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Quantum algorithms for approximate function loading
Quantum Physics · Submitted 15 Nov 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Here, we introduce two approximate quantum-state preparation methods for the NISQ era inspired by the Grover-Rudolph algorithm, which partially solve the problem of loading real functions.
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The abstract, sentence by sentence
Loading classical data into quantum computers represents an essential stage in many relevant quantum algorithms, especially in the field of quantum machine learning. Therefore, the inefficiency of this loading process means a major bottleneck for the application of these algorithms. Here, we introduce two approximate quantum-state preparation methods for the NISQ era inspired by the Grover-Rudolph algorithm, which partially solve the problem of loading real functions. Indeed, by allowing for an infidelity $\epsilon$ and under certain smoothness conditions, we prove that the complexity of the implementation of the Grover-Rudolph algorithm without ancillary qubits, first introduced by M\"ott\"onen $\textit{et al}$, results into $\mathcal{O}(2^{k_0(\epsilon)})$, with $n$ the number of qubits and $k_0(\epsilon)$ asymptotically independent of $n$. This leads to a dramatic reduction in the number of required two-qubit gates. Aroused by this result, we also propose a variational algorithm capable of loading functions beyond the aforementioned smoothness conditions. Our variational Ansatz is explicitly tailored to the landscape of the function, leading to a quasi-optimized number of hyperparameters. This allows us to achieve high fidelity in the loaded state with high speed convergence for the studied examples.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Loading classical data into quantum computers represents an essential stage in many relevant quantum algorithms, especially in the field of quantum machine learning. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s1 | Therefore, the inefficiency of this loading process means a major bottleneck for the application of these algorithms. | background 0.88 | 0.05 | 0.04 | asserted 1.00 |
| s2 | Here, we introduce two approximate quantum-state preparation methods for the NISQ era inspired by the Grover-Rudolph algorithm, which partially solve the problem of loading real functions. | contribution 1.00 | 0.91 | 0.24 | asserted 0.90 |
| s3 | Indeed, by allowing for an infidelity $\epsilon$ and under certain smoothness conditions, we prove that the complexity of the implementation of the Grover-Rudolph algorithm without ancillary qubits, first introduced by M\"ott\"onen $\textit{et al}$, results into $\mathcal{O}(2^{k_0(\epsilon)})$, with $n$ the number of qubits and $k_0(\epsilon)$ asymptotically independent of $n$. | result 0.96 | 0.68 | 0.32 | asserted 1.00 |
| s4 | This leads to a dramatic reduction in the number of required two-qubit gates. | interpretation 0.80 | 0.68 | 0.18 | asserted 1.00 |
| s5 | Aroused by this result, we also propose a variational algorithm capable of loading functions beyond the aforementioned smoothness conditions. | contribution 0.88 | 0.92 | 0.09 | asserted 0.94 |
| s6 | Our variational Ansatz is explicitly tailored to the landscape of the function, leading to a quasi-optimized number of hyperparameters. | method 0.92 | 0.88 | 0.07 | asserted 1.00 |
| s7 | This allows us to achieve high fidelity in the loaded state with high speed convergence for the studied examples. | result 0.99 | 0.91 | 0.24 | asserted 1.00 |
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The paper
Original abstract
Loading classical data into quantum computers represents an essential stage in many relevant quantum algorithms, especially in the field of quantum machine learning. Therefore, the inefficiency of this loading process means a major bottleneck for the application of these algorithms. Here, we introduce two approximate quantum-state preparation methods for the NISQ era inspired by the Grover-Rudolph algorithm, which partially solve the problem of loading real functions. Indeed, by allowing for an infidelity $\epsilon$ and under certain smoothness conditions, we prove that the complexity of the implementation of the Grover-Rudolph algorithm without ancillary qubits, first introduced by M\"ott\"onen $\textit{et al}$, results into $\mathcal{O}(2^{k_0(\epsilon)})$, with $n$ the number of qubits and $k_0(\epsilon)$ asymptotically independent of $n$. This leads to a dramatic reduction in the number of required two-qubit gates. Aroused by this result, we also propose a variational algorithm capable of loading functions beyond the aforementioned smoothness conditions. Our variational Ansatz is explicitly tailored to the landscape of the function, leading to a quasi-optimized number of hyperparameters. This allows us to achieve high fidelity in the loaded state with high speed convergence for the studied examples.
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How the literature uses it
13 arXiv papers cite this in 18 sentences
Papers citing it · 11 sentences without a confident label
Compares with it 1 paper
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Hence, the total gate count for this variational algorithm is $Ω(n^2)$ whereas the classical time and space complexity is exponential in $n$ [ref] .
Linear-depth quantum circuits for loading Fourier approximations of arbitrary functions · Comparison with Previous Work · 0.81
Cites it as background 5 papers
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We consider a dataset that has been amplitude encoded [ref] , such that the dataset is described by an ensemble ${p_i, |ψ_i⟩}$ of normalized quantum states $|ψ_i⟩$.
Covariance matrix preparation for quantum principal component analysis · Introduction · 0.86
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There are several previous methods that attempt to address the input problem [ref] ; we highlight some prominent approaches here.
Data is often loadable in short depth: Quantum circuits from tensor networks for finance, images, fluids, and proteins · Introduction · 0.96
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Multiple techniques utilising these approaches have been developed including: preparing Matrix Product States (MPS) [ref] , sparse states [ref] , truncated series expansions [ref] , training variational circuits [ref] , compressing repeated values [ref] , preparing states via coherent arithmetic [ref] , and QSVT polynomial approximation [ref] .
Quantum state preparation via piecewise QSVT · Introduction · 0.90
What it cites · 26 sentences without a confident label
Uses the method or tool of 1 paper
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The circuit proposed as an ansatz, $U(θ)$, for our model is based on the Grover-Rudolph circuit [ref] .
Creating superpositions that correspond to efficiently integrable probability distributions · Ansatz · 0.98
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