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arXiv:2607.15184

Backpropagating Pauli Propagation

Sheng-Hsuan Lin, Etienne Granet, Kevin Hémery, Henrik Dreyer

quant-phcond-mat.str-elSubmitted Jul 16, 2026• Announced Jul 17, 2026• 0 citations
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Abstract

We develop a backpropagation algorithm for evaluating parameter gradients in quantum circuits using Pauli propagation simulation. The method has computational complexity comparable to that of standard sparse Pauli simulation techniques, while producing gradients whose accuracy is of the same order as the corresponding observable expectation values. By exploiting the reversibility of quantum circuits, the algorithm reduces the memory cost by a factor of $\mathcal{O}(n_\text{param})$ compared with conventional reverse-mode automatic differentiation, where $n_\text{param}$ denotes the number of parameters in the circuit. Compared with finite difference methods, the algorithm is $\mathcal{O}(n_\text{param})$ more efficient in function evaluations. These features enable efficient and accurate classical optimization of quantum circuits for applications such as state preparation and time-evolution compression, while also allowing operator-complexity measures such as the operator stabilizer Rényi entropy to be monitored and regularized during optimization. We demonstrate the method by optimizing low-energy state-preparation circuits for transverse-field Ising models in one, two, and three dimensions and for the three-dimensional Heisenberg model, and by compressing two-dimensional time-evolution circuits.

Comments
14+6 pages, 10 figures

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We develop a backpropagation algorithm for evaluating parameter gradients in quantum circuits using Pauli propagation simulation. The method has computational complexity comparable to that of standard sparse Pauli simulation techniques, while producing gradients whose accuracy is of the same order as the corresponding observable expectation values. By exploiting the reversibility of quantum circuits, the algorithm reduces the memory cost by a factor of O(nparam)\mathcal{O}(n_\text{param})O(nparam​) compared with conventional reverse-mode automatic differentiation, where nparamn_\text{param}nparam​ denotes the number of parameters in the circuit. Compared with finite difference methods, the algorithm is O(nparam)\mathcal{O}(n_\text{param})O(nparam​) more efficient in function evaluations. These features enable efficient and accurate classical optimization of quantum circuits for applications such as state preparation and time-evolution compression, while also allowing operator-complexity measures such as the operator stabilizer Rényi entropy to be monitored and regularized during optimization. We demonstrate the method by optimizing low-energy state-preparation circuits for transverse-field Ising models in one, two, and three dimensions and for the three-dimensional Heisenberg model, and by compressing two-dimensional time-evolution circuits.
Efficient quantum circuit optimization schemes are central to quantum simulation of strongly interacting quantum many body systems. Here, we present an optimization algorithm which combines machine learning techniques and tensor network methods. The said algorithm runs on classical hardware and finds shallow, accurate quantum circuits by minimizing scalar cost functions. The gradients relevant for the optimization process are computed using the reverse mode automatic differentiation technique implemented on top of the time-evolved block decimation algorithm for matrix product states. A variation of the ADAM optimizer is utilized to perform a gradient descent on the manifolds of charge conserving unitary operators to find the optimal quantum circuit. The efficacy of this approach is demonstrated by finding the ground states of spin chain Hamiltonians for the Ising, three-state Potts and the massive Schwinger models for system sizes up to L=100. The first ten excited states of these models are also obtained for system sizes L=24. All circuits achieve high state fidelities within reasonable CPU time and modest memory requirements.
Decoherence of quantum hardware is currently limiting its practical applications. At the same time, classical algorithms for simulating quantum circuits have progressed substantially. Here, we demonstrate a hybrid framework that integrates classical simulations with quantum hardware to improve the computation of an observable’s expectation value by reducing the quantum circuit depth. In this framework, a quantum circuit is partitioned into two subcircuits: one that describes the backpropagated Heisenberg evolution of an observable, executed on a classical computer, while the other is a Schrödinger evolution run on quantum processors. The overall effect is to reduce the depths of the circuits executed on quantum devices and enable the recovery of expectation values at intermediate times throughout the classically backpropagated circuit, trading this with classical overhead and an increased number of circuit executions. We demonstrate the effectiveness of this method on a Hamiltonian simulation problem, achieving more accurate expectation value estimates compared to using quantum hardware alone.
Variational quantum algorithms utilize parameterized circuits many times during training, at new parameter values, over batches of data, and with gradients. Pauli propagation approximates the expectation values of such circuits classically by evolving the observable in the Heisenberg picture as a truncated sum of Pauli strings. We present PADO-Pauli, a GPU engine for Pauli propagation that compiles a circuit and its observables once into sparse matrices in which each rotation angle contributes only a cosine and a sine factor, so every later evaluation reuses the compiled program. It evaluates batches of parameters and data inputs embedded as rotation angles. By default its gradients come from a vector-Jacobian product registered with PyTorch's automatic differentiation, so they flow through the angles to any classical layer that produces them. PADO-Pauli also provides truncation by Pauli weight and coefficient size, and noise channels. Zero-filtering removes terms that do not contribute to an expectation value in the all-zero reference state and makes the batched forward pass about 1.5 times faster. Compared to cuPauliProp on an NVIDIA A100, PADO-Pauli evaluates up to 10 times faster on 4 to 12 qubits and uses up to 3.1 times less memory on 16 qubits. We check its accuracy against exact statevector simulation up to 28 qubits. These features make PADO-Pauli a practical toolkit for training variational quantum algorithms, including quantum machine learning models.
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The discovery of the backpropagation algorithm ranks among one of the most important moments in the history of machine learning, and has made possible the training of large-scale neural networks through its ability to compute gradients at roughly the same computational cost as model evaluation. Despite its importance, a similar backpropagation-like scaling for gradient evaluation of parameterised quantum circuits has remained elusive. Currently, the most popular method requires sampling from a number of circuits that scales with the number of circuit parameters, making training of large-scale quantum circuits prohibitively expensive in practice. Here we address this problem by introducing a class of structured circuits that are not known to be classically simulable and admit gradient estimation with significantly fewer circuits. In the simplest case – for which the parameters feed into commuting quantum gates – these circuits allow for fast estimation of the gradient, higher order partial derivatives and the Fisher information matrix. Moreover, specific families of parameterised circuits exist for which the scaling of gradient estimation is in line with classical backpropagation, and can thus be trained at scale. In a toy classification problem on 16 qubits, such circuits show competitive performance with other methods, while reducing the training cost by about two orders of magnitude.
An efficient implementation of quantum algorithms is often hindered by the lack of efficient primitives for operators and state preparation. This limits both the ability of near-term quantum hardware to simulate complex problems and the potential of fault-tolerant algorithms to achieve practical quantum advantage. To address this, we propose a framework that maps discrete operators onto compact, structured quantum circuits. The resulting variational circuits can be tailored to the connectivity and long-range interaction of the target hardware. The learning process employs backpropagation together with a cost function that efficiently optimizes unitary operators and non-unitary --dense or sparse-- operators using only a single ancilla qubit for block encoding. Additionally, we introduce a regularization term that reduces the approximation error. The approach is validated for both quantum mechanical and engineering applications. In the former case, we learn propagators that arise in native quantum problems --such as quantum simulation and quantum chemistry-- and achieve improved resource scaling in comparison to standard Suzuki--Trotter expansions. In the latter case, we demonstrate the approach's ability to implement the second-order central finite difference approximation of the Laplace operator --relevant for solving partial differential equations-- while improving upon current error metrics. The final example deals with learning a dense, non-unitary operator that arises in the analysis of inviscid potential flow around an airfoil. These results motivate the extension of the framework to a broader class of operator-synthesis problems beyond the prototypical engineering and quantum applications considered here.