Testing matrix product states

TL;DR

This work analyzes the problem of testing whether an unknown pure quantum state is a matrix product state (MPS) of bond dimension $r$ within the property-testing framework. It first refines the two-copy product test for product states ($r=1$), providing a simpler, optimal bound that sharpens prior results and demonstrates constant-sample testing properties for constant entanglement-distance. For general $r\ge 2$, it introduces the MPS tester, which uses a concurrently-run set of rank testers via weak Schur sampling to certify $r$-rank reductions across all prefixes, achieving a copy complexity of $m=O(n r^2/\delta^2)$ with perfect completeness, and proving a lower bound of $\Omega(\sqrt{n}/\delta^2)$ copies. The results emphasize a fundamental $n$-dependence for $r\ge 2$, contrasting with the constant-copy regime for product states, and they leverage representation theory and low-rank approximations to connect spectral properties to MPS-structure. Overall, the paper advances practical testing methods for structured quantum states and clarifies the sample-complexity landscape in relation to entanglement and tensor-network representations.

Abstract

Devising schemes for testing the amount of entanglement in quantum systems has played a crucial role in quantum computing and information theory. Here, we study the problem of testing whether an unknown state $|ψ\rangle$ is a matrix product state (MPS) in the property testing model. MPS are a class of physically-relevant quantum states which arise in the study of quantum many-body systems. A quantum state $|ψ_{1,...,n}\rangle$ comprised of $n$ qudits is said to be an MPS of bond dimension $r$ if the reduced density matrix $ψ_{1,...,k}$ has rank $r$ for each $k \in \{1,...,n\}$. When $r=1$, this corresponds to the set of product states. For larger values of $r$, this yields a more expressive class of quantum states, which are allowed to possess limited amounts of entanglement. In the property testing model, one is given $m$ identical copies of $|ψ\rangle$, and the goal is to determine whether $|ψ\rangle$ is an MPS of bond dimension $r$ or whether $|ψ\rangle$ is far from all such states. For the case of product states, we study the product test, a simple two-copy test previously analyzed by Harrow and Montanaro (FOCS 2010), and a key ingredient in their proof that $\mathsf{QMA(2)}=\mathsf{QMA}(k)$ for $k \geq 2$. We give a new and simpler analysis of the product test which achieves an optimal bound for a wide range of parameters, answering open problems of Harrow and Montanaro (FOCS 2010) and Montanaro and de Wolf (2016). For the case of $r\geq 2$, we give an efficient algorithm for testing whether $|ψ\rangle$ is an MPS of bond dimension $r$ using $m = O(n r^2)$ copies, independent of the dimensions of the qudits, and we show that $Ω(n^{1/2})$ copies are necessary for this task. This lower bound shows that a dependence on the number of qudits $n$ is necessary, in sharp contrast to the case of product states where a constant number of copies suffices.

Figures (2)

• Figure 1: The product test and MPS tester.
• Figure 2: Upper bounds on $\operatorname{PT}(\omega)$ as a function of $\omega = 1-\epsilon$. The red line is the function $\tfrac{1}{3} \omega^2 + \tfrac{2}{3}$ and the magenta line is the function $\omega^2 - \omega+1$. The thick pink line is the minimum of the two. This is the upper bound we prove.The blue line is the function $1-\epsilon + \epsilon^2 + \epsilon^{3/2}$ and the cyan line is the function $1 - \tfrac{11}{512} \epsilon$. The thick light blue line is the minimum of the two. This is the upper bound of Harrow and Montanaro Harrow2013_product_testing.

Theorems & Definitions (50)

• Definition 1: Matrix product states
• Definition 2: Distance to $\mathop{\mathrm{\mathsf{MPS}}}\nolimits(r)$
• Definition 3: $\mathop{\mathrm{\mathsf{MPS}}}\nolimits(r)$ tester
• Definition 4: The SWAP test
• Definition 5: The product test
• Definition 6
• Theorem 7: Harrow2013_product_testing
• Theorem 8: Product test upper-bound
• Proposition 9: Product test lower-bound
• Corollary 10: Product test, tight bound