# Source code for pennylane.transforms.metric_tensor

# you may not use this file except in compliance with the License.
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"""
Contains the metric_tensor batch_transform which wraps multiple
methods of computing the metric tensor.
"""
import functools
import warnings

import pennylane as qml
from pennylane import numpy as np
from pennylane.circuit_graph import LayerData

from .batch_transform import batch_transform

def expand_fn(tape, approx=None, allow_nonunitary=True, aux_wire=None, device_wires=None):
"""Set the metric tensor based on whether non-unitary gates are allowed."""
# pylint: disable=unused-argument,too-many-arguments
if not allow_nonunitary and approx is None:  # pragma: no cover
return qml.transforms.expand_nonunitary_gen(tape)
return qml.transforms.expand_multipar(tape)

[docs]@functools.partial(batch_transform, expand_fn=expand_fn) def metric_tensor(tape, approx=None, allow_nonunitary=True, aux_wire=None, device_wires=None): r"""Returns a function that computes the metric tensor of a given QNode or quantum tape. The metric tensor convention we employ here has the following form: .. math:: \text{metric_tensor}_{i, j} = \text{Re}\left[ \langle \partial_i \psi(\bm{\theta}) | \partial_j \psi(\bm{\theta}) \rangle - \langle \partial_i \psi(\bm{\theta}) | \psi(\bm{\theta}) \rangle \langle \psi(\bm{\theta}) | \partial_j \psi(\bm{\theta}) \rangle \right] with short notation :math:| \partial_j \psi(\bm{\theta}) \rangle := \frac{\partial}{\partial \theta_j}| \psi(\bm{\theta}) \rangle. It is closely related to the quantum fisher information matrix, see :func:~.pennylane.qinfo.transforms.quantum_fisher and eq. (27) in arxiv:2103.15191 <https://arxiv.org/abs/2103.15191>_. .. note:: Only gates that have a single parameter and define a generator are supported. All other parametrized gates will be decomposed if possible. The generator of all parametrized operations, with respect to which the tensor is computed, are assumed to be Hermitian. This is the case for unitary single-parameter operations. Args: tape (pennylane.QNode or .QuantumTape): quantum tape or QNode to find the metric tensor of approx (str): Which approximation of the metric tensor to compute. - If None, the full metric tensor is computed - If "block-diag", the block-diagonal approximation is computed, reducing the number of evaluated circuits significantly. - If "diag", only the diagonal approximation is computed, slightly reducing the classical overhead but not the quantum resources (compared to "block-diag"). allow_nonunitary (bool): Whether non-unitary operations are allowed in circuits created by the transform. Only relevant if approx is None. Should be set to True if possible to reduce cost. aux_wire (int or str or pennylane.wires.Wires): Auxiliary wire to be used for Hadamard tests. If None (the default), a suitable wire is inferred from the (number of) used wires in the original circuit and device_wires. device_wires (.wires.Wires): Wires of the device that is going to be used for the metric tensor. Facilitates finding a default for aux_wire if aux_wire is None. hybrid (bool): Specifies whether classical processing inside a QNode should be taken into account when transforming a QNode. - If True, and classical processing is detected, the Jacobian of the classical processing will be computed and included. When evaluated, the returned metric tensor will be with respect to the QNode arguments. The output shape can vary widely. - If False, any internal QNode classical processing will be **ignored**. When evaluated, the returned metric tensor will be with respect to the **gate** arguments, and not the QNode arguments. The output shape is a single two-dimensional tensor. Returns: func: Function which accepts the same arguments as the QNode. When called, this function will return the metric tensor. The block-diagonal part of the metric tensor always is computed using the covariance-based approach. If no approximation is selected, the off block-diagonal is computed using Hadamard tests. .. warning:: Performing the Hadamard tests requires a device that has an additional wire as compared to the wires on which the original circuit was defined. This wire may be specified via aux_wire. The available wires on the device may be specified via device_wires. By default (that is, if device_wires=None ), contiguous wire numbering and usage is assumed and the additional wire is set to the next wire of the device after the circuit wires. If the given or inferred aux_wire does not exist on the device, a warning is raised and the block-diagonal approximation is computed instead. It is significantly cheaper in this case to explicitly set approx="block-diag" . The flag allow_nonunitary should be set to True whenever the device with which the metric tensor is computed supports non-unitary operations. This will avoid additional decompositions of gates, in turn avoiding a potentially large number of additional Hadamard test circuits to be run. State vector simulators, for example, often allow applying operations that are not unitary. On a real QPU, setting this flag to True may cause exceptions because the computation of the metric tensor will request invalid operations on a quantum device. **Example** Consider the following QNode: .. code-block:: python dev = qml.device("default.qubit", wires=3) @qml.qnode(dev, interface="autograd") def circuit(weights): qml.RX(weights[0], wires=0) qml.RY(weights[1], wires=0) qml.CNOT(wires=[0, 1]) qml.RZ(weights[2], wires=1) qml.RZ(weights[3], wires=0) return qml.expval(qml.PauliZ(0) @ qml.PauliZ(1)), qml.expval(qml.PauliY(1)) We can use the metric_tensor transform to generate a new function that returns the metric tensor of this QNode: >>> mt_fn = qml.metric_tensor(circuit) >>> weights = np.array([0.1, 0.2, 0.4, 0.5], requires_grad=True) >>> mt_fn(weights) tensor([[ 0.25 , 0. , -0.0497, -0.0497], [ 0. , 0.2475, 0.0243, 0.0243], [-0.0497, 0.0243, 0.0123, 0.0123], [-0.0497, 0.0243, 0.0123, 0.0123]], requires_grad=True) In order to save cost, one might want to compute only the block-diagonal part of the metric tensor, which requires significantly fewer executions of quantum functions and does not need an auxiliary wire on the device. This can be done using the approx keyword: >>> mt_fn = qml.metric_tensor(circuit, approx="block-diag") >>> weights = np.array([0.1, 0.2, 0.4, 0.5], requires_grad=True) >>> mt_fn(weights) tensor([[0.25 , 0. , 0. , 0. ], [0. , 0.2475, 0. , 0. ], [0. , 0. , 0.0123, 0.0123], [0. , 0. , 0.0123, 0.0123]], requires_grad=True) These blocks are given by parameter groups that belong to groups of commuting gates. The tensor can be further restricted to the diagonal via approx="diag". However, this will not save further quantum function evolutions but only classical postprocessing. The returned metric tensor is also fully differentiable in all interfaces. For example, we can compute the gradient of the Frobenius norm of the metric tensor with respect to the QNode weights : >>> norm_fn = lambda x: qml.math.linalg.norm(mt_fn(x), ord="fro") >>> grad_fn = qml.grad(norm_fn) >>> grad_fn(weights) array([-0.0282246 , 0.01340413, 0. , 0. ]) .. details:: :title: Usage Details This transform can also be applied to low-level :class:~.QuantumTape objects. This will result in no implicit quantum device evaluation. Instead, the processed tapes, and post-processing function, which together define the metric tensor are directly returned: >>> params = np.array([1.7, 1.0, 0.5], requires_grad=True) >>> with qml.tape.QuantumTape() as tape: ... qml.RX(params[0], wires=0) ... qml.RY(params[1], wires=0) ... qml.CNOT(wires=[0, 1]) ... qml.PhaseShift(params[2], wires=1) ... qml.expval(qml.PauliX(0)) >>> tapes, fn = qml.metric_tensor(tape) >>> tapes [<QuantumTape: wires=[0, 1], params=0>, <QuantumTape: wires=[0, 1], params=1>, <QuantumTape: wires=[0, 1], params=3>, <QuantumTape: wires=[2, 0], params=1>, <QuantumTape: wires=[2, 0, 1], params=2>, <QuantumTape: wires=[2, 0, 1], params=2>] This can be useful if the underlying circuits representing the metric tensor computation need to be analyzed. We clearly can distinguish the first three tapes used for the block-diagonal from the last three tapes that use the auxiliary wire 2 , which was not used by the original tape. The output tapes can then be evaluated and post-processed to retrieve the metric tensor: >>> dev = qml.device("default.qubit", wires=2) >>> fn(qml.execute(tapes, dev, None)) array([[ 0.25 , 0. , 0.42073549], [ 0. , 0.00415023, -0.26517488], [ 0.42073549, -0.26517488, 0.24878844]]) The first term of the off block-diagonal entries of the full metric tensor are computed with Hadamard tests. This first term reads .. math :: \mathfrak{Re}\left\{\langle \partial_i\psi|\partial_j\psi\rangle\right\} and can be computed using an augmented circuit with an additional qubit. See for example the appendix of McArdle et al. (2019) <https://doi.org/10.1038/s41534-019-0187-2>__ for details. The block-diagonal of the tensor is computed using the covariance matrix approach. In addition, we may extract the factors for the second terms :math:\langle \psi|\partial_j\psi\rangle of the *off block-diagonal* tensor from the quantum function output for the covariance matrix! This means that in total only the tapes for the first terms of the off block-diagonal are required in addition to the circuits for the block diagonal. """ if not tape.trainable_params: warnings.warn( "Attempted to compute the metric tensor of a tape with no trainable parameters. " "If this is unintended, please mark trainable parameters in accordance with the " "chosen auto differentiation framework, or via the 'tape.trainable_params' property." ) return [], lambda _: () # pylint: disable=too-many-arguments if approx in {"diag", "block-diag"}: # Only require covariance matrix based transform diag_approx = approx == "diag" return _metric_tensor_cov_matrix(tape, diag_approx)[:2] if approx is None: return _metric_tensor_hadamard(tape, allow_nonunitary, aux_wire, device_wires) raise ValueError( f"Unknown value {approx} for keyword argument approx. " "Valid values are 'diag', 'block-diag' and None." )
def _contract_metric_tensor_with_cjac(mt, cjac): """Execute the contraction of pre-computed classical Jacobian(s) and the metric tensor of a tape in order to obtain the hybrid metric tensor of a QNode. Args: mt (array): Metric tensor of a tape (2-dimensional) cjac (array or tuple[array]): The classical Jacobian of a QNode Returns: array or tuple[array]: Hybrid metric tensor(s) of the QNode. The number of metric tensors depends on the number of QNode arguments for which the classical Jacobian was computed, the tensor shape(s) depend on the shape of these QNode arguments. """ if isinstance(cjac, tuple): # Classical processing of multiple arguments is present. Return cjac.T @ mt @ cjac # as a tuple of contractions. metric_tensors = tuple( qml.math.tensordot(c, qml.math.tensordot(mt, c, axes=[[-1], [0]]), axes=[[0], [0]]) for c in cjac if c is not None ) if len(metric_tensors) == 1: return metric_tensors[0] return metric_tensors is_square = cjac.shape == (1,) or (cjac.ndim == 2 and cjac.shape[0] == cjac.shape[1]) if is_square and qml.math.allclose(cjac, qml.numpy.eye(cjac.shape[0])): # Classical Jacobian is the identity. No classical processing # is present inside the QNode. return mt mt = qml.math.tensordot(cjac, qml.math.tensordot(mt, cjac, axes=[[-1], [0]]), axes=[[0], [0]]) return mt @metric_tensor.custom_qnode_wrapper def qnode_execution_wrapper(self, qnode, targs, tkwargs): """Here, we overwrite the QNode execution wrapper in order to take into account that classical processing may be present inside the QNode.""" hybrid = tkwargs.pop("hybrid", True) if isinstance(qnode, qml.ExpvalCost): if qnode._multiple_devices: # pylint: disable=protected-access warnings.warn( "ExpvalCost was instantiated with multiple devices. Only the first device " "will be used to evaluate the metric tensor.", UserWarning, ) qnode = qnode.qnodes.qnodes[0] tkwargs.setdefault("device_wires", qnode.device.wires) mt_fn = self.default_qnode_wrapper(qnode, targs, tkwargs) _expand_fn = lambda tape: self.expand_fn(tape, *targs, **tkwargs) cjac_fn = qml.transforms.classical_jacobian(qnode, expand_fn=_expand_fn) def wrapper(*args, **kwargs): if not qml.math.get_trainable_indices(args): warnings.warn( "Attempted to compute the metric tensor of a QNode with no trainable parameters. " "If this is unintended, please add trainable parameters in accordance with the " "chosen auto differentiation framework." ) return () try: mt = mt_fn(*args, **kwargs) except qml.wires.WireError as e: if str(e) == "No device wires are unused by the tape.": warnings.warn( "The device does not have a wire that is not used by the tape." "\n\nReverting to the block-diagonal approximation. It will often be " "much more efficient to request the block-diagonal approximation directly!" ) else: warnings.warn( "An auxiliary wire is not available." "\n\nThis can occur when computing the full metric tensor via the " "Hadamard test, and the device does not provide an " "additional wire or the requested auxiliary wire does not exist " "on the device." "\n\nReverting to the block-diagonal approximation. It will often be " "much more efficient to request the block-diagonal approximation directly!" ) tkwargs["approx"] = "block-diag" return self.__call__(qnode, *targs, **tkwargs)(*args, **kwargs) if not hybrid: return mt kwargs.pop("shots", False) cjac = cjac_fn(*args, **kwargs) return _contract_metric_tensor_with_cjac(mt, cjac) return wrapper def _metric_tensor_cov_matrix(tape, diag_approx): r"""This is the metric tensor method for the block diagonal, using the covariance matrix of the generators of each layer. Args: tape (pennylane.QNode or .QuantumTape): quantum tape or QNode to find the metric tensor of diag_approx (bool): if True, use the diagonal approximation. If False , a block-diagonal approximation of the metric tensor is computed. Returns: list[pennylane.tape.QuantumTape]: Transformed tapes that compute the probabilities required for the covariance matrix callable: Post-processing function that computes the covariance matrix from the results of the tapes in the first return value list[list[.Observable]]: Observables measured in each tape, one inner list corresponding to one tape in the first return value list[list[float]]: Coefficients to scale the results for each observable, one inner list corresponding to one tape in the first return value This method assumes the generator of all parametrized operations with respect to which the tensor is computed to be Hermitian. This is the case for unitary single-parameter operations. """ # get the circuit graph graph = tape.graph metric_tensor_tapes = [] obs_list = [] coeffs_list = [] params_list = [] for queue, curr_ops, param_idx, _ in graph.iterate_parametrized_layers(): params_list.append(param_idx) coeffs_list.append([]) obs_list.append([]) # for each operation in the layer, get the generator for op in curr_ops: obs, s = qml.generator(op) obs_list[-1].append(obs) coeffs_list[-1].append(s) # Create a quantum tape with all operations # prior to the parametrized layer, and the rotations # to measure in the basis of the parametrized layer generators. with tape.__class__() as layer_tape: for op in queue: qml.apply(op) for o in obs_list[-1]: o.diagonalizing_gates() qml.probs(wires=tape.wires) metric_tensor_tapes.append(layer_tape) def processing_fn(probs): gs = [] for prob, obs, coeffs in zip(probs, obs_list, coeffs_list): # calculate the covariance matrix of this layer scale = qml.math.convert_like(qml.math.outer(coeffs, coeffs), prob) scale = qml.math.cast_like(scale, prob) g = scale * qml.math.cov_matrix(prob, obs, wires=tape.wires, diag_approx=diag_approx) gs.append(g) # create the block diagonal metric tensor return qml.math.block_diag(gs) return metric_tensor_tapes, processing_fn, obs_list, coeffs_list @functools.lru_cache() def _get_gen_op(op, allow_nonunitary, aux_wire): r"""Get the controlled-generator operation for a given operation. Args: op (pennylane.operation.Operation): Operation from which to extract the generator allow_nonunitary (bool): Whether non-unitary gates are allowed in the circuit aux_wire (int or pennylane.wires.Wires): Auxiliary wire on which to control the operation Returns qml.Operation: Controlled-generator operation of the generator of op, controlled on wire aux_wire. Raises ValueError: If the generator of op is not known or it is non-unitary while allow_nonunitary=False. If allow_nonunitary=True, a general :class:~.pennylane.ControlledQubitUnitary is returned, otherwise only controlled Pauli operations are used. If the operation has a non-unitary generator but allow_nonunitary=False, the operation op should have been decomposed before, leading to a ValueError. """ op_to_cgen = { qml.RX: qml.CNOT, qml.RY: qml.CY, qml.RZ: qml.CZ, qml.PhaseShift: qml.CZ, # PhaseShift is the same as RZ up to a global phase } try: cgen = op_to_cgen[op.__class__] return cgen(wires=[aux_wire, *op.wires]) except KeyError as e: if allow_nonunitary: mat = qml.matrix(qml.generator(op, format="observable")) return qml.ControlledQubitUnitary(mat, control_wires=aux_wire, wires=op.wires) raise ValueError( f"Generator for operation {op} not known and non-unitary operations " "deactivated via allow_nonunitary=False." ) from e def _get_first_term_tapes(tape, layer_i, layer_j, allow_nonunitary, aux_wire): r"""Obtain the tapes for the first term of all tensor entries belonging to an off-diagonal block. Args: tape (pennylane.tape.QuantumTape): Tape that is being transformed layer_i (list): The first layer of parametrized ops, of the format of the layers generated by iterate_parametrized_layers layer_j (list): The second layer of parametrized ops allow_nonunitary (bool): Whether non-unitary operations are allowed in the circuit; passed to _get_gen_op aux_wire (object or pennylane.wires.Wires): Auxiliary wire on which to control the controlled-generator operations Returns: list[pennylane.tape.QuantumTape]: Transformed tapes that compute the first term of the metric tensor for the off-diagonal block belonging to the input layers list[tuple[int]]: 2-tuple indices assigning the tapes to metric tensor entries """ tapes = [] ids = [] # Exclude the backwards cone of layer_i from the backwards cone of layer_j ops_between_cgens = [op for op in layer_j.pre_ops if op not in layer_i.pre_ops] # Iterate over differentiated operation in first layer for diffed_op_i, par_idx_i in zip(layer_i.ops, layer_i.param_inds): gen_op_i = _get_gen_op(diffed_op_i, allow_nonunitary, aux_wire) # Iterate over differentiated operation in second layer # There will be one tape per pair of differentiated operations for diffed_op_j, par_idx_j in zip(layer_j.ops, layer_j.param_inds): gen_op_j = _get_gen_op(diffed_op_j, allow_nonunitary, aux_wire) with tape.__class__() as new_tape: # Initialize auxiliary wire qml.Hadamard(wires=aux_wire) # Apply backward cone of first layer for op in layer_i.pre_ops: qml.apply(op) # Controlled-generator operation of first diff'ed op qml.apply(gen_op_i) # Apply first layer and operations between layers for op in ops_between_cgens: qml.apply(op) # Controlled-generator operation of second diff'ed op qml.apply(gen_op_j) # Measure X on auxiliary wire qml.expval(qml.PauliX(aux_wire)) tapes.append(new_tape) # Memorize to which metric entry this tape belongs ids.append((par_idx_i, par_idx_j)) return tapes, ids def _metric_tensor_hadamard(tape, allow_nonunitary, aux_wire, device_wires): r"""Generate the quantum tapes that execute the Hadamard tests to compute the first term of off block-diagonal metric entries and combine them with the covariance matrix-based block-diagonal tapes. Args: tape (pennylane.QNode or .QuantumTape): quantum tape or QNode to find the metric tensor of allow_nonunitary (bool): Whether non-unitary operations are allowed in circuits created by the transform. Only relevant if approx is None Should be set to True if possible to reduce cost. aux_wire (int or .wires.Wires): Auxiliary wire to be used for Hadamard tests. By default, a suitable wire is inferred from the number of used wires in the original circuit. device_wires (.wires.Wires): Wires of the device that is going to be used for the metric tensor. Facilitates finding a default for aux_wire if aux_wire is None . Returns: list[pennylane.tape.QuantumTape]: Tapes to evaluate the metric tensor callable: processing function to obtain the metric tensor from the tape results """ # Get tapes and processing function for the block-diagonal metric tensor, # as well as the generator observables and generator coefficients for each diff'ed operation diag_tapes, diag_proc_fn, obs_list, coeffs = _metric_tensor_cov_matrix(tape, diag_approx=False) # Obtain layers of parametrized operations and account for the discrepancy between trainable # and non-trainable parameter indices graph = tape.graph par_idx_to_trainable_idx = {idx: i for i, idx in enumerate(sorted(tape.trainable_params))} layers = [ LayerData( layer.pre_ops, layer.ops, tuple(par_idx_to_trainable_idx[idx] for idx in layer[2]), layer.post_ops, ) for layer in graph.iterate_parametrized_layers() ] if len(layers) <= 1: return diag_tapes, diag_proc_fn # Get default for aux_wire aux_wire = _get_aux_wire(aux_wire, tape, device_wires) # Get all tapes for the first term of the metric tensor and memorize which # entry they belong to first_term_tapes = [] ids = [] block_sizes = [] for idx_i, layer_i in enumerate(layers): block_sizes.append(len(layer_i.param_inds)) for layer_j in layers[idx_i + 1 :]: _tapes, _ids = _get_first_term_tapes(tape, layer_i, layer_j, allow_nonunitary, aux_wire) first_term_tapes.extend(_tapes) ids.extend(_ids) # Combine block-diagonal and off block-diagonal tapes tapes = diag_tapes + first_term_tapes # prepare off block-diagonal mask mask = 1 - qml.math.block_diag([qml.math.ones((bsize, bsize)) for bsize in block_sizes]) # Required for slicing in processing_fn num_diag_tapes = len(diag_tapes) def processing_fn(results): """Postprocessing function for the full metric tensor.""" nonlocal mask # Split results diag_res, off_diag_res = results[:num_diag_tapes], results[num_diag_tapes:] # Get full block-diagonal tensor diag_mt = diag_proc_fn(diag_res) # Prepare the mask to match the used interface mask = qml.math.convert_like(mask, diag_mt) # Initialize off block-diagonal tensor using the stored ids first_term = qml.math.zeros_like(diag_mt) if ids: off_diag_res = qml.math.stack(off_diag_res, 1)[0] inv_ids = [_id[::-1] for _id in ids] first_term = qml.math.scatter_element_add(first_term, list(zip(*ids)), off_diag_res) first_term = qml.math.scatter_element_add(first_term, list(zip(*inv_ids)), off_diag_res) # Second terms of off block-diagonal metric tensor expvals = qml.math.zeros_like(first_term[0]) idx = 0 for prob, obs in zip(diag_res, obs_list): for o in obs: l = qml.math.cast(o.eigvals(), dtype=np.float64) w = tape.wires.indices(o.wires) p = qml.math.marginal_prob(prob, w) expvals = qml.math.scatter_element_add(expvals, (idx,), qml.math.dot(l, p)) idx += 1 # Construct <\partial_i\psi|\psi><\psi|\partial_j\psi> and mask it second_term = qml.math.tensordot(expvals, expvals, axes=0) * mask # Subtract second term from first term off_diag_mt = first_term - second_term # Rescale first and second term _coeffs = qml.math.hstack(coeffs) scale = qml.math.convert_like(qml.math.tensordot(_coeffs, _coeffs, axes=0), results[0]) off_diag_mt = scale * off_diag_mt # Combine block-diagonal and off block-diagonal mt = off_diag_mt + diag_mt return mt return tapes, processing_fn def _get_aux_wire(aux_wire, tape, device_wires): r"""Determine an unused wire to be used as auxiliary wire for Hadamard tests. Args: aux_wire (object): Input auxiliary wire. Returned unmodified if not None tape (pennylane.tape.QuantumTape): Tape to infer the wire for device_wires (.wires.Wires): Wires of the device that is going to be used for the metric tensor. Facilitates finding a default for aux_wire if aux_wire is None . Returns: object: The auxiliary wire to be used. Equals aux_wire if it was not None , and an often reasonable choice else. """ if aux_wire is not None: if device_wires is None or aux_wire in device_wires: return aux_wire raise qml.wires.WireError("The requested aux_wire does not exist on the used device.") if device_wires is not None: unused_wires = qml.wires.Wires(device_wires.toset().difference(tape.wires.toset())) if not unused_wires: raise qml.wires.WireError("No device wires are unused by the tape.") return unused_wires[0] _wires = tape.wires for _aux in range(tape.num_wires): if _aux not in _wires: return _aux return tape.num_wires

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