Pauli Operators¶

Represent and manipulate Pauli operators. Evaluate Pauli expectation values using ideal statevector simulation. Refer to these Jupyter Notebook tutorials on Forge:

GettingStarted/Examples/Circuits/quasar_derivatives.ipynb
GettingStarted/Examples/Circuits/quasar_parameters.ipynb
GettingStarted/Examples/Circuits/quasar_pauli.ipynb

Classes

Pauli
PauliExpectation
PauliOperator
PauliString

Class Pauli¶

Math

conj()

dot()

norm2()

norminf()

zero()

zeros_like()

sieved()

Pauli Summary Attributes

max_order()
max_qubit()

min_qubit()
nqubit()

nqubit_sparse()
nterm()

qubits()
summary_str()

Hilbert Space Representation

from_matrix()

matrix_vector_product()

to_matrix()

OrderedDict Methods

get()

setdefault()

update()

Pauli Starter Objects

IXYZ()

Class PauliExpectation¶

zero()

zeros_like()

Class PauliOperator¶

char()

from_string()

qubit()

Class PauliString¶

chars()

from_string()

order()

qubits()

class quasar.pauli.Pauli(*args, **kwargs)¶

Class Pauli represents Pauli operators as an OrderedDict of PauliString : float/complex``pairs. Use ``dict access methods to add Pauli strings and access/modify the coefficients of existing strings.

static IXYZ()¶

Returns four PauliStarter objects corresponding to the I, X, Y, and Z Pauli operators respectively.

When PauliStarter objects are indexed by the [] operator, they generate a corresponding Pauli object on the given qubit index. For instance, after generating PauliStarter objects using I, X, Y, Z = quasar.Pauli.IXYZ(), X[1] generates an X Pauli operator object on qubit index 1. Since indices applied to I PauliStarter objects have no effect, it is convention to refer to the I operator using I[-`]. Use +, -, *, and /, as well as float and complex coefficients to build linear combinations of products of Pauli operators.

Returns: I, X, Y, Z PauliStarter objects, returned in that order
Return type: PauliStarter

property conj¶

Return a new version of the Pauli object with the values conjugated.

Type: Pauli

dot(other)¶

Calculate the dot product of coincident Pauli strings in self and another Pauli object.

Parameters: other (Pauli) – the other Pauli object in the dot product with self.
Returns: the dot product of coincident Pauli strings in self and other.
Return type: float

static from_matrix(matrix, qubit_indices=None, cutoff=1e-14)¶

Convert a computational-basis Hilbert space operator to an equivalent Pauli form.

Parameters

param_matrix (np.ndarray) – matrix representing a computational-basis Hilbert space operator.
qubit_indices (list of ints) – qubit indices to be occupied by Pauli operators. If default value None, all qubit indices are included in qubit_indices.
cutoff (float) – remove a Pauli string from the Pauli form if the absolute value of its coefficient is less than cutoff (default - 1.0E-14).

Returns

matrix expressed in an equivalent Pauli form.

Return type

Pauli

get(k[, d]) → D[k] if k in D, else d. d defaults to None.¶

matrix_vector_product(statevector, min_qubit=None, nqubit=None, dtype=<class 'numpy.complex128'>)¶

Evaluate how a Pauli object acts on a statevector in the computational-basis Hilbert space.

Parameters

statevector (np.ndarray of shape 2**N) – the statevector on which to evaluate how self acts in the computational-basis Hilbert space.
min_qubit (int) – the minimum occupied qubit index (default - None).
nqubit (int) – the total number of qubit indices in the circuit, including empty qubiit indices (default - None).
dtype (real or complex dtype) – the dtype to perform the computation at (default - None).

Returns

the result of self acting on statevector in the computational-basis Hilbert space.

Return type

np.ndarray, complex dtype

property max_order¶

The maximum number of one-qubit Pauli operators in the Pauli object.

Type: int

property max_qubit¶

The maximum occupied qubit index (or -1 if no occupied qubits)

Type: int

property min_qubit¶

The minimum occupied qubit index (or 0 if no occupied qubits)

Type: int

property norm2¶

Return the square root of the dot product of self with self.

Type: float

property norminf¶

Return the magnitude of the largest coefficient of self.

Type: float

property nqubit¶

The total number of qubit indices in the circuit (including empty qubit indices).

Type: int

property nqubit_sparse¶

The total number of occupied qubit indices in the circuit (excluding empty qubit indices).

Type: int

property nterm¶

The number of Pauli strings in the Pauli object.

Type: int

property qubits¶

The unique occupied qubit indices over all strings in the Pauli object.

Type: SortedSet

setdefault(key, default=None)¶

Return value for item identified by key in sorted dict.

If key is in the sorted dict then return its value. If key is not in the sorted dict then insert key with value default and return default.

Optional argument default defaults to none.

Runtime complexity: O(log(n)) – approximate.

>>> sd = SortedDict()
>>> sd.setdefault('a', 1)
1
>>> sd.setdefault('a', 10)
1
>>> sd
SortedDict({'a': 1})

Parameters

key – key for item
default – value for item (default None)

Returns

value for item identified by key

sieved(cutoff=1e-14)¶

Remove strings which have small or zero coefficients from the self.

Parameters: cutoff (float) – remove a Pauli string if the absolute value of its coefficient is less than cutoff (default - `1.0E-14).
Returns: a new Pauli object with strings removed from self if the absolute value of their coefficient is less than cutoff.
Return type: Pauli

property summary_str¶

A formatted summary string that displays the values of the nqubit, nterm, and max_order attributes of the Pauli object.

Type: str

to_matrix(min_qubit=None, nqubit=None, dtype=<class 'numpy.complex128'>)¶

Convert a Pauli object to the computational-basis Hilbert space.

Parameters

min_qubit (int) – the minimum occupied qubit index (default - None).
nqubit (int) – the total number of qubit indices in the circuit, including empty qubit indices (default - None).
dtype (complex dtype) – the dtype to perform the computation at (default - None).

Returns

self expressed in the computational-basis Hilbert space.

Return type

np.ndarray, complex dtype

update(*args, **kwargs)¶

Update sorted dict with items from args and kwargs.

Overwrites existing items.

Optional arguments args and kwargs may be a mapping, an iterable of pairs or keyword arguments. See SortedDict.__init__() for details.

Parameters

args – mapping or iterable of pairs
kwargs – keyword arguments mapping

static zero()¶

Returns: a new Pauli object initialized with no strings.
Return type: Pauli

static zeros_like(x)¶

Returns a new Pauli object initialized with the strings of another Pauli object, but with coefficient values initialized to zero.

Parameters: x (Pauli) – Pauli object whose strings will be used to initialize the new Pauli object.
Returns: a Pauli object initialized with the strings of x, but with coefficient values initialized to zero.
Return type: Pauli

class quasar.pauli.PauliExpectation(*args, **kwargs)¶

Class PauliExpectation represents the expectation value of a Pauli-sparse Hermitian operator.

static zero()¶: Returns: Pauli: a new Pauli object initialized with no strings.

static zeros_like(x)¶

Returns a new Pauli object initialized with the strings of another Pauli object, but with coefficient values initialized to zero.

Parameters: x (Pauli) – Pauli object whose strings will be used to initialize the new Pauli object.
Returns: a Pauli object initialized with the strings of x, but with coefficient values initialized to zero.
Return type: Pauli

class quasar.pauli.PauliOperator¶

Class PauliOperator represents a one-qubit Pauli operator with a given character (‘X’, ‘Y’, or ‘Z’) and qubit index.

property char¶

Return the character representing the type of Pauli operator (‘X’,’Y’, or ‘Z’).

Type: str

static from_string(string)¶

Parameters: string (str) – string to convert into a PauliOperator object
Returns: a PauliOperator object from a string composed of the character (‘X’,’Y’, or ‘Z’) representing the type of Pauli operator and the qubit index.
Return type: PauliOperator

property qubit¶

Return qubit index.

Type: int

class quasar.pauli.PauliString¶

Class PauliString represents a string of one or more PauliOperator objects as a tuple of PauliOperator objects.

property chars¶: The characters (‘X’, ‘Y’, or ‘Z’) corresponding to all of the PauliOperator objects in the PauliString.

static from_string(string)¶

Return a PauliString object from a string expressing a linear combination of products of one or more PauliOperator objects.

Parameters: string (str) – string to convert into a PauliOperator object
Returns: a PauliString object from a string expressing a linear combination of products of one or more PauliOperator objects.
Return type: PauliString

property order¶

The number of PauliOperator objects in the PauliString.

Type: int

property qubits¶

The qubit indices occupied by all of the PauliOperator objects in the PauliString.

Type: tuple of ints