q-Point: A Numeric Format for Quantum Circuit Simulation Using Polar Form Complex Numbers

Quantum circuit simulation is playing a critical role in the current era of quantum computing. However, quantum circuit simulation suffers from huge memory requirements that scale exponentially according to the number of qubits. Our observation reveals that the conventional complex number representation using real and imaginary values adds to the memory overhead beyond the intrinsic cost of simulating quantum states. Instead, using the radius and phase value of a complex number better reflects the properties of the complex values used in the quantum circuit simulation providing better memory efficiency. This paper proposes q-Point, a compact numeric format for quantum circuit simulation that utilizes polar form representation instead of rectangular form representation to store complex numbers. The proposed q-Point format consists of three fields: i) exponent bits for radius value ii) mantissa bits for radius value iii) mantissa bits for phase value. However, a naive application of the q-Point format has the potential to cause issues with both simulation accuracy and simulation speed. To preserve simulation accuracy with fewer bits, we use a multi-level encoding scheme that employs different mantissa bits depending on the exponent range. Additionally, to prevent possible slowdown due to the add operation in polar form complex numbers, we use a technique that adaptively applies both polar and rectangular forms. Equipped with these optimizations, the proposed q-Point format demonstrates reasonable simulation accuracy while using only half of the memory requirement using the baseline format. Additionally, the q-Point format enables an average of 1.37× and 1.16× faster simulation for QAOA and VQE benchmark circuits.