From 035836a9aadf1559f2b32796f835571142c42b6a Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Fri, 20 Sep 2024 14:34:48 +0400
Subject: [PATCH 01/16] backup

---
 src/qibo/quantum_info/linalg_operations.py | 30 ++++++++++++++++++++++
 1 file changed, 30 insertions(+)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index ae87ba4e12..ccb72e9be5 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -1,5 +1,6 @@
 """Module with common linear algebra operations for quantum information."""
 
+# %%
 import math
 from typing import List, Tuple, Union
 
@@ -196,3 +197,32 @@ def matrix_exponentiation(
     backend = _check_backend(backend)
 
     return backend.calculate_matrix_exp(phase, matrix, eigenvectors, eigenvalues)
+
+
+def schmidt_decomposition(state, backend=None):
+    decomposition = state
+    return decomposition
+
+
+# %%
+import numpy as np
+
+from qibo import set_backend
+from qibo.backends import NumpyBackend
+from qibo.quantum_info import random_statevector
+
+set_backend("numpy")
+backend = NumpyBackend()
+
+#%%
+nqubits = 2
+dims = 2**nqubits
+
+new_dims = int(np.sqrt(dims))
+
+state = random_statevector(dims, backend=backend)
+
+#%%
+new_state = state.reshape(-1, new_dims)
+U, S, V = np.linalg.svd(new_state)
+

From 858ed923ec10acf2ba7dbc99118c767caad91ae8 Mon Sep 17 00:00:00 2001
From: "pre-commit-ci[bot]"
 <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Date: Fri, 4 Oct 2024 04:55:22 +0000
Subject: [PATCH 02/16] [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci
---
 src/qibo/quantum_info/linalg_operations.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index 3275779a24..8eb120edae 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -232,7 +232,7 @@ def schmidt_decomposition(state, backend=None):
 set_backend("numpy")
 backend = NumpyBackend()
 
-#%%
+# %%
 nqubits = 2
 dims = 2**nqubits
 
@@ -240,6 +240,6 @@ def schmidt_decomposition(state, backend=None):
 
 state = random_statevector(dims, backend=backend)
 
-#%%
+# %%
 new_state = state.reshape(-1, new_dims)
 U, S, V = np.linalg.svd(new_state)

From bb290e50f1ca7e5e49ddfd4c228462bf8b0c29b5 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Fri, 4 Oct 2024 12:50:00 +0400
Subject: [PATCH 03/16] backup

---
 src/qibo/quantum_info/linalg_operations.py | 36 +++++++++-------------
 tests/test_quantum_info_operations.py      | 15 +++++++++
 2 files changed, 29 insertions(+), 22 deletions(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index 8eb120edae..488fecfc0e 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -1,6 +1,5 @@
 """Module with common linear algebra operations for quantum information."""
 
-# %%
 import math
 from typing import List, Tuple, Union
 
@@ -217,29 +216,22 @@ def matrix_power(matrix, power: Union[float, int], backend=None):
     return backend.calculate_matrix_power(matrix, power)
 
 
-def schmidt_decomposition(state, backend=None):
-    decomposition = state
-    return decomposition
-
-
-# %%
-import numpy as np
-
-from qibo import set_backend
-from qibo.backends import NumpyBackend
-from qibo.quantum_info import random_statevector
+def schmidt_decomposition(
+    state, partition: Union[List[int], Tuple[int, ...]], backend=None
+):
+    backend = _check_backend(backend)
 
-set_backend("numpy")
-backend = NumpyBackend()
+    nqubits = backend.np.log2(state.shape[-1])
+    if not nqubits.is_integer():
+        raise_error(ValueError, f"dimensions of ``state`` must be a power of 2.")
 
-# %%
-nqubits = 2
-dims = 2**nqubits
+    nqubits = int(nqubits)
+    partition_2 = set(list(range(nqubits))) ^ set(partition)
 
-new_dims = int(np.sqrt(dims))
+    tensor = backend.np.reshape(state, [2] * nqubits)
+    tensor = backend.np.transpose(tensor, partition + partition_2)
+    tensor = backend.np.reshape(tensor, 2 ** len(partition), -1)
 
-state = random_statevector(dims, backend=backend)
+    U, S, Vh = np.linalg.svd(tensor)
 
-# %%
-new_state = state.reshape(-1, new_dims)
-U, S, V = np.linalg.svd(new_state)
+    return U, S, Vh
diff --git a/tests/test_quantum_info_operations.py b/tests/test_quantum_info_operations.py
index 3d97f1f63a..0afb802d5c 100644
--- a/tests/test_quantum_info_operations.py
+++ b/tests/test_quantum_info_operations.py
@@ -7,6 +7,7 @@
     commutator,
     matrix_power,
     partial_trace,
+    schmidt_decomposition,
 )
 from qibo.quantum_info.metrics import purity
 from qibo.quantum_info.random_ensembles import random_density_matrix, random_statevector
@@ -130,3 +131,17 @@ def test_matrix_power(backend, power):
             float(backend.np.real(backend.np.trace(power))),
             purity(state, backend=backend),
         )
+
+
+def test_schmidt_decomposition(backend):
+    state_A = random_statevector(4, seed=10, backend=backend)
+    state_B = random_statevector(4, seed=11, backend=backend)
+    state = backend.np.kron(state_A, state_B)
+
+    U, S, Vh = schmidt_decomposition(state, [0, 1], backend=backend)
+
+    coeffs = backend.np.abs(S) ** 2
+    entropy = backend.np.where(np.abs(S) < 1e-10, 0.0, backend.np.log(coeffs))
+    entropy = -backend.np.sum(coeffs * entropy)
+
+    backend.assert_allclose(entropy, 0.0)

From 3469c9c540a6f4341e8413b0fa1b732b36b0ed10 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Mon, 7 Oct 2024 09:15:09 +0400
Subject: [PATCH 04/16] point to svd

---
 src/qibo/quantum_info/linalg_operations.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index 488fecfc0e..2435b6d4d2 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -232,6 +232,6 @@ def schmidt_decomposition(
     tensor = backend.np.transpose(tensor, partition + partition_2)
     tensor = backend.np.reshape(tensor, 2 ** len(partition), -1)
 
-    U, S, Vh = np.linalg.svd(tensor)
+    U, S, Vh = singular_value_decomposition(tensor, backend=backend)
 
     return U, S, Vh

From 248b9089b24461565aadbe0e0fedf96c99c69642 Mon Sep 17 00:00:00 2001
From: "pre-commit-ci[bot]"
 <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Date: Mon, 7 Oct 2024 05:25:25 +0000
Subject: [PATCH 05/16] [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci
---
 src/qibo/quantum_info/linalg_operations.py | 2 +-
 tests/test_quantum_info_operations.py      | 3 ++-
 2 files changed, 3 insertions(+), 2 deletions(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index 0b83d2df78..c8ac5c65a9 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -241,7 +241,7 @@ def singular_value_decomposition(matrix, backend=None):
 
     return backend.calculate_singular_value_decomposition(matrix)
 
-  
+
   def schmidt_decomposition(
 
     state, partition: Union[List[int], Tuple[int, ...]], backend=None
diff --git a/tests/test_quantum_info_operations.py b/tests/test_quantum_info_operations.py
index fcf49153b5..6a80965150 100644
--- a/tests/test_quantum_info_operations.py
+++ b/tests/test_quantum_info_operations.py
@@ -164,6 +164,7 @@ def test_singular_value_decomposition(backend):
 
     backend.assert_allclose(S_sorted, coeffs_sorted)
 
+
 def test_schmidt_decomposition(backend):
     state_A = random_statevector(4, seed=10, backend=backend)
     state_B = random_statevector(4, seed=11, backend=backend)
@@ -175,4 +176,4 @@ def test_schmidt_decomposition(backend):
     entropy = backend.np.where(np.abs(S) < 1e-10, 0.0, backend.np.log(coeffs))
     entropy = -backend.np.sum(coeffs * entropy)
 
-    backend.assert_allclose(entropy, 0.0)
\ No newline at end of file
+    backend.assert_allclose(entropy, 0.0)

From 0248926d4836559511e599a2ab373d1e4f5f07a6 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Mon, 7 Oct 2024 09:36:17 +0400
Subject: [PATCH 06/16] stash

---
 src/qibo/quantum_info/linalg_operations.py | 21 ++++++++++++++-------
 1 file changed, 14 insertions(+), 7 deletions(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index c8ac5c65a9..0b4961bc0a 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -242,19 +242,28 @@ def singular_value_decomposition(matrix, backend=None):
     return backend.calculate_singular_value_decomposition(matrix)
 
 
-  def schmidt_decomposition(
-
+def schmidt_decomposition(
     state, partition: Union[List[int], Tuple[int, ...]], backend=None
-
 ):
+    """Return the Schmidt decomposition of a :math:`n`-qubit bipartite pure quantum ``state``.
+
+    .. math::
+        \\ket{\\psi} = \\sum_{k} \\, c_{k} \\, \\ket{\\phi_{k}}\\!\\ket{\\psi_{k}}
+
+    Args:
+        state (_type_): _description_
+        partition (Union[List[int], Tuple[int, ...]]): _description_
+        backend (_type_, optional): _description_. Defaults to None.
+
+    Returns:
+        _type_: _description_
+    """
     backend = _check_backend(backend)
 
     nqubits = backend.np.log2(state.shape[-1])
     if not nqubits.is_integer():
         raise_error(ValueError, f"dimensions of ``state`` must be a power of 2.")
 
-
-
     nqubits = int(nqubits)
     partition_2 = set(list(range(nqubits))) ^ set(partition)
 
@@ -262,6 +271,4 @@ def schmidt_decomposition(
     tensor = backend.np.transpose(tensor, partition + partition_2)
     tensor = backend.np.reshape(tensor, 2 ** len(partition), -1)
 
-
-
     return singular_value_decomposition(tensor, backend=backend)

From 8be4f15b985ffa50af49bfa3bfaad45be67fe403 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Mon, 7 Oct 2024 09:36:22 +0400
Subject: [PATCH 07/16] api ref

---
 doc/source/api-reference/qibo.rst | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/doc/source/api-reference/qibo.rst b/doc/source/api-reference/qibo.rst
index e21f0cc19f..93cd0e66ec 100644
--- a/doc/source/api-reference/qibo.rst
+++ b/doc/source/api-reference/qibo.rst
@@ -1975,6 +1975,12 @@ Matrix power
 .. autofunction:: qibo.quantum_info.matrix_power
 
 
+Schmidt decomposition
+"""""""""""""""""""""
+
+.. autofunction:: qibo.quantum_info.schmidt_decomposition
+
+
 Quantum Networks
 ^^^^^^^^^^^^^^^^
 

From c924ad0adad2e8433b8df3f30659c12dee694afb Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Mon, 7 Oct 2024 09:59:13 +0400
Subject: [PATCH 08/16] docstring

---
 src/qibo/quantum_info/linalg_operations.py | 26 +++++++++++++++++-----
 1 file changed, 21 insertions(+), 5 deletions(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index 0b4961bc0a..0f386d49cf 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -247,16 +247,32 @@ def schmidt_decomposition(
 ):
     """Return the Schmidt decomposition of a :math:`n`-qubit bipartite pure quantum ``state``.
 
+    Given a bipartite pure state :math:`\\ket{\\psi}\\in\\mathcal{H}_{A}\\otimes\\mathcal{H}_{B}`,
+    its Schmidt decomposition is given by
+
     .. math::
-        \\ket{\\psi} = \\sum_{k} \\, c_{k} \\, \\ket{\\phi_{k}}\\!\\ket{\\psi_{k}}
+        \\ket{\\psi} = \\sum_{k = 1}^{\\min\\{a, \\, b\\}} \\, c_{k} \\, 
+            \\ket{\\phi_{k}} \\otimes \\ket{\\nu_{k}} \\, ,
+    
+    with :math:`a` and :math:`b` being the respective cardinalities of :math:`\\mathcal{H}_{A}`
+    and :math:`\\mathcal{H}_{B}`, and :math:`\\{\\phi_{k}\\}_{k\\in[\\min\\{a, \\, b\\}]} 
+    \\subset \\mathcal{H}_{A}` and :math:`\\{\\nu_{k}\\}_{k\\in[\\min\\{a, \\, b\\}]} 
+    \\subset \\mathcal{H}_{B}` being orthonormal sets. The coefficients 
+    :math:`\\{c_{k}\\}_{k\\in[\\min\\{a, \\, b\\}]}` are real, non-negative, and unique
+    up to re-ordering.
+
+    
 
     Args:
-        state (_type_): _description_
-        partition (Union[List[int], Tuple[int, ...]]): _description_
-        backend (_type_, optional): _description_. Defaults to None.
+        state (ndarray): stevector or density matrix.
+        partition (Union[List[int], Tuple[int, ...]]): indices of qubits in one of the two
+            partitions. The other partition is inferred as the remaining qubits.
+        backend (:class:`qibo.backends.abstract.Backend`, optional): backend
+            to be used in the execution. If ``None``, it uses
+            :class:`qibo.backends.GlobalBackend`. Defaults to ``None``.
 
     Returns:
-        _type_: _description_
+        ndarray, ndarray, ndarray: 
     """
     backend = _check_backend(backend)
 

From 9ae4dafd3b875169968d7fc7059ab042dd01ad61 Mon Sep 17 00:00:00 2001
From: "pre-commit-ci[bot]"
 <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Date: Mon, 7 Oct 2024 06:01:17 +0000
Subject: [PATCH 09/16] [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci
---
 src/qibo/quantum_info/linalg_operations.py | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index 0f386d49cf..2c26a2f857 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -251,17 +251,17 @@ def schmidt_decomposition(
     its Schmidt decomposition is given by
 
     .. math::
-        \\ket{\\psi} = \\sum_{k = 1}^{\\min\\{a, \\, b\\}} \\, c_{k} \\, 
+        \\ket{\\psi} = \\sum_{k = 1}^{\\min\\{a, \\, b\\}} \\, c_{k} \\,
             \\ket{\\phi_{k}} \\otimes \\ket{\\nu_{k}} \\, ,
-    
+
     with :math:`a` and :math:`b` being the respective cardinalities of :math:`\\mathcal{H}_{A}`
-    and :math:`\\mathcal{H}_{B}`, and :math:`\\{\\phi_{k}\\}_{k\\in[\\min\\{a, \\, b\\}]} 
-    \\subset \\mathcal{H}_{A}` and :math:`\\{\\nu_{k}\\}_{k\\in[\\min\\{a, \\, b\\}]} 
-    \\subset \\mathcal{H}_{B}` being orthonormal sets. The coefficients 
+    and :math:`\\mathcal{H}_{B}`, and :math:`\\{\\phi_{k}\\}_{k\\in[\\min\\{a, \\, b\\}]}
+    \\subset \\mathcal{H}_{A}` and :math:`\\{\\nu_{k}\\}_{k\\in[\\min\\{a, \\, b\\}]}
+    \\subset \\mathcal{H}_{B}` being orthonormal sets. The coefficients
     :math:`\\{c_{k}\\}_{k\\in[\\min\\{a, \\, b\\}]}` are real, non-negative, and unique
     up to re-ordering.
 
-    
+
 
     Args:
         state (ndarray): stevector or density matrix.
@@ -272,7 +272,7 @@ def schmidt_decomposition(
             :class:`qibo.backends.GlobalBackend`. Defaults to ``None``.
 
     Returns:
-        ndarray, ndarray, ndarray: 
+        ndarray, ndarray, ndarray:
     """
     backend = _check_backend(backend)
 

From b93529c98b7655013d65acd9f56a4bf007732d2b Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Mon, 7 Oct 2024 10:31:36 +0400
Subject: [PATCH 10/16] comprehensive testing

---
 tests/test_quantum_info_operations.py | 16 ++++++++++++++--
 1 file changed, 14 insertions(+), 2 deletions(-)

diff --git a/tests/test_quantum_info_operations.py b/tests/test_quantum_info_operations.py
index 6a80965150..60e32e516b 100644
--- a/tests/test_quantum_info_operations.py
+++ b/tests/test_quantum_info_operations.py
@@ -172,8 +172,20 @@ def test_schmidt_decomposition(backend):
 
     U, S, Vh = schmidt_decomposition(state, [0, 1], backend=backend)
 
+    # recovering original state
+    recovered = np.zeros_like(state.shape, dtype=complex)
+    recovered = backend.cast(recovered, dtype=recovered.dtype)
+    for coeff, u, vh in zip(S, U.T, Vh):
+        if abs(coeff) > 1e-10:
+            recovered = recovered + coeff * backend.np.kron(u, vh)
+
+    print(S)
+
+    backend.assert_allclose(recovered, state)
+
+    # entropy test
     coeffs = backend.np.abs(S) ** 2
-    entropy = backend.np.where(np.abs(S) < 1e-10, 0.0, backend.np.log(coeffs))
+    entropy = backend.np.where(backend.np.abs(S) < 1e-10, 0.0, backend.np.log(coeffs))
     entropy = -backend.np.sum(coeffs * entropy)
 
-    backend.assert_allclose(entropy, 0.0)
+    backend.assert_allclose(entropy, 0.0, atol=1e-14)

From 7c18032f43e7f2ecd337ff1f6199bd99dc18740a Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Mon, 7 Oct 2024 10:37:56 +0400
Subject: [PATCH 11/16] fix bug + docstring

---
 src/qibo/quantum_info/linalg_operations.py | 18 +++++++++++++++---
 1 file changed, 15 insertions(+), 3 deletions(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index 2c26a2f857..5bc1000886 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -261,7 +261,15 @@ def schmidt_decomposition(
     :math:`\\{c_{k}\\}_{k\\in[\\min\\{a, \\, b\\}]}` are real, non-negative, and unique
     up to re-ordering.
 
+    The decomposition is calculated using :func:`qibo.quantum_info.singular_value_decomposition`,
+    resulting in
 
+    .. math::
+        \\ketbra{\\psi}{\\psi} = U \\, S \\, V^{\\dagger} \\, ,
+    
+    where :math:`U` is an :math:`a \\times a` unitary matrix, :math:`V` is an :math:`b \\times b`
+    unitary matrix, and :math:`S` is an :math:`a \\times b` positive semidefinite diagonal matrix
+    that contains the singular values of :math:`\\ketbra{\\psi}{\\psi}`.
 
     Args:
         state (ndarray): stevector or density matrix.
@@ -272,19 +280,23 @@ def schmidt_decomposition(
             :class:`qibo.backends.GlobalBackend`. Defaults to ``None``.
 
     Returns:
-        ndarray, ndarray, ndarray:
+        ndarray, ndarray, ndarray: Respectively, the matrices :math:`U`, :math:`S`, 
+        and :math:`V^{\\dagger}`.
     """
     backend = _check_backend(backend)
 
-    nqubits = backend.np.log2(state.shape[-1])
+    nqubits = math.log2(state.shape[-1])
     if not nqubits.is_integer():
         raise_error(ValueError, f"dimensions of ``state`` must be a power of 2.")
 
     nqubits = int(nqubits)
     partition_2 = set(list(range(nqubits))) ^ set(partition)
+    partition_2 = (
+        list(partition_2) if isinstance(partition, list) else tuple(partition_2)
+    )
 
     tensor = backend.np.reshape(state, [2] * nqubits)
     tensor = backend.np.transpose(tensor, partition + partition_2)
-    tensor = backend.np.reshape(tensor, 2 ** len(partition), -1)
+    tensor = backend.np.reshape(tensor, (2 ** len(partition), -1))
 
     return singular_value_decomposition(tensor, backend=backend)

From 142dbdc36ca7393d45530fc9d47cca189b1aa6c2 Mon Sep 17 00:00:00 2001
From: "pre-commit-ci[bot]"
 <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Date: Mon, 7 Oct 2024 06:38:32 +0000
Subject: [PATCH 12/16] [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci
---
 src/qibo/quantum_info/linalg_operations.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index 5bc1000886..97e6662c47 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -266,7 +266,7 @@ def schmidt_decomposition(
 
     .. math::
         \\ketbra{\\psi}{\\psi} = U \\, S \\, V^{\\dagger} \\, ,
-    
+
     where :math:`U` is an :math:`a \\times a` unitary matrix, :math:`V` is an :math:`b \\times b`
     unitary matrix, and :math:`S` is an :math:`a \\times b` positive semidefinite diagonal matrix
     that contains the singular values of :math:`\\ketbra{\\psi}{\\psi}`.
@@ -280,7 +280,7 @@ def schmidt_decomposition(
             :class:`qibo.backends.GlobalBackend`. Defaults to ``None``.
 
     Returns:
-        ndarray, ndarray, ndarray: Respectively, the matrices :math:`U`, :math:`S`, 
+        ndarray, ndarray, ndarray: Respectively, the matrices :math:`U`, :math:`S`,
         and :math:`V^{\\dagger}`.
     """
     backend = _check_backend(backend)

From d67fb06995281b8dd474108e127562f91321551b Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Mon, 7 Oct 2024 13:34:12 +0400
Subject: [PATCH 13/16] coverage

---
 tests/test_quantum_info_operations.py | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/tests/test_quantum_info_operations.py b/tests/test_quantum_info_operations.py
index 60e32e516b..aeb7220844 100644
--- a/tests/test_quantum_info_operations.py
+++ b/tests/test_quantum_info_operations.py
@@ -166,6 +166,10 @@ def test_singular_value_decomposition(backend):
 
 
 def test_schmidt_decomposition(backend):
+    with pytest.raises(ValueError):
+        test = random_statevector(3, backend=backend)
+        test = schmidt_decomposition(test, backend=backend)
+
     state_A = random_statevector(4, seed=10, backend=backend)
     state_B = random_statevector(4, seed=11, backend=backend)
     state = backend.np.kron(state_A, state_B)
@@ -179,8 +183,6 @@ def test_schmidt_decomposition(backend):
         if abs(coeff) > 1e-10:
             recovered = recovered + coeff * backend.np.kron(u, vh)
 
-    print(S)
-
     backend.assert_allclose(recovered, state)
 
     # entropy test

From 51123a044afc4592835005d4789cbeccc7524c66 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Mon, 7 Oct 2024 13:44:11 +0400
Subject: [PATCH 14/16] replace transpiler function

---
 src/qibo/transpiler/unitary_decompositions.py | 31 +++----------------
 1 file changed, 5 insertions(+), 26 deletions(-)

diff --git a/src/qibo/transpiler/unitary_decompositions.py b/src/qibo/transpiler/unitary_decompositions.py
index ad892db734..2de58f2ab7 100644
--- a/src/qibo/transpiler/unitary_decompositions.py
+++ b/src/qibo/transpiler/unitary_decompositions.py
@@ -3,6 +3,7 @@
 from qibo import gates, matrices
 from qibo.backends import _check_backend
 from qibo.config import PRECISION_TOL, raise_error
+from qibo.quantum_info.linalg_operations import schmidt_decomposition
 
 magic_basis = np.array(
     [[1, -1j, 0, 0], [0, 0, 1, -1j], [0, 0, -1, -1j], [1, 1j, 0, 0]]
@@ -84,30 +85,6 @@ def calculate_psi(unitary, magic_basis=magic_basis, backend=None):
     return psi, eigvals
 
 
-def schmidt_decompose(state, backend=None):
-    """Decomposes a two-qubit product state to its single-qubit parts.
-
-    Args:
-        state (ndarray): product state to be decomposed.
-
-    Returns:
-        (ndarray, ndarray): decomposition
-
-    """
-    backend = _check_backend(backend)
-    # tf.linalg.svd has a different behaviour
-    if backend.__class__.__name__ == "TensorflowBackend":
-        u, d, v = np.linalg.svd(backend.np.reshape(state, (2, 2)))
-    else:
-        u, d, v = backend.np.linalg.svd(backend.np.reshape(state, (2, 2)))
-    if not np.allclose(backend.to_numpy(d), [1, 0]):  # pragma: no cover
-        raise_error(
-            ValueError,
-            f"Unexpected singular values: {d}\nCan only decompose product states.",
-        )
-    return u[:, 0], v[0]
-
-
 def calculate_single_qubit_unitaries(psi, backend=None):
     """Calculates local unitaries that maps a maximally entangled basis to the magic basis.
 
@@ -133,8 +110,10 @@ def calculate_single_qubit_unitaries(psi, backend=None):
     # find e and f by inverting (A3), (A4)
     ef = (psi_bar[0] + 1j * psi_bar[1]) / np.sqrt(2)
     e_f_ = (psi_bar[0] - 1j * psi_bar[1]) / np.sqrt(2)
-    e, f = schmidt_decompose(ef, backend=backend)
-    e_, f_ = schmidt_decompose(e_f_, backend=backend)
+    e, _, f = schmidt_decomposition(ef, [0], backend=backend)
+    e, f = e[:, 0], f[0]
+    e_, _, f_ = schmidt_decomposition(e_f_, [0], backend=backend)
+    e_, f_ = e_[:, 0], f_[0]
     # find exp(1j * delta) using (A5a)
     ef_ = backend.np.kron(e, f_)
     phase = (

From 8474a15ec52eaaec54fdd1b1accd03fb7becd178 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Mon, 7 Oct 2024 14:09:43 +0400
Subject: [PATCH 15/16] fix test

---
 tests/test_quantum_info_operations.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/tests/test_quantum_info_operations.py b/tests/test_quantum_info_operations.py
index aeb7220844..4a9d3c0513 100644
--- a/tests/test_quantum_info_operations.py
+++ b/tests/test_quantum_info_operations.py
@@ -168,7 +168,7 @@ def test_singular_value_decomposition(backend):
 def test_schmidt_decomposition(backend):
     with pytest.raises(ValueError):
         test = random_statevector(3, backend=backend)
-        test = schmidt_decomposition(test, backend=backend)
+        test = schmidt_decomposition(test, [0], backend=backend)
 
     state_A = random_statevector(4, seed=10, backend=backend)
     state_B = random_statevector(4, seed=11, backend=backend)

From dcee04a5f20fd080a140e5046beae78aa42f42b8 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 16 Oct 2024 07:00:17 +0000
Subject: [PATCH 16/16] Update src/qibo/quantum_info/linalg_operations.py

Co-authored-by: BrunoLiegiBastonLiegi <45011234+BrunoLiegiBastonLiegi@users.noreply.github.com>
---
 src/qibo/quantum_info/linalg_operations.py | 5 +----
 1 file changed, 1 insertion(+), 4 deletions(-)

diff --git a/src/qibo/quantum_info/linalg_operations.py b/src/qibo/quantum_info/linalg_operations.py
index ac3674aa3e..395f93b665 100644
--- a/src/qibo/quantum_info/linalg_operations.py
+++ b/src/qibo/quantum_info/linalg_operations.py
@@ -375,10 +375,7 @@ def schmidt_decomposition(
         raise_error(ValueError, f"dimensions of ``state`` must be a power of 2.")
 
     nqubits = int(nqubits)
-    partition_2 = set(list(range(nqubits))) ^ set(partition)
-    partition_2 = (
-        list(partition_2) if isinstance(partition, list) else tuple(partition_2)
-    )
+    partition_2 = partition.__class__(set(list(range(nqubits))) ^ set(partition))
 
     tensor = backend.np.reshape(state, [2] * nqubits)
     tensor = backend.np.transpose(tensor, partition + partition_2)