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Preparing Arbitrary Continuous Functions in Quantum Registers With Logarithmic Complexity
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Quantum computers will be able solve important problems with significant polynomial and exponential speedups over their classical counterparts, for instance in option pricing in finance, and in real-space molecular chemistry simulations. However, key applications can only achieve their potential speedup if their inputs are prepared efficiently. We effectively solve the important problem of efficiently preparing quantum states following arbitrary continuous (as well as more general) functions with complexity logarithmic in the desired resolution, and with rigorous error bounds. This is enabled by the development of a fundamental subroutine based off of the simulation of rank-1 projectors. Combined with diverse techniques from quantum information processing, this subroutine enables us to present a broad set of tools for solving practical tasks, such as state preparation, numerical integration of Lipschitz continuous functions, and superior sampling from probability density functions. As a result, our work has significant implications in a wide range of applications, for instance in financial forecasting, and in quantum simulation.
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Cited by 6 Pith papers
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Sparse Quantum State Preparation with Sublinear T-Count
An n-qubit state supported on at most s basis states can be prepared to constant error with Õ(min{s, n^{3/4}√s}) T gates, and some such states require Ω(min{s, √(ns)}) T gates.
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Space-time tradeoff for sparse quantum state preparation
Any n-qubit d-sparse state can be prepared with depth O(nd log m/(m log(m/n)) + log nd) using m >= 6n ancillas, improving prior depth-ancilla trade-offs.
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Quantum algorithm for solving nonlinear differential equations based on physics-informed effective Hamiltonians
Differential equations are solved on a quantum computer by preparing the ground state of a Hamiltonian constructed from Chebyshev differential and data constraints, bypassing variational search.
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The State Preparation of Multivariate Normal Distributions using Tree Tensor Network
Multivariate normal distributions with low-rank or exponentially decaying correlations can be prepared by quantum circuits whose size grows only polylogarithmically in the desired accuracy when represented as tree ten...
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A Shapley Value Estimation Speedup for Efficient Explainable Quantum AI
Quantum amplitude estimation can estimate Shapley values with O(1/ε) queries to the value function, a quadratic improvement over classical Monte Carlo's O(σ²/ε²).
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Application of an upsampling algorithm to quantum state preparation of continuous and discrete probability distributions
An upsampling-based divide-and-conquer circuit for quantum state preparation of probability distributions is derived, but the discrete-distribution extension is not correctly normalized and the method is not novel.
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