A Cryptographic Perspective on the Verifiability of Quantum Advantage

TL;DR

The paper investigates verifiable quantum advantage from a cryptographic lens, linking verifiability of sampling-based quantum tasks to primitives like $ extsf{EFI}$ and $ extsf{PRS}$ and to meta-complexity problems such as a variant of $ extsf{MCSP}$, $\mathsf{SampMCSP}$. It introduces formal verifiability notions (VQA, UVQA, DVQA, QVQA, UQVQA) and shows how efficient algorithms for meta-complexity problems would yield universal verifiers, while cryptographic priors constrain verifiability through EFI/PR S arguments. The work establishes dualities between EFI and QVQA, and demonstrates how classical or quantum verification modes relate to the structure of output states (e.g., PRS, EFI) and to designated verifiability, with implications for design and analysis of future verifiable quantum advantage experiments. Overall, the results suggest that verifiability and the cryptographic structure of quantum states are deeply intertwined, potentially guiding the construction of verifiable quantum advantage experiments and the development of quantum cryptographic primitives. The findings highlight a path where the quest for verifiable quantum advantages informs both quantum information science and cryptography, including the potential for quantum-secure primitives rooted in EFI and PRS.

Abstract

In recent years, achieving verifiable quantum advantage on a NISQ device has emerged as an important open problem in quantum information. The sampling-based quantum advantages are not known to have efficient verification methods. This paper investigates the verification of quantum advantage from a cryptographic perspective. We establish a strong connection between the verifiability of quantum advantage and cryptographic and complexity primitives, including efficiently samplable, statistically far but computationally indistinguishable pairs of (mixed) quantum states ($\mathsf{EFI}$), pseudorandom states ($\mathsf{PRS}$), and variants of minimum circuit size problems ($\mathsf{MCSP}$). Specifically, we prove that a) a sampling-based quantum advantage is either verifiable or can be used to build $\mathsf{EFI}$ and even $\mathsf{PRS}$ and b) polynomial-time algorithms for a variant of $\mathsf{MCSP}$ would imply efficient verification of quantum advantages. Our work shows that the quest for verifiable quantum advantages may lead to applications of quantum cryptography, and the construction of quantum primitives can provide new insights into the verifiability of quantum advantages.

Figures (2)

• Figure 1: Scott Aaronson's categorization of quantum advantage proposals. Random circuit sampling boixo2018characterizing and Boson sampling boson are NISQable and Classically hard. Cryptographic proof of quantumness (PoQ) mahadevPoQcomputationalCHSH and Shor's algorithm shor are classically hard and efficiently verifiable. QAOA qaoa and VQE vqe are NISQable and efficiently verifiable.
• Figure 2: Verification process for RCS: The verifier publishes a circuit family $\mathfrak{C}$. Then Alice sends $C \in \mathfrak{C}$ and samples $\pmb{z}_C$ obtained from measuring $C|0^n\rangle$ to the verifier, and sends $C$ to Bob. Bob sends the description of the sampler $\mathcal{S}_{C}$ for his classically samplable spoofing distribution ${D}_C$ which depends on $C$, along with samples $\pmb{z}_{{D}_C}$ to the verifier.

Theorems & Definitions (45)

• Definition 1.1: Verifiable quantum advantage (Informal)
• Theorem 1.2: Informal
• Theorem 1.3: Informal
• Definition 2.1: $\textsf{EFI}$ pairs BCQ23
• Lemma 2.2
• proof
• Definition 2.3: Pseudorandom states ($\textsf{PRS}$) JLS18
• Definition 2.4: Classically unidentifiable state family
• Remark 2.5
• Definition 3.1: Verifiable quantum advantages (($s,t,\varepsilon$)-VQA)