QMA-hardness of Consistency of Local Density Matrices with Applications to Quantum Zero-Knowledge

  title={QMA-hardness of Consistency of Local Density Matrices with Applications to Quantum Zero-Knowledge},
  author={Anne Broadbent and Alex Bredariol Grilo},
  journal={2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)},
  • A. Broadbent, A. B. Grilo
  • Published 2020
  • Computer Science
  • 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
We provide several advances to the understanding of the class of Quantum Merlin-Arthur proof systems (QMA), the quantum analogue of NP. Our central contribution is proving a longstanding conjecture that the Consistency of Local Density Matrices (CLDM) problem is QMA-hard under Karp reductions. The input of CLDM consists of local reduced density matrices on sets of at most $k$ qubits, and the problem asks if there is an n-qubit global quantum state that is locally consistent with all of the k… Expand
Electronic Structure in a Fixed Basis is QMA-complete
It is proved that this electronic-structure problem, when restricted to a fixed single-particle basis and fixed number of electrons, is QMA-complete, which is equivalent to being NP-complete for the ElectronicStructure Hamiltonian in a fixed basis. Expand
Classically Verifiable (Dual-Mode) NIZK for QMA with Preprocessing
This construction has the so-called dual-mode property, which means that there are two computationally indistinguishable modes of generating CRS, and it has information theoretical soundness in one mode and information theoretical zero-knowledge property in the other. Expand
Classically Verifiable NIZK for QMA with Preprocessing
We propose three constructions of classically verifiable non-interactive zero-knowledge proofs and arguments (CV-NIZK) for QMA in various preprocessing models. 1. We construct a CV-NIZK for QMA inExpand
Jordan products of quantum channels and their compatibility
Given two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channelExpand
Non-Destructive Zero-Knowledge Proofs on Quantum States, and Multi-Party Generation of Authorized Hidden GHZ States
This work proposes a different approach, and starts the study of Non-Destructive Zero-Knowledge Proofs on Quantum States, and shows how it can prove useful to distribute a GHZ state between different parties, in such a way that only parties knowing a secret can be part of this GHZ. Expand
Preparation and verification of tensor network states
Esther Cruz, Flavio Baccari, Jordi Tura, 2 Norbert Schuch, 3 and J. Ignacio Cirac Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Straße 1, 85748 Garching, Germany, and Munich Center forExpand
Self-Testing of a Single Quantum Device Under Computational Assumptions
A protocol is constructed that allows a classical verifier to robustly certify that a single computationally bounded quantum device must have prepared a Bell pair and performed single-qubit measurements on it, up to a change of basis applied to both the device's state and measurements. Expand
The Round Complexity of Quantum Zero-Knowledge
This is the first protocol for constant-round statistical zero-knowledge arguments for QMA, and matches the best round complexity known for the corresponding protocols for NP with security against classical adversaries. Expand
A Black-Box Approach to Post-Quantum Zero-Knowledge in Constant Round
A new quantum rewinding technique is introduced that enables a simulator to extract a committed message of a malicious verifier while simulating verifier's internal state in an appropriate sense. Expand


Consistency of Local Density Matrices Is QMA-Complete
  • Yi-Kai Liu
  • Computer Science, Mathematics
  • 2006
It is shown that deciding the consistency of local density matrices is QMA-complete (where QMA is the quantum analogue of NP), which gives an interesting example of a hard problem in QMA. Expand
Low-Degree Testing for Quantum States, and a Quantum Entangled Games PCP for QMA
We show that given an explicit description of a multiplayer game, with a classical verifier and a constant number of players, it is QMA-hard, under randomized reductions, to distinguish between theExpand
The Complexity of the Local Hamiltonian Problem
This paper settles the question and shows that the 2-LOCAL HAMILTONIAN problem is QMA-complete, and demonstrates that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation. Expand
Quantum versus Classical Proofs and Advice
  • S. Aaronson, G. Kuperberg
  • Computer Science, Mathematics
  • Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07)
  • 2007
This paper shows that any quantum algorithm needs Omega (radic2n-m+1) queries to find an n-qubit "marked state"Psi rang, and gives an explicit QCMA protocol that nearly achieves this lower bound. Expand
QMA with Subset State Witnesses
A definition of a new class, SQMA, is provided where the possible quantum witnesses are restricted to the “simpler” subset states, i.e. a uniform superposition over the elements of a subset of n-bit strings. Expand
Complexity Classification of Local Hamiltonian Problems
  • T. Cubitt, A. Montanaro
  • Mathematics, Computer Science
  • 2014 IEEE 55th Annual Symposium on Foundations of Computer Science
  • 2014
This work characterises the complexity of the k-local Hamiltonian problem for all 2-local qubit Hamiltonians and proves for the first time QMA-completeness of the Heisenberg and XY interactions in this setting. Expand
A quantum linearity test for robustly verifying entanglement
A simple two-player test which certifies that the players apply tensor products of Pauli σX and σZ observables on the tensor product of n EPR pairs is introduced, which is the first robust self-test for n E PR pairs. Expand
Zero-Knowledge Proof Systems for QMA
This work proves that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Expand
Perfect Zero Knowledge for Quantum Multiprover Interactive Proofs
The main result is that the two classes are equal, i.e., MIP* = PZK-MIP*. Expand
Non-interactive zero-knowledge arguments for QMA, with preprocessing
If Learning With Errors (LWE) is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing, and it is shown that any language that has an (interactive) proof of quantum knowledge has an AoQK. Expand