Wave-Style Token Machines and Quantum Lambda Calculi

Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum lambda-calculus with implicit qubits. This, however, requires generalising the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows us to give a simple operational semantics to the calculus, coherently with the principles of quantum computation.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441

The Geometry of Synchronization (Long Version)

We graft synchronization onto Girard's Geometry of Interaction in its most concrete form, namely token machines. This is realized by introducing proof-nets for SMLL, an extension of multiplicative linear logic with a specific construct modeling synchronization points, and of a multi-token abstract machine model for it. Interestingly, the correctness criterion ensures the absence of deadlocks along reduction and in the underlying machine, this way linking logical and operational properties.Comment: 26 page

Low rank matrix recovery from rank one measurements

We study the recovery of Hermitian low rank matrices $X \in \mathbb{C}^{n \times n}$ from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form $a_j a_j^*$ for some measurement vectors $a_1,...,a_m$, i.e., the measurements are given by $y_j = \mathrm{tr}(X a_j a_j^*)$. The case where the matrix $X=x x^*$ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements, $y_j = |\langle x,a_j\rangle|^2$ via the PhaseLift approach, which has been introduced recently. We derive bounds for the number $m$ of measurements that guarantee successful uniform recovery of Hermitian rank $r$ matrices, either for the vectors $a_j$, $j=1,...,m$, being chosen independently at random according to a standard Gaussian distribution, or $a_j$ being sampled independently from an (approximate) complex projective $t$-design with $t=4$. In the Gaussian case, we require $m \geq C r n$ measurements, while in the case of $4$-designs we need $m \geq Cr n \log(n)$. Our results are uniform in the sense that one random choice of the measurement vectors $a_j$ guarantees recovery of all rank $r$-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate $4$-designs generalizes and improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.Comment: 24 page

Measurements, Models, Systems and Design

531 s.

Efficient Quantum Pseudorandomness

Randomness is both a useful way to model natural systems and a useful tool for engineered systems, e.g. in computation, communication and control. Fully random transformations require exponential time for either classical or quantum systems, but in many case pseudorandom operations can emulate certain properties of truly random ones. Indeed in the classical realm there is by now a well-developed theory of such pseudorandom operations. However the construction of such objects turns out to be much harder in the quantum case. Here we show that random quantum circuits are a powerful source of quantum pseudorandomness. This gives the for the first time a polynomialtime construction of quantum unitary designs, which can replace fully random operations in most applications, and shows that generic quantum dynamics cannot be distinguished from truly random processes. We discuss applications of our result to quantum information science, cryptography and to understanding self-equilibration of closed quantum dynamics.Comment: 6 pages, 1 figure. Short version of http://arxiv.org/abs/1208.069