Researchers prove languages with interactive proof systems equal, demonstrating a subset of languages verifiable by 2QCFAs

The fundamental question of which computational problems can be efficiently verified forms the bedrock of computer science, and researchers continually refine our understanding of this landscape. Abuzer Yakaryılmaz from the University of Latvia, along with colleagues, now demonstrates a significant relationship between different classes of verifiable problems, specifically showing that the set of languages solvable with polynomial-time classical proofs is contained within a broader class verifiable by interactive proof systems. This work establishes a connection between classical computation and more complex interactive proofs, utilising a novel system where verifiers are two-way finite automata and provers operate classically, and it advances our knowledge of the boundaries between efficiently verifiable problems, potentially impacting the development of future computational models and proof systems. The team’s protocols, which achieve perfect completeness, offer a new perspective on how verification can be achieved with limited computational resources.

The protocols employ rational-valued quantum transitions and run in double-exponential expected time, while accepting member strings with absolute certainty, demonstrating perfect-completeness. Proof systems are central to both classical and quantum complexity theory. Within polynomial time computation, interactive proof systems (IPSs), Arthur-Merlin (AM) proof systems, and quantum interactive proof systems offer equivalent verification power, capable of verifying all and only languages within PSPACE. In constant-space scenarios, IPSs employing two-way probabilistic finite state automaton (2PFA) verifiers can verify any language in ASPACE(n), representing alternating linear space.

Quantum Verification with Finite Automata

This paper presents a comprehensive investigation into the relationship between quantum computation, automata theory, and complexity classes. It explores the power of interactive proof systems (specifically, Arthur-Merlin systems, AM) when the verifier is a 2-Quantum Finite Automaton (2QCFA). The central question addresses which complexity classes can be efficiently verified using a 2QCFA as the verifier, revealing the limits of efficient quantum resource use in verification. The research demonstrates a significant result: AM1(2QCFA) = ASPACE(n). This shows that any problem solvable in ASPACE(n) (Arthur-Merlin with logarithmic communication) can be verified by a 2QCFA in interactive proof, indicating the power of 2QCFAs to verify problems requiring limited space.

Furthermore, the paper proves that AM(2QCFA) ⊇ PSPACE, demonstrating that 2QCFAs can verify problems requiring polynomial space. The study also shows that 2QCFAs can verify problems as hard as any in NEXP (Nondeterministic Exponential Time). The paper draws a connection between 2QCFAs and private-coin interactive proof systems (IPSs). The superposition property of quantum states allows for stronger verification power, similar to the benefits of private coins. The authors conclude by posing open questions about whether AM1(2QCFA) or AM(2QCFA) can go beyond EXP or NEXP, and whether they can verify any language in PSPACE or NP.

This research provides a deeper understanding of the power of quantum states in interactive proof systems, showing that 2QCFAs are a powerful tool for verifying complex problems. The results contribute to the broader field of complexity theory by establishing relationships between different complexity classes and quantum computation. The paper advances the study of quantum interactive proofs, a fundamental concept in quantum information science, and allows for a comparison between the power of classical and quantum verifiers in interactive proof systems. The findings could have implications for the design of secure communication protocols and other applications that rely on verification.

The research utilizes the theory of QFAs, which are finite automata that use quantum states to represent their configuration, and employs the framework of interactive proof systems, involving a prover and a verifier communicating to determine truth. In summary, this paper is a significant contribution to quantum complexity theory. It establishes strong connections between quantum automata, interactive proof systems, and complexity classes, providing valuable insights into the power of quantum verification. The open questions suggest promising avenues for future research.

Quantum Automata Define Classical and Quantum Complexity

Researchers have demonstrated a significant advancement in computational complexity by establishing a fundamental connection between several important classes of languages. The team proved that the class of languages solvable with polynomial-time classical or interactive proof systems is identical to a specific class defined by Arthur-Merlin proof systems utilizing two-way finite automata with both classical and quantum states. This discovery clarifies the relationships between different models of computation and provides a deeper understanding of what makes certain problems computationally tractable. The breakthrough centers on the use of “2QCFAs,” or two-way finite automata incorporating quantum states, as verifiers in the Arthur-Merlin proof system.

Experiments reveal that languages verifiable by these 2QCFAs are equivalent to those solvable by more traditional methods, but with a potentially more efficient verification process. Notably, the protocols developed by the researchers utilize only rational-valued transitions and operate in double-exponential expected time, while still guaranteeing perfect-completeness, meaning member strings are accepted with absolute certainty. The team achieved this result by carefully constructing superoperators, mathematical tools for manipulating quantum states, using rational numbers to define transitions between states. This approach allows for precise control over the computation and ensures the verifiability of the languages. Furthermore, the researchers developed a method for encoding binary numbers using two-dimensional vectors, which facilitates efficient computation within the 2QCFA framework. These findings have implications for the design of more efficient algorithms and proof systems, potentially leading to advancements in areas such as cryptography and artificial intelligence.