Nonlinear transformation of complex amplitudes via quantum singular value transformation

• URL: http://arxiv.org/abs/2107.10764v2
• Date: Fri, 17 May 2024 04:39:58 GMT
• Title: Nonlinear transformation of complex amplitudes via quantum singular value transformation
• Authors: Naixu Guo, Kosuke Mitarai, Keisuke Fujii,
• Abstract summary: This paper defines a task called nonlinear transformation of complex amplitudes on a quantum computer. We construct a block-encoding of complex amplitudes from a state preparation unitary. We discuss its possible applications to quantum machine learning, where complex amplitudes encoding classical or quantum data are processed.
• Score: 0.8009842832476994
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Due to the linearity of quantum operations, it is not straightforward to implement nonlinear transformations on a quantum computer, making some practical tasks like a neural network hard to be achieved. In this work, we define a task called nonlinear transformation of complex amplitudes and provide an algorithm to achieve this task. Specifically, we construct a block-encoding of complex amplitudes from a state preparation unitary. This allows us to transform the complex amplitudes by using quantum singular value transformation. We evaluate the required overhead in terms of input dimension and precision, which reveals that the algorithm depends on the roughly square root of input dimension and achieves an exponential speedup on precision compared with previous work. We also discuss its possible applications to quantum machine learning, where complex amplitudes encoding classical or quantum data are processed by the proposed method. This paper provides a promising way to introduce highly complex nonlinearity of the quantum states, which is essentially missing in quantum mechanics.

Related papers

• Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits [63.733312560668274]
  Given a quantum circuit containing d tunable RZ gates and G-d Clifford gates, can a learner perform purely classical inference to efficiently predict its linear properties? We prove that the sample complexity scaling linearly in d is necessary and sufficient to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. We devise a kernel-based learning model capable of trading off prediction error and computational complexity, transitioning from exponential to scaling in many practical settings.
  arXiv  Detail & Related papers  (2024-08-22T08:21:28Z)
• Quantum algorithms: A survey of applications and end-to-end complexities [90.05272647148196]
  The anticipated applications of quantum computers span across science and industry. We present a survey of several potential application areas of quantum algorithms. We outline the challenges and opportunities in each area in an "end-to-end" fashion.
  arXiv  Detail & Related papers  (2023-10-04T17:53:55Z)
• Quantum Phase Processing and its Applications in Estimating Phase and Entropies [10.8525801756287]
  "quantum phase processing" can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. Quantum phase processing can extract the eigen-information of quantum systems by simply measuring the ancilla qubit. We propose a new quantum phase estimation algorithm without quantum Fourier transform, which requires the fewest ancilla qubits and matches the best performance so far.
  arXiv  Detail & Related papers  (2022-09-28T17:41:19Z)
• An Amplitude-Based Implementation of the Unit Step Function on a Quantum Computer [0.0]
  We introduce an amplitude-based implementation for approximating non-linearity in the form of the unit step function on a quantum computer. We describe two distinct circuit types which receive their input either directly from a classical computer, or as a quantum state when embedded in a more advanced quantum algorithm.
  arXiv  Detail & Related papers  (2022-06-07T07:14:12Z)
• Improved Quantum Algorithms for Fidelity Estimation [77.34726150561087]
  We develop new and efficient quantum algorithms for fidelity estimation with provable performance guarantees. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation. We prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general.
  arXiv  Detail & Related papers  (2022-03-30T02:02:16Z)
• On nonlinear transformations in quantum computation [0.0]
  We develop a series of basic subroutines for implementing nonlinear transformations of input quantum states. Our algorithms are framed around the concept of a weighted state.
  arXiv  Detail & Related papers  (2021-12-23T01:51:37Z)
• Quantum algorithms for quantum dynamics: A performance study on the spin-boson model [68.8204255655161]
  Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter-approximation of the time-evolution operator. variational quantum algorithms have become an indispensable alternative, enabling small-scale simulations on present-day hardware. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage.
  arXiv  Detail & Related papers  (2021-08-09T18:00:05Z)
• Detailed Account of Complexity for Implementation of Some Gate-Based Quantum Algorithms [55.41644538483948]
  In particular, some steps of the implementation, as state preparation and readout processes, can surpass the complexity aspects of the algorithm itself. We present the complexity involved in the full implementation of quantum algorithms for solving linear systems of equations and linear system of differential equations.
  arXiv  Detail & Related papers  (2021-06-23T16:33:33Z)
• Information Scrambling in Computationally Complex Quantum Circuits [56.22772134614514]
  We experimentally investigate the dynamics of quantum scrambling on a 53-qubit quantum processor. We show that while operator spreading is captured by an efficient classical model, operator entanglement requires exponentially scaled computational resources to simulate.
  arXiv  Detail & Related papers  (2021-01-21T22:18:49Z)
• Quantum Amplitude Arithmetic [20.84884678978409]
  We propose the notion of quantum amplitude arithmetic (QAA) that intent to evolve the quantum state by performing arithmetic operations on amplitude. QAA is expected to find applications in a variety of quantum algorithms.
  arXiv  Detail & Related papers  (2020-12-21T00:17:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.