Quasi-quantum states and the quasi-quantum PCP theorem

Abstract: We introduce $k$-local quasi-quantum states: a superset of the regular quantum states, defined by relaxing the positivity constraint. We show that a $k$-local quasi-quantum state on $n$ qubits can be 1-1 mapped to a distribution of assignments over $n$ variables with an alphabet of size $4$, which is subject to non-linear constraints over its $k$-local marginals. Therefore, solving the $k$-local Hamiltonian over the quasi-quantum states is equivalent to optimizing a distribution of assignment over a classical $k$-local CSP. We show that this optimization problem is essentially classical by proving it is NP-complete. Crucially, just as ordinary quantum states, these distributions lack a simple tensor-product structure and are therefore not determined straightforwardly by their local marginals. Consequently, our classical optimization problem shares some unique aspects of Hamiltonian complexity: it lacks an easy search-to-decision reduction, and it is not clear that its 1D version can be solved with dynamical programming (i.e., it could remain NP-hard). Our main result is a PCP theorem for the $k$-local Hamiltonian over the quasi-quantum states in the form of a hardness-of-approximation result. The proof suggests the existence of a subtle promise-gap amplification procedure in a model that shares many similarities with the quantum local Hamiltonian problem, thereby providing insights on the quantum PCP conjecture.

• The paper establishes a quasi-quantum PCP theorem by mapping the k-local Hamiltonian problem to a classical constraint satisfaction problem.
• It demonstrates that optimizing over k-local quasi-quantum states remains NP-complete, paralleling classical Hamiltonian complexity.
• The framework offers new insights into quantum information processing by providing tighter relaxations than traditional reduced density matrix methods.

Analyzing Quasi-Quantum States and the Quasi-Quantum PCP Theorem

This paper introduces the concept of $k$-local quasi-quantum (qq) states, an extension of regular quantum states, achieved by relaxing the positivity constraint. The paper's principal objective is to establish a new class of optimization problems over these states and explore their complexity, with an emphasis on an analogue to the PCP theorem for quasi-quantum states in the context of the $k$-local Hamiltonian problem.

Overview

The authors propose a framework where $k$-local qq states extend traditional quantum state definitions. These states can be mapped to a distribution of assignments over variables with size-four alphabets, constrained by non-linear conditions over their $k$-local marginals. This mapping allows the optimization problem of solving the $k$-local Hamiltonian over these states to reduce effectively to a classical $k$-local constraint satisfaction problem (CSP). Remarkably, this problem remains NP-complete, aligning with classical Hamiltonian complexity issues such as the absence of a straightforward search-to-decision reduction.

Key Contributions

1. PCP Theorem for Quasi-Quantum States: The central contribution is establishing a PCP theorem for the $k$-local Hamiltonian involving quasi-quantum states. The theorem demonstrates the complexity of approximating the ground energy up to system-size errors, implicating insights into the quantum PCP conjecture.
2. Optimization Problem Complexity: The optimization problem mapped to a classical $k$-local CSP retains an NP-complete status. This alignment underscores the complexity inherent in quasi-quantum systems, similar to quantum systems, but manageable using classical frameworks.
3. Theoretical Implications: The work suggests that quasi-quantum states can serve as a practical optimization framework, providing potentially tighter relaxations than existing approaches focusing on reduced density matrices consistency.

Practical and Theoretical Implications

The introduction of $k$-local qq states and their accompanying PCP theorem provides valuable insights into Hamiltonian complexity, potentially impacting both theoretical pursuits and practical applications. The exploration of quasi-quantum states as a tool for classical optimization suggests new avenues for efficient solutions to quantum problems, situating these new states as a vital part of future quantum information processing landscapes.

Further, the theoretical implications offer a nuanced understanding of quantum complexity issues and suggest that quasi-quantum frameworks can play a pivotal role in unraveling complex quantum phenomena, possibly bridging some gaps between classical and quantum complexities.

Future Directions

The research opens multiple paths for future exploration:

• Algorithmic Development: Developing algorithms leveraging quasi-quantum state frameworks for practical optimization problems in quantum computing.
• Extension to Higher Dimensions: Assessing the applicability and utility of quasi-quantum states in more complex, higher-dimensional quantum systems.
• Impact on Quantum PCP Conjecture: Further investigation into how this work informs the quantum PCP conjecture, particularly in defining and understanding complexity across varying promise gaps and implementations.

In conclusion, the paper presents a substantial advancement in understanding and optimizing quantum systems via quasi-quantum states, providing robust tools and frameworks for both theoretical study and practical application in quantum information science.