# cirq.ops.SingleQubitPauliStringGateOperation

A Pauli operation applied to a qubit.

Inherits From: GateOperation, PauliString, Operation

Satisfies the contract of both GateOperation and PauliString. Relies implicitly on the fact that PauliString({q: X}) compares as equal to GateOperation(X, [q]).

coefficient

gate The gate applied by the operation.
pauli

qubit

qubits The qubits targeted by the operation.
tags Returns a tuple of the operation's tags.
untagged Returns the underlying operation without any tags.

## Methods

### after

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Determines the equivalent pauli string after some operations.

If the PauliString is $P$ and the Clifford operation is $C$, then the result is $C P C^\dagger$.

Args
ops A stabilizer operation or nested collection of stabilizer operations.

Returns
The result of propagating this pauli string from before to after the given operations.

### before

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Determines the equivalent pauli string before some operations.

If the PauliString is $P$ and the Clifford operation is $C$, then the result is $C^\dagger P C$.

Args
ops A stabilizer operation or nested collection of stabilizer operations.

Returns
The result of propagating this pauli string from after to before the given operations.

### conjugated_by

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Returns the Pauli string conjugated by a clifford operation.

The product-of-Paulis $P$ conjugated by the Clifford operation $C$ is

$$C^\dagger P C$$

For example, conjugating a +Y operation by an S operation results in a +X operation (as opposed to a -X operation).

In a circuit diagram where P is a pauli string observable immediately after a Clifford operation C, the pauli string P.conjugated_by(C) is the equivalent pauli string observable just before C.

--------------------------C---P---

= ---C---P------------------------

= ---C---P---------C^-1---C-------

= ---C---P---C^-1---------C-------

= --(C^-1 · P · C)--------C-------

= ---P.conjugated_by(C)---C-------


Analogously, a Pauli product P can be moved from before a Clifford C in a circuit diagram to after the Clifford C by conjugating P by the inverse of C:

---P---C---------------------------

= -----C---P.conjugated_by(C^-1)---


Args
clifford The Clifford operation to conjugate by. This can be an individual operation, or a tree of operations.

Note that the composite Clifford operation defined by a sequence of operations is equivalent to a circuit containing those operations in the given order. Somewhat counter-intuitively, this means that the operations in the sequence are conjugated onto the Pauli string in reverse order. For example, P.conjugated_by([C1, C2]) is equivalent to P.conjugated_by(C2).conjugated_by(C1).

#### Examples:

a, b = cirq.LineQubit.range(2)
print(cirq.X(a).conjugated_by(cirq.CZ(a, b)))
X(0)*Z(1)
print(cirq.X(a).conjugated_by(cirq.S(a)))
-Y(0)
print(cirq.X(a).conjugated_by([cirq.H(a), cirq.CNOT(a, b)]))
Z(0)*X(1)


Returns
The Pauli string conjugated by the given Clifford operation.

### controlled_by

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Returns a controlled version of this operation. If no control_qubits are specified, returns self.

Args
control_qubits Qubits to control the operation by. Required.
control_values For which control qubit values to apply the operation. A sequence of the same length as control_qubits where each entry is an integer (or set of integers) corresponding to the qubit value (or set of possible values) where that control is enabled. When all controls are enabled, the operation is applied. If unspecified, control values default to 1.

### dense

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Returns a cirq.DensePauliString version of this Pauli string.

This method satisfies the invariant P.dense(qubits).on(*qubits) == P.

Args
qubits The implicit sequence of qubits used by the dense pauli string. Specifically, if the returned dense Pauli string was applied to these qubits (via its on method) then the result would be a Pauli string equivalent to the receiving Pauli string.

Returns
A cirq.DensePauliString instance D such that D.on(*qubits) equals the receiving cirq.PauliString instance P.

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### expectation_from_density_matrix

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Evaluate the expectation of this PauliString given a density matrix.

Compute the expectation value of this PauliString with respect to an array representing a density matrix. By convention expectation values are defined for Hermitian operators, and so this method will fail if this PauliString is non-Hermitian.

state must be an array representation of a density matrix and have shape (2 ** n, 2 ** n) or (2, 2, ..., 2) (2*n entries), where state is expressed over n qubits.

qubit_map must assign an integer index to each qubit in this PauliString that determines which bit position of a computational basis state that qubit corresponds to. For example if state represents $|0\rangle |+\rangle$ and q0, q1 = cirq.LineQubit.range(2) then:

cirq.X(q0).expectation(state, qubit_map={q0: 0, q1: 1}) = 0
cirq.X(q0).expectation(state, qubit_map={q0: 1, q1: 0}) = 1


Args
state An array representing a valid density matrix.
qubit_map A map from all qubits used in this PauliString to the indices of the qubits that state is defined over.
atol Absolute numerical tolerance.
check_preconditions Whether to check that state represents a valid density matrix.

Returns
The expectation value of the input state.

Raises
NotImplementedError if this PauliString is non-Hermitian.

### expectation_from_state_vector

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Evaluate the expectation of this PauliString given a state vector.

Compute the expectation value of this PauliString with respect to a state vector. By convention expectation values are defined for Hermitian operators, and so this method will fail if this PauliString is non-Hermitian.

state must be an array representation of a state vector and have shape (2 ** n, ) or (2, 2, ..., 2) (n entries) where state is expressed over n qubits.

qubit_map must assign an integer index to each qubit in this PauliString that determines which bit position of a computational basis state that qubit corresponds to. For example if state represents $|0\rangle |+\rangle$ and q0, q1 = cirq.LineQubit.range(2) then:

cirq.X(q0).expectation(state, qubit_map={q0: 0, q1: 1}) = 0
cirq.X(q0).expectation(state, qubit_map={q0: 1, q1: 0}) = 1


Args
state_vector An array representing a valid state vector.
qubit_map A map from all qubits used in this PauliString to the indices of the qubits that state_vector is defined over.
atol Absolute numerical tolerance.
check_preconditions Whether to check that state_vector represents a valid state vector.

Returns
The expectation value of the input state.

Raises
NotImplementedError if this PauliString is non-Hermitian.

### expectation_from_wavefunction

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THIS FUNCTION IS DEPRECATED.

IT WILL BE REMOVED IN cirq v0.10.0.

### frozen

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Returns a cirq.PauliString with the same contents.

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### matrix

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Returns the matrix of self in computational basis of qubits.

Args
qubits Ordered collection of qubits that determine the subspace in which the matrix representation of the Pauli string is to be computed. Qubits absent from self.qubits are acted on by the identity. Defaults to self.qubits.

### mutable_copy

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Returns a new cirq.MutablePauliString with the same contents.

### pass_operations_over

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Determines how the Pauli string changes when conjugated by Cliffords.

The output and input pauli strings are related by a circuit equivalence. In particular, this circuit:

───ops───INPUT_PAULI_STRING───


will be equivalent to this circuit:

───OUTPUT_PAULI_STRING───ops───


up to global phase (assuming after_to_before is not set).

If ops together have matrix C, the Pauli string has matrix P, and the output Pauli string has matrix P', then P' == C^-1 P C up to global phase.

Setting after_to_before inverts the relationship, so that the output is the input and the input is the output. Equivalently, it inverts C.

Args
ops The operations to move over the string.
after_to_before Determines whether the operations start after the pauli string, instead of before (and so are moving in the opposite direction).

### to_z_basis_ops

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Returns operations to convert the qubits to the computational basis.

### transform_qubits

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Returns the same operation, but with different qubits.

Args
func The function to use to turn each current qubit into a desired new qubit.

Returns
The receiving operation but with qubits transformed by the given function.

### validate_args

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Raises an exception if the qubits don't match this operation's qid shape.

Call this method from a subclass's with_qubits method.

Args
qubits The new qids for the operation.

Raises
ValueError The operation had qids that don't match it's qid shape.

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### with_qubits

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Returns the same operation, but applied to different qubits.

Args
new_qubits The new qubits to apply the operation to. The order must exactly match the order of qubits returned from the operation's qubits property.

### with_tags

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Creates a new TaggedOperation, with this op and the specified tags.

This method can be used to attach meta-data to specific operations without affecting their functionality. The intended usage is to attach classes intended for this purpose or strings to mark operations for specific usage that will be recognized by consumers. Specific examples include ignoring this operation in optimization passes, hardware-specific functionality, or circuit diagram customizability.

Tags can be a list of any type of object that is useful to identify this operation as long as the type is hashable. If you wish the resulting operation to be eventually serialized into JSON, you should also restrict the operation to be JSON serializable.

Args
new_tags The tags to wrap this operation in.

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### __pow__

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Raise gate to a power, then reapply to the same qubits.

Only works if the gate implements cirq.ExtrapolatableEffect. For extrapolatable gate G this means the following two are equivalent:

(G ** 1.5)(qubit)  or  G(qubit) ** 1.5


Args
exponent The amount to scale the gate's effect by.

Returns
A new operation on the same qubits with the scaled gate.

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