Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space

Current and near-term quantum hardware is constrained by limited qubit counts, circuit depth, and the high cost of repeated measurements. We address these challenges for solid-state Hamiltonians by introducing a logarithmic-qubit encoding that maps a system with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> physical sites onto only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo fence="false" stretchy="false">&amp;#x2308;</mml:mo> <mml:msub> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>&amp;#x2061;</mml:mo> <mml:mi>N</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x2309;</mml:mo> </mml:math> qubits while maintaining a clear correspondence with the underlying physical model. Within this reduced register, we construct a compatible variational circuit and a Gray-code-inspired measurement strategy whose number of global settings grows only logarithmically with system size. To quantify the overall hardware load, we introduce a volumetric efficiency metric that combines the number of qubits, circuit depth, and the number of measurement settings into a single measure, expressing the overall computation costs. Using this metric, we show that the total space--time sampling volume required in a variational loop can be reduced dramatically from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mo>&amp;#x2061;</mml:mo> <mml:mi>N</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mn>3</mml:mn> </mml:msup> </mml:math> for a hardware-efficient ansatz, allowing an exponential reduction in time and size of the quantum hardware. These results demonstrate that large, structured solid-state Hamiltonians can be simulated on substantially smaller quantum registers with controlled sampling overhead and manageable circuit complexity, extending the reach of variational quantum algorithms on near-term devices.