Pauli-resolved virtual distillation
Abstract
Learning the full Pauli profile of the virtually distilled quantum state ρm/tr(ρm) has so far required exponentially many copies of ρ. We show that all 4n squared Pauli moments [tr(Pρm)]2 can be learned to additive error ε with confidence 1−δ from one 2m-replica measurement setting using O(m[n+log(1/δ)]/ε2) copies. This is an exponential speedup in the system size n over previous protocols. For m = 2, we propose the coherent Bell difference sampling circuit that realizes this measurement on current devices. The speedup originates from paired replicas that cancel the anticommutation signs of Pauli operators, collapsing the incompatibility of the Pauli family. We certify this collapse by introducing the quantum Bernstein norm, a computable incompatibility measure for nonlinear functionals. A further information-theoretic m-replica protocol recovers the signed moments tr(Pρm) using O(m[n+log(1/δ)]/ε4) copies. For fixed m in the stated lower-bound regime, it attains the optimal replica number, since any protocol with fewer replicas requires exponentially many copies.
Related
- Finite-Shot Operating Windows for Virtual Distillation and Symmetry Verification
- Pauli-ball discrepancy and stability of quantum codes
- Unitary fault-tolerant encoding of Pauli states in surface codes
- Pauli propagation enables fast classical simulation of strongly correlated quantum systems
- Sublinear Copies Suffice for Fidelity Estimation with Pauli Measurements
BibTeX
@misc{chen2026pauliresolved,
title = {{Pauli-resolved virtual distillation}},
author = {Si-Yuan Chen and Congcong Zheng and Kun Wang and Ming-Cheng Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24132},
eprint = {2609.24132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24132},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pauli-resolved virtual distillation
AU - Si-Yuan Chen
AU - Congcong Zheng
AU - Kun Wang
AU - Ming-Cheng Chen
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24132
UR - https://arxiv.org/abs/2609.24132
AB - Learning the full Pauli profile of the virtually distilled quantum state ρ^m/tr(ρ^m) has so far required exponentially many copies of ρ. We show that all 4^n squared Pauli moments [tr(Pρ^m)]^2 can be learned to additive error ε with confidence 1−δ from one 2m-replica measurement setting using O(m[n+log(1/δ)]/ε^2) copies. This is an exponential speedup in the system size n over previous protocols. For m = 2, we propose the coherent Bell difference sampling circuit that realizes this measurement on current devices. The speedup originates from paired replicas that cancel the anticommutation signs of Pauli operators, collapsing the incompatibility of the Pauli family. We certify this collapse by introducing the quantum Bernstein norm, a computable incompatibility measure for nonlinear functionals. A further information-theoretic m-replica protocol recovers the signed moments tr(Pρ^m) using O(m[n+log(1/δ)]/ε^4) copies. For fixed m in the stated lower-bound regime, it attains the optimal replica number, since any protocol with fewer replicas requires exponentially many copies.
ER -