Editing Pseudo-Secret Random Qubit Generator (PSQRG) (section)

== Outline ==
The general idea is that a classical Client gives instructions to a quantum Server to perform certain actions (quantum computation). Those actions lead to the Server having as output a single qubit, which is randomly chosen from within a set of chosen (by the Client) states. On the other hand, Client is supposed to know the classical description of Server's output qubit. To achieve this task, the instructions/quantum computation the Client uses are based on a family of trapdoor, two regular, one-way functions with certain extra properties (see [[Pseudo-Secret Random Qubit Generator (PSQRG)#Properties|Properties]] and [[Pseudo-Secret Random Qubit Generator (PSQRG)#Definitions|Definitions]]). Trapdoor one-way functions are hard to invert (e.g. for the Server) unless someone (the Client in this case) has some extra “trapdoor” information. Two-regular functions have two pre-images for every value in the range of the function. This extra information helps the Client classically reproduce the quantum computation to recover the classical description of the single qubit state, while it is still hard to classically reproduce for the Server, the same information as Client. Simple modifications to the protocol could achieve other similar sets of states.<br/><br/>
The protocol can be divided into two stages, Pre-images Superposition, where Client instructs the Server to generate superposition using the function with above properties and, Squeezing, where the Server is instructed by the Client to measure his output qubits and deliver outcomes, which she (Client) would use to classically compute the value of r.
*'''Preparation.''' Client randomly selects a function with required properties, which is public (Server knows), but the trapdoor information needed to invert the function is known only to the Client.
*'''Preimages Superposition.''' Server prepares two quantum registers (system comprising multiple qubits), first being control (containing inputs) and second being target (containing output of the function). Client instructs Server to create a [[Glossary#Superposition|superposition]] of input states by applying [[Glossary#Unitary Operations|Hadamard gate]] (quantum fourier transform) on control register. She then instructs Server to apply a [[Glossary#Unitary Operations|unitary gate]] (all quantum gates are represented by unitary matrices) which computes output of the function in the target register, taking input from the control register, thus yielding an entangled state from the Server's superposition state. Server is required to measure the target register in the computational basis (along Z axis) and get an outcome. This action would reduce the control register into a superposition of two pre-images corresponding to the measurement outcome of the target register. He conveys this outcome to the Client who computes, classically, the two pre-images using her trapdoor. This pair of pre-image would have some isolated similar qubits (without superposition) and a superposition of dissimilar qubits. The dissimilar qubits can be written as a superposition of isolated 0s and isolated 1s (a GHZ state), with [[Glossary#Unitary Operations|X (NOT) gates]] applied to qubits depending on the state of qubit in both the pre-images. If the last qubit belongs to the set of similar qubits, then Client aborts and this Stage is repeated.
*'''Squeezing.''' Client instructs Server to measure all the qubits of the control register in some basis chosen randomly by the Client, except the last one, and return to her the outcomes. The last unmeasured state contains the randomly prepared qubit hidden from the Server. Client can then compute the value of r by an equation (see [[Pseudo-Secret Random Qubit Generator (PSQRG)#Pseudo Code|Pseudo Code]]). This equation depends only on Client’s measurement basis angles, Server’s measurement outcome and the location of random X’s (unknown to the Server). Thus, the Client knows the state of her secret qubit prepared by the Server.