[Paper Review] Tensor network non-zero testing

Sevag Gharibian, Zeph Landau+2 more|arXiv (Cornell University)|Jun 20, 2014

Quantum many-body systems33 references3 citations

TL;DR

This paper investigates the computational complexity of tensor network non-zero testing (TNZ), proving that general TNZ is unlikely to be in the Polynomial Hierarchy unless it collapses. It shows that non-negative and injective tensor networks make TNZ tractable in NP, enabling the first known NP upper bound for commuting stoquastic $k$-SAT with $k \in O(\log n)$ and constant local dimension $D$. This reveals a new class of quantum Hamiltonians whose commuting variants are in NP under broad conditions.

ABSTRACT

Tensor networks are a central tool in condensed matter physics. In this paper, we study the task of tensor network non-zero testing (TNZ): Given a tensor network T, does T represent a non-zero vector? We show that TNZ is not in the Polynomial-Time Hierarchy unless the hierarchy collapses. We next show (among other results) that the special cases of TNZ on non-negative and injective tensor networks are in NP. Using this, we make a simple observation: The commuting variant of the MA-complete stoquastic k-SAT problem on D-dimensional qudits is in NP for logarithmic k and constant D. This reveals the first class of quantum Hamiltonians whose commuting variant is known to be in NP for all (1) logarithmic k, (2) constant D, and (3) for arbitrary interaction graphs.

Motivation & Objective

• To determine the computational complexity of deciding whether a tensor network represents a non-zero vector.
• To analyze the complexity of special cases of tensor network non-zero testing, particularly non-negative and injective tensor networks.
• To establish implications for the commuting $k$-local Hamiltonian problem and stoquastic $k$-SAT in quantum complexity theory.
• To identify conditions under which the commuting $k$-local Hamiltonian problem lies in NP, particularly for logarithmic $k$ and constant local dimension $D$.

Proposed method

• Formalizing the generalized tensor network non-zero testing (gTNZ) problem with input parameters $\alpha \geq \beta \geq 0$ and gap $\alpha - \beta \geq 1$.
• Proving that general gTNZ is $\#P$-hard and that TNZ is not in $\Sigma_i^p$ unless the Polynomial Hierarchy collapses.
• Demonstrating that non-negative tensor networks reduce to NP via a non-deterministic verification of a witness state with non-zero norm.
• Constructing a non-deterministic polynomial-time reduction from the commuting $k$-local Hamiltonian problem to TNZ, using spectral projectors of commuting terms.
• Leveraging the structure of stoquastic Hamiltonians with non-negative entries to show their ground state existence problem is in NP.
• Using injective tensor network properties to characterize non-zero networks and derive complexity bounds.

Experimental results

Research questions

• RQ1 Is the general tensor network non-zero testing problem in the Polynomial Hierarchy, and what are the consequences if it is not?
• RQ2 Can the non-negative tensor network case be solved in NP, and what structural properties enable this?
• RQ3 Does the ability to test non-zero tensor networks in NP imply that the commuting $k$-local Hamiltonian problem lies in NP for $k \in O(\log n)$ and $D \in O(1)$?
• RQ4 Can the specific structure of tensor networks derived from commuting Hamiltonians be exploited to place $k$-CLH in NP?
• RQ5 What is the complexity of TNZ for $G$-injective tensor networks, and how does it compare to injective or non-negative cases?

Key findings

• General tensor network non-zero testing (TNZ) is not in $\Sigma_i^p$ for any $i$ unless the Polynomial Hierarchy collapses to $\Sigma_{i+2}^p$, indicating strong intractability.
• The special case of non-negative tensor networks makes TNZ NP-complete, even for 3-regular graphs with bond dimension 3.
• The commuting variant of stoquastic $k$-SAT is in NP when $k \in O(\log n)$ and the local dimension $D \in O(1)$, due to the non-negative structure of the tensor network.
• A non-deterministic polynomial-time reduction from the commuting $k$-local Hamiltonian problem to TNZ exists for $k \in O\left(\log n\right)$ and $D \in O(1)$.
• Injective tensor networks are guaranteed to represent non-zero vectors, but the converse does not hold, indicating injectivity is sufficient but not necessary for non-zero output.
• The results establish the first known class of quantum Hamiltonians whose commuting variant is in NP for all logarithmic $k$, constant $D$, and arbitrary interaction graphs.