# Source code for qiskit.synthesis.linear_phase.cx_cz_depth_lnn

```# This code is part of Qiskit.
#
#
# obtain a copy of this license in the LICENSE.txt file in the root directory
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.

"""
Given -CZ-CX- transformation (a layer consisting only CNOT gates
followed by a layer consisting only CZ gates)
Return a depth-5n circuit implementation of the -CZ-CX- transformation over LNN.

Args:
mat_z: n*n symmetric binary matrix representing a -CZ- circuit
mat_x: n*n invertable binary matrix representing a -CX- transformation

Output:
QuantumCircuit: QuantumCircuit object containing a depth-5n circuit to implement -CZ-CX-

References:
 S. A. Kutin, D. P. Moulton, and L. M. Smithline, "Computation at a distance," 2007.
 D. Maslov and W. Yang, "CNOT circuits need little help to implement arbitrary
Hadamard-free Clifford transformations they generate," 2022.
"""

from copy import deepcopy
import numpy as np

from qiskit.circuit import QuantumCircuit
from qiskit.synthesis.linear.linear_matrix_utils import calc_inverse_matrix
from qiskit.synthesis.linear.linear_depth_lnn import _optimize_cx_circ_depth_5n_line

def _initialize_phase_schedule(mat_z):
"""
Given a CZ layer (represented as an n*n CZ matrix Mz)
Return a scheudle of phase gates implementing Mz in a SWAP-only netwrok
(c.f. Alg 1, )
"""
n = len(mat_z)
phase_schedule = np.zeros((n, n), dtype=int)
for i, j in zip(*np.where(mat_z)):
if i >= j:
continue

phase_schedule[i, j] = 3
phase_schedule[i, i] += 1
phase_schedule[j, j] += 1

return phase_schedule

def _shuffle(labels, odd):
"""
Args:
labels : a list of indices
odd : a boolean indicating whether this layer is odd or even,
Shuffle the indices in labels by swapping adjacent elements
(c.f. Fig.2, )
"""
swapped = [v for p in zip(labels[1::2], labels[::2]) for v in p]
return swapped + labels[-1:] if odd else swapped

def _make_seq(n):
"""
Given the width of the circuit n,
Return the labels of the boxes in order from left to right, top to bottom
(c.f. Fig.2, )
"""
seq = []
wire_labels = list(range(n - 1, -1, -1))

for i in range(n):
wire_labels_new = (
_shuffle(wire_labels, n % 2)
if i % 2 == 0
else wire_labels[0:1] + _shuffle(wire_labels[1:], (n + 1) % 2)
)
seq += [
(min(i), max(i)) for i in zip(wire_labels[::2], wire_labels_new[::2]) if i != i
]
wire_labels = wire_labels_new

return seq

def _swap_plus(instructions, seq):
"""
Given CX instructions (c.f. Thm 7.1, ) and the labels of all boxes,
Return a list of labels of the boxes that is SWAP+ in descending order
* Assumes the instruction gives gates in the order from top to bottom,
from left to right
* SWAP+ is defined in section 3.A. of . Note the northwest
diagonalization procedure of  consists exactly n layers of boxes,
each being either a SWAP or a SWAP+. That is, each northwest
diagonalization circuit can be uniquely represented by which of its
n(n-1)/2 boxes are SWAP+ and which are SWAP.
"""
instr = deepcopy(instructions)
swap_plus = set()
for i, j in reversed(seq):
cnot_1 = instr.pop()
instr.pop()

if instr == [] or instr[-1] != cnot_1:
# Only two CNOTs on same set of controls -> this box is SWAP+
else:
instr.pop()
return swap_plus

def _update_phase_schedule(n, phase_schedule, swap_plus):
"""
Given phase_schedule initialized to induce a CZ circuit in SWAP-only network and list of SWAP+ boxes
Update phase_schedule for each SWAP+ according to Algorithm 2, 
"""
layer_order = list(range(n))[-3::-2] + list(range(n))[-2::-2][::-1]
order_comp = np.argsort(layer_order[::-1])

# Go through each box by descending layer order

for i in layer_order:
for j in range(i + 1, n):
if (i, j) not in swap_plus:
continue
# we need to correct for the effected linear functions:

# We first correct type 1 and type 2 by switching
# the phase applied to c_j and c_i+c_j
phase_schedule[j, j], phase_schedule[i, j] = phase_schedule[i, j], phase_schedule[j, j]

# Then, we go through all the boxes that permutes j BEFORE box(i,j) and update:

for k in range(n):  # all boxes that permutes j
if k in (i, j):
continue
if (
order_comp[min(k, j)] < order_comp[i]
and phase_schedule[min(k, j), max(k, j)] % 4 != 0
):
phase = phase_schedule[min(k, j), max(k, j)]
phase_schedule[min(k, j), max(k, j)] = 0

# Step 1, apply phase to c_i, c_j, c_k
for l_s in (i, j, k):
phase_schedule[l_s, l_s] = (phase_schedule[l_s, l_s] + phase * 3) % 4

# Step 2, apply phase to c_i+ c_j, c_i+c_k, c_j+c_k:
for l1, l2 in [(i, j), (i, k), (j, k)]:
ls = min(l1, l2)
lb = max(l1, l2)
phase_schedule[ls, lb] = (phase_schedule[ls, lb] + phase * 3) % 4
return phase_schedule

def _apply_phase_to_nw_circuit(n, phase_schedule, seq, swap_plus):
"""
Given
Width of the circuit (int n)
A CZ circuit, represented by the n*n phase schedule phase_schedule
A CX circuit, represented by box-labels (seq) and whether the box is SWAP+ (swap_plus)
*   This circuit corresponds to the CX tranformation that tranforms a matrix to
a NW matrix (c.f. Prop.7.4, )
*   SWAP+ is defined in section 3.A. of .
*   As previously noted, the northwest diagonalization procedure of  consists
of exactly n layers of boxes, each being either a SWAP or a SWAP+. That is,
each northwest diagonalization circuit can be uniquely represented by which
of its n(n-1)/2 boxes are SWAP+ and which are SWAP.
Return a QuantumCircuit that computes the phase scheudle S inside CX
"""
cir = QuantumCircuit(n)

wires = list(zip(range(n), range(1, n)))
wires = wires[::2] + wires[1::2]

for i, (j, k) in zip(range(len(seq) - 1, -1, -1), reversed(seq)):
w1, w2 = wires[i % (n - 1)]

p = phase_schedule[j, k]

if (j, k) not in swap_plus:
cir.cx(w1, w2)

cir.cx(w2, w1)

if p % 4 == 0:
pass
elif p % 4 == 1:
cir.sdg(w2)
elif p % 4 == 2:
cir.z(w2)
else:
cir.s(w2)

cir.cx(w1, w2)

for i in range(n):
p = phase_schedule[n - 1 - i, n - 1 - i]
if p % 4 == 0:
continue
if p % 4 == 1:
cir.sdg(i)
elif p % 4 == 2:
cir.z(i)
else:
cir.s(i)

return cir

[docs]def synth_cx_cz_depth_line_my(mat_x: np.ndarray, mat_z: np.ndarray):
"""
Joint synthesis of a -CZ-CX- circuit for linear nearest neighbour (LNN) connectivity,
with 2-qubit depth at most 5n, based on Maslov and Yang.
This method computes the CZ circuit inside the CX circuit via phase gate insertions.

Args:
mat_z : a boolean symmetric matrix representing a CZ circuit.
Mz[i][j]=1 represents a CZ(i,j) gate

mat_x : a boolean invertible matrix representing a CX circuit.

Return:
QuantumCircuit : a circuit implementation of a CX circuit following a CZ circuit,
denoted as a -CZ-CX- circuit,in two-qubit depth at most 5n, for LNN connectivity.

Reference:
1. Kutin, S., Moulton, D. P., Smithline, L.,
*Computation at a distance*, Chicago J. Theor. Comput. Sci., vol. 2007, (2007),
`arXiv:quant-ph/0701194 <https://arxiv.org/abs/quant-ph/0701194>`_
2. Dmitri Maslov, Willers Yang, *CNOT circuits need little help to implement arbitrary
`arXiv:2210.16195 <https://arxiv.org/abs/2210.16195>`_.
"""

# First, find circuits implementing mat_x by Proposition 7.3 and Proposition 7.4 of 

n = len(mat_x)
mat_x = calc_inverse_matrix(mat_x)

cx_instructions_rows_m2nw, cx_instructions_rows_nw2id = _optimize_cx_circ_depth_5n_line(mat_x)

# Meanwhile, also build the -CZ- circuit via Phase gate insertions as per Algorithm 2 
phase_schedule = _initialize_phase_schedule(mat_z)
seq = _make_seq(n)
swap_plus = _swap_plus(cx_instructions_rows_nw2id, seq)

_update_phase_schedule(n, phase_schedule, swap_plus)

qc = _apply_phase_to_nw_circuit(n, phase_schedule, seq, swap_plus)

for i, j in reversed(cx_instructions_rows_m2nw):
qc.cx(i, j)

return qc
```