Estimating Fidelity to a Reference Quantum State

Abstract: We consider the problem of estimating the fidelity of an unknown quantum state to a known reference state to within additive error $\varepsilon$. We show that the sample complexity is $O(r^2/\varepsilon^2)$ with optimal $\varepsilon$-dependence when the reference state is of rank $r$, improving the previous best $O(r^2\log^2(1/\varepsilon)/\varepsilon^4)$ due to Utsumi, Nakata, Wang, and Takagi (QIP 2026). We also provide a lower bound of $Ω(r/\varepsilon^2)$, improving the previous best $Ω(r/\varepsilon+1/\varepsilon^2)$, with implications to quantum query complexity. Moreover, we further consider the case where the unknown state is of rank at most $r$ while the reference state can be arbitrary, for which the sample complexity is shown to be $O(r^2/\varepsilon^4)$. As an application, we present an approach to tolerant quantum state certification, generalizing the exact certification studied in Bădescu, O'Donnell, and Wright (STOC 2019).

• The paper provides rigorous upper and lower bounds for sample complexity in estimating fidelity, achieving optimal O(r/ε²) scaling for a classically specified reference state.
• It introduces a two-stage approach that projects samples onto the support of the reference state and applies subspace tomography using Bures distance to reduce effective dimensionality.
• The work has significant implications for quantum state certification and learning, bridging complexity gaps in quantum property testing and enhancing sample-efficient protocols.

Estimating Fidelity to a Reference Quantum State: Complexity-Theoretic and Algorithmic Advances

Problem Statement and Context

Estimating the fidelity between quantum states—especially when a classical description of a reference state is available—constitutes a fundamental task in quantum information science, underlying quantum state tomography and quantum state certification. Given a known reference quantum state $\sigma$ (with explicit classical specification) and multiple identical copies of an unknown quantum state $\rho$, the objective is to estimate the fidelity $F(\rho,\sigma)$ to within a specified additive accuracy $\epsilon$ using a minimal number of samples or queries.

While practical protocols exist for special cases (such as pure states), the general scenario (especially for mixed or high-rank states, or in high-dimensional settings) is computationally and statistically demanding, with established lower bounds that match the sample complexity of the hardest known quantum property testing problems.

Main Results

This work presents rigorous upper and lower bounds for sample complexity in estimating fidelity to a reference quantum state, significantly improving prior state-of-the-art results, and providing (up to polylogarithmic factors) optimal dependences on key parameters such as rank and target accuracy, for both the sample and quantum query complexity regimes.

Key Theorems:

• For a reference state $\sigma$ of rank $r$, the sample complexity to estimate $F(\rho, \sigma)$ within additive error $\epsilon$ is $O(r/\epsilon^2)$, and any quantum algorithm with access to $\rho$ (with known $\rho$0) requires $\rho$1 samples (Theorem 1.1, Theorem 4.1).
• If the unknown state $\rho$2 has rank at most $\rho$3 (with arbitrary reference $\rho$4), $\rho$5 samples suffice for achieving error $\rho$6 (Theorem 1.2).
• These bounds remove polylogarithmic (in $\rho$7) factors present in earlier protocols [UNWT25], thus achieving optimal $\rho$8-dependence in both settings.

Strong connections to tolerant quantum state certification and quantum property testing are established, along with instance-optimal lower bounds derived by reductions to well-known quantum spectrum testing tasks.

Technical Approach

A crucial structural insight exploited by the proposed algorithms is that, when estimating $\rho$9 with a known $F(\rho,\sigma)$0 of limited rank, the computation can be localized to the support of $F(\rho,\sigma)$1, yielding a drastic reduction in effective dimension. The two-stage algorithm comprises:

1. Filtration: Each sample of $F(\rho,\sigma)$2 is projected onto the support of $F(\rho,\sigma)$3 (using projective measurement), retaining only outcomes with nonzero overlap.
2. Subspace Tomography: State tomography is then conducted only within this low-dimensional subspace, utilizing optimal Bures distance tomography [PSW26], leading to improved dependence on error parameters compared to previous approaches.

The error contribution outside the reference subspace is shown to be negligible due to properties of quantum fidelity and homogeneity, so the fidelity can be well-approximated by only estimating in the projected subspace. For the low-rank unknown state scenario, the reference $F(\rho,\sigma)$4 is truncated to its largest $F(\rho,\sigma)$5 eigencomponents, so tomography again proceeds in a reduced subspace without significant loss in fidelity accuracy.

The optimality of the Bures distance metric for tomography is essential to achieving dimension-independence; previous trace-norm-based protocols incur extra undesirable scaling.

Lower Bounds and Query Complexity

Lower bounds on sample complexity are established by reductions to the quantum property testing problem UnifDis, demonstrating that distinguishing spectral support up to rank $F(\rho,\sigma)$6 requires $F(\rho,\sigma)$7 samples, and thus the scaling $F(\rho,\sigma)$8 for additive error $F(\rho,\sigma)$9 is optimal. These reductions are constructed carefully to ensure robust distinguishability and use combinatorial arguments to derandomize the hard instance generation.

The work also tightens quantum query complexity lower bounds in the "purified access" model, showing that quantum property testers must make at least $\epsilon$0 queries—a direct consequence via quantum sample-to-query lifting techniques [CWZ25], closing a gap between sample complexity and query complexity for this class of problems.

Application: Tolerant Quantum State Certification

As a direct application, the protocols yield a dimension- and rank-adaptive framework for tolerant quantum state certification: deciding whether an unknown state $\epsilon$1 is $\epsilon$2-close or $\epsilon$3-far in fidelity from a reference $\epsilon$4. The sample complexity for this tolerant version is shown to depend only on the smaller of the ranks of the two states, rather than the full ambient Hilbert space dimension, subsuming and strictly generalizing the exact certification model of [BOW19].

Implications and Open Questions

The presented complexity-theoretic separations and algorithmic results have several implications for quantum learning theory and efficient quantum benchmarking:

• For practical state engineering and verification tasks, the dimension-independent procedures enable scaling of quantum protocols to higher dimensional systems, as long as rank structure can be exploited.
• The rank-optimal scaling with respect to $\epsilon$5 and $\epsilon$6 sets the definitive statistical cost for fidelity estimation and tolerant certification protocols, benchmarking any future alternative approaches.

Several natural questions are proposed for further investigation:

• Can the $\epsilon$7 upper bound in the low-rank $\epsilon$8 setting be reduced to match the lower bound?
• Are there similarly optimal algorithms for trace distance estimation (and thus tolerant quantum state certification with respect to trace distance), especially for low-rank or classically known reference states?
• What are the limits of quantum sample-to-query lifting in related quantum hypothesis testing or entropy estimation tasks?

Position Relative to Prior and Simultaneous Work

A notable distinction is made with the simultaneous work [LT26], which considers fidelity estimation between two unknown states and achieves $\epsilon$9 sample complexity, whereas the focus here is on scenarios with a classically specified reference state, allowing strictly better complexity under the respective assumptions. Additionally, the methods herein subsume the best previously known results for this task [UNWT25, GP22], improving or removing all residual logarithmic factors.

Conclusion

This work provides nearly tight bounds and new optimal protocols for fidelity estimation with a known reference quantum state, and establishes the true complexity of tolerant quantum state certification in the sample-efficient regime. The results underscore the value of subspace reduction, fidelity decomposition, and Bures-distance-based tomography in the design of sample-efficient quantum property testing protocols, and clarify the complexity landscape of quantum state estimation and property testing.