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Toward speedup without quantum coherent access

Nhat A. Nghiem

TL;DR

The protocol provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm, and suggests there are certain matrices/Hamiltonians where the method can provide exponential improvement compared to the existing ones with respect to the sparsity.

Abstract

Along with the development of quantum technology, finding useful applications of quantum computers has been a central pursuit. Despite various quantum algorithms have been developed, many of them often require strong input assumptions, which is hardware demanding. In particular, recent advances on dequantization have revealed that the quantum advantage is more of a mere artifact of strong input assumptions. In this work, we propose a variant of these algorithms, leveraging both classical and quantum resources. Provided the classical knowledge (the entries) of the matrix/vector of interest, a classical procedure is used to pre-process this information. Then they are fed into a quantum circuit which is shown to be a block encoding of the matrix of interest. From this block-encoding, we show how to use it to tackle a wide range of problems, including principal component analysis, linear equation solving, Hamiltonian simulation, preparing ground state, and data fitting. We also analyze our protocol, showing that both the classical and quantum procedure can achieve logarithmic complexity in the input dimension, thus implying its potential for near term realization. We then discuss several implications and corollaries of our result. First,, our results suggest there are certain matrices/Hamiltonians where our method can provide exponential improvement compared to the existing ones with respect to the sparsity. Regarding dense linear systems, our method achieves exponential speed-up with respect to the inverse of error tolerance, compared to the best previously known quantum algorithm for dense systems. Last, and most importantly, regarding quantum data fitting, we show how the output of our quantum algorithms can be leveraged to predict unseen data. Thus, it provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm.

Toward speedup without quantum coherent access

TL;DR

The protocol provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm, and suggests there are certain matrices/Hamiltonians where the method can provide exponential improvement compared to the existing ones with respect to the sparsity.

Abstract

Along with the development of quantum technology, finding useful applications of quantum computers has been a central pursuit. Despite various quantum algorithms have been developed, many of them often require strong input assumptions, which is hardware demanding. In particular, recent advances on dequantization have revealed that the quantum advantage is more of a mere artifact of strong input assumptions. In this work, we propose a variant of these algorithms, leveraging both classical and quantum resources. Provided the classical knowledge (the entries) of the matrix/vector of interest, a classical procedure is used to pre-process this information. Then they are fed into a quantum circuit which is shown to be a block encoding of the matrix of interest. From this block-encoding, we show how to use it to tackle a wide range of problems, including principal component analysis, linear equation solving, Hamiltonian simulation, preparing ground state, and data fitting. We also analyze our protocol, showing that both the classical and quantum procedure can achieve logarithmic complexity in the input dimension, thus implying its potential for near term realization. We then discuss several implications and corollaries of our result. First,, our results suggest there are certain matrices/Hamiltonians where our method can provide exponential improvement compared to the existing ones with respect to the sparsity. Regarding dense linear systems, our method achieves exponential speed-up with respect to the inverse of error tolerance, compared to the best previously known quantum algorithm for dense systems. Last, and most importantly, regarding quantum data fitting, we show how the output of our quantum algorithms can be leveraged to predict unseen data. Thus, it provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm.
Paper Structure (28 sections, 26 theorems, 130 equations, 5 tables, 5 algorithms)

This paper contains 28 sections, 26 theorems, 130 equations, 5 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Organization.
  3. Overview of key technique
  4. Main Results
  5. Another type of state that admits efficient preparation procedure
  6. Principal component analysis
  7. System of linear algebraic equations
  8. Quantum simulation
  9. Preparing ground state
  10. Quantum data fitting
  11. Outlook and Conclusion
  12. Preliminaries
  13. Proof of Lemma \ref{['lemma: blockencodingknownmatrix']}
  14. More on quantum state preparation (Lemma \ref{['lemma: stateprepration']})
  15. Finding principal components based on the power method
  16. ...and 13 more sections

Key Result

Lemma 2.1

Provided the classical knowledge of entries of a matrix $\mathcal{A} \in \mathbb{R}^{m \times n}$ having $s_\mathcal{A}$ nonzero entries with a promise $||\mathcal{A}|| \leq 1$ where $||.||$ is the operator norm. Defining $\ket{\mathcal{A} } =\frac{1}{||\mathcal{A}||_F} \sum_{i=1}^n \sum_{j=1}^m \ma

Theorems & Definitions (28)

  • Lemma 2.1: Block-encoding known matrix; Appendix \ref{['sec: proofofLemmablockencodingknownmatrix']}
  • Definition A.1: Block Encoding Unitary
  • Lemma A.1: gilyen2019quantum Product
  • Lemma A.2: camps2020approximate Tensor Product
  • Lemma A.3: gilyen2019quantum Block Encoding of a Matrix
  • Lemma A.4: gilyen2019quantum Linear combination
  • Lemma A.5: Scaling Block encoding
  • Lemma A.6: gilyen2019quantum Theorem 30
  • Lemma A.7: Projector
  • Lemma A.8: guo2024nonlinear, or Theorem 2 in rattew2023non
  • ...and 18 more