Towards Practical Quantum Phase Estimation: A Modular, Scalable, and Adaptive Approach

TL;DR

This paper tackles the high resource demands of standard quantum phase estimation (QPE) on NISQ devices by introducing Adaptive Windowed Quantum Phase Estimation (AWQPE), a modular approach that partitions phase estimation into overlapping windows using blocks of size $m_i>1$. Each block estimates a chunk of $m_i$ bits via $U^{2^{k+p}}$ operations and an inverse QFT, with a robust classical post-processing stage (LSB-to-MSB ambiguity resolution) that reconciles tentative chunk decisions into a single high-precision phase estimate $\phi_{\text{est}}$. The authors provide a rigorous correctness framework, including lemmas for phase reconstruction, probabilistic guarantees on identifying the most likely outcomes, and handling of special ambiguity cases, all supported by Hoeffding-type bounds on the number of shots $N_{\text{shots}}$. They show that AWQPE reduces per-block qubit requirements and circuit depth while preserving the total unitary effort (still $2^n-1$ applications) and enabling parallel execution, which substantially improves practicality for near-term hardware. Numerical simulations in Qiskit and Dirichlet-kernel based tests validate AWQPE's accuracy across non-uniform chunk sizes and various phase values, highlighting its potential for reliable, scalable QPE and applications like Shor's algorithm on distributed or resource-constrained quantum platforms.

Abstract

Quantum Phase Estimation (QPE) is a cornerstone algorithm in quantum computing, with applications ranging from integer factorization to quantum chemistry simulations. However, the resource demands of standard QPE, which require a large number of coherent qubits and deep circuits, pose significant challenges for current Noisy Intermediate Scale Quantum (NISQ) devices. In this work, we introduce the Adaptive Windowed Quantum Phase Estimation (AWQPE) algorithm, a novel method designed to address the limitations of standard QPE. AWQPE utilizes small, independent blocks of $m > 1$ control qubits to estimate multiple phase bits simultaneously within a "window,'' thereby significantly reducing the number of iterations required to achieve a desired precision. These independent blocks are amenable to parallelization and, when combined with a robust least-significant-bit (LSB) to most-significant-bit (MSB) ambiguity resolution mechanism, enhance the algorithm's accuracy while mitigating the risk of error propagation. Our numerical simulations demonstrate AWQPE's accuracy and robustness, showcasing a distinct balance between resource efficiency and computational speed. This makes AWQPE particularly well-suited for near-term quantum platforms.

Figures (2)

• Figure 1: Quantum circuit diagrams for the non-uniform chunk size test case ($\phi = 0.8203125$ with $[m_1, m_2, \dots, m_B] = [3, 2, 3]$). Each circuit corresponds to an iterative block of the Adaptive Windowed Quantum Phase Estimation algorithm.
• Figure 2: Quantum circuit for the standard QPE for the test case ($\phi = 0.8203125$ with $n = 8$ control qubits).

Theorems & Definitions (15)

• Remark 2.1: Special Chunk
• Remark 2.2: Enhanced Confidence and Ambiguity Handling
• Remark 2.3: Alternative Ambiguity Resolution Criteria
• Remark 2.4: Relevance to Shor's Algorithm
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3: Probabilistic Outcome for a AWQPE Window
• Remark 3.4