Rotates Hamming-weight 2 states into their bitwise complements.

Inherits From: InteractionOperatorFermionicGate, ParityPreservingFermionicGate

With weights \((w_0, w_1, w_2)\) and exponent \(t\), this gate's matrix is defined as

\[ e^{-i t H}, \]


\[ H = \left(w_0 \left| 1001 \right\rangle\left\langle 0110 \right| + \text{h.c.}\right) + \left(w_1 \left| 1010 \right\rangle\left\langle 0101 \right| + \text{h.c.}\right) + \left(w_2 \left| 1100 \right\rangle\left\langle 0011 \right| + \text{h.c.}\right) \]

This corresponds to the Jordan-Wigner transform of

\[ H = -\left(w_0 a^{\dagger}_i a^{\dagger}_{i+3} a_{i+1} a_{i+2} + \text{h.c.}\right) - \left(w_1 a^{\dagger}_i a^{\dagger}_{i+2} a_{i+1} a_{i+3} + \text{h.c.}\right) - \left(w_2 a^{\dagger}_i a^{\dagger}_{i+1} a_{i+2} a_{i+3} + \text{h.c.}\right), \]

where \(a_i\), ..., \(a_{i+3}\) are the annihilation operators for the fermionic modes \(i\), ..., \((i+3)\), respectively mapped to the four qubits on which this gate acts.

weights The weights of the terms in the Hamiltonian.


fermion_generator The FermionOperator G such that the gate's unitary is exp(-i t G) with exponent t using the Jordan-Wigner transformation.

qubit_generator_matrix The (Hermitian) matrix G such that the gate's unitary is exp(-i * G).



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Returns a controlled version of this gate. If no arguments are specified, defaults to a single qubit control.

num_controls: Total number of control qubits. control_values: For which control qubit values to apply the sub gate. A sequence of length num_controls 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 sub gate is applied. If unspecified, control values default to 1. control_qid_shape: The qid shape of the controls. A tuple of the expected dimension of each control qid. Defaults to (2,) * num_controls. Specify this argument when using qudits.


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The FermionOperators \((G_i)_i\) such that the gate's fermionic generator is \(\sum_i w_i G_i + \text{h.c.}\) where \((w_i)_i\) are the gate's weights.


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Constructs the gate corresponding to the specified term in the Hamiltonian.


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Update the weights of the gate to effect conjugation by an FSWAP on the i-th and (i+1)th qubits.


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Constructs the Hamiltonian corresponding to the gate's generator.


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The number of qubits this gate acts on.


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The number of parameters (weights) in the generator.


Returns an application of this gate to the given qubits.

*qubits The collection of qubits to potentially apply the gate to.


Returns a list of operations applying the gate to all targets.

*targets The qubits to apply this gate to. For single-qubit gates this can be provided as varargs or a combination of nested iterables. For multi-qubit gates this must be provided as an Iterable[Sequence[Qid]], where each sequence has num_qubits qubits.

Operations applying this gate to the target qubits.

ValueError If targets are not instances of Qid or Iterable[Qid]. If the gate qubit number is incompatible.
TypeError If a single target is supplied and it is not iterable.


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An in-place version of permuted.


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Returns a gate with the Jordan-Wigner ordering changed.

If the Jordan-Wigner ordering of the original gate is given by init_pos, then the returned gate has Jordan-Wigner ordering (0, ..., n - 1), where n is the number of qubits on which the gate acts.

init_pos A permutation of (0, ..., n - 1).


Checks if this gate can be applied to the given qubits.

By default checks that:

  • inputs are of type Qid
  • len(qubits) == num_qubits()
  • qubit_i.dimension == qid_shape[i] for all qubits

Child classes can override. The child implementation should call super().validate_args(qubits) then do custom checks.

qubits The sequence of qubits to potentially apply the gate to.

ValueError The gate can't be applied to the qubits.


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The symbol to use in circuit diagrams.





Call self as a function.