Efficient implementation of quantum signal processing via the adiabatic-impulse model

TL;DR

The paper addresses efficient implementation of Quantum Signal Processing (QSP) by leveraging the Adiabatic-Impulse Model (AIM) for fast, nonadiabatic single-qubit gates. It develops a parameter-mapping strategy that aligns QSP phase sequences with AIM driving parameters, and introduces double Landau-Zener-Stückelberg-Majorana (LZSM) transitions to realize QSP rotations with reduced time variability and higher fidelity. The work further extends the framework to multi-level systems via qubitization and Quantum Singular Value Transformation (QSVT), showing how invariant two-level subspaces and phase-modulated projectors yield scalable polynomial transformations. Overall, the AIM-QSP correspondence provides a practical pathway for implementing QSP and QSVT on quantum hardware, with potential Hamiltonian-simulation applications, while acknowledging hardware-imposed constraints on driving and coherence.

Abstract

Here we investigate analogy between quantum signal processing (QSP) and the adiabatic-impulse model (AIM) in order to implement the QSP algorithm with fast quantum logic gates. QSP is an algorithm that uses single-qubit dynamics to perform a polynomial function transformation. AIM effectively describes the evolution of a two-level quantum system under strong external driving field. We can map parameters from QSP to AIM to implement QSP-like evolution with nonadiabatic, high-amplitude external drives. By choosing AIM parameters that control non-adiabatic transition parameters (such as driving amplitude $A$, frequency $ω$, and signal timing), one can achieve polynomial approximations and increase robustness in quantum circuits. The analogy presented here between QSP and AIM can be useful as a way to directly implement the QSP algorithm on quantum systems and obtain all the benefits from the fast Landau-Zener-Stuckelberg-Majotana (LZSM) quantum logic gates.

Figures (5)

• Figure 1: Occupation probability $P_-$ after the QSP sequence is applied to the qubit with ground state initial conditions. Chebyshev polynomials with Eqs. (\ref{['Cheb2']},\ref{['Cheb3']}) are shown in panel (a). The BB1 sequence, corresponding to the polynomial in Eq. \ref{['BB1Plynom']}, is shown in panel (b) (solid curve). The dots in (a,b) correspond to the probability obtained by performing the respective QSP sequences on an IBM-Q quantum processing unit (QPU). The scheme for the BB1 sequence is shown (b) in terms of QSP operators, where the green boxes correspond to the signal processing operator $S(\phi)$ and blue boxes correspond to the signal operator $W(a)$.
• Figure 2: Signal shape for the BB1 sequence realized with AIM and a large amplitude $A>\Delta$. The blue line corresponds to the harmonic-driven evolution described by the cosine in Eq. (\ref{['epsilon_sequence']}), the orange line corresponds to the adjustable phase gain while the driving is constant, which is used to match the QSP phases with the AIM phases, and the red crosses are the transition points. Time is normalized to the qubit resonant excitation period $T_\text{q}={2\pi}/{\Delta}$.
• Figure 3: (a) Adjustable rotation gate $R_x(\theta)$ with double LZSM transition. (b) Analogy for realizing an $R_x(\theta)$ rotation gate with a Mach-Zehnder interferometer.
• Figure 4: Panel (a) shows the resulting occupation probability $P_-$ for the BB1 sequence in Eq. \ref{['BB1Plynom']} compared to the qubit dynamics simulation result with the driving signal defined by AIM. Panel (b) presents a comparison of the hardware execution time $t/T_\text{q}$ for the BB1 sequence using different approaches: (1) IBM-Q time (blue line), computed as the difference between the execution time with zero gates and the time with the BB1 sequence. This isolates the time required solely for the QSP technique, enabling direct comparison with other methods. (2) IBM-Q with the optimization (orange line), where the execution time does not depend on the number of single-qubit gates—i.e., the entire BB1 sequence is replaced with a single compiled gate. (3) Direct realization of QSP via AIM (green curve). (4) Double LZSM transitions (purple line). To compare execution times we divide all of them by the corresponding period of the qubit resonance driving $T_\text{q}={2\pi}/{\Delta}$.
• Figure 5: Schematic diagrams for two ways to apply the adiabatic-impulse model for multi-level systems. Panel (a) shows the qubitization technique, which allows decomposing a multi-level system by $K=N(N-1)/2$ single-qubit evolution algorithms. The decomposed evolution operator can be realized with the QSP sequences, and then fast LZSM gates can be applied to perform QSP. Panel (b) demonstrates a way to directly realize multi-level evolution with generalized AIM, which consists of consecutive LZSM transitions with respective energy levels. Dashed lines represent diabatic energy levels in the two-qubit system; the orange line represents one of the possible trajectories for the occupation probability with transitions between levels on each anti-crossing point.