quantum-fidelity-estimation

name	quantum-fidelity-estimation
description	Quantum state fidelity estimation methodology with optimal sample complexity bounds. Covers O(r²/ε²) upper and Ω(r/ε²) lower bounds for rank-r reference states, tolerant certification, and quantum query complexity implications. Use when estimating quantum state fidelity, designing certification protocols, or analyzing quantum sample complexity.
metadata	{"arxiv_id":"2606.26034","published":"2026-06-24","authors":"Qisheng Wang","tags":["quantum","statistics","fidelity","sample-complexity","certification","quantum-query-complexity"]}

Quantum Fidelity Estimation

Description

Methodology for estimating the fidelity of an unknown quantum state to a known reference state with optimal sample complexity bounds. Provides tight upper bound O(r²/ε²) and lower bound Ω(r/ε²) for rank-r reference states.

Activation Keywords

• quantum state fidelity estimation
• fidelity to reference state
• quantum sample complexity
• quantum state certification
• tolerant certification
• 量子态保真度估计
• 量子样本复杂度
• quantum query complexity fidelity
• fidelity estimation bounds

Core Theory

Problem Statement

Given an unknown quantum state ρ and a known reference state σ of rank r, estimate F(ρ, σ) to within additive error ε.

Sample Complexity Bounds

• Upper bound: O(r²/ε²) when reference state has rank r (improves prior O(r²log²(1/ε)/ε⁴))
• Lower bound: Ω(r/ε²) (improves prior Ω(r/ε + 1/ε²))
• Generalized case: Unknown state rank ≤ r, arbitrary reference → O(r²/ε⁴)

Key Improvement

The ε-dependence is now optimal (1/ε² in upper bound, matching lower bound), removing the logarithmic factors from previous work (Utsumi et al., QIP 2026).

Usage Patterns

Pattern 1: Fidelity Estimation Protocol Design

1. Identify reference state rank r
2. Choose error tolerance ε
3. Compute sample budget: O(r²/ε²) copies
4. Implement parity-based observable measurements
5. Apply tolerant certification framework if needed

Pattern 2: Quantum Query Complexity Analysis

1. Use Ω(r/ε²) lower bound to establish query complexity limits
2. Compare algorithm performance against optimal bounds
3. Identify gap between upper and lower bounds (factor of r)

Pattern 3: Tolerant Quantum State Certification

1. Generalize exact certification (Badescu-O'Donnell-Wright, STOC 2019)
2. Define acceptance/rejection thresholds based on fidelity
3. Use O(r²/ε⁴) sample complexity for unknown low-rank states

Error Handling

• If ε is too small relative to available copies, increase ε or use amplitude estimation techniques
• If reference state is full-rank (r = d), sample complexity becomes O(d²/ε²) — may be infeasible for large d
• Lower bound gap of factor r indicates room for algorithm improvement

References

• arXiv:2606.26034 - Estimating Fidelity to a Reference Quantum State (Qisheng Wang, 2026)
• Utsumi, Nakata, Wang, Takagi (QIP 2026) - previous best O(r²log²(1/ε)/ε⁴)
• Badescu, O'Donnell, Wright (STOC 2019) - exact quantum state certification