Communicating Properties of Quantum States over Classical Noisy Channels

TL;DR

This work tackles transmitting properties of quantum states over classical noisy channels by introducing STT-UEP, a protocol that leverages shadow tomography to efficiently predict multiple observables and employs unequal error protection to protect measurement bases more than outcomes. The approach yields observable-agnostic communication with bit-cost scaling as $O(\log M)$ and exponential dependence on the maximum observable weight $w$, along with theoretical guarantees on estimation accuracy under channel errors. Empirical results show STT-UEP can outperform conventional state-quantization and standard shadow-tomography coding in terms of bit efficiency and reliability, particularly when protecting bases is crucial. The method advances quantum semantic communications by enabling task-focused transmission of state properties in noisy classical channels with practical applicability to distributed quantum sensing and computing.

Abstract

Transmitting information about quantum states over classical noisy channels is an important problem with applications to science, computing, and sensing. This task, however, poses fundamental challenges due to the exponential scaling of state space with system size. We introduce shadow tomography-based transmission with unequal error protection (STT-UEP), a novel communication protocol that enables efficient transmission of properties of quantum states, allowing decoder-side estimation of arbitrary observables. Unlike conventional approaches requiring the transmission of a number of bits that is exponential in the number of qubits, STT-UEP achieves communication complexity that scales logarithmically with the number of observables, depending on the observable weight. The protocol exploits classical shadow tomography for measurement efficiency, and applies unequal error protection by encoding measurement bases with stronger channel codes than measurement outcomes. We provide theoretical guarantees on estimation accuracy as a function of the bit error probability of the classical channel, and validate the approach against several benchmarks via numerical results.

Figures (4)

• Figure 1: Quantum semantic communication system: The encoder has access to $N$ copies of a quantum state $\ket{\psi}$. Upon measuring these $N$ copies, the encoder produces classical bits, which are transmitted, upon channel encoding through a classical channel. The decoder produces estimates $\{\hat{o}_1, \ldots,\hat{o}_M\}$ of the expected values of $M$ observables $\{O_1, \ldots, O_M\}$ that are a priori unknown to the encoder.
• Figure 2: To estimate the expected value $\langle O_m \rangle$, the decoder retains only measurements corresponding to compatible bases $P_i$. A measurement basis $P_i$ is compatible with an observable $O_m$ only if the condition $\sum_{j \in \mathcal{S}_m} \mathbbm{1}(P_{i,j}=O_{m,j}) = |\mathcal{S}_m|$ in \ref{['coms']} holds, so that all non-trivial Pauli matrices in $O_m$ are present in the corresponding position in $P_i$.
• Figure 3: Success probability $P_{\rm succ}$ versus the number of transmitted bit $B$ for STT-UEP with code rates $R_b=0.4$ and $R_b=1$, STT-CC, and CQCR with quantization resolutions $b=2$ bits, $b=4$ bits and $b=8$ bits. The code rate $R_u$ in STT-UEP, and the code rate $R$ for STT-CC and CQCR are adjust so that all schemes transmit the same total number of bits $B$ for a fair comparison.
• Figure 4: Success probability $P_{\rm succ}$ versus the number of copies $N$ for the STT-UEP, with $R_u=0.4$ and $R_b=1$.

Theorems & Definitions (1)

• Proposition 1