uses of an optical channel. Our achievability proof uses photonic dual-rail qubit encoding, which has been proposed for long-range repeater-based quantum communication and entanglement distribution. Our converse employs prior covert signal power limit results and adapts well-known methods to upper bound quantum capacity of optical channels. Finally, we believe that the gap between our lower and upper bounds for the number of reliable covert qubits can be mitigated by improving the quantum error correction codes and quantum channel capacity bounds.

DOI: 10.48550/arxiv.2401.06764

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\propto\sqrt{n}

qubits can be transmitted covertly and reliably over

n

\propto\sqrt{n}

qubits can be transmitted covertly and reliably over

n

L_{\rm EG}\sqrt{n}

entangled bits (ebits) can be generated covertly and reliably over

n

uses of a bosonic channel. We report a single-letter expression for optimal

L_{\rm EG}

as well as an achievable method. We additionally analyze the performance of covert entanglement generation using single- and dual-rail photonic qubits, which may be more practical for physical implementation.

L_{\text{sec }}

. The achievable construction relies on position-based coding, sequential decoding, and quadrature phase-shift keying. Furthermore, we adapt the code used in covert secret communication to show that a square root law (SRL) for covert entanglement generation similar to that for classical:

L_{\text{EG}} \sqrt{n}

entangled bits (ebits) can be generated covertly and reliably over

n

uses of a bosonic channel. We report a single-letter expression for optimal

L_{\text{EG}}

as well as an achievable method. We additionally analyze the performance of covert entanglement generation using single- and dual-rail photonic qubits for varying channel parameters. Despite a performance gap compared to the optimal strategy, these encodings may be more practical for physical implementation.

M

receivers. We demonstrate that coherent-state encoding achieves these capacities and establish that time-division is optimal in the covert regime. These results rely on a strong minimum entropy output conjecture, a direct result of the entropy photon-number inequality (EPnI) conjecture. Building on covert communication, we develop a system construction for reliable transmission of covert qubits over an arbitrary quantum channel using single- and dual-rail encodings. We apply this construction to the lossy thermal-noise bosonic channel, thereby deriving achievable rates. We further explore the impact of access to a two-way classical communication link. While a tight upper bound remains an open problem, we provide a converse upper bound by decomposing the channel into a concatenation of a pure-loss channel and a quantum-limited amplifier and discarding the former. These bounds demonstrate a \emph{square root law} (SRL) for covert quantum communication similar to that for classical:

M(n)\propto\sqrt{n}

qubits can be transmitted covertly and reliably over

n

uses of the channel. We conjecture that the gap between our lower and upper bounds on the number of reliable covert qubits can be reduced by improving quantum error-correction codes and quantum channel capacity bounds. We then shift our focus to \emph{covert quantum computing}, introducing and formalizing a setting in which an adversary attempts to detect whether computation is occurring on a shared quantum processing unit (QPU) by monitoring a subset of the qubits or qumodes. For superconducting architectures with a nearest-neighbor crosstalk noise model and planar qubit topologies, a circuit remains covert by inserting idle qubits between the qubits running the computation and the remaining qubits on the QPU. Using discrete isoperimetric inequalities, we show that a circuit placement exists such that only

\mathcal{O}(\sqrt{n})

spectator qubits bordering the circuit provide detection information to an adversary, where

n

is the number of qubits used for computation. Thus, in this setting, the cost of maintaining covertness while also maintaining reliability of the computation is an additional

\mathcal{O}(\sqrt{n})

idle border qubits. We experimentally explore this model on IQM's Emerald processor, using Ramsey experiments on the adversary's qubits to detect gate operations. While nearest-neighbor crosstalk was detected as expected, long-range coupling effects were also observed beyond the border qubits, identifying a side channel that an adversary can exploit for detection. Finally, we discuss the outlook, future research directions, and open problems.

M(n) \sqrt{n}

qubits can be transmitted covertly and reliably over

n

uses of a general quantum channel. We lower bound

M (n)

with and without assistance from a two-way covert classical channel. In the former case, we quantify the number of classical covert bits sufficient for our protocol.