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arXiv:quant-ph/0301087 · International Journal of Quantum Information

THE 2-LOCAL HAMILTONIAN PROBLEM ENCOMPASSES NP

Paweł Wocjan, Thomas Beth

Published Sep 1, 2003• 16 citations
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Abstract

We show that the NP-complete problems max cut and independent set can be formulated as the 2-local Hamiltonian problem as defined by Kitaev. The 5-local Hamiltonian problem was the first problem to be shown to be complete for the quantum complexity class QMA — the quantum analog of NP. Subsequently, it was shown that 3-locality is already sufficient for QMA-completeness. It is still not known whether the 2-local Hamiltonian problem is QMA-complete. Therefore it is interesting to determine what problems can be reduced to the 2-local Hamiltonian problem. Kitaev showed that 3-SAT can be formulated as a 3-local Hamiltonian problem. We extend his result by showing that 2-locality is sufficient in order to encompass NP.

DOI
10.1142/s021974990300022x

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We show that the NP-complete problems max cut and independent set can be formulated as the 2-local Hamiltonian problem as defined by Kitaev. The 5-local Hamiltonian problem was the first problem to be shown to be complete for the quantum complexity class QMA — the quantum analog of NP. Subsequently, it was shown that 3-locality is already sufficient for QMA-completeness. It is still not known whether the 2-local Hamiltonian problem is QMA-complete. Therefore it is interesting to determine what problems can be reduced to the 2-local Hamiltonian problem. Kitaev showed that 3-SAT can be formulated as a 3-local Hamiltonian problem. We extend his result by showing that 2-locality is sufficient in order to encompass NP.
We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum analog of the class NP, and shows that a natural extension of 3-SAT, namely local Hamiltonians, is QMA complete. The result builds upon the classical Cook-Levin proof of the NP completeness of SAT, but differs from it in several fundamental ways, which we highlight. This result raises a rich array of open problems related to quantum complexity, algorithms and entanglement, which we state at the end of this survey. This survey is the extension of lecture notes taken by Naveh for Aharonov's quantum computation course, held in Tel Aviv University, 2001.
We present a new way of encoding a quantum computation into a three-local Hamiltonian. Our construction is novel in that it does not include any terms that induce legal-illegal clock transitions. Therefore, the weights of the terms in the Hamiltonian do not scale with the size of the problem as in previous constructions. This improves the construction by Kempe and Regev [Quantum Inf. Comput. 3, 258–264 (2003); e-print quant-ph∕0302079] who were the first to prove that three-local Hamiltonian is complete for the complexity class QMA, the quantum analog of NP. Quantum k-SAT, a restricted version of the local Hamiltonian problem using only projector terms, was introduced by Bravyi (e-print quant-ph∕0602108) as an analog of the classical k-SAT problem. Bravyi proved that quantum 4-SAT is complete for the class QMA with one-sided error (QMA1) and that quantum 2-SAT is in P. We give an encoding of a quantum circuit into a quantum 4-SAT Hamiltonian using only three-local terms. As an intermediate step to this three-local construction, we show that quantum 3-SAT for particles with dimensions 3×2×2 (a qutrit and two qubits) is QMA1 complete. The complexity of quantum 3-SAT with qubits remains an open question.
We introduce the quantum complexity class FQMA. This class describes the complexity of generating a quantum state that serves as a witness for a given QMA problem. In a certain sense, FQMA is the quantum analogue of FNP (function problems associated with NP). The latter describes the complexity of finding a succinct proof for a NP decision problem. Whereas all FNP problems can be reduced to NP, there is no obvious reduction of FQMA to QMA since the solution of FQMA is a quantum state and the solution of QMA the answer yes or no. We consider quantum state generators that get classical descriptions of 3-local Hamiltonians on n qubits as input and prepare low energy states for these systems as output. We show that such state generators can be used to prepare witnesses for QMA problems. Hence low energy state preparation is FQMA-complete. Our proofs are extensions of the proofs by Kitaev et al. and Kempe and Regev for the QMA-completeness of k-local Hamiltonian problems. We show that FQMA can be solved by preparing thermal equilibrium states with an appropriate temperature decreasing as the reciprocal of a polynomial in n.
The canonical hard problems for NP and its quantum analogue, Quantum Merlin-Arthur (QMA), are MAX-k-SAT and the k-local Hamiltonian problem (k-LH), the quantum generalization of MAX-k-SAT, respectively. In recent years, however, an arguably even more physically motivated problem than k-LH has been formalized - the problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM). Perhaps surprisingly, [Ambainis, CCC 2014] showed that APX-SIM is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^{QMA[log]}-complete, for P^{QMA[log]} the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^{QMA[log]}-complete even when restricted to physically motivated Hamiltonians, obtaining as intermediate steps a variety of related complexity-theoretic results. Specifically, we first give a sequence of results which together yield P^{QMA[log]}-hardness for APX-SIM on well-motivated Hamiltonians such as the 2D Heisenberg model: - We show that for NP, StoqMA, and QMA oracles, a logarithmic number of adaptive queries is equivalent to polynomially many parallel queries. Formally, P^{NP[log]}=P^{||NP}, P^{StoqMA[log]}=P^{||StoqMA}, and P^{QMA[log]}=P^{||QMA}. (The result for NP was previously shown using a different proof technique.) These equalities simplify the proofs of our subsequent results. - Next, we show that the hardness of APX-SIM is preserved under Hamiltonian simulations (à la [Cubitt, Montanaro, Piddock, 2017]) by studying a seemingly weaker problem, ∀-APX-SIM. As a byproduct, we obtain a full complexity classification of APX-SIM, showing it is complete for P, P^{||NP},P^{||StoqMA}, or P^{||QMA} depending on the Hamiltonians employed. - Leveraging the above, we show that APX-SIM is P^{QMA[log]}-complete for any family of Hamiltonians which can efficiently simulate spatially sparse Hamiltonians. This implies APX-SIM is P^{QMA[log]}-complete even on physically motivated models such as the 2D Heisenberg model. Our second focus considers 1D systems: We show that APX-SIM remains P^{QMA[log]}-complete even for local Hamiltonians on a 1D line of 8-dimensional qudits. This uses a number of ideas from above, along with replacing the "query Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.