Resource-Efficient Hadamard Test Tailored Variational Framework for Nonlinear Dynamics on Quantum Computers

TL;DR

A low-depth implementation of a class of Hadamard test circuits is proposed, complemented by the development of a parameterized quantum ansatz specifically tailored for variational algorithms that exploit the underlying Hadamard test framework, suggesting a reliable circuit architecture for noisy intermediate-scale quantum devices.

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 (NISQ) 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.

Figures (9)

• Figure 1: Low-depth implementation of the Hadamard test circuits. Panel (a) shows the general construction of the Hadamard test circuits used to evaluate matrix elements or state overlaps. Here, the absence (presence) of the $S^{\dagger}$ gate allows the estimation of the real (imaginary) component of the complex amplitude. In panel (b), we illustrate our low-depth construction, achieved by systematically omitting redundant operations from the original quantum circuit.
• Figure 2: Ansatz structure tailored to the Hadamard test construction. Panel (a) shows the ansatz structure with only controlled-$\hat{\tilde{U}}(\lambda_{i})$ operations between the qubits. The ansatz consists of $d$ number of layers where the first layer is highlighted with the beige color. Panel (b) shows one possible controlled-$\hat{\tilde{U}}(\lambda_{i})$ gate, where $\hat{\tilde{U}} \in \{\hat{R}_{x}, \hat{R}_{y}, \hat{R}_{z}\}$. Here, each controlled-$\hat{\tilde{U}}(\lambda_{i})$ gate decomposes into two controlled-NOT gates and two single-qubit rotations. Panel (c) shows an alternate selection for the controlled-$\hat{\overline{U}}(\lambda_{i})$ operation that only consist of one CNOT gate and two single-qubit rotations $\hat{\overline{U}}_{s}(\pm\lambda/2)$. It is worth highlighting that the panel (b) and (c) show different selections of controlled-unitary operations, which are not equivalent to each other.
• Figure 3: Low-depth design of quantum circuits for evaluating various components of the Burgers' equation cost function. Quantum circuits in panels (a-c) measure the cost function constituents ${\rm Re}\{\bra{0} \hat{U}_{t}^{\dagger}\hat{U}^{\lambda_{j_0}}_{t + \tau}\ket{0}\}$, ${\rm Re}\{\bra{0} \hat{U}_{t}^{\dagger}\hat{A}\hat{U}^{\lambda_{j_0}}_{t + \tau}\ket{0}\}$, and ${\rm Re}\{\bra{0} \hat{U}_{t}^{\dagger}\hat{A}{D_{t}^{\dagger}}\hat{U}^{\lambda_{j_0}}_{t + \tau}\ket{0}\}$, respectively. To evaluate ${\rm Re}\{\bra{0} \hat{U}_{t}^{\dagger}\hat{A}^{\dagger}\hat{U}^{\lambda_{j_0}}_{t + \tau}\ket{0}\}$, and ${\rm Re}\{\bra{0} \hat{U}_{t}^{\dagger}\hat{A}^{\dagger}{D_{t}^{\dagger}}\hat{U}^{\lambda_{j_0}}_{t + \tau}\ket{0}\}$, $\hat{A}$ in panel (b) and (c) is inverted to implement $\hat{A}^{\dagger}$. $H$ is the Hadamard gate, $\hat{A}$ is the adder circuit, and $\hat{U}^{\lambda_{j_0}}_{t+\tau}$ is the unitary ansatz operator with $\lambda_{j} = \lambda_{j_{0}}$, while all other variational parameters remain unchanged.
• Figure 4: Nonlinear dynamics of fluid velocity fields governed by the Burgers' equation, with a focus on the analysis of shockwave behavior. Panel (a) illustrates the infidelity $F'(t) = 1 - \vert\braket{\Psi_{\rm classical}(t)|\Psi_{\rm opt}(t)}\vert^{2}$, characterizing evolved states within turbulent regimes with kinematic viscosity $\nu = 10^{-3}$. This metric, $F'(t)$, serves to quantify the discrepancy between states derived from classical simulations and those produced via VQAs using SGEO optimizer. Panel (b), (c), and (d) showcase the fluid velocity field in turbulent regimes for $n = 3$, $n = 4$, and $n = 5$ for $t = 0.7$, $t = 0.9$ and $t = 0.25$, respectively. Here, black color denotes the initial state which is a Gaussian function and purple (blue) color shows the fluid configuration obtained via classical (VQA) simulations.
• Figure 5: Comparison of two-qubit gate count $g_{2}$ and circuit depth for conventional and low-depth Hadamard test QCs for Burgers' dynamics. Panel a (b) and c (d) show the two-qubit gate count and circuit depth for superconducting (trapped-ion) based devices, respectively, where black and purple colors indicate the gate count $g_{2}$ for convectional and low-depth Hadamard test QCs.