# Qubit#

Qubits for quantum computing.

Todo: * Finish implementing measurement logic. This should include POVM. * Update docstrings. * Update tests.

class sympy.physics.quantum.qubit.IntQubit(*args, **kwargs)[source]#

A qubit ket that store integers as binary numbers in qubit values.

The differences between this class and `Qubit` are:

• The form of the constructor.

• The qubit values are printed as their corresponding integer, rather than the raw qubit values. The internal storage format of the qubit values in the same as `Qubit`.

Parameters:

values : int, tuple

If a single argument, the integer we want to represent in the qubit values. This integer will be represented using the fewest possible number of qubits. If a pair of integers and the second value is more than one, the first integer gives the integer to represent in binary form and the second integer gives the number of qubits to use. List of zeros and ones is also accepted to generate qubit by bit pattern.

nqubits : int

The integer that represents the number of qubits. This number should be passed with keyword `nqubits=N`. You can use this in order to avoid ambiguity of Qubit-style tuple of bits. Please see the example below for more details.

Examples

Create a qubit for the integer 5:

```>>> from sympy.physics.quantum.qubit import IntQubit
>>> from sympy.physics.quantum.qubit import Qubit
>>> q = IntQubit(5)
>>> q
|5>
```

We can also create an `IntQubit` by passing a `Qubit` instance.

```>>> q = IntQubit(Qubit('101'))
>>> q
|5>
>>> q.as_int()
5
>>> q.nqubits
3
>>> q.qubit_values
(1, 0, 1)
```

We can go back to the regular qubit form.

```>>> Qubit(q)
|101>
```

Please note that `IntQubit` also accepts a `Qubit`-style list of bits. So, the code below yields qubits 3, not a single bit `1`.

```>>> IntQubit(1, 1)
|3>
```

To avoid ambiguity, use `nqubits` parameter. Use of this keyword is recommended especially when you provide the values by variables.

```>>> IntQubit(1, nqubits=1)
|1>
>>> a = 1
>>> IntQubit(a, nqubits=1)
|1>
```
class sympy.physics.quantum.qubit.IntQubitBra(*args, **kwargs)[source]#

A qubit bra that store integers as binary numbers in qubit values.

class sympy.physics.quantum.qubit.Qubit(*args, **kwargs)[source]#

A multi-qubit ket in the computational (z) basis.

We use the normal convention that the least significant qubit is on the right, so `|00001>` has a 1 in the least significant qubit.

Parameters:

values : list, str

The qubit values as a list of ints ([0,0,0,1,1,]) or a string (‘011’).

Examples

Create a qubit in a couple of different ways and look at their attributes:

```>>> from sympy.physics.quantum.qubit import Qubit
>>> Qubit(0,0,0)
|000>
>>> q = Qubit('0101')
>>> q
|0101>
```
```>>> q.nqubits
4
>>> len(q)
4
>>> q.dimension
4
>>> q.qubit_values
(0, 1, 0, 1)
```

We can flip the value of an individual qubit:

```>>> q.flip(1)
|0111>
```

We can take the dagger of a Qubit to get a bra:

```>>> from sympy.physics.quantum.dagger import Dagger
>>> Dagger(q)
<0101|
>>> type(Dagger(q))
<class 'sympy.physics.quantum.qubit.QubitBra'>
```

Inner products work as expected:

```>>> ip = Dagger(q)*q
>>> ip
<0101|0101>
>>> ip.doit()
1
```
class sympy.physics.quantum.qubit.QubitBra(*args, **kwargs)[source]#

A multi-qubit bra in the computational (z) basis.

We use the normal convention that the least significant qubit is on the right, so `|00001>` has a 1 in the least significant qubit.

Parameters:

values : list, str

The qubit values as a list of ints ([0,0,0,1,1,]) or a string (‘011’).

`Qubit`

Examples using qubits

sympy.physics.quantum.qubit.matrix_to_density(mat)[source]#

Works by finding the eigenvectors and eigenvalues of the matrix. We know we can decompose rho by doing: sum(EigenVal*|Eigenvect><Eigenvect|)

sympy.physics.quantum.qubit.matrix_to_qubit(matrix)[source]#

Convert from the matrix repr. to a sum of Qubit objects.

Parameters:

matrix : Matrix, numpy.matrix, scipy.sparse

The matrix to build the Qubit representation of. This works with SymPy matrices, numpy matrices and scipy.sparse sparse matrices.

Examples

Represent a state and then go back to its qubit form:

```>>> from sympy.physics.quantum.qubit import matrix_to_qubit, Qubit
>>> from sympy.physics.quantum.represent import represent
>>> q = Qubit('01')
>>> matrix_to_qubit(represent(q))
|01>
```
sympy.physics.quantum.qubit.measure_all(qubit, format='sympy', normalize=True)[source]#

Perform an ensemble measurement of all qubits.

Parameters:

The qubit to measure. This can be any Qubit or a linear combination of them.

format : str

The format of the intermediate matrices to use. Possible values are (‘sympy’,’numpy’,’scipy.sparse’). Currently only ‘sympy’ is implemented.

Returns:

result : list

A list that consists of primitive states and their probabilities.

Examples

```>>> from sympy.physics.quantum.qubit import Qubit, measure_all
>>> from sympy.physics.quantum.gate import H
>>> from sympy.physics.quantum.qapply import qapply
```
```>>> c = H(0)*H(1)*Qubit('00')
>>> c
H(0)*H(1)*|00>
>>> q = qapply(c)
>>> measure_all(q)
[(|00>, 1/4), (|01>, 1/4), (|10>, 1/4), (|11>, 1/4)]
```
sympy.physics.quantum.qubit.measure_all_oneshot(qubit, format='sympy')[source]#

Perform a oneshot ensemble measurement on all qubits.

A oneshot measurement is equivalent to performing a measurement on a quantum system. This type of measurement does not return the probabilities like an ensemble measurement does, but rather returns one of the possible resulting states. The exact state that is returned is determined by picking a state randomly according to the ensemble probabilities.

Parameters:

qubits : Qubit

The qubit to measure. This can be any Qubit or a linear combination of them.

format : str

The format of the intermediate matrices to use. Possible values are (‘sympy’,’numpy’,’scipy.sparse’). Currently only ‘sympy’ is implemented.

Returns:

result : Qubit

The qubit that the system collapsed to upon measurement.

sympy.physics.quantum.qubit.measure_partial(qubit, bits, format='sympy', normalize=True)[source]#

Perform a partial ensemble measure on the specified qubits.

Parameters:

qubits : Qubit

The qubit to measure. This can be any Qubit or a linear combination of them.

bits : tuple

The qubits to measure.

format : str

The format of the intermediate matrices to use. Possible values are (‘sympy’,’numpy’,’scipy.sparse’). Currently only ‘sympy’ is implemented.

Returns:

result : list

A list that consists of primitive states and their probabilities.

Examples

```>>> from sympy.physics.quantum.qubit import Qubit, measure_partial
>>> from sympy.physics.quantum.gate import H
>>> from sympy.physics.quantum.qapply import qapply
```
```>>> c = H(0)*H(1)*Qubit('00')
>>> c
H(0)*H(1)*|00>
>>> q = qapply(c)
>>> measure_partial(q, (0,))
[(sqrt(2)*|00>/2 + sqrt(2)*|10>/2, 1/2), (sqrt(2)*|01>/2 + sqrt(2)*|11>/2, 1/2)]
```
sympy.physics.quantum.qubit.measure_partial_oneshot(qubit, bits, format='sympy')[source]#

Perform a partial oneshot measurement on the specified qubits.

A oneshot measurement is equivalent to performing a measurement on a quantum system. This type of measurement does not return the probabilities like an ensemble measurement does, but rather returns one of the possible resulting states. The exact state that is returned is determined by picking a state randomly according to the ensemble probabilities.

Parameters:

qubits : Qubit

The qubit to measure. This can be any Qubit or a linear combination of them.

bits : tuple

The qubits to measure.

format : str

The format of the intermediate matrices to use. Possible values are (‘sympy’,’numpy’,’scipy.sparse’). Currently only ‘sympy’ is implemented.

Returns:

result : Qubit

The qubit that the system collapsed to upon measurement.

sympy.physics.quantum.qubit.qubit_to_matrix(qubit, format='sympy')[source]#

Converts an Add/Mul of Qubit objects into it’s matrix representation

This function is the inverse of `matrix_to_qubit` and is a shorthand for `represent(qubit)`.