684 research outputs found
Encoding Hamiltonian circuits into multiplicative linear logic
10 pagesInternational audienceWe give a new proof of the NP-completeness of multiplicative linear logic without constants by a direct encoding of the Hamiltonian circuit decision problem
Quantum Commuting Circuits and Complexity of Ising Partition Functions
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum
computation consisting only of commuting two-qubit gates and is not universal
in the sense of standard quantum computation. Nevertheless, it has been shown
that if there is a classical algorithm that can simulate IQP efficiently, the
polynomial hierarchy (PH) collapses at the third level, which is highly
implausible. However, the origin of the classical intractability is still less
understood. Here we establish a relationship between IQP and computational
complexity of the partition functions of Ising models. We apply the established
relationship in two opposite directions. One direction is to find subclasses of
IQP that are classically efficiently simulatable in the strong sense, by using
exact solvability of certain types of Ising models. Another direction is
applying quantum computational complexity of IQP to investigate (im)possibility
of efficient classical approximations of Ising models with imaginary coupling
constants. Specifically, we show that there is no fully polynomial randomized
approximation scheme (FPRAS) for Ising models with almost all imaginary
coupling constants even on a planar graph of a bounded degree, unless the PH
collapses at the third level. Furthermore, we also show a multiplicative
approximation of such a class of Ising partition functions is at least as hard
as a multiplicative approximation for the output distribution of an arbitrary
quantum circuit.Comment: 36 pages, 5 figure
Complexity classification of two-qubit commuting hamiltonians
We classify two-qubit commuting Hamiltonians in terms of their computational
complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can
apply to any pair of qubits, starting in a computational basis state. We prove
a dichotomy theorem: either this model is efficiently classically simulable or
it allows one to sample from probability distributions which cannot be sampled
from classically unless the polynomial hierarchy collapses. Furthermore, the
only simulable Hamiltonians are those which fail to generate entanglement. This
shows that generic two-qubit commuting Hamiltonians can be used to perform
computational tasks which are intractable for classical computers under
plausible assumptions. Our proof makes use of new postselection gadgets and Lie
theory.Comment: 34 page
Symmetry-protected adiabatic quantum transistors
The standard circuit model of quantum computation differs in principle from a modern day computer chip in that computation is brought to stationary qubits in the former whereas information is routed spatially across a chip by transistors in the latter. Recently a model was proposed that addresses this key difference in implementation, it was dubbed the adiabatic quantum transistor model to emphasise its similarity to a classical transistor. Here we generalise this model to the setting of spin chains in inherently quantum phases of matter with a property called symmetry-protected order. Our generalisation is significant as it shows the computational properties of the model persist robustly throughout each symmetry-protected quantum phase of matter
Concrete resource analysis of the quantum linear system algorithm used to compute the electromagnetic scattering cross section of a 2D target
We provide a detailed estimate for the logical resource requirements of the
quantum linear system algorithm (QLSA) [Phys. Rev. Lett. 103, 150502 (2009)]
including the recently described elaborations [Phys. Rev. Lett. 110, 250504
(2013)]. Our resource estimates are based on the standard quantum-circuit model
of quantum computation; they comprise circuit width, circuit depth, the number
of qubits and ancilla qubits employed, and the overall number of elementary
quantum gate operations as well as more specific gate counts for each
elementary fault-tolerant gate from the standard set {X, Y, Z, H, S, T, CNOT}.
To perform these estimates, we used an approach that combines manual analysis
with automated estimates generated via the Quipper quantum programming language
and compiler. Our estimates pertain to the example problem size N=332,020,680
beyond which, according to a crude big-O complexity comparison, QLSA is
expected to run faster than the best known classical linear-system solving
algorithm. For this problem size, a desired calculation accuracy 0.01 requires
an approximate circuit width 340 and circuit depth of order if oracle
costs are excluded, and a circuit width and depth of order and
, respectively, if oracle costs are included, indicating that the
commonly ignored oracle resources are considerable. In addition to providing
detailed logical resource estimates, it is also the purpose of this paper to
demonstrate explicitly how these impressively large numbers arise with an
actual circuit implementation of a quantum algorithm. While our estimates may
prove to be conservative as more efficient advanced quantum-computation
techniques are developed, they nevertheless provide a valid baseline for
research targeting a reduction of the resource requirements, implying that a
reduction by many orders of magnitude is necessary for the algorithm to become
practical.Comment: 37 pages, 40 figure
- …