Tamper Detection against Unitary Operators

We consider (Enc, Dec) schemes which are used to encode a classical/quantum message $m$ and derive an $n$-qubit quantum codeword $\psi_m$. The quantum codeword $\psi_m$ can adversarially tamper via a unitary $U \in \mathcal{U}$ from some known tampering unitary family $\mathcal{U}$, resulting in $U \psi_m U^\dagger$. Firstly, we initiate the general study of quantum tamper detection codes, which must detect that tampering occurred with high probability. In case there was no tampering, we would like to output the message $m$ with a probability of $1$. We show that quantum tamper detection codes exist for both classical messages and quantum messages for any family of unitaries $\mathcal{U}$, such that $|\mathcal{U}| < 2^{2^{\alpha n}}$ for some known constant $\alpha \in (0,1)$ and all the unitaries satisfy one additional condition : \begin{itemize} \item Far from Identity : For each $U \in \mathcal{U}$, we require that its modulus of trace value isn't too much i.e. $ |Trace(U)| \leq \phi N$, where $N=2^n.$ \end{itemize} Quantum tamper-detection codes are quantum generalizations of classical tamper detection codes studied by Jafargholi et al. \cite{JW15}. Additionally for classical message $m$, if we must either output message $m$ or detect that tampering occurred and output $\perp$ with high probability, we show that it is possible without the restriction of Far from Identity condition for any family of unitaries $\mathcal{U}$, such that $|\mathcal{U} | < 2^{2^{\alpha n}}$. We also provide efficient (Enc, Dec) schemes when the family of tampering unitaries are from Pauli group $\mathcal{P}_n$, which can be thought of as a quantum version of the algebraic manipulation detection (AMD) codes of Cramer et al. \cite{CDFPW08}.