openfermioncirq.QuadraticFermionicSimulationGate

class openfermioncirq.QuadraticFermionicSimulationGate(weights: Tuple[float, float] = (1, 1), **kwargs)[source]

(w0 |10><01| + h.c.) + w1 * |11><11| interaction.

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

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

where

\[H = \left(w_0 \left| 10 \right\rangle\left\langle 01 \right| + \text{h.c.}\right) - w_1 \left| 11 \right\rangle \left\langle 11 \right|.\]

This corresponds to the Jordan-Wigner transform of

\[H = (w_0 a^{\dagger}_i a_{i+1} + \text{h.c.}) + w_1 a_{i}^{\dagger} a_{i+1}^{\dagger} a_{i} a_{i+1},\]

where \(a_i\) and \(a_{i+1}\) are the annihilation operators for the fermionic modes \(i\) and \((i+1)\), respectively mapped to the first and second qubits on which this gate acts.

Parameters:weights – The weights of the terms in the Hamiltonian.
__init__(weights: Tuple[float, float] = (1, 1), **kwargs) → None[source]

Initializes the parameters used to compute the gate’s matrix.

The eigenvalue of each eigenspace of a gate is computed by

1. Starting with an angle in half turns as returned by the gate’s _eigen_components method:

θ

  1. Shifting the angle by global_shift:

    θ + s

  2. Scaling the angle by exponent:

    (θ + s) * e

  3. Converting from half turns to a complex number on the unit circle:

    exp(i * pi * (θ + s) * e)

Parameters:
  • exponent – The t in gate**t. Determines how much the eigenvalues of the gate are scaled by. For example, eigenvectors phased by -1 when gate**1 is applied will gain a relative phase of e^{i pi exponent} when gate**exponent is applied (relative to eigenvectors unaffected by gate**1).
  • global_shift –

    Offsets the eigenvalues of the gate at exponent=1. In effect, this controls a global phase factor on the gate’s unitary matrix. The factor is:

    exp(i * pi * global_shift * exponent)

    For example, cirq.X**t uses a global_shift of 0 but cirq.rx(t) uses a global_shift of -0.5, which is why cirq.unitary(cirq.rx(pi)) equals -iX instead of X.

Methods

__init__(weights, float] = (1, 1), **kwargs) Initializes the parameters used to compute the gate’s matrix.
controlled(num_controls, control_values, …) Returns a controlled version of this gate.
num_qubits() The number of qubits this gate acts on.
on(*qubits) Returns an application of this gate to the given qubits.
qubit_index_to_equivalence_group_key(index) Returns a key that differs between non-interchangeable qubits.
validate_args(qubits) Checks if this gate can be applied to the given qubits.
wrap_in_linear_combination(coefficient, …)

Attributes

exponent
global_shift