Evaluating analytic gradients on quantum hardware

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.

Figure 1

“Parameter shift rule” in the larger context of hybrid optimization. A quantum node, in which a variational quantum algorithm is executed, can compute derivatives of its outputs with respect to gate parameters by running the original circuit twice, but with a shift in the parameter in question.

Figure 2

Quantum circuit illustrating the “linear combination of unitaries” technique [13]. Between interfering Hadamards, two unitary circuits or gates $A$ and $\mathcal{G}$ are applied conditioned on an ancilla. Depending on the state of the ancilla qubit, the effect is equivalent to applying a sum or difference of $A$ and $\mathcal{G}$.