Tight Success Probabilities for Quantum Period Finding and Phase Estimation

TL;DR

This work provides tight, general-purpose bounds on the success probability ${\cal P}(M)$ for quantum period finding, and hence for quantum phase estimation, under a broad class of classical post-processing rules. By introducing and exploiting the sinusoid-ratio function $H_L(x;M)$ and its perturbations, the authors obtain exact expressions and near-tight lower and upper bounds that converge to 1 as the quantum circuit size grows or as the classical post-processing is enhanced. A key contribution is the explicit accounting of the offset distribution $\delta_k$ when translating the quantum Fourier sampling into a successful recovery of the period, yielding bounds that are superior to prior results (notably those in Ekeräa et al. 2024) for large $M$ and $r$. The results enable principled tradeoffs between quantum resources (via $n$) and classical effort (via $M$), and include practical implications for continued-fraction post-processing and related lattice/brute-force enhancements, supported by a quantum-simulation demonstration. Overall, the paper advances understanding of how to achieve near-certain success in a single quantum run by carefully balancing quantum and classical computation in Shor-type period-finding and phase-estimation tasks.

Abstract

Period finding and phase estimation are fundamental in quantum computing. Prior work has established lower bounds on their success probabilities. Such quantum algorithms measure a state $|\hat\ell\rangle$ in an $n$-qubit computational basis, $\hat\ell \in [0, 2^n - 1]$, and then post-process this measurement to produce the final output, in the case of period finding, a divisor of the period $r$. We consider a general post-processing algorithm which succeeds whenever the measured $\hat\ell$ is within some tolerance $M$ of a positive integer multiple of $2^n / r$. We give new (tight) lower and upper bounds on the success probability that converge to 1. The parameter $n$ captures the complexity of the quantum circuit. The parameter $M$ can be tuned by varying the post-processing algorithm (e.g., additional brute-force search, lattice methods). Our tight analysis allows for the careful exploitation of the tradeoffs between the complexity of the quantum circuit and the effort spent in classical processing when optimizing the probability of success. We note that the most recent prior work in most recent work does not give tight bounds for general $M$.

Figures (2)

• Figure 1: In (a) we give the quantum circuit for phase estimation. The top $t$ qubits estimate the $t$ most significant bits in the phase $\varphi$. In (b) we illustrate non-trivial behvior of the function $H_L(x;M,\varepsilon)$. In all cases, $x=1/2$ is a critical point. In one case it is not the unique critical point. In the other case it is a global maximum, not minimum.
• Figure 2: Quantum circuit for period finding. $\rm H_n$ is the Hadamaard and $\rm F_n$ the Discrete Fourier Transform.

Theorems & Definitions (41)

• Theorem 1: Lower Bound
• Theorem 2: Upper Bound
• Theorem 3
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5: Mittag-Leffler
• Lemma 6: DiGamma