Parameter Shift Gradient

Algorithm Overview

Property	Detail
Category	Gradient Estimation
Primitives	ParamShiftEstimatorGradient, ParamShiftSamplerGradient
Framework	Qiskit Algorithms (qiskit_algorithms.gradients)
Use Case	Analytically compute parameter gradients of parameterized quantum circuits

The parameter shift rule computes exact analytic gradients of parameterized quantum circuits by evaluating the circuit at two shifted parameter values ($\theta \pm \pi/2$). Unlike numerical methods, it introduces no approximation error and is directly executable on quantum hardware. It is restricted to circuits whose gates belong to a supported set of single-parameter rotation and controlled gates.

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Mathematical Principles

Let $f(\theta)$ be the expectation value or sampling probability of a parameterized circuit with parameter vector $\theta \in \mathbb{R}^n$. For any gate generator with eigenvalues $\pm\frac{1}{2}$, the exact gradient with respect to parameter $\theta_i$ is:

$$\frac{\partial f}{\partial \theta_i} = \frac{f!\left(\theta + \frac{\pi}{2}\hat{e}_i\right) - f!\left(\theta - \frac{\pi}{2}\hat{e}_i\right)}{2}$$

where $\hat{e}_i$ is the $i$-th standard basis vector.

Circuit evaluations per gradient call (for $n$ parameters to differentiate):

Quantity	Value
Shifted evaluations per parameter	2
Total circuit evaluations	$2n$
Shift magnitude	$\pi/2$ (fixed)

The rule is exact — it does not depend on step size and carries no truncation error.

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Core Parameters

ParamShiftEstimatorGradient

Parameter	Type	Required	Default	Description
estimator	BaseEstimatorV2	Yes	—	Estimator primitive for expectation value computation
precision	float | None	No	None	Overrides default primitive precision; uses primitive default if None
transpiler	Transpiler | None	No	None	Optional transpiler with a .run() method
transpiler_options	dict[str, Any] | None	No	None	Keyword arguments forwarded to transpiler.run()

ParamShiftSamplerGradient

Parameter	Type	Required	Default	Description
sampler	BaseSamplerV2	Yes	—	Sampler primitive for probability distribution computation
shots	int | None	No	None	Overrides default primitive shot count; uses primitive default if None
transpiler	Transpiler | None	No	None	Optional transpiler with a .run() method
transpiler_options	dict[str, Any] | None	No	None	Keyword arguments forwarded to transpiler.run()

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Inputs and Outputs

ParamShiftEstimatorGradient

Input (via .run()):

Argument	Type	Description
circuits	Sequence[QuantumCircuit]	Parameterized circuits to differentiate
observables	Sequence[BaseOperator]	Observables whose expectation value gradients are computed
parameter_values	Sequence[Sequence[float]]	Current parameter values for each circuit
parameters	Sequence[Sequence[Parameter]] | None	Subset of parameters to differentiate; None differentiates all

Output: EstimatorGradientResult

Field	Type	Description
gradients	list[np.ndarray]	Gradient arrays, one per circuit; shape (num_params,)
metadata	list[dict]	Per-circuit metadata, includes "parameters" key
precision	float | list[float]	Precision used (resolved from primitive if not explicitly set)

ParamShiftSamplerGradient

Input (via .run()):

Argument	Type	Description
circuits	Sequence[QuantumCircuit]	Parameterized circuits to differentiate
parameter_values	Sequence[Sequence[float]]	Current parameter values for each circuit
parameters	Sequence[Sequence[Parameter] | None] | None	Subset of parameters to differentiate; None differentiates all

Output: SamplerGradientResult

Field	Type	Description
gradients	list[list[dict[int, float]]]	Per-circuit gradient distributions; each element is a list of dicts mapping bitstring int to gradient value
metadata	list[dict]	Per-circuit metadata, includes "parameters" key
shots	int | list[int]	Shot count used (resolved from primitive if not explicitly set)

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Implementation Notes

• Both classes extend BaseEstimatorGradient / BaseSamplerGradient from qiskit_algorithms.gradients.base.
• Circuits are preprocessed via _preprocess(), which decomposes unsupported gates into the SUPPORTED_GATES set before differentiation.
• Shifted parameter values are generated by _make_param_shift_parameter_values() from qiskit_algorithms.gradients.utils.
• All $2n$ shifted circuit evaluations for a single gradient call are batched into one primitive job.
• Gradient is computed as: gradient[i] = (evs[:n//2] - evs[n//2:]) / 2 over the batched result array.
• If a transpiler is provided, circuits are transpiled before execution. For ParamShiftEstimatorGradient, observables are re-laid-out via observable.apply_layout(circuit.layout).
• Gates outside SUPPORTED_GATES are decomposed during preprocessing; circuits using unsupported gates that cannot be decomposed will raise an error.

Supported gates:

x, y, z, h, rx, ry, rz, p, cx, cy, cz, ryy, rxx, rzz, rzx

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Sample Code

Estimator Gradient

from qiskit.circuit import QuantumCircuit, ParameterVector
from qiskit.quantum_info import SparsePauliOp
from qiskit.primitives import StatevectorEstimator
from qiskit_algorithms.gradients import ParamShiftEstimatorGradient

# Build a simple parameterized circuit
theta = ParameterVector("θ", 2)
qc = QuantumCircuit(2)
qc.ry(theta[0], 0)
qc.ry(theta[1], 1)
qc.cx(0, 1)

observable = SparsePauliOp("ZZ")
parameter_values = [0.5, 1.0]

estimator = StatevectorEstimator()
gradient = ParamShiftEstimatorGradient(estimator)

result = gradient.run([qc], [observable], [parameter_values]).result()
print(result.gradients)   # [array([dE/dθ₀, dE/dθ₁])]

Sampler Gradient

from qiskit.circuit import QuantumCircuit, ParameterVector
from qiskit.primitives import StatevectorSampler
from qiskit_algorithms.gradients import ParamShiftSamplerGradient

theta = ParameterVector("θ", 1)
qc = QuantumCircuit(1)
qc.ry(theta[0], 0)
qc.measure_all()

parameter_values = [0.5]

sampler = StatevectorSampler()
gradient = ParamShiftSamplerGradient(sampler)

result = gradient.run([qc], [parameter_values]).result()
print(result.gradients)   # [[{0: dP(0)/dθ₀, 1: dP(1)/dθ₀}]]