768 research outputs found
Oracle Complexity Classes and Local Measurements on Physical Hamiltonians
The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of
estimating ground state energies of local Hamiltonians. Perhaps surprisingly,
[Ambainis, CCC 2014] showed that the related, but arguably more natural,
problem of simulating local measurements on ground states of local Hamiltonians
(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 more physical Hamiltonians, obtaining as intermediate steps a variety of
related complexity-theoretic results.
We first give a sequence of results which together yield P^QMA[log]-hardness
for APX-SIM on well-motivated Hamiltonians: (1) We show that for NP, StoqMA,
and QMA oracles, a logarithmic number of adaptive queries is equivalent to
polynomially many parallel queries. These equalities simplify the proofs of our
subsequent results. (2) Next, we show that the hardness of APX-SIM is preserved
under Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). 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. (3) 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, including 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.Comment: 38 pages, 3 figure
A new construction for a QMA complete 3-local Hamiltonian
We present a new way of encoding a quantum computation into a 3-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, who
were the first to prove that 3-local Hamiltonian is complete for the complexity
class QMA, the quantum analogue of NP.
Quantum k-SAT, a restricted version of the local Hamiltonian problem using
only projector terms, was introduced by Bravyi as an analogue of the classical
k-SAT problem. Bravyi proved that quantum 4-SAT is complete for the class QMA
with one-sided error (QMA_1) and that quantum 2-SAT is in P. We give an
encoding of a quantum circuit into a quantum 4-SAT Hamiltonian using only
3-local terms. As an intermediate step to this 3-local construction, we show
that quantum 3-SAT for particles with dimensions 3x2x2 (a qutrit and two
qubits) is QMA_1 complete. The complexity of quantum 3-SAT with qubits remains
an open question.Comment: 11 pages, 4 figure
QMA-complete problems for stoquastic Hamiltonians and Markov matrices
We show that finding the lowest eigenvalue of a 3-local symmetric stochastic
matrix is QMA-complete. We also show that finding the highest energy of a
stoquastic Hamiltonian is QMA-complete and that adiabatic quantum computation
using certain excited states of a stoquastic Hamiltonian is universal. We also
show that adiabatic evolution in the ground state of a stochastic frustration
free Hamiltonian is universal. Our results give a new QMA-complete problem
arising in the classical setting of Markov chains, and new adiabatically
universal Hamiltonians that arise in many physical systems.Comment: 11 pages. Contains several new results not present in version 1
An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem
We present a classical approximation algorithm for the MAX-2-Local Hamiltonian problem. This is a maximization version of the QMA-complete 2-Local Hamiltonian problem in quantum computing, with the additional assumption that each local term is positive semidefinite. The MAX-2-Local Hamiltonian problem generalizes NP-hard constraint satisfaction problems, and our results may be viewed as generalizations of approximation approaches for the MAX-2-CSP problem. We work in the product state space and extend the framework of Goemans and Williamson for approximating MAX-2-CSPs. The key difference is that in the product state setting, a solution consists of a set of normalized 3-dimensional vectors rather than boolean numbers, and we leverage approximation results for rank-constrained Grothendieck inequalities. For MAX-2-Local Hamiltonian we achieve an approximation ratio of 0.328. This is the first example of an approximation algorithm beating the random quantum assignment ratio of 0.25 by a constant factor
Realizable Hamiltonians for Universal Adiabatic Quantum Computers
It has been established that local lattice spin Hamiltonians can be used for
universal adiabatic quantum computation. However, the 2-local model
Hamiltonians used in these proofs are general and hence do not limit the types
of interactions required between spins. To address this concern, the present
paper provides two simple model Hamiltonians that are of practical interest to
experimentalists working towards the realization of a universal adiabatic
quantum computer. The model Hamiltonians presented are the simplest known
QMA-complete 2-local Hamiltonians. The 2-local Ising model with 1-local
transverse field which has been realized using an array of technologies, is
perhaps the simplest quantum spin model but is unlikely to be universal for
adiabatic quantum computation. We demonstrate that this model can be rendered
universal and QMA-complete by adding a tunable 2-local transverse XX coupling.
We also show the universality and QMA-completeness of spin models with only
1-local Z and X fields and 2-local ZX interactions.Comment: Paper revised and extended to improve clarity; to appear in Physical
Review
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