Evaluating analytic gradients on quantum hardware

By Maria Schuld, Ville Bergholm, Christian Gogolin, Josh A. Izaac, Nathan Killoran. Physical Review A, 2018

1,140 citations recorded by the source corpus.

An important application for near-term quantum computing lies in optimization tasks, with applications ranging from quantum chemistry and drug discovery to machine learning. In many settings --- most prominently in so-called parametrized or variational algorithms --- the objective function is a result of hybrid quantum-classical processing. To optimize the objective, it is useful to have access to exact gradients of quantum circuits with respect to gate parameters. This paper shows how gradients of expectation values of quantum measurements can be estimated using the same, or almost the same, architecture that executes the original circuit. It generalizes previous results for qubit-based platforms, and proposes recipes for the computation of gradients of continuous-variable circuits. Interestingly, in many important instances it is sufficient to run the original quantum circuit twice while shifting a single gate parameter to obtain the corresponding component of the gradient. More general cases can be solved by conditioning a single gate on an ancilla.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/efb241f6d4209b1202d2b64072ed89dc9e93f73a

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Citations in the corpus

Absence of Barren Plateaus in Quantum Convolutional Neural Networks (2020, background): “Moreover, if the operator Hη has eigenvalues ±1 (or more generally, if it only has two distinct eigenvalues) one can evaluate the cost partial derivative via the parameter-shift rule [53, 54].”
JAX, M.D.: End-to-End Differentiable, Hardware Accelerated, Molecular Dynamics in Pure Python (2019, background): “…finance (Capriotti, 2010), atmospheric modelling (Charpentier & Ghemires, 2000; Carmichael et al., 1997), optimal control (Walther, 2007), physics engines (de Avila Belbute-Peres et al., 2018), protein modelling (Ingraham et al., 2018; AlQuraishi, 2019), and quantum circuits (Schuld et al., 2019).”
Estimating the gradient and higher-order derivatives on quantum hardware (2020, methodology): “[10–13], but extended to the Hessian.”
Quantum convolutional neural network for classical data classification (2021, methodology): “The cost function gradient can be calculated classically or by using a quantum computer via parameter-shift rule (Li et al. 2017; Mitarai et al. 2018; Schuld et al. 2019).”
Quantum variational algorithms are swamped with traps (2022, neutral): “Instead, we focus on the joint distribution of the loss function, gradient norm, and Hessian determinant, where the gradient and Hessian have been normalized by the number of parameters q in the reverse light cone of each term in the Pauli expansion of the problem Hamiltonian; by the parameter shift”
End-to-End Differentiable , Hardware Accelerated , Molecular Dynamics in Pure Python (2019, background): “…finance (Capriotti, 2010), atmospheric modelling (Charpentier & Ghemires, 2000; Carmichael et al., 1997), optimal control (Walther, 2007), physics engines (de Avila Belbute-Peres et al., 2018), protein modelling (Ingraham et al., 2018; AlQuraishi, 2019), and quantum circuits (Schuld et al., 2019).”
Quantum Natural Gradient (2019, methodology): “when optimizers are compared in terms of both wall time and number of required quantum evaluations. These experiments were performed with the open-source quantummachinelearningsoftwarelibraryPennyLane[2,31]. Newfunctionalitywasaddedfor eﬃcientlycomputingtheblock-diagonalg(l) ij and diagonal g ii app”
Diagnosing Barren Plateaus with Tools from Quantum Optimal Control (2021, background): “[33] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran.”
Variational Quantum Linear Solver (2019, neutral): “[59] M.”
Parameterized quantum circuits as machine learning models (2019, methodology): “…j as a function of the derivatives of the expectation values Each of these quantities can be estimated on quantum hardware using the parameter shift rule where subscripts θ ± π 2 e j indicate the shifted parameter vector to use for the evaluation (see Schuld et al. [45] for a detailed derivation).”
Training Variational Quantum Algorithms Is NP-Hard. (2021, methodology): “This is convenient, as with the parameter shift rule [5]- the gradient can be calculated efficiently.”
PennyLane: Automatic differentiation of hybrid quantum-classical computations (2018, methodology): “These rules are summarized and extended in a companion paper [ 42 ] , which provides the theoretical foundation for derivative computations in Pen-nyLane.”
Variational quantum algorithms (2020, methodology): “Surprisingly, there is a hardware-friendly protocol to evaluate the partial derivative of C(θ) with respect to θl often referred to as the parameter-shift rule [63–66].”
Continuous-variable quantum neural networks (2018, methodology): “This method does not require the use of a simulator; instead, gradients of the cost function can be calculated using the same quantum neural network hardware that is being trained [118].”
Variational Quantum Circuits for Deep Reinforcement Learning (2019, background): “The process of optimization needs to calculate gradients of expectation values of quantum measurements, which can be conducted by the same circuit architecture and slightly different parameters, respectively [28].”
General parameter-shift rules for quantum gradients (2021, methodology): “, G has two eigenvalues), the derivative can be computed via the parameter-shift rule [16, 23, 38]”