Two Variations of Quantum Phase Estimation for Reducing Circuit Error Rates: Application to the Harrow--Hassidim--Lloyd Algorithm

TL;DR

The paper tackles the practical challenge of circuit error in QPE-driven quantum algorithms by introducing two variations, QSPE and QPPE, which respectively shift and puncture phase-estimation qubits based on available phase information. These techniques are integrated into a hybrid quantum–classical HHL framework, yielding significantly reduced qubit counts and gate depths while maintaining correct eigenvalue processing. Hardware experiments on IBM superconducting devices corroborate reduced error rates and closer alignment to ideal solutions for the hybrid approach (Hybrid25) compared to prior methods (Hybrid19). The work provides a generalizable strategy for mitigating circuit errors in QPE-based workflows and lays groundwork for extending these ideas to related spectral and SVD-based quantum algorithms.

Abstract

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, our method reduces qubit and gate counts compared to previous implementations, leading to lower overall circuit error rates on current hardware. Experimental demonstrations on IBM superconducting hardware confirm the error-mitigation effectiveness of the proposed hybrid method.

Figures (11)

• Figure 1: Circuit for the $n$-qubit QFT. This circuit comprises only Hadamard and controlled-$R_k$ gates. The shaded region highlights the sequence of a Hadamard gate followed by controlled-$R_k$ gates that encodes the input labels $j_1,\dots,j_n$ into the phase of the target qubit. The final layer of SWAP gates, which effects the qubit reversal, has been omitted. In this figure, the bottom wire corresponds to the first qubit and the top wire to the last, which reverses the usual top-to-bottom numbering. The same convention applies throughout.
• Figure 2: Circuit for the $n$-qubit IQFT$'$. This circuit is obtained by omitting the SWAP gates originally appended to the original IQFT circuit. As a result, the qubit ordering is reversed, and the assignment of subsequent gates to each qubit is correspondingly adjusted.
• Figure 3: Circuit of the QPE algorithm. The input state $\ket{0}^{\otimes n}\otimes\ket{u}$ is prepared on registers $F$ and $G$. For each $k$, a controlled-$U^{2^{k-1}}$ gate is applied to register $G$ using the $k$-th qubit of $F$ as the control. Register $F$ is then measured to extract the $n$-bit estimate $\widetilde{\varphi}=0.\varphi_1\varphi_2\cdots\varphi_n$ of the phase $\varphi$. The phase-estimation subcircuit (PE step) is used as a subroutine in the HHL algorithm.
• Figure 4: Left-shifting circuit for phase bits. This circuit is derived by omitting the shaded region from the original 4-qubit QPE circuit. More precisely, the first and second qubits of register $F$ are unused, and all associated gates and measurements are removed. The third and fourth bits of the phase estimate $\widetilde{\varphi}$ are obtained by executing the circuit on the remaining third and fourth qubits and measuring them. This example demonstrates that the deletion of the leading qubits of register $F$ induces a left-shifting operation on the phase bit string.
• Figure 5: Punctured circuit for known phase bits. This circuit is derived from the original 4-qubit QPE circuit under the assumption of prior knowledge of certain phase bits. In this example, we assume $\varphi_2 = 0$ and $\varphi_4 = 1$. Based on this information, the second and fourth qubits of register $F$ are rendered inactive, and all associated gates and measurements are removed. The original circuit contains gates acting on the first and third qubits, which are controlled by the second and fourth qubits. These gates are then modified accordingly. The first and third bits of the phase estimate $\widetilde{\varphi}$ are obtained by executing the circuit on the remaining first and third qubits and measuring them. This example demonstrates how puncturing the circuit to remove known-qubit operations enables phase estimation over the remaining unknown bits.