We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal $\mathcal{\widetilde{O}}(d\kappa\log(1/\epsilon))$ query complexity for a $d$-sparse matrix, where $\kappa$ is the condition number, and $\epsilon$ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.
We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal O(dκlog(1/ϵ)) query complexity for a d-sparse matrix, where κ is the condition number, and ϵ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.
We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal Oe(dκlog(1/∊)) query complexity for a d-sparse matrix, where κ is the condition number, and ∊ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.
In the quantum linear systems problem (QLSP), we are given query access to a d-sparse N×N matrix A with condition number κ, and the ability to prepare a quantum state proportional to a vector b. The goal is to output an ε-approximation to the quantum state proportional to the solution x of Ax=b. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities O(κdlog(1/ε)) and κd(κd/ε)o(1), while the best known lower bounds were Ω(κlog(1/ε)) and Ω(κd). We improve these bounds and show that the complexity of the QLSP is Θ(κdlog(1/ε)). We also resolve an open problem of Berry and Childs by showing that any N×N unitary can be implemented with bounded error using O(N) queries to its matrix entries.
We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal O ~ ( d κ log ( 1 / ϵ ) ) query complexity for a d -sparse matrix, where κ is the condition number, and ϵ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.
Given a linear system of equations Ax=b, quantum linear system solvers (QLSSs) approximately prepare a quantum state ∣x⟩ for which the amplitudes are proportional to the solution vector x. Asymptotically optimal QLSSs have query complexity O(κlog(1/ε)), where κ is the condition number of A, and ε is the approximation error. However, runtime guarantees for existing optimal and near-optimal QLSSs do not have favorable constant prefactors, in part because they rely on complex or difficult-to-analyze techniques like variable-time amplitude amplification and adiabatic path-following. Here, we give a conceptually simple QLSS that does not use these techniques. If the solution norm ∥x∥ is known exactly, our QLSS requires only a single application of kernel reflection (a straightforward extension of the eigenstate filtering (EF) technique of previous work) and the query complexity of the QLSS is (1+O(ε))κln(22/ε). If the norm is unknown, our method allows it to be estimated up to a constant factor using O(loglog(κ)) applications of kernel projection (a direct generalization of EF) yielding a straightforward QLSS with near-optimal O(κloglog(κ)logloglog(κ)+κlog(1/ε)) total complexity. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that O(κ) complexity can be achieved for norm estimation, yielding an optimal QLSS with O(κlog(1/ε)) complexity while still avoiding the need to invoke the adiabatic theorem. Finally, we compute an explicit upper bound of 56κ+1.05κln(1/ε)+o(κ) for the complexity of our optimal QLSS.
Quantum simulation is expected to be a main application of quantum computers with realistic utility in quantum chemistry, materials science and beyond. However, preparing excited or general eigenstates is a central challenge, particularly when the desired eigenvalue is not known in advance, or when the overlap with the initial state is insufficient. We introduce the Dominant Eigenstate Filtering via Eigenprobability Amplification and Thresholding (DEFEAT) algorithm that identifies and filters the eigenstate with the largest overlap with the supplied initial state. Our key observation is that we do not need prior knowledge of the target eigenvalue, as we construct efficient twirling superoperators that map initial states to eigenprobability density operators ρ, diagonal in the Hamiltonian eigenbasis and encoding its spectral weights, alongside its block-encoding implementation. Our crucial innovation is the quadratic amplification of probabilities via the factorisation ρ=ρsqrt†ρsqrt, analogous to the recently introduced sum-of-squares spectral amplification (SOSSA), which we use here to amplify the separation between dominant and subdominant components. Thresholding then yields the dominant-eigenstate projector. Compared with conventional phase estimation, DEFEAT improves the dependence on the overlap with the initial state while requiring substantially fewer ancillary qubits. We prove that the query complexity of the filtering step is optimal up to logarithmic factors and establish a complementary lower bound for eigenstate preparation under purified query access to ρ. We validate in numerical simulations that the convergence rate of DEFEAT matches our theoretical results. Our results provide a general eigenvalue-agnostic primitive for dominant eigenstate filtering and preparation, and for estimating properties of dominant eigenstates.
We transform the problem of solving linear system of equations Ax=b to a problem of finding the right singular vector with singular value zero of an augmented matrix C, and present two quantum algorithms for solving this problem. The first algorithm solves the problem directly by applying the quantum eigenstate filtering algorithm with query complexity of $O\left( sκ\log \left( 1/ε\right) \right) $ for a s-sparse matrix C, where κ is the condition number of the matrix A, and ε is the desired precision. The second algorithm uses the quantum resonant transition approach, the query complexity scales as $O\left[sκ+ \log\left( 1/ε\right)/\log \log \left( 1/ε\right) \right] $. Both algorithms meet the optimal query complexity in κ, and are simpler than previous algorithms.