qpe_toolbox.estimation.hadamard_test

Functions

build_hadamard_test_circuit(init_mps, U_gate, theta)

Construct the quantum circuit implementing the Hadamard test.

run_hadamard_test(init_mps, U_gate, theta, n_shots, *)

Run the Hadamard test circuit and estimate the expectation value \(Z(\theta)\).

Module Contents

qpe_toolbox.estimation.hadamard_test.build_hadamard_test_circuit(init_mps, U_gate, theta)[source]

Construct the quantum circuit implementing the Hadamard test.

The circuit prepares an ancilla qubit, applies a controlled unitary \(U\), applies a phase rotation on the ancilla, and measures the ancilla in the X basis.

This circuit can be used to estimate the real or imaginary part of \(\bra{\psi} U \ket{\psi}\) by choosing appropriate values of theta.

Parameters:

  • init_mps (MatrixProductState) – Initial state \(\ket{\psi}\) of the data register.

  • U_gate (Gate or list) – Unitary operator to be tested. If a Gate instance is provided, it must act on all data qubits and be controlled by the ancilla qubit. If a list is provided, it is interpreted as a sequence of Trotter slices, each slice being a list of gate specifications.

  • theta (float) – Phase angle applied to the ancilla qubit. Typical values: - 0 for estimating the real part - -π/2 for estimating the imaginary part

Returns:

circ – Circuit implementing the Hadamard test.

Return type:

CircuitMPS

qpe_toolbox.estimation.hadamard_test.run_hadamard_test(init_mps, U_gate, theta, n_shots, *, seed=None)[source]

Run the Hadamard test circuit and estimate the expectation value \(Z(\theta)\).

The returned value is

\[Z(\theta) = P(0) - P(1) = \mathrm{Re} \left[e^{i\theta} \bra{\psi} U \ket{\psi} \right]\]

where \(P(0)\) and \(P(1)\) are the probabilities of measuring the ancilla qubit in states \(\ket{0}\) and \(\ket{1}\).

Parameters:

  • init_mps (MatrixProductState) – Initial state \(\ket{\psi}\) of the data register.

  • U_gate (Gate or list) – Unitary operator used in the Hadamard test. See build_circuit method for accepted formats.

  • theta (float) – Phase angle applied to the ancilla qubit.

  • n_shots (int or qpe_toolbox.EXACT) – Number of measurement shots. If EXACT, probabilities are computed exactly, else probabilities are estimated by sampling.

  • seed (None or int, optional) – A random seed, passed to numpy.random.seed if given.

Returns:

Z – Estimated value of \(Z(\theta) = P(0) - P(1)\).

Return type:

float