# Gradients (qiskit.aqua.operators.gradients)¶

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 parametrized Ansatz V(θ), and an Operator O(ω).

Gradients

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}

ham_grad = Gradient(grad_method='param_shift').convert(operator=op, params=[x])
ham_grad.assign_parameters(value_dict).eval()

state_grad = Gradient(grad_method='lin_comb').convert(operator=op, params=[a])
state_grad.assign_parameters(value_dict).eval()

prob_grad = Gradient(grad_method='fin_diff').convert(
operator=CircuitStateFn(primitive=qc, coeff=1.), params=[a]
)
prob_grad.assign_parameters(value_dict).eval()


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 parametrized 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()


NaturalGradients

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.)
nat_grad = NaturalGradient(grad_method='lin_comb,
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 from qiskit.aqua.components.optimizers.

# Base Classes¶

 DerivativeBase Base class for differentiating opflow objects. GradientBase Base class for first-order operator gradient. HessianBase Base class for the Hessian of an expected value. QFIBase Base class for Quantum Fisher Information (QFI).

# Converters¶

 CircuitGradient Circuit to gradient operator converter. CircuitQFI Circuit to Quantum Fisher Information operator converter.

# Derivatives¶

 Gradient Convert an operator expression to the first-order gradient. Hessian Compute the Hessian of an expected value. NaturalGradient Convert an operator expression to the first-order gradient. QFI Compute the Quantum Fisher Information (QFI).