Optimal single-copy estimation of quantum state moments: why Tr(rho 3) and Tr(rho 4) are equally hard

arXiv:2609.37781 · quant-ph · Submitted 2026-09-29 · Read on arXiv

quant-ph

Submitted: 2026-09-29

Updated: 2026-09-29

Terminology

• Trace — 27× in this paper · explained in 2 episodes
• Accuracy — 17× in this paper · explained in 1 episode
• Sample Complexity — 17× in this paper · explained in 15 episodes
• Entanglement — 13× in this paper · explained in 3 episodes
• Adaptivity — 7× in this paper · explained in 1 episode
• Fidelity — 4× in this paper · explained in 2 episodes
• Hamiltonian Learning — 3× in this paper · explained in 2 episodes
• Adaptation — 2× in this paper · explained in 1 episode
• Distillation — 2× in this paper · explained in 2 episodes
• Entanglement Entropy — 2× in this paper · explained in 2 episodes
• Fisher Information — 2× in this paper · explained in 3 episodes
• Generative AI — 2× in this paper · explained in 2 episodes
• Hilbert Space — 2× in this paper · explained in 2 episodes
• Symposium on Foundations of Computer Science — 2× in this paper
• Adam — 1× in this paper · explained in 2 episodes
• Ancillary Qubits — 1× in this paper · explained in 1 episode
• Generalization — 1× in this paper · explained in 13 episodes
• Regularization — 1× in this paper · explained in 3 episodes
• Verification — 1× in this paper · explained in 2 episodes

Sources

• Trace Estimation of Quantum State Powers: Sample Complexity and Computational Hardness
• The Church of the Symmetric Subspace
• Quantum Nonlinear Properties from a Single Measurement Setting
• Exponential Separations between Quantum Learning with and without Purification
• Exponential speedup in measurement property learning with post-measurement states
• Near-Optimal Simultaneous Estimation of Quantum State Moments
• Exponential Advantage from One More Replica in Estimating Nonlinear Properties of Quantum States
• Optimal Ansatz-free Hamiltonian Learning In Situ

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