7 research outputs found
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
Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction
In this work we propose a many-body Hamiltonian construction which introduces
only a single separate energy scale of order , for a
small parameter , and for terms in the target Hamiltonian. In its
low-energy subspace, the construction can approximate any normalized target
Hamiltonian with norm ratios
to within relative precision . This comes at the expense
of increasing the locality by at most one, and adding an at most poly-sized
ancilliary system for each coupling; interactions on the ancilliary system are
geometrically local, and can be translationally-invariant.
As an application, we discuss implications for QMA-hardness of the local
Hamiltonian problem, and argue that "almost" translational invariance-defined
as arbitrarily small relative variations of the strength of the local terms-is
as good as non-translational-invariance in many of the constructions used
throughout Hamiltonian complexity theory. We furthermore show that the choice
of geared limit of many-body systems, where e.g. width and height of a lattice
are taken to infinity in a specific relation, can have different
complexity-theoretic implications: even for translationally-invariant models,
changing the geared limit can vary the hardness of finding the ground state
energy with respect to a given promise gap from computationally trivial, to
QMAEXP-, or even BQEXPSPACE-complete
stateQIP = statePSPACE
Complexity theory traditionally studies the hardness of solving classical
computational problems. In the quantum setting, it is also natural to consider
a different notion of complexity, namely the complexity of physically preparing
a certain quantum state. We study the relation between two such state
complexity classes: statePSPACE, which contains states that can be generated by
space-uniform polynomial-space quantum circuits, and stateQIP, which contains
states that a polynomial-time quantum verifier can generate by interacting with
an all-powerful untrusted quantum prover. The latter class was recently
introduced by Rosenthal and Yuen (ITCS 2022), who proved that statePSPACE
stateQIP.
Our main result is the reverse inclusion, stateQIP statePSPACE,
thereby establishing equality of the two classes and providing a natural
state-complexity analogue to the celebrated QIP = PSPACE theorem of Jain, et
al. (J. ACM 2011). To prove this, we develop a polynomial-space quantum
algorithm for solving a large class of exponentially large "PSPACE-computable"
semidefinite programs (SDPs), which also prepares an optimiser encoded in a
quantum state. Our SDP solver relies on recent block-encoding techniques from
quantum algorithms, demonstrating that these techniques are also useful for
complexity theory.
Using similar techniques, we also show that optimal prover strategies for
general quantum interactive protocols can be implemented in quantum polynomial
space. We prove this by studying an algorithmic version of Uhlmann's theorem
and establishing an upper bound on the complexity of implementing Uhlmann
transformations.Comment: 61 page