openfermioncirq.CubicFermionicSimulationGate

class openfermioncirq.CubicFermionicSimulationGate(weights: Tuple[complex, complex, complex] = (1.0, 1.0, 1.0), **kwargs)[source]

w0 * |110><101| + w1 * |110><011| + w2 * |101><011| + hc interaction.

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

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

where

\[H = \left(w_0 \left| 110 \right\rangle\left\langle 101 \right| + \text{h.c.}\right) + \left(w_1 \left| 110 \right\rangle\left\langle 011 \right| + \text{h.c.}\right) + \left(w_2 \left| 101 \right\rangle\left\langle 011 \right| + \text{h.c.}\right)\]

This corresponds to the Jordan-Wigner transform of

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

where \(a_i\), \(a_{i+1}\), \(a_{i+2}\) are the annihilation operators for the fermionic modes \(i\), \((i+1)\) \((i+2)\), respectively mapped to the three qubits on which this gate acts.

Parameters:weights – The weights of the terms in the Hamiltonian.
__init__(weights: Tuple[complex, complex, complex] = (1.0, 1.0, 1.0), **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, complex, complex] = (1.0, …) 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.
validate_args(qubits) Checks if this gate can be applied to the given qubits.
wrap_in_linear_combination(coefficient, …)

Attributes

exponent
global_shift