Pauli measurements are not optimal for single-copy tomography
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910843893972992 |
|---|---|
| author | Acharya, Jayadev Dharmavarapu, Abhilash Liu, Yuhan Yu, Nengkun |
| author_facet | Acharya, Jayadev Dharmavarapu, Abhilash Liu, Yuhan Yu, Nengkun |
| contents | Quantum state tomography is a fundamental problem in quantum computing. Given $n$ copies of an unknown $N$-qubit state $ρ\in \mathbb{C}^{d \times d},d=2^N$, the goal is to learn the state up to an accuracy $ε$ in trace distance, with at least probability 0.99. We are interested in the copy complexity, the minimum number of copies of $ρ$ needed to fulfill the task.
Pauli measurements have attracted significant attention due to their ease of implementation in limited settings. The best-known upper bound is $O(\frac{N \cdot 12^N}{ε^2})$, and no non-trivial lower bound is known besides the general single-copy lower bound $Ω(\frac{8^n}{ε^2})$, achieved by hard-to-implement structured POVMs such as MUB, SIC-POVM, and uniform POVM.
We have made significant progress on this long-standing problem. We first prove a stronger upper bound of $O(\frac{10^N}{ε^2})$. To complement it with a lower bound of $Ω(\frac{9.118^N}{ε^2})$, which holds under adaptivity. To our knowledge, this demonstrates the first known separation between Pauli measurements and structured POVMs.
The new lower bound is a consequence of a novel framework for adaptive quantum state tomography with measurement constraints. The main advantage over prior methods is that we can use measurement-dependent hard instances to prove tight lower bounds for Pauli measurements. Moreover, we connect the copy-complexity lower bound to the eigenvalues of the measurement information channel, which governs the measurement's capacity to distinguish states. To demonstrate the generality of the new framework, we obtain tight-bounds for adaptive quantum tomography with $k$-outcome measurements, where we recover existing results and establish new ones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18170 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pauli measurements are not optimal for single-copy tomography Acharya, Jayadev Dharmavarapu, Abhilash Liu, Yuhan Yu, Nengkun Quantum Physics Computational Complexity Information Theory E.4; F.2.0; G.3 Quantum state tomography is a fundamental problem in quantum computing. Given $n$ copies of an unknown $N$-qubit state $ρ\in \mathbb{C}^{d \times d},d=2^N$, the goal is to learn the state up to an accuracy $ε$ in trace distance, with at least probability 0.99. We are interested in the copy complexity, the minimum number of copies of $ρ$ needed to fulfill the task. Pauli measurements have attracted significant attention due to their ease of implementation in limited settings. The best-known upper bound is $O(\frac{N \cdot 12^N}{ε^2})$, and no non-trivial lower bound is known besides the general single-copy lower bound $Ω(\frac{8^n}{ε^2})$, achieved by hard-to-implement structured POVMs such as MUB, SIC-POVM, and uniform POVM. We have made significant progress on this long-standing problem. We first prove a stronger upper bound of $O(\frac{10^N}{ε^2})$. To complement it with a lower bound of $Ω(\frac{9.118^N}{ε^2})$, which holds under adaptivity. To our knowledge, this demonstrates the first known separation between Pauli measurements and structured POVMs. The new lower bound is a consequence of a novel framework for adaptive quantum state tomography with measurement constraints. The main advantage over prior methods is that we can use measurement-dependent hard instances to prove tight lower bounds for Pauli measurements. Moreover, we connect the copy-complexity lower bound to the eigenvalues of the measurement information channel, which governs the measurement's capacity to distinguish states. To demonstrate the generality of the new framework, we obtain tight-bounds for adaptive quantum tomography with $k$-outcome measurements, where we recover existing results and establish new ones. |
| title | Pauli measurements are not optimal for single-copy tomography |
| topic | Quantum Physics Computational Complexity Information Theory E.4; F.2.0; G.3 |
| url | https://arxiv.org/abs/2502.18170 |