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

Converters

Derivatives