EPSR: Extended Parameter Shift Rule

• EPSR is a family of analytic and algorithmic techniques that generalizes the PSR to compute gradients and higher derivatives for complex quantum circuits.
• It leverages polynomial expansion, Fourier analysis, and convex optimization to enable exact derivative extraction without ancillary gates while reducing resource overhead.
• EPSR finds practical applications in variational quantum algorithms, quantum control, and photonic platforms, offering robust and scalable gradient evaluation.

The Extended Parameter Shift Rule (EPSR) is a family of analytic and algorithmic techniques for evaluating gradients and higher derivatives of quantum circuit cost functions—such as expectation values or probabilities—with respect to circuit parameters. EPSR extends the original parameter-shift rule (PSR), which is exact for gates generated by Hermitian operators with only two distinct eigenvalues, to more general gates, arbitrary generator spectra, multi-parameter evolutions, and even black-box or photonic settings. These extensions are central to scalable variational quantum algorithms, quantum control, hybrid quantum-classical machine learning, black-box optimization, and photonic platforms. EPSR allows for analytic derivative extraction, robust resource scaling, and implementation without ancillary operations or controlled gates, and can be formulated for optimal efficiency and minimal noise amplification.

1. Foundational Principles and the Standard Parameter-Shift Rule

The classic PSR applies to a parameterized quantum gate of the form $U_G(\theta) = \exp(-i a\,\theta\,G)$, where $G$ is a Hermitian generator with two distinct eigenvalues $(e_0, e_1)$. In this case, an expectation function

$$f(\theta) = \langle \psi | U_G^\dagger(\theta) A U_G(\theta) | \psi \rangle$$

has derivative

$$\frac{d}{d\theta} f(\theta) = r \left[ f\left( \theta + \frac{\pi}{4r} \right) - f\left( \theta - \frac{\pi}{4r} \right) \right]$$

with $r = \frac{a}{2}(e_1 - e_0)$ (Crooks, 2019). This two-shift formula is exact, uses only forward-evaluated circuits at shifted parameter values, and requires neither ancillary qubits nor controlled unitaries.

In variational quantum algorithms, this property is exploited to compute analytic parameter gradients for optimization, facilitating gradient-based training without the need for finite-difference approximations or extra quantum resources.

2. Algebraic and Spectral Generalizations: EPSR for Multi-Eigenvalue Generators

For general gates where the generator $G$ has more than two distinct eigenvalues, the standard PSR’s two-term analytic expression becomes invalid. EPSR addresses this in two algebraic directions (Izmaylov et al., 2021):

• Polynomial expansion of the exponential: Given $G$ with $L$ eigenvalues $\{\lambda_k\}$, write

$$e^{i\theta G} = \sum_{n=0}^{L-1} a_n(\theta) (iG)^n,$$

where $a_n(\theta)$ is determined by solving $$\sum_{n=0}^{L-1} (i\lambda_k)^n a_n(\theta) = e^{i\theta\lambda_k}$$ for each $k$.

The cost function, a finite trigonometric series in $\theta$, can be differentiated exactly by summing over $L^2$ shifted evaluations, with precise coefficients. For symmetric spectra (e.g., $\{-1,0,1\}$), the shift points and weights simplify, reducing the quantum resource requirement.

• Generator decomposition: Decompose $G = \sum_l d_l O_l$, where each $O_l$ has a simple spectrum (e.g., two or three distinct eigenvalues). The compound gate is expressed as a product of exponentials of the $O_l$: $e^{i \theta G} = \prod_l e^{i d_l \theta O_l}$ when the $O_l$ commute. The gradient is then computed as a linear combination of expectation values at various parameter shifts for the $O_l$’s, scaling as $O(\log L)$ in the best cases, with no ancillary qubits required.

This algebraic extension allows efficient and exact derivative extraction even for gates such as transmon, fSim, match-gate, and $S^2$-conserving fermionic operators, where the spectral structure would otherwise render PSR inapplicable or inefficient (Izmaylov et al., 2021).

3. EPSR via Trigonometric Interpolation, Fourier Analysis, and Convex Optimization

EPSR can be rigorously deduced by exploiting the finite Fourier (trigonometric polynomial) structure of expectation values in circuit parameters. For a single-parameter gate,

$$E(x) = a_0 + \sum_{l=1}^R [ a_l \cos(\Omega_l x) + b_l \sin(\Omega_l x) ]$$

with $\Omega_l$ determined by eigenvalue spacings (Wierichs et al., 2021, Theis, 2021). The derivative is reconstructed analytically via Dirichlet kernel interpolation:

$E'(0) = \sum_{\mu=1}^{2R} E(x_\mu) y_\mu$

with $x_\mu$ as equispaced shifts and weights $y_\mu$ given in closed form (e.g., [(Wierichs et al., 2021), Eq. (A6)]). For higher derivatives and multi-parameter gates, EPSR composes this analytic kernel with stochastic parameter-shift rules or shift-based convolution formulas.

The systematic construction and optimization of shift rules is formalized via convex analysis and duality (Theis, 2021):

• One finds shift vectors $A$ and weights $u$ such that, for any $f$ with support in Fourier spectrum $\Xi$,

$$\sum_{a \in A} u_a e^{-2\pi i a \cdot \xi} = (2\pi i \xi)^\alpha, \quad \forall \xi \in \Xi.$$

• Optimal finite-support EPSRs minimize the total error amplification (L1 norm of $u$), leading to primal-dual convex programs, strong duality results, and explicit construction of minimum-variance shift rules.

In more general scenarios, including “proper” shift rules for perturbed Hamiltonians $e^{ixA + B}$, convolution-based EPSRs provide analytic, unbiased derivative estimators; their optimality and tail behavior are determined by the band-limited nature of the expectation (Theis, 2022).

4. Practical Realization: Applications in Quantum and Classical Optimization

EPSR enables analytic and resource-efficient gradient evaluations in a range of quantum-classical optimization settings:

• Hybrid Quantum–Classical Optimization: EPSR is directly relevant for VQE, QAOA, quantum machine learning, and quantum control workflows. It allows for analytic differentiation through arbitrary-depth, multi-parameter, and structured parameterized circuits without the need for tomography, ancillae, or post-selection (Banchi et al., 2020, Wierichs et al., 2021).
• Quantum Machine Learning: EPSR and stochastic generalizations (SPSA, guided-SPSA hybrids) enable efficient and robust gradient computation in high-dimensional parameter spaces, reducing the number of quantum resources, mitigating noise, and improving solution optimality (Hoffmann et al., 2022, Periyasamy et al., 24 Apr 2024).
• Black-Box Classical Optimization: Quantum-inspired PSR/EPSR forms are applicable to classical black-box optimization (Hai, 16 Mar 2025), especially by adapting shift parameters and multi-term rules to the function landscape, substantially reducing query complexity compared to finite-difference schemes.

For generalized device Hamiltonians or noisy hardware, approximate EPSR variants (e.g., aGPSR) balance the cost-accuracy tradeoff by solving reduced-size systems for pseudo-spectral shifts, achieving exponential reductions in circuit evaluations with controlled approximation error (Abramavicius et al., 23 May 2025).

5. EPSR for Photonic and Optical Neural Systems

EPSR has been adapted to photonic quantum processors and hardware optical neural networks, where the device physics naturally presents a Fourier series dependence on phase-shift parameters:

• Photonic Parameter-Shift Rule: The derivative of an expectation value with respect to a phase-shifter acting on Fock states is written as an exact linear combination of measurements at $2n$ shifted phase settings (for $n$ photons), with coefficients given by the inverse discrete Fourier transform of the number operator sequence. This rule is robust to finite sampling, photon distinguishability, and experimental imperfections, and shows favorable scaling (Pappalardo et al., 3 Oct 2024, Hoch et al., 9 Oct 2024, Jiang et al., 13 Jun 2025).
• Unitary Optical Neural Networks (UONNs): The parameter-shift rule leverages the inherent Fourier structure of Mach–Zehnder interferometers, enabling direct hardware-based analytic gradient computation in optical neural networks. This approach bypasses the fundamental difficulties of error back-propagation in analog photonic systems and is extensible to intensity and complex field objectives.

These adaptations guarantee that EPSR can be practically executed on photonic platforms, in situ, with gate resources scaling linearly in the number of photons or circuit width.

6. Resource, Optimality, and Noise Considerations

A central aspect of EPSR is its impact on circuit evaluation resources and noise amplification:

• Resource Scaling: The number of required parameter shift evaluations in basic PSR is two per parameter. In EPSR, this scales with the number of unique eigenvalues (algebraic methods), the order of the Fourier expansion (trigonometric methods), or the number of unique spectral gaps (generalized approaches). Optimal shift rules, engineered for ill-posed or nearly degenerate spectra, use regularization and condition number analysis to ensure numerical stability and minimal variance (Markovich et al., 2023).
• Noise and Statistical Error: The L1 norm of the EPSR coefficients determines the worst-case variance amplification in shot-based quantum measurements. By formulating the shift rule optimization as a convex program (Theis, 2021), one can achieve provably minimum-variance derivatives and systematic trade-offs between measurement cost and statistical error.
• Approximate and Adaptive EPSR: When device constraints or spectral complexity prevent full exact shift rules, reduced aGPSR or hybrid EPSR-SPSA approaches (e.g. guided-SPSA (Periyasamy et al., 24 Apr 2024)) balance circuit evaluations, robustness, and convergence quality by mixing analytic and stochastic estimators and dynamically allocating gradient measurement resources.

7. Limitations, Open Questions, and Future Directions

Despite the broad applicability and efficiency of EPSR, several limitations and open avenues remain:

• Spectral Complexity: The measurement cost of algebraic and trigonometric EPSRs can scale quadratically with the number of generator eigenvalues. For systems with a rapidly growing set of spectral gaps, further heuristic or adaptive gap selection strategies are needed (Abramavicius et al., 23 May 2025).
• Infinite Support and Truncation in Fourier-Based EPSR: Optimal proper shift rules typically have infinite support; practical implementation requires truncation or folding, introducing controllable but nonzero bias (Theis, 2022).
• Optimization of Shift Parameters: Systematic selection and robust optimization of shift points (phases, pseudo-gaps) and corresponding weights are nontrivial for ill-posed, nearly degenerate, or highly clustered spectra. Regularization (e.g., Tikhonov–Arsenin) and convex analytic frameworks are crucial for ensuring robust estimators (Markovich et al., 2023, Theis, 2021).
• Noise and Hardware Constraints: While designed for NISQ applications, further studies are necessary to fully characterize and optimize EPSR schemes under hardware constraints, decoherence, and limited shot budgets, particularly in non-standard platforms (e.g., superconducting, neutral atom, photonic).

Future research is directed toward adaptive, learning-based, or hybrid optimization of EPSR parameters, integration with model-based optimization (Rotosolve, quantum analytic descent), and further theoretical exploration of universality and efficiency limits for variational quantum and classical black-box learning.

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Table: EPSR Algorithmic Classes and Features

EPSR Variant	Generator Spectrum	Resource Scaling	Exactness	Key References
Standard PSR	2 eigenvalues	2 evals/parameter	Exact	(Crooks, 2019)
Algebraic EPSR (poly. exp.)	L eigenvalues	$O(L^2)$ evals	Exact	(Izmaylov et al., 2021)
Generator decomposition EPSR	Decomposable	$O(\log L)$–$O(L)$	Exact	(Izmaylov et al., 2021)
Fourier/Kernal EPSR	Fourier spectrum R	$2R$ evals	Exact	(Wierichs et al., 2021)
Convex-optimized EPSR	General	Min. for constraint	Exact	(Theis, 2021)
Approximate GPSR (aGPSR)	Arbitrary	User-chosen $K$	Approx.	(Abramavicius et al., 23 May 2025)
Photonic/Optical EPSR	Number states/mesh	$2n$ (photons)	Exact	(Pappalardo et al., 3 Oct 2024), etc.
Hybrid EPSR–SPSA	General	Adaptive	Approx.	(Periyasamy et al., 24 Apr 2024)