[Paper Review] Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition

Gavin E. Crooks|arXiv (Cornell University)|May 30, 2019

Quantum Computing Algorithms and Architecture1 references93 citations

TL;DR

The paper broadens the parameter-shift rule to a wider class of parameterized quantum gates by decomposing gates into shift-differentiable components, enabling gradient evaluation for multi-qubit gates. It provides concrete decompositions (e.g., CAN via XX/YY/ZZ) and discusses efficient gradient computation on classical hardware.

ABSTRACT

The parameter-shift rule is an approach to measuring gradients of quantum circuits with respect to their parameters, which does not require ancilla qubits or controlled operations. Here, I discuss applying this approach to a wider range of parameterize quantum gates by decomposing gates into a product of standard gates, each of which is parameter-shift rule differentiable.

Motivation & Objective

• Motivate and enable gradient evaluation for variational quantum algorithms without ancilla qubits.
• Derive and apply the parameter-shift rule to gates with two eigenvalues and extend to general gates via decomposition.
• Provide concrete gate decompositions (e.g., CAN into XX/YY/ZZ) to enable gradient computation.
• Demonstrate gradient computation for challenging gates like cross-resonance (CR) and discuss practical considerations for classical simulation.

Proposed method

• Derive the parameter-shift rule for gates with two unique eigenvalues and show how to express UG(θ) in a form that yields the shift-difference of circuit expectations.
• Show how to decompose a general multi-parameter gate into a product of shift-differentiable gates, then apply the product rule to obtain gradients (e.g., CAN(tx, ty, tz) = XX(tx) YY(ty) ZZ(tz)).
• Provide explicit shift constants r (e.g., r = π/2 for Pauli rotation gates and r = π/2 for XX/YY/ZZ components).
• Work through the cross-resonance (CR) gate and derive its gradient via decomposition into simpler, shift-differentiable parts with analytic expressions for parameter derivatives.
• Discuss middle-out gradient computation on classical hardware to reduce memory and provide a practical gradient evaluation strategy.

Experimental results

Research questions

• RQ1 Can the parameter-shift rule be extended to gates whose generators have more than two eigenvalues by decomposing the gate into shift-differentiable components?
• RQ2 What are concrete decomposition strategies for common 2-qubit gates (e.g., CAN, CR) that allow efficient gradient evaluation?”,
• RQ3 Can gradients be computed efficiently on classical hardware using middle-out or back-propagation-style methods without excessive memory usage?

Key findings

• A gate with two unique eigenvalues can be differentiated exactly with a shift rule; gates with more eigenvalues can be handled by decomposing into shift-differentiable components.
• A 2-qubit canonical gate CAN(tx, ty, tz) can be decomposed into XX, YY, and ZZ components, each with r = π/2, enabling gradients for all three parameters.
• Specific decompositions enable gradient evaluation for the cross-resonance (CR) gate, including an XX- and a ZZ-based parametrization with analytic expressions for the parameters.
• The CR gate can be decomposed so that its gradient with respect to the parameters is computable via the product rule, with explicit derivative formulas provided.
• In classical simulations, middle-out gradients can reduce memory usage to about twice the forward circuit storage while increasing time by roughly a factor of six, compared to back-propagation-style storage.