ZPow Rotation via Phase Gradient

from qualtran import Bloq, CompositeBloq, BloqBuilder, Signature, Register
from qualtran import QBit, QInt, QUInt, QAny
from qualtran.drawing import show_bloq, show_call_graph, show_counts_sigma
from typing import *
import numpy as np
import sympy
import cirq

ZPowConstViaPhaseGradient

Apply an \(Z**t\) on a qubit using a phase gradient state.

This bloq implements a Z**t by conditionally loading t/2 into a quantum register, conditioned on the qubit q (rotation target), and then adding this value to the phase gradient to get a phase kickback, and uncomputes the load. This controlled-load trick is taken from Ref. [2], Fig 2a.

See :class:PhaseGradientState for details on phase gradients.

It loads an approximation of t/2 to phase_grad_bitsize bits, which is loaded using phase_grad_bitsize clean ancilla.

The total Tofolli cost is phase_grad_bitsize - 2.

Parameters

• exponent: value of t to apply Z**t

• phase_grad_bitsize: number of qubits of the phase gradient state.

Registers

• q: qubit to apply rotation on.

• phase_grad: phase gradient state of type QFxp with phase_grad_bitsize fractional bits.

References

• Improved quantum circuits for elliptic curve discrete logarithms. Haner et al. 2020. Section 3: Components. “Integer addition” and Fig 2a.

from qualtran.bloqs.rotations import ZPowConstViaPhaseGradient

Example Instances

zpow_const_via_phase_grad = ZPowConstViaPhaseGradient.from_precision(3 / 8, eps=1e-11)

eps = sympy.symbols("eps")
zpow_const_via_phase_grad_symb_prec = ZPowConstViaPhaseGradient.from_precision(3 / 8, eps=eps)

t = sympy.symbols("t")
zpow_const_via_phase_grad_symb_angle = ZPowConstViaPhaseGradient.from_precision(t, eps=1e-11)

Graphical Signature

from qualtran.drawing import show_bloqs
show_bloqs([zpow_const_via_phase_grad, zpow_const_via_phase_grad_symb_prec, zpow_const_via_phase_grad_symb_angle],
           ['`zpow_const_via_phase_grad`', '`zpow_const_via_phase_grad_symb_prec`', '`zpow_const_via_phase_grad_symb_angle`'])

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
zpow_const_via_phase_grad_g, zpow_const_via_phase_grad_sigma = zpow_const_via_phase_grad.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(zpow_const_via_phase_grad_g)
show_counts_sigma(zpow_const_via_phase_grad_sigma)

Counts totals:

• AddIntoPhaseGrad: 1

• C[XorK]: 2