11 research outputs found
Quantum Computing for Financial Mathematics
Quantum computing has recently appeared in the headlines of many scientific
and popular publications. In the context of quantitative finance, we provide
here an overview of its potential.Comment: 5 page
A coherence quantifier based on the quantum optimal transport cost
In this manuscript, we present a coherence measure based on the quantum
optimal transport cost in terms of convex roof extended method. We also obtain
the analytical solutions of the quantifier for pure states. At last, we propose
an operational interpretation of the coherence measure for pure states
Improving the speed of variational quantum algorithms for quantum error correction
We consider the problem of devising a suitable Quantum Error Correction (QEC)
procedures for a generic quantum noise acting on a quantum circuit. In general,
there is no analytic universal procedure to obtain the encoding and correction
unitary gates, and the problem is even harder if the noise is unknown and has
to be reconstructed. The existing procedures rely on Variational Quantum
Algorithms (VQAs) and are very difficult to train since the size of the
gradient of the cost function decays exponentially with the number of qubits.
We address this problem using a cost function based on the Quantum Wasserstein
distance of order 1 (). At variance with other quantum distances
typically adopted in quantum information processing, lacks the unitary
invariance property which makes it a suitable tool to avoid to get trapped in
local minima. Focusing on a simple noise model for which an exact QEC solution
is known and can be used as a theoretical benchmark, we run a series of
numerical tests that show how, guiding the VQA search through the , can
indeed significantly increase both the probability of a successful training and
the fidelity of the recovered state, with respect to the results one obtains
when using conventional approaches
Quantum Monge-Kantorovich problem and transport distance between density matrices
A quantum version of the Monge-Kantorovich optimal transport problem is
analyzed. The transport cost is minimized over the set of all bipartite
coupling states , such that both of its reduced density matrices
and of size are fixed. We show that, selecting the
quantum cost matrix to be proportional to the projector on the antisymmetric
subspace, the minimal transport cost leads to a semidistance between
and , which can be bounded from below by the rescaled Bures distance
and from above by the root infidelity. In the single qubit case we provide a
semi-analytic expression for the optimal transport cost between any two states,
and prove that its square root satisfies the triangle inequality and yields an
analogue of the Wasserstein distance of order two on the set of density
matrices. Assuming that the cost matrix suffers decoherence, we study the
quantum-to-classical transition of the Earth Mover's distance, propose a
continuous family of interpolating distances, and demonstrate in the case of
diagonal mixed states that the quantum transport is cheaper than the classical
one.Comment: 14 pages including appendices, 4 figures. Comments are welcome
The Quantum Wasserstein Distance of Order 1
We propose a generalization of the Wasserstein distance of order 1 to the
quantum states of qudits. The proposal recovers the Hamming distance for
the vectors of the canonical basis, and more generally the classical
Wasserstein distance for quantum states diagonal in the canonical basis. The
proposed distance is invariant with respect to permutations of the qudits and
unitary operations acting on one qudit and is additive with respect to the
tensor product. Our main result is a continuity bound for the von Neumann
entropy with respect to the proposed distance, which significantly strengthens
the best continuity bound with respect to the trace distance. We also propose a
generalization of the Lipschitz constant to quantum observables. The notion of
quantum Lipschitz constant allows us to compute the proposed distance with a
semidefinite program. We prove a quantum version of Marton's transportation
inequality and a quantum Gaussian concentration inequality for the spectrum of
quantum Lipschitz Moreover, we derive bounds on the contraction coefficients of
shallow quantum circuits and of the tensor product of one-qudit quantum
channels with respect to the proposed distance. We discuss other possible
applications in quantum machine learning, quantum Shannon theory, and quantum
many-body systems
Quantum Earth Mover's Distance: A New Approach to Learning Quantum Data
Quantifying how far the output of a learning algorithm is from its target is
an essential task in machine learning. However, in quantum settings, the loss
landscapes of commonly used distance metrics often produce undesirable outcomes
such as poor local minima and exponentially decaying gradients. As a new
approach, we consider here the quantum earth mover's (EM) or Wasserstein-1
distance, recently proposed in [De Palma et al., arXiv:2009.04469] as a quantum
analog to the classical EM distance. We show that the quantum EM distance
possesses unique properties, not found in other commonly used quantum distance
metrics, that make quantum learning more stable and efficient. We propose a
quantum Wasserstein generative adversarial network (qWGAN) which takes
advantage of the quantum EM distance and provides an efficient means of
performing learning on quantum data. Our qWGAN requires resources polynomial in
the number of qubits, and our numerical experiments demonstrate that it is
capable of learning a diverse set of quantum data