5,977 research outputs found
Exponential Quantum Speed-ups are Generic
A central problem in quantum computation is to understand which quantum
circuits are useful for exponential speed-ups over classical computation. We
address this question in the setting of query complexity and show that for
almost any sufficiently long quantum circuit one can construct a black-box
problem which is solved by the circuit with a constant number of quantum
queries, but which requires exponentially many classical queries, even if the
classical machine has the ability to postselect.
We prove the result in two steps. In the first, we show that almost any
element of an approximate unitary 3-design is useful to solve a certain
black-box problem efficiently. The problem is based on a recent oracle
construction of Aaronson and gives an exponential separation between quantum
and classical bounded-error with postselection query complexities.
In the second step, which may be of independent interest, we prove that
linear-sized random quantum circuits give an approximate unitary 3-design. The
key ingredient in the proof is a technique from quantum many-body theory to
lower bound the spectral gap of local quantum Hamiltonians.Comment: 24 pages. v2 minor correction
The quantum correlation between the selection of the problem and that of the solution sheds light on the mechanism of the quantum speed up
In classical problem solving, there is of course correlation between the
selection of the problem on the part of Bob (the problem setter) and that of
the solution on the part of Alice (the problem solver). In quantum problem
solving, this correlation becomes quantum. This means that Alice contributes to
selecting 50% of the information that specifies the problem. As the solution is
a function of the problem, this gives to Alice advanced knowledge of 50% of the
information that specifies the solution. Both the quadratic and exponential
speed ups are explained by the fact that quantum algorithms start from this
advanced knowledge.Comment: Earlier version submitted to QIP 2011. Further clarified section 1,
"Outline of the argument", submitted to Phys Rev A, 16 page
On the role of entanglement and correlations in mixed-state quantum computation
In a quantum computation with pure states, the generation of large amounts of
entanglement is known to be necessary for a speedup with respect to classical
computations. However, examples of quantum computations with mixed states are
known, such as the deterministic computation with one quantum qubit (DQC1)
model [Knill and Laflamme, Phys. Rev. Lett. 81, 5672 (1998)], in which
entanglement is at most marginally present, and yet a computational speedup is
believed to occur. Correlations, and not entanglement, have been identified as
a necessary ingredient for mixed-state quantum computation speedups. Here we
show that correlations, as measured through the operator Schmidt rank, are
indeed present in large amounts in the DQC1 circuit. This provides evidence for
the preclusion of efficient classical simulation of DQC1 by means of a whole
class of classical simulation algorithms, thereby reinforcing the conjecture
that DQC1 leads to a genuine quantum computational speedup
Efficient classical simulation of slightly entangled quantum computations
We present a scheme to efficiently simulate, with a classical computer, the
dynamics of multipartite quantum systems on which the amount of entanglement
(or of correlations in the case of mixed-state dynamics) is conveniently
restricted. The evolution of a pure state of n qubits can be simulated by using
computational resources that grow linearly in n and exponentially in the
entanglement. We show that a pure-state quantum computation can only yield an
exponential speed-up with respect to classical computations if the entanglement
increases with the size n of the computation, and gives a lower bound on the
required growth.Comment: 4 pages. Major changes. Significantly improved simulation schem
Universal quantum computation with little entanglement
We show that universal quantum computation can be achieved in the standard
pure-state circuit model while, at any time, the entanglement entropy of all
bipartitions is small---even tending to zero with growing system size. The
result is obtained by showing that a quantum computer operating within a small
region around the set of unentangled states still has universal computational
power, and by using continuity of entanglement entropy. In fact an analogous
conclusion applies to every entanglement measure which is continuous in a
certain natural sense, which amounts to a large class. Other examples include
the geometric measure, localizable entanglement, smooth epsilon-measures,
multipartite concurrence, squashed entanglement, and several others. We discuss
implications of these results for the believed role of entanglement as a key
necessary resource for quantum speed-ups
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