# Source code for pennylane.templates.subroutines.hilbert_schmidt

# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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"""
This submodule contains the templates for the Hilbert-Schmidt tests.
"""
# pylint: disable-msg=too-many-arguments
import pennylane as qml
from pennylane.operation import AnyWires, Operation

[docs]class HilbertSchmidt(Operation): r"""Create a Hilbert-Schmidt template that can be used to compute the Hilbert-Schmidt Test (HST). The HST is a useful quantity used when we want to compile an unitary U with an approximate unitary V. The HST is used as a distance between U and V, the result of executing the HST is 0 if and only if V is equal to U (up to a global phase). Therefore we can define a cost by: .. math:: C_{HST} = 1 - \frac{1}{d^2} \left|Tr(V^{\dagger}U)\right|^2, where the quantity :math:\frac{1}{d^2} \left|Tr(V^{\dagger}U)\right|^2 is obtained by executing the Hilbert-Schmidt Test. It is equivalent to taking the outcome probability of the state :math:|0 ... 0\rangle for the following circuit: .. figure:: ../../_static/templates/subroutines/hst.png :align: center :width: 80% :target: javascript:void(0); It defines our decomposition for the Hilbert-Schmidt Test template. Args: params (array): Parameters for the quantum function V. v_function (callable): Quantum function that represents the approximate compiled unitary V. v_wires (int or Iterable[Number, str]]): The wire(s) the approximate compiled unitary act on. u_tape (.QuantumTape): U, the unitary to be compiled as a qml.tape.QuantumTape. Raises: QuantumFunctionError: The argument u_tape must be a QuantumTape. QuantumFunctionError: v_function is not a valid quantum function. QuantumFunctionError: U and V do not have the same number of wires. QuantumFunctionError: The wires v_wires are a subset of V wires. QuantumFunctionError: u_tape and v_tape must act on distinct wires. **Reference** [1] Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T. Sornborger and Patrick J. Coles Quantum-assisted Quantum Compiling. arxiv/1807.00800 <https://arxiv.org/pdf/1807.00800.pdf>_ .. seealso:: :class:~.LocalHilbertSchmidt .. details:: :title: Usage Details Consider that we want to evaluate the Hilbert-Schmidt Test cost between the unitary U and an approximate unitary V. We need to define some functions where it is possible to use the :class:~.HilbertSchmidt template. Here the considered unitary is Hadamard and we try to compute the cost for the approximate unitary RZ. For an angle that is equal to 0 (Identity), we have the maximal cost which is 1. .. code-block:: python with qml.QueuingManager.stop_recording(): with qml.tape.QuantumTape() as u_tape: qml.Hadamard(wires=0) def v_function(params): qml.RZ(params[0], wires=1) dev = qml.device("default.qubit", wires=2) @qml.qnode(dev) def hilbert_test(v_params, v_function, v_wires, u_tape): qml.HilbertSchmidt(v_params, v_function=v_function, v_wires=v_wires, u_tape=u_tape) return qml.probs(u_tape.wires + v_wires) def cost_hst(parameters, v_function, v_wires, u_tape): return (1 - hilbert_test(v_params=parameters, v_function=v_function, v_wires=v_wires, u_tape=u_tape)[0]) Now that the cost function has been defined it can be called for specific parameters: >>> cost_hst([0], v_function = v_function, v_wires = [1], u_tape = u_tape) 1 """ num_wires = AnyWires grad_method = None def __init__(self, *params, v_function, v_wires, u_tape, do_queue=None, id=None): self._num_params = len(params) if not isinstance(u_tape, qml.tape.QuantumScript): raise qml.QuantumFunctionError("The argument u_tape must be a QuantumTape.") u_wires = u_tape.wires self.hyperparameters["u_tape"] = u_tape if not callable(v_function): raise qml.QuantumFunctionError( "The argument v_function must be a callable quantum function." ) self.hyperparameters["v_function"] = v_function v_tape = qml.tape.make_qscript(v_function)(*params) self.hyperparameters["v_tape"] = v_tape self.hyperparameters["v_wires"] = v_tape.wires if len(u_wires) != len(v_wires): raise qml.QuantumFunctionError("U and V must have the same number of wires.") if not qml.wires.Wires(v_wires).contains_wires(v_tape.wires): raise qml.QuantumFunctionError("All wires in v_tape must be in v_wires.") # Intersection of wires if len(qml.wires.Wires.shared_wires([u_tape.wires, v_tape.wires])) != 0: raise qml.QuantumFunctionError("u_tape and v_tape must act on distinct wires.") wires = qml.wires.Wires(u_wires + v_wires) super().__init__(*params, wires=wires, do_queue=do_queue, id=id) @property def num_params(self): return self._num_params
[docs] @staticmethod def compute_decomposition( params, wires, u_tape, v_tape, v_function=None, v_wires=None ): # pylint: disable=arguments-differ,unused-argument r"""Representation of the operator as a product of other operators.""" n_wires = len(u_tape.wires + v_tape.wires) first_range = range(n_wires // 2) second_range = range(n_wires // 2, n_wires) decomp_ops = [qml.Hadamard(wires[i]) for i in first_range] # CNOT first layer decomp_ops.extend( qml.CNOT(wires=[wires[i], wires[j]]) for i, j in zip(first_range, second_range) ) # Unitary U for op_u in u_tape.operations: # The operation has been defined outside of this function, to queue it we call qml.apply. qml.apply(op_u) decomp_ops.append(op_u) # Unitary V conjugate decomp_ops.extend(qml.adjoint(op_v, lazy=False) for op_v in v_tape.operations) # CNOT second layer decomp_ops.extend( qml.CNOT(wires=[wires[i], wires[j]]) for i, j in zip(reversed(first_range), reversed(second_range)) ) # Hadamard second layer decomp_ops.extend(qml.Hadamard(wires[i]) for i in first_range) return decomp_ops
[docs]class LocalHilbertSchmidt(HilbertSchmidt): r"""Create a Local Hilbert-Schmidt template that can be used to compute the Local Hilbert-Schmidt Test (LHST). The result of the LHST is a useful quantity for compiling a unitary U with an approximate unitary V. The LHST is used as a distance between U and V, it is similar to the Hilbert-Schmidt test, but the measurement is made only on one qubit at the end of the circuit. The LHST cost is always smaller than the HST cost and is useful for large unitaries. .. figure:: ../../_static/templates/subroutines/lhst.png :align: center :width: 80% :target: javascript:void(0); Args: params (array): Parameters for the quantum function V. v_function (Callable): Quantum function that represents the approximate compiled unitary V. v_wires (int or Iterable[Number, str]]): the wire(s) the approximate compiled unitary act on. u_tape (.QuantumTape): U, the unitary to be compiled as a qml.tape.QuantumTape. Raises: QuantumFunctionError: The argument u_tape must be a QuantumTape QuantumFunctionError: v_function is not a valid Quantum function. QuantumFunctionError: U and V do not have the same number of wires. QuantumFunctionError: The wires v_wires are a subset of V wires. QuantumFunctionError: u_tape and v_tape must act on distinct wires. **Reference** [1] Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T. Sornborger and Patrick J. Coles Quantum-assisted Quantum Compiling. arxiv/1807.00800 <https://arxiv.org/pdf/1807.00800.pdf>_ .. seealso:: :class:~.HilbertSchmidt .. details:: :title: Usage Details Consider that we want to evaluate the Local Hilbert-Schmidt Test cost between the unitary U and an approximate unitary V. We need to define some functions where it is possible to use the :class:~.LocalHilbertSchmidt template. Here the considered unitary is CZ and we try to compute the cost for the approximate unitary. .. code-block:: python import numpy as np with qml.QueuingManager.stop_recording(): with qml.tape.QuantumTape() as u_tape: qml.CZ(wires=[0,1]) def v_function(params): qml.RZ(params[0], wires=2) qml.RZ(params[1], wires=3) qml.CNOT(wires=[2, 3]) qml.RZ(params[2], wires=3) qml.CNOT(wires=[2, 3]) dev = qml.device("default.qubit", wires=4) @qml.qnode(dev) def local_hilbert_test(v_params, v_function, v_wires, u_tape): qml.LocalHilbertSchmidt(v_params, v_function=v_function, v_wires=v_wires, u_tape=u_tape) return qml.probs(u_tape.wires + v_wires) def cost_lhst(parameters, v_function, v_wires, u_tape): return (1 - local_hilbert_test(v_params=parameters, v_function=v_function, v_wires=v_wires, u_tape=u_tape)[0]) Now that the cost function has been defined it can be called for specific parameters: >>> cost_lhst([3*np.pi/2, 3*np.pi/2, np.pi/2], v_function = v_function, v_wires = [1], u_tape = u_tape) 0.5 """
[docs] @staticmethod def compute_decomposition( params, wires, u_tape, v_tape, v_function=None, v_wires=None ): # pylint: disable=arguments-differ,unused-argument r"""Representation of the operator as a product of other operators (static method).""" n_wires = len(u_tape.wires + v_tape.wires) first_range = range(n_wires // 2) second_range = range(n_wires // 2, n_wires) decomp_ops = [qml.Hadamard(wires[i]) for i in first_range] # CNOT first layer decomp_ops.extend( qml.CNOT(wires=[wires[i], wires[j]]) for i, j in zip(first_range, second_range) ) # Unitary U for op_u in u_tape.operations: qml.apply(op_u) decomp_ops.append(op_u) # Unitary V conjugate decomp_ops.extend(qml.adjoint(qml.apply, lazy=False)(op_v) for op_v in v_tape.operations) decomp_ops.extend((qml.CNOT(wires=[wires[0], wires[n_wires // 2]]), qml.Hadamard(wires[0]))) return decomp_ops

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