Optimal sample sizes found for quantum state tomography with joint measurements
Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements
Data Structures and AlgorithmsInformation TheoryMachine Learning
Summary
Quantum state tomography is a way to figure out the exact state of a quantum system by making measurements. This paper finds the best number of samples needed to accurately estimate a low-rank quantum state when one can measure a few samples together at once. The authors show that measuring samples jointly can reduce the number of measurements needed, but only up to a certain point. Their results provide exact formulas and methods to reach this optimal efficiency.
What this means in practice
- •For quantum computing engineers: Determine the minimal joint sample measurements needed for efficient quantum state verification in hardware calibration.
- •For quantum communications system designers: Optimize protocols for reconstructing quantum states from limited noisy measurements when only small joint sample blocks can be measured.
A theory result. No direct application yet.
Authors
Ashwin Nayak, Xingyu Zhou
Abstract
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\varepsilon$, estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $$ Θ\left( \frac{dr}{\varepsilon^2} \max\left\{1,\frac r{\sqrt t}\right\} \right)$$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt t$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.