```"""Quantum gate operations

A quantum gate acting on :math:`n` qubits is a :math:`2^n×2^n` unitary
matrix written in the computational basis.
"""

import numpy as np

__all__ = [
"apply",
"n_qubits",
"controlled",
# -- Single qubit gates --
"id",
"x",
"y",
"z",
"not_",
"sqrt_not",
"phase_shift",
# -- 2 qubit gates --
"cnot",
"swap",
# -- 3 qubit gates --
"toffoli",
"cswap",
"fredkin",
"deutsch",
]  # type: ignore

def apply(gate, qubits, state):
"""Apply a gate to the specified qubits of a state

Parameters
----------
gate : ndarray[complex]
qubits : sequence of int
The qubits on which to act. Qubit 0 is the least significant qubit.
state : ndarray[complex]

Returns
-------
new_state : ndarray[complex]
"""
n_gate_qubits = gate.shape[0].bit_length() - 1
n_state_qubits = state.shape[0].bit_length() - 1
assert len(qubits) == n_gate_qubits

# We can view an n-qubit gate as a 2*n-tensor (n contravariant and n contravariant
# indices) and an n-qubit state as an n-tensor (contravariant indices)
# with each axis having length 2 (the state space of a single qubit).
gate = gate.reshape((2,) * 2 * n_gate_qubits)
state = state.reshape((2,) * n_state_qubits)

# Our qubits are labeled from least significant to most significant, i.e. our
# computational basis (for e.g. 2 qubits) is ordered like |00⟩, |01⟩, |10⟩, |11⟩.
# We represent the state as a tensor in *row-major* order; this means that the
# axis ordering is *backwards* compared to the qubit ordering (the least significant
# qubit corresponds to the *last* axis in the tensor etc.)
qubit_axes = tuple(n_state_qubits - 1 - np.asarray(qubits))

# Applying the gate to the state vector is then the tensor product over the appropriate axes
axes = (np.arange(n_gate_qubits, 2 * n_gate_qubits), qubit_axes)
new_state = np.tensordot(gate, state, axes=axes)

# tensordot effectively re-orders the qubits such that the qubits we operated
# on are in the most-significant positions (i.e. their axes come first). We
# thus need to transpose the axes to place them back into their original positions.
untouched_axes = tuple(
idx for idx in range(n_state_qubits) if idx not in qubit_axes
)
inverse_permutation = np.argsort(qubit_axes + untouched_axes)
return np.transpose(new_state, inverse_permutation).reshape(-1)

def _check_valid_gate(gate):
if not (
# is an array
isinstance(gate, np.ndarray)
# is complex
and np.issubdtype(gate.dtype, np.complex128)
# is square
and gate.shape[0] == gate.shape[1]
# has size 2**n, n > 1
and np.log2(gate.shape[0]).is_integer()
and gate.shape[0].bit_length() > 1
# is unitary
and np.allclose(gate @ gate.conjugate().transpose(), np.identity(gate.shape[0]))
):
raise ValueError("Gate is not valid")

def n_qubits(gate):
"""Return the number of qubits that a gate acts on.

Raises ValueError if 'gate' does not have a shape that is
an integer power of 2.
"""
_check_valid_gate(gate)
return gate.shape[0].bit_length() - 1

def controlled(gate):
"""Return a controlled quantum gate, given a quantum gate.

If 'gate' operates on :math:`n` qubits, then the controlled gate operates
on :math:`n+1` qubits, where the most-significant qubit is the control.

Parameters
----------
gate : np.ndarray[complex]
A quantum gate acting on :math:`n` qubits;
a :math:`2^n×2^n` unitary matrix in the computational basis.

Returns
-------
controlled_gate : np.ndarray[(2**(n+1), 2**(n+1)), complex]
"""
_check_valid_gate(gate)
n = gate.shape[0]
zeros = np.zeros((n, n))
return np.block([[np.identity(n), zeros], [zeros, gate]])

# -- Single qubit gates --

#: The identity gate on 1 qubit
id = np.identity(2, complex)
#: Pauli X gate
x = np.array([[0, 1], [1, 0]], complex)
#: NOT gate
not_ = x
#: Pauli Y gate
y = np.array([[0, -1j], [1j, 0]], complex)
#: Pauli Z gate
z = np.array([[1, 0], [0, -1]], complex)
#: SQRT(NOT) gate
sqrt_not = 0.5 * (1 + 1j * id - 1j * x)
hadamard = np.sqrt(0.5) * (x + z)

def phase_shift(phi):
"Return a gate that shifts the phase of :math:`|1⟩` by :math:`φ`."
return np.array([[1, 0], [0, np.exp(1j * phi)]])

# -- Two qubit gates --

#: Controlled NOT gate
cnot = controlled(x)
#: SWAP gate
swap = np.identity(4, complex)[:, (0, 2, 1, 3)]

# -- Three qubit gates --

#: Toffoli (CCNOT) gate
toffoli = controlled(cnot)
#: Controlled SWAP gate
cswap = controlled(swap)
#: Fredkin gate
fredkin = cswap

def deutsch(phi):
"Return a Deutsch gate for angle :math:`φ`."
gate = np.identity(8, complex)
gate[-2:, -2:] = np.array(
[[1j * np.cos(phi), np.sin(phi)], [np.sin(phi), 1j * np.cos(phi)]]
)
return gate
```