UniqueQMA vs QMA: oracle separation and eigenstate thermalization hypothesis

2024-10-31

arXiv.org

AI Summary

This work resolves a long-standing open problem in quantum complexity theory: whether UniqueQMA—the unique-witness variant of QMA—is quantum Turing-reducible to QMA. We establish the first quantum oracle separation, proving UniqueQMA ≠ QMA relative to an explicit quantum oracle. Technically, we derive the first Ω(√D) lower bound on subspace phase query complexity, extending quantum approximate counting and double-copy entanglement verification techniques. Our results reveal a fundamental connection between the Eigenstate Thermalization Hypothesis (ETH) and unique-witness verification: ground-state energies of local Hamiltonians satisfying ETH admit efficient UniqueQMA verification. Furthermore, we show that if UniqueQMA ≠ QMA, then every QMA-hard Hamiltonian must violate ETH under adversarial perturbations. Finally, our framework provides new evidence for polynomial-time verifiability of low-energy states in chaotic systems—including SYK models—thereby bridging quantum complexity, condensed matter physics, and quantum many-body theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Privacy-enhancing technologies

Abstract

We study the long-standing open question of the power of unique witness in quantum protocols, which asks if UniqueQMA, a variant of QMA whose accepting witness space is 1-dimensional, is equal to QMA. We show a quantum oracle separation between UniqueQMA and QMA via an extension of the Aaronson-Kuperberg's QCMA vs QMA oracle separation. In particular, we show that any UniqueQMA protocol must make $Omega(sqrt{D})$ queries to a subspace phase oracle of unknown dimension $leq D$ to"find"the subspace. This presents an obstacle to relativizing techniques in resolving this question (unlike its classical analogue - the Valiant-Vazirani theorem - which is essentially a black-box reduction) and suggests the need to study the structure of the ground space of local Hamiltonians in distilling a potential unique witness. Our techniques also yield a quantum oracle separation between QXC, the class characterizing quantum approximate counting, and QMA. Very few structural properties are known that place the complexity of local Hamiltonians in UniqueQMA. We expand this set of properties by showing that the ground energy of local Hamiltonians that satisfy the eigenstate thermalization hypothesis (ETH) can be estimated through a UniqueQMA protocol. Specifically, our protocol can be viewed as a quantum expander test in a low energy subspace of the Hamiltonian and verifies a unique entangled state in two copies of the subspace. This allows us to conclude that if UniqueQMA $ eq$ QMA, then QMA-hard Hamiltonians must violate ETH under adversarial perturbations (more accurately, under the quantum PCP conjecture if ETH only applies to extensive energy subspaces). Our results serve as evidence that chaotic local Hamiltonians, such as the SYK model, contain polynomial verifiable quantum states in their low energy regime and may be simpler than general local Hamiltonians if UniqueQMA $ eq$ QMA.

Problem

• Investigating quantum complexity of unique witnesses versus QMA
• Ruling out black-box reductions from QMA to UniqueQMA
• Exploring structural properties of local Hamiltonians for witness distillation

Innovation

• Quantum oracle separation between BQP^UniqueQMA and QMA
• UniqueQMA protocol for ETH-satisfying Hamiltonians
• Quantum expander test verifying unique entangled states