Quantum Copy-Protection and Quantum Money (1110.5353v1)

Abstract: Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner's scheme required a central bank to verify the money, and the question of whether there can be unclonable quantum money that anyone can verify has remained open since. One can also ask a related question, which seems to be new: can quantum states be used as copy-protected programs, which let the user evaluate some function f, but not create more programs for f? This paper tackles both questions using the arsenal of modern computational complexity. Our main result is that there exist quantum oracles relative to which publicly-verifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its input-output behavior can be quantumly copy-protected. This provides the first formal evidence that these tasks are achievable. The technical core of our result is a "Complexity-Theoretic No-Cloning Theorem," which generalizes both the standard No-Cloning Theorem and the optimality of Grover search, and might be of independent interest. Our security argument also requires explicit constructions of quantum t-designs. Moving beyond the oracle world, we also present an explicit candidate scheme for publicly-verifiable quantum money, based on random stabilizer states; as well as two explicit schemes for copy-protecting the family of point functions. We do not know how to base the security of these schemes on any existing cryptographic assumption. (Note that without an oracle, we can only hope for security under some computational assumption.)

• The paper explores using quantum mechanics to create publicly verifiable quantum money and copy-protected programs, leveraging the unclonable nature of quantum states.
• It investigates extending Wiesner's quantum money scheme to be publicly verifiable, proposing constructions based on quantum oracles and a candidate scheme using random stabilizer states.
• The paper argues that quantum states can copy-protect functions difficult to learn from input-output data, basing this on the Complexity-Theoretic No-Cloning Theorem and utilizing technical constructs like quantum t-designs.

Quantum Copy-Protection and Quantum Money: An Overview

The paper "Quantum Copy-Protection and Quantum Money" by Scott Aaronson tackles two compelling questions in the field of quantum computing and information: whether quantum states can be used to create publicly verifiable money and whether programs can be quantumly copy-protected. The exploration of these questions is rooted in classical notions where making information unclonable is challenging due to the inherent ability to copy readable information indefinitely. The paper leverages quantum mechanics to propose solutions that are not feasible in classical computing.

Publicly Verifiable Quantum Money

The paper revisits the idea initially proposed by Wiesner, 40 years ago, using quantum states to create money that defies counterfeiting. Wiesner's scheme was limited as a central bank was required for verification. Aaronson extends this notion by investigating if there can be unclonable quantum money that anyone can verify. The paper presents evidence via the construction of quantum oracles where such publicly-verifiable quantum money schemes are posited to be plausible. The approach demonstrates that if quantum money is possible, proving its impossibility would require non-relativizing techniques, hence beyond current methodologies. Additionally, a candidate scheme utilizing random stabilizer states is proposed, albeit the security of the scheme is not cemented on any cryptographic assumption.

Quantum Copy-Protection

The task of quantum copy-protection is likened to the challenge of distributing software in a manner that it can be utilized to compute functions without enabling efficient reproduction of the software. Here, the paper identifies families of functions that cannot be efficiently learned from input-output behavior and argues that quantum states can quantumly protect such families. The Complexity-Theoretic No-Cloning Theorem is central to this argument, generalizing existing quantum mechanics and search optimizations to formulate a basis for secure copy-protection.

Technical Constructs and Implications

The security propositions within the paper also hinge upon the explicit constructions of quantum $t$-designs, which provide approximations of Haar-random states, thus enabling the mock-up of quantum states necessary for applications like copy-protection.

The implications of this research are multifaceted. On a theoretical level, it establishes a correlation between quantum mechanics and computational complexity foundations, opening avenues to cryptographic primitives previously deemed unattainable. Practically, realizing untamable quantum money could transform financial and digital transaction security paradigms. The paper also touches on speculative future developments where, despite not depending on cryptographic assumptions, quantum strategies might necessitate new hardware or computational models, such as quantum computers, to fully realize monetary and copy-protection applications.

In conclusion, while Aaronson's paper sets a promising foundation—highlighting possibilities that quantum mechanics could offer in terms of copy-protection and digital currency—it leaves open several challenging problems for future exploration. These include developing more explicit schemes and proving their security, exploring solutions for unclonable identity cards or proofs, and refining complexity-theoretic frameworks to better understand quantum information potentials.