12,438 research outputs found
Detecting Violations of Differential Privacy for Quantum Algorithms
Quantum algorithms for solving a wide range of practical problems have been
proposed in the last ten years, such as data search and analysis, product
recommendation, and credit scoring. The concern about privacy and other ethical
issues in quantum computing naturally rises up. In this paper, we define a
formal framework for detecting violations of differential privacy for quantum
algorithms. A detection algorithm is developed to verify whether a (noisy)
quantum algorithm is differentially private and automatically generate bugging
information when the violation of differential privacy is reported. The
information consists of a pair of quantum states that violate the privacy, to
illustrate the cause of the violation. Our algorithm is equipped with Tensor
Networks, a highly efficient data structure, and executed both on TensorFlow
Quantum and TorchQuantum which are the quantum extensions of famous machine
learning platforms -- TensorFlow and PyTorch, respectively. The effectiveness
and efficiency of our algorithm are confirmed by the experimental results of
almost all types of quantum algorithms already implemented on realistic quantum
computers, including quantum supremacy algorithms (beyond the capability of
classical algorithms), quantum machine learning models, quantum approximate
optimization algorithms, and variational quantum eigensolvers with up to 21
quantum bits
Algorithmic Polynomials
The approximate degree of a Boolean function is
the minimum degree of a real polynomial that approximates pointwise within
. Upper bounds on approximate degree have a variety of applications in
learning theory, differential privacy, and algorithm design in general. Nearly
all known upper bounds on approximate degree arise in an existential manner
from bounds on quantum query complexity. We develop a first-principles,
classical approach to the polynomial approximation of Boolean functions. We use
it to give the first constructive upper bounds on the approximate degree of
several fundamental problems:
- for the -element
distinctness problem;
- for the -subset sum problem;
- for any -DNF or -CNF formula;
- for the surjectivity problem.
In all cases, we obtain explicit, closed-form approximating polynomials that
are unrelated to the quantum arguments from previous work. Our first three
results match the bounds from quantum query complexity. Our fourth result
improves polynomially on the quantum query complexity of the
problem and refutes the conjecture by several experts that surjectivity has
approximate degree . In particular, we exhibit the first natural
problem with a polynomial gap between approximate degree and quantum query
complexity
Blind Quantum Computing with Weak Coherent Pulses
The recently proposed Universal Blind Quantum Computation (UBQC) protocol
allows a client to perform an arbitrary quantum computation on a remote server
such that perfect privacy is guaranteed if the client is capable of producing
random separable single qubit states. While from a theoretical point of view,
this arguably constitutes the lowest possible quantum requirement, from a
pragmatic point of view, generation of random single qubits which can be sent
along long distances without loss is quite challenging and can never be
achieved perfectly.
In analogy to the concept of approximate security developed for other
cryptographic protocols, we introduce here the concept of approximate blindness
for UBQC, allowing us to characterize the robustness of the protocol to
possible imperfections.
Following this, we present a remote blind single qubit preparation protocol,
by which a client with access to realistic quantum devices (such as coherent
laser light) can in a delegated fashion prepare quantum states arbitrarily
close to perfect random single qubit states. We finally prove that access to
coherent states is sufficient to efficiently achieve approximate blindness with
arbitrary small security parameter.Comment: 16 pages, 1 figur
Duality of privacy amplification against quantum adversaries and data compression with quantum side information
We show that the tasks of privacy amplification against quantum adversaries
and data compression with quantum side information are dual in the sense that
the ability to perform one implies the ability to perform the other. These are
two of the most important primitives in classical information theory, and are
shown to be connected by complementarity and the uncertainty principle in the
quantum setting. Applications include a new uncertainty principle formulated in
terms of smooth min- and max-entropies, as well as new conditions for
approximate quantum error correction.Comment: v2: Includes a derivation of an entropic uncertainty principle for
smooth min- and max-entropies. Discussion of the
Holevo-Schumacher-Westmoreland theorem remove
Converse bounds for private communication over quantum channels
This paper establishes several converse bounds on the private transmission
capabilities of a quantum channel. The main conceptual development builds
firmly on the notion of a private state, which is a powerful, uniquely quantum
method for simplifying the tripartite picture of privacy involving local
operations and public classical communication to a bipartite picture of quantum
privacy involving local operations and classical communication. This approach
has previously led to some of the strongest upper bounds on secret key rates,
including the squashed entanglement and the relative entropy of entanglement.
Here we use this approach along with a "privacy test" to establish a general
meta-converse bound for private communication, which has a number of
applications. The meta-converse allows for proving that any quantum channel's
relative entropy of entanglement is a strong converse rate for private
communication. For covariant channels, the meta-converse also leads to
second-order expansions of relative entropy of entanglement bounds for private
communication rates. For such channels, the bounds also apply to the private
communication setting in which the sender and receiver are assisted by
unlimited public classical communication, and as such, they are relevant for
establishing various converse bounds for quantum key distribution protocols
conducted over these channels. We find precise characterizations for several
channels of interest and apply the methods to establish several converse bounds
on the private transmission capabilities of all phase-insensitive bosonic
channels.Comment: v3: 53 pages, 3 figures, final version accepted for publication in
IEEE Transactions on Information Theor
General paradigm for distilling classical key from quantum states
We develop a formalism for distilling a classical key from a quantum state in
a systematic way, expanding on our previous work on secure key from bound
entanglement [K. Horodecki et. al., Phys. Rev. Lett. 94 (2005)]. More detailed
proofs, discussion and examples are provided of the main results. Namely, we
demonstrate that all quantum cryptographic protocols can be recast in a way
which looks like entanglement theory, with the only change being that instead
of distilling EPR pairs, the parties distill private states. The form of these
general private states are given, and we show that there are a number of useful
ways of expressing them. Some of the private states can be approximated by
certain states which are bound entangled. Thus distillable entanglement is not
a requirement for a private key. We find that such bound entangled states are
useful for a cryptographic primitive we call a controlled private quantum
channel. We also find a general class of states which have negative partial
transpose (are NPT), but which appear to be bound entangled. The relative
entropy distance is shown to be an upper bound on the rate of key. This allows
us to compute the exact value of distillable key for a certain class of private
states.Comment: 41 pages, ReVTeX4, improved version, resubmitted to IEE
Secure key from bound entanglement
We characterize the set of shared quantum states which contain a
cryptographically private key. This allows us to recast the theory of privacy
as a paradigm closely related to that used in entanglement manipulation. It is
shown that one can distill an arbitrarily secure key from bound entangled
states. There are also states which have less distillable private key than the
entanglement cost of the state. In general the amount of distillable key is
bounded from above by the relative entropy of entanglement. Relationships
between distillability and distinguishability are found for a class of states
which have Bell states correlated to separable hiding states. We also describe
a technique for finding states exhibiting irreversibility in entanglement
distillation.Comment: 4 pages, no figures, to appear in PR
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