101 research outputs found
A Full Characterization of Quantum Advice
We prove the following surprising result: given any quantum state rho on n
qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of
two-qubit interactions), such that any ground state of H can be used to
simulate rho on all quantum circuits of fixed polynomial size. In terms of
complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which
supersedes the previous result of Aaronson that BQP/qpoly is contained in
PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in
power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools --
including a result of Alon et al. on learning of real-valued concept classes, a
result of Aaronson on the learnability of quantum states, and a result of
Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new
ones. The main new tool is a so-called majority-certificates lemma, which is
closely related to boosting in machine learning, and which seems likely to find
independent applications. In its simplest version, this lemma says the
following. Given any set S of Boolean functions on n variables, any function f
in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm
in S, such that each fi is the unique function in S compatible with O(log|S|)
input/output constraints.Comment: We fixed two significant issues: 1. The definition of YQP machines
needed to be changed to preserve our results. The revised definition is more
natural and has the same intuitive interpretation. 2. We needed properties of
Local Hamiltonian reductions going beyond those proved in previous works
(whose results we'd misstated). We now prove the needed properties. See p. 6
for more on both point
A Survey of Quantum Learning Theory
This paper surveys quantum learning theory: the theoretical aspects of
machine learning using quantum computers. We describe the main results known
for three models of learning: exact learning from membership queries, and
Probably Approximately Correct (PAC) and agnostic learning from classical or
quantum examples.Comment: 26 pages LaTeX. v2: many small changes to improve the presentation.
This version will appear as Complexity Theory Column in SIGACT News in June
2017. v3: fixed a small ambiguity in the definition of gamma(C) and updated a
referenc
Unbounded-error One-way Classical and Quantum Communication Complexity
This paper studies the gap between quantum one-way communication complexity
and its classical counterpart , under the {\em unbounded-error}
setting, i.e., it is enough that the success probability is strictly greater
than 1/2. It is proved that for {\em any} (total or partial) Boolean function
, , i.e., the former is always exactly one half
as large as the latter. The result has an application to obtaining (again an
exact) bound for the existence of -QRAC which is the -qubit random
access coding that can recover any one of original bits with success
probability . We can prove that -QRAC exists if and only if
. Previously, only the construction of QRAC using one qubit,
the existence of -RAC, and the non-existence of
-QRAC were known.Comment: 9 pages. To appear in Proc. ICALP 200
Learning hard quantum distributions with variational autoencoders
Studying general quantum many-body systems is one of the major challenges in
modern physics because it requires an amount of computational resources that
scales exponentially with the size of the system.Simulating the evolution of a
state, or even storing its description, rapidly becomes intractable for exact
classical algorithms. Recently, machine learning techniques, in the form of
restricted Boltzmann machines, have been proposed as a way to efficiently
represent certain quantum states with applications in state tomography and
ground state estimation. Here, we introduce a new representation of states
based on variational autoencoders. Variational autoencoders are a type of
generative model in the form of a neural network. We probe the power of this
representation by encoding probability distributions associated with states
from different classes. Our simulations show that deep networks give a better
representation for states that are hard to sample from, while providing no
benefit for random states. This suggests that the probability distributions
associated to hard quantum states might have a compositional structure that can
be exploited by layered neural networks. Specifically, we consider the
learnability of a class of quantum states introduced by Fefferman and Umans.
Such states are provably hard to sample for classical computers, but not for
quantum ones, under plausible computational complexity assumptions. The good
level of compression achieved for hard states suggests these methods can be
suitable for characterising states of the size expected in first generation
quantum hardware.Comment: v2: 9 pages, 3 figures, journal version with major edits with respect
to v1 (rewriting of section "hard and easy quantum states", extended
discussion on comparison with tensor networks
Two Results about Quantum Messages
We show two results about the relationship between quantum and classical
messages. Our first contribution is to show how to replace a quantum message in
a one-way communication protocol by a deterministic message, establishing that
for all partial Boolean functions we
have . This bound was previously
known for total functions, while for partial functions this improves on results
by Aaronson, in which either a log-factor on the right hand is present, or the
left hand side is , and in which also no entanglement is
allowed.
In our second contribution we investigate the power of quantum proofs over
classical proofs. We give the first example of a scenario, where quantum proofs
lead to exponential savings in computing a Boolean function. The previously
only known separation between the power of quantum and classical proofs is in a
setting where the input is also quantum.
We exhibit a partial Boolean function , such that there is a one-way
quantum communication protocol receiving a quantum proof (i.e., a protocol of
type QMA) that has cost for , whereas every one-way quantum
protocol for receiving a classical proof (protocol of type QCMA) requires
communication
Approximating Fractional Time Quantum Evolution
An algorithm is presented for approximating arbitrary powers of a black box
unitary operation, , where is a real number, and
is a black box implementing an unknown unitary. The complexity of
this algorithm is calculated in terms of the number of calls to the black box,
the errors in the approximation, and a certain `gap' parameter. For general
and large , one should apply a total of times followed by our procedure for approximating the fractional
power . An example is also given where for
large integers this method is more efficient than direct application of
copies of . Further applications and related algorithms are also
discussed.Comment: 13 pages, 2 figure
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