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arXiv:2608.17249

Low-Depth and Noise-Resilient Quantum State Preparation for Partial Differential Equations via Virtual Rz

Toru Fujii

quant-phSubmitted Aug 18, 2026• Announced Aug 19, 2026• 0 citations
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Abstract

Preparing smooth real-amplitude quantum states is a key subroutine in quantum solvers for dissipative PDEs, such as LCHS, where discretized positive weights must be encoded into amplitudes. Exact state-preparation decompositions can reach unit fidelity ideally, but their two-qubit depth grows quickly and causes severe fidelity loss on NISQ hardware. We propose a low-depth, hardware-aware variational ansatz tailored to smooth, weakly entangled, near-real target distributions typical of damped PDE dynamics. The circuit uses one layer of local Ry rotations to generate real amplitudes, a nearest-neighbor CZ entangling layer to introduce limited entanglement, and additional Rz rotations implemented virtually as frame updates. Virtual Rz operations add no physical pulses and do not increase circuit duration, providing extra degrees of freedom without enlarging the gate footprint; in simulation they are treated as ideal to isolate their benefit. From a tensor-network viewpoint, the alternating structure restricts the state to a low-bond-dimension MPS, matching the target smoothness (for 3 qubits, bond dimension <= 2). We optimize parameters with COBYLA to minimize infidelity and benchmark against exact state preparation (Qiskit) and a RealAmplitudes (CZ) baseline. Under depolarizing noise representative of NISQ and early fault-tolerant regimes, the proposed single-layer circuit achieves high ideal fidelity with O(n) depth and substantially higher noisy fidelity than deeper exact constructions. In coherent-noise sweeps, virtual Rz parameters absorb systematic phase errors and axis mismatch, maintaining near-unity fidelity over a wide error range. These results indicate that virtual-Rz-enabled, low-depth circuits provide a practical, noise-resilient state-preparation primitive for PDE solvers on NISQ and early FTQC hardware.

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Preparing smooth real-amplitude quantum states is a key subroutine in quantum solvers for dissipative PDEs, such as LCHS, where discretized positive weights must be encoded into amplitudes. Exact state-preparation decompositions can reach unit fidelity ideally, but their two-qubit depth grows quickly and causes severe fidelity loss on NISQ hardware. We propose a low-depth, hardware-aware variational ansatz tailored to smooth, weakly entangled, near-real target distributions typical of damped PDE dynamics. The circuit uses one layer of local Ry rotations to generate real amplitudes, a nearest-neighbor CZ entangling layer to introduce limited entanglement, and additional Rz rotations implemented virtually as frame updates. Virtual Rz operations add no physical pulses and do not increase circuit duration, providing extra degrees of freedom without enlarging the gate footprint; in simulation they are treated as ideal to isolate their benefit. From a tensor-network viewpoint, the alternating structure restricts the state to a low-bond-dimension MPS, matching the target smoothness (for 3 qubits, bond dimension <= 2). We optimize parameters with COBYLA to minimize infidelity and benchmark against exact state preparation (Qiskit) and a RealAmplitudes (CZ) baseline. Under depolarizing noise representative of NISQ and early fault-tolerant regimes, the proposed single-layer circuit achieves high ideal fidelity with O(n) depth and substantially higher noisy fidelity than deeper exact constructions. In coherent-noise sweeps, virtual Rz parameters absorb systematic phase errors and axis mismatch, maintaining near-unity fidelity over a wide error range. These results indicate that virtual-Rz-enabled, low-depth circuits provide a practical, noise-resilient state-preparation primitive for PDE solvers on NISQ and early FTQC hardware.
Variational quantum circuits (VQCs) solving partial differential equations (PDEs) on near-term quantum hardware face a critical challenge: hardware noise degrades solution fidelity and disrupts convergence. We present a systematic study of three noise channels; depolarizing, amplitude damping, and bit-flip on VQCs constrained by PDE residual loss functions for the heat equation, Burgers' equation, and the Saint-Venant shallow water equations. We benchmark three error mitigation strategies: zero-noise extrapolation (ZNE) via Richardson polynomial fitting, probabilistic error cancellation (PEC), and measurement error mitigation through inverse confusion matrices. Our numerical experiments on 6-qubit, 4-layer circuits demonstrate that ZNE reduces absolute error by 82-96% at low noise (p = 0.001), with effectiveness degrading gracefully at higher noise strengths. We prove analytically and confirm numerically that physics-constrained circuits exhibit inherent noise resilience: at p = 0.01, constrained circuits maintain 25-47% higher fidelity than unconstrained counterparts, with the advantage scaling with PDE complexity. PEC provides near-exact correction at low gate counts but incurs exponential sampling overhead, rendering it impractical beyond ~60 gates at p >= 0.02. Error budget decomposition reveals that systematic errors dominate at all noise levels (43-58%), while the PDE residual component grows from ~10% to ~31% as noise increases, indicating that physics constraints absorb noise through structured gradient information. These results establish practical guidelines for deploying variational PDE solvers on NISQ hardware.
Abstract Resource-efficient, low-depth implementations of quantum circuits remain a promising strategy for achieving reliable and scalable computation on quantum hardware, as they reduce gate resources and limit the accumulation of noisy operations. Here, we propose a low-depth implementation of a class of Hadamard test circuits, complemented by the development of a parameterized quantum ansatz specifically tailored for variational algorithms that exploit the underlying Hadamard test framework. Our findings demonstrate a significant reduction in single- and two-qubit gate counts, suggesting a reliable circuit architecture for noisy intermediate-scale quantum devices. Building on this foundation, we tested our low-depth scheme to investigate the expressive capacity of the proposed parameterized ansatz in simulating nonlinear Burgers’ dynamics. The resulting variational quantum states faithfully capture the shockwave feature of the turbulent regime and maintain high overlaps with classical benchmarks, underscoring the practical effectiveness of our framework. Furthermore, we evaluate the effect of hardware noise by modeling the error properties of real quantum processors and by executing the variational algorithm on a trapped-ion-based IBEX Q1 device. The outcomes of our demonstrations highlight the resilience of our low-depth scheme in the turbulent regime, consistently preparing high-fidelity variational states that exhibit strong agreement with classical benchmarks. Our work contributes to the advancement of resource-efficient strategies for quantum computation, offering a robust framework for tackling a range of computationally intensive problems across numerous applications.
Preparation and verification of specific quantum states is an important capability for quantum devices to realise advantages over classical computations and algorithms. In this work, we have demonstrated an end-to-end framework that combines resource-efficient quantum state preparation with rapid, robust fidelity verification on near-term quantum hardware. By experimentally preparing and validating a structured complex quantum state encoding a digitized acoustic signal on the Quantinuum H2-1 trapped-ion platform, we achieved a high hardware fidelity of Fhw=0.929F_{\mathrm{hw}} = 0.929Fhw​=0.929. Crucially, this milestone was realized without relying on idealized assumptions or deep fault-tolerant overhead, but rather through resource-minimal circuits optimized for NISQ-era and early fault-tolerant devices. Furthermore, we addressed a key limitation in current quantum state certification. While validation methods like shadow overlap work well for random states, their sample complexity can become prohibitively high for the structured states used in practical algorithms. We mitigate this by introducing a pre-measurement basis-change technique that reduces the verification parameter, τττ, by over 10 orders of magnitude for structured targets. This approach tightens the theoretical certification guarantees of the shadow overlap method and integrates tensor-network preparation and shadow validation into a unified workflow. These results shift the paradigm of how structured classical data can be mapped to and verified on quantum hardware under realistic noise and measurement budgets. By compressing a robust verification procedure to just 1,000 measurement shots, this framework offers an immediate, scalable benchmarking standard.
Quantum state preparation, a crucial subroutine in quantum computing, involves generating a target quantum state from initialized qubits. Arbitrary state preparation algorithms can be broadly categorized into arithmetic decomposition (AD) and variational quantum state preparation (VQSP). AD employs a predefined procedure to decompose the target state into a series of gates, whereas VQSP iteratively tunes ansatz parameters to approximate target state. VQSP is particularly apt for Noisy-Intermediate Scale Quantum (NISQ) machines due to its shorter circuits. However, achieving noise-robust parameter optimization still remains challenging. We present RobustState, a novel VQSP training methodology that combines high robustness with high training efficiency. The core idea involves utilizing measurement outcomes from real machines to perform back-propagation through classical simulators, thus incorporating real quantum noise into gradient calculations. RobustState serves as a versatile, plug-and-play technique applicable for training parameters from scratch or fine-tuning existing parameters to enhance fidelity on target machines. It is adaptable to various ansatzes at both gate and pulse levels and can even benefit other variational algorithms, such as variational unitary synthesis. Comprehensive evaluation of RobustState on state preparation tasks for 4 distinct quantum algorithms using 10 real quantum machines demonstrates a coherent error reduction of up to 7.1 ×\times× and state fidelity improvement of up to 96\% and 81\% for 4-Q and 5-Q states, respectively. On average, RobustState improves fidelity by 50\% and 72\% for 4-Q and 5-Q states compared to baseline approaches.
Distributed quantum processors could scale variational algorithms beyond single devices, but circuit depth, communication overhead, and noise limit their performance. We compare ladder, mixed-canonical, and brick-wall realizations of matrix-product-state (MPS) pretraining with matched per-layer two-qubit-block resources. Despite comparable ideal variational quantum eigensolver performance and gradient scales, the shallower brick-wall architecture reduces circuit duration, idle-time decoherence, and zero-noise-extrapolation (ZNE) overhead, yielding superior noisy and ZNE-assisted performance. We then extend MPS-pretrained circuits to modular processors with one communication qubit per quantum processing unit (QPU); a nearest-neighbor QPU-path schedule keeps the per-layer depth constant as QPUs are added. Distributed circuits whose inter-QPU links realize long-range interactions of the target Hamiltonian match or outperform the single-processor brick-wall under noise when communication idle time is short compared with the coherence time. These results establish hardware-aware co-design of tensor-network pretraining, circuit scheduling, and communication topology as a principle for variational quantum computation on noisy modular hardware.
Gate-based quantum algorithms are typically implemented by circuits consisting of many single-qubit and multi-qubit gates operating on a quantum input state, followed by measurements of the output state. For certain quantum subroutines, such as initial state preparation and quantum Fourier transforms, explicit decompositions of the circuit in terms of single-qubit and two-qubit maximally entangling gates may exist. However, they often lead to large-depth circuits that are challenging for noisy intermediate-scale quantum (NISQ) hardware. Additionally, exact decompositions might only exist for some modular quantum circuits. Therefore, it is essential to find gate combinations that approximate these circuits to high fidelity with potentially low depth. Gradient-based optimization has been used to find such approximate decompositions. Still, these traditional optimizers often run into problems of slow convergence requiring many iterations, and performing poorly in the presence of noise, a factor that is especially relevant for NISQ hardware. Here we present iteratively preconditioned gradient descent (IPG) for optimizing quantum circuits and demonstrate performance speedups for state preparation and implementation of quantum algorithmic subroutines. IPG is a noise-resilient, higher-order algorithm that has shown promising gains in convergence speed for classical optimizations, converging locally at a linear rate for convex problems and superlinearly when the solution is unique. Specifically, we show an improvement in fidelity by a factor of 104for preparing a 4-qubit W state and a maximally entangled 5-qubit GHZ state with compared to other commonly used classical optimizers tuning the same ansatz. Such faster convergence with promise for noise-resilience could provide advantages for quantum algorithms on NISQ hardware, especially since the cost of running each iteration on a quantum computer is substantially higher than the classical optimizer step.