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arXiv:quant-ph/0210077

Quantum NP - A Survey

Dorit Aharonov, Tomer Naveh

Published Oct 11, 2002• 116 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0210077

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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 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 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.
QMA and QCMA are possible quantum analogues of the complexity class NP. In QMA the proof is a quantum state and the verification is a quantum circuit. In contrast, in QCMA the proof is restricted to be a classical state. It is not known whether QMA strictly contains QCMA. Here we show that two known QMA-complete problems can be modified to QCMA-complete problems in a natural way: (1) Deciding whether a 3-local Hamiltonian has low energy states (with energy smaller than a given value) that can be prepared with at most k elementary gates is QCMA-complete, whereas it is QMA-complete when the restriction on the complexity of preparation is dropped. (2) Deciding whether a (classically described) quantum circuit does not act as the identity on all basis states is QCMA-complete. It is QMA-complete to decide whether it does not act on all states as the identity.
QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most notably the ``Local Hamiltonian'' problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems. The first one is ``Consistency of Local Density Matrices'': given several density matrices describing different (constant-size) subsets of an n-qubit system, decide whether these are consistent with a single global state. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. This uses algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, ``Fermionic Local Hamiltonian'' and ``N-representability,'' are QMA-complete. These problems arise in calculating the ground state energies of molecular systems. N-representability is a key component in recently developed numerical methods using the contracted Schrodinger equation. Although these problems have been studied since the 1960's, it is only recently that the theory of quantum computation has allowed us to properly characterize their complexity. Finally, we study some special cases of the Consistency problem, pertaining to 1-dimensional and ``stoquastic'' systems. We also give an alternative proof of a result due to Jaynes: whenever local density matrices are consistent, they are consistent with a Gibbs state.
Quantum mechanics has important consequences for machines that store and manipulate information. In particular, quantum computers might be more powerful than classical computers; examples of this include Shor's algorithm for factoring and discrete logarithms, and Grover's algorithm for black-box search. Because of these theoretical results, and the possibility that we may eventually succeed in building scalable quantum computers, it is interesting to study complexity classes based on quantum computation. QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most of which are variants of the Local Hamiltonian problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is Consistency of Local Density Matrices : given a collection of density matrices describing different subsets of an n-qubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, Fermionic Local Hamiltonian and N- representability, are QMA-complete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. N-representability is particularly interesting, as it is a key component in recently developed numerical methods using the contracted Schrodinger equation. Although these problems have been studied since the 1960's, it is only recently that the theory of quantum computation has provided the right tools to properly characterize their complexity. Finally, we study some special cases of the Consistency problem, pertaining to 1-dimensional and stoquastic systems. We also give an alternative proof of a result due to Jaynes : whenever local density matrices are consistent, they are consistent with a Gibbs state
Quantum 3-SAT Is QMA1_11​-Complete
Quantum satisfiability is a constraint satisfaction problem that generalizes classical boolean satisfiability. In the quantum kkk-SAT problem, each constraint is specified by a kkk-local projector and is satisfied by any state in its nullspace. Bravyi showed that quantum 2-SAT can be solved efficiently on a classical computer and that quantum kkk-SAT with k≥4k\geq 4k≥4 is QMA1_11​-complete [S. Bravyi, Efficient Algorithm for a Quantum Analogue of 2-SAT, eprint arXiv:quant-ph/0602108, 2006]. Quantum 3-SAT was known to be contained in QMA1_11​ [Bravyi, 2006], but its computational hardness was unknown until now. We prove that quantum 3-SAT is QMA1_11​-hard, and therefore complete for this complexity class.