PhaseEstimationScale

class PhaseEstimationScale(bound)

Bases: object

Set and use a bound on eigenvalues of a Hermitian operator in order to ensure phases are in the desired range and to convert measured phases into eigenvectors.

The bound is set when constructing this class. Then the method scale is used to find the factor by which to scale the operator.

If bound is equal exactly to the largest eigenvalue, and the smallest eigenvalue is minus the largest, then these two eigenvalues will not be distinguished. For example, if the Hermitian operator is the Pauli Z operator with eigenvalues \(1\) and \(-1\), and bound is \(1\), then both eigenvalues will be mapped to \(1\). This can be avoided by making bound a bit larger.

Increasing bound decreases the part of the interval \([0, 1)\) that is used to map eigenvalues to phi. However, sometimes this results in a better determination of the eigenvalues, because 1) although there are fewer discrete phases in the useful range, it may shift one of the discrete phases closer to the actual phase. And, 2) If one of the discrete phases is close to, or exactly equal to the actual phase, then artifacts (probability) in neighboring phases will be reduced. This is important because the artifacts may be larger than the probability in a phase representing another eigenvalue of interest whose corresponding eigenstate has a relatively small weight in the input state.

Parameters:

bound (float) – an upper bound on the absolute value of the eigenvalues of a Hermitian operator. (The operator is not needed here.)

Attributes

scale

Return the Hamiltonian scaling factor.

Return the scale factor by which a Hermitian operator must be multiplied so that the phase of the corresponding unitary is restricted to \([-\pi, \pi]\). This factor is computed from the bound on the absolute values of the eigenvalues of the operator. The methods scale_phase and scale_phases are used recover the eigenvalues corresponding the original (unscaled) Hermitian operator.

Returns:

The scale factor.

Methods

classmethod from_pauli_sum(pauli_sum)

Create a PhaseEstimationScale from a SummedOp representing a sum of Pauli Operators.

It is assumed that the pauli_sum is the sum of PauliOp objects. The bound on the absolute value of the eigenvalues of the sum is obtained as the sum of the absolute values of the coefficients of the terms. This is the best bound available in the generic case. A PhaseEstimationScale object is instantiated using this bound.

Parameters:

pauli_sum (SparsePauliOp | Operator | BaseOperator) – A SparsePauliOp or other suitable quantum info Operator

Raises:

ValueError – if pauli_sum is not a sum of Pauli operators.

Returns:

A PhaseEstimationScale object

Return type:

PhaseEstimationScale

scale_phase(phi, id_coefficient=0.0)

Convert a phase into an eigenvalue.

The input phase phi corresponds to the eigenvalue of a unitary obtained by exponentiating a scaled Hermitian operator. Recall that the phase is obtained from phi as \(2\pi\phi\). Furthermore, the Hermitian operator was scaled so that phi is restricted to \([-1/2, 1/2]\), corresponding to phases in \([-\pi, \pi]\). But the values of phi read from the phase-readout register are in \([0, 1)\). Any value of phi greater than \(1/2\) corresponds to a raw phase of minus the complement with respect to 1. After this possible shift, the phase is scaled by the inverse of the factor by which the Hermitian operator was scaled to recover the eigenvalue of the Hermitian operator.

Parameters:

• phi (float) – Normalized phase in \([0, 1)\) to be converted to an eigenvalue.

• id_coefficient (float) – All eigenvalues are shifted by this value.

Returns:

An eigenvalue computed from the input phase.

Return type:

float

scale_phases(phases, id_coefficient=0.0)

Convert a list or dict of phases to eigenvalues.

The values in the list, or keys in the dict, are values of phi` and are converted as described in the description of ``scale_phase. In case phases is a dict, the values of the dict are passed unchanged.

Parameters:

• phases (list | dict) – a list or dict of values of phi.

• id_coefficient (float) – All eigenvalues are shifted by this value.

Returns:

Eigenvalues computed from phases.

Return type:

dict | list