[Paper Review] 3-Local Hamiltonian is QMA-complete

Julia Kempe, Oded Regev|ArXiv.org|Feb 10, 2003

Advanced Algebra and Geometry2 references20 citations

TL;DR

This paper proves that the 3-local Hamiltonian problem is QMA-complete, reducing the locality required for QMA-completeness from Kitaev's original 5-local result. By constructing a quantum verifier Hamiltonian that encodes a quantum circuit's computation in a 3-local interaction structure, the authors show that the ground state energy problem for 3-local Hamiltonians captures the full power of quantum Merlin-Arthur proof systems.

ABSTRACT

It has been shown by Kitaev that the 5-local Hamiltonian problem is QMA-complete. Here we reduce the locality of the problem by showing that 3-local Hamiltonian is already QMA-complete.

Motivation & Objective

• To close the gap between classical and quantum complexity by determining the minimal locality for QMA-completeness in the local Hamiltonian problem.
• To show that 3-local Hamiltonian is sufficient to capture the full expressive power of QMA, improving upon Kitaev's 5-local construction.
• To provide a constructive reduction from any QMA problem to a 3-local Hamiltonian instance, preserving the promise gap and quantum verification structure.
• To demonstrate that the locality can be reduced to 3 while maintaining QMA-completeness, thereby tightening the complexity-theoretic understanding of quantum constraint satisfaction.
• To leave open the 2-local case, suggesting it may require higher-dimensional qudits or a different approach for completeness.

Proposed method

• Construct a quantum circuit verifier for a QMA problem and embed its computation into a Hamiltonian using clock states and computational history states.
• Define a 3-local Hamiltonian $ H = H_{clock} + H_{comp} $, where $ H_{clock} $ enforces valid clock evolution and $ H_{comp} $ enforces correct gate application and input/output conditions.
• Use a projection onto the legal computational history space $ \Pi $ to restrict the Hamiltonian to valid states, ensuring that only correct computation paths contribute to the energy.
• Bound the operator norm of $ H_{comp} $ by $ O(T) $, where $ T $ is the circuit size, and show that any state with significant amplitude in illegal configurations has high energy.
• Prove that for any state in the legal subspace, the energy of $ H_{comp} $ is at least $ \Omega(1/T^3) $, using spectral analysis and the structure of the propagation terms.
• Establish that the ground state energy is below a threshold $ a $ if and only if the original QMA instance has a valid witness, thus completing the QMA-completeness reduction.

Experimental results

Research questions

• RQ1 Is the 3-local Hamiltonian problem QMA-complete, or is 5-local the minimal possible locality for QMA-completeness?
• RQ2 Can the locality of the Hamiltonian in Kitaev's QMA-completeness construction be reduced from 5 to 3 without losing completeness?
• RQ3 What structural properties of quantum circuits and their energy landscapes allow for a 3-local encoding of quantum computation?
• RQ4 Does the 2-local Hamiltonian problem remain QMA-complete, or is it strictly weaker than 3-local?
• RQ5 Can the QMA-completeness of the local Hamiltonian problem be established using only qubits, or are higher-dimensional systems necessary for 2-local completeness?

Key findings

• The 3-local Hamiltonian problem is QMA-complete, establishing that quantum constraint satisfaction with only three-body interactions captures the full power of quantum interactive proof systems.
• The reduction constructs a 3-local Hamiltonian whose ground state energy is below a threshold $ a $ if and only if there exists a quantum witness that satisfies the verification circuit with high probability.
• The energy gap between yes and no instances is inverse polynomial in the input size, satisfying the promise problem requirement with $ b - a > 1/poly(n) $.
• The proof uses a clock register and computational history states to encode the evolution of a quantum circuit, with 3-local terms enforcing gate application, clock progression, and initialization/finalization.
• Any state with significant amplitude in the illegal subspace (e.g., invalid clock states or incorrect gate sequences) has energy at least $ \Omega(T^{12}) $, which dominates the $ O(T) $ norm of the computation Hamiltonian.
• The analysis shows that the minimal energy of the full Hamiltonian is $ \Omega(1/T^3) $ for legal states, which is bounded away from zero and sufficient to distinguish yes and no instances.