Evaluating analytic gradients on quantum hardware

Abstract: 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.

• The paper proposes methods for evaluating analytic gradients on quantum hardware to optimize quantum objective functions, extending techniques to continuous-variable circuits.
• Key methodologies for qubit circuits include the parameter shift rule for gates with two eigenvalues and an ancilla-based method for more complex gates.
• For continuous-variable circuits, the study demonstrates parameter shift rules for Gaussian gates, enabling gradient computation under certain conditions.

Insights into Quantum Gradient Evaluation for Optimization Tasks

The paper under consideration addresses the evaluation of analytic gradients on quantum hardware, a critical aspect of optimizing quantum objective functions in hybrid quantum-classical algorithms. The study extends known results for qubit-based quantum computing and introduces methods for continuous-variable (CV) circuits, thereby broadening the utility of such techniques across different quantum computing paradigms.

Summary and Key Points

The principal contribution of this paper lies in proposing methods to calculate the gradients of expectation values in quantum circuits, which are crucial for gradient-based optimization techniques. The authors emphasize the relevance of these methods in several application fields, such as quantum chemistry, drug discovery, and machine learning. The research specifically targets parametric or variational quantum circuits, which feature gates with tunable parameters. The computation of gradients on quantum hardware traditionally faces challenges, particularly when the direct derivative of a quantum gate is not itself realizable as a quantum gate.

The paper presents several key methodologies:

• Parameter Shift Rule for Discrete-Variable (Qubit) Circuits: This technique applies to gates where the generator has two distinct eigenvalues. The authors leverage a strategy whereby the original quantum circuit is run twice with a shift in the gate parameters to compute the gradient component effectively.
• Ancilla-based Method for General Qubit Gates: For cases where the parameter shift rule is not applicable due to a more complex eigenvalue structure of the gate generator, the authors describe an alternative involving a linear combination of unitary operations mediated by an ancilla qubit.
• Parameter Shift Rules for Continuous-Variable Circuits: The authors extend the parameter shift rule paradigm to continuous-variable circuits, focusing on Gaussian gates. Such gates are common in CV quantum computing systems. The study finds that gradients can be computed using parameter shifts, given that subsequent operations remain Gaussian and observables are low-degree polynomials of creation and annihilation operators.

Implications and Future Developments

The work significantly impacts the practical deployment of optimization algorithms on quantum computers by reducing computational overhead and enhancing accuracy in gradient estimation. The methods proposed allow for direct implementation on current quantum hardware, ensuring compatibility and integration with existing quantum computing architectures.

From a theoretical standpoint, this paper contributes to a better understanding of the derivative properties of quantum operations, which may pave the way for further innovations in quantum algorithm design. Practically, these insights facilitate more efficient hybrid quantum-classical algorithms, potentially leading to advancements in solving complex optimization problems that are currently infeasible using classical computation techniques alone.

Looking forward, the challenge remains to extend these derivative computation techniques to more complex quantum operations, including non-Gaussian elements in CV quantum computing, or operations characterized by complex or unbounded eigenvalue spectra. Explorations in this direction might involve developing more sophisticated mathematical frameworks or novel quantum operations.

In conclusion, this research offers a valuable foundation for implementing gradient-based optimization strategies in quantum computing. The examination of both qubit-based and continuous-variable architectures enriches the adaptability and robustness of the techniques, marking a step forward in the effective utilization of quantum hardware for complex computational tasks.