Solovay-Kitaev Decomposition Algorithm (#4801)

**Context:** Implements the `Solovay-Kitaev` decomposition algorithm
using the description mentioned in
[arXiv:0505030](https://arxiv.org/abs/quant-ph/0505030)

**Description of the Change:** This adds a Solovay-Kitaev decomposition
for the Clifford+T workflow in the `solovay_kitaev.py` file.

1. `sk_decomposition` technique is the parent method, which takes in any
single-qubit operator and uses its `SU(2)` representation for finding
its decomposition in the specified basis up to a global phase via
recursive implementation contained in `_solovay_kitaev` function.
2. `_approximate_set` method allows us to build an approximate set of
Clifford sequences that are used in the base case return for the above
`_solovay_kitaev` recursive call.
3. `_group_commutator_decompose` allows us to perform the group
commutation decomposition and uses the recipe mentioned in Sec 4.1
[arXiv:0505030](https://arxiv.org/abs/quant-ph/0505030).

**Benefits:** Should allow one to do the Clifford+T decomposition.

**Possible Drawbacks:** Better algorithms exist in the literature,
specifically for decomposing rotation gates.

**Related GitHub Issues:**

---------

Co-authored-by: Matthew Silverman <[email protected]>
Co-authored-by: Tom Bromley <[email protected]>
Co-authored-by: soranjh <[email protected]>

doc/releases/changelog-dev.md

@@ -6,6 +6,10 @@
 
 <h4>Decompose circuits into the Clifford+T gateset 🧩</h4>
 
+* `qml.transforms.decompositions.sk_decomposition` method implements the Solovay-Kitaev algorithm for
+  approximately decomposing any single-qubit operation to Clifford+T basis.
+  [(#4801)](https://github.com/PennyLaneAI/pennylane/pull/4801)
+
 <h4>Transforms (TODO: better title) 🤖</h4>
 
 <h4>Use an iterative approach for quantum phase estimation 🔄</h4>

pennylane/ops/functions/map_wires.py

@@ -100,17 +100,17 @@ def map_wires(
             return new_op
         return input.map_wires(wire_map=wire_map)
     if isinstance(input, (QuantumScript, QNode)) or callable(input):
-        return _map_wires_transform(input, wire_map=wire_map)
+        return _map_wires_transform(input, wire_map=wire_map, queue=queue)
 
     raise ValueError(f"Cannot map wires of object {input} of type {type(input)}.")
 
 
 @partial(transform)
 def _map_wires_transform(
-    tape: qml.tape.QuantumTape, wire_map=None
+    tape: qml.tape.QuantumTape, wire_map=None, queue=False
 ) -> (Sequence[qml.tape.QuantumTape], Callable):
-    ops = [map_wires(op, wire_map) for op in tape.operations]
-    measurements = [map_wires(m, wire_map) for m in tape.measurements]
+    ops = [map_wires(op, wire_map, queue=queue) for op in tape.operations]
+    measurements = [map_wires(m, wire_map, queue=queue) for m in tape.measurements]
 
     out = tape.__class__(ops=ops, measurements=measurements, shots=tape.shots)
     out.trainable_params = tape.trainable_params

pennylane/transforms/__init__.py

@@ -74,6 +74,16 @@
     ~transforms.CommutationDAG
     ~transforms.CommutationDAGNode
 
+Transform for Clifford+T compilation
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The following functions assist in decomposing operations to the Clifford+T basis.
+
+.. autosummary::
+    :toctree: api
+
+    ~transforms.sk_decomposition
+
 Transform for circuit cutting
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
@@ -230,6 +240,7 @@
 from .decompositions import (
     one_qubit_decomposition,
     two_qubit_decomposition,
+    sk_decomposition,
 )
 from .defer_measurements import defer_measurements
 from .sign_expand import sign_expand

pennylane/transforms/decompositions/__init__.py

@@ -17,3 +17,5 @@
 
 from .single_qubit_unitary import one_qubit_decomposition
 from .two_qubit_unitary import two_qubit_decomposition
+
+from .clifford_t import sk_decomposition

pennylane/transforms/decompositions/clifford_t/__init__.py

@@ -0,0 +1,16 @@
+# Copyright 2018-2023 Xanadu Quantum Technologies Inc.
+
+# 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
+
+#     http://www.apache.org/licenses/LICENSE-2.0
+
+# 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
+# limitations under the License.
+"""This module contains methods related to Clifford+T decomposition"""
+
+from .solovay_kitaev import sk_decomposition