Sample- and Hardware-Efficient Fidelity Estimation by Stripping Phase-Dominated Magic

TL;DR

Direct fidelity estimation (DFE) of a pure target state $\ket{\psi}$ typically requires exponential sampling in the number of qubits $n$ due to magic; the paper proposes phase-stripping to obtain $\ket{\breve{\psi}}$ and a Hadamard-test–based fidelity estimator that replaces heavy diagonal gates with nonlinear classical post-processing, enabling a single $n$-qubit fan-out gate. The FOFE framework achieves sampling costs of $\mathcal{O}\left(\dfrac{\|\breve{\psi}\|^{1/\alpha}_{2-2\alpha}}{\epsilon^2}\log(M\delta_f^{-1})\right)$, reducing to $\mathcal{O}(1)$ for phase states, and is extended by nonlinear direct fidelity estimation (NLDFE) with a divide-and-conquer (DNC) variant that further lowers sampling while preserving Pauli measurements. The work also develops extensions to quantum-state tomography via mutually unbiased bases and Pauli shadows and provides analytical bounds on the phase-stripped norm via stabilizer Rényi entropies. Taken together, these results offer a practical path to noise-resilient quantum algorithms on near-term devices by significantly reducing sampling overhead under restricted gate resources, while clarifying the resource gap between observing properties and generating them.

Abstract

Direct fidelity estimation (DFE) is a famous tool for estimating the fidelity with a target pure state. However, such a method generally requires exponentially many sampling copies due to the large magic of the target state. This work proposes a sample- and gate-efficient fidelity estimation algorithm that is affordable within feasible quantum devices. We show that the fidelity estimation with pure states close to the structure of phase states, for which sample-efficient DFE is limited by their strong entanglement and magic, can be done by using $\mathcal{O}(\mathrm{poly}(n))$ sampling copies, with a single $n$-qubit fan-out gate. As the target state becomes a phase state, the sampling complexity reaches $\mathcal{O}(1)$. Such a drastic improvement stems from a crucial step in our scheme, the so-called phase stripping, which can significantly reduce the target-state magic. Furthermore, we convert a complex diagonal gate resource, which is needed to design a phase-stripping-adapted algorithm, into nonlinear classical post-processing of Pauli measurements so that we only require a single fan-out gate. Additionally, as another variant using the nonlinear post-processing, we propose a nonlinear extension of the conventional DFE scheme. Here, the sampling reduction compared to DFE is also guaranteed, while preserving the Pauli measurement as the only circuit resource. We expect our work to contribute to establishing noise-resilient quantum algorithms by enabling a significant reduction in sampling overhead for fidelity estimation under the restricted gate resources, and ultimately to clarifying a fundamental gap between the resource overhead required to understand complex physical properties and that required to generate them.

Figures (7)

• Figure 1: Schematic illustration of the $6$-qubit fan-out-based fidelity estimation (FOFE). We assume that the sampled $T_{\mathbf{a}}$ is of full Pauli weight. Here, $\ket{+}$ is an ancilla state and $\rho$ is an input. The conjugation of single-qubit Clifford $V_i\;(i\in [6])$ is such that $VXV^{\dag}=T_{\mathbf{a}_i}$.
• Figure 2: (a) The ($n=7$)-qubit estimation variance of fidelity estimation using $n$ CNOTs and one qubit ancilla. We compared our result with DFE flammia2011julia2025. For both methods, the target is a $3$rd-order complete hypergraph state $\ket{K_7}$, and the input state is $\ket{K_7}$ with a depolarizing noise. (analytical fidelity $\simeq0.8955$), and $5000$ sampling copies were used. (b) The $l_1$-norm scaling of Haar-random pure states and their phase-stripped states. $10000$ copies are used to estimate $\mathbb{E}_{\psi\in {\rm Haar}} \|\breve{\psi}\|_1$supple for each qubit size.
• Figure 3: (a) Schematic illustration of DNC-based algorithm for NLDFE. (b) Improvement in $l_1$-norm of $100$-copies of Haar random pure states via the DNC-based algorithm.
• Figure 4: Graph representation (pink edges) corresponding to the adjacency matrix $N_{G}(\mathbf{x})$ from the $25$ qubit Union Jack state zhu2019. Here, $\mathbf{x}$ is chosen following each qubit's $\mathbb{F}_2$ value (number next to each vertex).
• Figure 5: Upper and lower bounds of the variance of $\frac{1}{2}$-DFE for random- and complete-third-ordered hypergraph states. First graph used Eq. \ref{['eq:var_graph_bound']} with $2000$-random copies, and the second used Eq. \ref{['eq:complete_graph_var']}.

Theorems & Definitions (38)

• Theorem 1
• Definition 1
• Lemma 1
• proof
• Corollary 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof