Product Decomposition of Periodic Functions in Quantum Signal Processing

Haah, Jeongwan

doi:10.22331/q-2019-10-07-190

Cited by 78 publications

(101 citation statements)

References 22 publications

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“…Constructing the quantum circuit for QSP requires computing a sequence of phase factors beforehand, and there are classical algorithms capable of doing this [25]. Some recent progress has been made to efficiently compute phase factors for high-degree polynomials to high precision [13,19].…”

Section: Remark 2 [23 Theorem 2] Provides a Singular Value Transformentioning

confidence: 99%

Near-optimal ground state preparation

Lin

2020

Quantum

127

105

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Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

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Section: Remark 2 [23 Theorem 2] Provides a Singular Value Transformentioning

confidence: 99%

Near-optimal ground state preparation

Lin

2020

Quantum

127

105

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“…and τ max = 1000 [55]. Within each segment, we choose q to be the smallest positive integer satisfying…”

Section: Appendix D: Proof Of Lemma 2 In Higher Dimensionsmentioning

confidence: 99%

Locality and Digital Quantum Simulation of Power-Law Interactions

et al. 2019

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The propagation of information in non-relativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1/r α . The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al., FOCS'18. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α > 3D (where D is the number of dimensions).

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“…However, this procedure requires solution of all roots of a high degree polynomial, which can be unstable for the range of polynomials ∼ 100 considered here. The stability of such procedure has recently been improved by Haah [37], though the number of bits of precision needed still scales as O( log( / )). Significant progress has been achieved recently, enabling robust computation of phase factors for polynomials of degrees ranging from thousands to tens of thousands [18,23].…”

Section: Discussionmentioning

confidence: 99%

“…We also remark that in QSP, the construction of the block-encoding U involves a sequence of parameters called phase factors. For a given polynomial P (x), the computation of the phase factors can be efficiently performed on classical computers [32,37]. There are however difficulties in computing such phase factors, which will be discussed in Section 6.…”

Section: Algorithmmentioning

confidence: 99%

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

Lin

Tong

2020

Quantum

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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~(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.

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Product Decomposition of Periodic Functions in Quantum Signal Processing

Cited by 78 publications

References 22 publications

Near-optimal ground state preparation

Near-optimal ground state preparation

Locality and Digital Quantum Simulation of Power-Law Interactions

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

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