# Source code for pyquil.unitary_tools

```
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# Copyright 2016-2017 Rigetti Computing
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
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# http://www.apache.org/licenses/LICENSE-2.0
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# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
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from typing import Union, List
import numpy as np
from pyquil.gate_matrices import SWAP, QUANTUM_GATES, STATES
from pyquil.operator_estimation import TensorProductState
from pyquil.paulis import PauliSum, PauliTerm
from pyquil.quilbase import Gate
[docs]def all_bitstrings(n_bits):
"""All bitstrings in lexicographical order as a 2d np.ndarray.
This should be the same as ``np.array(list(itertools.product([0,1], repeat=n_bits)))``
but faster.
"""
n_bitstrings = 2 ** n_bits
out = np.zeros(shape=(n_bitstrings, n_bits), dtype=np.int8)
tf = np.array([False, True])
for i in range(n_bits):
# Lexicographical ordering gives a pattern of 1's
# where runs of 1s of length 2**j are tiled 2**i times
# i indexes from the *left*
# j indexes from the *right*
j = n_bits - i - 1
out[np.tile(np.repeat(tf, 2 ** j), 2 ** i), i] = 1
return out
def qubit_adjacent_lifted_gate(i, matrix, n_qubits):
"""
Lifts input k-qubit gate on adjacent qubits starting from qubit i
to complete Hilbert space of dimension 2 ** num_qubits.
Ex: 1-qubit gate, lifts from qubit i
Ex: 2-qubit gate, lifts from qubits (i+1, i)
Ex: 3-qubit gate, lifts from qubits (i+2, i+1, i), operating in that order
In general, this takes a k-qubit gate (2D matrix 2^k x 2^k) and lifts
it to the complete Hilbert space of dim 2^num_qubits, as defined by
the right-to-left tensor product (1) in arXiv:1608.03355.
Developer note: Quil and the QVM like qubits to be ordered such that qubit 0 is on the right.
Therefore, in ``qubit_adjacent_lifted_gate``, ``lifted_pauli``, and ``lifted_state_operator``,
we build up the lifted matrix by performing the kronecker product from right to left.
Note that while the qubits are addressed in decreasing order,
starting with num_qubit - 1 on the left and ending with qubit 0 on the
right (in a little-endian fashion), gates are still lifted to apply
on qubits in increasing index (right-to-left) order.
:param int i: starting qubit to lift matrix from (incr. index order)
:param np.array matrix: the matrix to be lifted
:param int n_qubits: number of overall qubits present in space
:return: matrix representation of operator acting on the
complete Hilbert space of all num_qubits.
:rtype: sparse_array
"""
n_rows, n_cols = matrix.shape
assert n_rows == n_cols, 'Matrix must be square'
gate_size = np.log2(n_rows)
assert gate_size == int(gate_size), 'Matrix must be 2^n by 2^n'
gate_size = int(gate_size)
# Outer-product to lift gate to complete Hilbert space
# bottom: i qubits below target
bottom_matrix = np.eye(2 ** i, dtype=np.complex128)
# top: Nq - i (bottom) - gate_size (gate) qubits above target
top_qubits = n_qubits - i - gate_size
top_matrix = np.eye(2 ** top_qubits, dtype=np.complex128)
return np.kron(top_matrix, np.kron(matrix, bottom_matrix))
def two_swap_helper(j, k, num_qubits, qubit_map):
"""
Generate the permutation matrix that permutes two single-particle Hilbert
spaces into adjacent positions.
ALWAYS swaps j TO k. Recall that Hilbert spaces are ordered in decreasing
qubit index order. Hence, j > k implies that j is to the left of k.
End results:
j == k: nothing happens
j > k: Swap j right to k, until j at ind (k) and k at ind (k+1).
j < k: Swap j left to k, until j at ind (k) and k at ind (k-1).
Done in preparation for arbitrary 2-qubit gate application on ADJACENT
qubits.
:param int j: starting qubit index
:param int k: ending qubit index
:param int num_qubits: number of qubits in Hilbert space
:param np.array qubit_map: current index mapping of qubits
:return: tuple of swap matrix for the specified permutation,
and the new qubit_map, after permutation is made
:rtype: tuple (np.array, np.array)
"""
if not (0 <= j < num_qubits and 0 <= k < num_qubits):
raise ValueError("Permutation SWAP index not valid")
perm = np.eye(2 ** num_qubits, dtype=np.complex128)
new_qubit_map = np.copy(qubit_map)
if j == k:
# nothing happens
return perm, new_qubit_map
elif j > k:
# swap j right to k, until j at ind (k) and k at ind (k+1)
for i in range(j, k, -1):
perm = qubit_adjacent_lifted_gate(i - 1, SWAP, num_qubits).dot(perm)
new_qubit_map[i - 1], new_qubit_map[i] = new_qubit_map[i], new_qubit_map[i - 1]
elif j < k:
# swap j left to k, until j at ind (k) and k at ind (k-1)
for i in range(j, k, 1):
perm = qubit_adjacent_lifted_gate(i, SWAP, num_qubits).dot(perm)
new_qubit_map[i], new_qubit_map[i + 1] = new_qubit_map[i + 1], new_qubit_map[i]
return perm, new_qubit_map
def permutation_arbitrary(qubit_inds, n_qubits):
"""
Generate the permutation matrix that permutes an arbitrary number of
single-particle Hilbert spaces into adjacent positions.
Transposes the qubit indices in the order they are passed to a
contiguous region in the complete Hilbert space, in increasing
qubit index order (preserving the order they are passed in).
Gates are usually defined as `GATE 0 1 2`, with such an argument ordering
dictating the layout of the matrix corresponding to GATE. If such an
instruction is given, actual qubits (0, 1, 2) need to be swapped into the
positions (2, 1, 0), because the lifting operation taking the 8 x 8 matrix
of GATE is done in the little-endian (reverse) addressed qubit space.
For example, suppose I have a Quil command CCNOT 20 15 10.
The median of the qubit indices is 15 - hence, we permute qubits
[20, 15, 10] into the final map [16, 15, 14] to minimize the number of
swaps needed, and so we can directly operate with the final CCNOT, when
lifted from indices [16, 15, 14] to the complete Hilbert space.
Notes: assumes qubit indices are unique (assured in parent call).
See documentation for further details and explanation.
Done in preparation for arbitrary gate application on
adjacent qubits.
:param qubit_inds: (int) Qubit indices in the order the gate is
applied to.
:param int n_qubits: Number of qubits in system
:return:
perm - permutation matrix providing the desired qubit reordering
qubit_arr - new indexing of qubits presented in left to right
decreasing index order. Should be identical to passed 'args'.
start_i - starting index to lift gate from
:rtype: tuple (sparse_array, np.array, int)
"""
# Begin construction of permutation
perm = np.eye(2 ** n_qubits, dtype=np.complex128)
# First, sort the list and find the median.
sorted_inds = np.sort(qubit_inds)
med_i = len(qubit_inds) // 2
med = sorted_inds[med_i]
# The starting position of all specified Hilbert spaces begins at
# the qubit at (median - med_i)
start = med - med_i
# Array of final indices the arguments are mapped to, from
# high index to low index, left to right ordering
final_map = np.arange(start, start + len(qubit_inds))[::-1]
start_i = final_map[-1]
# Note that the lifting operation takes a k-qubit gate operating
# on the qubits i+k-1, i+k-2, ... i (left to right).
# two_swap_helper can be used to build the
# permutation matrix by filling out the final map by sweeping over
# the qubit_inds from left to right and back again, swapping qubits into
# position. we loop over the qubit_inds until the final mapping matches
# the argument.
qubit_arr = np.arange(n_qubits) # current qubit indexing
made_it = False
right = True
while not made_it:
array = range(len(qubit_inds)) if right else range(len(qubit_inds))[::-1]
for i in array:
pmod, qubit_arr = two_swap_helper(np.where(qubit_arr == qubit_inds[i])[0][0],
final_map[i], n_qubits,
qubit_arr)
# update permutation matrix
perm = pmod.dot(perm)
if np.allclose(qubit_arr[final_map[-1]:final_map[0] + 1][::-1], qubit_inds):
made_it = True
break
# for next iteration, go in opposite direction
right = not right
assert np.allclose(qubit_arr[final_map[-1]:final_map[0] + 1][::-1], qubit_inds)
return perm, qubit_arr[::-1], start_i
def lifted_gate_matrix(matrix: np.ndarray, qubit_inds: List[int], n_qubits: int):
"""
Lift a unitary matrix to act on the specified qubits in a full ``n_qubits``-qubit
Hilbert space.
For 1-qubit gates, this is easy and can be achieved with appropriate kronning of identity
matrices. For 2-qubit gates acting on adjacent qubit indices, it is also easy. However,
for a multiqubit gate acting on non-adjactent qubit indices, we must first apply a permutation
matrix to make the qubits adjacent and then apply the inverse permutation.
:param matrix: A 2^k by 2^k matrix encoding an n-qubit operation, where ``k == len(qubit_inds)``
:param qubit_inds: The qubit indices we wish the matrix to act on.
:param n_qubits: The total number of qubits.
:return: A 2^n by 2^n lifted version of the unitary matrix acting on the specified qubits.
"""
n_rows, n_cols = matrix.shape
assert n_rows == n_cols, 'Matrix must be square'
gate_size = np.log2(n_rows)
assert gate_size == int(gate_size), 'Matrix must be 2^n by 2^n'
gate_size = int(gate_size)
pi_permutation_matrix, final_map, start_i = permutation_arbitrary(qubit_inds, n_qubits)
if start_i > 0:
check = final_map[-gate_size - start_i:-start_i]
else:
# Python can't deal with `arr[:-0]`
check = final_map[-gate_size - start_i:]
np.testing.assert_allclose(check, qubit_inds)
v_matrix = qubit_adjacent_lifted_gate(start_i, matrix, n_qubits)
return np.dot(np.conj(pi_permutation_matrix.T),
np.dot(v_matrix, pi_permutation_matrix))
[docs]def lifted_gate(gate: Gate, n_qubits: int):
"""
Lift a pyquil :py:class:`Gate` in a full ``n_qubits``-qubit Hilbert space.
This function looks up the matrix form of the gate and then dispatches to
:py:func:`lifted_gate_matrix` with the target qubits.
:param gate: A gate
:param n_qubits: The total number of qubits.
:return: A 2^n by 2^n lifted version of the gate acting on its specified qubits.
"""
if len(gate.params) > 0:
matrix = QUANTUM_GATES[gate.name](*gate.params)
else:
matrix = QUANTUM_GATES[gate.name]
for mod in reversed(gate.modifiers):
if mod == 'DAGGER':
matrix = matrix.conj().T
if mod == 'CONTROLLED':
zero = np.eye(2)
zero[1, 1] = 0
one = np.eye(2)
one[0, 0] = 0
matrix = np.kron(zero, np.eye(*matrix.shape)) + np.kron(one, matrix)
return lifted_gate_matrix(matrix=matrix,
qubit_inds=[q.index for q in gate.qubits],
n_qubits=n_qubits)
[docs]def program_unitary(program, n_qubits):
"""
Return the unitary of a pyQuil program.
:param program: A program consisting only of :py:class:`Gate`.:
:return: a unitary corresponding to the composition of the program's gates.
"""
umat = np.eye(2 ** n_qubits)
for instruction in program:
if isinstance(instruction, Gate):
unitary = lifted_gate(gate=instruction, n_qubits=n_qubits)
umat = unitary.dot(umat)
else:
raise ValueError("Can only compute program unitary for programs composed of `Gate`s")
return umat
[docs]def lifted_pauli(pauli_sum: Union[PauliSum, PauliTerm], qubits: List[int]):
"""
Takes a PauliSum object along with a list of
qubits and returns a matrix corresponding the tensor representation of the
object.
Useful for generating the full Hamiltonian after a particular fermion to
pauli transformation. For example:
Converting a PauliSum X0Y1 + Y1X0 into the matrix
.. code-block:: python
[[ 0.+0.j, 0.+0.j, 0.+0.j, 0.-2.j],
[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[ 0.+2.j, 0.+0.j, 0.+0.j, 0.+0.j]]
Developer note: Quil and the QVM like qubits to be ordered such that qubit 0 is on the right.
Therefore, in ``qubit_adjacent_lifted_gate``, ``lifted_pauli``, and ``lifted_state_operator``,
we build up the lifted matrix by performing the kronecker product from right to left.
:param pauli_sum: Pauli representation of an operator
:param qubits: list of qubits in the order they will be represented in the resultant matrix.
:returns: matrix representation of the pauli_sum operator
"""
if isinstance(pauli_sum, PauliTerm):
pauli_sum = PauliSum([pauli_sum])
n_qubits = len(qubits)
result_hilbert = np.zeros((2 ** n_qubits, 2 ** n_qubits), dtype=np.complex128)
# left kronecker product corresponds to the correct basis ordering
for term in pauli_sum.terms:
term_hilbert = np.array([1])
for qubit in qubits:
term_hilbert = np.kron(QUANTUM_GATES[term[qubit]], term_hilbert)
result_hilbert += term_hilbert * term.coefficient
return result_hilbert
def tensor_up(pauli_sum: Union[PauliSum, PauliTerm], qubits: List[int]):
"""
Takes a PauliSum object along with a list of
qubits and returns a matrix corresponding the tensor representation of the
object.
This is the same as :py:func:`lifted_pauli`. Nick R originally wrote this functionality
and really likes the name ``tensor_up``. Who can blame him?
:param pauli_sum: Pauli representation of an operator
:param qubits: list of qubits in the order they will be represented in the resultant matrix.
:returns: matrix representation of the pauli_sum operator
"""
return lifted_pauli(pauli_sum=pauli_sum, qubits=qubits)
def lifted_state_operator(state: TensorProductState, qubits: List[int]):
"""Take a TensorProductState along with a list of qubits and return a matrix
corresponding to the tensored-up representation of the states' density operator form.
Developer note: Quil and the QVM like qubits to be ordered such that qubit 0 is on the right.
Therefore, in ``qubit_adjacent_lifted_gate``, ``lifted_pauli``, and ``lifted_state_operator``,
we build up the lifted matrix by using the *left* kronecker product.
:param state: The state
:param qubits: list of qubits in the order they will be represented in the resultant matrix.
"""
mat = 1.0
for qubit in qubits:
oneq_state = state[qubit]
assert oneq_state.qubit == qubit
state_vector = STATES[oneq_state.label][oneq_state.index][:, np.newaxis]
state_matrix = state_vector @ state_vector.conj().T
mat = np.kron(state_matrix, mat)
return mat
```