8 research outputs found
LWPP and WPP are not uniformly gap-definable
AbstractResolving an issue open since Fenner, Fortnow, and Kurtz raised it in [S. Fenner, L. Fortnow, S. Kurtz, Gap-definable counting classes, J. Comput. System Sci. 48 (1) (1994) 116–148], we prove that LWPP is not uniformly gap-definable and that WPP is not uniformly gap-definable. We do so in the context of a broader investigation, via the polynomial degree bound technique, of the lowness, Turing hardness, and inclusion relationships of counting and other central complexity classes
Computation in generalised probabilistic theories
From the existence of an efficient quantum algorithm for factoring, it is
likely that quantum computation is intrinsically more powerful than classical
computation. At present, the best upper bound known for the power of quantum
computation is that BQP is in AWPP. This work investigates limits on
computational power that are imposed by physical principles. To this end, we
define a circuit-based model of computation in a class of operationally-defined
theories more general than quantum theory, and ask: what is the minimal set of
physical assumptions under which the above inclusion still holds? We show that
given only an assumption of tomographic locality (roughly, that multipartite
states can be characterised by local measurements), efficient computations are
contained in AWPP. This inclusion still holds even without assuming a basic
notion of causality (where the notion is, roughly, that probabilities for
outcomes cannot depend on future measurement choices). Following Aaronson, we
extend the computational model by allowing post-selection on measurement
outcomes. Aaronson showed that the corresponding quantum complexity class is
equal to PP. Given only the assumption of tomographic locality, the inclusion
in PP still holds for post-selected computation in general theories. Thus in a
world with post-selection, quantum theory is optimal for computation in the
space of all general theories. We then consider if relativised complexity
results can be obtained for general theories. It is not clear how to define a
sensible notion of an oracle in the general framework that reduces to the
standard notion in the quantum case. Nevertheless, it is possible to define
computation relative to a `classical oracle'. Then, we show there exists a
classical oracle relative to which efficient computation in any theory
satisfying the causality assumption and tomographic locality does not include
NP.Comment: 14+9 pages. Comments welcom
Nonunitary quantum computation in the ground space of local Hamiltonians
A central result in the study of quantum Hamiltonian complexity is that the k-local Hamiltonian problem is
quantum-Merlin-Arthur–complete. In that problem, we must decide if the lowest eigenvalue of a Hamiltonian
is bounded below some value, or above another, promised one of these is true. Given the ground state of the
Hamiltonian, a quantum computer can determine this question, even if the ground state itself may not be efficiently
quantum preparable. Kitaev’s proof of QMA-completeness encodes a unitary quantum circuit in QMA into the
ground space of a Hamiltonian. However, we now have quantum computing models based on measurement
instead of unitary evolution; furthermore, we can use postselected measurement as an additional computational
tool. In this work, we generalize Kitaev’s construction to allow for nonunitary evolution including postselection.
Furthermore, we consider a type of postselection under which the construction is consistent, which we call tame
postselection. We consider the computational complexity consequences of this construction and then consider
how the probability of an event upon which we are postselecting affects the gap between the ground-state energy
and the energy of the first excited state of its corresponding Hamiltonian. We provide numerical evidence that the
two are not immediately related by giving a family of circuits where the probability of an event upon which we
postselect is exponentially small, but the gap in the energy levels of the Hamiltonian decreases as a polynomial