# Quantum Information Processing

## Introduction

The Quantum Information Processing (QIP) module aims at providing basic tools for quantum computing simulation both for simple quantum algorithm design and for experimental realization. It offers two different approaches, one with `QubitCircuit` calculating unitary evolution under quantum gates by matrix product, another called `Processor` using open system solvers in QuTiP to simulate noisy quantum device.

## Quantum Circuit

The most common model for quantum computing is the quantum circuit model. In QuTiP, we use `QubitCircuit` to represent a quantum circuit. The circuit is characterized by registers and gates:

- Registers: The argument `N` specifies the number of qubit registers in the circuit and the argument `num_cbits` (optional) specifies the number of classical bits available for measurement and control.

- Gates: Each quantum gate is saved as a class object `Gate` with information such as gate name, target qubits and arguments. Gates can also be controlled on a classical bit by specifying the register number with the argument `classical_controls`.

- Measurements: We can also carry out measurements on individual qubit (both in the middle and at the end of the circuit). Each measurement is saved as a class object `Measurement` with parameters such as targets, the target qubit on which the measurement will be carried out, and classical_store, the index of the classical register which stores the result of the measurement.

A circuit with the various gates and registers available is demonstrated below:

    from qutip.qip.circuit import QubitCircuit, Gate
    from qutip import tensor, basis

    qc = QubitCircuit(N=2, num_cbits=1)
    swap_gate = Gate(name="SWAP", targets=[0, 1])

    qc.add_gate(swap_gate)
    qc.add_measurement("M0", targets=[1], classical_store=0) # measurement gate
    qc.add_gate("CNOT", controls=0, targets=1)
    qc.add_gate("X", targets=0, classical_controls=[0]) # classically controlled gate
    qc.add_gate(swap_gate)

    print(qc.gates)

Output:

    [Gate(SWAP, targets=[0, 1], controls=None, classical controls=None, control_value=None),
     Measurement(M0, target=[1], classical_store=0),
     Gate(CNOT, targets=[1], controls=[0], classical controls=None, control_value=None),
     Gate(X, targets=[0], controls=None, classical controls=[0], control_value=None),
     Gate(SWAP, targets=[0, 1], controls=None, classical controls=None, control_value=None)]

## Unitaries

There are a few useful functions associated with the circuit object. For example, the `propagators` method returns a list of the unitaries associated with the sequence of gates in the circuit. By default, the unitaries are expanded to the full dimension of the circuit:

    U_list = qc.propagators()
    print(U_list)

Output:

    [Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
    Qobj data =
    [[1. 0. 0. 0.]
     [0. 0. 1. 0.]
     [0. 1. 0. 0.]
     [0. 0. 0. 1.]], Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
    Qobj data =
    [[1. 0. 0. 0.]
     [0. 1. 0. 0.]
     [0. 0. 0. 1.]
     [0. 0. 1. 0.]], Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
    Qobj data =
    [[0. 0. 1. 0.]
     [0. 0. 0. 1.]
     [1. 0. 0. 0.]
     [0. 1. 0. 0.]], Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
    Qobj data =
    [[1. 0. 0. 0.]
     [0. 0. 1. 0.]
     [0. 1. 0. 0.]
     [0. 0. 0. 1.]]]

Another option is to only return the unitaries in their original dimension. This can be achieved with the argument `expand=False` specified to the `propagators`.

    U_list = qc.propagators(expand=False)
    print(U_list)

Output:

    [Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
    Qobj data =
    [[1. 0. 0. 0.]
     [0. 0. 1. 0.]
     [0. 1. 0. 0.]
     [0. 0. 0. 1.]], Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
    Qobj data =
    [[1. 0. 0. 0.]
     [0. 1. 0. 0.]
     [0. 0. 0. 1.]
     [0. 0. 1. 0.]], Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
    Qobj data =
    [[0. 1.]
     [1. 0.]], Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
    Qobj data =
    [[1. 0. 0. 0.]
     [0. 0. 1. 0.]
     [0. 1. 0. 0.]
     [0. 0. 0. 1.]]]

## Gates

The pre-defined gates for the class `qutip.qip.Gate` are shown in the table below:

Gate name | Description
--- | ---
“RX” | Rotation around x axis
“RY” | Rotation around y axis
“RZ” | Rotation around z axis
“X” | Pauli-X gate
“Y” | Pauli-Y gate
“Z” | Pauli-Z gate
“S” | Single-qubit rotation or Z90
“T” | Square root of S gate
“SQRTNOT” | Square root of NOT gate
“SNOT” | Hardmard gate
“PHASEGATE” | Add a phase one the state 1
“CRX” | Controlled rotation around x axis
“CRY” | Controlled rotation around y axis
“CRZ” | Controlled rotation around z axis
“CX” | Controlled X gate
“CY” | Controlled Y gate
“CZ” | Controlled Z gate
“CS” | Controlled S gate
“CT” | Controlled T gate
“CPHASE” | Controlled phase gate
“CNOT” | Controlled NOT gate
“CSIGN” | Same as CPHASE
“QASMU” | U rotation gate used as a primitive in the QASM standard
“BERKELEY” | Berkeley gate
“SWAPalpha” | SWAPalpha gate
“SWAP” | Swap the states of two qubits
“ISWAP” | Swap gate with additional phase for 01 and 10 states
“SQRTSWAP” | Square root of the SWAP gate
“SQRTISWAP” | Square root of the ISWAP gate
“FREDKIN” | Fredkin gate
“TOFFOLI” | Toffoli gate
“GLOBALPHASE” | Global phase

For some of the gates listed above, `QubitCircuit` also has a primitive `resolve_gates` method that decomposes them into elementary gate sets such as CNOT or SWAP with single-qubit gates (RX, RY and RZ). However, this method is not fully optimized. It is very likely that the depth of the circuit can be further reduced by merging quantum gates. It is required that the gate resolution be carried out before the measurements to the circuit are added.

Custom Gates

In addition to these pre-defined gates, QuTiP also allows the user to define their own gate. The following example shows how to define a customized gate. The key step is to define a gate function returning a `qutip.Qobj` and save it in the attribute `user_gates`.

    from qutip.qip.circuit import Gate
    from qutip.qip.operations import rx

    def user_gate1(arg_value):
         # controlled rotation X
         mat = np.zeros((4, 4), dtype=np.complex)
         mat[0, 0] = mat[1, 1] = 1.
         mat[2:4, 2:4] = rx(arg_value)
         return Qobj(mat, dims=[[2, 2], [2, 2]])

    def user_gate2():
         # S gate
         mat = np.array([[1.,   0],
                         [0., 1.j]])
         return Qobj(mat, dims=[[2], [2]])

    qc = QubitCircuit(2)
    qc.user_gates = {"CTRLRX": user_gate1,
                     "S"     : user_gate2}

    # qubit 0 controls qubit 1
    qc.add_gate("CTRLRX", targets=[0,1], arg_value=np.pi/2)

    # qubit 1 controls qubit 0
    qc.add_gate("CTRLRX", targets=[1,0], arg_value=np.pi/2)

    # we also add a gate using a predefined Gate object
    g_T = Gate("S", targets=[1])
    qc.add_gate(g_T)
    props = qc.propagators()

    print(props[0])

Output:

    Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False
    Qobj data =
    [[1.        +0.j         0.        +0.j         0.        +0.j
      0.        +0.j        ]
     [0.        +0.j         1.        +0.j         0.        +0.j
      0.        +0.j        ]
     [0.        +0.j         0.        +0.j         0.70710678+0.j
      0.        -0.70710678j]
     [0.        +0.j         0.        +0.j         0.        -0.70710678j
      0.70710678+0.j        ]]

    print(props[1])

Output:

    Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False
    Qobj data =
    [[1.        +0.j         0.        +0.j         0.        +0.j
      0.        +0.j        ]
     [0.        +0.j         0.70710678+0.j         0.        +0.j
      0.        -0.70710678j]
     [0.        +0.j         0.        +0.j         1.        +0.j
      0.        +0.j        ]
     [0.        +0.j         0.        -0.70710678j 0.        +0.j
      0.70710678+0.j        ]]

    print(props[2])

Output:

    Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False
    Qobj data =
    [[1.+0.j 0.+0.j 0.+0.j 0.+0.j]
     [0.+0.j 0.+1.j 0.+0.j 0.+0.j]
     [0.+0.j 0.+0.j 1.+0.j 0.+0.j]
     [0.+0.j 0.+0.j 0.+0.j 0.+1.j]]

## Plotting a Quantum Circuit

A quantum circuit (described above) can directly be plotted using the QCircuit library (https://github.com/CQuIC/qcircuit). QCiruit is a quantum circuit drawing application and is implemented directly into QuTiP.

The circuit image visualization requires LaTeX and ImageMagick for display. The module automatically generates the LaTeX code for plotting the circuit, produces the pdf and converts it to the png format. On Mac and Linux, ImageMagick can be easily installed with the command conda install imagemagick if you have conda installed. Otherwise, please follow the installation instructions on the ImageMagick documentation.

On windows, you need to download and install ImageMagick installer. In addition, you also need perl (for `pdfcrop`) and Ghostscript (additional dependency of ImageMagick for png conversion).

If you want to check whether all dependencies are installed, see if the following three commands work correctly: `pdflatex`, `pdfcrop` and `magick anypdf.pdf anypdf.png`, where `anypdf.pdf` is any pdf file you have.

An example code for plotting the example quantum circuit from above is given:

    from qutip.qip.circuit import QubitCircuit, Gate
    # create the quantum circuit
    qc = QubitCircuit(2, num_cbits=1)
    qc.add_gate("CNOT", controls=0, targets=1)
    qc.add_gate("H", targets=1)
    qc.add_gate("ISWAP", targets=[0,1])
    qc.add_measurement("M0", targets=1, classical_store=0)
    # plot the quantum circuit
    qc.png

## Circuit simulation

There are two different ways to simulate the action of quantum circuits using QuTiP:

- The first method utilizes unitary application through matrix products on the input states. This method simulates circuits exactly in a deterministic manner. This is achieved through `CircuitSimulator`. A short guide to exact simulation can be found at Operator-level circuit simulation. The teleportation notebook is also useful as an example.

- A different method of circuit simulation employs driving Hamiltonians with the ability to simulate circuits in the presence of noise. This can be achieved through the various classes in `qutip.qip.device`. A short guide to processors for QIP simulation can be found at Pulse-level circuit simulation.