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## Gradients (qiskit.opflow.gradients)¶

Obsoleto desde a versão 0.24.0: The qiskit.opflow module is deprecated and will be removed no earlier than 3 months after the release date. For code migration guidelines, visit https://qisk.it/opflow_migration.

Given an operator that represents either a quantum state resp. an expectation value, the gradient framework enables the evaluation of gradients, natural gradients, Hessians, as well as the Quantum Fisher Information.

Suppose a parameterized quantum state |ψ(θ)〉 = V(θ)|ψ〉 with input state |ψ〉 and parameterized Ansatz V(θ), and an Operator O(ω).

We want to compute one of: * $$d⟨ψ(θ)|O(ω)|ψ(θ)〉/ dω$$ * $$d⟨ψ(θ)|O(ω)|ψ(θ)〉/ dθ$$ * $$d⟨ψ(θ)|i〉⟨i|ψ(θ)〉/ dθ$$

The last case corresponds to the gradient w.r.t. the sampling probabilities of |ψ(θ). These gradients can be computed with different methods, i.e. a parameter shift, a linear combination of unitaries and a finite difference method.

Examples

x = Parameter('x')
ham = x * X
a = Parameter('a')

q = QuantumRegister(1)
qc = QuantumCircuit(q)
qc.h(q)
qc.p(params[0], q[0])
op = ~StateFn(ham) @ CircuitStateFn(primitive=qc, coeff=1.)

value_dict = {x: 0.1, a: np.pi / 4}

operator=CircuitStateFn(primitive=qc, coeff=1.), params=[a]
)


Hessians

We want to compute one of: * $$d^2⟨ψ(θ)|O(ω)|ψ(θ)〉/ dω^2$$ * $$d^2⟨ψ(θ)|O(ω)|ψ(θ)〉/ dθ^2$$ * $$d^2⟨ψ(θ)|O(ω)|ψ(θ)〉/ dθ dω$$ * $$d^2⟨ψ(θ)|i〉⟨i|ψ(θ)〉/ dθ^2$$

The last case corresponds to the Hessian w.r.t. the sampling probabilities of |ψ(θ)〉. Just as the first order gradients, the Hessians can be evaluated with different methods, i.e. a parameter shift, a linear combination of unitaries and a finite difference method. Given a tuple of parameters Hessian().convert(op, param_tuple) returns the value for the second order derivative. If a list of parameters is given Hessian().convert(op, param_list) returns the full Hessian for all the given parameters according to the given parameter order.

QFI

The Quantum Fisher Information QFI is a metric tensor which is representative for the representation capacity of a parameterized quantum state |ψ(θ)〉 = V(θ)|ψ〉 generated by an input state |ψ〉 and a parameterized Ansatz V(θ). The entries of the QFI for a pure state read $$\mathrm{QFI}_{kl} = 4 \mathrm{Re}[〈∂kψ|∂lψ〉−〈∂kψ|ψ〉〈ψ|∂lψ〉]$$.

Just as for the previous derivative types, the QFI can be computed using different methods: a full representation based on a linear combination of unitaries implementation, a block-diagonal and a diagonal representation based on an overlap method.

Examples

q = QuantumRegister(1)
qc = QuantumCircuit(q)
qc.h(q)
qc.p(params[0], q[0])
op = ~StateFn(ham) @ CircuitStateFn(primitive=qc, coeff=1.)

value_dict = {x: 0.1, a: np.pi / 4}

qfi = QFI('lin_comb_full').convert(
operator=CircuitStateFn(primitive=qc, coeff=1.), params=[a]
)
qfi.assign_parameters(value_dict).eval()


The natural gradient is a special gradient method which re-scales a gradient w.r.t. a state parameter with the inverse of the corresponding Quantum Fisher Information (QFI) $$\mathrm{QFI}^{-1} d⟨ψ(θ)|O(ω)|ψ(θ)〉/ dθ$$. Hereby, we can choose a gradient as well as a QFI method and a regularization method which is used together with a least square solver instead of exact inversion of the QFI:

Examples

op = ~StateFn(ham) @ CircuitStateFn(primitive=qc, coeff=1.)
qfi_method='lin_comb_full',
regularization='ridge').convert(operator=op, params=params)


The derivative classes come with a gradient_wrapper() function which returns the corresponding callable and are thus compatible with the optimizers.

### Base Classes¶

 Deprecated: Base class for differentiating opflow objects. GradientBase([grad_method]) Deprecated: Base class for first-order operator gradient. HessianBase([hess_method]) Deprecated: Base class for the Hessian of an expected value. QFIBase([qfi_method]) Deprecated: Base class for Quantum Fisher Information (QFI).

### Converters¶

 Deprecated: Circuit to gradient operator converter. Deprecated: Circuit to Quantum Fisher Information operator converter.

### Derivatives¶

 Gradient([grad_method]) Deprecated: Convert an operator expression to the first-order gradient. Hessian([hess_method]) Deprecated: Compute the Hessian of an expected value. NaturalGradient([grad_method, qfi_method, ...]) Deprecated: Convert an operator expression to the first-order gradient. QFI([qfi_method]) Deprecated: Compute the Quantum Fisher Information (QFI).