Approximate Quantum Compiler (#6727)

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* Update qiskit/transpiler/synthesis/aqc/approximate.py

* Update qiskit/transpiler/synthesis/aqc/cnot_unit_objective.py

Co-authored-by: Luciano Bello <bel@zurich.ibm.com>

* Update test/python/transpiler/aqc/sample_data.py

Co-authored-by: Luciano Bello <bel@zurich.ibm.com>

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* default configuration, aqc plugin setup

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Co-authored-by: Matthew Treinish <mtreinish@kortar.org>

* added a test where AQC is invoked via the plugin discovery interface

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* Update qiskit/transpiler/synthesis/aqc/aqc.py

Co-authored-by: Luciano Bello <bel@zurich.ibm.com>

* added tests for cnot networks

* added support for plugin configuration

Co-authored-by: Liam Madden <liam.madden2@ibm.com>
Co-authored-by: Luciano Bello <bel@zurich.ibm.com>
Co-authored-by: Matthew Treinish <mtreinish@kortar.org>

qiskit/transpiler/synthesis/aqc/__init__.py

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+# This code is part of Qiskit.
+#
+# (C) Copyright IBM 2021.
+#
+# This code is licensed under the Apache License, Version 2.0. You may
+# obtain a copy of this license in the LICENSE.txt file in the root directory
+# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
+#
+# 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.
+r"""
+Implementation of Approximate Quantum Compiler as described in the paper [1].
+
+We are interested in compiling a quantum circuit, which we formalize as finding the best
+circuit representation in terms of an ordered gate sequence of a target unitary matrix
+:math:`U\in U(d)`, with some additional hardware constraints. In particular, we look at
+representations that could be constrained in terms of hardware connectivity, as well
+as gate depth, and we choose a gate basis in terms of CNOT and rotation gates.
+We recall that the combination of CNOT and rotation gates is universal in :math:`SU(d)` and
+therefore it does not limit compilation.
+
+To properly define what we mean by best circuit representation, we define the metric
+as the Frobenius norm between the unitary matrix of the compiled circuit :math:`V` and
+the target unitary matrix :math:`U`, i.e., :math:`\|V - U\|_{\mathrm{F}}`. This choice
+is motivated by mathematical programming considerations, and it is related to other
+formulations that appear in the literature. Let's take a look at the problem in more details.
+
+Let :math:`n` be the number of qubits and :math:`d=2^n`. Given a CNOT structure :math:`ct`
+and a vector of rotation angles :math:`\theta`, the parametric circuit forms a matrix
+:math:`Vct(\theta)\in SU(d)`. If we are given a target circuit forming a matrix
+:math:`U\in SU(d)`, then we would like to compute
+
+.. math::
+
+    argmax_{\theta}\frac{1}{d}|\langle Vct(\theta),U\rangle|
+
+where the inner product is the Frobenius inner product. Note that
+:math:`|\langle V,U\rangle|\leq d` for all unitaries :math:`U` and :math:`V`, so the objective
+has range in :math:`[0,1]`.
+
+Our strategy is to maximize
+
+.. math::
+
+    \frac{1}{d}\Re \langle Vct(\theta),U\rangle
+
+using its gradient. We will now discuss the specifics by going through an example.
+
+While the range of :math:`Vct` is a subset of :math:`SU(d)` by construction, the target
+circuit may form a general unitary matrix. However, for any :math:`U\in U(d)`,
+
+.. math::
+
+    \frac{\exp(2\pi i k/d)}{\det(U)^{1/d}}U\in SU(d)\text{ for all }k\in\{0,\ldots,d-1\}.
+
+Thus, we should normalize the target circuit by its global phase and then approximately
+compile the normalized circuit. We can add the global phase back in afterwards.
+
+In the algorithm let :math:`U'` denote the un-normalized target matrix and :math:`U`
+the normalized  target matrix. Now that we have :math:`U`, we give the gradient function
+to the Nesterov's method optimizer and compute :math:`\theta`.
+
+To add the global phase back in, we can form the control circuit as
+
+.. math::
+
+    \frac{\langle Vct(\theta),U'\rangle}{|\langle Vct(\theta),U'\rangle|}Vct(\theta).
+
+Note that while we optimized using Nesterov's method in the paper, this was for its convergence
+guarantees, not its speed in practice. It is much faster to use L-BFGS which is used as a
+default optimizer in this implementation.
+
+A basic usage of the AQC algorithm should consist of the following steps::
+
+    # Define a target circuit as a unitary matrix
+    unitary = ...
+
+    # Define a number of qubits for the algorithm, at least 3 qubits
+    num_qubits = int(round(np.log2(unitary.shape[0])))
+
+    # Choose a layout of the CNOT structure for the approximate circuit, e.g. ``spin`` for
+    # a linear layout.
+    layout = options.get("layout") or "spin"
+
+    # Choose a connectivity type, e.g. ``full`` for full connectivity between qubits.
+    connectivity = options.get("connectivity") or "full"
+
+    # Define a targeted depth of the approximate circuit in the number of CNOT units.
+    depth = int(options.get("depth") or 0)
+
+    # Generate a network made of CNOT units
+    cnots = make_cnot_network(
+        num_qubits=num_qubits,
+        network_layout=layout,
+        connectivity_type=connectivity,
+        depth=depth,
+    )
+
+    # Create an optimizer to be used by AQC
+    optimizer = L_BFGS_B()
+
+    # Create an instance
+    aqc = AQC(optimizer)
+
+    # Create a template circuit that will approximate our target circuit
+    approximate_circuit = CNOTUnitCircuit(num_qubits=num_qubits, cnots=cnots)
+
+    # Create an objective that defines our optimization problem
+    approximating_objective = DefaultCNOTUnitObjective(num_qubits=num_qubits, cnots=cnots)
+
+    # Run optimization process to compile the unitary
+    aqc.compile_unitary(
+        target_matrix=unitary,
+        approximate_circuit=approximate_circuit,
+        approximating_objective=approximating_objective,
+    )
+
+Now ``approximate_circuit`` is a circuit that approximates the target unitary to a certain
+degree and can be used instead of the original matrix.
+
+References:
+
+    [1]: Liam Madden, Andrea Simonetto, Best Approximate Quantum Compiling Problems.
+        `arXiv:2106.05649 <https://arxiv.org/abs/2106.05649>`_
+"""
+
+from .approximate import ApproximateCircuit, ApproximatingObjective
+from .aqc import AQC
+from .aqc_plugin import AQCSynthesisPlugin
+from .cnot_structures import make_cnot_network
+from .cnot_unit_circuit import CNOTUnitCircuit
+from .cnot_unit_objective import CNOTUnitObjective, DefaultCNOTUnitObjective

qiskit/transpiler/synthesis/aqc/approximate.py

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+# This code is part of Qiskit.
+#
+# (C) Copyright IBM 2021.
+#
+# This code is licensed under the Apache License, Version 2.0. You may
+# obtain a copy of this license in the LICENSE.txt file in the root directory
+# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
+#
+# 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.
+"""Base classes for an approximate circuit definition."""
+
+from abc import ABC, abstractmethod
+from typing import Optional
+import numpy as np
+
+from qiskit import QuantumCircuit
+
+
+class ApproximateCircuit(QuantumCircuit, ABC):
+    """A base class that represents an approximate circuit."""
+
+    def __init__(self, num_qubits: int, name: Optional[str] = None) -> None:
+        """
+        Args:
+            num_qubits: number of qubit this circuit will span.
+            name: a name of the circuit.
+        """
+        super().__init__(num_qubits, name=name)
+
+    @property
+    @abstractmethod
+    def thetas(self) -> np.ndarray:
+        """
+        The property is not implemented and raises a ``NotImplementedException`` exception.
+
+        Returns:
+            a vector of parameters of this circuit.
+        """
+        raise NotImplementedError
+
+    @abstractmethod
+    def build(self, thetas: np.ndarray) -> None:
+        """
+        Constructs this circuit out of the parameters(thetas). Parameter values must be set before
+            constructing the circuit.
+
+        Args:
+            thetas: a vector of parameters to be set in this circuit.
+        """
+        raise NotImplementedError
+
+
+class ApproximatingObjective(ABC):
+    """
+    A base class for an optimization problem definition. An implementing class must provide at least
+    an implementation of the ``objective`` method. In such case only gradient free optimizers can
+    be used. Both method, ``objective`` and ``gradient``, preferable to have in an implementation.
+    """
+
+    def __init__(self) -> None:
+        # must be set before optimization
+        self._target_matrix = None
+
+    @abstractmethod
+    def objective(self, param_values: np.ndarray) -> float:
+        """
+        Computes a value of the objective function given a vector of parameter values.
+
+        Args:
+            param_values: a vector of parameter values for the optimization problem.
+
+        Returns:
+            a float value of the objective function.
+        """
+        raise NotImplementedError
+
+    @abstractmethod
+    def gradient(self, param_values: np.ndarray) -> np.ndarray:
+        """
+        Computes a gradient with respect to parameters given a vector of parameter values.
+
+        Args:
+            param_values: a vector of parameter values for the optimization problem.
+
+        Returns:
+            an array of gradient values.
+        """
+        raise NotImplementedError
+
+    @property
+    def target_matrix(self) -> np.ndarray:
+        """
+        Returns:
+            a matrix being approximated
+        """
+        return self._target_matrix
+
+    @target_matrix.setter
+    def target_matrix(self, target_matrix: np.ndarray) -> None:
+        """
+        Args:
+            target_matrix: a matrix to approximate in the optimization procedure.
+        """
+        self._target_matrix = target_matrix
+
+    @property
+    @abstractmethod
+    def num_thetas(self) -> int:
+        """
+
+        Returns:
+            the number of parameters in this optimization problem.
+        """
+        raise NotImplementedError

qiskit/transpiler/synthesis/aqc/aqc.py

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+# This code is part of Qiskit.
+#
+# (C) Copyright IBM 2021.
+#
+# This code is licensed under the Apache License, Version 2.0. You may
+# obtain a copy of this license in the LICENSE.txt file in the root directory
+# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
+#
+# 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.
+"""A generic implementation of Approximate Quantum Compiler."""
+from typing import Optional
+
+import numpy as np
+
+from qiskit.algorithms.optimizers import L_BFGS_B, Optimizer
+from qiskit.quantum_info import Operator
+from .approximate import ApproximateCircuit, ApproximatingObjective
+
+
+class AQC:
+    """
+    A generic implementation of Approximate Quantum Compiler. This implementation is agnostic of
+    the underlying implementation of the approximate circuit, objective, and optimizer. Users may
+    pass corresponding implementations of the abstract classes:
+
+        * Optimizer is an instance of :class:`~qiskit.algorithms.optimizer.Optimizer` and
+          used to run the optimization process. A choice of optimizer may affect overall
+          convergence, required time for the optimization process and achieved objective value.
+        * Approximate circuit represents a template which parameters we want to optimize.
+          Currently, there's only one implementation based on 4-rotations CNOT unit blocks:
+          :class:`~qiskit.transpiler.aqc.CNOTUnitCircuit`. See the paper for more details.
+        * Approximate objective is tightly coupled with the approximate circuit implementation
+          and provides two methods for computing objective function and gradient with respect to
+          approximate circuit parameters. This objective is passed to the optimizer. Currently,
+          there's only one implementation based on 4-rotations CNOT unit blocks:
+          :class:`~qiskit.transpiler.aqc.DefaultCNOTUnitObjective`. This is a naive implementation
+          of the objective function and gradient and may suffer from performance issues.
+    """
+
+    def __init__(
+        self,
+        optimizer: Optional[Optimizer] = None,
+        seed: Optional[int] = None,
+    ):
+        """
+        Args:
+            optimizer: an optimizer to be used in the optimization procedure of the search for
+                the best approximate circuit. By default ``L_BFGS_B`` is used with max iterations
+                is set to 1000.
+            seed: a seed value to be user by a random number generator.
+        """
+        super().__init__()
+        self._optimizer = optimizer
+        self._seed = seed
+
+    def compile_unitary(
+        self,
+        target_matrix: np.ndarray,
+        approximate_circuit: ApproximateCircuit,
+        approximating_objective: ApproximatingObjective,
+        initial_point: Optional[np.ndarray] = None,
+    ) -> None:
+        """
+        Approximately compiles a circuit represented as a unitary matrix by solving an optimization
+        problem defined by ``approximating_objective`` and using ``approximate_circuit`` as a
+        template for the approximate circuit.
+
+        Args:
+            target_matrix: a unitary matrix to approximate.
+            approximate_circuit: a template circuit that will be filled with the parameter values
+                obtained in the optimization procedure.
+            approximating_objective: a definition of the optimization problem.
+            initial_point: initial values of angles/parameters to start optimization from.
+        """
+        matrix_dim = target_matrix.shape[0]
+        # check if it is actually a special unitary matrix
+        target_det = np.linalg.det(target_matrix)
+        if not np.isclose(target_det, 1):
+            su_matrix = target_matrix / np.power(target_det, (1 / matrix_dim))
+            global_phase_required = True
+        else:
+            su_matrix = target_matrix
+            global_phase_required = False
+
+        # set the matrix to approximate in the algorithm
+        approximating_objective.target_matrix = su_matrix
+
+        optimizer = self._optimizer or L_BFGS_B(maxiter=1000)
+
+        if initial_point is None:
+            np.random.seed(self._seed)
+            initial_point = np.random.uniform(0, 2 * np.pi, approximating_objective.num_thetas)
+
+        opt_result = optimizer.minimize(
+            fun=approximating_objective.objective,
+            x0=initial_point,
+            jac=approximating_objective.gradient,
+        )
+
+        approximate_circuit.build(opt_result.x)
+
+        approx_matrix = Operator(approximate_circuit).data
+
+        if global_phase_required:
+            alpha = np.angle(np.trace(np.dot(approx_matrix.conj().T, target_matrix)))
+            approximate_circuit.global_phase = alpha