# 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.
# See the License for the specific language governing permissions and
# pylint: disable=too-many-instance-attributes,too-many-arguments,too-many-branches
from scipy.stats import multinomial

import pennylane as qml
from pennylane import numpy as np

[docs]class ShotAdaptiveOptimizer(GradientDescentOptimizer): r"""Optimizer where the shot rate is adaptively calculated using the variances of the parameter-shift gradient. By keeping a running average of the parameter-shift gradient and the *variance* of the parameter-shift gradient, this optimizer frugally distributes a shot budget across the partial derivatives of each parameter. In addition, if computing the expectation value of a Hamiltonian using :class:~.ExpvalCost, weighted random sampling can be used to further distribute the shot budget across the local terms from which the Hamiltonian is constructed. .. note:: The shot adaptive optimizer only supports single QNodes or :class:~.ExpvalCost objects as objective functions. The bound device must also be instantiated with a finite number of shots. Args: min_shots (int): The minimum number of shots used to estimate the expectations of each term in the Hamiltonian. Note that this must be larger than 2 for the variance of the gradients to be computed. mu (float): The running average constant :math:\mu \in [0, 1]. Used to control how quickly the number of shots recommended for each gradient component changes. b (float): Regularization bias. The bias should be kept small, but non-zero. term_sampling (str): The random sampling algorithm to multinomially distribute the shot budget across terms in the Hamiltonian expectation value. Currently, only "weighted_random_sampling" is supported. Only takes effect if the objective function provided is an instance of :class:~.ExpvalCost. Set this argument to None to turn off random sampling of Hamiltonian terms. stepsize (float): The learning rate :math:\eta. The learning rate *must* be such that :math:\eta < 2/L = 2/\sum_i|c_i|, where: * :math:L \leq \sum_i|c_i| is the bound on the Lipschitz constant <https://en.wikipedia.org/wiki/Lipschitz_continuity>__ of the variational quantum algorithm objective function, and * :math:c_i are the coefficients of the Hamiltonian used in the objective function. **Example** For VQE/VQE-like problems, the objective function for the optimizer can be realized as an :class:~.ExpvalCost object, constructed using a :class:~.Hamiltonian. >>> coeffs = [2, 4, -1, 5, 2] >>> obs = [ ... qml.PauliX(1), ... qml.PauliZ(1), ... qml.PauliX(0) @ qml.PauliX(1), ... qml.PauliY(0) @ qml.PauliY(1), ... qml.PauliZ(0) @ qml.PauliZ(1) ... ] >>> H = qml.Hamiltonian(coeffs, obs) >>> dev = qml.device("default.qubit", wires=2, shots=100) >>> cost = qml.ExpvalCost(qml.templates.StronglyEntanglingLayers, H, dev) Once constructed, the cost function can be passed directly to the optimizer's step method. The attributes opt.shots_used and opt.total_shots_used can be used to track the number of shots per iteration, and across the life of the optimizer, respectively. >>> shape = qml.templates.StronglyEntanglingLayers.shape(n_layers=2, n_wires=2) >>> params = np.random.random(shape) >>> opt = qml.ShotAdaptiveOptimizer(min_shots=10) >>> for i in range(60): ... params = opt.step(cost, params) ... print(f"Step {i}: cost = {cost(params):.2f}, shots_used = {opt.total_shots_used}") Step 0: cost = -5.69, shots_used = 240 Step 1: cost = -2.98, shots_used = 336 Step 2: cost = -4.97, shots_used = 624 Step 3: cost = -5.53, shots_used = 1054 Step 4: cost = -6.50, shots_used = 1798 Step 5: cost = -6.68, shots_used = 2942 Step 6: cost = -6.99, shots_used = 4350 Step 7: cost = -6.97, shots_used = 5814 Step 8: cost = -7.00, shots_used = 7230 Step 9: cost = -6.69, shots_used = 9006 Step 10: cost = -6.85, shots_used = 11286 Step 11: cost = -6.63, shots_used = 14934 Step 12: cost = -6.86, shots_used = 17934 Step 13: cost = -7.19, shots_used = 22950 Step 14: cost = -6.99, shots_used = 28302 Step 15: cost = -7.38, shots_used = 34134 Step 16: cost = -7.66, shots_used = 41022 Step 17: cost = -7.21, shots_used = 48918 Step 18: cost = -7.53, shots_used = 56286 Step 19: cost = -7.46, shots_used = 63822 Step 20: cost = -7.31, shots_used = 72534 Step 21: cost = -7.23, shots_used = 82014 Step 22: cost = -7.31, shots_used = 92838 .. details:: :title: Usage Details The shot adaptive optimizer is based on the iCANS1 optimizer by Kübler et al. (2020) <https://quantum-journal.org/papers/q-2020-05-11-263/>__, and works as follows: 1. The initial step of the optimizer is performed with some specified minimum number of shots, :math:s_{min}, for all partial derivatives. 2. The parameter-shift rule is then used to estimate the gradient :math:g_i with :math:s_i shots for each parameter :math:\theta_i, parameters, as well as the variances :math:v_i of the estimated gradients. 3. Gradient descent is performed for each parameter :math:\theta_i, using the pre-defined learning rate :math:\eta and the gradient information :math:g_i: :math:\theta_i \rightarrow \theta_i - \eta g_i. 4. A maximum shot number is set by maximizing the improvement in the expected gain per shot. For a specific parameter value, the improvement in the expected gain per shot is then calculated via .. math:: \gamma_i = \frac{1}{s_i} \left[ \left(\eta - \frac{1}{2} L\eta^2\right) g_i^2 - \frac{L\eta^2}{2s_i}v_i \right], where: * :math:L \leq \sum_i|c_i| is the bound on the Lipschitz constant <https://en.wikipedia.org/wiki/Lipschitz_continuity>__ of the variational quantum algorithm objective function, * :math:c_i are the coefficients of the Hamiltonian, and * :math:\eta is the learning rate, and *must* be bound such that :math:\eta < 2/L for the above expression to hold. 5. Finally, the new values of :math:s_{i+1} (shots for partial derivative of parameter :math:\theta_i) is given by: .. math:: s_{i+1} = \frac{2L\eta}{2-L\eta}\left(\frac{v_i}{g_i^2}\right)\propto \frac{v_i}{g_i^2}. In addition to the above, to counteract the presence of noise in the system, a running average of :math:g_i and :math:s_i (:math:\chi_i and :math:\xi_i respectively) are used when computing :math:\gamma_i and :math:s_i. For more details, see: * Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles. "Operator Sampling for Shot-frugal Optimization in Variational Algorithms." arXiv:2004.06252 <https://arxiv.org/abs/2004.06252>__ (2020). * Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles. "An Adaptive Optimizer for Measurement-Frugal Variational Algorithms." Quantum 4, 263 <https://quantum-journal.org/papers/q-2020-05-11-263/>__ (2020). """ def __init__( self, min_shots, term_sampling="weighted_random_sampling", mu=0.99, b=1e-6, stepsize=0.07 ): self.term_sampling = term_sampling self.trainable_args = set() # hyperparameters self.min_shots = min_shots self.mu = mu # running average constant self.b = b # regularization bias self.lipschitz = None self.shots_used = 0 """int: number of shots used on the current iteration""" self.total_shots_used = 0 """int: total number of shots used across all iterations""" # total number of iterations self.k = 0 # Number of shots per parameter self.s = None # maximum number of shots required to evaluate across all parameters self.max_shots = None # Running average of the parameter gradients self.chi = None # Running average of the variance of the parameter gradients self.xi = None super().__init__(stepsize=stepsize)
[docs] @staticmethod def weighted_random_sampling(qnodes, coeffs, shots, argnums, *args, **kwargs): """Returns an array of length shots containing single-shot estimates of the Hamiltonian gradient. The shots are distributed randomly over the terms in the Hamiltonian, as per a multinomial distribution. Args: qnodes (Sequence[.QNode]): Sequence of QNodes, each one when evaluated returning the corresponding expectation value of a term in the Hamiltonian. coeffs (Sequence[float]): Sequences of coefficients corresponding to each term in the Hamiltonian. Must be the same length as qnodes. shots (int): The number of shots used to estimate the Hamiltonian expectation value. These shots are distributed over the terms in the Hamiltonian, as per a Multinomial distribution. argnums (Sequence[int]): the QNode argument indices which are trainable *args: Arguments to the QNodes **kwargs: Keyword arguments to the QNodes Returns: array[float]: the single-shot gradients of the Hamiltonian expectation value """ # determine the shot probability per term prob_shots = np.abs(coeffs) / np.sum(np.abs(coeffs)) # construct the multinomial distribution, and sample # from it to determine how many shots to apply per term si = multinomial(n=shots, p=prob_shots) shots_per_term = si.rvs()[0] grads = [] for h, c, p, s in zip(qnodes, coeffs, prob_shots, shots_per_term): # if the number of shots is 0, do nothing if s == 0: continue # set the QNode device shots h.device.shots = [(1, s)] jacs = [] for i in argnums: j = qml.jacobian(h, argnum=i)(*args, **kwargs) if s == 1: j = np.expand_dims(j, 0) # Divide each term by the probability per shot. This is # because we are sampling one at a time. jacs.append(c * j / p) grads.append(jacs) return [np.concatenate(i) for i in zip(*grads)]
[docs] @staticmethod def check_device(dev): r"""Verifies that the device used by the objective function is non-analytic. Args: dev (.Device): the device to verify Raises: ValueError: if the device is analytic """ if dev.analytic: raise ValueError( "The Rosalin optimizer can only be used with devices " "that estimate expectation values with a finite number of shots." )
[docs] def check_learning_rate(self, coeffs): r"""Verifies that the learning rate is less than 2 over the Lipschitz constant, where the Lipschitz constant is given by :math:\sum |c_i| for Hamiltonian coefficients :math:c_i. Args: coeffs (Sequence[float]): the coefficients of the terms in the Hamiltonian Raises: ValueError: if the learning rate is large than :math:2/\sum |c_i| """ self.lipschitz = np.sum(np.abs(coeffs)) if self.stepsize > 2 / self.lipschitz: raise ValueError(f"The learning rate must be less than {2 / self.lipschitz}")
def _single_shot_expval_gradients(self, expval_cost, args, kwargs): """Compute the single shot gradients of an ExpvalCost object""" qnodes = expval_cost.qnodes coeffs = expval_cost.hamiltonian.coeffs device = qnodes[0].device self.check_device(device) original_shots = device.shots if self.lipschitz is None: self.check_learning_rate(coeffs) try: if self.term_sampling == "weighted_random_sampling": grads = self.weighted_random_sampling( qnodes, coeffs, self.max_shots, self.trainable_args, *args, **kwargs ) elif self.term_sampling is None: device.shots = [(1, self.max_shots)] # We iterate over each trainable argument, rather than using # qml.jacobian(expval_cost), to take into account the edge case where # different arguments have different shapes and cannot be stacked. grads = [ qml.jacobian(expval_cost, argnum=i)(*args, **kwargs) for i in self.trainable_args ] else: raise ValueError( f"Unknown Hamiltonian term sampling method {self.term_sampling}. " "Only term_sampling='weighted_random_sampling' and " "term_sampling=None currently supported." ) finally: device.shots = original_shots return grads def _single_shot_qnode_gradients(self, qnode, args, kwargs): """Compute the single shot gradients of a QNode.""" device = qnode.device self.check_device(qnode.device) original_shots = device.shots if self.lipschitz is None: self.check_learning_rate(1) try: device.shots = [(1, self.max_shots)] grads = [qml.jacobian(qnode, argnum=i)(*args, **kwargs) for i in self.trainable_args] finally: device.shots = original_shots return grads
[docs] def compute_grad( self, objective_fn, args, kwargs ): # pylint: disable=signature-differs,arguments-differ,arguments-renamed r"""Compute gradient of the objective function, as well as the variance of the gradient, at the given point. Args: objective_fn (function): the objective function for optimization args: arguments to the objective function kwargs: keyword arguments to the objective function Returns: tuple[array[float], array[float]]: a tuple of NumPy arrays containing the gradient :math:\nabla f(x^{(t)}) and the variance of the gradient """ if isinstance(objective_fn, qml.ExpvalCost): grads = self._single_shot_expval_gradients(objective_fn, args, kwargs) elif isinstance(objective_fn, qml.QNode) or hasattr(objective_fn, "device"): grads = self._single_shot_qnode_gradients(objective_fn, args, kwargs) else: raise ValueError( "The objective function must either be encoded as a single QNode or " "an ExpvalCost object for the Shot adaptive optimizer. " ) # grads will have dimension [max(self.s), *params.shape] # For each parameter, we want to truncate the number of shots to self.s[idx], # and take the mean and the variance. gradients = [] gradient_variances = [] for i, grad in enumerate(grads): p_ind = np.ndindex(*grad.shape[1:]) g = np.zeros_like(grad[0]) s = np.zeros_like(grad[0]) for idx in p_ind: grad_slice = grad[(slice(0, self.s[i][idx]),) + idx] g[idx] = np.mean(grad_slice) s[idx] = np.var(grad_slice, ddof=1) gradients.append(g) gradient_variances.append(s) return gradients, gradient_variances
[docs] def step(self, objective_fn, *args, **kwargs): """Update trainable arguments with one step of the optimizer. Args: objective_fn (function): the objective function for optimization *args: variable length argument list for objective function **kwargs: variable length of keyword arguments for the objective function Returns: list[array]: The new variable values :math:x^{(t+1)}. If single arg is provided, list[array] is replaced by array. """ self.trainable_args = set() for index, arg in enumerate(args): if getattr(arg, "requires_grad", False): self.trainable_args |= {index} if self.s is None: # Number of shots per parameter self.s = [ np.zeros_like(a, dtype=np.int64) + self.min_shots for i, a in enumerate(args) if i in self.trainable_args ] # keep track of the number of shots run s = np.concatenate([i.flatten() for i in self.s]) self.max_shots = max(s) self.shots_used = int(2 * np.sum(s)) self.total_shots_used += self.shots_used # compute the gradient, as well as the variance in the gradient, # using the number of shots determined by the array s. grads, grad_variances = self.compute_grad(objective_fn, args, kwargs) new_args = self.apply_grad(grads, args) if self.xi is None: self.chi = [np.zeros_like(g, dtype=np.float64) for g in grads] self.xi = [np.zeros_like(g, dtype=np.float64) for g in grads] # running average of the gradient self.chi = [self.mu * c + (1 - self.mu) * g for c, g in zip(self.chi, grads)] # running average of the gradient variance self.xi = [self.mu * x + (1 - self.mu) * v for x, v in zip(self.xi, grad_variances)] for idx, (c, x) in enumerate(zip(self.chi, self.xi)): xi = x / (1 - self.mu ** (self.k + 1)) chi = c / (1 - self.mu ** (self.k + 1)) # determine the new optimum shots distribution for the next # iteration of the optimizer s = np.ceil( (2 * self.lipschitz * self.stepsize * xi) / ((2 - self.lipschitz * self.stepsize) * (chi**2 + self.b * (self.mu**self.k))) ) # apply an upper and lower bound on the new shot distributions, # to avoid the number of shots reducing below min(2, min_shots), # or growing too significantly. gamma = ( (self.stepsize - self.lipschitz * self.stepsize**2 / 2) * chi**2 - xi * self.lipschitz * self.stepsize**2 / (2 * s) ) / s argmax_gamma = np.unravel_index(np.argmax(gamma), gamma.shape) smax = max(s[argmax_gamma], 2) self.s[idx] = np.squeeze(np.int64(np.clip(s, min(2, self.min_shots), smax))) self.k += 1 # unwrap from list if one argument, cleaner return if len(new_args) == 1: return new_args[0] return new_args
[docs] def step_and_cost(self, objective_fn, *args, **kwargs): """Update trainable arguments with one step of the optimizer and return the corresponding objective function value prior to the step. The objective function will be evaluated using the maximum number of shots across all parameters as determined by the optimizer during the optimization step. .. warning:: Unlike other gradient descent optimizers, the objective function will be evaluated **separately** to the gradient computation, and will result in extra device evaluations. Args: objective_fn (function): the objective function for optimization *args : variable length argument list for objective function **kwargs : variable length of keyword arguments for the objective function Returns: tuple[list [array], float]: the new variable values :math:x^{(t+1)} and the objective function output prior to the step. If single arg is provided, list [array] is replaced by array. """ new_args = self.step(objective_fn, *args, **kwargs) if isinstance(objective_fn, qml.ExpvalCost): device = objective_fn.qnodes[0].device elif isinstance(objective_fn, qml.QNode) or hasattr(objective_fn, "device"): device = objective_fn.device original_shots = device.shots try: device.shots = int(self.max_shots) forward = objective_fn(*args, **kwargs) finally: device.shots = original_shots return new_args, forward

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