Copy Protection is a functionality first defined by Aaronson [1] that enables a Vendor to send a program (a circuit) to a Client so that the Client cannot duplicate it. Classically, this functionality has been proven impossible. However, it is possible to copy protect some families of programs using quantum computation.

• Any kind of software licence protection.

A Copy Protection protocol for a family of circuits is made of two algorithms:

• Protect, which takes as input a classical description of a circuit $C$ and outputs a quantum encoding $\rho _{C}$ of this circuit.
• Eval, which takes as input a quantum state and an classical input, and returns a classical output.

A Copy Protection scheme for a family of circuits has $\varepsilon -correctness$ if for any circuit $C$ of this family and for any input $x$ for this circuit
$$ Pr[\mathbf {Eval} (\rho _{C},x)=f(x); \rho _{C}\gets \mathbf {Protect} (C)]\geq 1-\varepsilon $$
A Copy Protection scheme for a family of circuits has $\delta -security $ if no polynomially bounded quantum adversary can efficiently copy a protected program, more formally if for any such adversary, her probability of winning the following game is lower than $1-\delta$ :

• Challenger samples a circuit C in the family and sends Protect(C) to the Adversary;
• The Adversary runs any polynomial computation she wants on Protect(C) and sends two quantum states, respectively $\psi _{A}$ and $\psi _{B}$ to two of her agents, respectively Alice and Bob;
• The Challenger samples two inputs $x_{A},x_{B}$ for the circuit and sends $x_{A}$ to Alice and $x_{B}$ to Bob;
• Alice sends $y_{A}$ to the Challenger and Bob sends $y_{B}$ to the Challenger;
• The Adversary wins iff $C(x_{A})=y_{A}$ and $C(x_{B})=y_{B}$.

We assume that Alice and Bob cannot communicate with each other.

Even with quantum computation, Copy Protection is not possible for all families of circuits. Currently, it has been proven impossible for all learnable functions and de-quantumizable functions [2].

1. Aaronson, Scott. “Quantum copy-protection and quantum money.” In 2009 24th Annual IEEE Conference on Computational Complexity, pp. 229-242. IEEE, 2009.
2. Ananth, Prabhanjan, and Rolando L. La Placa. “Secure software leasing.” In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pp. 501-530. Cham: Springer International Publishing, 2021.