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).