7 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
Sparse Selfreducible Sets and Nonuniform Lower Bounds
It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in (Formula presented.), or even in (Formula presented.) that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are selfreducible? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that (Formula presented.) does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that (Formula presented.) does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of (Formula presented.) is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for (Formula presented.)
A novel characterization of the complexity class based on counting and comparison
This is the author's accepted versionFinal version available from Elsevier via the DOI in this recordThe complexity class Θ2P, which is the class of languages recognizable by deterministic Turing machines in polynomial time with at most logarithmic many calls to an NP oracle, received extensive attention in the literature. Its complete problems can be characterized by different specific tasks, such as deciding whether the optimum solution of an NP problem is unique, or whether it is in some sense “odd” (e.g., whether its size is an odd number). In this paper, we introduce a new characterization of this class and its generalization ΘkP to the k-th level of the polynomial hierarchy. We show that problems in ΘkP are also those whose solution involves deciding, for two given sets A and B of instances of two Σk−1P-complete (or Πk−1P-complete) problems, whether the number of “yes”-instances in A is greater than those in B. Moreover, based on this new characterization, we provide a novel sufficient condition for ΘkP-hardness. We also define the general problem Comp-Validk, which is proven here Θk+1P-complete. Comp-Validk is the problem of deciding, given two sets A and B of quantified Boolean formulas with at most k alternating quantifiers, whether the number of valid formulas in A is greater than those in B. Notably, the problem Comp-Sat of deciding whether a set contains more satisfiable Boolean formulas than another set, which is a particular case of Comp-Valid1, demonstrates itself as a very intuitive Θ2P-complete problem. Nonetheless, to our knowledge, it eluded its formal definition to date. In fact, given its strict adherence to the count-and-compare semantics here introduced, Comp-Validk is among the most suitable tools to prove ΘkP-hardness of problems involving the counting and comparison of the number of “yes”-instances in two sets. We support this by showing that the Θ2P-hardness of the Max voting scheme over mCP-nets is easily obtained via the new characterization of ΘkP introduced in this paper.This work was supported by the UK EPSRC grants EP/J008346/1, EP/L012138/1, and EP/M025268/1, and by The Alan Turing Institute under the EPSRC grant EP/N510129/1. We thank Dominik Peters and the anonymous reviewers for their helpful comments on a preliminary version of the paper
On Polynomially Many Queries to NP or QMA Oracles
We study the complexity of problems solvable in deterministic polynomial time
with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as
and , respectively. The former allows one to classify problems more
finely than the Polynomial-Time Hierarchy (PH), whereas the latter
characterizes physically motivated problems such as Approximate Simulation
(APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by
the classes and , defined identically to
and , except that only logarithmically many oracle queries are
allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by
a machine have a "query graph" which is a tree, then this computation
can be simulated in .
In this work, we first show that for any verification class
, any machine with a query
graph of "separator number" can be simulated using deterministic time
and queries to a -oracle. When (which
includes the case of -treewidth, and thus also of trees), this gives an
upper bound of , and when , this yields bound
(QP meaning quasi-polynomial time). We next show how to
combine Gottlob's "admissible-weighting function" framework with the
"flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a
unified approach for embedding computations directly into APX-SIM
instances in a black-box fashion. Finally, we formalize a simple no-go
statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear
polynomial specified via an arithmetic circuit, if one can "weakly
compress" so that its optimal value requires bits to represent, then
can be decided with only queries to an NP-oracle.Comment: 46 pages pages, 5 figures, to appear in ITCS 202
Average Dependence and Random Oracles (Preliminary Report)
This paper is a technical investigation of issues in computational complexity theory relative to a random oracle. We introduce “average dependence,” an alternative method to Bennett and Gill’s “measure preserving map technique and illustrate our technique by the following results