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arXiv:2303.05533 · npj Quantum Information

Realization of quantum signal processing on a noisy quantum computer

Yuta Kikuchi, Conor Mc Keever, Luuk Coopmans, Michael Lubasch, Marcello Benedetti

Published Sep 23, 2023• 41 citations
arXivHTMLPDF

Abstract

Abstract Quantum signal processing (QSP) is a powerful toolbox for the design of quantum algorithms and can lead to asymptotically optimal computational costs. Its realization on noisy quantum computers without fault tolerance, however, is challenging because it requires a deep quantum circuit in general. We propose a strategy to run an entire QSP protocol on noisy quantum hardware by carefully reducing overhead costs at each step. To illustrate the approach, we consider the application of Hamiltonian simulation for which QSP implements a polynomial approximation of the time evolution operator. We test the protocol by running the algorithm on the Quantinuum H1-1 trapped-ion quantum computer powered by Honeywell. In particular, we compute the time dependence of bipartite entanglement entropies for Ising spin chains and find good agreements with exact numerical simulations. To make the best use of the device, we determine optimal experimental parameters by using a simplified error model for the hardware and numerically studying the trade-off between Hamiltonian simulation time, polynomial degree, and total accuracy. Our results are the first step in the experimental realization of QSP-based quantum algorithms.

DOI
10.1038/s41534-023-00762-0

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Abstract Quantum signal processing (QSP) is a powerful toolbox for the design of quantum algorithms and can lead to asymptotically optimal computational costs. Its realization on noisy quantum computers without fault tolerance, however, is challenging because it requires a deep quantum circuit in general. We propose a strategy to run an entire QSP protocol on noisy quantum hardware by carefully reducing overhead costs at each step. To illustrate the approach, we consider the application of Hamiltonian simulation for which QSP implements a polynomial approximation of the time evolution operator. We test the protocol by running the algorithm on the Quantinuum H1-1 trapped-ion quantum computer powered by Honeywell. In particular, we compute the time dependence of bipartite entanglement entropies for Ising spin chains and find good agreements with exact numerical simulations. To make the best use of the device, we determine optimal experimental parameters by using a simplified error model for the hardware and numerically studying the trade-off between Hamiltonian simulation time, polynomial degree, and total accuracy. Our results are the first step in the experimental realization of QSP-based quantum algorithms.
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Quantum signal processing (QSP), which enables systematic polynomial transformations on quantum data through sequences of qubit rotations, has emerged as a fundamental building block for quantum algorithms and data reuploading quantum neural networks. While recent experiments have demonstrated the feasibility of shallow QSP circuits, the inherent limitations in scaling QSP to achieve complex transformations on quantum hardware remain an open and critical question. Here we report the first experimental realization of deep QSP circuits in a trapped-ion quantum simulator. By manipulating the qubit encoded in a trapped 43Ca+{}^{43}{\mathrm{Ca}}^{+}43Ca+ ion, we demonstrate high-precision simulation of some prominent functions used in quantum algorithms and machine learning, with circuit depths ranging from 15 to 360 layers and implementation times significantly longer than the coherence time of the qubit. Our results reveal a crucial trade-off between the precision of function simulation and the concomitant accumulation of hardware noise, highlighting the importance of striking a balance between circuit depth and accuracy in practical QSP implementation. This work addresses a key gap in understanding the scalability and limitations of QSP-based algorithms on quantum hardware, providing valuable insights for developing quantum algorithms as well as practically realizing quantum singular value transformation and data reuploading quantum machine learning models.
Quantum signal processing (QSP) serves as the asymptotically optimal technique for Hamiltonian simulation on a quantum computer. By approximating the time evolution operator via the Jacobi-Anger expansion, the Hamiltonian simulation problem reduces to a problem in polynomial approximation theory: find a sufficient degree-ddd polynomial series to approximate e−iτxe^{-iτx}e−iτx on [−1,1][-1,1][−1,1] within error εεε. While d∈O~(τ)d\in\tilde{\mathcal{O}}(τ)d∈O~(τ) is known to be asymptotically optimal, there exists a gap between state-of-the-art bounds and the optimal constant multiplicative factor, which is approximately equal to 1. Here, we close this gap almost entirely, to the point where possible future improvements will not be of practical significance. Our improvement resides in a careful treatment of the Bessel tail in the Jacobi-Anger series using Kapteyn's and Watson's inequalities, thereby reducing the overhead estimates for all Hamiltonian simulation tasks on quantum computers by a factor of ≈e/2\approx e/2≈e/2.
Quantum signal processing (QSP) is a template for designing algorithms in the quantum circuit model. We consider three facets of this template. First, how to compute the exact circuit for a given algorithm and problem parameters. QSP requires two interrelated pre-processing steps. We give an algorithm for the first step, and compare its performance to recent alternatives. We find that our algorithm, using the Wilson method, is competitive for sparse polynomials away from the near-coherent regime. Next, we give an algorithm for quantum phase or energy estimation (QPE and QEE), designed with QSP. QEE replaces the common oracle access model in QPE with a block-encoding. We use von Neumann's measurement protocol and Hamiltonian simulation with QSP to show that QEE can be solved efficiently. We then show that when the standard QPE oracle is block-encoded, QPE algorithm with logarithmic overhead. We also show that whenever the system of interest is described by a linear combination of Pauli terms, the gate complexity of the entire algorithm is efficient in Clifford and T gates. We then consider how QSP relates to time-evolution. We show that QSP circuits can be understood as the circuits generated by a piecewise continuous time-evolution. Specifically, this evolution fits the bilinear control model, a common model in quantum control theory. Finally, we engineer a continuous time-dependent system for which the quantum adiabatic theorem will hold, which syntactically resembles QSP. We show that in the high-degree and small-signal limit, when the adiabatic schedule is Lipschitz continuous, the quantum adiabatic evolution of an eigenstate of H(0) is close to a QSP circuit applied to the same eigenstate. Crucially, our formalism establishes a direct functional relationship between the adiabatic schedule and the set of phase factors that determine the QSP circuit.