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class HamiltonianPhaseEstimation(num_evaluation_qubits, quantum_instance=None, sampler=None)

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Bases: object

Run the Quantum Phase Estimation algorithm to find the eigenvalues of a Hermitian operator.

This class is nearly the same as PhaseEstimation, differing only in that the input in that class is a unitary operator, whereas here the input is a Hermitian operator from which a unitary will be obtained by scaling and exponentiating. The scaling is performed in order to prevent the phases from wrapping around 2π2\pi. The problem of estimating eigenvalues λj\lambda_j of the Hermitian operator HH is solved by running a circuit representing

exp(ibH)ψ=jexp(ibλj)cjλj,\exp(i b H) |\psi\rangle = \sum_j \exp(i b \lambda_j) c_j |\lambda_j\rangle,

where the input state is

ψ=jcjλj,|\psi\rangle = \sum_j c_j |\lambda_j\rangle,

and λj\lambda_j are the eigenvalues of HH.

Here, bb is a scaling factor sufficiently large to map positive λ\lambda to [0,π)[0,\pi) and negative λ\lambda to [π,2π)[\pi,2\pi). Each time the circuit is run, one measures a phase corresponding to lambdajlambda_j with probability cj2|c_j|^2.

If HH is a Pauli sum, the bound bb is computed from the sum of the absolute values of the coefficients of the terms. There is no way to reliably recover eigenvalues from phases very near the endpoints of these intervals. Because of this you should be aware that for degenerate cases, such as H=ZH=Z, the eigenvalues ±1\pm 1 will be mapped to the same phase, π\pi, and so cannot be distinguished. In this case, you need to specify a larger bound as an argument to the method estimate.

This class uses and works together with PhaseEstimationScale to manage scaling the Hamiltonian and the phases that are obtained by the QPE algorithm. This includes setting, or computing, a bound on the eigenvalues of the operator, using this bound to obtain a scale factor, scaling the operator, and shifting and scaling the measured phases to recover the eigenvalues.

Note that, although we speak of “evolving” the state according the Hamiltonian, in the present algorithm, we are not actually considering time evolution. Rather, the role of time is played by the scaling factor, which is chosen to best extract the eigenvalues of the Hamiltonian.

A few of the ideas in the algorithm may be found in Ref. [1].


[1]: Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments

T.E. O’Brien, B. Tarasinski, B.M. Terhal arXiv:1809.09697(opens in a new tab)


  • num_evaluation_qubits (int) – The number of qubits used in estimating the phase. The phase will be estimated as a binary string with this many bits.
  • quantum_instance (QuantumInstance |Backend | None) – Pending deprecation: The quantum instance on which the circuit will be run.
  • sampler (BaseSampler | None) – The sampler primitive on which the circuit will be sampled.



HamiltonianPhaseEstimation.estimate(hamiltonian, state_preparation=None, evolution=None, bound=None)

Run the Hamiltonian phase estimation algorithm.


  • hamiltonian (PauliOp |MatrixOp |SummedOp |Pauli |SparsePauliOp |PauliSumOp) – A Hermitian operator. If the algorithm is used with a Sampler primitive, the allowed types are Pauli, SparsePauliOp, and PauliSumOp. If the algorithm is used with a QuantumInstance, PauliOp, ``MatrixOp, PauliSumOp, and SummedOp types are allowed.
  • state_preparation (StateFn |QuantumCircuit |Statevector | None) – The StateFn to be prepared, whose eigenphase will be measured. If this parameter is omitted, no preparation circuit will be run and input state will be the all-zero state in the computational basis.
  • evolution (EvolutionSynthesis |EvolutionBase | None) – An evolution converter that generates a unitary from hamiltonian. If None, then the default PauliTrotterEvolution is used.
  • bound (float | None) – An upper bound on the absolute value of the eigenvalues of hamiltonian. If omitted, then hamiltonian must be a Pauli sum, or a PauliOp, in which case a bound will be computed. If hamiltonian is a MatrixOp, then bound may not be None. The tighter the bound, the higher the resolution of computed phases.

Return type



HamiltonianPhaseEstimationResult instance containing the result of the estimation and diagnostic information.


  • TypeError – If evolution is not of type EvolutionSynthesis when a Sampler is provided.
  • TypeError – If hamiltonian type is not Pauli or SparsePauliOp or PauliSumOp when a Sampler is provided.
  • ValueError – If bound is None and hamiltonian is not a Pauli sum, i.e. a PauliSumOp or a SummedOp whose terms are of type PauliOp.
  • TypeError – If evolution is not of type EvolutionBase when no Sampler is provided.
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