qpe_toolbox.estimation.lcu_walk_operator¶

Functions¶

`get_lcu_weights`(hamiltonian)	Compute the LCU weights, normalization factor, and ancilla register size for a Hamiltonian.
`build_lcu_prepare_state_mps`(hamiltonian, *[, cutoff])	Construct the normalized MPS representing the L register state \(\ket{\mathcal{L}}\).
`build_lcu_prepare_mpo`(hamiltonian, *[, cutoff])	Construct the PREPARE oracle MPO \(\ket{0}\bra{\mathcal{L}}\).
`lcu_select_gates`(hamiltonian)	Construct the full list of gate instructions for the SELECT oracle.
`build_lcu_select_mpo`(hamiltonian, *[, cutoff])	Construct the MPO implementing the SELECT oracle.
`build_lcu_reflection_mpo`(hamiltonian, *[, cutoff])	Construct the reflection operator \(\mathcal{R}_L\) for the L register.
`run_qpe_lcu_walk_operator`(H, psi0_mps, m_ph, *[, ...])	Perform LCU and quantum phase estimation (QPE) using the walk operator.
`qpe_first_stage_walk`(H, psi0_mps, m_ph, regs, *[, ...])	LCU first stage of QPE: apply Hadamard wall and controlled-W sequence.
`get_energy_from_lcu_walk_phase`(theta, lmb)	Get the energy from the eigenphase of the LCU Walk operator.
`estimate_lcu_error`(m_ph, E0, lmb)	Estimate the error bound in energy in LCU QPE from finite number of phase qubits.

Module Contents¶

qpe_toolbox.estimation.lcu_walk_operator.get_lcu_weights(hamiltonian)[source]¶

Compute the LCU weights, normalization factor, and ancilla register size for a Hamiltonian.

Parameters:

hamiltonian (Hamiltonian) – Hamiltonian from the QPE-Toolbox Hamiltonian class.

Returns:

weights (list of float) – Absolute values of the Hamiltonian coefficients, extended with zeros to match a power-of-two length.
lmb (float) – Sum of absolute values of Hamiltonian coefficients (normalization factor).
L (int) – Number of original Hamiltonian terms.
m_L (int) – Number of qubits required for the auxiliary L-register, i.e., ceil(log2(L)).

qpe_toolbox.estimation.lcu_walk_operator.build_lcu_prepare_state_mps(hamiltonian, *, cutoff=1e-10)[source]¶

Construct the normalized MPS representing the L register state \(\ket{\mathcal{L}}\).

\[\ket{\mathcal{L}} = \sum_\ell \sqrt{\frac{w_\ell}{\lambda}} \ket{\ell}\]

Parameters:

hamiltonian (Hamiltonian) – Hamiltonian to encode with LCU.
cutoff (float, default 1e-10) – Singular value cutoff for MPS compression.

Returns:

L_mps – MPS representing the state \(\ket{\mathcal{L}}\).

Return type:

MatrixProductState

qpe_toolbox.estimation.lcu_walk_operator.build_lcu_prepare_mpo(hamiltonian, *, cutoff=1e-10)[source]¶

Construct the PREPARE oracle MPO \(\ket{0}\bra{\mathcal{L}}\).

Parameters:

hamiltonian (Hamiltonian) – Hamiltonian to encode with LCU.
cutoff (float, default 1e-10) – Cutoff for MPO compression.

Returns:

prep_mpo – MPO implementing the PREPARE oracle.

Return type:

MatrixProductOperator

qpe_toolbox.estimation.lcu_walk_operator.lcu_select_gates(hamiltonian)[source]¶

Construct the full list of gate instructions for the SELECT oracle.

Parameters:: hamiltonian (Hamiltonian) – Hamiltonian object from the QPE-Toolbox Hamiltonian class.
Returns:: gates – Gate sequence implementing the SELECT oracle.
Return type:: list of Gate

qpe_toolbox.estimation.lcu_walk_operator.build_lcu_select_mpo(hamiltonian, *, cutoff=1e-10)[source]¶

Construct the MPO implementing the SELECT oracle.

Parameters:

hamiltonian (Hamiltonian) – Hamiltonian describing the system.
cutoff (float, default 1e-10) – MPO compression cutoff.

Returns:

select_mpo – MPO implementing \(SELECT = \sum_\ell \ket{\ell}\bra{\ell} \otimes H_\ell\).

Return type:

MatrixProductOperator

qpe_toolbox.estimation.lcu_walk_operator.build_lcu_reflection_mpo(hamiltonian, *, cutoff=1e-10)[source]¶

Construct the reflection operator \(\mathcal{R}_L\) for the L register.

\[\mathcal{R}_L = 2 \ket{\mathcal{L}}\bra{\mathcal{L}}\otimes\mathbb{1} - \mathbb{1}\]

Parameters:

hamiltonian (Hamiltonian) – Hamiltonian object from the QPE-Toolbox Hamiltonian class.
cutoff (float, default 1e-10) – MPO compression cutoff.

Returns:

R_L – MPO representing the reflection.

Return type:

MatrixProductOperator

qpe_toolbox.estimation.lcu_walk_operator.run_qpe_lcu_walk_operator(H, psi0_mps, m_ph, *, max_bond=0, cutoff=1e-10, verbosity=0)[source]¶

Perform LCU and quantum phase estimation (QPE) using the walk operator.

Parameters:

H (Hamiltonian) – Hamiltonian object from the QPE-Toolbox Hamiltonian class.
psi0_mps (MatrixProductState) – Initial state of the physical register.
m_ph (int) – Number of phase estimation qubits.
max_bond (int, default 0) – Maximum MPS bond dimension.
cutoff (float, default 1e-10) – Truncation cutoff for MPS compression.
verbosity (int, default 0) – Verbosity level. If >= 1, print result summary. If >= 2, print detailed progress messages.

Returns:

traces (dict) – Contains timing information: {‘ctimes’: […]}.
theta (float) – Estimated phase from the walk operator.

qpe_toolbox.estimation.lcu_walk_operator.qpe_first_stage_walk(H, psi0_mps, m_ph, regs, *, max_bond=0, cutoff=1e-10, verbosity=0)[source]¶

LCU first stage of QPE: apply Hadamard wall and controlled-W sequence.

Parameters:

H (Hamiltonian) – Hamiltonian from the QPE-Toolbox Hamiltonian class.
psi0_mps (MatrixProductState) – Initial state of the physical register.
m_ph (int) – Number of phase qubits.
regs (dict) – Dictionary with registers: {‘phase’:…, ‘L’:…, ‘phys’:…}.
max_bond (int, default 0) – Maximum MPS bond dimension.
cutoff (float, default 1e-10) – MPO/MPS compression cutoff.
verbosity (int, default 0) – Verbosity level. If >= 2, print progress messages.

Returns:

traces (dict) – Contains timing information: {‘ctimes’: […]}.
circ (CircuitMPS) – Circuit representing the applied QPE first stage.

qpe_toolbox.estimation.lcu_walk_operator.get_energy_from_lcu_walk_phase(theta, lmb)[source]¶

Get the energy from the eigenphase of the LCU Walk operator.

\[E = \lambda \cos(2 \pi \theta)\]

Parameters:

theta (float) – Eigenphase of the Walk operator.
lmb (float) – One-norm of LCU weights.

Returns:

energy – Estimated energy.

Return type:

float

qpe_toolbox.estimation.lcu_walk_operator.estimate_lcu_error(m_ph, E0, lmb)[source]¶

Estimate the error bound in energy in LCU QPE from finite number of phase qubits.

Parameters:

m_ph (int) – Number of phase estimation qubits.
E0 (float) – Ground state energy estimate.
lmb (float) – LCU normalization factor.

Returns:

delta_E – Estimated upper bound on the energy error.

Return type:

float