></a:math> qubits can simulate <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>N</c:mi><c:mo>=</c:mo><c:msup><c:mn>2</c:mn><c:mi mathvariant="normal">Γ</c:mi></c:msup></c:mrow></c:math> number of sites, in comparison to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>N</e:mi></e:math> qubits for <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>N</f:mi></f:math> sites via the Jordan-Wigner approach. A two-band diatomic tight-binding model is used to demonstrate the effectiveness of the state-vector basis representation. Both one-particle and two-particle quantum circuits are constructed, and some numerical tests on IBM hardware are presented.