GF(\(2^m\)) Square

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

GF2Square

In place squaring for elements in GF(\(2^m\))

The bloq implements in-place squaring of a quantum registers storing elements from GF(\(2^m\)). Specifically, it implements the transformation

\[ |a\rangle \rightarrow |a^2\rangle \]

The key insight is that for elements in GF(\(2^m\)),

\[ a^2 =a_0 + a_1 x^2 + a_2 x^4 + ... + a_{n-1} x^{2(n - 1)} \]

Thus, squaring can be implemented via a linear reversible circuit using only CNOT gates.

Parameters

• bitsize: The degree \(m\) of the galois field \(GF(2^m)\). Also corresponds to the number of qubits in the input register to be squared.

Registers

• x: Input THRU register of size \(m\) that stores elements from \(GF(2^m)\).

from qualtran.bloqs.gf_arithmetic import GF2Square

Example Instances

gf16_square = GF2Square(4)

import sympy

m = sympy.Symbol('m')
gf2_square_symbolic = GF2Square(m)

Graphical Signature

from qualtran.drawing import show_bloqs
show_bloqs([gf16_square, gf2_square_symbolic],
           ['`gf16_square`', '`gf2_square_symbolic`'])

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
gf16_square_g, gf16_square_sigma = gf16_square.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(gf16_square_g)
show_counts_sigma(gf16_square_sigma)

Counts totals:

• SynthesizeLRCircuit: 1