Quantum operators

PennyLane supports a wide variety of quantum operators—such as gates, noisy channels, state preparations and measurements. These operators can be used in quantum functions, like shown in the following example:

import pennylane as qp

def my_quantum_function(x, y):
    qp.RZ(x, wires=0)
    qp.CNOT(wires=[0,1])
    qp.RY(y, wires=1)
    qp.AmplitudeDamping(0.1, wires=0)
    return qp.expval(qp.PauliZ(1))

This quantum function uses the RZ, CNOT, RY gates, the AmplitudeDamping noisy channel as well as the PauliZ observable.

Functions applied to operators extract information (such as the matrix representation) or transform operators (like turning a gate into a controlled gate).

PennyLane supports the following operators and operator functions:

Operator functions

Various functions and transforms are available for manipulating operators, and extracting information. These can be broken down into two main categories:

Operator to Operator functions

adjoint(fn[, lazy])	Create the adjoint of an Operator or a function that applies the adjoint of the provided function.
ctrl(op, control[, control_values, ...])	Create a method that applies a controlled version of the provided op.
cond(condition[, true_fn, false_fn, elifs])	Quantum-compatible if-else conditionals --- condition quantum operations on parameters such as the results of mid-circuit qubit measurements.
exp(op[, coeff, id])	Take the exponential of an Operator times a coefficient.
sum(*summands[, grouping_type, method, id, lazy])	Construct an operator which is the sum of the given operators.
pow(base[, z, lazy, id])	Raise an Operator to a power.
prod(*ops[, id, lazy])	Construct an operator which represents the generalized product of the operators provided.
s_prod(scalar, operator[, lazy, id])	Construct an operator which is the scalar product of the given scalar and operator provided.
generator(op[, format])	Returns the generator of an operation.
map_wires(input, wire_map[, queue, replace])	Changes the wires of an operator, tape, qnode or quantum function according to the given wire map.
dot(coeffs, ops[, pauli, grouping_type, method])	Returns the dot product between the coeffs vector and the ops list of operators.
evolve()	This method is dispatched and its functionality depends on the type of the input op.
simplify(input)	Simplifies an operator, tape, qnode or quantum function by reducing its arithmetic depth or number of rotation parameters.

These operator functions act on operators to produce new operators.

>>> op = qp.prod(qp.PauliX(0), qp.PauliZ(1))
>>> op = qp.sum(qp.Hadamard(0), op)
>>> op = qp.s_prod(1.2, op)
>>> op
1.2 * (H(0) + X(0) @ Z(1))

Operator to Other functions

matrix(op[, wire_order])	The dense matrix representation of an operation or quantum circuit.
eigvals(op[, k, which])	The eigenvalues of one or more operations.
is_commuting(operation1, operation2)	Check if two operations are commuting using a lookup table.
is_hermitian(op)	Check if the operation is hermitian.
is_unitary(op)	Check if the operation is unitary.
iterative_qpe(base, aux_wire, iters)	Performs the iterative quantum phase estimation circuit.

These operator functions act on operators and return other data types. All operator functions can be used on instantiated operators.

>>> op = qp.RX(0.54, wires=0)
>>> qp.matrix(op)
array([[0.9637709+0.j        , 0.       -0.26673144j],
       [0.       -0.26673144j, 0.9637709+0.j        ]])

Some operator functions can also be used in a functional form:

>>> x = torch.tensor(0.6, requires_grad=True)
>>> matrix_fn = qp.matrix(qp.RX)
>>> matrix_fn(x, wires=0)
tensor([[0.9553+0.0000j, 0.0000-0.2955j],
        [0.0000-0.2955j, 0.9553+0.0000j]], grad_fn=<StackBackward0>)

In the functional form, they are usually differentiable with respect to gate arguments:

>>> loss = torch.real(torch.trace(matrix_fn(x, wires=0)))
>>> loss.backward()
>>> x.grad
tensor(-0.2955)

Some operator transforms can also act on multiple operators, by passing quantum functions, QNodes or tapes:

>>> def circuit(theta):
...     qp.RX(theta, wires=1)
...     qp.Z(wires=0)
>>> qp.matrix(circuit, wire_order=[0,1])(np.pi / 4)
array([[ 0.92387953+0.j        ,  0.        -0.38268343j,
         0.        +0.j        ,  0.        +0.j        ],
       [ 0.        -0.38268343j,  0.92387953+0.j        ,
         0.        +0.j        ,  0.        +0.j        ],
       [ 0.        +0.j        ,  0.        +0.j        ,
        -0.92387953+0.j        ,  0.        +0.38268343j],
       [ 0.        +0.j        ,  0.        +0.j        ,
         0.        +0.38268343j, -0.92387953+0.j        ]])

Matrix to Operator functions

pauli_decompose(H[, hide_identity, ...])	Decomposes a Hermitian matrix into a linear combination of Pauli operators.
qsvt(A, poly, encoding_wires[, ...])	Implements the Quantum Singular Value Transformation (QSVT) for a matrix or Hamiltonian A, using a polynomial defined by poly and a block encoding specified by block_encoding.

These functions take a matrix and return an associated native PennyLane operator. For example:

>>> mat = np.array([[1, 1], [1, -1]])
>>> h = qp.pauli_decompose(mat)
>>> type(h)
<class 'pennylane.ops.op_math.linear_combination.LinearCombination'>
>>> print(h)
1.0 * X(0) + 1.0 * Z(0)

Qubit operators

Non-parametrized gates

Identity	The Identity operator
Hadamard	The Hadamard operator
PauliX	The Pauli X operator
PauliY	The Pauli Y operator
PauliZ	The Pauli Z operator
S	The single-qubit phase gate
T	The single-qubit T gate
SX	The single-qubit Square-Root X operator.
CNOT	The controlled-NOT operator
CZ	The controlled-Z operator
CY	The controlled-Y operator
CH	The controlled-Hadamard operator
SWAP	The swap operator
ISWAP	The i-swap operator
ECR	An echoed RZX(\(\pi/2\)) gate.
SISWAP	The square root of i-swap operator.
SQISW	alias of SISWAP
CSWAP	The controlled-swap operator
Toffoli	Toffoli (controlled-controlled-X) gate.
MultiControlledX	Apply a PauliX gate controlled on an arbitrary computational basis state.
Barrier	The Barrier operator, used to separate the compilation process into blocks or as a visual tool.
WireCut	The wire cut operation, used to manually mark locations for wire cuts.

Parametrized gates

Rot	Arbitrary single qubit rotation
RX	The single qubit X rotation
RY	The single qubit Y rotation
RZ	The single qubit Z rotation
MultiRZ	Arbitrary multi Z rotation.
PauliRot	Arbitrary Pauli word rotation.
PhaseShift	Arbitrary single qubit local phase shift
ControlledPhaseShift	A qubit controlled phase shift.
CPhase	alias of ControlledPhaseShift
PCPhase	A projector-controlled phase gate.
CPhaseShift00	A qubit controlled phase shift.
CPhaseShift01	A qubit controlled phase shift.
CPhaseShift10	A qubit controlled phase shift.
CRX	The controlled-RX operator
CRY	The controlled-RY operator
CRZ	The controlled-RZ operator
CRot	The controlled-Rot operator
U1	U1 gate.
U2	U2 gate.
U3	Arbitrary single qubit unitary.
IsingXX	Ising XX coupling gate
IsingXY	Ising (XX + YY) coupling gate
IsingYY	Ising YY coupling gate
IsingZZ	Ising ZZ coupling gate
PSWAP	Phase SWAP gate
GlobalPhase	A global phase operation that multiplies all components of the state by \(e^{-i \phi}\).

Quantum chemistry gates

SingleExcitation	Single excitation rotation.
SingleExcitationPlus	Single excitation rotation with positive phase-shift outside the rotation subspace.
SingleExcitationMinus	Single excitation rotation with negative phase-shift outside the rotation subspace.
DoubleExcitation	Double excitation rotation.
DoubleExcitationPlus	Double excitation rotation with positive phase-shift outside the rotation subspace.
DoubleExcitationMinus	Double excitation rotation with negative phase-shift outside the rotation subspace.
OrbitalRotation	Spin-adapted spatial orbital rotation.
FermionicSWAP	Fermionic SWAP rotation.

Electronic Hamiltonians built independently using OpenFermion tools can be readily converted to a PennyLane observable using the import_operator() function.

Gates constructed from a matrix

QubitUnitary	Apply an arbitrary unitary matrix with a dimension that is a power of two.
ControlledQubitUnitary	Apply an arbitrary fixed unitary matrix U to wires.
DiagonalQubitUnitary	Apply an arbitrary diagonal unitary matrix with a dimension that is a power of two.
SpecialUnitary	Gate from the group \(SU(N)\) with \(N=2^n\) for \(n\) qubits.
BlockEncode	Construct a unitary \(U(A)\) such that an arbitrary matrix \(A\) is encoded in the top-left block.

Gates performing arithmetics

QubitCarry	Apply the QubitCarry operation to four input wires.
QubitSum	Apply a QubitSum operation on three input wires.
IntegerComparator	Apply a controlled Pauli X gate using integer comparison as the condition.

State preparation

BasisState	Prepares a single computational basis state.
StatePrep	Prepare subsystems using a state vector in the computational basis.
QubitDensityMatrix	Prepare subsystems using the given density matrix.

Noisy channels

AmplitudeDamping	Single-qubit amplitude damping error channel.
GeneralizedAmplitudeDamping	Single-qubit generalized amplitude damping error channel.
PhaseDamping	Single-qubit phase damping error channel.
DepolarizingChannel	Single-qubit symmetrically depolarizing error channel.
BitFlip	Single-qubit bit flip (Pauli \(X\)) error channel.
PhaseFlip	Single-qubit bit flip (Pauli \(Z\)) error channel.
ResetError	Single-qubit Reset error channel.
PauliError	Pauli operator error channel for an arbitrary number of qubits.
QubitChannel	Apply an arbitrary fixed quantum channel.
ThermalRelaxationError	Thermal relaxation error channel.

Observables

Hadamard	The Hadamard operator
Hermitian	An arbitrary Hermitian observable.
Identity	The Identity operator
PauliX	The Pauli X operator
PauliY	The Pauli Y operator
PauliZ	The Pauli Z operator
Projector	Observable corresponding to the state projector \(P=\ket{\phi}\bra{\phi}\).
Hamiltonian	alias of LinearCombination
SparseHamiltonian	A Hamiltonian represented directly as a sparse matrix in Compressed Sparse Row (CSR) format.

Continuous-Variable (CV) operators

If you would like to learn more about the CV model of quantum computing, check out the quantum photonics page of the Strawberry Fields documentation.

CV gates

Identity	The Identity operator
Beamsplitter	Beamsplitter interaction.
ControlledAddition	Controlled addition operation.
ControlledPhase	Controlled phase operation.
CrossKerr	Cross-Kerr interaction.
CubicPhase	Cubic phase shift.
Displacement	Phase space displacement.
InterferometerUnitary	A linear interferometer transforming the bosonic operators according to the unitary matrix \(U\).
Kerr	Kerr interaction.
QuadraticPhase	Quadratic phase shift.
Rotation	Phase space rotation.
Squeezing	Phase space squeezing.
TwoModeSqueezing	Phase space two-mode squeezing.

CV state preparation

CatState	Prepares a cat state.
CoherentState	Prepares a coherent state.
DisplacedSqueezedState	Prepares a displaced squeezed vacuum state.
FockDensityMatrix	Prepare subsystems using the given density matrix in the Fock basis.
FockState	Prepares a single Fock state.
FockStateVector	Prepare subsystems using the given ket vector in the Fock basis.
GaussianState	Prepare subsystems in a given Gaussian state.
SqueezedState	Prepares a squeezed vacuum state.
ThermalState	Prepares a thermal state.

CV observables

FockStateProjector	The number state observable \(\ket{n}\bra{n}\).
Identity	The Identity operator
NumberOperator	The photon number observable \(\langle \hat{n}\rangle\).
TensorN	The tensor product of the NumberOperator acting on different wires.
QuadP	The momentum quadrature observable \(\hat{p}\).
PolyXP	An arbitrary second-order polynomial observable.
QuadOperator	The generalized quadrature observable \(\x_\phi = \x cos\phi+\p\sin\phi\).
QuadX	The position quadrature observable \(\hat{x}\).

Qutrit operators

Qutrit non-parametrized gates

TShift	The qutrit shift operator
TClock	Ternary Clock gate
THadamard	The ternary Hadamard operator
TAdd	The 2-qutrit controlled add gate
TSWAP	The ternary swap operator.

Qutrit gates constructed from a matrix

QutritUnitary	Apply an arbitrary, fixed unitary matrix.
ControlledQutritUnitary	Apply an arbitrary fixed unitary to wires with control from the control_wires.

Qutrit parametrized gates

TRX	The single qutrit X rotation
TRY	The single qutrit Y rotation
TRZ	The single qutrit Z rotation

Qutrit State preparation

QutritBasisState

Prepares a single computational basis state for a qutrit system.

Qutrit noisy channels

QutritDepolarizingChannel	Single-qutrit symmetrically depolarizing error channel.
QutritAmplitudeDamping	Single-qutrit amplitude damping error channel.
TritFlip	Single-qutrit trit flip error channel, used for applying "bit flips" on each qutrit subspace.
QutritChannel	Apply an arbitrary fixed qutrit channel.

Qutrit Observables

THermitian	An arbitrary Hermitian observable for qutrits.
GellMann	The Gell-Mann observables for qutrits

Pulse-level operators

If you would like to learn more about the implementation of pulse-level control in PennyLane, see the pulse module documentation.

Pulse operator

ParametrizedEvolution	Parametrized evolution gate, created by passing a ParametrizedHamiltonian to the evolve() function
ParametrizedHamiltonian	Callable object holding the information representing a parametrized Hamiltonian.

introduction/operations

 

Download Python script

 

Download Notebook

 

View on GitHub