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‎README.md‎

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--help Show this message and exit.
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```
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## Development
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# Discovery for Diverse Experimental Resources
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Our package allows for the discovery of quantum experiments for a range of experimental goals, constraints and
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resources. Experiments that can be produced include:
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* state creation (heralded or post-selected)
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* quantum gates (heralded or post-selected)
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* measurements of quantum states
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* entanglement swapping
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* (covered elsewhere: mixed state creation)
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Sources for photons in these experiments can be SPDC sources, deterministic single-photon sources or a mix of the two.
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Detectors can be photon-number-resolving or not.
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Each of these experiments can be described with a graph. The interpretation of nodes and edges varies with the kind of
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experiment.
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### Rules for Loss Functions
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With these varying interpretations (e.g. for single photon sources, input photons, entanglement swapping), different
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constraints apply on what kind of graph can correspond to an experiment (Topological Rules).
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With the different ways of performing the experiments (heralded/post-selected & number-resolving/non-number-resolving),
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different events are selected out of all possibilities (post-selection rules).
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#### Topological Rules
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All experiments that our package is applied to can be described by a graph. When describing state creation using SPDC
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each edge can be interpreted as a pair-creation. In this case all edges of the complete graph can be considered
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physically legitimate. When describing other experiments edges can be interpreted differently. Not every edge will be
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physically meaningful. Consequentially there are constraints on which connections of the complete graph are used in the
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optimization.
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*(A) Single Photon Sources and Input Photons*
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Deterministic single photon sources and input photons (such as in gates) are described as (input) vertices in a graph.
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An edge connecting an input vertex to a detector describes a path in which a photon can travel from the input into the
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detector. This interpretation stems from the [Klyshko picture](https://arxiv.org/pdf/1805.06484.pdf). From this a
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constraint on the graph follows. Two input vertices can not be connected by an edge. It could not be interpreted
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physically.
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*(B) Entanglement Swapping and Teleportation*
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In entanglement swapping, photons are entangled that have not interacted before. If we want to design an entanglement
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swapping experiment of two photons, the target is to discover a graph that produces an entangled state between the two
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photons. However any edge between the corresponding vertices would translate into a common source crystal. A constraint
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that ensures legitimate entanglement swapping is to remove any edge between the two parties
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#### Post-Selection Rules
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The rules for post-selecting coincidence events have been described in the
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[Theseus paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.031044). Here, post-selection projects the
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space of possibilities containing arbitrary combinations of crystals firing into the space of possibilities where only
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crystals fire for which all detectors at the end of the experiment click. In the graph picture these combinations
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correspond to the perfect matchings. A state is produced with fidelity one in post-selection if all possibilities of
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coincidence events contribute to that state.
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Other experimental settings (such as heralding) and additional experimental resources (such as number-resolving
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detectors)
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perform a different kind of projection on the space of possibilities by selecting for different events. This different
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selection is reflected in the fidelity of the state. The products of the edge weights belonging to each possibility
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contribute to the norm of the fidelity.
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*(A) Heralding*
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Heralding is a less strict form of selecting events. Instead of putting a detector in every path and selecting for
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coincidence, only a subset of the paths are detected _heralding_ an output state in the unmeasured paths. This selection
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rule not only allows for possibilities where one photon is in every path (perfect matchings) but also for other
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possibilities (edge covers) as long as they cover the heralding detectors. This can lead to cross-terms that are not
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present when post-selecting for coincidence in all paths. Consequentially it is more difficult to find a graph with
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fidelity one, also requiring more experimental resources.
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*(B) Single Photon Sources and Input Photons*
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When describing heralded experiments (above) one has to consider edge covers instead of perfect matchings in the graph
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for possible events. These possibilities include one edge being included twice in an edge cover, corresponding to a
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crystal firing twice in an experiment. For single photon sources and other deterministic input photons such
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possibilities do not exist. Only edge covers that cover the input vertices exactly once are considered for the norm of
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the fidelity.
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*(C) Photon Number-Resolving Detectors*
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Photon number-resolving detectors are a valuable resource that can restrict the space of possibilities more than a
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regular detector. When one can be certain that exactly one photon, and not two, has entered a detector it reduces the
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number of events that could have led to this outcome, eliminating cross terms.
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*(D) States in Fock Basis*
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...
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## Loss Functions For Target State Optimization
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As explained above, the loss function depend largely on the different experimental conditions. Independent of these
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conditions they fall into two categories.
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* Fidelity
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* Count Rate
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A Fidelity of one ensures that an experiment has no unwanted cross terms. Every possibility that is selected for
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contributes directly to the target outcome.
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However, we have come to find that optimizing exclusively for fidelity in some cases can lead the optimization to scale
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down the weights of the entire graph to minimize the contributions of crossterms. While the fidelity will be very close
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to one in those cases the generally low edge weights would lead to very low count rates of successful events in actual
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experiments.
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To find solutions with higher weights we have introduced the _simplified count rate_ as a loss function.
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# Config Examples
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This section gives examples for config files showcasing the features for different kinds of experiments that can
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searched for.
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## Target State Optimization
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This is used when the in- and out-going states can be clearly defined by a state functions. For those, the loss
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functions `cr` and `fid` are used.
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Below are some examples to give an idea of the scope.
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For further details, refer to the definition of the function `setup_for_target` in `main.py`.
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### Post-selected State creation
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Here is an example for a config file optimizing for a graph that creates a three particle four-dimensional GHZ state.
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```json
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{
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"description": "Finding a setup for the creation of the three-particle four-dimensional GHZ state. It can be realized with three ancillary particles",
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"foldername": "ghz_346",
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"target_state": [
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"000",
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"111",
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"222",
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"333"
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],
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"num_anc": 3,
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"loss_func": "cr",
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"thresholds": [
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0.25,
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0.1
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],
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"samples": 10,
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"optimizer": "L-BFGS-B",
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"ftol": 1e-06,
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"edges_tried": 20,
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"tries_per_edge": 5
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}
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```
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General info is given by `description`. With `foldername` one can give a custom name for the subfolder where solutions
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are saved.
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In the simple case of post-selected state creation with SPDC crystals, `target_state`, `num_anc` and `loss_func` are all
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that are needed to define the objective of the optimization.
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`thresholds` is necessary to decide whether a topological optimization step has been successful. In this case,
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when `1-countrate < 0.25` and `1-fidelity < 0.1`.
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### Post-Selected Quantum Gates
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For an example how topological constraints on the starting graph of the optimization are dealt with, consider the
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example of a post-selected CNOT(2,3) quantum gate.
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```json
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{
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"description": "Postselected CNOT between a qubit (control) and a qutrit (target). Two ancillary photons from SPDC.",
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"foldername": "cnot_23",
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"target_state": [
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"0000",
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"0101",
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"0202",
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"1011",
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"1112",
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"1210"
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],
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"in_nodes": [
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0,
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1
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],
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"out_nodes": [
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2,
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3
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],
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"num_anc": 2,
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"loss_func": "cr",
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"thresholds": [
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0.3,
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0.1
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],
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"samples": 10,
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"optimizer": "L-BFGS-B",
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"ftol": 1e-06,
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"edges_tried": 30,
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"tries_per_edge": 5
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}
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```
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Here, `target_state` defines a logic table. It is defined through `in_nodes` and `out_nodes`, which entries belong to
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incoming photons and which belong to outgoing photons. These definitions suffice to automatically put constraints on the
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starting graph.
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### Heralded Quantum Gates with Single Photon Sources
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Two additional features are heralding and single photon sources. A simple example is a CNOT gate between two qubits
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heralding on two ancillary detectors.
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```json
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{
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"description": "Heralded CNOT gate between two qubits with two single photon sources. Similar has been done: https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.126.140501 could this have a better success probability?",
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"foldername": "cnot22sp",
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"target_state": [
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"0000",
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"0101",
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"1011",
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"1110"
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],
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"num_anc": 2,
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"in_nodes": [
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0,
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1
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],
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"out_nodes": [
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2,
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3
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],
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"single_emitters": [
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4,
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5
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],
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"heralding_out": true,
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"loss_func": "cr",
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"thresholds": [
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1,
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0.1
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],
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"samples": 10,
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"optimizer": "L-BFGS-B",
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"ftol": 1e-06,
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"edges_tried": 30,
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"tries_per_edge": 5
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}
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```
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Here, the two single photon sources given by `single_emitters` introduce further topological constraints on the starting
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graph. As they have the role of ancillary photons in this case, we need to set `num_anc` accordingly. The graph
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corresponding to this optimization will have a total of eight nodes. Two for incoming, two for outgoing, two for single
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photon sources and two for ancillary detectors.
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If `num_anc` is larger than the sum of lengths of `in_nodes` and `single_emitters`, the necessary amount of particles is
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created through SPDC.
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Additionaly `heralding_out` is set to `true` here. The photons corresponding to `out_nodes` will not be detected.
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### Measurement
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```json
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{
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"description": "Measurement for three particle W state",
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"foldername": "W_measurement",
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"target_state": [
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"001",
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"010",
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"100"
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],
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"in_nodes": [
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0,
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1,
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2
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],
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"num_anc": 0,
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"loss_func": "cr",
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"samples": 10,
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"optimizer": "L-BFGS-B",
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"ftol": 1e-06,
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"thresholds": [
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0.3,
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0.1
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],
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"tries_per_edge": 5,
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"edges_tried": 30
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}
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```
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## Entanglement Optimization
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When `loss_func` is set to `"ent"`, no target state is set. Instead the optimizer maximizes the entanglement that can be
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achieved by a graph with the local dimensions given by `dim`.
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```json
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{
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"description": "Maximizing entanglement in k=2 bi-partitions for four qubits.",
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"K": 2,
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"dim": 2222,
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"ftol": 1e-07,
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"loss_func": "ent",
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"min_edge": 4,
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"num_pre": 5,
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"optimizer": "SLSQP",
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"imaginary": false,
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"samples": 10,
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"thresholds": [
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0.000001
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],
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"tries_per_edge": 3,
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"var_factor": 0
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}
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```
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## Optimizing for Arbitrary Functions of the Graph
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There is also the option to define an arbitrary loss function, which should be defined or imported in `lossfunctions.py`
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.
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Here is an example for optimizing the assembly index of the graph.
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```json
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{
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"foldername": "assembly",
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"loss_func": "lff",
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"lff_name": "top_n_assembly",
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"dimensions": [
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2,
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2,
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2,
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2
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],
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"num_vertices": 4,
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"num_cols": 2,
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"size_of_graph": 8,
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"optimizer": "L-BFGS-B",
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"ftol": 1e-06,
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"samples": 1,
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"thresholds": [
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99999
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],
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"topopt": false,
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"edges_tried": 30,
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"tries_per_edge": 5,
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"unicolor": false,
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"imaginary": false,
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"num_pre": 1,
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"save_history": true
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}
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```
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To use a custom defined loss function `loss_func` should be set to `"lff"` (loss from function).
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The name of the loss function is given as a string to `lff_name`. This function should be defined or imported
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in `lossfunctions.py`. It should take a `Graph` object and a `cnfg` dictionary as arguments and return a real number.
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# Development
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