Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space

TL;DR

This paper introduces a logarithmic-qubit encoding for solid state Hamiltonians, significantly reducing hardware requirements and enabling efficient variational quantum simulations on near-term devices.

Contribution

It presents a novel logarithmic-qubit encoding and measurement strategy that drastically reduces the quantum resources needed for simulating large solid state Hamiltonians.

Findings

Reduced the total computation volume from N^2 to (log N)^3.

Developed a variational circuit compatible with the reduced encoding.

Achieved exponential reduction in hardware size and time for simulations.

Abstract

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 $N$ physical sites onto only $\lceil \log_2 N \rceil$ 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 qubit, 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 $N^2$ to $(logN)^3$ for 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.