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

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.

💡 Research Summary

The paper tackles the long‑standing inefficiency of Direct Fidelity Estimation (DFE) when the target pure state possesses a large amount of “magic”, i.e., a large Pauli l₁‑norm that typically grows exponentially with the number of qubits. This situation arises for phase‑dominated states such as hypergraph states or general phase states, for which DFE requires O(2ⁿ) copies of the prepared state to achieve a constant additive error.

The authors introduce a “phase‑stripping” operation that removes all complex phases from a state |ψ⟩, leaving a real‑amplitude state |ψ̃⟩ whose coefficients are the absolute values of those of |ψ⟩. Formally, |ψ⟩ = D(ϕ) |ψ̃⟩ where D(ϕ) is a diagonal unitary encoding the phase function ϕ(x). For many states of interest the magic is almost entirely carried by D(ϕ); after stripping, the l₁‑norm ∥ψ̃∥₁ collapses from exponential to O(poly n) and becomes exactly 1 for genuine phase states (|ψ̃⟩ = |+⟩^{⊗n}).

The key technical contribution is an algorithm that estimates the fidelity ⟨ψ|ρ|ψ⟩ using only a single n‑qubit fan‑out gate (n CNOTs with a common control) together with classical nonlinear post‑processing of Pauli measurement outcomes. The procedure is:

1. Sample a Pauli index a according to the distribution proportional to |c̃_a|, where c̃_a are the Pauli coefficients of |ψ̃⟩ (this is an l₂‑α‑sampling with α∈