# Basics

Perceval allows to simulate a wide range of optical devices and circuits.

## Basic component Circuits

Perceval allows to build photonic quantum circuits. A circuit in Perceval is an arrangement of devices – such as beam splitters, waveplates and so on – or sub-circuits, with a fixed number $$m$$ of spatial modes (which can be seen as “tracks” for photons).

These photonic quantum circuits are capable of performing a wide range of quantum computations. For example, the well-known Boson Sampling proposal can be implemented by passing n single photons through a passive linear optical circuit designed to implement a so-called Haar random m-mode unitary transformation, then measuring the m output modes of this linear optical circuit. On the other hand, path encoding each single photon by using two spatial modes per each photon essentially defines a qubit, the basic unit of quantum information processing. Thus, using i photons and 2i modes, we can define an i-qubit input quantum state. Passing this input state through the m-mode linear optical circuit, and using the remaining n-i single photons, and m-2i remaining modes as ancillary modes and photons (to perform appropriate post-selections), one can implement any i-qubit unitary transformation on the input state, thereby allowing universal quantum computation. What we have just described is essentially the famous Knill-Laflamme-Milburn protocol for universal qubit quantum computation. Alternatively, qubits can also be defined by performing a polarization encoding of the photons. Perceval supports both types of qubit encodings.

How to construct and simulate circuits is explained in more details in section Circuits.

## Sources

While Perceval allows to simulate optical circuits alone by providing an explicit input state – i.e. specifying how many photons are sent in each spatial mode – it also allows to design single photon sources to generate these photons.

Single photon sources are defined by specifying parameters such as brightness, purity or indistinguishability.

Once a source has been defined, it can be plugged into a circuit, and used to simulate the whole setup.

## Encoding

An optical circuit can be used to implement a quantum circuit. Perceval does not impose any specific encoding for the qu-bits. Many possible encodings exist, here we present two commonly used encodings: Spatial Modes encoding and Polarization Modes encoding. It is also however possible to conceive other types of encodings.

### Spatial Modes encoding

In Spatial Modes encoding (also called Path encoding), each qubit of a quantum circuit with a pair of spatial modes. Each state of the qubit corresponds to a Fock state where one photon is in one of the spatial modes:

• qubit state $$|0\rangle$$ corresponds to having one photon in mode 0, and no photon in mode 1. Hence the corresponding Fock state is: $$|1,0\rangle$$,

• similarly, qubit state $$|1\rangle$$ corresponds to having no photon in mode 0, and one photon in mode 1; the corresponding Fock state is: $$|0,1\rangle$$.

Note

In spatial encoding, some Fock states don’t correspond to any qubit state. An example of such a Fock state is $$|2,0\rangle$$ where two photons are sent in path 0 and no photon in path 1.

More generally, any state which isn’t a superposition of Fock states $$|0,1\rangle$$ and $$|1,0\rangle$$ can’t be associated with a qubit state.

### Polarization Modes encoding

In Polarization Modes encoding, each qubit is associated with one spatial mode containing only one photon. A state of the qubit is encoded using the polarization of the photon.

After choosing a basis for photon polarization, each state in the qubit basis $$\{|0\rangle, |1\rangle\}$$ is associated with a state in the polarization basis. For example, with polarization basis $$\{|{P:H}\rangle, |{P:V}\rangle\}$$:

• qubit state $$|0\rangle$$ corresponds to Fock state $$|{P:H}\rangle$$ – which denotes the state where one photon is present in spatial mode 0 with horizontal polarization

• qubit state $$|1\rangle$$ corresponds to Fock state $$|{P:V}\rangle$$ – which denotes the state where one photon is present in spatial mode 0 with vertical polarization

Note

In polarization encoding – as in path encoding – some Fock states can’t be associated with a qubit state. In polarization encoding this is the case when either no photon or more than one photons are present in the spatial mode, or if the polarization is unspecified (see Polarization)

## Simulation and Sampling

Consider a quantum circuit C which, when measured, outputs bitstrings x with a probability p_x. A classical algorithm providing a strong simulation of C is an algorithm which can approximate, up to very good precision (or more precisely relative error), p_x for all x. On the other hand, a classical algorithm that weakly simulates a quantum circuit is only required to sample outputs x with probabilities close to p_x (or in more precise terms up to a small error in the total variational distance ). For generic quantum circuits C, both these tasks become quickly unfeasible as the size of these circuits increases. The optimized classical algorithms for simulating linear optical quantum circuits which are at the heart of Perceval allow us to perform both weak and strong simulations of photonic quantum circuits, with sizes comparable to those of circuit currently being implemented on quantum hardware. Thus, Perceval is a powerful tool for both experimentalists and theorists wishing to explore the capabilities of current and near-term photonic quantum hardware. Perceval allows for strong and weak simulation of tasks such as Boson Sampling, Quantum machine learning, variational quantum algorithms, as well as small instances of Shors algorithm and Grover’s search !

## Numeric and Symbolic computation

One of the key feature built-in most of the Perceval module is the ability to produce numeric and symbolic computation. Symbolic computation use the excellent sympy library and enable, when working on smaller dimension problem, to get analytical solution of a problem. Selection of the feature is enabled with the use_symbolic boolean parameter available on numerous object methods.