post-quantum hardness amplification

This work addresses the verifiability of complexity analysis and termination guarantees in quantum computation. It introduces, for the first time, a physically realizable Quantum Term Rewriting System (QTRS) and establishes its equivalence with uniform families of quantum circuits, thereby precisely characterizing the class of functions FBQP. By incorporating termination proof techniques, the authors define a subclass of terminating QTRS whose reduction lengths directly bound the size of the corresponding quantum circuits. This framework not only enables static complexity analysis and certification but is also proven complete for functions computable in quantum polynomial time, providing a foundational theoretical basis for ensuring both reliability and efficiency in quantum programs.

This work addresses three central questions in quantum Probabilistically Checkable Proofs (QPCP): the equivalence of adaptive and nonadaptive QPCP, the complexity characterization of multiprover QPCP, and its connection to local Hamiltonian reductions. We unify the modeling of adaptive and multiprover QPCP, design novel quantum interactive proof protocols, and integrate relativization/de-relativization techniques with constant-query local Hamiltonian reductions. Our contributions are: (1) the first proof that nonadaptive QPCP with constant query complexity can simulate adaptive QPCP; (2) a sufficient condition for QPCP[q] ⊆ QCMA; (3) a proof that if QMA(k) admits a PCP characterization, then QMA(2) = QMA; and (4) an explicit oracle separation demonstrating that QPCP inherently requires nonrelativizing techniques. These results establish tight connections between two major open problems—whether QMA(2) = QMA and whether QPCP is QCMA-solvable—advancing the foundational understanding of quantum proof systems.

Can the local reduced density matrices (e.g., $k$-body marginals) of an approximately optimal quantum witness for a $mathsf{QMA}$ problem be efficiently approximated using only classical polynomial-time access to a $mathsf{QMA}$ decision oracle—without preparing or manipulating the full quantum state? Method: We introduce the first adaptive classical algorithm that integrates a circuit-to-Hamiltonian mapping preserving near-optimal witnesses, local consistency checks, and density matrix reconstruction techniques. Contributions: (1) For any constant locality $k$ and inverse-polynomial precision, the $k$-body reduced density matrices of an approximately optimal $mathsf{QMA}$ witness can be computed in classical polynomial time via adaptive queries to a $mathsf{QMA}$ oracle; (2) we define a new $mathsf{QMA}$-complete problem, “Low-Energy Density Matrix Verification”; (3) we establish the first rigorous classical reduction framework from a $mathsf{QMA}$ decision oracle to extraction of local quantum information.

This work addresses the Uhlmann transformation problem—determining the computational complexity of converting one entangled state into another via local operations. Methodologically, it establishes the first unified framework of unitary synthesis complexity, algorithmizes Uhlmann’s theorem, and constructs a rigorous reduction theory. Leveraging tools from unitary circuit synthesis, quantum interactive proofs, and quantum channel decoding, it precisely characterizes the complexity of over ten fundamental quantum tasks—including noisy channel decoding, quantum commitment breaking, and Hawking radiation decoding—proving that all are BQPSPACE-complete. Crucially, it uncovers deep equivalences between the Uhlmann transformation problem and central complexity classes such as BQPSPACE and quantum zero-knowledge protocols. This work provides the first universal, reducibility-based paradigm for characterizing the unitary complexity of quantum information processing tasks.

This work addresses the challenge of high T-count in Clifford+T circuits, which significantly increases resource overhead in fault-tolerant quantum computing. Existing local synthesis methods are constrained by circuit representations and struggle to achieve optimal T-count and depth. To overcome this limitation, the paper introduces Q-PreSyn, the first approach to integrate reinforcement learning into the pre-synthesis phase of quantum circuit compilation. By training an agent to learn sequences of function-preserving local editing operations, Q-PreSyn produces circuit representations that are more amenable to downstream synthesis. Without introducing any approximation error, the method achieves up to a 20% reduction in T-count compared to state-of-the-art techniques on circuits with up to 25 qubits, substantially improving synthesis efficiency.

This work proposes a parameterized parallel solving algorithm to address the inefficiency of existing approaches on hard CircuitSAT instances, such as those arising in logic equivalence verification and preimage attacks on cryptographic hash functions. The method decomposes the original problem into multiple weakened subformulas by introducing specialized constraints and dynamically guides the generation of high-quality decompositions through a hardness estimation mechanism tailored for parallel computing environments. By innovatively integrating parameterized decomposition strategies with runtime difficulty assessment, the approach achieves efficient structural decoupling of complex CircuitSAT instances. Experimental results demonstrate that the algorithm significantly improves solving efficiency on representative hard instances, exhibiting both practical utility and strong scalability.

This work investigates whether QMA admits quantum interactive oracle proof systems (qIOPs) as a relaxed formulation of the quantum PCP conjecture. By restricting the verifier’s quantum resources—specifically, the number of queries and circuit complexity—the paper presents two unconditional qIOP constructions: one requiring only a constant number of queried qubits, and another using only a constant number of qubits throughout the entire verification process. Key technical ingredients include shared EPR pairs, multi-round quantum interactive protocols, constant-query mechanisms, and a novel single-prover multi-qubit test. The results establish various trade-offs between communication complexity and verifier resource requirements, offering new tools for quantum complexity theory.

This work investigates the cryptographic underpinnings of anti-commuting operator tests in classically verifiable quantum computation and introduces a framework termed the Test of Non-Commutativity (ToNC). By establishing rigorous reductions between ToNC and classical key agreement (KA) as well as oblivious transfer (OT) based on one-way functions, the paper demonstrates for the first time that ToNC can be used to construct KA and implement OT. Furthermore, it introduces a post-quantum hardcore measure theorem and an interactive XOR lemma, enabling hardness amplification for KA and OT in post-quantum settings. These results uncover a deep connection between quantum verification protocols and classical cryptographic primitives, thereby laying a new theoretical foundation for classically verifiable quantum computation.

This work proposes the first three-tier hierarchical explanatory framework for lattice-based post-quantum cryptography (PQC), such as ML-KEM and ML-DSA, to enhance the technical intelligibility and transparency of PQC security assumptions. The theoretical tier characterizes security boundaries through computational complexity classifications; the mathematical tier deepens structural understanding of lattice problems by integrating combinatorial Hodge theory and polyhedral geometry; and the experimental tier implements an empirical Julia-based platform to quantitatively analyze the behavior of lattice basis reduction algorithms like LLL and BKZ in low dimensions. While introducing no new attacks or hardness results, this framework systematically bridges formal proofs, mathematical foundations, and implementation characteristics, substantially strengthening the structural interpretability of PQC security assumptions.