[Paper Review] Pseudorandom States, Non-Cloning Theorems and Quantum Money

Zhengfeng Ji, Yi-Kai Liu+1 more|arXiv (Cornell University)|Nov 1, 2017

Quantum Computing Algorithms and Architecture42 references3 citations

TL;DR

This paper introduces pseudorandom quantum states (PRS) as a quantum analogue of classical pseudorandomness, constructing them from quantum-secure one-way functions. It proves that no efficient quantum algorithm can clone multiple copies of a PRS, leading to a private-key quantum money scheme secure under computational assumptions.

ABSTRACT

We propose the concept of pseudorandom states and study their constructions, properties, and applications. Under the assumption that quantum-secure one-way functions exist, we present concrete and efficient constructions of pseudorandom states. The non-cloning theorem plays a central role in our study---it motivates the proper definition and characterizes one of the important properties of pseudorandom quantum states. Namely, there is no efficient quantum algorithm that can create more copies of the state from a given number of pseudorandom states. As the main application, we prove that any family of pseudorandom states naturally gives rise to a private-key quantum money scheme.

Motivation & Objective

• To formalize the concept of pseudorandom quantum states (PRS) as a computational analogue of Haar-random states.
• To establish a connection between the quantum non-cloning theorem and the security of PRS, ensuring no efficient cloning is possible.
• To construct efficient, quantum-secure PRS using quantum-secure one-way functions and the phase kick-back technique.
• To demonstrate that any family of PRS naturally yields a private-key quantum money scheme.
• To extend the framework to pseudorandom unitary operators (PRUs), showing their utility and implications for PRS construction.

Proposed method

• Define PRS via computational indistinguishability: a quantum algorithm cannot distinguish a PRS from a Haar-random state with more than negligible advantage.
• Construct PRS using a pseudorandom function (PRF) via the phase kick-back operation: $ T_k |x\rangle = \omega_N^{\textsf{PRF}_k(x)} |x\rangle $, followed by the Hadamard transform.
• Form the PRS as $ |\phi_k\rangle = T_k H^{\otimes n} |0\rangle $, where $ H^{\otimes n} $ creates a uniform superposition.
• Prove that no efficient quantum algorithm can clone $ q+1 $ copies of a PRS from $ q $ copies, leveraging the cryptographic non-cloning theorem.
• Construct candidate PRUs by repeating the $ T_k H^{\otimes n} $ circuit polynomially many times or using a controlled-PRF construction on $ 2n $ qubits.
• Show that any PRU induces a PRS via $ U_k |0\rangle $, establishing a direct link between PRUs and PRS.

Experimental results

Research questions

• RQ1 What is a suitable computational definition of pseudorandom quantum states that captures quantum indistinguishability from Haar randomness?
• RQ2 How can pseudorandom quantum states be efficiently constructed under standard quantum-secure assumptions like one-way functions?
• RQ3 To what extent does the quantum non-cloning theorem enforce the security of pseudorandom states against cloning attacks?
• RQ4 Can pseudorandom states be used to construct practical quantum money schemes with efficient verification and strong security?
• RQ5 What is the relationship between pseudorandom unitary operators and pseudorandom quantum states, and can PRUs serve as a building block for PRS?

Key findings

• Under the assumption of quantum-secure one-way functions, the paper constructs efficient pseudorandom states using a phase-kickback-based unitary transformation and the Hadamard gate.
• The non-cloning theorem is formally linked to PRS security: no efficient quantum algorithm can produce $ q+1 $ copies of a PRS from $ q $ copies with more than negligible success probability.
• Any family of pseudorandom states gives rise to a private-key quantum money scheme, where states serve as unforgeable banknotes.
• The security of the quantum money scheme follows directly from the non-cloning property of PRS, without requiring complex adversary methods like the inner-product method.
• Pseudorandom unitary operators (PRUs) can be constructed from PRFs via repeated application of phase-kickback circuits, and any PRU induces a PRS via application to the $ |0\rangle $ state.
• The construction of PRS via $ T_k H^{\otimes n} |0\rangle $ yields states that are computationally indistinguishable from Haar random states, satisfying the core definition of pseudorandomness.