Efficient Fully-Coherent Quantum Signal Processing Algorithms for Real-Time Dynamics Simulation

Abstract

Hamiltonian simulation is a fundamental problem at the heart of quantum computation, and the associated simulation algorithms are useful building blocks for designing larger quantum algorithms. In order to be successfully concatenated into a larger quantum algorithm, a Hamiltonian simulation algorithm must succeed with arbitrarily high success probability 1 − δ while only requiring a single copy of the initial state, a property which we call fully-coherent. Although optimal Hamiltonian simulation has been achieved by quantum signal processing (QSP), with query complexity linear in time t and logarithmic in inverse error ln(1/ ), the corresponding algorithm is not fully-coherent as it only succeeds with probability close to 1/4. While this simulation algorithm can be made fully-coherent by employing amplitude amplification at the expense of appending a ln(1/δ) multiplicative factor to the query complexity, here we develop a new fully-coherent Hamiltonian simulation algorithm that achieves a query complexity additive in ln(1/δ): Θ H |t| + ln(1/ ) + ln(1/δ) . We accomplish this by compressing the spectrum of the Hamiltonian with an affine transformation, and applying to it a QSP polynomial that approximates the complex exponential only over the range of the compressed spectrum. We further numerically analyze the complexity of this algorithm and demonstrate its application to the simulation of the Heisenberg model in constant and time-dependent external magnetic fields. We believe that this efficient fully-coherent Hamiltonian simulation algorithm can serve as a useful subroutine in quantum algorithms where maintaining coherence is paramount.

Quantum Phase Processing: Transform and Extract Eigen-Information of Quantum Systems

“…Here we show that our results also match the optimal query complexity. In particular, compared to the Hamiltonian simulation based on QSVT [4,5,9], our approach does not need the implementation of linear-combination-of-unitaries [26] and amplitude amplification [27,41].…”

“…Quantum computing has been applied in many important tasks, including breaking encryption [1], searching databases [2], and simulating quantum evolution [3]. Recent advances in quantum computing show that quantum singular value transformation (QSVT) introduced by Gilyén et al [4] has led to a unified framework of the most known quantum algorithms [5], including amplitude amplification [4], quantum walks [4], phase estimation [5,6], and Hamiltonian simulations [7][8][9][10]. This framework can further be used to develop new quantum algorithms such as quantum entropies estimation [11][12][13], fidelity estimation [14], ground state preparation and ground energy estimation [15][16][17].…”

Towards Quantum Advantage in Financial Market Risk using Quantum Gradient Algorithms

“…On the other hand, Ref. [17] introduces a coherent Hamiltonian simulation method which does not require postselection and instead succeeds with arbitrarily high probability 1−δ, scaling as O(|t|+ln(1/ phase )+ ln(1/δ)). For a target t, approximation error phase and δ = 2 phase , this coherent Hamiltonian simulation algorithm queries U and its inverse a total number of times…”

“…(19), which corresponds to m = 1 in Eq. (17). Because in the QAE setting we are computing the gradients of θ(x) = sin −1 f (x) instead of f (x), we need to construct an oracle which performs…”

Quantum Simulation of Light-Front QCD for Jet Quenching in Nuclear Environments

“…Physical results are obtained by averaging over multiple trajectories. An interesting question is whether one can simulate the diffusion Hamiltonian evolution more efficiently by using some random quantum circuit [139] or modifying the Quantum Signal Processing algorithm [140,141]. This is left for future studies.…”