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The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2)

Authors Jonas Kamminga , Dorian Rudolph

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LIPIcs.ITCS.2026.83.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum Complexity Theory
  • PSPACE
  • QMA(2)
  • Consistency of Local Density Matrices
  • Polynomial Optimization

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Abstract

In this work we investigate the computational complexity of the pure consistency of local density matrices (PureCLDM) and pure N-representability (Pure-N-Representability; analog of PureCLDM for bosonic or fermionic systems) problems. In these problems the input is a set of reduced density matrices and the task is to determine whether there exists a global pure state consistent with these reduced density matrices. While mixed CLDM, i.e. where the global state can be mixed, was proven to be QMA-complete by Broadbent and Grilo [JoC 2022], almost nothing was known about the complexity of the pure version. Before our work the best upper and lower bounds were QMA(2) and QMA. Our contribution to the understanding of these problems is twofold.
Firstly, we define a pure state analogue of the complexity class QMA^+ of Aharanov and Regev [FOCS 2003], which we call PureSuperQMA. We prove that both pure-N-Representability and PureCLDM are complete for this new class. Along the way we supplement Broadbent and Grilo by proving hardness for 2-qubit reduced density matrices and showing that mixed N-Representability is QMA-complete.
Secondly, we improve the upper bound on PureCLDM. Using methods from algebraic geometry, we prove that PureSuperQMA ⊆ PSPACE. Our methods, and the PSPACE upper bound, are also valid for PureCLDM with exponential or even perfect precision, hence precisePureCLDM is not preciseQMA(2) = NEXP-complete, unless PSPACE = NEXP. We view this as evidence for a negative answer to the longstanding open question whether PureCLDM is QMA(2)-complete.
The techniques we develop for our PSPACE upper bound are quite general. We are able to use them for various applications: from proving PSPACE upper bounds on other quantum problems to giving an efficient parallel (NC) algorithm for (non-convex) quadratically constrained quadratic programs with few constraints.

Cite As Get BibTex

Jonas Kamminga and Dorian Rudolph. The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2). In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 83:1-83:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.83

Author Details

Jonas Kamminga
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany
Dorian Rudolph
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany

Funding

  • Kamminga, Jonas: supported by the DFG under grant number 450041824.
  • Rudolph, Dorian: supported by the DFG under grant number 432788384 and 563388236.

Acknowledgements

The authors would like to thank Dima Grigoryev for suggesting that the GP algorithm could be parallelized, pointing to helpful references, and giving us advice on how this could be proven.

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