Double-bracket algorithm for quantum signal processing without post-selection
arXiv:2504.01077 · doi:10.22331/q-2025-12-23-1954
Abstract
Quantum signal processing (QSP), a framework for implementing matrix-valued polynomials, is a fundamental primitive in various quantum algorithms. Despite its versatility, a potentially underappreciated challenge is that all systematic protocols for implementing QSP rely on post-selection. This can impose prohibitive costs for tasks when amplitude amplification cannot sufficiently improve the success probability. For example, in the context of ground-state preparation, this occurs when using a too poor initial state. In this work, we introduce a new formula for implementing QSP transformations of Hermitian matrices, which requires neither auxiliary qubits nor post-selection. Rather, using approximation to the exact unitary synthesis, we leverage the theory of the double-bracket quantum algorithms to provide a new quantum algorithm for QSP, termed Double-Bracket QSP (DB-QSP). The algorithm requires the energy and energetic variance of the state to be measured at each step and has a recursive structure, which leads to circuit depths that can grow super exponentially with the degree of the polynomial. With these strengths and caveats in mind, DB-QSP should be viewed as complementing the established QSP toolkit. In particular, DB-QSP can deterministically implement low-degree polynomials to "warm start" QSP methods involving post-selection.
References in corpus (42)
- Quantum algorithm for solving linear systems of equations
- Variational Quantum Algorithms
- Barren plateaus in quantum neural network training landscapes
- Quantum principal component analysis
- Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
- Hamiltonian Simulation by Qubitization
- Optimal Hamiltonian Simulation by Quantum Signal Processing
- Simulating Hamiltonian dynamics with a truncated Taylor series
- Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
- Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
- A Theory of Trotter Error
- The General Quantum Interference Principle and the Duality Computer
- Exponential improvement in precision for simulating sparse Hamiltonians
- Heisenberg-limited ground state energy estimation for early fault-tolerant quantum computers
- Nearly optimal lattice simulation by product formulas
- Near-optimal ground state preparation
- Ground state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices
- The methodology of resonant equiangular composite quantum gates
- Application of fermionic marginal constraints to hybrid quantum algorithms
- Geometric Optimization Methods for Adaptive Filtering
- Optimal Control for Generating Quantum Gates in Open Dissipative Systems
- Preparing ground states of quantum many-body systems on a quantum computer
- Hamiltonian Simulation with Optimal Sample Complexity
- Does provable absence of barren plateaus imply classical simulability?
- The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation
- Duality and Recycling Computing in Quantum Computers
- Single-ancilla ground state preparation via Lindbladians
- Computing Ground State Properties with Early Fault-Tolerant Quantum Computers
- Gradient Flows for Optimisation and Quantum Control: Foundations and Applications
- Realization of quantum signal processing on a noisy quantum computer
- An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models
- Optimizing quantum circuits with Riemannian gradient flow
- Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers
- Efficient Product Formulas for Commutators and Applications to Quantum Simulation
- Variational quantum simulation: a case study for understanding warm starts
- Fragmented imaginary-time evolution for early-stage quantum signal processors
- Classically estimating observables of noiseless quantum circuits
- Double-bracket quantum algorithms for diagonalization
- The Solovay-Kitaev algorithm
- Limitations on the simulation of non-sparse Hamiltonians
- Quantum Dynamic Programming
- Molecular Properties from Quantum Krylov Subspace Diagonalization