[gradients] Add adjoint differentiation for simulator backends

[masa-whitestone]

Required prerequisites

• Search the issue tracker to check if your feature has already been mentioned or rejected in other issues.

Describe the feature

Describe the feature

I would like to request support for adjoint differentiation as a gradient strategy in CUDA-Q, with an initial focus on simulator backends.

CUDA-Q currently documents the following gradient strategies:

• CentralDifference
• ForwardDifference
• ParameterShift

For simulator-based workflows, especially VQE / QAOA / QML workloads with many parameters, an adjoint / reverse-mode gradient method could substantially reduce gradient-evaluation cost compared with parameter-shift.

Motivation

This would be useful for:

• large parameterized circuits evaluated on simulators
• variational workloads where gradient cost dominates runtime
• training and optimization loops that currently require many shifted circuit evaluations

CUDA-Q already exposes a gradient abstraction in C++, so this seems like a natural extension point for adding another gradient strategy.

Proposed initial scope

To keep the request realistic, I am not asking for hardware-backend support in the first step.

A good initial scope might be:

• simulator backends only
• first-order gradients
• expectation-value objectives (e.g. observe)
• continuous gate parameters

A narrower first implementation (for example, statevector simulation first) would also be perfectly reasonable.

Possible API direction

One possible direction could be:

• C++: a new cudaq::gradients::adjoint (or similar) strategy
• Python: an AdjointGradient option alongside the existing gradient strategies

I am not attached to the exact API shape; this is only a suggestion.

Prior search / why this is not a duplicate

Before opening this issue, I searched the public issue tracker and discussions for:

• adjoint gradient
• adjoint differentiation
• autodiff

I found related issues around broadcasting / vectorization / tensor inputs, but I did not find a dedicated feature request for adjoint differentiation as a CUDA-Q gradient strategy.

Related but not duplicate:

• [RFC] cudaq::observe_n(...), spin_op observe broadcasting #62 discusses batched / broadcasted observe evaluation
• Request: Native Vectorized Batch Evaluation for Parameterized Quantum Circuits (Similar to PennyLane's Parameter Broadcasting) #3664 requests native vectorized batch evaluation for parameterized circuits

This request is instead about adding adjoint differentiation as a gradient strategy for simulator backends.

Why this would help CUDA-Q

CUDA-Q is already strong for simulator-heavy hybrid quantum-classical workflows. Adding adjoint differentiation would make it more attractive for variational optimization and training workloads where simulator performance matters.

Contribution

I would be happy to help discuss scope and contribute an implementation if this direction aligns with the maintainers’ roadmap.

I am happy to start with a simulator-only implementation if that is the preferred first step.