Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup

At its core a $t$-design is a method for sampling from a set of unitaries in a way which mimics sampling randomly from the Haar measure on the unitary group, with applications across quantum information processing and physics. We construct new families of quantum circuits on $n$-qubits giving rise to $\varepsilon$-approximate unitary $t$-designs efficiently in $O(n^3t^{12})$ depth. These quantum circuits are based on a relaxation of technical requirements in previous constructions. In particular, the construction of circuits which give efficient approximate $t$-designs by Brandao, Harrow, and Horodecki (F.G.S.L Brandao, A.W Harrow, and M. Horodecki, Commun. Math. Phys. (2016).) required choosing gates from ensembles which contained inverses for all elements, and that the entries of the unitaries are algebraic. We reduce these requirements, to sets that contain elements without inverses in the set, and non-algebraic entries, which we dub partially invertible universal sets. We then adapt this circuit construction to the framework of measurement based quantum computation(MBQC) and give new explicit examples of $n$-qubit graph states with fixed assignments of measurements (graph gadgets) giving rise to unitary $t$-designs based on partially invertible universal sets, in a natural way. We further show that these graph gadgets demonstrate a quantum speedup, up to standard complexity theoretic conjectures. We provide numerical and analytical evidence that almost any assignment of fixed measurement angles on an $n$-qubit cluster state give efficient $t$-designs and demonstrate a quantum speedup.Comment: 25 pages,7 figures. Comments are welcome. Some typos corrected in newest version. new References added.Proofs unchanged. Results unchange

Distribution multi-utilisateur de paires de photons intriqués aux longueurs d'onde des télécommunications

In order to progress towards a quantum communication network, it is vital to limit the necessary resources. In particular, we show here that we can use a single source of entangled photons pairs, to serve a large number of users. This implies that the considered source has to be spectrally broadband, and compatible with the current optical telecommunications’ infrastructure. At first, we studied a correlated photon source, based on SPDC. From the experimental values measured (counts and coincidences), an upper bound of the visibility that can be achieved in entanglement is calculated as well as the source brightness. A quality factor is established in order to compare and rank the demultiplexers tested in relation to the compromise between the quality of the quantum correlations and the source brightness.We then characterized a source of polarization entangled photon pairs. The entanglement is obtained by performing a double pumping path in a crystal of PPLN, generating a state Φ> =(1/√2)(|HH> +|VV>) . The generated photon pairs are split by using the same demultiplexers. Measurements of visibilities and the Bell parameter S are performed. The performances of the demultiplexers are compared to those obtained with the twin photons.We show that it is possible with a single source, to establish a quantum communication channel with at least 3 couples of users simultaneously. We propose at the end of this study, various methods of improvement.Afin de pouvoir œuvrer vers un réseau de communication quantique, il est primordial de limiter les ressources nécessaires. En particulier, nous montrons ici que l’on peut utiliser une source unique de paires de photons intriqués, pour desservir un grand nombre d’utilisateurs. Ceci implique que la source en question soit à large bande spectrale d’émission, et compatible avec les infrastructures actuelles de télécommunication optique. Dans un premier temps nous avons étudié une source de photons corrélés, basée sur la fluorescence paramétrique. A partir des valeurs expérimentales mesurées (coups et coïncidences), une borne maximale de la visibilité que l’on pourra obtenir en intrication est calculée ainsi que la brillance. Un facteur de qualité est établi afin de comparer et classer les démultiplexeurs testés par rapport au compromis entre la qualité des corrélations quantiques et la brillance de la source. La deuxième partie consiste à caractériser une source de photons intriqués en polarisation. L’intrication a été réalisée en effectuant un double passage dans un cristal de PPLN, générant un état de type |Φ> =(1/√2)(|HH> +|VV>) . Les photons des paires générées sont aussi séparés par démultiplexage en longueur d’onde. Des mesures de visibilités et du paramètre de Bell S sont effectuées. Les performances des démultiplexeurs sont comparées à celles prévues avec les photons jumeaux. Nous montrons qu’il est possible à partir d’une source unique, d’établir un canal quantique de communication avec au moins 3 couples d’utilisateurs en simultané. Nous proposons à la fin de cette étude, diverses méthodes d’amélioration

Experimental wavelength division multiplexed photon pair distribution

We have experimentally implemented the distribution of photon pairs produced by spontaneous parametric down conversion through telecom dense wavelength division multiplexing filters. Using the measured counts and coincidences between symmetric channels, we evaluate the maximum fringe visibility that can be obtained with polarization entangled photons and compare different filter technologies.Comment: 3 pages, 4 figures, submitted to Optics Letter

On Unitary <i>t</i>-Designs from Relaxed Seeds

The capacity to randomly pick a unitary across the whole unitary group is a powerful tool across physics and quantum information. A unitary $t$-design is designed to tackle this challenge in an efficient way, yet constructions to date rely on heavy constraints. In particular, they are composed of ensembles of unitaries which, for technical reasons, must contain inverses and whose entries are algebraic. In this work, we reduce the requirements for generating an $\varepsilon$-approximate unitary $t$-design. To do so, we first construct a specific $n$-qubit random quantum circuit composed of a sequence of, randomly chosen, 2-qubit gates, chosen from a set of unitaries which is approximately universal on $U(4)$, yet need not contain unitaries and their inverses, nor are in general composed of unitaries whose entries are algebraic; dubbed $relaxed$ seed. We then show that this relaxed seed, when used as a basis for our construction, gives rise to an $\varepsilon$-approximate unitary $t$-design efficiently, where the depth of our random circuit scales as $poly(n, t, log(1/\varepsilon))$, thereby overcoming the two requirements which limited previous constructions. We suspect the result found here is not optimal, and can be improved. Particularly because the number of gates in the relaxed seeds introduced here grows with $n$ and $t$. We conjecture that constant sized seeds such as those in ( Brand\~ao, Harrow, and Horodecki; Commun. Math. Phys. (2016) 346: 397) are sufficient.Comment: Typos corrected. Readability improved. Results unchanged. Proofs unchange

Simple performance evaluation of pulsed spontaneous parametric down-conversion sources for quantum communications

Fast and complete characterization of pulsed spontaneous parametric down conversion (SPDC) sources is important for applications in quantum information processing and communications. We propose a simple method to perform this task, which only requires measuring the counts on the two output channels and the coincidences between them, as well as modeling the filter used to reduce the source bandwidth. The proposed method is experimentally tested and used for a complete evaluation of SPDC sources (pair emission probability, total losses, and fidelity) of different bandwidths. This method can find applications in the setting up of SPDC sources and in the continuous verification of the quality of quantum communication links

Multiuser distribution of entangled photon pairs at telecommunication wavelength

Afin de pouvoir œuvrer vers un réseau de communication quantique, il est primordial de limiter les ressources nécessaires. En particulier, nous montrons ici que l’on peut utiliser une source unique de paires de photons intriqués, pour desservir un grand nombre d’utilisateurs. Ceci implique que la source en question soit à large bande spectrale d’émission, et compatible avec les infrastructures actuelles de télécommunication optique. Dans un premier temps nous avons étudié une source de photons corrélés, basée sur la fluorescence paramétrique. A partir des valeurs expérimentales mesurées (coups et coïncidences), une borne maximale de la visibilité que l’on pourra obtenir en intrication est calculée ainsi que la brillance. Un facteur de qualité est établi afin de comparer et classer les démultiplexeurs testés par rapport au compromis entre la qualité des corrélations quantiques et la brillance de la source. La deuxième partie consiste à caractériser une source de photons intriqués en polarisation. L’intrication a été réalisée en effectuant un double passage dans un cristal de PPLN, générant un état de type |Φ> =(1/√2)(|HH> +|VV>) . Les photons des paires générées sont aussi séparés par démultiplexage en longueur d’onde. Des mesures de visibilités et du paramètre de Bell S sont effectuées. Les performances des démultiplexeurs sont comparées à celles prévues avec les photons jumeaux. Nous montrons qu’il est possible à partir d’une source unique, d’établir un canal quantique de communication avec au moins 3 couples d’utilisateurs en simultané. Nous proposons à la fin de cette étude, diverses méthodes d’amélioration.In order to progress towards a quantum communication network, it is vital to limit the necessary resources. In particular, we show here that we can use a single source of entangled photons pairs, to serve a large number of users. This implies that the considered source has to be spectrally broadband, and compatible with the current optical telecommunications’ infrastructure. At first, we studied a correlated photon source, based on SPDC. From the experimental values measured (counts and coincidences), an upper bound of the visibility that can be achieved in entanglement is calculated as well as the source brightness. A quality factor is established in order to compare and rank the demultiplexers tested in relation to the compromise between the quality of the quantum correlations and the source brightness.We then characterized a source of polarization entangled photon pairs. The entanglement is obtained by performing a double pumping path in a crystal of PPLN, generating a state Φ> =(1/√2)(|HH> +|VV>) . The generated photon pairs are split by using the same demultiplexers. Measurements of visibilities and the Bell parameter S are performed. The performances of the demultiplexers are compared to those obtained with the twin photons.We show that it is possible with a single source, to establish a quantum communication channel with at least 3 couples of users simultaneously. We propose at the end of this study, various methods of improvement

Unitary tt-designs from relaxedrelaxed seeds

In this work we reduce the requirements for generating $t$-designs, an important tool for randomisation with applications across quantum information and physics. We show that random quantum circuits with support over families of $relaxed$ finite sets of unitaries which are approximately universal in $U(4)$ (we call such sets $seeds$), converge towards approximate unitary $t$-designs efficiently in $poly(n,t)$ depth, where $n$ is the number of inputs of the random quantum circuit, and $t$ is the order of the design. We show this convergence for seeds which are relaxed in the sense that every unitary matrix in the seed need not have an inverse in the seed, nor be composed entirely of algebraic entries in general, two requirements which have restricited previous constructions. We suspect the result found here is not optimal, and can be improved. Particularly because the number of gates in the relaxed seeds introduced here grows with $n$ and $t$. We conjecture that constant sized seeds such as those in (Brand\~ao, Harrow, and Horodecki, Commun. Math. Phys. 2016) are sufficient