Two variations of quantum phase estimation for reducing circuit error rates: Application to the Harrow-Hassidim-Lloyd algorithm

We introduce two variations of the quantum phase estimation algorithm: quantum shifted phase estimation and quantum punctured phase estimation. The shifted method employs a bit-string left shift to discard the most significant bit and focus on lower-order phase components, and the punctured method removes qubits corresponding to known phase bits, thereby streamlining the circuit. To demonstrate the effectiveness of the two variations, we integrate them into a hybrid quantum-classical implementation of the Harrow--Hassidim--Lloyd algorithm for solving linear systems. The hybrid method leverages both quantum and classical processors to identify and remove unnecessary qubits and gates. As a result, for small-scale linear systems, our method reduces qubit and gate counts relative to previous implementations, yielding lower overall circuit error rates on current hardware. Experimental demonstrations on IBM superconducting hardware confirm the error-mitigation effectiveness on small problem instances.