Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

MIPco=coRE

Junqiao Lin

TL;DR

This work proves that the commuting-operator variant of multiprover interactive proofs, MIPco, is exactly the coRE class, complementing the established result MIP* = RE. Central to the argument is a generalized gap compression theorem adapted to the commuting-operator model via tracially embeddable strategies, enabling the same compression machinery used in the tensor-product case to apply here. The authors introduce the weakly compressible condition and a conditional-linear verifier framework, then decompose the compression into question reduction, answer reduction, and parallel repetition, while leveraging synchronous-Correlated/PAULI-based self-testing and the NPA hierarchy. They also streamline the compression proof with modern Pauli-basis tests and introspection, and demonstrate broad consequences for operator algebras, noncommutative polynomials, and complexity of non-local games, including the nonexistence of computable separations between tensor-product and commuting-operator models. The results imply deep uncomputability phenomena for quantum values and provide alternative routes to proofs of MIP* = RE, as well as potential applications to other operator-algebra problems and zero-knowledge interactive proofs in the quantum regime.

Abstract

In 2020, a landmark result by Ji, Natarajan, Vidick, Wright, and Yuen showed that MIP*, the class of languages that can be decided by a classical verifier interacting with multiple computationally unbounded provers sharing entanglement in the tensor product model, is equal to RE. We show that the class MIPco, a complexity class defined similarly to MIP* except with provers sharing the commuting operator model of entanglement, is equal to the class coRE. This shows that giving the provers two different models of entanglement leads to two completely different computational powers for interactive proof systems. Our proof builds upon the compression theorem used in the proof of MIP*=RE, and we use the tracially embeddable strategies framework to show that the same compression procedure in MIP* =RE also has the same desired property in the commuting operator setting. We also give a more streamlined proof of the compression theorem for non-local games by incorporating the synchronous framework used by Mousavi et al. [STOC 2022], as well as the improved Pauli basis test introduced by de la Salle [ArXiv:2204.07084]. We introduce a new equivalence condition for RE/coRE-complete problems, which we call the weakly compressible condition. We show that both MIP* and MIPco satisfy this condition through the compression theorem, and thereby establish that the uncomputability for MIP* and MIPco can be proved under a unified framework (despite these two complexity classes being different). Notably, this approach also gives an alternative proof of the MIP*=RE theorem, which does not rely on the preservation of the entanglement bound. In addition to non-local games, this new condition could also potentially be applicable to other decision problems.

MIPco=coRE

TL;DR

This work proves that the commuting-operator variant of multiprover interactive proofs, MIPco, is exactly the coRE class, complementing the established result MIP* = RE. Central to the argument is a generalized gap compression theorem adapted to the commuting-operator model via tracially embeddable strategies, enabling the same compression machinery used in the tensor-product case to apply here. The authors introduce the weakly compressible condition and a conditional-linear verifier framework, then decompose the compression into question reduction, answer reduction, and parallel repetition, while leveraging synchronous-Correlated/PAULI-based self-testing and the NPA hierarchy. They also streamline the compression proof with modern Pauli-basis tests and introspection, and demonstrate broad consequences for operator algebras, noncommutative polynomials, and complexity of non-local games, including the nonexistence of computable separations between tensor-product and commuting-operator models. The results imply deep uncomputability phenomena for quantum values and provide alternative routes to proofs of MIP* = RE, as well as potential applications to other operator-algebra problems and zero-knowledge interactive proofs in the quantum regime.

Abstract

In 2020, a landmark result by Ji, Natarajan, Vidick, Wright, and Yuen showed that MIP*, the class of languages that can be decided by a classical verifier interacting with multiple computationally unbounded provers sharing entanglement in the tensor product model, is equal to RE. We show that the class MIPco, a complexity class defined similarly to MIP* except with provers sharing the commuting operator model of entanglement, is equal to the class coRE. This shows that giving the provers two different models of entanglement leads to two completely different computational powers for interactive proof systems. Our proof builds upon the compression theorem used in the proof of MIP*=RE, and we use the tracially embeddable strategies framework to show that the same compression procedure in MIP* =RE also has the same desired property in the commuting operator setting. We also give a more streamlined proof of the compression theorem for non-local games by incorporating the synchronous framework used by Mousavi et al. [STOC 2022], as well as the improved Pauli basis test introduced by de la Salle [ArXiv:2204.07084]. We introduce a new equivalence condition for RE/coRE-complete problems, which we call the weakly compressible condition. We show that both MIP* and MIPco satisfy this condition through the compression theorem, and thereby establish that the uncomputability for MIP* and MIPco can be proved under a unified framework (despite these two complexity classes being different). Notably, this approach also gives an alternative proof of the MIP*=RE theorem, which does not rely on the preservation of the entanglement bound. In addition to non-local games, this new condition could also potentially be applicable to other decision problems.

Paper Structure

This paper contains 94 sections, 82 theorems, 363 equations, 14 figures, 3 tables, 12 algorithms.

Table of Contents

  1. Introduction
  2. The complexity of non-local games.
  3. The compressible condition.
  4. Explicit separation between the models of entanglement.
  5. Parallel repetition for the commuting operator model.
  6. Technical overview
  7. The compressible condition.
  8. The compressible condition for decision problems.
  9. Applying the compressible condition to non-local games.
  10. Towards proving the gap compression theorem
  11. Question Reduction.
  12. Answer Reduction.
  13. Parallel repetition.
  14. From $\mathsf{MIP}\xspace^*$ to $\mathsf{MIP}\xspace^{\mathrm{co}}$.
  15. Consequences
  16. ...and 79 more sections

Key Result

Theorem 1.1

$\mathsf{MIP}\xspace^{\mathrm{co}} = \mathsf{coRE}\xspace$.

Figures (14)

  • Figure 1: A diagram representation of the conditionally linear function $\mathtt{L}$.
  • Figure 2: The sampling/decision procedure for the $(p,m,d)$-quantum low-individual degree test, the only change is the distribution of the synchronicity test, which only changes the constant in \ref{['thm:soundnessQLDT']} (see jiQuantumSoundnessTesting2022 for more details).
  • Figure 3: The typed graph for the quantum low-individual degree test
  • Figure 4: Left: The description for the magic square game, where each row and column corresponds to an equation. Right: A oracularizable perfect strategy for the magic square game.
  • Figure 5: The typed graph $(\mathtt{T}^{\text{PB}}, \mathtt{E}^{\text{PB}})$ for the $n$ qubit Pauli basis test, where each of the vertices above also contains a self-loop (this is a black edge). Similar to the definition of the CL distribution, each vertices in the graph represents a potential question label and each edge represents a potential question pair. Each of the edges are colour coded in order to better explain the game procedure and we refer to \ref{['fig:PauliBasis']} for more details.
  • ...and 9 more figures

Theorems & Definitions (179)

  • Theorem 1.1
  • Definition 1.2: Compressible language
  • Theorem 1.3: Gap compression, informal
  • Lemma 2.1: Computability of finite fields, Lemma 3.16 of jiMIPRE2022a
  • Lemma 2.2: Lemma 3.14 of jiMIPRE2022a
  • Definition 2.3: Canonical representation of an affine line
  • Lemma 2.4: Schwartz-Zippel schwartzFastProbabilisticAlgorithms1980zippelProbabilisticAlgorithmsSparse1979
  • Lemma 3.1: Orthogonalization lemma, Theorem 1.2 of delasalleOrthogonalizationPositiveOperator2022
  • Lemma 3.2: Lemma 3.26 of jiMIPRE2022a
  • Lemma 3.3: Lemma 8.5 of jiMIPRE2022a
  • ...and 169 more