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arXiv:1910.14596 · Quantum

Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems

Lin Lin, Yu Tong

Published Nov 11, 2020• 106 citations
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Abstract

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.

DOI
10.22331/q-2020-11-11-361

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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.
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.
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))\mathcal{\widetilde{O}}(d\kappa\log(1/\epsilon))O(dκlog(1/ϵ)) query complexity for a ddd-sparse matrix, where κ\kappaκ is the condition number, and ϵ\epsilonϵ 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 ddd-sparse N×NN\times NN×N matrix AAA with condition number κκκ, and the ability to prepare a quantum state proportional to a vector b⃗\vec bb. The goal is to output an εεε-approximation to the quantum state proportional to the solution x⃗\vec xx of Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities O(κdlog⁡(1/ε))O(κd\log(1/ε))O(κdlog(1/ε)) and κd(κd/ε)o(1)κ\sqrt d(κd/ε)^{o(1)}κd​(κd/ε)o(1), while the best known lower bounds were Ω(κlog⁡(1/ε))Ω(κ\log(1/ε))Ω(κlog(1/ε)) and Ω(κd)Ω(κ\sqrt d)Ω(κd​). We improve these bounds and show that the complexity of the QLSP is Θ(κdlog⁡(1/ε))Θ(κ\sqrt d\log(1/ε))Θ(κd​log(1/ε)). We also resolve an open problem of Berry and Childs by showing that any N×NN\times NN×N unitary can be implemented with bounded error using O(N)O(\sqrt N)O(N​) queries to its matrix entries.
Quantum algorithms for linear systems produce the solution state A − 1 | b ⟩ by querying two oracles: O A that block encodes the coefficient matrix and O b that prepares the initial state. We present a quantum linear system algorithm making Θ ( 1 / p ) queries to O b , which is optimal in the success probability, and O ( κ log ⁡ ( 1 / p ) ( log ⁡ log ⁡ ( 1 / p ) + log ⁡ ( 1 / ϵ ) ) ) queries to O A , nearly optimal in all parameters including the condition number and accuracy. Notably, our complexity scaling of initial state preparation holds even when p is not known a priori . This contrasts with recent results achieving O ( κ log ⁡ ( 1 / ϵ ) ) complexity to both oracles, which, while optimal in O A , is highly suboptimal in O b as κ can be arbitrarily larger than 1 / p . In various applications such as solving differential equations, preparing ground states of operators with real spectra, and estimating and transforming eigenvalues of non-normal matrices, we can further improve the dependence on p using a block preconditioning scheme to nearly match or outperform best p
We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with O (κ poly(log (κ ε))) runtime, where κ is the condition number, and ε is the target accuracy. This is near optimal with respect to both κ and ε, and is achieved without relying on complicated amplitude amplification procedures that are difficult to implement. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.
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 ρρρ, 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†ρsqrtρ=ρ_{\rm sqrt}^\daggerρ_{\rm sqrt}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 ρρρ. 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.